% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% The Project Gutenberg EBook of The Tides and Kindred Phenomena in the   %
% Solar System, by Sir George Howard Darwin                               %
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
% with this eBook or online at www.gutenberg.net                          %
%                                                                         %
%                                                                         %
% Title: The Tides and Kindred Phenomena in the Solar System              %
%        The Substance of Lectures Delivered in 1897 at the Lowell        %
%        Institute, Boston, Massachusetts                                 %
%                                                                         %
% Author: Sir George Howard Darwin                                        %
%                                                                         %
% Release Date: January 31, 2012 [EBook #38722]                           %
% Most recently updated: June 11, 2021                         %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: UTF-8                                           %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK TIDES AND KINDRED PHENOMENA ***
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%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{PGtext}
The Project Gutenberg EBook of The Tides and Kindred Phenomena in the
Solar System, by Sir George Howard Darwin

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.net


Title: The Tides and Kindred Phenomena in the Solar System
       The Substance of Lectures Delivered in 1897 at the Lowell
       Institute, Boston, Massachusetts

Author: Sir George Howard Darwin

Release Date: January 31, 2012 [EBook #38722]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK TIDES AND KINDRED PHENOMENA ***
\end{PGtext}
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%%%% Credits and transcriber's note %%%%
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Produced by Andrew D. Hwang, Bryan Ness and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive/Canadian Libraries)
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\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
\cleardoublepage
\pagenumbering{roman}
\PageSep{iii}
\begin{center}
\textbf{\Huge THE TIDES}
\vfil
\textbf{\large AND KINDRED PHENOMENA IN THE \\[4pt]
SOLAR SYSTEM}
\vfil
{\footnotesize THE SUBSTANCE OF LECTURES DELIVERED \\[4pt]
IN 1897 AT THE LOWELL INSTITUTE, \\[4pt]
BOSTON, MASSACHUSETTS}
\vfil
BY
\vfil
\textbf{\Large GEORGE HOWARD DARWIN}
\vfil
\scriptsize PLUMIAN PROFESSOR AND FELLOW OF TRINITY COLLEGE IN THE \\[4pt]
UNIVERSITY OF CAMBRIDGE
\vfil\vfil
\Graphic[png]{1.25in}{riverside}
\vfil\vfil\vfil
\footnotesize BOSTON AND NEW YORK \\
\normalsize HOUGHTON, MIFFLIN AND COMPANY \\
\textgoth{The Riverside Press, Cambridge} \\
1899
\end{center}
\PageSep{iv}
\clearpage
\null\vfill
\begin{center}
\footnotesize
COPYRIGHT, 1898, BY GEORGE HOWARD DARWIN \\
ALL RIGHTS RESERVED
\end{center}
\vfill
\PageSep{v}


\Preface

\First{In} 1897 I delivered a course of lectures on
the Tides at the Lowell Institute in Boston,
Massachusetts, and this book contains the substance
of what I then said. The personal form
of address appropriate to a lecture is, I think,
apt to be rather tiresome in a book, and I have
therefore taken pains to eliminate all traces of
the lecture from what I have written.

A mathematical argument is, after all, only
organized common sense, and it is well that men
of science should not always expound their work
to the few behind a veil of technical language,
but should from time to time explain to a larger
public the reasoning which lies behind their
mathematical notation. To a man unversed in
popular exposition it needs a great effort to shell
away the apparatus of investigation and the
technical mode of speech from the thing behind
it, and I owe a debt of gratitude to Mr.~Lowell,
trustee of the Institute, for having afforded me
the occasion for making that effort.
\PageSep{vi}

It is not unlikely that the first remark of
many who see my title will be that so small a
subject as the Tides cannot demand a whole volume;
but, in fact, the subject branches out in
so many directions that the difficulty has been to
attain to the requisite compression of my matter.
Many popular works on astronomy devote a few
pages to the Tides, but, as far as I know, none
of these books contain explanations of the practical
methods of observing and predicting the
Tides, or give any details as to the degree of
success attained by tidal predictions. If these
matters are of interest, I invite my readers not
to confine their reading to this preface. The
later chapters of this book are devoted to the
consideration of several branches of speculative
Astronomy, with which the theory of the Tides
has an intimate relationship. The problems involved
in the origin and history of the solar
and of other celestial systems have little bearing
upon our life on the earth, yet these questions
can hardly fail to be of interest to all those
whose minds are in any degree permeated by
the scientific spirit.

I think that there are many who would like to
understand the Tides, and will make the attempt
to do so provided the exposition be sufficiently
\PageSep{vii}
simple and clear; it is to such readers I address
this volume. It is for them to say how far I
have succeeded in rendering these intricate subjects
interesting and intelligible, but if I have
failed it has not been for lack of pains.

The figures and diagrams have, for the most
part, been made by Mr.~Edwin Wilson of Cambridge,
but I have to acknowledge the courtesy
of the proprietors of \Title{Harper's}, the \Title{Century},
and the \Title{Atlantic Monthly} magazines, in supplying
me with some important illustrations.

A considerable portion of \Ref{Chapter}{III}.\ on the
``Bore'' is to appear as an article in the \Title{Century
Magazine} for October, 1898, and the reproductions
of Captain Moore's photographs of the
``Bore'' in the Tsien-Tang-Kiang have been
prepared for that article. The \Title{Century} has also
kindly furnished the block of Dr.~Isaac Roberts's
remarkable photograph of the great nebula in
the constellation of Andromeda; it originally
appeared in an article on Meteorites in the number
for October,~1890. The greater portion of
the text and the whole of the illustrations of
\Ref{Chapter}{XX}.\ were originally published in \Title{Harper's
Magazine} for June,~1889. Lastly, portions
of Chapters \Ref{}{XV}.~and~\Ref{}{XVI}.\ appeared in
the \Title{Atlantic Monthly} for April, 1898, published
\PageSep{viii}
by Messrs.\ Houghton, Mifflin~\&~Co., who also
make themselves responsible for the publication
of the American edition of this book.

In conclusion, I wish to take this opportunity
of thanking my American audience for the cordiality
of their reception, and my many friends
across the Atlantic for their abundant hospitality
and kindness.

\Signature{G. H. DARWIN.}
{\textsc{Cambridge}, \textit{August}, 1898.}
\PageSep{ix}


\Contents

\ToCChap{I}
{TIDES AND METHODS OF OBSERVATION}

\ToCSect{Definition of tide}{\PgNos{1}{3}}
\ToCSect{Oceanic tides}{\PgNos[,]{4}{5}}
\ToCSect{Methods of observation}{\PgNo{6}}
\ToCSect{Tide-gauge}{\PgNos{7}{12}}
\ToCSect{Tide-curve}{\PgNo{12}}
\ToCSect{Site for tide-gauge}{\PgNo{13}}
\ToCSect{Irregularities in tide-curve}{\PgNos[,]{14}{15}}
\ToCSect{Authorities}{\PgNo{16}}


\ToCChap{II}
{SEICHES IN LAKES}

\ToCSect{Meaning of seiche}{\PgNo{17}}
\ToCSect{Uses of scientific apparatus}{\PgNo{18}}
\ToCSect{Forel's plemyrameter}{\PgNos[,]{19}{20}}
\ToCSect{Records of the level of the lake}{\PgNo{20}}
\ToCSect{Interpretation of record}{\PgNos{21}{23}}
\ToCSect{Limnimeter}{\PgNo{24}}
\ToCSect{Mode of oscillation in seiches}{\PgNos{25}{28}}
\ToCSect{Wave motion in deep and in shallow water}{\PgNos{29}{32}}
\ToCSect{Composition of waves}{\PgNos{32}{36}}
\ToCSect{Periods of seiches}{\PgNos[,]{37}{38}}
\ToCSect{Causes of seiches}{\PgNos[,]{39}{40}}
\ToCSect{Vibrations due to wind and to steamers}{\PgNos{41}{47}}
\ToCSect{Aerial waves and their action on lakes and on the sea}{\PgNos{48}{53}}
\ToCSect{Authorities}{\PgNos[,]{53}{54}}
\PageSep{x}


\ToCChap{III}
{TIDES IN RIVERS---TIDE MILLS}

\ToCSect{Definition of ebb and flow}{\PgNo{56}}
\ToCSect{Tidal currents in rivers}{\PgNos{56}{58}}
\ToCSect{Progressive change of wave in shallow water}{\PgNos[,]{58}{59}}
\ToCSect{The bore}{\PgNo{59}}
\ToCSect{Captain Moore's survey of the Tsien-Tang-Kiang}{\PgNos{60}{64}}
\ToCSect{Diagram of water-levels during the flow}{\PgNos[,]{64}{65}}
\ToCSect{Chinese superstition}{\PgNos{68}{71}}
\ToCSect{Pictures of the bore}{\PgNo{69}}
\ToCSect{Other cases of bores}{\PgNo{71}}
\ToCSect{Causes of the bore}{\PgNo{72}}
\ToCSect{Tidal energy}{\PgNos[,]{73}{74}}
\ToCSect{Tide mills}{\PgNos[,]{74}{75}}
\ToCSect{Authorities}{\PgNo{75}}


\ToCChap{IV}
{HISTORICAL SKETCH}

\ToCSect{Theories of the Chinese}{\PgNos[,]{76}{77}}
\ToCSect{Theories of the Arabs}{\PgNos{77}{79}}
\ToCSect{Theories of the Norsemen}{\PgNos[,]{79}{80}}
\ToCSect{Writings of Posidonius and Strabo}{\PgNos{80}{84}}
\ToCSect{Seleucus the Babylonian on the diurnal inequality}{\PgNos[,]{84}{85}}
\ToCSect{Galileo and Kepler}{\PgNo{85}}
\ToCSect{Newton and his successors}{\PgNos{86}{88}}
\ToCSect{Empirical method of tidal prediction}{\PgNos{88}{90}}
\ToCSect{Authorities}{\PgNo{90}}


\ToCChap{V}
{TIDE-GENERATING FORCE}

\ToCSect{Inertia and centrifugal force}{\PgNos{91}{93}}
\ToCSect{Orbital motion of earth and moon}{\PgNos{93}{95}}
\ToCSect{Tide-generating force}{\PgNos{96}{100}}
\ToCSect{Law of its dependence on the moon's distance}{\PgNos{101}{103}}
\ToCSect{Earth's rotation}{\PgNos[,]{103}{104}}
\PageSep{xi}
\ToCSect{Second explanation of tide-generating force}{\PgNos[,]{104}{105}}
\ToCSect{Horizontal tide-generating force}{\PgNos[,]{105}{106}}
\ToCSect{Successive changes of force in the course of a day}{\PgNos[,]{107}{108}}
\ToCSect{Authorities}{\PgNo{108}}


\ToCChap{VI}
{DEFLECTION OF THE VERTICAL}

\ToCSect{Deflection of a pendulum by horizontal tidal force}{\PgNos{109}{111}}
\ToCSect{Path pursued by a pendulum under tidal force}{\PgNos{111}{113}}
\ToCSect{Object of measuring the deflection of a pendulum}{\PgNos{113}{115}}
\ToCSect{Attempt to measure deflection by bifilar pendulum}{\PgNos{115}{125}}
\ToCSect{Microsisms}{\PgNos{125}{127}}
\ToCSect{The microphone as a seismological instrument}{\PgNos{127}{130}}
\ToCSect{Paschwitz's work with the horizontal pendulum}{\PgNos[,]{131}{132}}
\ToCSect{Supposed measurement of the lunar deflection of gravity}{\PgNo{132}}
\ToCSect{Authorities}{\PgNo{133}}


\ToCChap{VII}
{THE ELASTIC DISTORTION OF THE EARTH'S SURFACE BY
VARYING LOADS}

\ToCSect{Distortion of an elastic surface by superincumbent load}{\PgNos{134}{137}}
\ToCSect{Application of the theory to the earth}{\PgNos[,]{137}{138}}
\ToCSect{Effects of tidal load}{\PgNos{138}{140}}
\ToCSect{Probable deflections at various distances from the coast}{\PgNos{140}{143}}
\ToCSect{Deflections observed by M.~d'Abbadie and by Dr.~Paschwitz}{\PgNos{143}{145}}
\ToCSect{Effects of atmospheric pressure on the earth's surface}{\PgNos{145}{147}}
\ToCSect{Authorities}{\PgNo{148}}


\ToCChap{VIII}
{EQUILIBRIUM THEORY OF TIDES}

\ToCSect{Explanation of the figure of equilibrium}{\PgNos{149}{151}}
\ToCSect{Map of equilibrium tide}{\PgNos{151}{153}}
\ToCSect{Tides according to the equilibrium theory}{\PgNos{153}{156}}
\ToCSect{Solar tidal force compared with lunar}{\PgNos{156}{158}}
\PageSep{xii}
\ToCSect{Composition of lunar and solar tides}{\PgNos[,]{158}{159}}
\ToCSect{Points of disagreement between theory and fact}{\PgNos{159}{162}}
\ToCSect{Authorities}{\PgNo{162}}


\ToCChap{IX}
{DYNAMICAL THEORY OF THE TIDE WAVE}

\ToCSect{Free and forced waves in an equatorial canal}{\PgNos{163}{165}}
\ToCSect{Critical depth of canal}{\PgNos{165}{167}}
\ToCSect{General principle as to free and forced oscillations}{\PgNos{167}{174}}
\ToCSect{Inverted and direct oscillation}{\PgNos[,]{172}{173}}
\ToCSect{Canal in latitude~$60°$}{\PgNos[,]{174}{175}}
\ToCSect{Tides where the planet is partitioned into canals}{\PgNo{175}}
\ToCSect{Removal of partitions; vortical motion of the water}{\PgNos[,]{176}{177}}
\ToCSect{Critical latitude where the rise and fall vanish}{\PgNos[,]{177}{178}}
\ToCSect{Diurnal inequality}{\PgNos{178}{180}}
\ToCSect{Authorities}{\PgNo{181}}


\ToCChap{X}
{TIDES IN LAKES---COTIDAL CHART}

\ToCSect{The tide in a lake}{\PgNos{182}{185}}
\ToCSect{The Mediterranean Sea}{\PgNos[,]{185}{186}}
\ToCSect{Derived tide of the Atlantic}{\PgNos{186}{188}}
\ToCSect{Cotidal chart}{\PgNos{188}{192}}
\ToCSect{Authorities}{\PgNo{192}}


\ToCChap{XI}
{HARMONIC ANALYSIS OF THE TIDE}

\ToCSect{Tide in actual oceans due to single equatorial satellite}{\PgNos{193}{196}}
\ToCSect{Substitution of ideal satellites for the moon}{\PgNos{197}{199}}
\ToCSect{Partial tide due to each ideal satellite}{\PgNos{199}{201}}
\ToCSect{Three groups of partial tides}{\PgNo{201}}
\ToCSect{Semidiurnal group}{\PgNos{201}{204}}
\ToCSect{Diurnal group}{\PgNos{204}{206}}
\ToCSect{Meteorological tides}{\PgNos[,]{206}{207}}
\ToCSect{Shallow water tides}{\PgNos{207}{210}}
\ToCSect{Authorities}{\PgNo{210}}
\PageSep{xiii}


\ToCChap{XII}
{REDUCTION OF TIDAL OBSERVATIONS}

\ToCSect{Method of singling out a single partial tide}{\PgNos{211}{214}}
\ToCSect{Variety of plans adopted}{\PgNos{214}{217}}
\ToCSect{Tidal abacus}{\PgNos{217}{220}}
\ToCSect{Authorities}{\PgNo{220}}


\ToCChap{XIII}
{TIDE TABLES}

\ToCSect{Definition of special and general tables}{\PgNo{221}}
\ToCSect{Reference to moon's transit}{\PgNos[,]{222}{223}}
\ToCSect{Examples at Portsmouth and at Aden}{\PgNos{223}{228}}
\ToCSect{General inadequacy of tidal information}{\PgNos[,]{229}{230}}
\ToCSect{Method of calculating tide tables}{\PgNos{230}{233}}
\ToCSect{Tide-predicting machine}{\PgNos{233}{241}}
\ToCSect{Authorities}{\PgNo{241}}


\ToCChap{XIV}
{THE DEGREE OF ACCURACY OF TIDAL PREDICTION}

\ToCSect{Effects of wind and barometric pressure}{\PgNos[,]{242}{243}}
\ToCSect{Errors at Portsmouth}{\PgNos[,]{243}{244}}
\ToCSect{Errors at Aden}{\PgNos{245}{249}}
\ToCSect{Authorities}{\PgNo{250}}


\ToCChap{XV}
{CHANDLER'S NUTATION---THE RIGIDITY OF THE EARTH}

\ToCSect{Nutation of the earth and variation of latitude}{\PgNos{251}{254}}
\ToCSect{Elasticity of the earth}{\PgNos[,]{254}{255}}
\ToCSect{Tide due to variation of latitude}{\PgNos[,]{255}{256}}
\ToCSect{Rigidity of the earth}{\PgNos{256}{260}}
\ToCSect{Transmission of earthquake shocks}{\PgNos[,]{261}{262}}
\ToCSect{Authorities}{\PgNos[,]{262}{263}}
\PageSep{xiv}


\ToCChap{XVI}
{TIDAL FRICTION}

\ToCSect{Friction retards the tide}{\PgNos{264}{267}}
\ToCSect{Retardation of planetary rotation}{\PgNos{267}{269}}
\ToCSect{Reaction on the satellite}{\PgNos{269}{272}}
\ToCSect{Ancient eclipses of the sun}{\PgNos[,]{272}{273}}
\ToCSect{Law of variation of tidal friction with moon's distance}{\PgNos{273}{275}}
\ToCSect{Transformations of the month and of the day}{\PgNos{275}{280}}
\ToCSect{Initial and final conditions of motion}{\PgNos[,]{280}{281}}
\ToCSect{Genesis of the moon}{\PgNos{281}{285}}
\ToCSect{Minimum time requisite for the evolution}{\PgNos[,]{285}{286}}
\ToCSect{Rotation of the moon}{\PgNos[,]{286}{287}}
\ToCSect{The month ultimately to be shorter than the day}{\PgNos{287}{289}}


\ToCChap{XVII}
{TIDAL FRICTION (\textit{continued})}

\ToCSect{Discovery of the Martian satellites}{\PgNos{290}{298}}
\ToCSect{Rotation of Mercury, of Venus, and of the Jovian satellites}{\PgNos[,]{298}{299}}
\ToCSect{Adaptation of the earth's figure to changed rotation}{\PgNos{299}{302}}
\ToCSect{Ellipticity of the internal strata of the earth}{\PgNos{302}{304}}
\ToCSect{Geological evidence}{\PgNos{304}{306}}
\ToCSect{Distortion of a plastic planet and trend of continents}{\PgNos{306}{308}}
\ToCSect{Obliquity of the ecliptic}{\PgNos{308}{312}}
\ToCSect{Eccentricity of lunar orbit}{\PgNos[,]{312}{313}}
\ToCSect{Eccentricity of the orbits of double stars}{\PgNo{313}}
\ToCSect{Plane of the lunar orbit}{\PgNos[,]{313}{314}}
\ToCSect{Short summary}{\PgNos[,]{314}{315}}
\ToCSect{Authorities}{\PgNo{315}}


\ToCChap{XVIII}
{THE FIGURES OF EQUILIBRIUM OF A ROTATING MASS OF
LIQUID}

\ToCSect{Plateau's experiment}{\PgNos{316}{319}}
\ToCSect{Stability of a celestial sphere of liquid}{\PgNos{319}{321}}
\ToCSect{The two ellipsoids of Maclaurin and that of Jacobi}{\PgNos{321}{323}}
\PageSep{xv}
\ToCSect{Transitions with change of rotation}{\PgNos[,]{323}{324}}
\ToCSect{Coalescence of Jacobi's with Maclaurin's ellipsoid}{\PgNos{324}{326}}
\ToCSect{Poincaré's law of stability and coalescence}{\PgNos[,]{326}{327}}
\ToCSect{Poincaré's pear-shaped figure}{\PgNos[,]{327}{328}}
\ToCSect{Hour-glass figure of equilibrium}{\PgNos{328}{332}}
\ToCSect{Figures of planets}{\PgNos[,]{332}{333}}
\ToCSect{Authorities}{\PgNo{333}}


\ToCChap{XIX}
{THE EVOLUTION OF CELESTIAL SYSTEMS}

\ToCSect{The Nebular Hypothesis}{\PgNos{334}{338}}
\ToCSect{Nebula in Andromeda}{\PgNos[,]{338}{339}}
\ToCSect{Distribution of satellites in the solar system}{\PgNos{339}{341}}
\ToCSect{Genesis of celestial bodies by fission}{\PgNo{342}}
\ToCSect{Dr.~See's speculations as to systems of double stars}{\PgNos{342}{344}}
\ToCSect{Diversity of celestial bodies}{\PgNos{344}{346}}
\ToCSect{Authorities}{\PgNo{346}}


\ToCChap{XX}
{SATURN'S RINGS}

\ToCSect{Description}{\PgNos[,]{347}{348}}
\ToCSect{Discovery of Saturn's rings}{\PgNos{348}{352}}
\ToCSect{Diagram of the rings}{\PgNos{353}{356}}
\ToCSect{Roche's investigation}{\PgNos[,]{356}{357}}
\ToCSect{Roche's limit}{\PgNos{358}{360}}
\ToCSect{The limit for the several planets}{\PgNos[,]{360}{361}}
\ToCSect{Meteoric constitution of Saturn's rings}{\PgNos[,]{361}{362}}
\ToCSect{Maxwell's investigations}{\PgNos{362}{367}}
\ToCSect{Spectroscopic examination of the rings}{\PgNos{367}{369}}
\ToCSect{Authorities}{\PgNo{369}}
\PageSep{xvi}
%[Blank Page]
\PageSep{xvii}


% [** TN: Plates (Figs. 23, 33, and 40) listed separately in the original;
% omitted separate "FULL-PAGE" and "IN TEXT" headings, folded plate
% references into main illustration list.]
% LIST OF ILLUSTRATIONS
\Illustrations

\LoFFig{1}{Well for Tide-Gauge}{7}
\LoFFig{2}{Pipe of Tide-Gauge}{9}
\LoFFig{3}{Indian Tide-Gauge}{10}
\LoFFig{4}{Légé's Tide-Gauge}{11}
\LoFFig{5}{Bombay Tide-Curve from Noon, April 22, to Noon, April 30, 1884}{14}
\LoFFig{6}{Sites for a Tide-Gauge}{15}
\LoFFig{7}{Plemyrameter}{20}
\LoFFig{8, 9}{Records of Seiches at Évian}{23}
\LoFFig{10}{Map of Lake of Geneva}{26}
\LoFFig{11}{Wave in Deep Water}{30}
\LoFFig{12}{Wave in Shallow Water}{31}
\LoFFig{13}{Simple Wave}{33}
\LoFFig{}{Composition of Two Equal and Opposite Waves}{34}
\LoFFig{14}{Vibrations due to Steamers}{45}
\LoFFig{15}{Progressive Change of a Wave in Shallow Water}{59}
\LoFFig{16}{Chart of the Estuary of the Tsien-Tang-Kiang}{61}
\LoFFig{17}{Bore-Shelter on the Tsien-Tang-Kiang}{64}
\LoFFig{18}{Diagram of the Flow of the Tide on the Tsien-Tang-Kiang}{66}
\LoFFig{19}{Pictures of the Bore on the Tsien-Tang-Kiang}{69}
\LoFFig{20}{Earth and Moon}{93}
\PageSep{xviii}
\LoFFig{21}{Revolution of a Body without Rotation}{98}
\LoFFig{22}{Tide-generating Force}{100}
\LoFFig{23}{Horizontal Tide-generating Force}{106}
\LoFFig{24}{Deflection of a Pendulum; the Moon and Observer on the Equator}{111}
\LoFFig{25}{Deflection of a Pendulum; the Moon in N. Declination~$15°$,
the Observer in N. Latitude~$30°$}{112}
\LoFFig{26}{Bifilar Pendulum}{115}
\LoFFig{27}{Form of Dimple in an Elastic Surface}{135}
\LoFFig{28}{Distortion of Land and Sea-Bed by Tidal Load}{139}
\LoFFig{29}{Chart of Equilibrium Tides}{152}
\LoFFig{30}{Forced Oscillations of a Pendulum}{171}
\LoFFig{31}{The Tide in a Lake}{184}
\LoFFig{32}{Chart of Cotidal Lines}{190}
\LoFFig{33}{Tidal Abacus}{218}
\LoFFig{34}{Curves of Intervals and Heights at Portsmouth and at Aden}{227}
\LoFFig{35}{Diagram of Tide-predicting Instrument}{235}
\LoFFig{36}{Frictionally Retarded Tide}{266}
\LoFFig{37}{Maclaurin's and Jacobi's Ellipsoids of Equilibrium}{323}
\LoFFig{38}{Figures of Equilibrium}{325}
\LoFFig{39}{Hour-glass Figure of Equilibrium}{329}
\LoFFig{40}{Nebula in Andromeda}{339}
\LoFFig{41}{The Planet Saturn}{349}
\LoFFig{42}{Diagram of Saturn and his Rings}{353}
\LoFFig{43}{Roche's Figure of a Satellite when elongated to the utmost}{357}

\MainMatter
\PageSep{1}
\index{Barometric pressure|see{Atmospheric pressure}}%
\index{Bifilar|see{Pendulum}}%
\index{D'Abbadie|see{Abbadie}}%
\index{Horizontal tide-generating force|see{also Pendulum}}%
\index{Rebeur|see{Paschwitz}}%
\index{Thomson, Sir W.|see{Kelvin}}%
\index{Tidal problem|see{Laplace, Harmonic Analysis, etc.}}%
\index{Tide|see{also other headings; \eg\ for tide-generating force \textit{see} Force}}%
\index{Vertical|see{Deflection}}%
% [** TN: Text printed by \Chapter macro]
% THE TIDES


\Chapter{I}
{Tides and Methods of Observation}

\First{The} great wave caused by an earthquake is
\index{Tide!definition|(}%
often described in the newspapers as a tidal
wave, and the same name is not unfrequently
applied to such a short series of enormous waves
as is occasionally encountered by a ship in the
open sea. We must of course use our language
in the manner which is most convenient, but as
in this connection the adjective ``tidal'' implies
simply greatness and uncommonness, the use of
the term in such a sense cannot be regarded as
appropriate.

The word ``tidal'' should, I think, only be
used when we are referring to regular and persistent
alternations of rise and fall of sea-level.
Even in this case the term may perhaps be used
in too wide a sense, for in many places there is a
regular alternation of the wind, which blows in-shore
during the day and out during the night
with approximate regularity, and such breezes
\PageSep{2}
alternately raise and depress the sea-level, and
thus produce a sort of tide. Then in the Tropics
there is a regularly alternating, though small,
periodicity in the pressure of the atmosphere,
which is betrayed by an oscillation in the height
of the barometer. Now the ocean will respond
to the atmospheric pressure, so that the sea-level
will fall with a rising barometer, and rise
with a falling barometer. Thus a regularly
periodic rise and fall of the sea-level must result
from this cause also. Again, the melting of the
snows in great mountain ranges, and the annual
variability in rainfall and evaporation, produce
approximately periodic changes of level in the
estuaries of rivers, and although the period of
these changes is very long, extending as they do
over the whole year, yet from their periodicity
they partake of the tidal character.

These changes of water level are not, however,
tides in the proper sense of the term, and a true
tide can only be adequately defined by reference
to the causes which produce it. A tide, in fact,
means a rising and falling of the water of the
ocean caused by the attractions of the sun and
moon.

Although true tides are due to astronomical
causes, yet the effects of regularly periodic winds,
variation of atmospheric pressure, and rainfall
are so closely interlaced with the true tide that
in actual observation of the sea it is necessary to
\PageSep{3}
consider them both together. It is accordingly
practically convenient to speak of any regular
alternation of sea-level, due to the wind and to
the other influences to which I have referred, as
a Meteorological Tide. The addition of the adjective
``meteorological'' justifies the use of the
term ``tide'' in this connection.

We live at the bottom of an immense sea of
air, and if the attractions of the sun and moon
affect the ocean, they must also affect the air.
This effect will be shown by a regular rise and
fall in the height of the barometer. Although
such an effect is undoubtedly very small, yet it
is measurable. The daily heating of the air by
the sun, and its cooling at night, produce marked
alternations in the atmospheric pressure, and this
effect may by analogy be called an atmospheric
meteorological tide.

The attractions of the moon and sun must certainly
act not only on the sea, but also on the
solid earth; and, since the earth is not perfectly
rigid or stiff, they must produce an alternating
change in its shape. Even if the earth is now
so stiff that the changes in its shape escape
detection through their minuteness, yet such
changes of shape must exist. There is much
evidence to show that in the early stages of their
histories the planets consisted largely or entirely
of molten rock, which must have yielded to tidal
influences. I shall, then, extend the term ``tide''
\index{Tide!definition|)}%
\PageSep{4}
so as to include such alternating deformations of
a solid and elastic, or of a molten and plastic,
globe. These corporeal tides will be found to
lead us on to some far-reaching astronomical
speculations. The tide, in the sense which I
\index{Tide!general description|(}%
have attributed to the term, covers a wide field
of inquiry, and forms the subject of the present
volume.

I now turn to the simplest and best known
form of tidal phenomena. When we are at the
seashore, or on an estuary, we see that the water
rises and falls nearly twice a day. To be more
exact, the average interval from one high water
to the next is twelve hours twenty-five minutes,
and so high water falls later, according to the
clock, by twice twenty-five minutes, or by fifty
minutes, on each successive day. Thus if high
water falls to-day at noon, it will occur to-morrow
at ten minutes to one. Before proceeding, it
may be well to remark that I use high water and
low water as technical terms. In common parlance
the level of water may be called high or
low, according as whether it is higher or lower
than usual. But when the level varies periodically,
there are certain moments when it is highest
and lowest, and these will be referred to as
the times of high and low water, or of high and
of low tide. In the same way I shall speak of
the heights at high and low water, as denoting
the water-level at the moments in question.
\PageSep{5}

The most elementary observations would show
that the time of high water has an intimate relationship
to the moon's position. The moon, in
fact, passes the meridian on the average fifty
minutes later on each succeeding day, so that if
high water occurs so many hours after the moon
is due south on any day, it will occur on any
other day about the same number of hours after
the moon was south. This rule is far from being
exact, for it would be found that the interval
from the moon's passage to high water differs
considerably according to the age of the moon.
I shall not, however, attempt to explain at present
how this rough rule as to the time of high
water must be qualified, so as to convert it into
an accurate statement.

But it is not only the hour of high water which
changes from day to day, for the height to which
the water rises varies so conspicuously that the
fact could not escape the notice of even the most
casual observer. It would have been necessary
to consult a clock to discover the law by which
the hour of high water changes from day to day;
but at the seashore it would be impossible to
avoid noticing that some rocks or shoals which
are continuously covered by the sea at one part
of a fortnight are laid bare at others. It is, in
fact, about full and new moon that the range
from low to high water is greatest, and at the
moon's first or third quarter that the range is
\PageSep{6}
least. The greater tides are called ``springs,''
and the smaller ``neaps.''

The currents produced in the sea by tides are
\index{Gauge, tide!description of|(}%
often very complicated where the open sea is
broken by islands and headlands, and the knowledge
of tidal currents at each place is only to be
gained by the practical experience of the pilot.
Indeed, in the language of sailors, the word
``tide'' is not unfrequently used as meaning
tidal current, without reference to rise and fall.
These currents are often of great violence, and
vary from hour to hour as the water rises and
falls, so that the pilot requires to know how the
water stands in-shore in order to avail himself of
his practical knowledge of how the currents will
make in each place. A tide table is then of
\index{Tide!general description|)}%
much use, even at places where the access to a
harbor is not obstructed by a bar or shoal. It
is, of course, still more important for ships to
have a correct forecast of the tides where the
entrance to the harbor is shallow.

I have now sketched in rough outline some
of the peculiarities of the tides, and it will have
become clear that the subject is a complicated
one, not to be unraveled without regular observation.
\index{Observation!methods of tidal|(}%
I shall, therefore, explain how tides are
observed scientifically, and how the facts are collected
upon which the scientific treatment of the
tides is based.

The rise and fall of the sea may, of course, be
\PageSep{7}
roughly estimated by observing the height of the
water on posts or at jetties, which jut out into
moderately deep water. But as the sea is continually
disturbed by waves, observations of this
kind are not susceptible of accuracy, and for
\Figure{1}{Well for Tide-Gauge}{png}
scientific purposes more elaborate apparatus is
required. The exact height of the water can
only be observed in a place to which the sea has
a moderately free access, but where the channel
is so narrow as to prevent the waves from sensibly
disturbing the level of the water. This result
\PageSep{8}
is obtained in a considerable variety of ways,
but one of them may be described as typical of
all.

A well (\fig{1}) about two feet in diameter is
dug to a depth of several feet below the lowest
tide and in the neighborhood of deep water. The
well is lined with iron, and a two-inch iron pipe
runs into the well very near its bottom, and passes
down the shore to the low-water line. Here it is
joined to a flexible pipe running out into deep
water, and ending with a large rose pierced with
many holes, like that of a watering can. The
rose (\fig{2}) is anchored to the bottom of the
sea, and is suspended by means of a buoy, so as
to be clear of the bottom. The tidal water can
thus enter pretty freely into the well, but the
passage is so narrow that the wave motion is not
transmitted into the well. Inside the well there
floats a water-tight copper cylinder, weighted at
the bottom so that it floats upright, and counter-poised
so that it only just keeps its top clear of
the water. To the top of the float there is fastened
a copper tape or wire, which runs up to
the top of the well and there passes round a
wheel. Thus as the water rises and falls this
wheel turns backwards and forwards.

It is hardly necessary to describe in detail the
simple mechanism by which the turning of this
wheel causes a pencil to move backwards and forwards
in a straight line. The mechanism is,
\PageSep{9}
however, such that the pencil moves horizontally
backwards and forwards by exactly the same
amount as the water rises or falls in the well; or,
\Figure{2}{Pipe of Tide-Gauge}{png}
if the rise and fall of the tide is considerable,
the pencil only moves by half as much, or one
third, or even one tenth as much as the water.
At each place a scale of reduction is so chosen
as to bring the range of motion of the pencil
within convenient limits. We thus have a pencil
which will draw the rise and fall of the tide
on the desired scale.

It remains to show how the times of the rise
and fall are indicated. The end of the pencil
touches a sheet of paper which is wrapped round
a drum about five feet long and twenty-four
\PageSep{10}
\index{Curve, tide!irregularities in|(}%
inches in circumference. If the drum were kept
still the pencil would simply draw a straight line
to and fro along the length of the drum as the
water rises and falls. But the drum is kept
turning by clockwork, so that it makes exactly
one revolution in twenty-four hours. Since the
drum is twenty-four inches round, each inch of
circumference corresponds to one hour. If the
water were at rest the pencil would simply draw
a circle round the paper, and the beginning and
ending of the line would join, whilst if the drum
remained still and the water moved, the pencil
\Figure{3}{Indian Tide-Gauge}{jpg}
would draw a straight line along the length of
the cylinder; but when both drum and water
are in motion, the pencil draws a curve on the
cylinder from which the height of water may be
read off at any time in each day and night. At
the end of twenty-four hours the pencil has returned
to the same part of the paper from which
\PageSep{11}
\Figure[0.7]{4}{Légé's Tide-Gauge}{jpg}
\index{Gauge, tide!description of|)}%
\PageSep{12}
it started, and it might be thought that there
would be risk of confusion between the tides of
to-day and those of yesterday. But since to-day
the tides happen about three quarters of an hour
later than yesterday, it is found that the lines
keep clear of one another, and, in fact, it is
usual to allow the drum to run for a fortnight
before changing the paper, and when the old
sheet is unwrapped from the drum, there is
written on it a tidal record for a fortnight.

The instrument which I have described is
called a ``tide-gauge,'' and the paper a ``tide-curve.''
As I have already said, tide-gauges
may differ in many details, but this description
will serve as typical of all. Another form of
tide-gauge is shown in \fig{4}; here a continuous
sheet of paper is placed over the drum, so that
there is no crossing of the curves, as in the first
example. Yet another form, designed by Lord
Kelvin, is shown on p.~170 of vol.~iii.\ of his
``Popular Lectures.''

The actual record for a week is exhibited in
\index{Curve, tide!at Bombay}%
\fig{5}, on a reduced scale. This tide-curve was
drawn at Bombay by a tide-gauge of the pattern
first described. When the paper was wrapped
on the drum, the right edge was joined to the
left, and now that it is unwrapped the curve
must be followed out of the paper on the left
and into it again on the right. The figure
shows that spring tide occurred on April~26,
\PageSep{13}
1884; the preceding neap tide was on the~18th,
and is not shown. It may be noticed that the
law of the tide is conspicuously different from
that which holds good on the coast of England,
for the two successive high or low waters which
occur on any day have very different heights.
Thus, for example, on April~26 low water occurred
at 5.50~\PM, and the water fell to $5$~ft.\
$2$~in., whereas the next low water, occurring at
5.45~\AM\ of the~27th, fell to $1$~ft.\ $3$~in., the
heights being in both cases measured from a
certain datum. When we come to consider the
theory of the tides the nature of this irregularity
will be examined.

The position near the seashore to be chosen
for the erection of the tide-gauge is a matter of
much importance. The choice of a site is generally
limited by nature, for it should be near
the open sea, should be sheltered from heavy
weather, and deep water must be close at hand
even at low tide.

In the sketch map shown in \fig{6} a site such
as~$A$ is a good one when the prevailing wind
blows in the direction of the arrow. A position
such as~$B$, although well sheltered from heavy
seas, is not so good, because it is found that
tide-curves drawn at~$B$ would be much zigzagged.
These zigzags appear in the Bombay curves,
although at Bombay they are usually very
smooth ones.
\PageSep{14}

These irregularities in the tide-curve are not
due to tides, and as the object of the observation
\index{Observation!methods of tidal|)}%
is to determine the nature of the tides it is
\Figure{5}{Bombay Tide-Curve from Noon, April~22,
to Noon, April~30, 1884}{jpg}
desirable to choose a site for the gauge where
\index{Gauge, tide!site for}%
the zigzags shall not be troublesome; but it is
not always easy to foresee the places which will
furnish smooth tide-curves.

Most of us have probably at some time or
other made a scratch on the sand by the seashore,
\PageSep{15}
and watched the water rise over it. We
generally make our mark on the sand at the
furthest point, where the wash of a rather large
wave has brought up the water. For perhaps
five or ten minutes no wave brings the water up
as far as the mark, and one begins to think that
it was really an extraordinarily large wave which
was marked, although it did not seem so at the
time. Then a wave brings up the water far over
the mark, and immediately all the waves submerge
it. This little observation simply points
\Figure{6}{Sites for a Tide-Gauge}{png}
to the fact that the tide is apt to rise by jerks,
and it is this irregularity of rise and fall which
marks the notches in the tide-curves to which I
have drawn attention.

Now in scientific matters it is well to follow
up the clues afforded by such apparently insignificant
facts as this. An interesting light is
indeed thrown on the origin of these notches on
tide-curves by an investigation, not very directly
\PageSep{16}
connected with tidal observation, on which I shall
make a digression in the following chapter.
\index{Curve, tide!irregularities in|)}%

\begin{Authorities}
Baird's \Title{Manual for Tidal Observations} (Taylor \& Francis,
\index{Baird, \Title{Manual for Tidal Observation}}%
1886). Price 7\textit{s.}~6\textit{d.} Figs.\ \figref{1},~\figref{2},~\figref{3},~\figref{6} are reproduced from this
work.

The second form of tide-gauge shown in \fig{4} is made by
Messrs.\ Légé, and is reproduced from a woodcut kindly provided
by them.

Sir William Thomson's (Lord Kelvin's) \Title{Popular Lectures and
Addresses}, vol.~iii. (Macmillan,~1891).
\end{Authorities}
\PageSep{17}


\Chapter{II}
{Seiches in Lakes}

\First{It} has been known for nearly three centuries
\index{Lakes!seiches in|(}%
\index{Seiches!definition}%
that the water of the Lake of Geneva is apt to
\index{Geneva!seiches in lake|(}%
rise and fall by a few inches, sometimes irregularly
and sometimes with more or less regularity;
and the same sort of oscillation has been observed
in other Swiss lakes. These quasi-tides,
called seiches, were until recently supposed only
to occur in stormy weather, but it is now known
that small seiches are of almost daily occurrence.\footnote
  {The word ``seiche'' is a purely local one. It has been alleged
  to be derived from ``sèche,'' but I can see no reason for
  associating dryness with the phenomenon.}

Observations were made by Vaucher in the
\index{Vaucher, record of a great seiche at Geneva}%
last century on the oscillations of the Lake of
Geneva, and he gave an account of a celebrated
seiche in the year~1600, when the water oscillated
through three or four feet; but hardly any
systematic observation had been undertaken when
Professor Forel, of Lausanne, attacked the subject,
\index{Forel!on seiches|(}%
and it is his very interesting observations
which I propose to describe.

Doctor Forel is not a mathematician, but is
\PageSep{18}
rather a naturalist of the old school, who notes any
interesting fact and then proceeds carefully to
investigate its origin. His papers have a special
charm in that he allows one to see all the workings
of his mind, and tells of each difficulty as it
arose and how he met it. To those who like to
read of such work, almost in the form of a narrative,
I can strongly recommend these papers,
which afford an admirable example of research
thoroughly carried out with simple appliances.

People are nowadays too apt to think that
science can only be carried to perfection with
elaborate appliances, and yet it is the fact that
many of the finest experiments have been made
with cardboard, cork, and sealing-wax. The principal
reason for elaborate appliances in the laboratories
of universities is that a teacher could not
deal with a large number of students if he had
to show each of them how to make and set up
his apparatus, and a student would not be able
to go through a large field of study if he had to
spend days in preparation. Great laboratories
have, indeed, a rather serious defect, in that they
tend to make all but the very best students helpless,
and thus to dwarf their powers of resource
and inventiveness. The mass of scientific work
is undoubtedly enormously increased by these
institutions, but the number of really great investigators
seems to remain almost unaffected by
them. But I must not convey the impression
\PageSep{19}
that, in my opinion, great laboratories are not
useful. It is obvious, indeed, that without them
science could not be taught to large numbers of
students, and, besides, there are many investigations
in which every possible refinement of apparatus
is necessary. But I do say that the
number of great investigators is but little increased
by laboratories, and that those who are
interested in science, but yet have not access to
laboratories, should not give up their study in
despair.

Doctor Forel's object was, in the first instance,
to note the variations of the level of the lake, after
obliterating the small ripple of the waves on the
surface. The instrument used in his earlier investigations
was both simple and delicate. Its principle
was founded on casual observation at the port
of Morges, where there happens to be a breakwater,
pierced by a large ingress for ships and a small
one for rowing boats. He accidentally noticed
that at the small passage there was always a current
setting either inwards or outwards, and it
occurred to him that such a current would form
a very sensitive index of the rise and fall of the
water in the lake. He therefore devised an instrument,
illustrated in \fig{7}, and called by him
a plemyrameter, for noting currents of even the
\index{Plemyrameter, observation of seiches with|(}%
most sluggish character. Near the shore he
made a small tank, and he connected it with the
lake by means of an india-rubber siphon pipe of
\PageSep{20}
small bore. Where the pipe crossed the edge
of the tank he inserted a horizontal glass tube
of seven millimetres diameter, and in that tube
he put a float of cork, weighted with lead so that
it should be of the same density as water. At
the ends of the glass tube there were stops, so
that the float could not pass out of it. When
the lake was higher than the tank, the water ran
\Figure{7}{Plemyrameter}{png}
through the siphon pipe from lake to tank, and
the float remained jammed in the glass tube
against the stop on the side towards the tank;
and when the lake fell lower than the tank, the
float traveled slowly to the other end and remained
there. The siphon pipe being small, the
only sign of the waves in the lake was that the
float moved with slight jerks, instead of uniformly.
Another consequence of the smallness
of the tube was that the amount of water which
could be delivered into the tank or drawn out of
it in one or two hours was so small that it might
\PageSep{21}
practically be neglected, so that the water level
in the tank might be considered as invariable.

This apparatus enabled Forel to note the rise
and fall of the water, and he did not at first
attempt to measure the height of rise and fall,
as it was the periodicity in which he was principally
interested.

In order to understand the record of observations,
\index{Seiches!records of}%
it must be remembered that when the
float is towards the lake, the water in the tank
stands at the higher level, and when the float
is towards the tank the lake is the higher. In
the diagrams, of which \fig{8} is an example, the
straight line is divided into a scale of hours and
minutes. The zigzag line gives the record, and
the lower portions represent that the water of
the lake was below the tank, and the upper line
that it was above the tank. The fact that the
float only moved slowly across from end to end
of the glass tube, is indicated by the slope of
the lines, which join the lower and upper portions
of the zigzags. Then on reading \fig{8} we
see that from $2$~hrs.\ $1$~min.\ to $2$~hrs.\ $4$~mins.\ the
water was high and the float was jammed against
the tank end of the tube, because there was a
current from the lake to the tank. The float
then slowly left the tank end and traveled
across, so that at $2$~hrs.\ $5$~mins.\ the water was
low in the lake. It continued, save for transient
changes of level, to be low until $2$~hrs.\ $30$~mins.,
\PageSep{22}
when it rose again. Further explanation seems
unnecessary, as it should now be easy to read
this diagram, and that shown in~\fig{9}.
\index{Plemyrameter, observation of seiches with|)}%

The sharp pinnacles indicate alternations of
level so transient that the float had not time to
travel across from one end of the glass tube
to the other, before the current was reversed.
These pinnacles may be disregarded for the
present, since we are only considering seiches of
considerable period.

These two diagrams are samples of hundreds
which were obtained at various points on the
shores of Geneva, and of other lakes in Switzerland.
In order to render intelligible the method
by which Forel analyzed and interpreted these
records, I must consider \fig{8} more closely. In
this case it will be noticed that the record shows
a long high water separated from a long low
water by two pinnacles with flat tops. These
pieces at the ends have an interesting significance.
When the water of the lake is simply
oscillating with a period of about an hour we
have a trace of the form shown in~\fig{9}. But
when there exists concurrently with this another
oscillation, of much smaller range and of short
period, the form of the trace will be changed.
When the water is high in consequence of the
large and slow oscillation, the level of the lake
cannot be reduced below that of the tank by
the small short oscillation, and the water merely
\PageSep{23}
stands a little higher or a little lower, but always
remains above the level of the tank, so that the
trace continues on the higher level. But when,
in course of the changes of the large oscillation,
the water has sunk to near the mean level of the
%[** TN: Figures 8 and 9 combined; cross-refs handled by preamble code]
\Figure{8}{Records of Seiches at Évian}{png}
lake, the short oscillation will become manifest,
and so it is only at the ends of the long flat
pieces that we shall find evidence of the quick
oscillation.

Thus, in these two figures there was in one
case only one sort of wave, and in the other
there were two simultaneous waves. These
records are amongst the simplest of those obtained
by Forel, and yet even here the oscillations
of the water were sufficiently complicated.
It needed, indeed, the careful analysis of many
records to disentangle the several waves and to
determine their periods.

After having studied seiches with a plemyrameter
for some time, Forel used another form of
\PageSep{24}
apparatus, by which he could observe the amplitude
of the waves as well as their period. His
apparatus was, in fact, a very delicate tide-gauge,
which he called a limnimeter. The only
\index{Limnimeter, a form of tide gauge}%
difference between this instrument and the one
already described as a tide-gauge is that the
drum turned much more rapidly, so that five
feet of paper passed over the drum in twenty-four
hours, and that the paper was comparatively
narrow, the range of the oscillation being
small. The curve was usually drawn on the full
scale, but it could be quickly reduced to half
scale when large seiches were under observation.

It would be impossible in a book of this kind
\index{Lakes!mode of rocking in seiches}%
to follow Forel in the long analysis by which he
interpreted his curves. He speaks thus of the
complication of simultaneous waves: ``All these
oscillations are embroidered one on the other
and interlace their changes of level. There is
here matter to disturb the calmest mind. I
must have a very stout faith in the truth of my
hypothesis to persist in maintaining that, in the
midst of all these waves which cross and mingle,
there is, nevertheless, a recognizable rhythm.
This is, however, what I shall try to prove.''\footnote
  {\Title{Deuxième Étude}, p.~544.}
The hypothesis to which he here refers, and
triumphantly proves, is that seiches consist of a
rocking of the whole water of the lake about
fixed lines, just as by tilting a trough the water
\PageSep{25}
in it may be set swinging, so that the level at
the middle remains unchanged, while at the two
ends the water rises and falls alternately.

In another paper he remarks: ``If you will
\index{Lakes!mode of rocking in seiches}%
follow and study with me these movements you
will find a great charm in the investigation.
When I see the water rising and falling on the
shore at the end of my garden I have not before
me a simple wave which disturbs the water of
the bay of Morges, but I am observing the manifestation
of a far more important phenomenon.
It is the whole water of the lake which is rocking.
It is a gigantic impulse which moves the
whole liquid mass of Leman throughout its
length, breadth, and depth\dots. It is probable
that the same thing would be observed in far
larger basins of water, and I feel bound to
recognize in the phenomenon of seiches the
\index{Seiches!longitudinal and transverse|(}%
grandest oscillatory movement which man can
study on the face of our globe.''\footnote
  {\Title{Les Seiches, Vagues d'Oscillation}, p.~11.}

It will now be well to consider the map of
Geneva in~\fig{10}. Although the lake somewhat
resembles the arc of a circle, the curvature of
its shores will make so little difference in the
nature of the swinging of the water that we
may, in the first instance, consider it as practically
straight.

Forel's analysis of seiches led him to conclude
that the oscillations were of two kinds, the longitudinal
\PageSep{26}
and the transverse. In the longitudinal
seiche the water rocks about a line drawn across
the lake nearly through Morges, and the water
at the east end of the lake rises when that at
the west falls, and vice versa. The line about
which the water rocks is called a node, so that in
this case there is one node at the middle of the
lake. This sort of seiche is therefore called a
uninodal longitudinal seiche. The period of the
\Figure{10}{Map of Lake of Geneva}{png}
oscillation is the time between two successive
high waters at any place, and it was found to be
seventy-three minutes, but the range of rise and
fall was very variable. There are also longitudinal
seiches in which there are two nodes,
dividing the lake into three parts, of which the
central one is twice as long as the extreme parts;
such an oscillation is called a binodal longitudinal
seiche. In this mode the water at the middle
\PageSep{27}
of the lake is high when that at the two ends
is low, and vice versa; the period is thirty-five
minutes.

Other seiches of various periods were observed,
\index{Seiches!periods of}%
some of which were no doubt multinodal. Thus
in a trinodal seiche, the nodes divide the lake
into four parts, of which the two central ones
are each twice as long as the extreme parts. If
there are any number of nodes, their positions
are such that the central portion of the lake is
divided into equal lengths, and the terminal
parts are each of half the length of the central
part or parts. This condition is necessary in
order that the ends of the lake may fall at places
where there is no horizontal current. In all such
modes of oscillation the places where the horizontal
current is evanescent are called loops, and
these are always halfway between the nodes,
where there is no rise and fall.

A trinodal seiche should have a period of
about twenty-four minutes, and a quadrinodal
seiche should oscillate in about eighteen minutes.
The periods of these quicker seiches would, no
doubt, be affected by the irregularity in the form
and depth of the lake, and it is worthy of notice
that Forel observed at Morges seiches with
periods of about twenty minutes and thirty minutes,
which he conjectured to be multinodal.

The second group of seiches were transverse,
\index{Seiches!longitudinal and transverse|)}%
being observable at Morges and Évian. It was
\PageSep{28}
clear that these oscillations, of which the period
was about ten minutes, were transversal, because
at the moment when the water was highest at
Morges it was lowest at Évian, and vice versa.
As in the case of the longitudinal seiches, the
principal oscillation of this class was uninodal,
but the node was, of course, now longitudinal to
the lake. The irregularity in the width and
depth of the lake must lead to great diversity of
period in the transverse seiches appropriate to
the various parts of the lake. The transverse
seiches at one part of the lake must also be
transmitted elsewhere, and must confuse the
seiches appropriate to other parts. Accordingly
there is abundant reason to expect oscillations of
such complexity as to elude complete explanation.

The great difficulty of applying deductive
reasoning to the oscillations of a sheet of water
of irregular outline and depth led Forel to construct
a model of the lake. By studying the
waves in his model he was able to recognize
many of the oscillations occurring in the real
lake, and so obtained an experimental confirmation
of his theories, although the periods of
oscillation in the model of course differed enormously
from those observed in actuality.

The theory of seiches cannot be considered as
demonstrated, unless we can show that the water
of such a basin as that of Geneva is capable of
\index{Geneva!seiches in lake|)}%
\index{Geneva!model of lake}%
\PageSep{29}
swinging at the rates observed. I must, therefore,
now explain how it may be proved that the
periods of the actual oscillations agree with the
facts of the case.

As a preliminary let us consider the nature of
wave motion. There are two very distinct cases
of the undulatory motion of water, which nevertheless
graduate into one another. The distinction
lies in the depth of the water compared with
the length of the wave, measured from crest to
crest, in the direction of wave propagation. The
wave-length may be used as a measuring rod,
and if the depth of the water is a small fraction
of the wave-length, it must be considered shallow,
but if its depth is a multiple of the wave-length,
it will be deep. The two extremes of
course graduate into one another.

In a wave in deep water the motion dies out
pretty rapidly as we go below the surface, so that
when we have gone down half a wave-length
below the surface, the motion is very small. In
shallow water, on the other hand, the motion extends
quite to the bottom, and in water which is
neither deep nor shallow, the condition of affairs
is intermediate. The two figures, \figref{11}~and~\figref{12},
show the nature of the movement in the two
classes of waves. In both cases the dotted lines
\index{Waves!in deep and shallow water}%
show the position of the water when at rest, and
the full lines show the shapes assumed by the
rectangular blocks marked out by the dotted
\PageSep{30}
lines, when wave motion is disturbing the water.
It will be observed that in the deep water, as
shown in \fig{11}, the rectangular blocks change
their shape, rise and fall, and move to and fro.
Taking the topmost row of rectangles, each block
of water passes successively in time through all
the forms and positions shown by the top row
of quasi-parallelograms. So also the successive
changes of the second row of blocks are indicated
by the second strip, and the third and the fourth
indicate the same. The changes in the bottom
\Figure[0.7]{11}{Wave in Deep Water}{png}
row are relatively very small both as to shape
and as to displacement, so that it did not seem
worth while to extend the figure to a greater
depth.

Turning now to the wave in shallow water in~\fig{12},
we see that each of the blocks is simply
displaced sideways and gets thinner or more
\PageSep{31}
squat as the wave passes along. Now, I say that
we may roughly classify the water as being deep
with respect to wave motion when its depth is
more than half a wave-length, and as being shallow
when it is less. Thus the same water may
be shallow for long waves and deep for short
\index{Waves!speed of}%
ones. For example, the sea is very shallow for
\Figure[0.7]{12}{Wave in Shallow Water}{png}
the great wave of the oceanic tide, but it is very
deep even for the largest waves of other kinds.
Deepness and shallowness are thus merely relative
to wave-length.

The rate at which a wave moves can be exactly
calculated from mathematical formulæ,
from which it appears that in the deep sea a
wave $63$~metres in length travels at $36$~kilometres
per hour, or, in British measure, a wave of
$68$~yards in length travels $22\frac{1}{2}$~miles an hour.
Then, the rule for other waves is that the speed
varies as the square root of the wave-length, so
that a wave $16$~metres long---that is, one quarter
of $63$~metres---travels at $18$~kilometres an
hour, which is half of $36$~kilometres an hour.
Or if its length were $7$~metres, or one ninth as
\PageSep{32}
long, it would travel at $12$~kilometres an hour,
or one third as quick.

Although the speed of waves in deep water
depends on wave-length, yet in shallow water the
speed is identical for waves of all lengths, and
depends only on the depth of the water. In
water $10$~metres deep, the calculated velocity of
a wave is $36$~kilometres an hour; or if the water
were $2\frac{1}{2}$~metres deep (quarter of $10$~metres), it
would travel $18$~kilometres (half of $36$~kilometres)
an hour; the law of variation being that
the speed of the wave varies as the square root
of the depth. For water that is neither deep nor
shallow, the rate of wave propagation depends
both on depth and on wave-length, according to
a law which is somewhat complicated.

In the case of seiches, the waves are very long
compared with the depth, so that the water is to
be considered as shallow; and here we know
that the speed of propagation of the wave depends
only on depth. The average depth of the
Lake of Geneva may be taken as about $150$~metres,
and it follows that the speed of a long wave
in the lake is about $120$~kilometres an hour.

In order to apply this conclusion to the study
of seiches, we have to consider what is meant by
the composition of two waves. If I take the
series of numbers
\[
\text{\&c.}\quad
100\quad 71\quad 0\quad -71\quad -100\quad -71\quad 0\quad 71\quad 100\quad
\text{\&c.}
\]
and plot out, at equal distances, a figure of
\PageSep{33}
\index{Waves!composition of|(}%
heights proportional to these numbers, setting
off the positive numbers above and the negative
numbers below a horizontal line, I get the simple
wave line shown in~\fig{13}. Now, if this
wave is traveling to the right, the same series of
numbers will represent the wave at a later time,
when they are all displaced towards the right, as
in the dotted line.

Now turn to the following schedule of numbers,
and consider those which are written in the
top row of each successive group of three rows.
The columns represent equidistant spaces, and
the rows equidistant times. The first set of
numbers, $-100$,~$-71$,~$0$,~\&c., are those which
were plotted out as a wave in~\fig{13}; in the top
\Figure[0.7]{13}{Simple Wave}{png}
row of the second group they are the same, but
moved one space to the right, so that they represent
the movement of the wave to the right in
one interval of time. In the top row of each
successive group the numbers are the same, but
always displaced one more space to the right;
they thus represent the successive positions of a
\PageSep{34}
\begin{figure}[hp!]
\centering
\ifthenelse{\boolean{ForPrinting}}{%
  \makebox[0pt][c]{\includegraphics[height=0.9\textheight]{./images/fig13b.pdf}}
}{
  \includegraphics[width=0.75\textwidth]{./images/fig13b.pdf}
}
\caption{Composition of Two Equal and Opposite Waves}
\label{fig:13b}
\iffalse
-100    -71     0    71    100    71     0    -71    -100

-100    -71     0    71    100    71     0    -71    -100
[*--]
-200   -142     0   142    200   142     0   -142    -200
[*--]
 -71   -100   -71     0     71   100    71      0     -71

 -71      0    71   100     71     0   -71   -100     -71
[*--]
-142   -100     0   100    142   100     0   -100    -142
[*--]
   0    -71  -100   -71      0    71   100     71       0

   0     71   100    71      0   -71  -100    -71       0
[*--]
   0      0     0     0      0     0     0      0       0
[*--]
  71      0   -71  -100    -71     0    71    100      71

  71    100    71     0    -71  -100   -71      0      71
[*--]
 142    100     0  -100   -142  -100     0    100     142
[*--]
 100     71     0   -71   -100   -71     0     71     100

 100     71     0   -71   -100   -71     0     71     100
[*--]
 200    142     0  -142   -200  -142     0    142     200
[*--]
  71    100    71     0    -71  -100   -71      0      71

  71      0   -71  -100    -71     0    71    100      71
[*--]
 142    100     0  -100   -142  -100     0    100     142
[*--]
   0     71   100    71      0   -71  -100     -71      0

   0    -71  -100   -71      0    71   100      71      0
[*--]
   0      0     0     0      0     0     0       0      0
[*--]
 -71      0    71   100     71     0   -71    -100    -71

 -71   -100   -71     0     71   100    71       0    -71
[*--]
-142   -100     0   100    142   100     0    -100   -142
[*--]
-100 \DPtypo{71}{-71} 0    71    100    71     0     -71   -100

-100    -71     0    71    100    71     0     -71   -100
[*--]
-200   -142     0   142    200   142     0    -142   -200
[*--]
\fi
\end{figure}
\PageSep{35}
wave moving to the right. The table ends in
the same way as it begins, so that in eight of
these intervals of time the wave has advanced
through a space equal to its own length.

If we were to invert these upper figures, so
that the numbers on the right are exchanged
with those on the left, we should have a series of
numbers representing a wave traveling to the
left. Such numbers are shown in the second
row in each group.

When these two waves coëxist, the numbers
must be compounded together by addition, and
then the result is the series of numbers written
in the third rows. These numbers represent the
resultant of a wave traveling to the right, and of
an equal wave traveling simultaneously to the
left.

It may be well to repeat that the first row of
each group represents a wave moving to the
right, the second row represents a wave moving
to the left, and the third represents the resultant
of the two. Now let us consider the nature
of this resultant motion; the third and the
seventh columns of figures are always zero, and
therefore at these two places the water neither
rises nor falls,---they are, in fact, nodes. If the
schedule were extended indefinitely both ways,
exactly halfway between any pairs of nodes
there would be a loop, or line across which there
is no horizontal motion. In the schedule, as it
\PageSep{36}
stands, the first, fifth, and ninth columns are
loops.

At the extreme right and at the extreme left
the resultant numbers are the same, and represent
a rise of the water from $-200$ to~$+200$,
and a subsequent fall to~$-200$ again. If these
nine columns represent the length of the lake,
the motion is that which was described as binodal,
for there are two nodes dividing the lake
into three parts, there is a loop at each end, and
when the water is high in the middle it is low at
the ends, and vice versa. It follows that two
equal waves, each as long as the lake, traveling
in opposite directions, when compounded together
give the motion which is described as the
binodal longitudinal seiche.

Now let us suppose that only five columns of
the table represent the length of the lake. The
resultant numbers, which again terminate at
each end with a loop, are:---
\begin{align*}
-200&& -142&&  0&&  142&&  200& \displaybreak[0] \\
-142&& -100&&  0&&  100&&  142& \displaybreak[0] \\
   0&&    0&&  0&&    0&&    0& \displaybreak[0] \\
 142&&  100&&  0&& -100&& -142& \displaybreak[0] \\
 200&&  142&&  0&& -142&& -200& \displaybreak[0] \\
 142&&  100&&  0&& -100&& -142& \displaybreak[0] \\
   0&&    0&&  0&&    0&&    0& \displaybreak[0] \\
-142&& -100&&  0&&  100&&  142& \displaybreak[0] \\
-200&& -142&&  0&&  142&&  200&
\end{align*}

Since the middle column consists of zero
throughout, the water neither rises nor falls
\PageSep{37}
there, and there is a node at the middle. Again,
since the numbers at one end are just the same
as those at the other, but reversed as to positive
and negative, when the water is high at one end
it is low at the other. The motion is, in fact, a
simple rocking about the central line, and is that
described as the uninodal longitudinal seiche.

The motion is here again the resultant of two
equal waves moving in opposite directions, and
the period of the oscillation is equal to the time
which either simple wave takes to travel through
its own length. But the length of the wave is
now twice that of the lake. Hence it follows
that the period of the rocking motion is the
time occupied by a wave in traveling twice the
length of the lake. We have already seen that
in shallow water the rate at which a wave moves
is independent of its length and depends only
on the depth of the water, and that in water of
the same depth as the Lake of Geneva the wave
travels $120$~kilometres an hour. The Lake of
Geneva is $70$~kilometres long, so that the two
waves, whose composition produces a simple rocking
\index{Waves!composition of|)}%
of the water, must each of them have a
length of $140$~kilometres. Hence it follows that
the period of a simple rocking motion, with one
node in the middle of the Lake of Geneva, will
be almost exactly $\frac{140}{120}$~of an hour, or $70$~minutes.
Forel, in fact, found the period to be $73$~minutes.
He expresses this result by saying that
\PageSep{38}
a uninodal longitudinal seiche in the Lake of
Geneva has a period of $73$~minutes. His observations
also showed him that the period of a
binodal seiche was $35$~minutes. It follows from
the previous discussion that when there are two
nodes the period of the oscillation should be
half as long as when there is one node. Hence,
we should expect that the period would be
about $36$~or $37$~minutes, and the discrepancy
between these two results may be due to the
fact that the formula by which we calculate the
period of a binodal seiche would require some
correction, because the depth of the lake is not
so very small compared with the length of these
shorter waves.

It is proper to remark that the agreement
between the theoretical and observed periods is
suspiciously exact. The lake differs much in
depth in different parts, and it is not quite certain
what is the proper method of computing
the average depth for the determination of the
period of a seiche. It is pretty clear, in fact,
that the extreme closeness of the agreement is
accidentally due to the assumption of a round
number of metres as the average depth of the
lake. The concordance between theory and observation
must not, however, be depreciated too
much, for it is certain that the facts of the case
agree well with what is known of the depth of
the lake.
\index{Forel!on seiches|)}%
\PageSep{39}

The height of the waves called ``seiches'' is
\index{Seiches!causes of}%
\index{Storms a cause of seiches}%
very various. I have mentioned an historical
seiche which had a range of as much as four
feet, and Forel was able by his delicate instruments
still to detect them when they were only a
millimetre or a twenty-fifth of an inch in height.
It is obvious, therefore, that whatever be the
cause of seiches, that cause must vary widely
in intensity. According to Forel, seiches arise
from several causes. It is clear that anything
which heaps up the water at one end of the
lake, and then ceases to act, must tend to produce
an oscillation of the whole. Now, a rise
of water level at one end or at one side of the
lake may be produced in various ways. Some,
and perhaps many, seiches are due to the tilting
of the whole lake bed by minute earthquakes.
\index{Earthquakes!a cause of seiches}%
Modern investigations seem to show that this is
a more fertile cause than Forel was disposed to
allow, and it would therefore be interesting to
see the investigation of seiches repeated with the
aid of delicate instruments for the study of
earthquakes, some of which will be described in
\Ref{Chapter}{VI}. I suspect that seiches would be
observed at times when the surface of the earth
is much disturbed.

The wind is doubtless another cause of seiches.
\index{Wind!a cause of seiches}%
When it blows along the lake for many hours in
one direction, it produces a superficial current,
and heaps up the water at the end towards
\PageSep{40}
\index{Atmospheric pressure!cause of seiches}%
which it is blowing. If such a wind ceases
somewhat suddenly, a seiche will certainly be
started, and will continue for hours until it dies
out from the effects of the friction of the water
on the lake bottom. Again, the height of the
barometer will often differ slightly at different
parts of the lake, and the water will respond, just
as does the mercury, to variations of atmospheric
pressure. About a foot of rise of water should
correspond to an inch of difference in the height
of barometer. The barometric pressure cannot
be quite uniform all over the Lake of Geneva,
and although the differences must always be
exceedingly small, yet it is impossible to doubt
that this cause, combined probably with wind,
will produce many seiches. I shall return later
\index{Seiches!causes of}%
to the consideration of an interesting speculation
as to the effects of barometric pressure on
the oscillation of lakes and of the sea. Lastly,
Forel was of opinion that sudden squalls or local
storms were the most frequent causes of seiches.
\index{Storms a cause of seiches}%
I think that he much overestimated the efficiency
of this cause, because his theory of the path of
the wind in sudden and local storms is one that
would hardly be acceptable to most meteorologists.

Although, then, it is possible to indicate causes
competent to produce seiches, yet we cannot as
yet point out the particular cause for any individual
seiche. The complication of causes is so
\PageSep{41}
\index{Lakes!vibrations|(}%
\index{Vibration of lakes|(}%
great that this degree of uncertainty will probably
never be entirely removed.

But I have not yet referred to the point which
justifies this long digression on seiches in a book
on the tides. The subject was introduced by
the irregularities in the line traced by the tide-gauge
at Bombay, which indicated that there
are oscillations of the water with periods ranging
from two minutes to a quarter of an hour or
somewhat longer. Now these zigzags are not
found in the sea alone, for Forel observed on
the lake oscillations of short period, which resembled
seiches in all but the fact of their more
rapid alternations. Some of these waves are
perhaps multinodal seiches, but it seems that
they are usually too local to be true seiches
affecting the whole body of the lake at one time.
Forel calls these shorter oscillations ``vibrations,''
thus distinguishing them from proper seiches.
A complete theory of the so-called vibrations
has not yet been formulated, although, as I shall
show below, a theory is now under trial which
serves to explain, at least in part, the origin of
vibrations.

Forel observed with his limnimeter or tide-gauge
that when there is much wind, especially
\index{Wind!vibrations of lakes due to}%
from certain quarters, vibrations arise which are
quite distinct from the ordinary visible wave
motion. The period of the visible waves on the
\PageSep{42}
Lake of Geneva is from $4$~to $5$~seconds,\footnote
  {I observed when it was blowing half a gale on Ullswater, in
  Cumberland, that the waves had a period of about a second.}
whereas
vibrations have periods ranging from $45$~seconds
to $4$~minutes. Thus there is a clear line separating
waves from vibrations. Forel was unable
to determine what proportion of the area of the
lake is disturbed by vibrations at any one time,
and although their velocity was not directly observed,
there can be no doubt that these waves
are propagated at a rate which corresponds to
their length and to the depth of the water. I
have little doubt but that the inequalities which
produce notches in a tide-curve have the same
origin as vibrations on lakes.

It is difficult to understand how a wind, whose
\index{Wind!vibrations of lakes due to}%
only visible effect is short waves, can be responsible
for raising waves of a length as great as a
thousand yards or a mile, and yet we are driven
to believe that this is the case. But Forel also
found that steamers produce vibrations exactly
like those due to wind. The resemblance was
indeed so exact that vibrations due to wind
could only be studied at night, when it was
known that no steamers were traveling on the
lake, and, further, the vibrations due to steamers
could only be studied when there was no wind.

His observations on the steamer vibrations are
amongst the most curious of all his results.
When a boat arrives at the pier at Morges, the
\PageSep{43}
water rises slowly by about $5$~to $8$~millimetres,
and then falls in about $20$~to $30$~seconds. The
amount and the rapidity of the rise and fall
vary with the tonnage of the boat and with the
rate of her approach. After the boat has passed,
the trace of the limnimeter shows irregularities
with sharp points, the variations of height ranging
from about two to five millimetres, with a
period of about two minutes. These vibrations
continue to be visible during two to three hours
after the boat has passed. As these boats travel
at a speed of $20$~kilometres an hour, the vibrations
persist for a long time after any renewal
of them by the boat has ceased. These vibrations
are called by Forel ``the subsequent steamer
vibrations.''

That the agitation of the water should continue
for more than two hours is very remarkable,
and shows the delicacy of the method of
observation. But it seems yet more strange
that, when a boat is approaching Morges, the
vibrations should be visible during $25$~minutes
before she reaches the pier. These he calls
``antecedent steamer vibrations.'' They are
more rapid than the subsequent ones, having a
period of a minute to a minute and a quarter.
Their height is sometimes two millimetres (a
twelfth of an inch), but they are easily detected
when less than one millimetre in height. It
appears that these antecedent vibrations are first
\PageSep{44}
noticeable when the steamer rounds the mole of
Ouchy, when she is still at a distance of $10$~kilometres.
As far as one can judge from the speed
at which waves are transmitted in the Lake of
Geneva, the antecedent vibrations, which are
noticed $25$~minutes before the arrival of the
boat, must have been generated when she was at
a distance of $12$~kilometres from Morges. \Fig{14}
gives an admirable tracing of these steamer
vibrations.\footnote
  {From \Title{Les Seiches, Vagues d'Oscillation fixe des Lacs}, 1876.}

In this figure the line~$\Seg{a}{a'}$ was traced between
two and three o'clock in the morning, and shows
scarcely any sign of perturbation. Between
three and eight o'clock in the morning no observations
were taken, but the record begins again
at eight o'clock. The portion marked~$\Seg{b}{b'}$ shows
weak vibrations, probably due to steamers passing
along the coast of Savoy. The antecedent
vibrations, produced by a steamer approaching
Morges, began about the time of its departure
from Ouchy, and are shown at~$\Seg{c}{c'}$. The point~$d$
shows the arrival of this boat at Morges, and
$d'$~shows the effect of another boat coming from
Geneva. The portion marked~$\Seg[e]{e}{e}$ shows the
subsequent steamer vibrations, which were very
clear during more than two hours after the boats
had passed.

Dr.~Forel was aware that similar vibrations occur
in the sea, for he says: ``What are these
\index{Sea!vibrations of}%
\PageSep{45}
oscillations with periods of
$5$,~$10$, $20$, or $100$~minutes,
which are sometimes irregular?
Are they analogous to
our seiches? Not if we define
seiches as uninodal oscillations,
for it is clear that
if, in a closed basin of $70$~kilometres
in length, uninodal
seiches have a period of
$73$~minutes, in the far greater
basin of the Mediterranean,
or of the ocean, a uninodal
wave of oscillation must have
a much longer period. They
resemble much more closely
what I have called vibrations,
and, provisionally, I
shall call them by the name
of `vibrations of the sea.' I
\index{Sea!vibrations of}%
venture to invite men of science
who live on the seacoast
to follow this study.
It presents a fine subject for
research, either in the interpretation
of the phenomenon
or in the establishment of
the relations between these
movements and meteorological
conditions.''\footnote
  {\Title{Seiches et Vibrations des Lacs et de la Mer}, 1879, p.~5.}
%[** TN: Figure wrapped in the original]
\TallFig{14}{Vibrations due to Steamers}{png}
\PageSep{46}

These vibrations are obviously due to the wind
or to steamers, but it is a matter of no little surprise
that such insignificant causes should produce
even very small waves of half a mile to a
mile in length.

The manner in which this is brought about is
undoubtedly obscure, yet it is possible to obtain
some sort of insight into the way in which these
long waves arise. When a stone falls into calm
water waves of all sorts of lengths are instantaneously
generated, and the same is true of
any other isolated disturbance. Out of all these
waves the very long ones and the very short
ones are very small in height. Theoretically,
waves of infinitely great and of infinitely small
lengths, yet in both cases of infinitely small
heights, are generated at the instant of the impulse,
but the waves of enormous length and
those of very small length are of no practical
importance, and we need only consider the moderate
waves. For the shorter of these the water
is virtually deep, and so they will each travel
outwards at a pace dependent on length, the
longer ones outstripping the shorter ones. But
for the longer waves the water will be shallow,
and they will all travel together. Thus the general
effect at a distance is the arrival of a long
wave first, followed by an agitated rippling.
The point which we have to note is that an isolated
disturbance will generate long waves and
\PageSep{47}
that they will run ahead of the small ones. It
is important also to observe that the friction of
the water annuls the oscillation in the shorter
waves more rapidly than it does that of the
longer ones, and therefore the long waves are
more persistent. Now we may look at the disturbance
due to a steamer or to the wind as consisting
of a succession of isolated disturbances,
each of which will create long waves outstripping
the shorter ones. These considerations afford a
sort of explanation of what is observed, but I do
not understand how it is that the separation of
the long from the short waves is so complete, nor
what governs the length of the waves, nor have
I made any attempt to evaluate the greater rapidity
of decrease of short waves than long ones.\footnote
  {See, however, S.~S. Hough, \Title{Proc.\ Lond.\ Math.\ Soc.}, xxviii.\
\index{Hough, S. S.!frictional extinction of waves}%
  p.~276.}
It must then be left to future investigators to
elucidate these points.

\TB

The subject of seiches and vibrations clearly
affords an interesting field for further research.
The seiches of Lake George in New South Wales
have been observed by Mr.~Russell, the government
\index{Russell, observation of seiches in New South Wales}%
astronomer at Sydney; but until last year
they do not seem to have been much studied on
any lakes outside of Switzerland. The great
lakes of North America are no doubt agitated by
seiches on a much larger scale than those on the
\PageSep{48}
comparatively small basin of Geneva. This idea
appears to have struck Mr.~Napier Denison of
\index{Denison, F. Napier, vibrations and seiches on lakes|(}%
Toronto, and he has been so fortunate as to enlist
the interest of Mr.~Bell Dawson, the chief of
\index{Dawson coöperates in investigation of seiches}%
the Canadian Tidal Survey, and of Mr.~Stupart,
\index{Stupart coöperates in investigation of seiches}%
the director of the Meteorological Department.
Mr.~Denison's attention has been, in the first instance,
principally directed towards those notches
in tide-curves which have afforded the occasion
for the present discussion of this subject. He
has made an interesting suggestion as to the
origin of these oscillations, which I will now
explain.

The wind generally consists of a rather shallow
\index{Waves!in atmosphere|(}%
current, so that when it is calm at the earth's
surface there is often a strong wind at the top
of a neighboring mountain; or the wind aloft
may blow from a different quarter from that below.
If we ascend a mountain or go up in a
balloon, the temperature of the air falls on the
average by a certain definite number of degrees
per thousand feet. But the normal rate of fall
of temperature is generally interrupted on passing
into an upper current, which blows from a
different direction. This abrupt change of temperature
corresponds with a sudden change of
density, so that the upper layer of air must be
regarded as a fluid of different density from that
of the lower air, over which it slides.

Now Helmholtz has pointed out that one layer
\index{Atmospheric waves, Helmholtz on|(}%
\index{Helmholtz!on atmospheric waves|(}%
\PageSep{49}
of fluid cannot slide over another, without generating
waves at the surface of separation. We
are familiar with this fact in the case of sea-waves
generated by wind. A mackerel sky
\index{Mackerel sky, evidence of air-waves}%
proves also the applicability to currents of air of
Helmholtz's observation. In this case the moisture
of the air is condensed into clouds at the
crests of the air waves, and reabsorbed in the
hollows, so that the clouds are arranged in a visible
ripple-mark. A mackerel sky is not seen in
stormy weather, for it affords proof of the existence
of an upper layer of air sliding with only
moderate velocity over a lower layer. The distance
from crest to crest must be considerable
as measured in yards, yet we must regard the
mackerel sky as a mere ripple formed by a slow
relative velocity of the two layers. If this is so,
it becomes of interest to consider what wave-lengths
may be expected to arise when the upper
current is moving over the lower with a speed of
perhaps a hundred miles an hour. The problem
is not directly soluble, for even in the case of
sea-waves it is impossible to predict the wave-lengths.
We do know, however, that the duration
of the wind and the size of the basin are
material circumstances, and that in gales in the
open ocean the waves attain a very definite magnitude.

Although the problem involved is not a soluble
one, yet Helmholtz has used the analogy of
\PageSep{50}
oceanic waves for an approximate determination
of the sizes of the atmospheric ones. His
method is a very fertile one in many complex
physical investigations, where an exact solution
is not attainable. The method may be best illustrated
by one or two simple cases.

It is easy for the mathematician to prove that
the period of a swing of a simple pendulum must
vary as the square root of its length. The proof
does not depend on the complete solution of the
problem, so that even if it were insoluble he
would still be sure of the correctness of his conclusion.
If, then, a given pendulum is observed
to swing in a certain period, it is certain that a
similar pendulum of four times the length will
take twice as long to perform its oscillation. In
the same way, the engine power required for a
ship is determinable from experiments on the
resistance suffered by a small model when towed
through the water. The correct conclusion is
discovered in this case, although it is altogether
impossible to discover the resistance of a ship
by \textit{à~priori} reasoning.

The wave motion at the surface separating
two fluids of different densities presents another
problem of the same kind, and if the result is
known in one case, it can be confidently predicted
in another. Now oceanic waves generated
\index{Waves!in atmosphere|)}%
by wind afford the known case, and Helmholtz
has thence determined by analogy the
\PageSep{51}
lengths of the atmospheric waves which must
exist aloft. By making plausible suppositions
as to the densities of the two layers of air and
as to their relative velocity, he has shown that
sea-waves of ten yards in length will correspond
with air-waves of as much as twenty miles. A
wave of this length would cover the whole sky,
and might have a period of half an hour. It is
clear then that mackerel sky will disappear in
stormy weather, because we are too near to the
crests and furrows to observe the orderly arrangement
of the clouds.

Although the waves are too long to be seen as
such, yet the unsteadiness of the barometer in a
gale of wind affords evidence of the correctness
of this theory. In fact, when the crest of denser
air is over the place of observation the barometer
rises, and it falls as the hollow passes. The
waves in the continuous trace of the barometer
have some tendency to regularity, and have
periods of from ten minutes to half an hour.
The analogy seems to be pretty close with the
confused and turbulent sea often seen in a gale
of wind in the open ocean.\footnote
  {A gust of wind will cause the barometer to vary, without a
  corresponding change in the density of the air. It is not therefore
  safe to interpret the oscillations of the barometer as being
  due entirely to true changes of pressure. If, however, the intermittent
  squalls in a gale are connected with the waves aloft,
  the waviness of the barometric trace would still afford signals
  of the passage of crests and hollows above.}
\index{Atmospheric waves, Helmholtz on|)}%
\index{Helmholtz!on atmospheric waves|)}%
\PageSep{52}

Mr.~Denison's application of this theory consists
in supposing that the vibrations of the sea
and of lakes are the response of the water to
variations in the atmospheric pressure. The sea,
being squeezed down by the greater pressure,
should fall as the barometer rises, and conversely
should rise as the barometer falls. He is engaged
in a systematic comparison of the simultaneous
excursions of the water and of the barometer
on Lake Huron. Thus far the evidence
seems decidedly favorable to the theory. He
concludes that when the water is least disturbed,
so also is the barometric trace; and that when
the undulations of the lake become large and
rapid, the atmospheric waves recorded by the
barometer have the same character. There is
also a considerable degree of correspondence
between the periods of the two oscillations. The
smaller undulations of the water correspond with
the shorter air-waves, and are magnified as they
run into narrower and shallower places, so as to
make conspicuous ``vibrations.''

It is interesting to note that the vibrations of
the water have a tendency to appear before those
in the barometer, so that they seem to give a
warning of approaching change of weather. It
is thus not impossible that we here have the
foreshadowing of a new form of meteorological
instrument, which may be of service in the forecasting
of the weather.
\PageSep{53}

I must, however, emphasize that these conclusions
are preliminary and tentative, and that
much observation will be needed before they can
be established as definite truths. Whatever
may be the outcome, the investigation appears
promising, and it is certainly already interesting.
\index{Denison, F. Napier, vibrations and seiches on lakes|)}%
\index{Lakes!vibrations|)}%
\index{Vibration of lakes|)}%

\begin{Authorities}
Papers by Dr.~Forel on Seiches.
\index{Forel!list of papers}%

\Journal ``Bibliothèque Universelle, Archives des Sciences physiques
et naturelles,'' Geneva:---

\Paper{\Title{Formule des Seiches}, 1876.}

\Paper{\Title{Limnimètre Enregistreur}, 1876.}

\Paper{\Title{Essai monographique}, 1877.}

\Paper{\Title{Causes des Seiches}, \DPchg{Sept.~15}{15~Sept.}, 1878.}

\Paper{\Title{Limnographe}, 15~Déc., 1878.}

\Paper{\Title{Seiche du \emph{20}~Février, \emph{1879}}, 15~Avril, 1879.}

\Paper{\Title{Seiches dicrotes}, 15~Jan., 1880.}

\Paper{\Title{Formules des Seiches}, 15~Sept., 1885.}

\Journal ``Bulletin de la Soc.\ Vaudoise des Sciences naturelles:''---

\Paper{\Title{Première Étude}, 1873.}

\Paper{\Title{Deuxième Étude}, 1875.}

\Paper{\Title{Limnimétrie du Lac Léman}. I\iere~Série. Bull.~xiv.\ 1877.
II\ieme~Série. Bull.~xv. III\ieme~Série. Bull.~xv. 1879.}

\Journal ``Actes de la Soc.\ helv.\ Andermatt:''---

\Paper{\Title{Les Seiches, Vagues d'Oscillation}, 1875.}

\Journal ``Association Française pour l'avancement,'' etc.:---

\Paper{\Title{Seiches et Vibrations}, Congrès de Montpelier, 1879.}

\Journal ``Annales de Chimie et de Physique:''---

\Paper{\Title{Les Seiches, Vagues d'Oscillation}, 1876.}

\Paper{\Title{Un Limnimètre Enregistreur}, 1876.}

\TB

Helmholtz, Sitzungsberichte der Preuss.\ Akad.\ der Wissenschaft,
July~25, 1889; transl.\ by Abbe in \Title{Smithsonian Reports}.
\PageSep{54}

F.~Napier Denison:---

\Paper{\Title{Secondary Undulations~\dots\ found in Tide-Gauges.} ``Proc.\
Canadian Institute,'' Jan.~16, 1897.}

\Paper{\Title{The Great Lakes as a Sensitive Barometer.} ``Proc.\ Canadian
Institute,'' Feb.~6, 1897.}
\index{Lakes!seiches in|)}%

\Paper{Same title, but different paper, ``Canadian Engineer,'' Oct.\
and Nov., 1897.}
\end{Authorities}
\index{Forel!list of papers}%
\PageSep{55}


%[** TN: Footnote mark handled by \Chapter logic]
\Chapter{III}
{Tides in Rivers---Tide Mills}

\footnotetext{The account of the bore in this chapter appeared as an
  article in the \Title{Century Magazine} for August,~1898. The illustrations
  then used are now reproduced, through the courtesy of
  the proprietors.}

\First{Since} most important towns are situated on
\index{Rivers!tide wave in|(}%
rivers or on estuaries, a large proportion of our
tidal observations relates to such sites. I shall
therefore now consider the curious, and at times
very striking phenomena which attend the rise
and fall of the tide in rivers.

The sea resembles a large pond in which the
water rises and falls with the oceanic tide, and a
river is a canal which leads into it. The rhythmical
rise and fall of the sea generate waves
which would travel up the river, whatever were
the cause of the oscillation of the sea. Accordingly,
a tide wave in a river owes its origin
directly to the tide in the sea, which is itself
produced by the tidal attractions of the sun and
moon.

We have seen in \Ref{Chapter}{II}.\ that long waves
progress in shallow water at a speed which depends
only on the depth of the water, and that
\PageSep{56}
waves are to be considered as long when their
length is at least twice the depth of the water.
Now the tide wave in a river is many hundreds
of times as long as the depth, and it must therefore
progress at a speed dependent only on the
depth. That speed is very slow compared with
the motion of the great tide wave in the open
ocean.

The terms ``ebb'' and ``flow'' are applied to
\index{Ebb and flow defined}%
\index{Flow and ebb defined}%
tidal currents. The current ebbs when the
water is receding from the land seaward, and
flows when it is approaching the shore. On the
open seacoast the water ebbs as the water-level
falls, and it flows as the water rises. Thus at
high and low tide the water is neither flowing
landward nor ebbing seaward, and we say that
it is slack or dead. In this case ebb and flow
are simultaneous with rise and fall, and it is not
uncommon to hear the two terms used synonymously;
but we shall see that this usage is incorrect.

I begin by considering the tidal currents in a
river of uniform depth, so sluggish in its own
proper current that it may be considered as a
stagnant canal, and the only currents to be considered
are tidal currents. At any point on the
river bank there is a certain mean height of
water, such that the water rises as much above
that level at high water as it falls below it at
low water. The law of tidal current is, then,
\index{Currents, tidal, in rivers}%
\PageSep{57}
very simple. Whenever the water stands above
the mean level the current is up-stream and progresses
along with the tide wave; and whenever
it stands below mean level the current is down-stream
and progresses in the direction contrary
to the tide wave. Since the current is up-stream
when the water is higher than the mean, and
down-stream when it is lower, it is obvious that
when it stands exactly at mean level the current
is neither up nor down, and the water is slack
or dead. Also, at the moment of high water
the current is most rapid up-stream, and at low
water it is most rapid down-stream. Hence the
tidal current ``flows'' for a long time after high
water has passed and when the water-level is
falling, and ``ebbs'' for a long time after low
water and when the water-level is rising.

The law of tidal currents in a uniform canal
communicating with the sea is thus very different
from that which holds on an open seacoast,
where slack water occurs at high and at low
water, instead of at mean water. But rivers
gradually broaden and become deeper as they
approach the coast, and therefore the tidal currents
in actual estuaries must be intermediate
between the two cases of the open seacoast and
the uniform canal.

A river has also to deliver a large quantity of
water into the sea in the course of a single tidal
oscillation, and its own proper current is superposed
\PageSep{58}
on the tidal currents. Hence in actual
rivers the resultant current continues to flow up
stream after high water is reached, with falling
water-level, but ceases flowing before mean water-level
is reached, and the resultant current ebbs
down-stream after low water, and continues to
ebb with the rising tide until mean water is
reached, and usually for some time afterward.
The downward stream, in fact, lasts longer than
the upward one. The moments at which the
currents change will differ in each river according
to the depth, the rise and fall of the tide at
the mouth, and the amount of water delivered
by the river. An obvious consequence of this
is that in rivers the tide rises quicker than it
falls, so that a shorter time elapses between low
water and high water than between high water
and low water.

The tide wave in a river has another peculiarity
of which I have not yet spoken. The complete
theory of waves would be too technical for a book
of this sort, and I must ask the reader to accept
as a fact that a wave cannot progress along a
river without changing its shape. The change
is such that the front slope of the wave gradually
gets steeper, and the rear slope becomes more
gradual. This is illustrated in~\fig{15}, which
shows the progress of a train of waves in shallow
water as calculated theoretically. If the
steepening of the advancing slope of a wave
\PageSep{59}
were carried to an extreme, the wave would present
the form of a wall of water; but the mere
advance of a wave into shallow water would by
itself never suffice to produce so great a change
of form without the concurrence of the natural
\Figure{15}{Progressive Change of a Train
of Waves in Shallow Water}{png}
stream of the river. The downward current in
the river has, in fact, a very important influence
in heading the sea-water back, and this coöperates
with the natural change in the shape of a
wave as it runs into shallow water, so as to exaggerate
the steepness of the advancing slope of
the wave.

There are in the estuaries of many rivers
\index{Rivers!tide wave in|)}%
broad flats of mud or sand which are nearly dry
at low water, and in such situations the tide not
unfrequently rises with such great rapidity that
the wave assumes the form of a wall of water.
This sort of tide wave is called a ``bore,'' and in
\index{Bore!definition}%
French \textit{mascaret}. Notwithstanding the striking
nature of the phenomenon, very little has been
published on the subject, and I know of only one
series of systematic observations of the bore.
As the account to which I refer is contained in
the official publications of the English Admiralty,
it has probably come under the notice of only a
\PageSep{60}
small circle of readers. But the experiences of
the men engaged in making these observations
were so striking that an account of them should
prove of interest to the general public. I have,
moreover, through the kindness of Admiral Sir
William Wharton and of Captain Moore, the
\index{Moore, Captain!survey of Tsien-Tang-Kiang|(}%
advantage of supplementing verbal description
by photographs.

The estuary on which the observations were
made is that of the Tsien-Tang-Kiang, a considerable
\index{Tsien-Tang-Kiang, the bore in|(}%
river which flows into the China Sea about
sixty miles south of the great Yang-Tse-Kiang.
At most places the bore occurs only intermittently,
but in this case it travels up the river at
every tide. The bore may be observed within
seventy miles of Shanghai, and within an easy
walk of the great city of Hangchow; and yet
\index{Hangchow, the bore at|(}%
nothing more than a mere mention of it is to be
found in any previous publication.

In 1888 Captain Moore, R.~N., in command
of Her Majesty's surveying ship Rambler,
thought that it was desirable to make a thorough
survey of the river and estuary. He returned
to the same station in~1892; and the account
which I give of his survey is derived from reports
drawn up after his two visits. The annexed
sketch-map shows the estuary of the
Tsien-Tang, and the few places to which I shall
have occasion to refer are marked thereon.

On the morning of September~19, 1888, the
\PageSep{61}
Rambler was moored near an island, named
after the ship, to the southwest of Chapu Bay;
and on the~20th the two steam cutters Pandora
and Gulnare, towing the sailing cutter
\Figure{16}{Chart of the Estuary of the Tsien-Tang-Kiang}{png}
Brunswick, left the ship with instruments for
observing and a week's provisions.

Captain Moore had no reason to suspect that
the tidal currents would prove dangerous out
in the estuary, and he proposed to go up the
estuary about thirty miles to Haining, and then
follow the next succeeding bore up-stream to
Hangchow. Running up-stream with the flood,
all went well until about~11.30, when they were
about fifteen miles southwest by west of Kanpu.
The leading boat, the Pandora, here grounded,
and anchored quickly, but swung round violently
as far as the keel would let her. The other
boats, being unable to stop, came up rapidly;
and the Gulnare, casting off the Brunswick,
\PageSep{62}
struck the Pandora, and then drove on to and
over the bank, and anchored. The boats soon
floated in the rising flood, and although the engines
of the steam cutters were kept going
full speed, all three boats dragged their anchors
in an eleven-knot stream. When the flood
slackened, the three boats pursued their course
to the mouth of the river, where they arrived
about 4~\PM. The ebb was, however, so violent
that they were unable to anchor near one another.
Their positions were chosen by the advice of
some junkmen, who told Captain Moore, very
erroneously as it turned out, that they would be
safe from the night bore.

The night was calm, and at~11.29 the murmur
of the bore was heard to the eastward; it could
be seen at~11.55, and passed with a roar at~12.20,
well over toward the opposite bank, as predicted
by the Chinese. The danger was now supposed
to be past; but at~1~\AM\ a current of extreme
violence caught the Pandora, and she had
much difficulty to avoid shipwreck. In the
morning it was found that her rudder-post and
propeller-guard were broken, and the Brunswick
and Gulnare were nowhere to be seen.
They had, in fact, been in considerable danger,
and had dragged their anchors three miles up
the river. At 12.20~\AM\ they had been struck
by a violent rush of water in a succession of big
ripples. In a few moments they were afloat in
\PageSep{63}
an eight-knot current; in ten minutes the water
rose nine feet, and the boats began to drag their
anchors, although the engines of the Gulnare
were kept going full speed. After the boats had
dragged for three miles, the rush subsided, and
when the anchor was hove up the pea and the
greater part of the chain were as bright as polished
silver.

This account shows that all the boats were in
imminent danger, and that great skill was needed
to save them. After this experience and warning,
the survey was continued almost entirely
from the shore.

The junks which navigate the river are well
aware of the dangers to which the English boats
were exposed, and they have an ingenious method
of avoiding them. At various places on the
bank of the river there are shelter platforms, of
which I show an illustration in~\fig{17}. Immediately
after the passing of the bore the
\index{Bore!bore-shelter}%
junks run up-stream with the after-rush and
make for one of these shelters, where they allow
themselves to be left stranded on the raised
platform shown in the picture. At the end of
this platform there is a sort of round tower
jutting out into the stream. The object of this
is to deflect the main wave of the bore so as to
protect the junks from danger. After the passage
of the bore, the water rises on the platform
very rapidly, but the junks are just able to float
\PageSep{64}
in safety. Captain Moore gives a graphic account
of the spectacle afforded by the junks as
they go up-stream, and describes how on one
occasion he saw no less than thirty junks swept
\Figure{17}{Bore-Shelter on the Tsien-Tang-Kiang}{jpg}
up in the after-rush, at a rate of ten knots, past
the town of Haining toward Hangchow, with all
sail set but with their bows in every direction.

Measurements of the water-level were made
in the course of the survey, and the results, in
the form of a diagram, \fig{18}, exhibit the nature
of the bore with admirable clearness. The
observations of water-level were taken simultaneously
at three places, viz., Volcano Island
in the estuary, Rambler Island near the mouth
of the river, and Haining, twenty-six miles up
the river. In the figure, the distance between
\PageSep{65}
the lines marked Rambler and Volcano represents
fifty-one miles, and that between Rambler
and Haining twenty-six miles. The vertical
scales show the height of water, measured in
feet, above and below the mean level of the
water at these three points. The lines joining
these vertical scales, marked with the hours of
the clock, show the height of the water simultaneously.
The hour of~8.30 is indicated by
the lowest line it shows that the water was
one foot below mean level at Volcano Island,
twelve feet below at Rambler Island, and eight
feet below at Haining. Thus the water sloped
down from Haining to Rambler, and from Volcano
to Rambler; the water was running up the
estuary toward Rambler Island, and down the
estuary to the same point. At 9~and at~9.30
there was no great change, but the water had
risen two or three feet at Volcano Island and at
Rambler Island. By ten~o'clock the water was
rising rapidly at Rambler Island, so that there
was a nearly uniform slope up the river from
Volcano Island to Haining. The rise at Rambler
Island then continued to be very rapid,
while the water at Haining remained almost
stationary. This state of affairs went on until
midnight, by which time the water had risen
twenty-one feet at Rambler Island, and about
six feet at Volcano Island, but had not yet risen
at all at Haining. No doubt through the whole
\PageSep{66}
of this time the water was running down the
river from Haining towards its mouth. It is
clear that this was a state of strain which could
not continue long, for there was over twenty
feet of difference of level between Rambler
Island, outside, and Haining, in the river. Almost
exactly at midnight the strain broke down
and the bore started somewhere between Rambler
\index{Bore!diagram of rise in Tsien-Tang}%
Island and Kanpu, and rushed up the river
in a wall of water twelve feet high. This result
is indicated in the figure by the presence of two
lines marked ``midnight.'' After the bore had
\Figure{18}{Diagram of the Flow of the Tide on
the Tsien-Tang-Kiang}{jpg}
passed there was an after-rush that carried the
water up eight feet more. It was on this that
the junks were swept up the stream, as already
described. At~1.30 the after-rush was over,
\PageSep{67}
but the water was still somewhat higher at
Rambler Island than at Haining, and a gentle
current continued to set up-stream. The water
then began to fall at Rambler Island, while it
continued to rise at Haining up to three o'clock.
At this point the ebb of the tide sets in. I do
not reproduce the figure which exhibits the fall
of the water in the ebbing tide, for it may suffice
to say that there is no bore down-stream,
\index{Bore!pictures}%
although there is at one time a very violent
current.

In 1892 Captain Moore succeeded, with considerable
\index{Moore, Captain!illustrations of bore}%
difficulty, in obtaining photographs of
the bore as it passed Haining. They tell more
of the violence of the wave than could be conveyed
by any amount of description. The photographs,
reproduced in~\fig{19}, do not, however,
show that the broken water in the rear of the
crest is often disturbed by a secondary roller, or
miniature wave, which leaps up, from time to
time, as if struck by some unseen force, and disappears
in a cloud of spray. These breakers
were sometimes twenty to thirty feet above the
level of the river in front of the bore.

The upper of these pictures is from a photograph,
taken at a height of twenty-seven feet
above the river, as the bore passed Haining on
October~10, 1892. The height of this bore was
eleven feet. The lower pictures, also taken at
Haining, represent the passage of the bore on
\PageSep{68}
October~9, 1892. The first of these photographs
was taken at 1.29~\PM, and the second
represents the view only one minute later.

The Chinese regard the bore with superstitious
\index{Bore!Chinese superstition|(}%
\index{Chinese!superstition as to bore|(}%
reverence, and their explanation, which I quote
from Captain Moore's report, is as follows:
``Many hundred years ago there was a certain
general who had obtained many victories over
the enemies of the Emperor, and who, being
constantly successful and deservedly popular
among his countrymen, excited the jealousy of
his sovereign, who had for some time observed
with secret wrath his growing influence. The
Emperor accordingly caused him to be assassinated
and thrown into the Tsien-Tang-Kiang,
where his spirit conceived the idea of revenging
itself by bringing the tide in from the ocean in
such force as to overwhelm the city of Hangchow,
then the magnificent capital of the empire.
As my interpreter, who has been for some years
in America, put it, `his sowl felt a sort of ugly-like
arter the many battles he had got for the
Emperor.' The spirit so far succeeded as to
flood a large portion of the country, when the
Emperor, becoming alarmed at the distress and
loss of property occasioned, endeavored to enter
into a sort of compact with it by burning paper
and offering food upon the sea-wall. This, however,
did not have the desired effect, as the high
tide came in as before; and it was at last determined
\PageSep{69}
\ifthenelse{\boolean{ForPrinting}}{%
  \TallFig[0.7]{19}{Pictures of the Bore on the Tsien-Tang-Kiang}{jpg}
}{%
  \Figure[0.75]{19}{Pictures of the Bore on the Tsien-Tang-Kiang}{jpg}
}
\index{Wharton, Sir W. J., illustration of bore}%
\PageSep{70}
to erect a pagoda at the spot where the
worst breach in the embankment had been made.
Hence the origin of the Bhota Pagoda. A
pagoda induces the good \textit{fungshui}, or spirit.
After it was built the flood tide, though it still
continued to come in the shape of a bore, did
not flood the country as before.''

We ``foreign devils'' may take the liberty of
suspecting that the repairs to the embankment
had also some share in this beneficial result.

This story is remarkable in that it refers to
the reign of an Emperor whose historical existence
is undoubted. It thus differs from many
of the mythical stories which have been invented
by primitive peoples to explain great natural
phenomena. There is good reason to suppose,
in fact, that this bore had no existence some centuries
ago; for Marco Polo, in the thirteenth
\index{Marco Polo, resident of Hangchow}%
century, stayed about a year and a half at
Hangchow, and gives so faithful and minute
\index{Hangchow, the bore at|)}%
an account of that great town that it is almost
impossible to believe that he would have omitted
to notice a fact so striking. But the Emperor
referred to in the Chinese legend reigned some
centuries before the days of Marco Polo, so that
we have reason to believe that the bore is intermittent.
\index{Bore!Chinese superstition|)}%
I have also learned from Captain
Moore himself that at the time of the great
\index{Moore, Captain!survey of Tsien-Tang-Kiang|)}%
Taiping rebellion, the suppression of which was
principally due to ``Chinese'' Gordon, the intensity
\index{Chinese!superstition as to bore|)}%
\index{Tsien-Tang-Kiang, the bore in|)}%
\PageSep{71}
of the bore was far less than it is to-day.
\index{Bore!rivers where found}%
This shows that the bore is liable to great variability,
according as the silting of the estuary
changes.

The people at Haining still continue to pay
religious reverence to the bore, and on one of
the days when Captain Moore was making observations
some five or six thousand people assembled
on the river-wall to propitiate the god of
the waters by throwing in offerings. This was
the occasion of one of the highest bores at spring
tide, and the rebound of the bore from the sea-wall,
and the sudden heaping up of the waters
as the flood conformed to the narrow mouth of
the river, here barely a mile in width at low
water, was a magnificent spectacle. A series of
breakers were formed on the back of the advancing
flood, which for over five minutes were not
less than twenty-five feet above the level of the
river in front of the bore. On this occasion
Captain Moore made a rough estimate that a
million and three quarters of tons of water passed
the point of observation in one minute.

The bore of which I have given an account is
perhaps the largest known; but relatively small
ones are to be observed on the Severn and Wye
\index{Severn, bore in the}%
\index{Wye, bore in the}%
in England, on the Seine in France, on the Petitcodiac
\index{Petitcodiac, bore in the}%
\index{Seine, bore in the}%
in Canada, on the Hugli in India, and
\index{Hugli, bore on the}%
doubtless in many other places. In general,
however, it is only at spring tides and with certain
\PageSep{72}
winds that the phenomenon is at all striking.
In September,~1897, I was on the banks of the
Severn at spring tide; but there was no proper
bore, and only a succession of waves up-stream,
\index{Bore!causes}%
and a rapid rise of water-level.

I have shown, at the beginning of this chapter,
that the heading back of the sea water by
the natural current of a river, and the progressive
change of shape of a wave in shallow water combine
to produce a rapid rise of the tide in rivers.
But the explanation of the bore, as resulting
from these causes, is incomplete, because it leaves
their relative importance indeterminate, and
serves rather to explain a rapid rise than an absolutely
sudden one. I think that it would be
impossible, from the mere inspection of an estuary,
to say whether there would be a bore there;
we could only say that the situation looked
promising or the reverse.

The capriciousness of the appearance of the
bore proves in fact that it depends on a very nice
balance between conflicting forces, and the irregularity
in the depth and form of an estuary renders
the exact calculation of the form of the
rising tide an impossibility. It would be easy
to imitate the bore experimentally on a small
scale; but, as in many other physical problems,
we must rest satisfied with a general comprehension
of the causes which produce the observed
result.
\PageSep{73}

The manner in which the Chinese avail themselves
of the after-rush for ascending the river
affords an illustration of the utilization by mankind
of tidal energy. In going up-stream, a
\index{Energy, tidal, utilization of}%
barge, say of one hundred tons, may rise some
twenty or thirty feet. There has, then, been
done upon that barge a work of from two to
three thousand foot-tons. Whence does this
energy come? Now, I say that it comes from
the rotation of the earth; for we are making the
tide do the work for us, and thus resisting the
tidal movement. But resistance to the tide has
the effect of diminishing the rate at which the
earth is spinning round. Hence it is the earth's
rotation which carries the barge up the river, and
we are retarding the earth's rotation and making
the day infinitesimally longer by using the tide
in this way. This resistance is of an analogous
character to that due to tidal friction, the consideration
of which I must defer to a future
chapter, as my present object is to consider the
uses which may be made of tidal energy.

It has been supposed by many that when the
coal supply of the world has been exhausted we
shall fall back on the tides to do our work. But
a little consideration will show that although this
source of energy is boundless, there are other far
more accessible funds on which to draw.

I saw some years ago a suggestion that the
rise and fall of old hulks on the tide would afford
\PageSep{74}
\index{Energy, tidal, utilization of}%
serviceable power. If we picture to ourselves the
immense weight of a large ship, we may be deluded
for a moment into agreement with this
project, but numerical calculation soon shows its
futility. The tide takes about six hours to rise
from low water to high water, and the same
period to fall again. Let us suppose that the
water rises ten feet, and that a hulk of $10,000$
tons displacement is floating on it; then it is
easy to show that only twenty horse-power will
be developed by its rise and fall. We should
then require ten such hulks to develop as much
work as would be given by a steam engine of
very moderate size, and the expense of the installation
would be far better bestowed on water-wheels
in rivers or on wind-mills. I am glad to
\index{Mills worked by the tide}%
say that the projector of this scheme gave it up
when its relative insignificance was pointed out
to him. It is the only instance of which I ever
heard where an inventor was deterred by the impracticability
of his plan.

We may, then, fairly conclude that, with existing
mechanical appliances, the attempt to utilize
the tide on an open coast is futile. But
where a large area of tidal water can be easily
trapped at high water, its fall may be made to
work mill-wheels or turbines with advantage.
The expense of building long jetties to catch the
water is prohibitive, and therefore tide mills are
only practicable where there exists an easily
\PageSep{75}
adaptable configuration of shoals in an estuary.
There are, no doubt, many such mills in the
\index{Mills worked by the tide}%
world, but the only one which I happen to have
seen is at Bembridge, in the Isle of Wight. At
this place embankments formed on the natural
shoals are furnished with lock-gates, and inclose
many acres of tidal water. The gates open automatically
with the rising tide, and the incipient
outward current at the turn of the tide closes
the gates again, so that the water is trapped.
The water then works a mill wheel of moderate
size. When we reflect on the intermittence of
work from low water to high water and the great
inequality of work with springs and neaps, it
may be doubted whether this mill is worth the
expense of retaining the embankments and lock-gates.

We see then that, notwithstanding the boundless
energy of the tide, rivers and wind and fuel
are likely for all time to be incomparably more
important for the use of mankind.

\begin{Authorities}
On waves in rivers see Airy's article on \Title{Tides and Waves} in
\index{Airy, Sir G. B.!tides in rivers}%
\index{Rivers!Airy on tide in}%
the ``Encyclopædia Metropolitana.'' Some of his results will
also be found in the article \Title{Tides} in the ``Encyclopædia Britannica.''

Commander Moore, R.~N., \Title{Report on the Bore of the Tsien-Tang-Kiang.}
Sold by Potter, Poultry, London,~1888.

\Title{Further Report},~\&c., by the same author and publisher,~1893.
\end{Authorities}
\PageSep{76}


\Chapter{IV}
{Historical Sketch}

\First{I Cannot} claim to have made extensive investigations
\index{History!of tidal theories|(}%
as to the ideas of mankind at different
periods on the subject of the tides, but I propose
in the present chapter to tell what I have
been able to discover.

No doubt many mythologies contain stories
explanatory of the obvious connection between
the moon and the tide. But explanations, professing
at least to be scientific, would have been
brought forward at periods much later than
those when the mythological stories originated,
and I shall only speak of the former.

I have to thank my colleagues at Cambridge
for the translations from the Chinese, Arabic,
\index{Chinese!theories of tide}%
Icelandic, and classical literatures of such passages
as they were able to discover.

I learn from Professor Giles that Chinese
\index{Giles on Chinese theories of the tide}%
writers have suggested two causes for the tides:
first, that water is the blood of the earth, and
that the tides are the beating of its pulse; and
secondly, that the tides are caused by the earth
breathing. Ko~Hung, a writer of the fourth
century of our era, gives a somewhat obscure
\PageSep{77}
\index{Chinese!theories of tide}%
\index{Giles on Chinese theories of the tide}%
explanation of spring and neap tides. He says
that every month the sky moves eastward and
then westward, and hence the tides are greater
and smaller alternately. Summer tides are said
to be higher than winter tides, because in summer
the sun is in the south and the sky is $15,000$~li
($5,000$~miles) further off, and therefore in
summer the female or negative principle in nature
is weak, and the male or positive principle
strong.

In China the diurnal inequality is such that
in summer the tide rises higher in the daytime
than in the night, whilst the converse is true
in winter. I suggest that this fact affords the
justification for the statement that the summer
tides are great.

\TB

Mr.~E.~G. Browne has translated for me the
\index{Arabian theories of tide|(}%
\index{Browne, E. G., Arabian theories of tide|(}%
following passage from the ``Wonders of Creation''
of Zakariyy\bar{a} ibn Muhammad ibn Mahm\bar{u}d
al Qazv\bar{i}n\bar{i}, who died in \AD~1283.\footnote
  {Wüstenfeld's edition, pp.~103,~104.}

``Section treating of certain wonderful conditions
of the sea.

``Know that at different periods of the four
seasons, and on the first and last days of the
months, and at certain hours of the night and
day, the seas have certain conditions as to the
rising of their waters and the flow and agitation
thereof.
\PageSep{78}

``As to the rising of the waters, it is supposed
that when the sun acts on them it rarefies them,
and they expand and seek a space ampler than
that wherein they were before, and the one part
repels the other in the five directions eastwards,
westwards, southwards, northwards, and upwards,
and there arise at the same time various winds
on the shores of the sea. This is what is said
as to the cause of the rising of the waters.

``As for the flow of certain seas at the time
of the rising of the moon, it is supposed that at
the bottom of such seas there are solid rocks
and hard stones, and that when the moon rises
over the surface of such a sea, its penetrating
rays reach these rocks and stones which are at
the bottom, and are then reflected back thence;
and the waters are heated and rarefied and seek
an ampler space and roll in waves towards the
seashore~\dots\ and so it continues as long as
the moon shines in mid-heaven. But when she
begins to decline, the boiling of the waters
ceases, and the particles cool and become dense
and return to their state of rest, and the currents
run according to their wont. This goes
on until the moon reaches the western horizon,
when the flow begins again, as it did when the
moon was in the eastern horizon. And this
flow continues until the moon is at the middle
of the sky below the horizon, when it ceases.
Then when the moon comes upward, the flow
\PageSep{79}
begins again until she reaches the eastern horizon.
This is the account of the flow and ebb
of the sea.

``The agitation of the sea resembles the agitation
of the humours in men's bodies, for verily
as thou seest in the case of a sanguine or bilious
man,~\&c., the humours stirring in his body, and
then subsiding little by little; so likewise the
sea has matters which rise from time to time as
they gain strength, whereby it is thrown into
violent commotion which subsides little by little.
And this the Prophet (on whom be the blessings
of God and his peace) hath expressed in a poetical
manner, when he says: `Verily the Angel,
who is set over the seas, places his foot in the
sea and thence comes the flow; then he raises it
and thence comes the ebb.'\,''
\index{Arabian theories of tide|)}%
\index{Browne, E. G., Arabian theories of tide|)}%

\TB

Mr.~Magnússon has kindly searched the old
\index{Icelandic theory of tides}%
\index{Magnússon on Icelandic theories of tides}%
Icelandic literature for references to the tides.
In the Rimbegla he finds this passage:---

``Beda the priest says that the tides follow
the moon, and that they ebb through her blowing
on them, but wax in consequence of her
movement.''

And again:---

``(At new moon) the moon stands in the way
of the sun and prevents him from drying up the
sea; she also drops down her own moisture.
For both these reasons, at every new moon, the
\PageSep{80}
\index{Icelandic theory of tides}%
\index{Magnússon on Icelandic theories of tides}%
ocean swells and makes those tides which we call
spring tides. But when the moon gets past the
sun, he throws down some of his heat upon
the sea, and diminishes thereby the fluidity of
the water. In this way the tides of the sea
are diminished.''

In another passage the author writes:---

``But when the moon is opposite to the sun,
the sun heats the ocean greatly, and as nothing
impedes that warmth, the ocean boils and the
sea flood is more impetuous than before---just
as one may see water rise in a kettle when it
boils violently. This we call spring tide.''

There seems to be a considerable inconsistency
in explaining one spring tide by the interception
of the sun's heat by the moon, and the next one
by the excess of that heat.

But it is not necessary to search ancient literature
for grotesque theories of the tides. In
1722 E.~Barlow, gentleman, in ``An Exact Survey
of the Tide,''\footnote
  {``The Second Edition, with Curious Maps.'' (London: John
  Hooke, 1722.)}
attributes it to the pressure
of the moon on the atmosphere. And theories
not less absurd have been promulgated during
the last twenty years.

\TB

The Greeks and Romans, living on the shores
of the Mediterranean, had not much occasion to
learn about the tide, and the passages in classical
\PageSep{81}
\index{Greek!theory and description of tides|(}%
\index{Roman description of tides|(}%
literature which treat of this matter are but
few. But where the subject is touched on we
see clearly their great intellectual superiority over
those other peoples, whose ideas have just been
quoted.

The only author who treats of the tide in any
\index{Strabo on tides|(}%
detail is Posidonius, and we have to rely for our
\index{Posidonius on tides|(}%
knowledge of his work entirely on quotations
from him by Strabo.\footnote
  {My attention was drawn to Strabo by a passage in Sir W.
  Thomson's (Lord Kelvin's) Popular Lectures, \Title{The Tides}, vol.~ii.
  I have to thank Mr.~Duff for the translations which follow from
  Strabo and Posidonius. The work consulted was Bake's \Title{Posidonius}
  (Leiden,~1810), but Mr.~Duff tells me that the text is very
  corrupt in some places, and he has therefore also consulted a
  more recent text.}

Posidonius says that Aristotle attributed the
\index{Aristotle on tides}%
flow and ebb of the sea at Cadiz to the mountainous
formation of the coast, but he very justly
pronounces this to be nonsense, particularly as
the coast of Spain is flat and sandy. He himself
attributes the tides to the moon's influence, and
the accuracy of his observations is proved by the
following interesting passage from Strabo:\footnotemark---
\footnotetext{Teubner's \Title{Strabo},~i.\ p.~236.}

``Posidonius says that the movement of the
ocean observes a regular series like a heavenly
body, there being a daily, monthly, and yearly
movement according to the influence of the
moon. For when the moon is above the (eastern)
horizon by the distance of one sign of the
zodiac (\ie~$30°$) the sea begins to flow, and encroaches
\PageSep{82}
visibly on the land until the moon
reaches the meridian. When she has passed the
meridian, the sea in turn ebbs gradually, until
the moon is above the western horizon by the
distance of one sign of the zodiac. The sea then
remains motionless while the moon is actually
setting, and still more so (\textit{sic}) so long as the
moon is moving beneath the earth as far as a
sign of the zodiac beneath the horizon. Then
the sea again advances until the moon has
reached the meridian below the earth; and retreats
while the moon is moving towards the east,
until she is the distance of a sign of the zodiac
below the horizon; it remains at rest until the
moon is the same distance above the horizon, and
then begins to flow again. Such is the daily
movement of the tides, according to Posidonius.

``As to their monthly movement, he says that
the ebbs are greatest at the conjunctions [of
the sun and moon], and then grow less until the
time of half moon, and increase again until the
time of full moon, and grow less again until
the moon has waned to half. Then the increase
of the tide follows until the conjunction. But
the increases last longer and come quicker [this
phrase is very obscure].

``The yearly movements of the tides he says
he learned from the people of Cadiz. They told
him that the ebb and flow alike were greatest at
the summer solstice. He guesses for himself
\PageSep{83}
that the tides grow less from the solstice to the
equinox, and then increase between the equinox
and the winter solstice, and then grow less until
the spring equinox, and greater until the summer
solstice.''

This is an excellent account of the tides at
Cadiz, but I doubt whether there is any foundation
\index{Polibius on tides at Cadiz}%
for that part which was derived from hearsay.
Lord Kelvin remarks, however, that it is interesting
to note that inequalities extending over
the year should have been recognized.

Strabo also says that there was a spring near
Cadiz in which the water rose and fell, and that
this was believed by the inhabitants, and by
Polybius, to be due to the influence of the ocean
tide, but Posidonius was not of this opinion.
Strabo says:---

``Posidonius denies this explanation. He says
there are two wells in the precinct of Hercules at
Cadiz, and a third in the city. Of the two former
the smaller runs dry while people are drawing
water from it, and when they stop drawing water
it fills again; the larger continues to supply
water all day, but, like all other wells, it falls
during the day but is replenished at night, when
the drawing of water has ceased. But since the
ebb tide often coincides with the replenishing of
the well, therefore, says Posidonius, the idle story
of the tidal influence has been believed by the
inhabitants.''
\PageSep{84}

Since the wells follow the sun, whilst the tide
follows the moon, the criticism of Posidonius is
a very just one. But Strabo blames him for
distrusting the Cadizians in a simple matter of
everyday experience, whilst accepting their evidence
as to an annual inequality in the tides.

There is another very interesting passage in
Strabo, the meaning of which was obviously unknown
to the Dutch commentator Bake---and
indeed must necessarily have been unintelligible
to him at the time when he wrote, on account of
the then prevailing ignorance of tidal phenomena
in remoter parts of the world. Strabo
writes:---

``Anyhow Posidonius says that Seleucus of
\index{Diurnal inequality!observed by Seleucus}%
\index{Posidonius on tides|)}%
\index{Seleucus, observation of tides of Indian Ocean}%
the Red Sea [also called the Babylonian] declares
that there is a certain irregularity and regularity
in these phenomena [the tides], according
to the different positions [of the moon] in the
zodiac. While the moon is in the equinoctial
signs, the phenomena are regular; but while she
is in the signs of the solstices, there is irregularity
both in the height and speed of the tides,
and in the other signs there is regularity or the
reverse in proportion to their nearness to the solstices
or to the equinoxes.''

Now let us consider the meaning of this.
When the moon is in the equinoxes she is on
the equator, and when she is in the solstices she
is at her maximum distances to the north or
\PageSep{85}
south of the equator---or, as astronomers say, in
her greatest north or south declination. Hence
Seleucus means that, when the moon is on the
\index{Seleucus, observation of tides of Indian Ocean}%
equator, the tides follow one another, with two
equal high and low waters a day; but when she
is distant from the equator, the regular sequence
is interrupted. In other words, the diurnal
inequality (which I shall explain in a later chapter)
vanishes when the moon is on the equator,
and is at its maximum when the declination is
greatest. This is quite correct, and since the
diurnal inequality is almost evanescent in the
\index{Diurnal inequality!observed by Seleucus}%
Atlantic, whilst it is very great in the Indian
Ocean, especially about Aden, it is clear that
Seleucus had watched the sea there, just as we
should expect him to do from his place of origin.

\TB

Many centuries elapsed after the classical
period before any scientific thought was bestowed
on the tides. Kepler recognized the
\index{Kepler!ideas concerning tides}%
tendency of the water on the earth to move
towards the sun and the moon, but he was unable
to submit his theory to calculation. Galileo
\index{Galileo!blames Kepler for his tidal theory}%
expresses his regret that so acute a man as
Kepler should have produced a theory, which
appeared to him to reintroduce the occult qualities
of the ancient philosophers. His own explanation
referred the phenomenon to the rotation
of the earth, and he considered that it afforded
a principal proof of the Copernican system.
\index{Greek!theory and description of tides|)}%
\index{Roman description of tides|)}%
\index{Strabo on tides|)}%
\PageSep{86}
\index{Kepler!ideas concerning tides}%

The theory of tide-generating force which will
be set forth in \Ref{Chapter}{V}.\ is due to Newton,
\index{Newton!founder of tidal theory}%
who expounded it in his ``Principia'' in~1687.
His theory affords the firm basis on which all
subsequent work has been laid.

In 1738 the Academy of Sciences of Paris
offered the theory of the tides as the subject for
a prize. The authors of four essays received
prizes, viz., Daniel Bernoulli, Euler, Maclaurin,
\index{Bernoulli, Daniel, essay on tides}%
\index{Euler, essay on tides}%
\index{Maclaurin!essay on tides}%
and Cavalleri. The first three adopted, not only
\index{Cavalleri, essay on tides}%
the theory of gravitation, but also Newton's
theory to its fullest extent. A considerable
portion of Bernoulli's work is incorporated in
the account of the theory of the tides which I
shall give later. The essays of Euler and Maclaurin
contained remarkable advances in mathematical
knowledge, but did not add greatly to
the theory of the tides. The Jesuit priest
Cavalleri adopted the theory of vortices to explain
the tides, and it is not worth while to
follow him in his erroneous and obsolete speculations.

Nothing of importance was added to our
knowledge until the great French mathematician
Laplace took up the subject in~1774. It was he
\index{Laplace!theory of tides|(}%
who for the first time fully recognized the difficulty
of the problem, and showed that the earth's
rotation is an essential feature in the conditions.
The actual treatment of the tidal problem is in
effect due to Laplace, although the mode of
\PageSep{87}
presentment of the theory has come to differ
considerably from his.

Subsequently to Laplace, the most important
workers in this field have been Sir John Lubbock
\index{Lubbock, Sir J., senior, on tides}%
senior, Whewell, Airy, and Lord Kelvin.
\index{Kelvin, Lord!initiates harmonic analysis}%
\index{Whewell!on tides}%
\index{Whewell!empirical construction of tide tables|(}%
The work of Lubbock and Whewell is chiefly
remarkable for the coördination and analysis of
enormous masses of data at various ports, and
the construction of trustworthy tide tables.
Airy contributed an important review of the
whole tidal theory. He also studied profoundly
the theory of waves in canals, and considered
the effects of frictional resistances on the progress
of tidal and other waves.

Lord Kelvin initiated a new and powerful
method of considering tidal oscillations. His
method possesses a close analogy with that already
used in discussing the irregularities in the
motions of the moon and planets. His merit
consists in the clear conception that the plan of
procedure which has been so successful in the
one case would be applicable to the other. The
difference between the laws of the moon's motion
and those of tidal oscillations is, however,
so great that there is scarcely any superficial
resemblance between the two methods. This
so-called ``harmonic analysis'' of the tides is
\index{Harmonic analysis!initiated by Lord Kelvin}%
daily growing in favor in the eyes of men of
science, and is likely to supersede all the older
methods. I shall explain it in a future chapter.
\PageSep{88}

Amongst all the grand work which has been
bestowed on this difficult subject, Newton stands
out first, and next to him we must rank Laplace.
However original any future contribution to the
science of tides may be, it would seem as though
it must perforce be based on the work of these
two. The exposition which I shall give hereafter
of the theory of oceanic tides is based on
the work of Newton, Bernoulli, Laplace, and
\index{Bernoulli, Daniel, essay on tides}%
\index{Laplace!theory of tides|)}%
Kelvin, in proportions of which it would be
difficult to assign the relative importance.

\TB

The connection between the moon and the
tide is so obvious that long before the formulation
of a satisfactory theory fairly accurate predictions
of the tides were made and published.
On this head Whewell\footnote
  {\Title{History of the Inductive Sciences}, 1837, vol.~ii.\ p.~248 \textit{et~seq.}}
\index{History!of tidal theories|)}%
has the following interesting
passage:---

``The course which analogy would have recommended
for the cultivation of our knowledge of
tides would have been to ascertain by an analysis
of long series of observations, the effects of
changes in the time of transit, parallax, and
declination of the moon, and thus to obtain the
laws of phenomena; and then to proceed to
investigate the laws of causation.

``Though this was not the course followed by
mathematical theorists, it was really pursued by
those who practically calculated tide tables; and
\PageSep{89}
the application of knowledge to the useful purposes
of life, being thus separated from the
promotion of the theory, was naturally treated
as a gainful property, and preserved by secrecy.
\dots~Liverpool, London, and other places, had
their tide tables, constructed by undivulged
methods, which methods, in some instances at
least, were handed down from father to son for
several generations as a family possession; and
the publication of new tables accompanied by a
statement of the mode of calculation was resented
as an infringement of the rights of property.

``The mode in which these secret methods
were invented was that which we have pointed
out,---the analysis of a considerable series of
observations. Probably the best example of this
was afforded by the Liverpool tide tables. These
were deduced by a clergyman named Holden,
from observations made at that port by a harbor
master of the name of Hutchinson, who was
led, by a love of such pursuits, to observe the
tides for above twenty years, day and night.
Holden's tables, founded on four years of these
observations, were remarkably accurate.

``At length men of science began to perceive
that such calculations were part of their business;
and that they were called upon, as the
guardians of the established theory of the universe,
to compare it in the greatest possible
\PageSep{90}
detail with the facts. Mr.~Lubbock was the
first mathematician who undertook the extensive
labors which such a conviction suggested. Finding
that regular tide observations had been made
at the London docks from~1795, he took nineteen
years of these (purposely selecting the
length of the cycle of the motions of the lunar
orbit), and caused them (in~1831) to be analyzed
by Mr.~Dessiou, an expert calculator. He thus
obtained tables for the effect of the moon's
declination, parallax, and hour of transit, on the
tides; and was enabled to produce tide tables
founded upon the data thus obtained. Some
mistakes in these as first published (mistakes unimportant
as to the theoretical value of the work)
served to show the jealousy of the practical tide
table calculators, by the acrimony with which the
oversights were dwelt upon; but in a very few
years the tables thus produced by an open and scientific
process were more exact than those which
resulted from any of the secrets; and thus practice
was brought into its proper subordination to
theory.''
\index{Whewell!empirical construction of tide tables|)}%

\begin{Authorities}
The history from Galileo to Laplace is to be found in the
\Title{Mécanique Céleste} of Laplace, book~xiii, chapter~i.

The other authorities are quoted in the text or in footnotes.
\end{Authorities}
\PageSep{91}


\Chapter{V}
{Tide-generating Force}

\First{It} would need mathematical reasoning to fully
\index{Centripetal and centrifugal forces|(}%
explain how the attractions of the sun and moon
give rise to tide-generating forces. But as this
\index{Forces!centripetal and centrifugal|(}%
book is not intended for the mathematician, I
must endeavor to dispense with technical language.

A body in motion will move in a straight line,
unless it is deflected from its straight path by
some external force, and the resistance to the
deflection is said to be due to inertia. The motion
of the body then is equivalent in its effect
to a force which opposes the deflection due to
the external force, and in many cases it is permissible
to abstract our attention from the motion
of the system and to regard it as at rest, if
at the same time we introduce the proper ideal
forces, due to inertia, so that they shall balance
the action of the real external forces.

If I tie a string to a stone and whirl it round,
the string is thrown into a state of tension. The
natural tendency of the stone, at each instant, is
to move onward in a straight line, but it is continuously
deflected from its straight path by the
\PageSep{92}
tension of the string. In this case the ideal
force, due to inertia, whereby the stone resists
its continuous deflection, is called centrifugal
force. This force is in reality only a substitute
for the motion, but if we withdraw our attention
from the motion, it may be regarded as a reality.

The centrifugal force is transmitted to my
hand through the string, and I thus experience
an outward or centrifugal tendency. But the
stone itself is continually pulled inward by the
string, and the force is called centripetal. When
a string is under tension, as in this experiment,
it is subject to equal and opposite forces, so that
the tension implies the existence of a pair of
forces, one towards and the other away from the
centre of rotation. The force is to be regarded
as away from the centre when we consider the
sensation of the whirler, and as towards the centre
when we consider the thing whirled. A similar
double view occurs in commerce, where a
transaction which stands on the credit side in the
books of one merchant appears on the debit side
in the books of the other.

This simple experiment exemplifies the mechanism
by which the moon is kept revolving round
the earth. There is not of course any visible
connection between the two bodies, but an invisible
bond is provided by the attraction of gravity,
which replaces the string which unites the
stone to the hand. The moon, then, whirls
\PageSep{93}
\index{Forces!tide-generating|(}%
\index{Orbit!of moon and earth|(}%
round the earth at just such a rate and at just
\index{Earth and moon!diagram}%
such a distance, that her resistance to circular
motion, called centrifugal force, is counterbalanced
by the centripetal tendency of gravity. If
\index{Centripetal and centrifugal forces|)}%
she were nearer to us the attraction of gravity
would be greater, and she would have to go
round the earth faster, so as to make enough
centrifugal force to counterbalance the greater
\Figure[0.8]{20}{Earth and Moon}{png}
gravity. The converse would be true, and the
moon would go round slower, if she were further
from us.

The moon and the earth go round the sun in
companionship once in a year, but this annual
motion does not affect the interaction between
them, and we may put aside the orbital motion
of the earth, and suppose the moon and earth to
\index{Moon and earth!diagram}%
be the only pair of bodies in existence. When
the principle involved in a purely lunar tide is
grasped, the action of the sun in producing a
\index{Forces!centripetal and centrifugal|)}%
\PageSep{94}
solar tide will become obvious. But the analogy
of the string and stone is imperfect in one
respect where the distinction is important; the
moon, in fact, does not revolve exactly about
the earth, but about the centre of gravity of
the earth and moon. The earth is eighty times
as heavy as the moon, and so this centre of gravity
is not very far from the earth's centre. The
upper part of \fig{20} is intended to represent a
planet and its satellite; the lower part shows
the earth and the moon in their true proportions.
The upper figure is more convenient for
our present argument, and the planet and satellite
may be described as the earth and the moon,
notwithstanding the exaggeration of their relative
proportions. The point~$G$ is the centre of
gravity of the two, and the axis about which
they revolve passes through~$G$. This point is
sufficiently near to the centre of the earth to
permit us, for many purposes, to speak of the
moon as revolving round the earth. But in the
present case we must be more accurate and must
regard the moon and earth as revolving round~$G$,
their centre of gravity. The moon and earth
are on opposite sides of this point, and describe
circles round it. The distance of the moon's
centre from~$G$ is $237,000$~miles, whilst that of
the earth's centre is only $\DPchg{3000}{3,000}$~miles in the opposite
direction. The $\DPchg{3000}{3,000}$~and $237,000$~miles
together make up the $240,000$~miles which separate
the centres of the two bodies.
\PageSep{95}

A system may now be devised so as to resemble
the earth and moon more closely than that
of the string and stone with which I began. If
a large stone and a small one are attached to one
another by a light and stiff rod, the system can
be balanced horizontally about a point in the rod
called the centre of gravity~$G$. The two weights
may then be set whirling about a pivot at~$G$, so
that the rod shall always be horizontal. In consequence
of the rotation the rod is brought into
a state of stress, just as was the string in the
first example, and the centripetal stress in the
rod exactly counterbalances the centrifugal force.
The big and the little stones now correspond to
the earth and the moon, and the stress in the rod
plays the same part as the invisible bond of
gravity between the earth and the moon. Fixing
our attention on the smaller stone or moon
at the end of the longer arm of the rod, we see
that the total centrifugal force acting on the
moon, as it revolves round the centre of gravity,
is equal and opposite to the attraction of the
earth on the moon. On considering the short
arm of the rod between the pivot and the big
stone, we see also that the centrifugal force acting
on the earth is equal and opposite to the
attraction of the moon on it. In this experiment
as well as in the former one, we consider
the total of centrifugal force and of attraction,
but every particle of both the celestial bodies is
\index{Orbit!of moon and earth|)}%
\PageSep{96}
\index{Davis, method of presenting tide-generating force}%
acted on by these forces, and accordingly a
closer analysis is necessary.

It will now simplify matters if we make a supposition
which departs from actuality, introducing
the true conditions at a later stage in the
argument.

The earth's centre describes a circle about the
centre of gravity~$G$, with a radius of $\DPchg{3000}{3,000}$~miles,
and the period of the revolution is of course one
month. Now whilst this motion of revolution
of the earth's centre continues, let it be supposed
that the diurnal rotation is annulled. As this
is a mode of revolution which differs from that
of a wheel, it is well to explain exactly what is
meant by the annulment of the diurnal rotation.
This is illustrated in~\fig{21}, which shows the
successive positions assumed by an arrow in revolution
without rotation. The shaft of the arrow
always remains parallel to the same direction in
space, and therefore it does not rotate, although
the whole arrow revolves. It is obvious that every
particle of the arrow describes a circle of the
same radius, but that the circles described by
them are not concentric. The circles described
by the point and by the base of the arrow are
shown in the figure, and their centres are separated
by a distance equal to the length of the
arrow. Now the centrifugal force on a revolving
particle acts along the radius of the circle described,
and in this case the radii of the circles
\PageSep{97}
described by any two particles in the arrow are
always parallel. The parallelism of the centrifugal
forces at the two ends of the arrow is
indicated in the figure. Then again, the centrifugal
force must everywhere be equal as well
as parallel, because its intensity depends both on
the radius and on the speed of revolution, and
these are the same for every part. It follows
that if a body revolves without rotation, every
part of it is subject to equal and parallel centrifugal
forces. The same must therefore be
true of the earth when deprived of diurnal rotation.
Accordingly every particle of the idealized
non-rotating earth is continuously subject to
equal and parallel centrifugal forces, in consequence
of the revolution of the earth's centre
in its monthly orbit with a radius of $\DPchg{3000}{3,000}$~miles.\footnote
  {I owe the suggestion of this method of presenting the origin
  of tide-generating force to Professor Davis of Harvard
\index{Davis, method of presenting tide-generating force}%
  University.}

We have seen that the total of centrifugal
force acting on the whole earth must be just
such as to balance the total of the centripetal
forces due to the moon's attraction. If, then,
the attractional forces, acting on every particle
of the earth, were also equal and parallel, there
would be a perfect balance throughout. We
shall see, however, that although there is a perfect
balance on the whole, there is not that uniformity
\PageSep{98}
which would render the balance perfect
at every particle.

As far as concerns the totality of the attraction
the analogy is complete between the larger
stone, revolving at the end of the shorter arm
of the rod, and the earth revolving in its small
\Figure[0.7]{21}{Revolution of a Body without Rotation}{png}
orbit round~$G$. But a difference arises when we
compare the distribution of the tension of the
rod with that of the lunar attraction; for the
rod only pulls at the stone at the point where it
is attached to it, whereas the moon attracts every
particle of the earth. She does not, however,
attract every particle with equal force, for she
pulls the nearer parts more strongly than the
further, as is obvious from the nature of the law
of gravitation. The earth's centre is distant
sixty times its radius from the moon, so that the
nearest and furthest parts are distant fifty-nine
\PageSep{99}
and sixty-one radii respectively. Hence the attractions
at the nearest and furthest parts differ
only a little from the average, namely, that at
the centre; but it is just these small differences
which are important in this matter.

Since on the whole the attractions and the centrifugal
forces are equal and opposite, and since
the centrifugal forces acting on the non-rotating
earth are equal and parallel at every part, and
since the attraction at the earth's centre is the
average attraction, it follows that where the attraction
is stronger than the average it overbalances
the centrifugal force, and where it is weaker
it is overbalanced thereby.

The result of the contest between the two sets
of forces is illustrated in~\fig{22}. The circle
represents a section of the earth, and the moon
is a long way off in the direction~$M$.

Since the moon revolves round the earth,
whilst the earth is still deprived of rotation, the
figure only shows the state of affairs at a definite
instant of time. The face which the earth exhibits
to the moon is always changing, and the
moon returns to the same side of the earth only
at the end of the month. Hence the section of
the earth shown in this figure always passes
through the moon, while it is continually shifting
with respect to the solid earth. The arrows in
the figure show by their directions and lengths
the magnitudes and directions of the overbalance
\PageSep{100}
in the contest between centrifugal and centripetal
tendencies. The point~$V$ in the figure is
the middle of the hemisphere, which at the moment
portrayed faces full towards the moon. It
\Figure[0.7]{22}{Tide-Generating Force}{png}
is the middle of the round disk which the man in
the moon looks at. The middle of the face invisible
to the man in the moon is at~$I$. The
point of the earth which is only fifty-nine earth's
radii from the moon is at~$V$. Here attraction
overbalances centrifugal force, and this is indicated
by an arrow pointing towards the moon.
The point distant sixty-one earth's radii from
the moon is at~$I$, and attraction is here overbalanced,
as indicated by the arrow pointing away
from the moon.

I shall have to refer hereafter to the intensities
\PageSep{101}
of these forces, and will therefore here pause
to make some numerical calculations.

The moon is distant from the earth's centre
sixty times the earth's radius, and the attraction
of gravity varies inversely as the square of the
distance. Hence we may take $\frac{1}{60^{2}}$ or $\frac{1}{3,600}$ as a
measure of the intensity of the moon's attraction
at the earth's centre. The particle which occupies
the centre of the earth is also that particle
which is at the average distance of all the particles
constituting the earth's mass. Hence $\frac{1}{60^{2}}$ or
$\frac{1}{3,600}$ may be taken as a measure of the average
attraction of the moon on every particle of the
earth.

Now the point~$V$ is only distant fifty-nine
earth's radii from the moon, and therefore, on
the same scale, the moon attraction is measured
by $\frac{1}{59^{2}}$ or~$\frac{1}{3,481}$.

The attraction therefore at~$V$ exceeds the average
by $\frac{1}{59^{2}} - \frac{1}{60^{2}}$, or $\frac{1}{3,481} - \frac{1}{3,600}$. It will be well to
express these results in decimals; now $\frac{1}{3,481}$ is
$.000,287,27$, and $\frac{1}{3,600}$ is $.000,277,78$, so that the
difference is~$.000,009,49$. It is important to
notice that $\frac{2}{60^{3}}$ or $\frac{2}{216,000}$ is equal to~$.000,009,26$;
so that the difference is nearly equal to~$\frac{2}{60^{3}}$.

{\Loosen Again, the point~$I$ is distant sixty-one earth's
radii from the moon, and the moon's attraction
there is to be measured by $\frac{1}{61^{2}}$ or $\frac{1}{3,721}$. The attraction
at~$I$ therefore falls below the average by
$\frac{1}{60^{2}} - \frac{1}{61^{2}}$, or $\frac{1}{3,600} - \frac{1}{3,721}$; that is, by~$.000,277,78 - .000,268,75$,
\PageSep{102}
which is equal to~$.000,009,03$.
This again does not differ much from~$\frac{2}{60^{3}}$.}

These calculations show that the excess of the
actual attraction at~$V$ above the average attraction
is nearly equal to the excess of the average
above the actual attraction at~$I$. These two
excesses only differ from one another by $5$~per
cent.\ of either, and they are both approximately
equal to~$\frac{2}{60^{3}}$ on the adopted scale of measurement.

The use of any particular scale of measurement
is not material to this argument, and we
should always find that the two excesses are
nearly equal to one another. And further, if
the moon were distant from the earth by any
other number of earth's radii, we should find
that the two excesses are each nearly equal to $2$
divided by the cube of that number.\footnote
  {\Loosen This argument is very easily stated in algebraic notation.
  If $x$~be the number of earth's radii at which the moon is
  placed, the points $V$~and~$I$ are respectively distant $x - 1$ and
  $x + 1$ radii. Now $(x - 1)^{2}$~is nearly equal to~$x^{2} - 2x$ or to
  $x^{2}(1 - \frac{2}{x})$, and therefore $\frac{1}{(x - 1)^{2}}$~is nearly equal to~$\frac{1}{x^{2}(1 - \frac{2}{x})}$, which is
  nearly equal to~$\frac{1}{x^{2}}(1 + \frac{2}{x})$. Hence $\frac{1}{(x - 1)^{2}} - \frac{1}{x^{2}}$~is nearly equal to~$\frac{2}{x^{3}}$.
  By a similar argument $(x + 1)^{2}$~is nearly equal to~$x^{2}(1 + \frac{2}{x})$,
  and $\frac{1}{(x + 1)^{2}}$~is nearly equal to~$\frac{1}{x^{2}}(1 - \frac{2}{x})$; so that $\frac{1}{x^{2}} - \frac{1}{(x + 1)^{2}}$~is nearly
  equal to~$\frac{2}{x^{3}}$.}

We conclude then that the two overbalances
at $V$~and~$I$, which will be called tide-generating
forces, are nearly equal to one another, and vary
\PageSep{103}
inversely as the cube of the distance of the moon
from the earth.

The fact of the approximate equality of the
overbalance or excess on the two sides of the
earth is noted in the figure by two arrows at $V$
and $I$ of equal lengths. The argument would
be a little more complicated, if I were to attempt
to follow the mathematician in his examination
of the whole surface of the earth, and to trace
from point to point how the balance between
the opposing forces turns. The reader must
accept the results of such an analysis as shown
in \fig{22} by the directions and lengths of the
arrows.

We have already seen that the forces at $V$ and~$I$,
the middles of the faces of the earth which
are visible and invisible to the man in the moon,
are directed away from the earth's centre. The
edges of the earth's disk as seen from the moon
are at $D$ and~$D$, and here the arrows point inwards
to the earth's centre and are half as long
as those at $V$ and~$I$. At intermediate points,
they are intermediate both in size and direction.

The only point in which the system considered
differs from actuality is that the earth has
been deprived of rotation. But this restriction
may be removed, for, when the earth rotates
once in $24$~hours, no difference is made in the
forces which I have been trying to explain,
\PageSep{104}
although of course the force of gravity and the
shape of the planet are affected by the rotation.
This figure is called a diagram of tide-generating
forces, because the tides of the ocean are due to
the action of this system of forces.

The explanation of tide-generating force is
the very kernel of our subject, and, at the risk
of being tedious, I shall look at it from a slightly
different point of view. If every particle of the
earth and of the ocean were acted on by equal
and parallel forces, the whole system would
move together and the ocean would not be displaced
relatively to the earth; we should say
that the ocean was at rest. If the forces were
not quite equal and not quite parallel, there
would be a slight residual effect tending to make
the ocean move relatively to the solid earth. In
other words, any defect from equality and parallelism
in the forces would cause the ocean to
move on the earth's surface.

The forces which constitute the departure
from equality and parallelism are called ``tide-generating
forces,'' and it is this system which
is indicated by the arrows in~\fig{22}. Tide-generating
force is, in fact, that force which,
superposed on the average force, makes the actual
force. The average direction of the forces
which act on the earth, as due to the moon's
attraction, is along the line joining the earth's
centre to the moon's centre, and its average
\PageSep{105}
intensity is equal to the force at the earth's
centre.

Now at~$V$ the actual force is straight towards~$M$,
in the same direction as the average, but of
greater intensity. Hence we find an arrow
directed towards~$M$, the moon. At~$I$, the actual
force is again in the same direction as, but of
less intensity than, the average, and the arrow is
directed away from~$M$, the moon. At~$D$, the
actual force is almost exactly of the same intensity
as the average, but it is not parallel thereto,
and we must insert an inward force as shown by
the arrow, so that when this is compounded with
the average force we may get a total force in
the right direction.

Now let us consider how these forces tend to
affect an ocean lying on the surface of the earth.
The moon is directly over the head of an inhabitant
of the earth, that is to say in his zenith,
when he is at~$V$; she is right under his feet in
the nadir when he is at~$I$; and she is in the
observer's horizon, either rising or setting, when
he is anywhere on the circle~$D$. When the
inhabitant is at~$V$ or at~$I$ he finds that the tide-generating
force is towards the zenith; when he
is anywhere on the circle~$D$ he finds it towards
the nadir. At other places he finds it directed
towards or away from some point in the sky,
except along two circles halfway between $V$ and~$D$,
or between $I$ and~$D$, where the tide-generating
\PageSep{106}
force is level along the earth's surface, and may
be called horizontal.

A vertical force cannot make things move
sideways, and so the sea will not be moved horizontally
by it. The vertical part of the tide-generating
force is not sufficiently great to
overcome gravity, but will have the effect of
making the water appear lighter or heavier. It
will not, however, be effective in moving the
water, since the water must remain in contact
with the earth. We want, then, to omit the
vertical part of the force and leave behind only
the horizontal part, by which I mean a force
which, to an observer on the earth's surface, is
not directed either upwards or downwards, but
along the level to any point of the compass.

If there be a force acting at any point of the
earth's surface, and directed upwards or downwards
away from or towards some point in the
sky other than the zenith, it may be decomposed
into two forces, one vertically upwards or downwards,
and another along the horizontal surface.
Now as concerns the making of the tides,
no attention need be paid to that part which
is directed straight up or down, and the only
important part is that along the surface,---the
horizontal portion.

Taking then the diagram of tide-generating
forces in~\fig{22}, and obliterating the upward
and downward portions of the force, we are left
%% Plate 1
\Figure[0.9]{23}{Horizontal Tide-Generating Force}{jpg}
%% Facing page
%[Blank Page]
\PageSep{107}
\index{Horizontal tide-generating force}%
with a system of forces which may be represented
by the arrows in the perspective picture of horizontal
tide-generating force shown in~\fig{23}.

If we imagine an observer to wander over the
earth, $V$~is the place at which the moon is vertically
over his head, and the circle~$D$, shown by
the boundary of the shadow, passes through all
the places at which the moon is in the horizon,
just rising or setting. Then there is no horizontal
force where the moon is over his head or under
his feet, or where the moon is in his horizon
either rising or setting, but everywhere else there
is a force directed along the surface of the earth
in the direction of the point at which the moon
is straight overhead or underfoot.

Now suppose $P$ to be the north pole of the
earth, and that the circle $A_{1}$,~$A_{2}$, $A_{3}$, $A_{4}$,~$A_{5}$ is a
parallel of latitude---say the latitude of London.
Then if we watch our observer from external
space, he first puts in an appearance on the picture
at~$A_{1}$, and is gradually carried along to~$A_{2}$
by the earth's rotation, and so onwards. Just before
he comes to~$A_{2}$, the moon is due south of him,
and the tide-generating force is also south, but
not very large. It then increases, so that nearly
three hours later, when he has arrived at~$A_{3}$, it
is considerably greater. It then wanes, and
when he is at~$A_{4}$ the moon is setting and the
force is nil. After the moon has set, the force
is directed towards the moon's antipodes, and it
\PageSep{108}
is greatest about three hours after moonset, and
vanishes when the moon, still being invisible, is
on the meridian.

It must be obvious from this discussion that
the lunar horizontal tide-generating force will
differ, both as to direction and magnitude, according
to the position of the observer on the
earth and of the moon in the heavens, and that
it can only be adequately stated by means of
mathematical formulæ. I shall in the following
chapter consider the general nature of the
changes which the forces undergo at any point
\index{Forces!tide-generating|)}%
on the earth's surface.

But before passing on to that matter it should
be remarked that if the earth and sun had been
the only pair of bodies in existence the whole of
the argument would have applied equally well.
Hence it follows that there is also a solar tide-generating
force, which in actuality coëxists
with the lunar force. I shall hereafter show
how the relative importance of these two influences
is to be determined.

\begin{Authorities}
Any mathematical work on the theory of the tides; for example,
Thomson and Tait's \Title{Natural Philosophy}, Lamb's \Title{Hydrodynamics},
Bassett's \Title{Hydrodynamics}, article \Title{Tides}, ``Encycl.\ Britan.,''
Laplace's \Title{Mécanique Céleste},~\&c.
\end{Authorities}
\PageSep{109}


\Chapter{VI}
{Deflection of the Vertical}

\First{The} intensity of tide-generating force is to be
\index{Deflection of the vertical|(}%
estimated by comparison with some standard, and
it is natural to take as that standard the force of
gravity at the earth's surface. Gravity acts in a
vertical direction, whilst that portion of the tidal
force which is actually efficient in disturbing the
ocean is horizontal. Now the comparison between
a small horizontal force and gravity is
easily effected by means of a pendulum. For if
the horizontal force acts on a suspended weight,
the pendulum so formed will be deflected from
the vertical, and the amount of deflection will
measure the force in comparison with gravity.
A sufficiently sensitive spirit level would similarly
show the effect of a horizontal force by the
displacement of the bubble. When dealing with
tidal forces the displacements of either the pendulum
\index{Forces!numerical estimate|(}%
\index{Forces!deflection of vertical by|(}%
or the level must be exceedingly minute,
but, if measurable, they will show themselves as
a change in the apparent direction of gravity.
Accordingly a disturbance of this kind is often
described as a deflection of the vertical.

The maximum horizontal force due to the
\PageSep{110}
moon may be shown by a calculation, which involves
the mass and distance of the moon, to
have an intensity of $\frac{1}{11,660,000}$ of gravity.\footnote
  {It does not occur to me that there is any very elementary
  method of computing the maximum horizontal tidal force, but it
  is easy to calculate the vertical force at the points $V$~or~$I$ in~\fig{22}.

  {\Loosen The moon weighs $\frac{1}{80}$~of the earth, and has a radius $\frac{1}{4}$~as large.
  Hence lunar gravity on the moon's surface is~$\frac{1}{80} × 4^{2}$, or $\frac{1}{5}$~of
  terrestrial gravity at the earth's surface. The earth's radius is
  $4,000$~miles and the moon's distance from the earth's centre
  $240,000$~miles. Hence her distance from the nearer side of the
  earth is $236,000$~miles. Therefore lunar gravity at the earth's
  centre is $\frac{1}{5} × \frac{1}{240^{2}}$~of terrestrial gravity, and lunar gravity at the
  point~$V$ is $\frac{1}{5} × \frac{1}{236^{2}}$~of the same. Therefore the tidal force at~$V$
  is $\frac{1}{5} × \frac{1}{236^{2}} - \frac{1}{5} × \frac{1}{240^{2}}$~of terrestrial gravity. On multiplying the
  squares of~$236$ and of~$240$ by~$5$, we find that this difference is
  $\frac{1}{278,480} - \frac{1}{288,000}$. If these fractions are reduced to decimals
  and the subtraction is performed, we find that the force at~$V$
  is $.000,000,118,44$~of terrestrial gravity. When this decimal is
  written as a fraction, we find the result to be $\frac{1}{8,450,000}$~of
  gravity.}

  Now it is the fact, although I do not see how to prove it in an
  equally elementary manner, that the maximum horizontal tide-generating
  force has an intensity equal to $\frac{3}{4}$~of the vertical force
  at $V$~or~$I$. To find $\frac{3}{4}$~of the above fraction we must augment the
  denominator by one third part. Hence the maximum horizontal
  force is $\frac{1}{11,260,000}$~of gravity. This number does not agree exactly
  with that given in the text; the discrepancy is due to the
  fact that round numbers have been used to express the sizes and
  distance apart of the earth and the moon, and their relative
  masses.}
Such a
force must deflect the bob of a pendulum by the
same fraction of the length of the cord by which
it is suspended. If therefore the string were $10$~metres
or $33$~feet in length, the maximum deflection
of the weight would be $\frac{1}{11,660,000}$~of $10$~metres,
\PageSep{111}
or $\frac{1}{1,166}$~of a millimetre. In English measure this
is $\frac{1}{29,000}$~of an inch. But the tidal force is reversed
in direction about every six hours, so that the
pendulum will depart from its mean direction by
\index{Pendulum!curves traced by, under tidal force}%
as much in the opposite direction. Hence the
\Figure[0.8]{24}{Deflection of a Pendulum; the Moon and
Observer on the Equator}{png}
excursion to and fro of the pendulum under the
lunar influence will be $\frac{1}{14,500}$~of an inch. With a
pendulum one metre, or $3$~ft.\ $3$~in.\ in length,
the range of motion of the pendulum bob is
$\frac{1}{145,000}$~of an inch. For any pendulum of manageable
length this displacement is so small, that it
seems hopeless to attempt to measure it by direct
observation. Nevertheless the mass and distance
of the moon and the intensity of gravity being
known with a considerable degree of accuracy, it
is easy to calculate the deflection of the vertical
at any time.
\index{Forces!numerical estimate|)}%

The curves which are traced out by a pendulum
present an infinite variety of forms, corresponding
\PageSep{112}
to various positions of the observer on
the earth and of the moon in the heavens. Two
illustrations of these curves must suffice. \Fig{24}
shows the case when the moon is on the celestial
equator and the observer on the terrestrial
equator. The path is here a simple ellipse,
which is traversed twice over in the lunar day by
the pendulum. The hours of the lunar day at
\index{Pendulum!curves traced by, under tidal force}%
which the bob occupies successive positions are
marked on the curve.

If the larger ellipse be taken to show the displacement
of a pendulum when the sun and
\Figure[0.8]{25}{Deflection of a Pendulum; the Moon in N.
Declination~$15°$, the Observer in N. Latitude~$30°$}{png}
moon coöperate at spring tide, the smaller one
will show its path at the time of neap tide.

In \fig{25} the observer is supposed to be in
latitude~$30°$, whilst the moon stands $15°$~N. of
the equator; in this figure no account is taken
\PageSep{113}
of the sun's force. Here also the hours are
marked at the successive positions of the pendulum,
which traverses this more complex curve
only once in the lunar day. These curves are
somewhat idealized, for they are drawn on the
hypothesis that the moon does not shift her
position in the heavens. If this fact were taken
into account, we should find that the curve
would not end exactly where it began, and that
the character of the curve would change slowly
from day to day.

But even after the application of a correction
for the gradual shift of the moon in the heavens,
the curves would still be far simpler than in actuality,
because the sun's influence has been left
out of account. It has been remarked in the
last chapter that the sun produces a tide-generating
force, and it must therefore produce a
deflection of the vertical. Although the solar
deflection is considerably less than the lunar, yet
it would serve to complicate the curve to a great
degree, and it must be obvious then that when
the full conditions of actuality are introduced
the path of the pendulum will be so complicated,
that mathematical formulæ are necessary for
complete representation.

Although the direct observation of the tidal
deflection of the vertical would be impossible
even by aid of a powerful microscope, yet several
attempts have been made by more or less
\PageSep{114}
indirect methods. I have just pointed out that
the path of a pendulum, although drawn on an
ultra-microscopic scale, can be computed with a
high degree of accuracy. It may then occur to
the reader that it is foolish to take a great deal
of trouble to measure a displacement which is
scarcely measurable, and which is already known
with fair accuracy. To this it might be answered
that it would be interesting to watch the direct
gravitational effects of the moon on the earth's
surface. But such an interest does not afford
the principal grounds for thinking that this
attempted measurement is worth making. If the
solid earth were to yield to the lunar attraction
with the freedom of a perfect fluid, its surface
would always be perpendicular to the direction
of gravity at each instant of time. Accordingly
a pendulum would then always hang perpendicularly
to the average surface of the earth, and so
there would be no displacement of the pendulum
with reference to the earth's surface. If, then,
the solid earth yields partially to the lunar attraction,
the displacements of a pendulum must be
of smaller extent relatively to the earth than if
the solid earth were absolutely rigid. I must
therefore correct my statement as to our knowledge
of the path pursued by a pendulum, and
say that it is known if the earth is perfectly
unyielding. The accurate observation of the
movement of a pendulum under the influence of
\PageSep{115}
the moon, and the comparison of the observed
oscillation, with that computed on the supposition
that the earth is perfectly stiff, would afford
the means of determining to
what extent the solid earth is
yielding to tidal forces. Such
a result would be very interesting
as giving a measure of the
stiffness of the earth as a whole.

I must pass over the various
\index{Cambridge, experiments with bifilar pendulum at|(}%
\index{Darwin, G. H.!bifilar pendulum|(}%
\index{Darwin, Horace, bifilar pendulum|(}%
\index{Deflection of the vertical!experiments to measure|(}%
\index{Pendulum!bifilar|(}%
earlier attempts to measure the
lunar attraction, and will only
explain the plan, although it
was abortive, used in~1879 by
my brother Horace and myself.

Our object was to measure
the ultra-microscopic displacements
of a pendulum with reference
to the ground on which it
stood. The principle of the apparatus
used for this purpose is
due to Lord Kelvin; it is very
simple, although the practical
application of it was not easy.

%[** TN: Figure wrapped in the original]
\Figure[0.2]{26}{Bifilar Pendulum}{png}

\Fig{26} shows diagrammatically, and not drawn
to scale, a pendulum~$\Seg{A}{B}$ hanging by two wires.
At the foot of the pendulum there is a support~$C$
attached to the stand of the pendulum; $D$~is a
small mirror suspended by two silk fibres, one
being attached to the bottom of the pendulum~$B$
\PageSep{116}
and the other to the support~$C$. When the
two fibres are brought very close together, any
movement of the pendulum perpendicular to the
plane of the mirror causes the mirror to turn
through a considerable angle. The two silk
fibres diverge from one another, but if two vertical
lines passing through the two points of suspension
are $\frac{1}{1,000}$~of an inch apart, then when the
pendulum moves one of these points through a
millionth of an inch, whilst the other attached to~$C$
remains at rest, the mirror will turn through
an angle of more than three minutes of arc.
A lamp is placed opposite to the mirror, and
the image of the lamp formed by reflection in
the mirror is observed. A slight rotation of the
mirror corresponds to an almost infinitesimal
motion of the pendulum, and even excessively
small movements of the mirror are easily detected
by means of the reflected image of the light.

In our earlier experiments the pendulum was
hung on a solid stone gallows; and yet, when
the apparatus was made fairly sensitive, the image
of the light danced and wandered incessantly.
Indeed, the instability was so great that
the reflected image wandered all across the room.
We found subsequently that this instability was
due both to changes of temperature in the stone
gallows, and to currents in the air surrounding
the pendulum.

To tell of all the difficulties encountered
\PageSep{117}
might be as tedious as the difficulties themselves,
so I shall merely describe the apparatus in its
ultimate form. The pendulum was suspended,
as shown in~\fig{26}, by two wires; the two wires
being in an east and west plane, the pendulum
could only swing north and south. It was hung
inside a copper tube, just so wide that the solid
copper cylinder, forming the pendulum bob, did
not touch the sides of the tube. A spike projected
from the base of the pendulum bob
through a hole in the bottom of the tube. The
mirror was hung in a little box, with a plate-glass
front, which was fastened to the bottom of the
copper tube. The only communication between
the tube and the mirror-box was by the hole
through which the spike of the pendulum projected,
but the tube and mirror-box together
formed a water-tight vessel, which was filled with
a mixture of spirits of wine and boiled water.
The object of the fluid was to steady the
mirror and the pendulum, while allowing its
slower movements to take place. The water was
boiled to get rid of air in it, and the spirits of
wine was added to increase the resistance of the
fluid, for it is a remarkable fact that a mixture
of spirits and water has considerably more viscosity
or stickiness than either pure spirits or
pure water.

The copper tube, with the pendulum and mirror-box,
was supported on three legs resting on
\PageSep{118}
a block of stone weighing a ton, and this stood
on the native gravel in a north room in the laboratory
at Cambridge. The whole instrument
was immersed in a water-jacket, which was furnished
with a window near the bottom, so that
the little mirror could be seen from outside. A
water ditch also surrounded the stone pedestal,
and the water jacketing of the whole instrument
made the changes of temperature very slow.

A gas jet, only turned up at the moment of
observation, furnished the light to be observed
by reflection in the little mirror. The gas
burner could be made to travel to and fro along
a scale in front of the instrument. In the preliminary
description I have spoken of the motion
of the image of a fixed light, but it clearly
amounts to the same thing if we measure the
motion of the light, keeping the point of observation
fixed. In our instrument the image of
the movable gas jet was observed by a fixed telescope
placed outside of the room. A bright
light was unfortunately necessary, because there
was a very great loss of light in the passages to
and fro through two pieces of plate glass and a
considerable thickness of water.

Arrangements were made by which, without
entering the room, the gas jet could be turned
up and down, and could be made to move to and
fro in the room in an east and west direction,
until its image was observed in the telescope.
\PageSep{119}
There were also adjustments by which the two
silk fibres from which the mirror hung could be
brought closer together or further apart, thus
making the instrument more or less sensitive.
There was also an arrangement by which the image
of the light could be brought into the field
of view, when it had wandered away beyond the
limits allowed for by the traverse of the gas jet.

When the instrument was in adjustment, an
observation consisted of moving the gas jet until
its image was in the centre of the field of
view of the telescope; a reading of the scale, by
another telescope, determined the position of the
gas jet to within about a twentieth of an inch.

The whole of these arrangements were arrived
at only after laborious trials, but all the precautions
were shown by experience to be necessary,
and were possibly even insufficient to guard the
instrument from the effects of changes of temperature.
I shall not explain the manner in which
we were able to translate the displacements of
the gas jet into displacements of the pendulum.
It was not very satisfactory, and only gave approximate
results. A subsequent form of an
instrument of this kind, designed by my brother,
has been much improved in this respect. It was
he also who designed all the mechanical appliances
in the experiment of which I am speaking.

It may be well to reiterate that the pendulum
was only free to move north and south, and that
\PageSep{120}
our object was to find how much it swung. The
east and west motion of a pendulum is equally
interesting, but as we could not observe both
displacements at the same time, we confined our
attention in the first instance to the northerly
and southerly movements.

When properly adjusted the apparatus was so
sensitive that, if the bob of the pendulum moved
through $\frac{1}{40,000}$~of a millimetre, that is, a millionth
part of an inch, we could certainly detect the
movement, for it corresponded to a twentieth
of an inch in our scale of position of the gas
jet. When the pendulum bob moved through
this amount, the wires of the pendulum turned
through one two-hundredth of a second of arc;
this is the angle subtended by one inch at $770$~miles
distance. I do not say that we could actually
measure with this degree of refinement, but
we could detect a change of that amount. In
view of the instability of the pendulum, which
still continued to some extent, it may be hard to
gain credence for the statement that such a small
deflection was a reality, so I will explain how we
were sure of our correctness.

In setting up the apparatus, work had to be
conducted inside the room, and some preliminary
observations of the reflected image of a stationary
gas jet were made without the use of the telescope.
The scale on which the reflected spot
of light fell was laid on the ground at about
\PageSep{121}
seven feet from the instrument; in order to
watch it I knelt on the pavement behind the
scale, and leant over it. I was one day watching
on the scale the spot of light which revealed
the motion of the pendulum, and, being tired
with kneeling, supported part of my weight on
my hands a few inches in front of the scale.
The place where my hands rested was on the
bare earth, from which a paving stone had been
removed. I was surprised to find quite a large
change in the reading. It seemed at first incredible
that my change of position was the cause,
but after several trials I found that light pressure
with one hand was quite sufficient to produce
an effect. It must be remembered that this was
not simply a small pressure delivered on the bare
earth at, say, seven feet distance, but it was the
difference of effect produced by the same pressure
at seven feet and six feet; for, of course,
the change only consisted in the distribution of
the weight of a small portion of my body.

It is not very easy to catch the telescopic image
of a spot of light reflected from a mirror of
the size of a shilling. Accordingly, in setting
up our apparatus, we availed ourselves of this result,
for we found that the readiest way of bringing
the reflected image into the telescopic field
of view was for one of us to move slowly about
the room, until the image of the light was
brought, by the warping of the soil due to his
\PageSep{122}
weight, into the field of view of the telescope.
He then placed a heavy weight on the floor
where he had been standing; this of course
drove the image out of the field of view, but
after he had left the room the image of the flame
was found to be in the field.

We ultimately found, even when no special
pains had been taken to render the instrument
sensitive, that if one of us was in the room, and
stood at about sixteen feet south of the instrument
with his feet about a foot apart, and slowly
shifted his weight from one foot to the other, a
distinct change was produced in the image of the
gas flame, and of course in the position of the
little mirror, from which the image was derived
by reflection. It may be well to consider for
a moment the meaning of this result. If one
presses with a finger on a flat slab of jelly, a sort
of dimple is produced, and if a pin were sticking
upright in the jelly near the dimple, it would tilt
slightly towards the finger. Now this is like
what we were observing, for the jelly represents
the soil, and the tilt of the pin corresponds to
that of the pendulum. But the scale of the displacement
is very different, for our pendulum
stood on a block of stone weighing nearly a ton,
which rested on the native gravel at two feet below
the level of the floor, and the slabs of the
floor were removed from all round the pendulum.
The dimple produced by a weight of $140$~lbs.\ on
\PageSep{123}
\index{Distortion of soil!by weight}%
\index{Elastic distortion!of soil by weight}%
the stone paved floor must have been pretty
small, and the slope of the sides of that dimple
\index{Dimple!in soil, due to weight}%
at sixteen feet must have been excessively slight;
but we were here virtually observing the change
of slope at the instrument, when the centre of
the dimple was moved from a distance of fifteen
feet to sixteen feet.

It might perhaps be thought that all observation
would be rendered impossible by the street
traffic and by the ordinary work of the laboratory.
But such disturbances only make tremors
of very short period, and the spirits and water
damped out quick oscillations so thoroughly, that
no difference could be detected in the behavior
of the pendulum during the day and during the
night. Indeed, we found that a man could stand
close to the instrument and hit the tub and pedestal
smart blows with a stick, without producing
any sensible effect. But it was not quite easy to
try this experiment, because there was a considerable
disturbance on our first entering the room;
and when this had subsided small movements of
the body produced a sensible deflection, by slight
changes in the distribution of the experimenter's
weight.

It is clear that we had here an instrument of
amply sufficient delicacy to observe the lunar
tide-generating force, and yet we completely
failed to do so. The pendulum was, in fact,
always vacillating and changing its position by
\PageSep{124}
many times the amount of the lunar effect which
we sought to measure.

An example will explain how this was: A series
of frequent readings were taken from July
21st to~25th, 1881, with the pendulum arranged
to swing north and south. We found that there
was a distinct diurnal period, with a maximum at
noon, when the pendulum bob stood furthest
northward. The path of the pendulum was interrupted
by many minor zigzags, and it would
sometimes reverse its motion for an hour together.
But the diurnal oscillation was superposed on a
gradual drift of the pendulum, for the mean
diurnal position traveled slowly southward. Indeed,
in these four days the image disappeared
from the scale three times over, and was brought
back into the field of view three times by the
appliance for that purpose. On the night between
the 24th and~25th the pendulum took an
abrupt turn northward, and the scale reading
was found, on the morning of the~25th, nearly
at the opposite end of the scale from that towards
which it had been creeping for four days
previously.

Notwithstanding all our precautions the pendulum
was never at rest, and the image of the
flame was always trembling and dancing, or waving
slowly to and fro. In fact, every reading of
our scale had to be taken as the mean of the
excursions to right and left. Sometimes for two
\PageSep{125}
or three days together the dance of the image
would be very pronounced, and during other
days it would be remarkably quiescent.

The origin of these tremors and slower movements
\index{Earthquakes!microsisms and earth tremors|(}%
\index{Italian investigations in seismology|(}%
\index{Tremors, earth}%
is still to some extent uncertain. Quite
recent investigations by Professor Milne seem to
\index{Milne on seismology}%
show that part of them are produced by currents
in the fluid surrounding the pendulum, that
others are due to changes in the soil of a very
local character, and others again to changes
affecting a considerable tract of soil. But when
all possible allowance is made for these perturbations,
it remains certain that a large proportion
of these mysterious movements are due to minute
earthquakes.

Some part of the displacements of our pendulum
was undoubtedly due to the action of the
moon, but it was so small a fraction of the whole,
that we were completely foiled in our endeavor
to measure it.\footnote
  {Since the date of our experiment the bifilar pendulum has
  been perfected by my brother, and it is now giving continuous
  photographic records at several observatories. It is now made
  to be far less sensitive than in our original experiment, and no
  attempt is made to detect the direct effect of the moon.}
\index{Cambridge, experiments with bifilar pendulum at|)}%
\index{Darwin, G. H.!bifilar pendulum|)}%
\index{Darwin, Horace, bifilar pendulum|)}%
\index{Deflection of the vertical!experiments to measure|)}%
\index{Pendulum!bifilar|)}%

The minute earthquakes of which I have
\index{Microsisms, minute earthquakes|(}%
spoken are called by Italian observers microsisms,
and this name has been very generally
adopted. The literature on the subject of seismology
is now very extensive, and it would be
out of place to attempt to summarize here the
\PageSep{126}
conclusions which have been drawn from observation.
I may, however, permit myself to add a
few words to indicate the general lines of the research,
which is being carried on in many parts
of the world.

Italy is a volcanic country, and the Italians
have been the pioneers in seismology. Their
observations have been made by means of pendulums
of various lengths, and with instruments
of other forms, adapted for detecting vertical
movements of the soil. The conclusions at
which Father Bertelli arrived twenty years ago
\index{Bertelli on Italian seismology}%
may be summarized as follows:---

The oscillation of the pendulum is generally
\index{Pendulum!as seismological instrument}%
parallel to valleys or chains of mountains in the
neighborhood. The oscillations are independent
of local tremors, velocity and direction of wind,
rain, change of temperature, and atmospheric
electricity.

Pendulums of different lengths betray the
movements of the soil in different manners, according
to the agreement or disagreement of
their natural periods of oscillation with the period
of the terrestrial vibrations.

The disturbances are not strictly simultaneous
in the different towns of Italy, but succeed one
another at short intervals.

After earthquakes the ``tromometric'' or microseismic\DPnote{** [sic]}
\index{Tromometer, a seismological instrument}%
movements are especially apt to be in
a vertical direction. They are always so when
\PageSep{127}
\index{Bertelli on Italian seismology}%
the earthquake is local, but the vertical movements
are sometimes absent when the shock
occurs elsewhere. Sometimes there is no movement
at all, even when the shock occurs quite
close at hand.

The positions of the sun and moon appear to
have some influence on the movements of the
pendulum, but the disturbances are especially
\index{Pendulum!as seismological instrument}%
frequent when the barometer is low.

The curves of ``the monthly means of the
tromometric movement'' exhibit the same forms
in the various towns of Italy, even those which
are distant from one another.

The maximum of disturbance occurs near the
winter solstice and the minimum near the summer
solstice.

At Florence a period of earthquakes is presaged
\index{Earthquakes!microsisms and earth tremors|)}%
\index{Tromometer, a seismological instrument}%
by the magnitude and frequency of oscillatory
movements in a vertical direction. These
movements are observable at intervals and during
several hours after each shock.

Some very curious observations on microsisms
\index{Microsisms, minute earthquakes|)}%
have also been made in Italy with the microphone,
by which very slight movements of the
soil are rendered audible.

Cavaliere de Rossi, of Rome, has established a
``geodynamic'' observatory in a cave $700$~metres
above the sea at Rocca di Papa, on the external
slope of an extinct volcano.

At this place, remote from all carriages and
\PageSep{128}
\index{Rossi on Italian seismology|(}%
roads, he placed his microphone at a depth of $20$~metres
\index{Microphone as a seismological instrument|(}%
below the ground. It was protected
against insects by woolen wrappings. Carpet
was spread on the floor of the cave to deaden
the noise from particles of stone which might
possibly fall. Having established his microphone,
he waited till night, and then heard noises which
he says revealed ``natural telluric phenomena.''
The sounds which he heard he describes as
``roarings, explosions occurring isolated or in
volleys, and metallic or bell-like sounds'' (\textit{fremiti,
scopii isolati o di moschetteria, e suoni-metallici
o di campana}). They all occurred
mixed indiscriminately, and rose to maxima at
irregular intervals. By artificial means he was
able to cause noises which he calls ``rumbling (?)
or crackling'' (\textit{rullo o crepito}). The roaring
(\textit{fremito}) was the only noise which he could reproduce
artificially, and then only for a moment.
It was done by rubbing together the conducting
wires, ``in the same manner as the rocks must
rub against one another when there is an earthquake.''

A mine having been exploded in a quarry at
some distance, the tremors in the earth were
audible in the microphone for some seconds
subsequently.

There was some degree of coincidence between
the agitation of the pendulum-seismograph and
the noises heard with the microphone.
\PageSep{129}

At a time when Vesuvius became active,
Rocca di Papa was agitated by microsisms, and
the shocks were found to be accompanied by the
very same microphonic noises as before. The
noises sometimes became ``intolerably loud;''
especially on one occasion in the middle of the
night, half an hour before a sensible earthquake.
The agitation of the microphone corresponded
exactly with the activity of Vesuvius.

Rossi then transported his microphone to
Palmieri's Vesuvian observatory, and worked in
conjunction with him. He there found that
each class of shock had its corresponding noise.
The sussultorial shocks, in which I conceive the
movement of the ground is vertically up and
down, gave the volleys of musketry (\textit{i~colpi di
moschetteria}), and the undulatory shocks gave
the roarings (\textit{i~fremiti}). The two classes of
noises were sometimes mixed up together.

Rossi makes the following remarks: ``On
Vesuvius I was put in the way of discovering
that the simple fall and rise in the ticking which
occurs with the microphone [\textit{battito del orologio
unito al microfono}] (a phenomenon observed
by all, and remaining inexplicable to all) is a
consequence of the vibration of the ground.''
This passage alone might perhaps lead one to
suppose that clockwork was included in the circuit;
but that this was not the case, and that
``ticking'' is merely a mode of representing a
\PageSep{130}
\index{Paschwitz, von Rebeur!on horizontal pendulum|(}%
\index{Pendulum!horizontal|(}%
natural noise is proved by the fact that he subsequently
says that he considers the ticking to
be ``a telluric phenomenon.''

Rossi then took the microphone to the Solfatara
\index{Rossi on Italian seismology|)}%
of Pozzuoli, and here, although no sensible
tremors were felt, the noises were so loud as
to be heard simultaneously by all the people in
the room. The ticking was quite masked by
other natural noises. The noises at the Solfatara
were imitated by placing the microphone
on the lid of a vessel of boiling water. Other
seismic noises were then imitated by placing the
microphone on a marble slab, and scratching
\index{Microphone as a seismological instrument|)}%
and tapping the under surface of it.

The observations on Vesuvius led him to the
conclusion that the earthquake oscillations have
sometimes fixed ``nodes,'' for there were places
on the mountain where no effects were observed.
There were also places where the movement was
intensified, and hence it may be concluded that
the centre of disturbance may sometimes be very
distant, even when the observed agitation is
considerable.

At the present time perhaps the most distinguished
investigator in seismology is Professor
Milne, formerly of the Imperial College of Engineering
\index{Milne on seismology}%
at Tokyo. His residence in Japan gave
\index{Japan, frequency of earthquakes}%
him peculiar opportunities of studying earthquakes,
for there is, in that country, at least one
earthquake per diem of sufficient intensity to
\index{Italian investigations in seismology|)}%
\PageSep{131}
\index{Japan, frequency of earthquakes}%
affect a seismometer. The instrument of which
he now makes most use is called a horizontal
pendulum. The principle involved in it is old,
but it was first rendered practicable by von
Rebeur-Paschwitz, whose early death deprived
the world of a skillful and enthusiastic investigator.

The work of Paschwitz touches more closely
on our present subject than that of Milne, because
he made a gallant attempt to measure the
moon's tide-generating force, and almost persuaded
himself that he had done so.

The horizontal pendulum is like a door in its
mode of suspension. If a doorpost be absolutely
vertical, the door will clearly rest in any
position, but if the post be even infinitesimally
tilted the door naturally rests in one definite
position. A very small shift of the doorpost is
betrayed by a considerable change in the position
of the door. In the pendulum the door is
replaced by a horizontal boom, and the hinges
by steel points resting in agate cups, but the
principle is the same.

The movement of the boom is detected and
registered photographically by the image of a
light reflected from certain mirrors. Paschwitz
made systematic observations with his pendulum
at Wilhelmshaven, Potsdam, Strassburg, and
Orotava. He almost convinced himself at one
time that he could detect, amidst the wanderings
\PageSep{132}
of the curves of record, a periodicity corresponding
to the direct effect of the moon's action.
But a more searching analysis of his results left
the matter in doubt. Since his death the observations
at Strassburg have been continued by
M.~Ehlert. His results show an excellent consistency
\index{Ehlert, observation with horizontal pendulum}%
with those of Paschwitz, and are therefore
\index{Paschwitz, von Rebeur!on horizontal pendulum|)}%
confirmatory of the earlier opinion of the
latter. I am myself disposed to think that the
detection of the lunar attraction is a reality, but
the effect is so minute that it cannot yet be
relied on to furnish a trustworthy measurement
of the amount of the yielding of the solid earth
to tidal forces.

It might be supposed that doubt could hardly
arise as to whether or not the direct effect of
the moon's attraction had been detected. But
I shall show in the next chapter that at many
places the tidal forces must exercise in an indirect
manner an effect on the motion of a pendulum
\index{Pendulum!horizontal|)}%
much greater than the direct effect.

It was the consideration of this indirect effect,
and of other concomitants, which led us to
abandon our attempted measurement, and to
conclude that all endeavors in that direction
were doomed to remain for ever fruitless. I can
but hope that a falsification of our forecast by
M.~Ehlert and by others may be confirmed.
\PageSep{133}

\begin{Authorities}
G.~H. Darwin and Horace Darwin, ``Reports to the British
Association for the Advancement of Science:''---

\Title{Measurement of the Lunar Disturbance of Gravity.} York
meeting, 1881, pp.~93--126.

\Title{Second Report on the same}, with appendix. Southampton
meeting, 1882, pp.~95--119.

E.~von Rebeur-Paschwitz, \Title{Das Horizontalpendel}.

``Nova Acta Leop.\ Carol.\ Akad.,'' 1892, vol.~lx.\ no.~1, p.~213;
also ``Brit.\ Assoc.\ Reports,''~1893.

E.~von Rebeur-Paschwitz, \Title{Ueber Horizontalpendel-Beobach\-tungen
in Wilhelmshaven, Potsdam und Puerto Orotava auf Tenerifa}.

``Astron.\ Nachrichten,'' vol.~cxxx.\ pp.~194--215.

R.~Ehlert, \Title{Horizontalpendel-Beobachtungen}.

``Beiträge zur Geophysik,'' vol.~iii.\ Part~I., 1896.

C.~Davison, \Title{History of the Horizontal and Bifilar Pendulums}.
\index{Davison, history of bifilar and horizontal pendulums}%

``Appendix to Brit.\ Assoc.\ Report on Earth Tremors.'' Ipswich
meeting, 1895, pp.~184--192.

``British Association Reports of Committees.''

\Title{On Earth Tremors}, 1891--95 (the first being purely formal).

\Title{On Seismological Investigation}, 1896.

The literature on Seismology is very extensive, and would
\index{Seismology}%
need a considerable index; the reader may refer to \Title{Earthquakes}
and to \Title{Seismology} by John Milne. Both works form volumes in
the International Scientific Series, published by Kegan Paul,
Trench, Trübner \&~Co.
\end{Authorities}
\index{Deflection of the vertical|)}%
\index{Forces!deflection of vertical by|)}%
\PageSep{134}


\Chapter[Distortion of the Earth's Surface]{VII}
{The Elastic Distortion of the Earth's Surface
by Varying Loads}

\First{When} the tide rises and falls on the seacoast,
\index{Darwin, G. H.!distortion of earth's surface by varying loads|(}%
\index{Deflection of the vertical!due to tide|(}%
\index{Distortion of soil!by varying loads|(}%
\index{Elastic distortion!of earth by varying loads|(}%
many millions of tons of water are brought alternately
nearer and further from the land. Accordingly
a pendulum suspended within a hundred
miles or so of a seacoast should respond to the
attraction of the sea water, swinging towards the
sea at high water, and away from it at low water.
Since the rise and fall has a lunar periodicity the
pendulum should swing in the same period, even
if the direct attraction of the moon did not affect
it. But, as I shall now show, the problem is
further confused by another effect of the varying
tidal load.

We saw in \Ref{Chapter}{VI}.\ how a weight resting
on the floor in the neighborhood of our pendulum
produced a dimple by which the massive
stone pedestal of our instrument was tilted over.
Now as low tide changes to high tide the position
of an enormous mass of water is varied with
respect to the land. Accordingly the whole
coast line must rock to and fro with the varying
tide. We must now consider the nature of the
\PageSep{135}
distortion of the soil produced in this way. The
mathematical investigation of the form of the
dimple in a horizontal slab of jelly or other elastic
\index{Dimple!form of, in elastic slab}%
material, due to pressure at a single point,
shows that the slope at any place varies inversely
as the square of the distance from the centre.
That is to say, if starting from any point we
proceed to half our original distance, we shall
find four times as great a slope, and at one third
\Figure[0.7]{27}{Form of Dimple in an Elastic Surface}{png}
of the original distance the slope will be augmented
ninefold.

The theoretical form of dimple produced by
pressure at a single mathematical point is shown
in~\fig{27}. The slope is exaggerated so as to
render it visible, and since the figure is drawn on
the supposition that the pressure is delivered at
a mathematical point, the centre of the dimple
is infinitely deep. If the pressure be delivered
by a blunt point, the slope at a little distance
\PageSep{136}
will be as shown, but the centre will not be infinitely
deep. If therefore we pay no attention to
the very centre, this figure serves to illustrate
the state of the case. When the dimple is produced
by the pressure of a weight, that weight,
being endowed with gravitation, attracts any
other body with a force varying inversely as the
square of the distance. It follows, therefore,
that the slope of the dimple is everywhere exactly
\index{Slope of soil!due to elastic distortion}%
proportional to the gravitational attraction
\index{Attraction!of weight resting on elastic slab proportional to slope}%
of the weight. Since this is true of a single
weight, it is true of a group of weights, each
producing its own dimple by pressure and its own
attraction, strictly proportional to one another.
Thus the whole surface is deformed by the superposition
of dimples, and the total attraction is
the sum of all the partial attractions.

Let us then imagine a very thick horizontal
slab of glass supporting any weights at any parts
of its surface. The originally flat surface of the
slab will be distorted into shallow valleys and
low hills, and it is clear that the direct attraction
of the weights will everywhere be exactly proportional
to the slopes of the hillsides; also the
direction of the greatest slope at each place must
agree with the direction of the attraction. The
direct attraction of the weights will deflect a
pendulum from the vertical, and the deflection
must be exactly proportional to the slope produced
by the pressure of the weights. It may
\PageSep{137}
be proved that if the slab is made of a very stiff
glass the angular deflection of the pendulum
under the influence of attraction will be one fifth
\index{Attraction!of weight resting on elastic slab proportional to slope}%
of the slope of the hillside; if the glass were
of the most yielding kind, the fraction would be
one eighth. The fraction depends on the degree
of elasticity of the material, and the stiffer it is
the larger the fraction.

The observation of a pendulum consists in
noting its change of position with reference to
the surface of the soil; hence the slope of the
soil, and the direct attraction of the weight
which causes that slope, will be absolutely fused
together, and will be indistinguishable from one
another.

Now, this conclusion may be applied to the
tidal load, and we learn that, if rocks are of the
same degree of stiffness as glass of medium
quality, the direct attraction of the tidal load
produces one sixth of the apparent deflection of
a pendulum produced by the tilting of the soil.

If any one shall observe a pendulum, within
say a hundred miles of the seacoast, and shall
detect a lunar periodicity in its motion, he can
only conclude that what he observes is partly
due to the depression and tilting of the soil,
partly to attraction of the sea water, and partly
to the direct attraction of the moon. Calculation
indicates that, with the known average elasticity
of rock, the tilting of the soil is likely to
\PageSep{138}
\index{Elastic distortion!calculation and illustration|(}%
\index{Slope of soil!calculation and illustration of|(}%
be far greater than the other two effects combined.
Hence, if the direct attraction of the
moon is ever to be measured, it will first be
necessary to estimate and to allow for other important
oscillations with lunar periodicity. The
difficulty thus introduced into this problem is so
serious that it has not yet been successfully met.
It may perhaps some day be possible to distinguish
the direct effects of the moon's tidal attraction
from the indirect effects, but I am not
very hopeful of success in this respect. It was
pointed out in \Ref{Chapter}{VI}.\ that there is some
reason to think that a lunar periodicity in the
swing of a pendulum has been already detected,
and if this opinion is correct, the larger part of
the deflection was probably due to these indirect
effects.

The calculation of the actual tilting of the
coast line by the rising tide would be excessively
complex even if accurate estimates were obtainable
of the elasticity of the rock and of the tidal
load. It is, however, possible to formulate a
soluble problem of ideal simplicity, which will
afford us some idea of the magnitude of the
results occurring in nature.

In the first place, we may safely suppose the
earth to be flat, because the effect of the tidal
load is quite superficial, and the curvature of the
earth is not likely to make much difference in
the result. In the second place, it greatly simplifies
\PageSep{139}
the calculation to suppose the ocean to
consist of an indefinite number of broad canals,
separated from one another by broad strips of
land of equal breadth. Lastly, we shall suppose
that each strip of sea rocks about its middle line,
so that the water oscillates as in a seiche of the
Lake of Geneva; thus, when it is high water
on the right-hand coast of a strip of sea, it is
low water on the left-hand coast, and vice versa.
We have then to determine the change of shape
of the ocean-bed and of the land, as the tide
rises and falls. The problem as thus stated is
\Figure{28}{Distortion of Land and Sea-bed by Tidal Load}{png}
vastly simpler than in actuality, yet it will suffice
to give interesting indications of what must
occur in nature.

The figure~\figref{28} shows the calculated result, the
slopes being of course enormously exaggerated.
The straight line represents the level surface of
land and sea before the tidal oscillation begins,
the shaded part being the land and the dotted
part the sea. Then the curved line shows the
form of the land and of the sea-bed, when it is
low water at the right of the strip of land and high
\PageSep{140}
water at the left. The figure would be reversed
when the high water interchanges position
with the low water. Thus both land and sea
rock about their middle lines, but the figure
shows that the strip of land remains nearly flat
although not horizontal, whilst the sea-bed becomes
somewhat curved.

It will be noticed that there is a sharp nick at
the coast line. This arises from the fact that
deep water was assumed to extend quite up to
the shore line; if, however, the sea were given
a shelving shore, as in nature, the sharp nick
would disappear, although the form of the distorted
rocks would remain practically unchanged
elsewhere.

Thus far the results have been of a general
character, and we have made no assumptions as
to the degree of stiffness of the rock, or as to
the breadths of the oceans and continents. Let
us make hypotheses which are more or less
plausible. At many places on the seashore the
tide ranges through twenty or thirty feet, but
these great tides only represent the augmentation
of the tide-wave as it runs into shallow
water, and it would not be fair to suppose our
tide to be nearly so great. In order to be moderate,
I will suppose the tide to have a range of
$160$~centimetres, or, in round numbers, about $5$~feet.
Then, at the high-water side of the sea,
the water is raised by $80$~centimetres, and at the
\index{Elastic distortion!calculation and illustration|)}%
\index{Slope of soil!calculation and illustration of|)}%
\PageSep{141}
low-water side it is depressed by the same
amount. The breadth of the Atlantic is about
$4,000$~or $5,000$~miles. I take then, the breadth of
the oceans and of the continents as $3,900$~miles,
or $6,280$~kilometres. Lastly, as rocks are usually
stiffer than glass, I take the rock bed to
be twice as stiff as the most yielding glass, and
quarter as stiff again as the stiffest glass; this
assumption as to the elasticity of rock makes the
attraction at any place one quarter of the slope.
For a medium glass we found the fraction to be
about one sixth. These are all the data required
for determining the slope.

It is of course necessary to have a unit of
measurement for the slope of the surface. Now
a second of arc is the name for the angular
magnitude of an inch seen at $3\frac{1}{4}$~miles, and accordingly
a hundredth of a second of arc, usually
written~$0''.01$, is the angular magnitude of
an inch seen at $325$~miles; the angles will then
be measured in hundredths of seconds.

Before the tide rises, the land and sea-bed
are supposed to be perfectly flat and horizontal.
Then at high tides the slopes on the land are as
follows:---
\[
\begin{array}{r@{\ }lc}
\multicolumn{2}{c}{\ColHead{Distance from high-water mark}} &
\multicolumn{1}{c}{\ColHead[2in]{Slope of the land measured in hundredths of seconds of arc}} \\
 10 &\text{metres}     &  10 \\
100 &\text{metres}     & \Z8 \\
  1 &\text{kilometre}  & \Z6 \\
 10 &\text{kilometres} & \Z4 \\
 20 &\text{kilometres} & \Z3\rlap{$\frac{1}{2}$} \\
100 &\text{kilometres} & \Z2
\end{array}
\]
\PageSep{142}
The slope is here expressed in hundredths of a
second of arc, so that at $100$~kilometres from the
coast, where the slope is~$2$, the change of plane
amounts to the angle subtended by one inch at
$162$~miles.

When high water changes to low water, the
slopes are just reversed, hence the range of
change of slope is represented by the doubles of
these angles. If the change of slope is observed
by some form of pendulum, allowance must be
made for the direct attraction of the sea, and it
appears that with the supposed degree of stiffness
of rock these angles of slope must be augmented
in the proportion of $5$~to~$4$. Thus, we
double the angles to allow of change from high
to low water, and augment the numbers as $5$ is
to~$4$, to allow for the direct attraction of the sea.
Finally we find results which may be arranged
in the following tabular form:---
\[
\begin{array}{r@{\ }lc}
\multicolumn{2}{c}{\ColHead{Distance from high-water mark}} &
\multicolumn{1}{c}{\ColHead[1.75in]{Apparent range of deflection of the vertical}} \\
 10 &\text{metres}     & 0''.25\Z \\
100 &\text{metres}     & 0''.20\Z \\
  1 &\text{kilometre}  & 0''.15\Z \\
 10 &\text{kilometres} & 0''.10\Z \\
 20 &\text{kilometres} & 0''.084  \\
100 &\text{kilometres} & 0''.050
\end{array}
\]
At the centre of the continent, $1,950$~miles from
the coast, the range will be~$0''.012$.

If all the assumed data be varied, the ranges
of the slopes are easily calculable, but these
\PageSep{143}
results may be taken as fairly representative, although
perhaps somewhat underestimated. Lord
Kelvin has made an entirely independent estimate
\index{Kelvin, Lord!calculation of tidal attraction}%
of the probable deflection of a pendulum
by the direct attraction of the sea at high tide.
\index{Attraction!of tide calculated}%
He supposes the tide to have a range of $10$~feet
from low water to high water, and he then estimates
the attraction of a slab of water $10$~feet
thick, $50$~miles broad perpendicular to the coast,
and $100$~miles long parallel to the coast, on a
plummet $100$~yards from low-water mark and
opposite the middle of the $100$~miles. This
would, he thinks, very roughly represent the
state of things at St.~Alban's Head, in England.
He finds the attraction such as to deflect the
plumb-line, as high water changes to low water,
by a twentieth of a second of arc. The general
law as to the proportionality of slope to
attraction shows that, with our supposed degree
of stiffness of rock, the apparent deflection of a
plumb-line, due to the depression of the coast
and the attraction of the sea as high water
changes to low water, will then be a quarter of a
second of arc. Postulating a smaller tide, but
spread over a wider area, I found the result
would be a fifth of a second; thus the two results
present a satisfactory agreement.

This speculative investigation receives confirmation
from observation. The late M.~d'Abbadie
\index{Abbadie, tidal deflection of vertical}%
established an observatory at his château of
\index{Deflection of the vertical!due to tide|)}%
\PageSep{144}
\index{Abbadie, tidal deflection of vertical}%
Abbadia, close to the Spanish frontier and within
a quarter of a mile of the Bay of Biscay. Here
he constructed a special form of instrument for
detecting small changes in the direction of gravity.
Without going into details, it may suffice
to state that he compared a fixed mark with its
image formed by reflection from a pool of mercury.
He took $359$~special observations at the
times of high and low tide in order to see, as he
says, whether the water exercised an attraction
on the pool of mercury, for it had not occurred
to him that the larger effect would probably
arise from the bending of the rock. He found
that in $243$~cases the pool of mercury was tilted
towards the sea at high water or away from it at
low water; in $59$~cases there was no apparent
effect, and in the remaining $57$~cases the action
was inverted. The observations were repeated
later by his assistant in the case of $71$~successive
high waters\footnote
  {Presumably the observation at one high water was defective.}
and $73$~low waters, and he also
found that in about two thirds of the observations
the sea seemed to exercise its expected
influence. We may, I think, feel confident that
on the occasions where no effect or a reversal
was perceived, it was annulled or reversed by a
warping of the soil, such as is observed with
seismometers.

Dr.~von Rebeur-Paschwitz also noted deflections
\index{Paschwitz, von Rebeur!tidal deflection of vertical at Wilhelmshaven}%
due to the tide at Wilhelmshaven in Germany.
\PageSep{145}
\index{Atmospheric pressure!distortion of soil by}%
\index{Elastic distortion!by atmospheric pressure|(}%
\index{Pressure of atmosphere, elastic distortion of soil by}%
The deflection was indeed of unexpected
magnitude at this place, and this may probably
be due to the peaty nature of the soil, which
renders it far more yielding than if the observatory
were built on rock.

This investigation has another interesting application,
for the solid earth has to bear another
varying load besides that of the tide. The
atmosphere rests on the earth and exercises a
variable pressure, as shown by the varying
height of the barometer. The variation of
pressure is much more considerable than one
would be inclined to suspect off-hand. The
height of the barometer ranges through nearly
two inches, or say five centimetres; this means
that each square yard of soil supports a weight
greater by $1,260$~lbs.\ when the barometer is very
high, than when it is very low. If we picture
to ourselves a field loaded with half a ton to
each square yard, we may realize how enormous
is the difference of pressure in the two cases.

In order to obtain some estimate of the effects
of the changing pressure, I will assume, as before,
that the rocks are a quarter as stiff again
as the stiffest glass. On a thick slab of this
material let us imagine a train of parallel waves
of air, such that at the crests of the waves the
barometer is $5$~centimetres higher than at the
hollow. Our knowledge of the march of barometric
gradients on the earth's surface makes it
\PageSep{146}
plausible to assume that it is $1,500$~miles from
the line of highest to that of lowest pressure.
Calculation then shows that the slab is distorted
into parallel ridges and valleys, and that the
tops of the ridges are $9$~centimetres, or $3\frac{1}{2}$~inches,
higher than the hollows. Although the actual
distribution of barometric pressures is not of this
simple character, yet this calculation shows, with
a high degree of probability, that when the
barometer is very high we are at least $3$~inches
nearer the earth's centre than when it is very
low.

The consideration of the effects of atmospheric
\index{Atmospheric pressure!distortion of soil by}%
\index{Level of sea affected by atmospheric pressure}%
\index{Pressure of atmosphere, elastic distortion of soil by}%
pressure leads also to other curious conclusions.
I have remarked before that the sea must respond
\index{Sea!level affected by atmospheric pressure}%
to barometric pressure, being depressed
by high and elevated by low pressure. Since a
column of water $68$~centimetres ($2$~ft.\ $3$~in.)\ in
height weighs the same as a column, with the
same cross section, of mercury, and $5$~centimetres
in height, the sea should be depressed by $68$~centimetres
under the very high barometer as compared
with the very low barometer. But the
height of the water can only be determined with
reference to the land, and we have seen that the
land must be depressed by $9$~centimetres. Hence
the sea would be apparently depressed by only $59$~centimetres.

It is probable that, in reality, the larger barometric
inequalities do not linger quite long
\PageSep{147}
enough over particular areas to permit the sea to
attain everywhere its due slope, and therefore the
full difference of water level can only be attained
occasionally. On the other hand the elastic compression
\index{Elastic distortion!by atmospheric pressure|)}%
of the ground must take place without
sensible delay. Thus it seems probable that this
compression must exercise a very sensible effect
in modifying the apparent depression or elevation
of the sea under high and low barometer.

If delicate observations are made with some
form of pendulum, the air waves and the consequent
distortions of the soil should have a sensible
effect on the instrument. In the ideal case
which I have described above, it appears that
the maximum apparent deflection of the plumb-line
would be $\frac{1}{90}$~of a second of arc; this would
be augmented to $\frac{1}{70}$~of a second by the addition
of the true deflection, produced by the attraction
of the air. Lastly, since the slope and attraction
would be absolutely reversed when the air wave
assumed a different position with respect to the
observer, it is clear that the range of apparent
oscillation of the pendulum might amount to
$\frac{1}{35}$~of a second of arc.

This oscillation is actually greater than that
due to the direct tidal force of the moon acting
on a pendulum suspended on an ideally unyielding
earth. Accordingly we have yet another
reason why the direct measurement of the tidal
force presents a problem of the extremest difficulty.
\PageSep{148}

\begin{Authorities}
G.~H. Darwin, \Title{Appendix to the Second Report on Lunar Disturbance
of Gravity}. ``Brit.\ Assoc.\ Reports.'' Southampton,~1882.

Reprint of the same in the ``Philosophical Magazine.''

d'Abbadie, \Title{Recherches sur la verticale}. ``Ann.\ de~la Soc.\ Scient.\
de~Bruxelles,'' 1881.

von Rebeur-Paschwitz, \Title{Das Horizontalpendel}. ``Nova Acta
K.~Leop.\ Car.\ Akad.,'' Band~60, No.~1, 1892.
\end{Authorities}
\index{Darwin, G. H.!distortion of earth's surface by varying loads|)}%
\index{Distortion of soil!by varying loads|)}%
\index{Elastic distortion!of earth by varying loads|)}%
\PageSep{149}


\Chapter{VIII}
{Equilibrium Theory of Tides}

\First{It} is clearly necessary to proceed step by step
\index{Equilibrium theory of tides|(}%
towards the actual conditions of the tidal problem,
and I shall begin by supposing that the
oceans cover the whole earth, leaving no dry
land. It has been shown in \Ref{Chapter}{V}.\ that the
tidal force is the resultant of opposing centrifugal
and centripetal forces. The motion of the
system is therefore one of its most essential features.
We may however imagine a supernatural
being, who carries the moon round the earth and
makes the earth rotate at the actual relative
speeds, but with indefinite slowness as regards
absolute time. This supernatural being is further
to have the power of maintaining the tidal forces
at exactly their present intensities, and with their
actual relationship as regards the positions of
the moon and earth. Everything, in fact, is to
remain as in reality, except time, which is to be
indefinitely protracted. The question to be considered
is as to the manner in which the tidal
forces will cause the ocean to move on the slowly
revolving earth.

It appears from \fig{23} that the horizontal
\PageSep{150}
tidal force acts at right angles to the circle, where
the moon is in the horizon, just rising or just
setting, towards those two points, $V$~and~$I$, where
the moon is overhead in the zenith, or underfoot
in the nadir. The force will clearly generate
currents in the water away from the circle of
moonrise and moonset, and towards $V$~and~$I$.
The currents will continue to flow until the water
level is just so much raised above the primitive
surface at $V$~and~$I$, and depressed along the circle,
that the tendency to flow downhill towards
the circle is equal to the tendency to flow uphill
under the action of the tide-generating force.
When the currents have ceased to flow, the figure
of the ocean has become elongated, or egg-shaped
with the two ends alike, and the longer
axis of the egg is pointed at the moon. When
this condition is attained the system is at rest or
in equilibrium, and the technical name for the
egg-like form is a ``prolate ellipsoid of revolution''---``prolate''
because it is elongated, and
``of revolution'' because it is symmetrical with
respect to the line pointing at the moon. Accordingly
the mathematician says that the figure
of equilibrium under tide-generating force is a
prolate ellipsoid of revolution, with the major
axis directed to the moon.

It has been supposed that the earth rotates and
that the moon revolves, but with such extreme
slowness that the ocean currents have time
\PageSep{151}
\index{Figure of equilibrium!of ocean under tidal forces|(}%
\index{Forces!figure of equilibrium under tidal|(}%
enough to bring the surface to its form of equilibrium,
\index{Equilibrium theory of tides!chart and law of tide|(}%
at each moment of time. If the time be
sufficiently protracted, this is a possible condition
of affairs. It is true that with the earth spinning
at its actual rate, and with the moon revolving
as in nature, the form of equilibrium can
never be attained by the ocean; nevertheless it
is very important to master the equilibrium
theory.

\Fig{29} represents the world in two hemispheres,
as in an ordinary atlas, with parallels
of latitude drawn at $15°$~apart. At the moment
represented, the moon is supposed to be in the
zenith at $15°$~of north latitude, in the middle of
the right-hand hemisphere. The diametrically
opposite point is of course at $15°$~of south latitude,
in the middle of the other hemisphere.
These are the two points $V$~and~$I$ of figs.\ \figref{22}~and~\figref{23},
towards which the water is drawn, so that the
vertices of the ellipsoid are at these two spots.
A scale of measurement must be adopted for
estimating the elevation of the water above, and
its depression below the original undisturbed surface
of the globe. It will be convenient to measure
the elevation at these two spots by the
number~$2$. A series of circles are drawn round
these points, but one of them is, of necessity,
presented as partly in one hemisphere and partly
in the other. In the map they are not quite concentric
with the two spots, but on the actual
\PageSep{152}
\TallFig{29}{Chart of Equilibrium Tide}{png}
\PageSep{153}
\index{Semidiurnal tide!in equilibrium theory|(}%
globe they would be so. These circles show
where, on the adopted scale of measurement, the
elevation of height is successively $1\frac{1}{2}$,~$1$,~$\frac{1}{2}$. The
fourth circle, marked in chain dot, shows where
there is no elevation or depression above the original
surface. The next succeeding and dotted
circle shows where there is a depression of~$\frac{1}{2}$, and
the last dotted line is the circle of lowest water
where the depression is~$1$; it is the circle~$\Seg{D}{D}$ of~\fig{22},
and the circle of the shadow in~\fig{23}.

The elevation above the original spherical surface
at the vertices or highest points is just twice
as great as the greatest depression. But the
greatest elevation only occurs at two points,
whereas the greatest depression is found all along
a circle round the globe. The horizontal tide-generating
force is everywhere at right angles to
these circles, and the present figure is in effect a
reproduction, in the form of a map, of the perspective
picture in~\fig{23}.

Now as the earth turns from west to east, let
us imagine a man standing on an island in the
otherwise boundless sea, and let us consider what
he will observe. Although the earth is supposed
to be revolving very slowly, we may still call the
twenty-fourth part of the time of its rotation an
hour. The man will be carried by the earth's
rotation along some one of the parallels of latitude.
If, for example, his post of observation is
in latitude $30°$~N., he will pass along the second
\index{Equilibrium theory of tides!chart and law of tide|)}%
\index{Figure of equilibrium!of ocean under tidal forces|)}%
\index{Forces!figure of equilibrium under tidal|)}%
\PageSep{154}
parallel to the north of the equator. This parallel
cuts several of the circles which indicate the
elevation and depression of the water, and therefore
he will during his progress pass places where
the water is shallower and deeper alternately, and
he would say that the water was rising and falling
rhythmically. Let us watch his progress
across the two hemispheres, starting from the
extreme left. Shortly after coming into view he
is on the dotted circle of lowest water, and he
says it is low tide. As he proceeds the water
rises, slowly at first and more rapidly later, until
he is in the middle of the hemisphere; he arrives
there six hours later than when we first began to
watch him. It will have taken him about $5\frac{1}{2}$~hours
to pass from low water to high water. At
low water he was depressed by~$1$ below the original
level, and at high water he is raised by~$\frac{1}{2}$
above that level, so that the range from low
water to high water is represented by~$1\frac{1}{2}$. After
the passage across the middle of the hemisphere,
the water level falls, and after about $5\frac{1}{2}$~hours
more the water is again lowest, and the depression
is measured by~$1$ on the adopted scale.
Soon after this he passes out of this hemisphere
into the other one, and the water rises again
until he is in the middle of that hemisphere.
But this time he passes much nearer to the vertex
of highest water than was the case in the other
hemisphere, so that the water now rises to a
\PageSep{155}
height represented by about~$1\frac{4}{5}$. In this half of
his daily course the range of tide is from $1$~below
to $1\frac{4}{5}$~above, and is therefore~$2\frac{4}{5}$, whereas before
it was only~$1\frac{1}{2}$. The fact that the range of two
successive tides is not the same is of great importance
in tidal theory; it is called the diurnal
inequality of the tide.

It will have been noticed that in the left hemisphere
the range of fall below the original spherical
surface is greater than the range of rise
above it; whereas in the right hemisphere the
rise is greater than the fall. Mean water mark
is such that the tide falls on the average as much
below it as it rises above it, but in this case the
rise and fall have been measured from the originally
undisturbed surface. In fact the mean
level of the water, in the course of a day, is not
identical with the originally undisturbed surface,
although the two levels do not differ much from
one another.

The reader may trace an imaginary observer
in his daily progress along any other parallel of
latitude, and will find a similar series of oscillations
in the ocean; each latitude will, however,
present its own peculiarities. Then again the
moon moves in the heavens. In \fig{29} she has
been supposed to be $15°$~north of the equator,
but she might have been yet further northward,
or on the equator, or to the south of it. Her
extreme range is in fact $28°$~north or south of
\PageSep{156}
\index{Forces!those of sun and moon compared|(}%
\index{Lunar!tide-generating force compared with solar|(}%
\index{Moon and earth!tide-generating force compared with sun's|(}%
\index{Solar!tide-generating force compared with lunar|(}%
the equator. To represent each such case a new
map would be required, which would, however,
only differ from this one by the amount of displacement
of the central spots from the equator.

It is obvious that the two hemispheres in \fig{29}
are exactly alike, save that they are inverted
with respect to north and south; the right hemisphere
is in fact the same as the left upside down.
It is this inversion which causes the two successive
tides to be unlike one another, or, in other
words, gives rise to the diurnal inequality. But
\index{Diurnal inequality!according to equilibrium theory}%
there is one case where inversion makes no difference;
this is when the central spot is on the
equator in the left hemisphere, for its inversion
then makes the right hemisphere an exact reproduction
of the left one. In this case therefore
the two successive tides are exactly alike, and
there is no diurnal inequality. Hence the diurnal
inequality vanishes when the moon is on the
equator.

Our figure exhibits another important point,
for it shows that the tide has the greater range
in that hemisphere where the observer passes
nearest to one of the two central spots. That is
to say, the higher tide occurs in that half of the
daily circuit in which the moon passes nearest to
the zenith or to the nadir of the observer.

Thus far I have supposed the moon to exist
alone, but the sun also acts on the ocean according
\index{Sun!tide-generating force of, compared with that of moon|(}%
to similar laws, although with less intensity.
\index{Semidiurnal tide!in equilibrium theory|)}%
\PageSep{157}
We must now consider how the relative strengths
of the actions of the two bodies are to be determined.
It was indicated in \Ref{Chapter}{V}.\ that
tide-generating force varies inversely as the cube
of the distance from the earth of the tide-generating
body. The force of gravity varies inversely
as the square of the distance, so that, as
we change the distance of the attracting body,
tidal force varies with much greater rapidity than
does the direct gravitational attraction. Thus if
the moon stood at half her present distance from
the earth, her tide-generating force would be $8$~times
as great, whereas her direct attraction would
only be multiplied $4$~times. It is also obvious
that if the moon were twice as heavy as in reality,
her tide-generating force would be doubled;
and if she were half as heavy it would be halved.
Hence we conclude that tide-generating force
varies directly as the mass of the tide-generating
body, and inversely as the cube of the distance.

The application of this law enables us to compare
the sun's tidal force with that of the moon.
The sun is $25,500,000$~times as heavy as the
moon, so that, on the score of mass, the solar
tidal force should be $25\frac{1}{2}$~million times greater
than that of the moon. But the sun is $389$~times
as distant as the moon. And since the
cube of~$389$ is about $59$~millions, the solar tidal
force should be $59$~million times weaker than
that of the moon, on the score of distance.
\PageSep{158}

We have, then, a force which is $25\frac{1}{2}$~million
times stronger on account of the sun's greater
weight, and $59$~million times weaker on account
of his greater distance; it follows that the sun's
tide-generating force is $25\frac{1}{2}$-$59$ths, or a little
less than half of that of the moon.

We conclude then that if the sun acted alone
on the water, the degree of elongation or distortion
of the ocean, when in equilibrium, would
be a little less than half of that due to the moon
alone. When both bodies act together, the distortion
of the surface due to the sun is superposed
on that due to the moon, and a terrestrial
observer perceives only the total or sum of the
two effects.

When the sun and moon are together on the
same side of the earth, or when they are diametrically
opposite, the two distortions conspire
together, and the total tide will be half as great
again as that due to the moon alone, because
the solar tide is added to the lunar tide. And
when the sun and moon are at right angles to
\index{Sun!tide-generating force of, compared with that of moon|)}%
one another, the two distortions are at right
angles, and the low water of the solar tide conspires
with the high water of the lunar tide.
The composite tide has then a range only half as
great as that due to the moon alone, because the
solar tide, which has a range of about half that
\index{Solar!tide-generating force compared with lunar|)}%
of the lunar tide, is deducted from the lunar
\index{Lunar!tide-generating force compared with solar|)}%
tide. Since one and a half is three times a half,
\index{Forces!those of sun and moon compared|)}%
\index{Moon and earth!tide-generating force compared with sun's|)}%
\PageSep{159}
\index{Spring and neap tides!in equilibrium theory}%
it follows that when the moon and sun act together
the range of tide is three times as great
as when they act adversely. The two bodies
are together at change of moon and opposite at
full moon. In both of these positions their
actions conspire; hence at the change and the
full of moon the tides are at their largest, and
are called spring tides. When the two bodies
are at right angles to one another, it is half
moon, either waxing or waning, the tides have
their smallest range, and are called neap tides.
\index{Neap and spring tides!in equilibrium theory}%

The observed facts agree pretty closely with
this theory in several respects, for spring tide
occurs about the full and change of moon, neap
tide occurs at the half moon, and the range at
springs is usually about three times as great as
that at neaps. Moreover, the diurnal inequality
conforms to the theory in vanishing when the
moon is on the equator, and rising to a maximum
when the moon is furthest north or south. The
amount of the diurnal inequality does not, however,
agree with theory, and in many places the
tide which should be the greater is actually the
less.

The theory which I have sketched is called
the Equilibrium Theory of the Tides, because
it supposes that at each moment the ocean is
in that position of rest or equilibrium which it
would attain if indefinite time were allowed.
The general agreement with the real phenomena
\PageSep{160}
\index{Equilibrium theory of tides!defects of}%
proves the theory to have much truth about it,
but a detailed comparison with actuality shows
that it is terribly at fault. The lunar and solar
tidal ellipsoids were found to have their long
axes pointing straight towards the tide-generating
bodies, and, therefore, at the time when the
moon and sun pull together, it ought to be high
water just when they are due south. In other
words, at full and change of moon, it ought
to be high water exactly at noon and at midnight.
\index{High water!under moon in equilibrium theory}%
Now observation at spring tides shows
that at most places this is utterly contradictory
to fact.

It is a matter of rough observation that the
tides follow the moon's course, so that high
water always occurs about the same number of
hours after the moon is due south. This rule
has no pretension to accuracy, but it is better
than no rule at all. Now at change and full of
the moon, the moon crosses the meridian at the
same hour of the clock as the sun, for at change
of moon they are together, and at full moon
they are twelve hours apart. Hence the hour
of the clock at which high water occurs at
change and full of moon is in effect a statement
of the number of hours which elapse after the
moon's passage of the meridian up to high
water. This clock time affords a rough rule for
the time of high water at any other phase of the
moon; if, for example, it is high water at eight
\PageSep{161}
o'clock at full and change, approximately eight
hours will always elapse after the moon's passage
until high water occurs. Mariners call the clock
time of high water at change and full of moon
\index{Establishment of port!definition}%
\index{Establishment of port!zero in equilibrium theory}%
``the establishment of the port,'' because it
establishes a rough rule of the tide at all other
times.

According to the equilibrium theory, high
water falls at noon and midnight at full and
change of moon, or in the language of the mariner
the establishment of all ports should be
zero. But observation shows that the establishment
at actual ports has all sorts of values, and
that in the Pacific Ocean (where the tidal forces
have free scope) it is at least much nearer to six
hours than to zero. High water cannot be more
than six hours before or after noon or midnight
on the day of full or change of moon, because if
it occurs more than six hours after one noon, it
is less than six hours before the following midnight;
hence the establishment of any port
cannot possibly be more than six hours before or
after. Accordingly, the equilibrium theory is
nearly as much wrong as possible, in respect to
the time of high water. In fact, in many places
it is nearly low water at the time that the equilibrium
theory predicts high water.

It would seem then as if the tidal action of
the moon was actually to repel the water instead
of attracting it, and we are driven to ask whether
\PageSep{162}
\index{Establishment of port!definition}%
this result can possibly be consistent with the
theory of universal gravitation.

The existence of continental barriers across
the oceans must obviously exercise great influence
on the tides, but this fact can hardly be
responsible for a reversal of the previsions of the
equilibrium theory. It was Newton who showed
that a depression of the ocean under the moon
is entirely consistent with the theory of gravitation.
In the following chapter I shall explain
Newton's theory, and show how it explains the
discrepancy which we have found between the
equilibrium theory and actuality.

\begin{Authorities}
An exposition of the equilibrium theory will be found in any
mathematical work on the subject, or in the article \Title{Tides} in the
``Encyclopædia Britannica.''
\end{Authorities}
\index{Equilibrium theory of tides|)}%
\PageSep{163}


\Chapter{IX}
{Dynamical Theory of the Tide Wave}

\First{The} most serious difficulties in the complete
\index{Dynamical theory of tide-wave|(}%
tidal problem do not arise in a certain special
case which was considered by Newton. His supposition
was that the sea is confined to a canal
\index{Canal!critical depth|(}%
circling the equator, and that the moon and sun
move exactly in the equator.

An earthquake or any other gigantic impulse
may be supposed to generate a great wave in this
equatorial canal. The rate of progress of such
a wave is dependent on the depth of the canal
only, according to the laws sketched in \Ref{Chapter}{II}.,
and the earth's rotation and the moon's attraction
make no sensible difference in its speed
of transmission. If, for example, the canal were
$5$~kilometres ($3$~miles) in depth, such a great
wave would travel $796$~kilometres ($500$~miles)
per~hour. If the canal were shallower the speed
would be less than this; if deeper, greater.
Now there is one special depth which will be
found to have a peculiar importance in the theory
of the tide, namely, where the canal is $13\frac{3}{4}$~miles
deep. In this case the wave travels $1,042$~miles
an hour, so that it would complete the
\PageSep{164}
$25,000$~miles round the earth in exactly $24$~hours.
It is important to note that if the depth of the
equatorial canal be less than $13\frac{3}{4}$~miles, a wave
takes more than a day to complete the circuit of
the earth, and if the depth be greater the circuit
is performed in less than a day.

The great wave, produced by an earthquake or
other impulse, is called a ``free wave,'' because
\index{Free wave, explanation and contrast with forced}%
\index{Waves!forced and free}%
when once produced it travels free from the action
of external forces, and would persist forever,
were it not for the friction to which water is
necessarily subject. But the leading characteristic
of the tide wave is that it is generated and
kept in action by continuous forces, which act
on the fluid throughout all time. Such a wave
is called a ``forced wave,'' because it is due to
\index{Forced wave, explanation and contrast with free wave}%
the continuous action of external forces. The
rate at which the tide wave moves is moreover
dependent only on the rate at which the tidal
forces travel over the earth, and not in any degree
on the depth of the canal. It is true that
the depth of the canal exercises an influence on
the height of the wave generated by the tidal
forces, but the wave itself must always complete
the circuit of the earth in a day, because the
earth turns round in that period.

We must now contrast the progress of any
long ``free wave'' in the equatorial canal with
that of the ``forced'' tide wave. I may premise
that it will here be slightly more convenient to
\PageSep{165}
consider the solar instead of the lunar tide. The
lunar wave is due to a stronger tide-generating
force, and since the earth takes $24$~hours $50$~minutes
to turn round with respect to the moon, that
is the time which the lunar tide wave takes to
complete the circuit of the earth; but these differences
are not material to the present argument.
The earth turns with respect to the sun
in exactly one day, or as we may more conveniently
say, the sun completes the circuit of the
earth in that time. Therefore the solar tidal
influence travels over the surface of the earth
at the rate of $1,042$~miles an hour. Now this is
exactly the pace at which a ``free wave'' travels
in a canal of a depth of $13\frac{3}{4}$~miles; accordingly
\index{Canal!theory of tide wave in|(}%
in such a canal any long free wave just keeps
pace with the sun.

We have seen in \Ref{Chapter}{V}.\ that the solar
tide-generating force \emph{tends} to make a wave crest,
at those points of the earth's circumference where
it is noon and midnight. At each moment of
time the sun is generating a new wave, and after
it is generated that wave travels onwards as a
free wave. If therefore the canal has a depth
\index{Canal!critical depth|)}%
of $13\frac{3}{4}$~miles, each new wave, generated at each
moment of time, keeps pace with the sun, and
the summation of them all must build up two
enormous wave crests at opposite sides of the
earth.

If the velocity of a free wave were absolutely
\PageSep{166}
the same whatever were its height, the crests of
the two tide waves would become infinite in
height. As a fact the rate of progress of a wave
is somewhat influenced by its height, and therefore,
when the waves get very big, they will
cease to keep pace exactly with the sun, and
the cause for continuous exaggeration of their
heights will cease to exist. We may, however,
express this conclusion by saying that, when the
canal is $13\frac{3}{4}$~miles deep, the height of the tide
wave becomes mathematically infinite. This does
not mean that mathematicians assert that the
wave would really become infinite, but only that
the simple method of treatment which supposes
the wave velocity to depend only on the depth
of water becomes inadequate. If the ocean was
really confined to an equatorial canal, of this exact
depth, the tides would be of very great
height, and the theory would be even more complex
than it is. It is, however, hardly necessary
to consider this special case in further detail.

We conclude then that for the depth of $13\frac{3}{4}$~miles,
the wave becomes infinite in height, in
the qualified sense of infinity which I have described.
We may feel sure that the existence
of the quasi-infinite tide betokens that the behavior
of the water in a canal shallower than
$13\frac{3}{4}$~miles differs widely from that in a deeper
one. It is therefore necessary to examine into
the essential point in which the two cases differ
\PageSep{167}
from one another. In the shallower canal a free
wave covers less than $25,000$~miles a day, and
thus any wave generated by the sun would tend
to be left behind by him. On the other hand,
in the deeper canal a free wave would outstrip
the sun, and a wave generated by the sun tends
to run on in advance of him. But these are
only tendencies, for in both the shallower and
the deeper canal the actual tide wave exactly
keeps pace with the sun.

It would be troublesome to find out what
would happen if we had the water in the canal
at rest, and were suddenly to start the sun to
work at it; and it is fortunately not necessary
to attempt to do so. It is, however, certain that
for a long time the motion would be confused,
but that the friction of the water would finally
produce order out of chaos, and that ultimately
there would be a pair of antipodal tide crests
traveling at the same pace as the sun. Our
task, then, is to discover what that final state of
motion may be, without endeavoring to unravel
the preliminary chaos.

Let us take a concrete case, and suppose our
canal to be $3$~miles deep, in which we have seen
that a free wave will travel $500$~miles an hour.
Suppose, then, we start a long free wave in the
equatorial canal of $3$~miles deep, with two crests
\index{Canal!theory of tide wave in|)}%
$12,500$~miles apart, and therefore antipodal to
one another. The period of a wave is the time
\PageSep{168}
between the passage of two successive crests
past any fixed point. In this case the crests
are antipodal to one another, and therefore the
wave length is $12,500$~miles, and the wave
travels $500$~miles an hour, so that the period of a
free wave is $25$~hours. But the tide wave keeps
pace with the sun, traveling $1,042$~miles an
hour, and there are two antipodal crests, $12,500$~miles
apart; hence, the time between the passage
of successive tide crests is $12$~hours.

In this case a free wave would have a period
of $25$~hours, and the tide wave, resulting from
the action of solar tide-generating force, has a
period of $12$~hours. The contrast then lies between
the free wave, with a period of $25$~hours,
and the forced wave, with a period of $12$~hours.

For any other depth of ocean the free wave
will have another period depending on the depth,
but the period of the forced wave is always $12$~hours,
because it depends on the sun. If the
ocean be shallower than $3$~miles, the free period
will be greater than $25$~hours, and, if deeper,
less than $25$~hours. But if the ocean be deepened
to $13\frac{3}{4}$~miles, the free wave travels at the
same pace as the forced wave, and therefore the
two periods are coincident. For depths greater
than $13\frac{3}{4}$~miles, the period of the free wave is
less than that of the forced wave; and the
converse is true for canals less than $13\frac{3}{4}$~miles in
depth.
\PageSep{169}
\index{Forced oscillation!principle of}%
\index{Free oscillation contrasted with forced}%
\index{Principle of forced oscillations}%

Now let us generalize this conception; we
have a system which if disturbed and left to
itself will oscillate in a certain period, called the
free period. Periodic disturbing forces act on
this system and the period of the disturbance
is independent of the oscillating system itself.
The period of the disturbing forces is called the
forced period. How will such a system swing,
when disturbed with this forced periodicity?

A weight tied to the end of a string affords
an example of a very simple system capable of
oscillation, and the period of its free swing depends
on the length of the string only. I will
suppose the string to be $3$~feet, $3$~inches, or one
metre in length, so that the period of the swing
from right to left, or from left to right is one
second.\footnote
  {A pendulum of one metre in length is commonly called a
  seconds-pendulum, although its complete period is two seconds.}
If, holding the string, I move my
hand horizontally to and fro through a small
distance with a regular periodicity, I set the
pendulum a-swinging. The period of the movement
of my hand is the forced period, and the
free period is two seconds, being the time occupied
by a metre-long pendulum in moving from
right to left and back again to right. If I time
the to and fro motion of my hand so that its
period from right to left, or from left to right,
is exactly one second, the excursions of the pendulum
bob grow greater and greater without limit,
\PageSep{170}
\index{Forced oscillation!principle of}%
\index{Free oscillation contrasted with forced}%
\index{Principle of forced oscillations}%
because the successive impulses are stored up in
the pendulum, which swings further and further
with each successive impulse. This case is
exactly analogous with the quasi-infinite tides
which would arise in a canal $13\frac{3}{4}$~miles deep, and
here also this case is critical, separating two
modes of oscillation of the pendulum of different
characters.

Now when the hand occupies more than one
second in moving from right to left, the forced
period is greater than the free period of the
pendulum; and when the system is swinging
steadily, it will be observed that the excursion
of the hand agrees in direction with the excursion
of the pendulum, so that when the hand is
furthest to the right so is also the pendulum,
and vice versa. When the period of the force
is greater than the free period of the system, at
the time when the force tends to make the pendulum
move to the right, it is furthest to the
right. The excursions of the pendulum agree
in direction with that of the hand.

Next, when the hand occupies less than one
second to move from right to left or from left
to right, the forced period is less than the free
period, and it will be found that when the hand
is furthest to the right the pendulum is furthest
to the left. The excursions of the pendulum
are opposite in direction from those of the hand.
These two cases are illustrated by~\fig{30}, which
\PageSep{171}
\index{High water!position in shallow and deep canals in dynamical theory}%
will, perhaps, render my meaning more obvious.
We may sum up this discussion by saying that
in the case of a slowly varying disturbing force,
the oscillation and the force are consentaneous,
but that with a quickly oscillating force, the
oscillation is exactly inverted with respect to the
force.

Now, this simple case illustrates a general
dynamical principle, namely, that if a system
\Figure{30}{Forced Oscillations of a Pendulum}{png}
capable of oscillating with a certain period is
acted on by a periodic force, when the period of
the force is greater than the natural free period
of the system, the oscillations of the system
agree with the oscillations of the force; but if
the period of the force is less than the natural
free period of the system the oscillations are
inverted with reference to the force.

This principle may be applied to the case of
the tides in the canal. When the canal is more
than $13\frac{3}{4}$~miles deep, the period of the sun's disturbing
force is $12$~hours and is greater than the
\PageSep{172}
\index{Newton!theory of tide in equatorial canal}%
natural free period of the oscillation, because a
free wave would go more than half round the
earth in $12$~hours. We conclude, then, that when
the tide-generating forces are trying to make it
high water, it will be high water. It has been
\index{High water!position in shallow and deep canals in dynamical theory}%
shown that these forces are tending to make high
water immediately under the sun and at its antipodes,
and there accordingly will the high water
be. In this case the tide is said to be direct.

But when the canal is less than $13\frac{3}{4}$~miles
deep, the sun's disturbing force has, as before, a
period of $12$~hours, but the period of the free
wave is more than $12$~hours, because a free wave
would take more than $12$~hours to get half round
the earth. Thus the general principle shows
that where the forces are trying to make high
water, there will be low water, and vice versa.
Here, then, there will be low water under the sun
and at its antipodes, and such a tide is said to
be inverted, because the oscillation is the exact
inversion of what would be naturally expected.

All the oceans on the earth are very much
shallower than fourteen miles, and so, at least
near the equator, the tides ought to be inverted.
The conclusion of the equilibrium theory will
therefore be the exact opposite of the truth, near
the equator.

This argument as to the solar tide requires
but little alteration to make it applicable to the
lunar tide. In fact the only material difference
\PageSep{173}
\index{Waves!of tide in equatorial canal}%
in the conditions is that the period of the lunar
tide is $12$~hours $25$~minutes, instead of $12$~hours,
and so the critical depth of an equatorial canal,
\index{Equatorial canal, tide wave in}%
which would allow the lunar tide to become
quasi-infinite, is a little less than that for the
solar tide. This depth for the lunar tide is in
fact nearly $13$~miles.\footnote
  {It is worthy of remark that if the canal had a depth of between
  $13\frac{3}{4}$ and $13$~miles, the solar tides would be inverted, and
  the lunar tides would be direct. We should then, at the equator,
  have spring tide at half moon, when our actual neaps occur;
  and neap tide at full and change, when our actual springs occur.
  The tides would also be of enormous height, because the depth
  is nearly such as to make both tides quasi-infinite. If the depth
  of the canal were very nearly $13\frac{3}{4}$~miles the solar tide might be
  greater than the lunar. But these exceptional cases have only
  a theoretical interest.}

This discussion should have made it clear that
any tidal theory, worthy of the name, must take
account of motion, and it explains why the prediction
of the equilibrium theory is so wide from
the truth. Notwithstanding, however, this condemnation
of the equilibrium theory, it is of the
utmost service in the discussion of the tides,
because by far the most convenient and complete
way of specifying the forces which act on the
ocean at each instant is to determine the figure
which the ocean would assume, if the forces had
abundant time to act.

\TB

When the sea is confined to an equatorial
canal, the tidal problem is much simpler than
\PageSep{174}
when the ocean covers the whole planet, and
this is much simpler than when the sea is interrupted
by continents. Then again, we have
thus far supposed the sun and moon to be always
exactly over the equator, whereas they actually
range a long way both to the north and to the
south of the equator; and so here also the true
problem is more complicated than the one under
consideration. Let us next consider a case, still
far simpler than actuality, and suppose that
whilst the moon or sun still always move over
the equator, the ocean is confined to several
canals which run round the globe, following parallels
of latitude.
\index{Latitude!tidal wave in canal in high|(}%

The circumference of a canal in latitude~$60°$
\index{Canal!canal in high latitude|(}%
is only $12,500$~miles, instead of~$25,000$. If a
free wave were generated in such a canal with
two crests at opposite sides of the globe, the distance
from crest to crest would be $6,250$~miles.
Now if an equatorial canal and one in latitude~$60°$
have equal depths, a free wave will travel at
the same rate along each; and if in each canal
there be a wave with two antipodal crests, the
time occupied by the wave in latitude~$60°$ in
traveling through a space equal to its length will
be only half of the similar period for the equatorial
waves. The period of a free wave in latitude~$60°$
\index{Waves!in canal in high latitude|(}%
is therefore half what it is at the
equator, for a pair of canals of equal depths.
But there is only one sun, and it takes $12$~hours
\PageSep{175}
to go half round the planet, and therefore for
both canals the forced tide wave has a period of
$12$~hours. If, for example, both canals were
$8$~miles deep, in the equatorial canal the
\index{Canal!tides in ocean partitioned into canals}%
period of the free wave would be greater than
$12$~hours, whilst in the canal at $60°$~of latitude
it would be less than $12$~hours. It follows then
from the general principle as to forced and free
oscillations, that whilst the tide in the equatorial
canal would be inverted, that in latitude~$60°$
would be direct. Therefore, whilst it would be
low water under the moon at the equator, it
would be high water under the moon in latitude~$60°$.
Somewhere, between latitude~$60°$ and the
equator, there must be a place at which the free
period in a canal $8$~miles deep is the same as
the forced period, and in a canal at this latitude
the tide would be infinite in height, in the modified
sense explained earlier. It follows therefore
that there is for any given depth of canal, less
than $14$~miles, a critical latitude, at which the
tide tends to become infinite in height.

We conclude, that if the whole planet were
divided up into canals each partitioned off from
its neighbor, and if the canals were shallower
than $14$~miles, we should have inverted tides in
the equatorial region, and direct tides in the
polar regions, and, in one of the canals in some
middle latitude, very great tides the nature of
which cannot be specified exactly.
\PageSep{176}

The supposed partitions between neighboring
canals have introduced a limitation which must
be removed, if we are to approach actuality, but
I am unable by general reasoning to do more
than indicate what will be the effect of the removal
of the partitions. It is clear that when
the sea swells up to form the high water, the
water comes not only from the east and the west
of the place of high water, but also from the
north and south. The earth, as it rotates, carries
with it the ocean; the equatorial water is
carried over a space of $25,000$~miles in $24$~hours,
whereas the water in latitude~$60°$ is carried over
only $12,500$~miles in the same time. When, in
the northern hemisphere, water moves from north
to south it passes from a place where the surface
of the earth is moving slower, to where it is
moving quicker. Then, as the water goes to the
south, it carries with it only the velocity adapted
to the northern latitude, and so it gets left behind
by the earth. Since the earth spins from
west to east, a southerly current acquires a westward
trend. Conversely, when water is carried
northward of its proper latitude, it leaves the
\index{Latitude!tidal wave in canal in high|)}%
earth behind and is carried eastward. Hence
the water cannot oscillate northward and southward,
without at the same time oscillating eastward
and westward. Since in an ocean not
partitioned into canals, the water must necessarily
move not only east and west but also north
\index{Canal!canal in high latitude|)}%
\index{Waves!in canal in high latitude|)}%
\PageSep{177}
\index{Earth and moon!rotation of, effects on tides}%
\index{Rotation!of earth involved in tidal problem}%
\index{Vortical motion in oceanic tides}%
and south, it follows that tidal movements in the
ocean must result in eddies or vortices. The
\index{Eddies, tidal oscillation involves}%
eddying motion of the water must exist everywhere,
but it would be impossible, without mathematical
reasoning, to explain how all the eddies
fit into one another in time and place. It must
suffice for the present discussion for the reader
to know that the full mathematical treatment of
the problem shows this general conclusion to be
correct.

The very difficult mathematical problem of
the tides of an ocean covering the globe to a
uniform depth was first successfully attacked by
Laplace. He showed that whilst the tides of a
\index{Laplace!theory of tides|(}%
shallow ocean are inverted at the equator, as
proved by Newton, that they are direct towards
the pole. We have just arrived at the same
conclusion by considering the tide wave in a
canal in latitude~$60°$. But our reasoning indicated
that somewhere in between higher latitudes
and the equator, the tide would be of an undefined
character, with an enormous range of rise
and fall. The complete solution of the problem
shows, however, that this indication of the
canal theory is wrong, and that the tidal variation
of level absolutely vanishes in some latitude
intermediate between the equator and the pole.
The conclusion of the mathematician is that
there is a certain circle of latitude, whose position
depends on the depth of the sea, where
there is neither rise nor fall of tide.
\PageSep{178}
\index{Vortical motion in oceanic tides}%

At this circle the water flows northward and
southward, and to and fro between east and
west, but in such a way as never to raise or depress
the level of the sea. It is not true to say
that there is no tide at this circle, for there are
tidal currents without rise and fall. When the
ocean was supposed to be cut into canals, we
thereby obliterated the northerly and southerly
currents, and it is exactly these currents which
prevent the tides becoming very great, as we
were then led to suppose they would be.

It may seem strange that, whereas the first
rough solution of the problem indicates an oscillation
of infinite magnitude at a certain parallel
of latitude, the more accurate treatment of the
case should show that there is no oscillation of
level at all. Yet to the mathematician such a
result is not a cause of surprise. But whether
strange or not, it should be clear that if at the
equator it is low water under the moon, and if
near the pole it is high water under the moon,
there must in some intermediate latitude be a
place where the water is neither high nor low,
that is to say, where there is neither rise nor fall.\footnote
  {The mathematician knows that a quantity may change sign,
  either by passing through infinity or through zero. Where a
  change from positive to negative undoubtedly takes place, and
  where a passage through infinity can have no physical meaning,
  the change must take place by passage through zero.}

\TB

Now let us take one more step towards actuality,
and suppose the earth's equator to be
\PageSep{179}
oblique to the orbits of the moon and sun, so
that they may sometimes stand to the north and
sometimes to the south of the equator. We
have seen that in this case the equilibrium theory
indicates that the two successive tides on any
one day have unequal ranges. The mathematical
solution of the problem shows that this conclusion
is correct. It appears also that if the
ocean is deeper at the poles than at the equator,
that tide is the greater which is asserted to be
so by the equilibrium theory. If, however, the
ocean is shallower at the poles than at the equator,
it is found that the high water which the
equilibrium theory would make the larger is actually
the smaller and vice versa.

If the ocean is of the same depth everywhere,
we have a case intermediate between the two,
where it is shallower at the poles, and where it is
deeper at the poles. Now in one of these cases
it appears that the higher high water occurs
where in the other we find the lower high water
to occur; and so, when the depth is uniform,
the higher high water and the lower high water
must attain the same heights. We thus arrive
at the remarkable conclusion that, in an ocean
of uniform depth, the diurnal inequality of the
\index{Diurnal inequality!in Laplace's solution}%
tide is evanescent. There are, however, diurnal
inequalities in the tidal currents, which are so
adjusted as not to produce a rise or fall. This
result was first arrived at by the great mathematician
Laplace.
\PageSep{180}

According to the equilibrium theory, when the
moon stands some distance north of the equator,
the inequality between the successive tides on
the coasts of Europe should be very great, but
the difference is actually so small as to escape
ordinary observation. In the days of Laplace,
the knowledge of the tides in other parts of the
world was very imperfect, and it was naturally
thought that the European tides were fairly
representative of the whole world. When, then,
it was discovered that there would be no diurnal
inequality in an ocean of uniform depth covering
the whole globe, it was thought that a fair explanation
had been found for the absence of
that inequality in Europe. But since the days
of Laplace much has been learnt about the tides
\index{Laplace!theory of tides|)}%
in the Pacific and Indian oceans, and we now
know that a large diurnal inequality is almost
\index{Diurnal inequality!in Atlantic, Pacific, and Indian Oceans}%
universal, so that the tides of the North Atlantic
are exceptional in their simplicity. In fact, the
evanescence of the diurnal inequality is not much
closer to the truth than the large inequality
predicted by the equilibrium theory; and both
theories must be abandoned as satisfactory explanations
of the true condition of affairs. But
notwithstanding their deficiencies both these
theories are of importance in teaching us how
the tides are to be predicted. In the next chapter
I shall show how a further approximation to
the truth is attainable.
\PageSep{181}

\begin{Authorities}
The canal theory in its elementary form is treated in many
works on Hydrodynamics, and in \Title{Tides}, ``Encyclopædia Britannica.''

An elaborate treatment of the subject is contained in Airy's
\index{Airy, Sir G. B.!attack on Laplace}%
\Title{Tides and Waves}, ``Encyclopædia Metropolitana.'' Airy there
attacks Laplace for his treatment of the wider tidal problem,
but his strictures are now universally regarded as unsound.

Laplace's theory is contained in the \Title{Mécanique Céleste}, but it
is better studied in more recent works.

A full presentment of this theory is contained in Professor
Horace Lamb's \Title{Hydrodynamics}, Camb.\ Univ.\ Press, 1895, chapter~viii.
\index{Lamb, H., presentation of Laplace's theory}%

Important papers, extending Laplace's work, by Mr.~S.~S.
Hough, are contained in the \Title{Philosophical Transactions of the
\index{Hough, S. S.!dynamical solution of tidal problem}%
Royal Society}, A.~1897, pp.~201--258, and A.~1898, pp.~139--185.
\end{Authorities}
\index{Dynamical theory of tide-wave|)}%
\PageSep{182}


\Chapter{X}
{Tides in Lakes---Cotidal Chart}

\First{If} the conditions of the tidal problem are to
\index{Lakes!tides in|(}%
agree with reality, an ocean must be considered
which is interrupted by continental barriers of
land. The case of a sea or lake entirely surrounded
by land affords the simplest and most
complete limitation to the continuity of the
water. I shall therefore begin by considering
the tides in a lake.

The oscillations of a pendulum under the tidal
attraction of the moon were considered in \Ref{Chapter}{VI}.,
and we there saw that the pendulum
would swing to and fro, although the scale of
displacement would be too minute for actual
observation. Now a pendulum always hangs
perpendicularly to the surface of water, and
must therefore be regarded as a sort of level.
As it sways to and fro under the changing action
of the tidal force, so also must the surface
of water. If the water in question is a lake, the
rocking of the level of the lake is a true tide.

A lake of say a hundred miles in length is
very small compared with the size of the earth,
and its waters must respond almost instantaneously
\PageSep{183}
to the changes in the tidal force. Such
a lake is not large enough to introduce, to a
perceptible extent, those complications which
make the complete theory of oceanic tides so
difficult. The equilibrium theory is here actually
true, because the currents due to the changes in
the tidal force have not many yards to run before
equilibrium is established, and the lake may
be regarded as a level which responds almost
instantaneously to the tidal deflections of gravity.
The open ocean is a great level also, but sufficient
time is not allowed it to respond to the changes
in the direction of gravity, before that direction
has itself changed.

It was stated in \Ref{Chapter}{V}.\ that the maximum
horizontal force due to the moon has an intensity
equal to $\frac{1}{11,664,000}$~part of gravity, and that
therefore a pendulum $10$~metres long is deflected
through $\frac{1}{11,664,000}$~of $10$~metres, or through $\frac{1}{1,166}$~of
a millimetre. Now suppose our lake, $200$~kilometres
in length, runs east and west, and that
our pendulum is hung up at the middle of the
lake, $100$~kilometres from either end. In \fig{31}
let $\Seg{C}{D}$ represent the level of the lake as
undisturbed, and $\Seg{A}{B}$~an exaggerated pendulum.
When the tide-generating force displaces the
pendulum to~$\Seg{A}{B'}$, the surface of the lake must
assume the position~$\Seg{C'}{D'}$. Now $\Seg{A}{B}$~being $10$~metres,
$\Seg{B}{B'}$~may range as far as $\frac{1}{1,166}$~of a millimetre;
and it is obvious that $\Seg{C}{C'}$~must bear the
\PageSep{184}
same relation to~$\Seg{C}{B}$ that $\Seg{B}{B'}$ does to~$\Seg{A}{B}$.
Hence $\Seg{C}{C'}$ at its greatest may be $\frac{1}{11,664,000}$~of half
the length of the lake. The lake is supposed
to be twice $100$~kilometres in length, and $100$~kilometres
is $10$~million centimetres; thus $\Seg{C}{C'}$~is
$\frac{1}{1.1664}$~centimetre, or $\frac{9}{10}$~of a centimetre. When
the pendulum is deflected in the other direction
the lake rocks the other way, and $C'$~is just as
much above~$C$ as it was below it before. It
follows from this that the lunar tide at the ends
of a lake, $200$~kilometres or $120$~miles in length,
has a range of $1\frac{3}{4}$~centimetres or $\frac{2}{3}$~of an inch.
The solar tidal force is a little less than half as
strong as that due to the moon, and when the
two forces conspire together at the times of
spring tide, we should find a tide with a range
of $2\frac{1}{2}$~centimetres.
\Figure{31}{The Tide in a Lake}{png}

If the same rule were to apply to a lake $2,000$~kilometres
or $1,200$~miles in length, the range
of lunar tide would be about $17$~centimetres or
$7$~inches, and the addition of solar tides would
bring the range up to $25$~centimetres or $10$~inches.
\PageSep{185}
I dare say that, for a lake of such a
size, this rule would not be very largely in error.
But as we make the lake longer, the currents
set up by the tidal forces have not sufficient time
to produce their full effects before the intensity
and direction of the tidal forces change. Besides
this, if the lake were broad from north to south,
the earth's rotation would have an appreciable
effect, so that the water which flows from the
north to the south would be deflected westward,
and that which flows from south to north would
tend to flow eastward. The curvature of the
earth's surface must also begin to affect the
motion. For these reasons, such a simple rule
would then no longer suffice for calculating the
tide.

Mathematicians have not yet succeeded in
solving the tidal problem for a lake of large
dimensions, and so it is impossible to describe
the mode of oscillation. It may, however, be asserted
that the shape, dimensions, and depth of
the lake, and the latitudes of its boundaries will
affect the result. The tides on the northern and
southern shores will be different, and there will
be nodal lines, along which there will be no rise
and fall of the water.
\index{Lakes!tides in|)}%

The Straits of Gibraltar are so narrow, that
\index{Mediterranean Sea, tides in}%
the amount of water which can flow through
them in the six hours which elapse between
high and low water in the Atlantic is inconsiderable.
\PageSep{186}
\index{Waves!propagated northward in Atlantic|(}%
Hence the Mediterranean Sea is virtually
\index{Mediterranean Sea, tides in}%
a closed lake. The tides of this sea are
much complicated by the constriction formed
by the Sicilian and Tunisian promontories. Its
tides probably more nearly resemble those of two
lakes than of a single sheet of water. The tides
of the Mediterranean are, in most places, so inconspicuous
that it is usually, but incorrectly,
described as a tideless sea. Every visitor to
Venice must, however, have seen, or may we say
smelt, the tides, which at springs have a range of
some four feet. The considerable range of tide
at Venice appears to indicate that the Adriatic
\index{Adriatic, tide in}%
acts as a resonator for the tidal oscillation, in the
same way that a hollow vessel, tuned to a particular
note, picks out and resonates loudly when
that note is sounded.

We see, then, that whilst the tides of a small
lake are calculable by the equilibrium theory,
those of a large one, such as the Mediterranean,
remain intractable. It is clear, then, that the
tides of the ocean must present a problem yet
more complex than those of a large lake.

In the Pacific and Southern oceans the tidal
\index{Pacific Ocean, tide in, affects Atlantic}%
forces have almost uninterrupted sway, but the promontories
of Africa and of South America must
profoundly affect the progress of the tide wave
from east to west. The Atlantic Ocean forms a
\index{Atlantic, tide in|(}%
great bay in this vaster tract of water. If this
inlet were closed by a barrier from the Cape of
\PageSep{187}
\index{Pacific Ocean, tide in, affects Atlantic}%
Good Hope to Cape Horn, it would form a lake
large enough for the generation of much larger
tides than those of the Mediterranean Sea, although
probably much smaller than those which
we actually observe on our coasts. Let us now
suppose that the tides proper to the Atlantic are
non-existent, and let us remove the barrier between
the two capes. Then the great tide wave
sweeps across the Southern ocean from east to
west, and, on reaching the tract between Africa
and South America, generates a wave which
travels northward up the Atlantic inlet. This
secondary wave travels ``freely,'' at a rate dependent
only on the depth of the ocean. The
energy of the wave motion is concentrated, where
the channel narrows between North Africa and
Brazil, and the height of the wave must be augmented
in that region. Then the energy is
weakened by spreading, where the sea broadens
again, and it is again reconcentrated by the projection
of the North American coast line towards
Europe. Hence, even in this case, ideally simplified
as it is by the omission of the direct action
of the moon and sun, the range of tide would
differ at every portion of the coasts on each side
of the Atlantic.

The time of high water at any place must also
depend on the varying depth of the ocean, for it
is governed by the time occupied by the ``free
wave'' in traveling from the southern region to
\PageSep{188}
\index{Cotidal chart}%
the north. But in the south, between the two
capes of Africa and South America, the tidal
oscillation is constrained to keep regular time
with the moon, and so it will keep the same
rhythm at every place to the northward, at whatever
variable pace the wave may move. The
time of high water will of course differ at every
point, being later as we go northward. The
wave may indeed occupy so long on its journey,
that one high water may have only just arrived
at the northern coast of Africa, when another is
rounding the Cape of Good Hope.

Under the true conditions of the case, this
``free'' wave, generated in and propagated from
the southern ocean, is fused with the true
``forced'' tide wave generated in the Atlantic itself.
\index{Atlantic, tide in|)}%
It may be conjectured that on the coast of
Europe the latter is of less importance than the
\index{Europe, tides on coasts of}%
former. It is interesting to reflect that our tides
to-day depend even more on what occurred yesterday
or the day before in the Southern Pacific
and Indian oceans, than on the direct action of
the moon to-day. But the relative importance
of the two causes must remain a matter of conjecture,
for the problem is one of insoluble complexity.

Some sixty years ago Whewell, and after him
\index{Whewell!on cotidal charts}%
Airy, drew charts illustrative of what has just
\index{Airy, Sir G. B.!cotidal chart}%
been described. A map showing the march
of the tide wave is reproduced from Airy's
\index{Waves!propagated northward in Atlantic|)}%
\PageSep{189}
\index{Establishment of port!shown in cotidal chart}%
``Tides and Waves,'' in~\fig{32}. It claims to
show, from the observed times of high water at
the various parts of the earth, how the tide wave
travels over the oceans. Whewell and Airy were
\index{Whewell!on cotidal charts}%
well aware that their map could only be regarded
as the roughest approximation to reality. Much
has been learnt since their days, and the then
incomplete state of knowledge hardly permitted
them to fully realize how very rough was their
approximation to the truth. No more recent attempt
has been made to construct such a map,
and we must rest satisfied with this one. Even
if its lines may in places depart pretty widely
from the truth, it presents features of much interest.
I do not reproduce the Pacific Ocean,
because it is left almost blank, from deficiency
of data. Thus, in that part of the world where
the tides are most normal, and where the knowledge
of them would possess the greatest scientific
interest, we are compelled to admit an almost
total ignorance.

The lines on the map, \fig{32}, give the Greenwich
times of high water at full and change of
moon. They thus purport to represent the successive
positions of the crest of the tide wave.
For example, at noon and midnight (XII~o'clock),
at full and change of moon, the crest
of the tide wave runs from North Australia to
Sumatra, thence to Ceylon, whence it bends back
to the Island of Bourbon, and, passing some hundreds
\PageSep{190}
\TallFig[0.875]{32}{Chart of Cotidal Lines}{jpg}
\PageSep{191}
\index{Cotidal chart!for diurnal tide hitherto undetermined}%
of miles south of the Cape of Good Hope,
trends away towards the Antarctic Ocean. At
the same moment the previous tide crest has
traveled up the Atlantic, and is found running
across from Newfoundland to the Canary
Islands. A yet earlier crest has reached the
north of Norway. At this moment it is low
water from Brazil to the Gold Coast, and again
at Great Britain.

The successive lines then exhibit the progress
of the wave from hour to hour, and we see how
the wave is propagated into the Atlantic. The
crowding together of lines in places is the graphical
representation of the retardation of the
wave, as it runs into shallower water.

But even if this chart were perfectly trustworthy,
it would only tell us of the progress of
the ordinary semidiurnal wave, which produces
high water twice a day. We have, however, seen
reason to believe that two successive tides should
not rise to equal heights, and this figure does
not even profess to give any suggestion as to
how this inequality is propagated. In other
words, it is impossible to say whether two successive
tides of unequal heights tend to become
more or less unequal, as they run into any of
the great oceanic inlets. Thus the map affords
no indication of the law of the propagation of
the diurnal inequality.
\index{Diurnal inequality!not shown in cotidal chart}%

This sketch of the difficulties in the solution
\PageSep{192}
\index{Cotidal chart!for diurnal tide hitherto undetermined}%
of the full tidal problem might well lead to despair
of the possibility of tidal prediction on our
coasts. I shall, however, show in the next chapter
how such prediction is possible.

\begin{Authorities}
For cotidal charts see Whewell, \Title{Phil.\ Trans.\ Roy.\ Soc.}\ 1833,
or Airy's \Title{Tides and Waves}, ``Encyclopædia Metropolitana.''
\index{Airy, Sir G. B.!\Title{Tides and Waves}}%
\end{Authorities}
\PageSep{193}


\Chapter{XI}
{Harmonic Analysis of the Tide}

\First{It} is not probable that it will ever be possible
\index{Analysis, harmonic, of tide|(}%
\index{Harmonic analysis!account of|(}%
\index{Moon and earth!tide due to ideal, moving in equator}%
to determine the nature of the oceanic oscillation
as a whole with any accuracy. It is true that
we have already some knowledge of the general
march of the tide wave, and we shall doubtless
learn more in the future, but this can never suffice
for accurate prediction of the tide at any
place.

Although the equilibrium theory is totally
false as regards its prediction of the time of passage
and of the height of the tide wave, yet it
furnishes the stepping-stone leading towards the
truth, because it is in effect a compendious statement
of the infinite variety of the tidal force in
time and place.

I will begin my explanation of the practical
method of tidal prediction by obliterating the
sun, and by supposing that the moon revolves in
an equatorial circle round the earth. In this
case the equilibrium theory indicates that each
tide exactly resembles its predecessors and its
successors for all time, and that the successive
and simultaneous passages of the moon and of
\PageSep{194}
\index{Moon and earth!tide due to ideal, moving in equator}%
the wave crests across any place follow one
another at intervals of $12$~hours $25$~minutes. It
would always be exactly high water under or
opposite to the moon, and the height of high
water would be exactly determinate. In actual
oceans, even although only subject to the action
of such a single satellite, the motion of the water
would be so complex that it would be impossible
to predict the exact height or time of high or
of low water. But since the tidal forces operate
in a stereotyped fashion day after day, there will
be none of that variability which actually occurs
on the real earth under the actions of the real
sun and moon, and we may positively assert that
whatever the water does to-day it will do to-morrow.
Thus, if at a given place it is high water
at a definite number of hours after the equatorial
moon has crossed the meridian to-day, it will be
so to-morrow at the same number of hours after
the moon's passage, and the water will rise and
fall every day to the same height above and below
the mean sea level. If then we wanted to
know how the tide would rise and fall in a given
harbor, we need only watch the motion of the
sea at that place, for however the water may
move elsewhere its motion will always produce
the same result at the port of observation.
Thus, apart from the effects of wind, we should
only have to note the tide on any one day
to be able to predict it for all time. For by a
\PageSep{195}
\index{Satellites!tide due to single equatorial}%
single day of observation it would be easy to
note how many hours after the moon's passage
high water occurs, and how many feet it rises
and falls with reference to some fixed mark on
the shore. The delay after the moon's passage
and the amount of rise and fall would differ geographically,
but at each place there would be two
definite numbers giving the height of the tide
and the interval after the moon's passage until
high water. These two numbers are called the
tidal constants for the port; they would virtually
\index{Constants, tidal, explained}%
contain tidal predictions for all time.

Now if the moon were obliterated, leaving the
sun alone, and if he also always moved over
the equator, a similar rule would hold good,
but exactly $12$~hours would elapse from one
high water to the next, instead of $12$~hours $25$~minutes
as in the case of the moon's isolated
action. Thus two other tidal constants, expressive
of height and interval, would virtually contain
tidal prediction for the solar tide for all
time.

Theory here gives us some power of foreseeing
the relative importance of the purely lunar
and of the purely solar tide. The two waves
due to the sun alone or to the moon alone have
the same character, but the solar waves follow
one another a little quicker than the lunar waves,
and the sun's force is a little less than half the
moon's force. The close similarity between the
\PageSep{196}
\index{Satellites!tide due to single equatorial}%
actions of the sun and moon makes it safe to conclude
that the delay of the isolated solar wave
after the passage of the sun would not differ
much from the delay of the isolated lunar wave
after the passage of the moon, and that the
height of the solar wave would be about half of
that of the lunar wave. But theory can only be
trusted far enough to predict a rough proportionality
of the heights of the two tide waves to their
respective generating forces, and the approximate
equality of the intervals of retardation; but the
height and retardation of the solar wave could
not be accurately foretold from observation of
the lunar wave.

When the sun and moon coëxist, but still
move in equatorial circles, the two waves, which
we have considered separately, are combined.
The four tidal constants, two for the moon and
two for the sun, would contain the prediction of
the height of water for all time, for it is easy at
any future moment of time to discover the two
intervals of time since the moon and since the
sun have crossed the meridian of the place of
observation; we should then calculate the height
of the water above some mark on the shore on
the supposition that the moon exists alone, and,
again, on the supposition that the sun exists
alone, and adding the two results together, should
obtain the required height of the water at the
moment in question.
\PageSep{197}

But the real moon and sun do not move in
equatorial circles, but in planes which are oblique
to the earth's equator, and they are therefore
sometimes to the north and sometimes to the
south of the equator; they are also sometimes
nearer and sometimes further from the earth on
account of the eccentricity of the orbits in which
they move. Now the mathematician treats this
complication in the following way: he first considers
the moon alone and replaces it by a number
of satellites of various masses, which move
in various planes. It is a matter of indifference
that such a system of satellites could not maintain
the orbits assigned to them if they were allowed
to go free, but a mysterious being may be
postulated who compels the satellites to move in
the assigned orbits. One, and this is the largest
of these ideal satellites, has nearly the same mass
as the real moon and moves in a circle over the
equator; it is in fact the simple isolated moon
whose action I first considered. Another small
satellite stands still amongst the stars; others
move in such orbits that they are always vertically
overhead in latitude~$45°$; others repel instead
of attracting; and others move backwards
amongst the stars. Now all these satellites are
so arranged as to their masses and their orbits,
that the sum of their tidal forces is exactly the
same as those due to the real moon moving in
her actual orbit.
\PageSep{198}
\index{Interval from moon's transit to high water!in case of ideal satellite}%

So far the problem seems to be complicated
rather than simplified, for we have to consider a
dozen moons instead of one. The simplification,
however, arises from the fact that each satellite
either moves uniformly in an orbit parallel to the
equator, or else stands still amongst the stars.
It follows that each of the ideal satellites creates
a tide in the ocean which is of a simple character,
and repeats itself day after day in the same
way as the tide due to an isolated equatorial
moon. If all but one of these ideal satellites
were obliterated the observation of the tide for
a single day would enable us to predict the tide
for all time; because it would only be necessary
to note the time of high water after the ideal
satellite had crossed the meridian, and the height
\index{Height of tide!due to ideal satellite}%
of the high water, and then these two data would
virtually contain a tidal prediction for that tide
at the place of observation for all future time.
The interval and height are together a pair of
``tidal constants'' for the particular satellite in
question, and refer only to the particular place
at which the observation is made.

In actuality all the ideal satellites coëxist, and
the determination of the pair of tidal constants
appropriate to any one of them has to be made
by a complex method of analysis, of which I shall
say more hereafter. For the present it will suffice
to know that if we could at will annul all
the ideal satellites except one, and observe its
\PageSep{199}
\index{Moon and earth!ideal satellites replacing actual}%
\index{Partial tides in harmonic method}%
tide even for a single day, its pair of constants
could be easily determined. It would then only
be necessary to choose in succession all the satellites
\index{Satellites!ideal replacing sun and moon in harmonic analysis}%
as subjects of observation, and the materials
for a lunar tide table for all time would be obtained.

The motion of the sun round the earth is analogous
to that of the moon, and so the sun has
also to be replaced by a similar series of ideal
suns, and the partial tide due to each of them
has to be found. Finally at any harbor some
twenty pairs of numbers, corresponding to twenty
ideal moons and suns, give the materials for tidal
prediction for all time. Theoretically an infinite
number of ideal bodies is necessary for an absolutely
perfect representation of the tides, but
after we have taken some twenty of them, the
remainder are found to be excessively small in
mass, and therefore the tides raised by them are
so minute that they may be safely omitted. This
method of separating the tide wave into a number
of partial constituents is called ``harmonic
analysis.'' It was first suggested, and put into
practice as a practical treatment of the tidal
problem, by Sir William Thomson, now Lord
Kelvin, and it is in extensive use.
\index{Kelvin, Lord!initiates harmonic analysis}%

In this method the aggregate tide wave is considered
as the sum of a number of simple waves
following one another at exactly equal intervals
of time, and always presenting a constant rise
\PageSep{200}
\index{Moon and earth!ideal satellites replacing actual}%
\index{Prediction of tide!due to ideal satellite}%
and fall at the place of observation. When the
time of high water and the height of any one of
these constituent waves is known on any one
day, we can predict, with certainty, the height
of the water, as due to it alone, at any future
time however distant. The period of time which
elapses between the passage of one crest and of
the next is absolutely exact, for it is derived from
a study of the motions of the moon or sun, and
is determined to within a thousandth of a second.
The instant at which any one of the satellites
\index{Satellites!ideal replacing sun and moon in harmonic analysis}%
passes the meridian of the place is also
known with absolute accuracy, but the interval
after the passage of the satellite up to the high
water of any one of these constituent waves, and
the height to which the water will rise are only
derivable from observation at each port.

Since there are about twenty coëxistent waves
of sensible magnitude, a long series of observations
is requisite for disentangling any particular
wave from among the rest. The series must
also be so long that the disturbing influence of
the wind, both on height and time, may be eliminated
by the taking of averages. It may be
well to reiterate that each harbor has to be considered
by itself, and that a separate set of tidal
constants has to be found for each place. If it
is only required to predict the tides with moderate
accuracy some eight partial waves suffice, but
if high accuracy is to be attained, we have to
\PageSep{201}
consider a number of the smaller ones, bringing
the total up to $20$ or~$25$.

When the observed tidal motions of the sea
have been analyzed into partial tide waves, they
are found to fall naturally into three groups,
which correspond with the dissections of the sun
\index{Sun!ideal, replacing real sun in harmonic analysis}%
and moon into the ideal satellites. In the first
and most important group the crests follow one
another at intervals of somewhere about $12$~hours;
these are called the semidiurnal tides.
In the second group, the waves of which are in
most places of somewhat less height than those
of the semidiurnal group, the crests follow one
another at intervals of somewhere about $24$~hours,
and they are called diurnal. The tides
of the third group have a very slow periodicity,
for their periods are a fortnight, a month, half
a year, and a year; they are commonly of very
small height, and have scarcely any practical
importance; I shall therefore make no further
reference to them.

Let us now consider the semidiurnal group.
The most important of these is called ``the principal
lunar semidiurnal tide.'' It is the tide
\index{Lunar!tide, principal}%
\index{Semidiurnal tide!in harmonic method|(}%
raised by an ideal satellite, which moves in a circle
round the earth's equator. I began my explanation
of this method by a somewhat detailed
consideration of this wave. In this case, the
wave crests follow one another at intervals of
$12$~hours $25$~minutes $14\frac{1}{6}$~seconds. The average
\PageSep{202}
interval of time between the successive visible
transits of the moon over the meridian of the
place of observation is $24$~hours $50$~minutes $28\frac{1}{3}$~seconds;
and as the invisible transit corresponds
to a tide as well as the visible one, the interval
between the successive high waters is the time
between the successive transits, of which only
each alternate one is visible.

The tide next in importance is ``the principal
solar semidiurnal tide.'' This tide bears the
\index{Solar!principal tide}%
same relationship to the real sun that the principal
lunar semidiurnal tide bears to the real moon.
The crests follow one another at intervals of
exactly $12$~hours, which is the time from noon to
midnight and of midnight to noon. The height
of this partial wave is, at most places, a little less
than half of that of the principal lunar tide.

The interval between successive lunar tides is
$25\frac{1}{4}$~minutes longer than that between successive
solar tides, and as there are two tides a day, the
lunar tide falls behind the solar tide by $50\frac{1}{2}$~minutes
a day. If we imagine the two tides to start
together with simultaneous high waters, then in
about $7$~days the lunar tide will have fallen about
$6$~hours behind the solar tide, because $7$~times
$50\frac{1}{2}$~minutes is $5$~hours $54$~minutes. The period
from high water to low water of the principal
solar semidiurnal tide is $6$~hours, being half the
time between successive high waters. Accordingly,
when the lunar tide has fallen $6$~hours
\PageSep{203}
\index{Spring and neap tides!represented by principal lunar and solar tides}%
behind the solar tide, the low water of the solar
tide falls in with the high water of the lunar
tide. It may facilitate the comprehension of
this matter to take a numerical example; suppose
then that the lunar tide rises $4$~feet above
and falls by the same amount below the mean
level of the sea, and that the solar tide rises and
falls $2$~feet above and below the same level;
then if the two partial waves be started with their
high waters simultaneous, the joint wave will at
first rise and fall by $6$~feet. But after $7$~days it
is low solar tide when it is high lunar tide, and
so the solar tide is subtracted from the lunar
tide, and the compound wave has a height of
$4$~feet less $2$~feet, that is to say, of $2$~feet.
After nearly another $7$~days, or more exactly
after $14\frac{1}{2}$~days from the start, the lunar tide has
lost another $6$~hours, so that it has fallen back
$12$~hours in all, and the two high waters agree
together again, and the joint wave has again a
rise and fall of $6$~feet. When the two high
waters conspire it is called spring tide, and when
the low water of the solar tide conspires with the
high water of the lunar tide, it is called neap
tide. It thus appears that the principal lunar
and principal solar semidiurnal tides together
represent the most prominent feature of the tidal
oscillation.

The next in importance of the semidiurnal
waves is called the ``lunar elliptic tide,'' and here
\PageSep{204}
\index{Neap and spring tides!represented by principal lunar and solar tides}%
the crests follow one another at intervals of $12$~hours
$39$~minutes $30$~seconds. Now the interval
between the successive principal lunar tides was
\index{Lunar!elliptic tide}%
$12$~hours $25$~minutes $14$~seconds; hence, this
new tide falls behind the principal lunar tide by
$14\frac{1}{2}$~minutes in each half day. If this tide starts
so that its high water agrees with that of the
principal lunar tide, then after $13\frac{3}{4}$~days from
the start, its hollow falls in with the crest of the
former, and in $27\frac{1}{2}$~days from the start the two
crests agree again.

The moon moves round the earth in an ellipse,
and if to-day it is nearest to the earth, in $13\frac{3}{4}$~days
it will be furthest, and in $27\frac{1}{2}$~days it will
be nearest again. The moon must clearly exercise
a stronger tidal force and create higher
tides when she is near than when she is far;
hence every $27\frac{1}{2}$~days the tides must be larger,
and halfway between they must be smaller.
But the tide under consideration conspires with
the principal lunar tide every $27\frac{1}{2}$~days, and,
accordingly, the joint wave is larger every $27\frac{1}{2}$~days
and smaller in between. Thus this lunar
elliptic tide represents the principal effect of the
\index{Elliptic tide, lunar}%
elliptic motion of the moon round the earth.
There are other semidiurnal waves besides the
three which I have mentioned, but it would
hardly be in place to consider them further
here.
\index{Semidiurnal tide!in harmonic method|)}%

Now turning to the waves of the second kind,
\PageSep{205}
\index{Diurnal inequality!in harmonic method}%
which are diurnal in character, we find three, all
of great importance. In one of them the high
waters succeed one another at intervals of $25$~hours
$49$~minutes $9\frac{1}{2}$~seconds, and of the second
and third, one has a period of about $4$~minutes
less than $24$~hours and the other of about $4$~minutes
greater than the $24$~hours. It would
hardly be possible to show by general reasoning
how these three waves arise from the attraction
of three ideal satellites, and how these satellites
together are a substitute for the actions of the
true moon and sun. It must, however, be obvious
that the oscillation resulting from three coëxistent
waves will be very complicated.

All the semidiurnal tides result from waves of
essentially similar character, although some follow
one another a little more rapidly than others,
and some are higher and some are lower. An
accurate cotidal map, illustrating the progress of
any one of these semidiurnal waves over the
ocean, would certainly tell all that we care to
know about the progress of all the other waves
of the group.

Again, all the diurnal tides arise from waves
of the same character, but they are quite diverse
in origin from the semidiurnal waves, and have
only one high water a day instead of two. A
complete knowledge of the behavior of semidiurnal
waves would afford but little insight into
the behavior of the diurnal waves. At some
\PageSep{206}
time in the future the endeavor ought to be
made to draw a diurnal cotidal chart distinct
from the semidiurnal one, but our knowledge is
not yet sufficiently advanced to make the construction
of such a chart feasible.

\TB

All the waves of which I have spoken thus
far are generated by the attractions of the sun
and moon and are therefore called astronomical
tides, but the sea level is also affected by other
oscillations arising from other causes.

Most of the places, at which a knowledge of
the tides is practically important, are situated in
estuaries and in rivers. Now rain is more prevalent
\index{Rivers!annual meteorological tide in}%
at one season than at another, and mountain
snow melts in summer; hence rivers and
estuaries are subject to seasonal variability of
level. In many estuaries this kind of inequality
may amount to one or two feet, and such a considerable
change cannot be disregarded in tidal
prediction. It is represented by inequalities with
periods of a year and of half a year, which are
called the annual and semiannual meteorological
\index{Annual and semi-annual tides}%
\index{Meteorological!tides}%
tides.

Then again, at many places, especially in the
Tropics, there is a regular alternation of day and
night breezes, the effect of which is to heap up
% [** TN: "inland", "off-shore" on line breaks in the original; sole instances]
the water in-shore as long as the wind blows in-land,
\index{Wind!a cause of meteorological tides}%
and to lower it when the wind blows off-shore.
Hence there results a diurnal inequality
\PageSep{207}
\index{Estuary, annual meteorological tide in}%
of sea-level, which is taken into account in tidal
prediction by means of a ``solar diurnal meteorological
\index{Meteorological!tides}%
tide.'' Although these inequalities depend
entirely on meteorological influences and
have no astronomical counterpart, yet it is necessary
to take them into account in tidal prediction.

\TB

But besides their direct astronomical action,
the sun and moon exercise an influence on the
sea in a way of which I have not yet spoken.
We have seen how waves gradually change their
shape as they progress in a shallow river, so that
the crests become sharper and the hollows flatter,
while the advancing slope becomes steeper and
the receding one less steep. An extreme exaggeration
of this sort of change of shape was
found in the bore. Now it is an absolute rule,
in the harmonic analysis of the tide, that the
partial waves shall be of the simplest character,
and shall have a certain standard law of slope
on each side of their crests. If then any wave
ceases to present this standard simple form, it is
necessary to conceive of it as compound, and to
build it up out of several simple waves. By the
composition of a simple wave with other simple
waves of a half, a third, a quarter of the wave
length, a resultant wave can be built up which
shall assume any desired form. For a given
compound wave, there is no alternative of choice,
\PageSep{208}
for it can only be built up in one way. The
analogy with musical notes is here complete, for
a musical note of any quality is built up from
a fundamental, together with its octave and
twelfth, which are called overtones. So also the
distorted tide wave in a river is regarded as consisting
of simple fundamental tide, with over-tides
of half and third length. The periods of
these over-tides are also one half and one third
of that of the fundamental wave.

Out in the open ocean, the principal lunar
semidiurnal tide is a simple wave, but when it
runs into shallow water at the coast line, and
still more so in an estuary, it changes its shape.
\index{Estuary, annual meteorological tide in}%
The low water lasts longer than the high water,
and the time which elapses from low water to
high water is usually shorter than that from
high water to low water. The wave is in fact
no longer simple, and this is taken into account
by considering it to consist of a fundamental
lunar semidiurnal wave with a period of $12$~hours
$50$~minutes, of the first over-tide or octave
with a period of $6$~hours $25$~minutes, of the second
over-tide or twelfth with a period of $4$~hours
$17$~minutes, and of the third over-tide or
double octave with a period of $3$~hours $13$~minutes.
In estuaries, the first over-tide of the
lunar semidiurnal tide is often of great importance,
and even the second is considerable; the
third is usually very small, and the fourth and
\PageSep{209}
higher over-tides are imperceptible. In the same
way over-tides must be introduced, to represent
the change of form of the principal solar semidiurnal
tide. But it is not usually found necessary
to consider them in the cases of the less
important partial tides. The octave, the twelfth,
and the upper octave may be legitimately described
as tides, because they are due to the
attractions of the moon and of the sun, although
they arise indirectly through the distorting influence
of the shallowness of the water.

\TB

I have said above that about twenty different
simple waves afford a good representation of the
tides at any port. Out of these twenty waves,
some represent the seasonal change of level in
the water due to unequal rainfall and evaporation
at different times of the year, and others
represent the change of shape of the wave due
to shallowing of the water. Deducting these
quasi-tides, we are left with about twelve to
represent the true astronomical tide. It is not
possible to give an exact estimate of the number
of partial tides necessary to insure a good representation
of the aggregate tide wave, because
the characteristics of the motion are so different
at various places that partial waves, important
at one place, are insignificant at others. For
example, at an oceanic island the tides may be
more accurately represented as the sum of a
\PageSep{210}
dozen simple waves than by two dozen in a tidal
river.

The method of analyzing a tide into its constituent
parts, of which I have now given an
account, is not the only method by which the
tides may be treated, but as it is the most recent
and the best way, I shall not consider the older
methods in detail. The nature of the procedure
adopted formerly will, however, be indicated in
\Ref{Chapter}{XIII}.

\begin{Authorities}[Authority]
G.~H. Darwin, \Title{Harmonic Analysis of Tidal Observations}:
\index{Analysis, harmonic, of tide|)}%
\index{Darwin, G. H.!harmonic analysis}%
\index{Harmonic analysis!account of|)}%
``Report to British Association.'' Southport,~1883.

An outline of the method is also contained in \Title{Tides}, ``Encyclopædia
Britannica.''
\end{Authorities}
\PageSep{211}


\Chapter{XII}
{Reduction of Tidal Observations}

\First{I have} now to explain the process by which
\index{Reduction of tidal observations|(}%
the several partial tides may be disentangled
from one another.

The tide gauge furnishes a complete tidal record,
so that measurement of the tide curve gives
the height of the water at every instant of time
during the whole period of observation. The
\index{Observation!reduction of tidal|(}%
record may be supposed to begin at noon of a
given day, say of the first of January. The
longitude of the port of observation is of course
known, and the Nautical Almanack gives the
positions of the sun and moon on the day and
at the hour in question, with perfect accuracy.
The real moon has now to be replaced by a
series of ideal satellites, and the rules for the
substitution are absolutely precise. Accordingly,
the position in the heavens of each of
the ideal satellites is known at the moment of
time at which the observations begin. The
same is true of the ideal suns which replace the
actual sun.

I shall now refer to only a single one of the
ideal moons or suns, for, \textit{mutatis mutandis},
\PageSep{212}
what is true of one is true of all. It is easy to
calculate at what hour of the clock, measured in
the time of the place of observation, the satellite
in question will be due south. If the ideal
satellite under consideration were the one which
generates the principal lunar semidiurnal tide, it
would be due south very nearly when the real
moon is south, and the ideal sun which generates
the principal solar tide is south exactly at noon.
But there is no such obvious celestial phenomenon
associated with the transit of any other of
the satellites, although it is easy to calculate the
time of the southing of each of them. We have
then to discover how many hours elapse after
the passage of the particular satellite up to the
high water of its tide wave. The height of
the wave crest above, and the depression of the
wave hollow below the mean water mark must
also be determined. When this problem has
been solved for all the ideal satellites and suns,
the tides are said to be reduced, and the reduction
furnishes the materials for a tide table for
the place of observation.

The difficulty of finding the time of passage
and the height of the wave due to any one of
the satellites arises from the fact that all the
waves really coëxist, and are not separately
manifest. The nature of the disentanglement
may be most easily explained from a special
case, say for example that of the principal lunar
\PageSep{213}
semidiurnal tide, of which the crests follow one
another at intervals of $12$~hours $25$~minutes $14\frac{1}{6}$~seconds.

Since the waves follow one another at intervals
of approximately, but not exactly, a half-day, it
is convenient to manipulate the time scale so as
\index{Time!lunar}%
to make them exactly semidiurnal. Accordingly
we describe $24$~hours $50$~minutes $28\frac{1}{3}$~seconds as
a lunar day, so that there are exactly two waves
\index{Lunar!time}%
following one another in the lunar day.

The tide curve furnishes the height of the
\index{Curve, tide!partitioned into lunar time}%
water at every moment of time, but the time
having been registered by the clock of the tide
gauge is partitioned into ordinary days and
hours. It may, however, be partitioned at intervals
of $24$~hours $50$~minutes $28\frac{1}{3}$~seconds, and
into the twenty-fourth parts of that period, and
it will then be divided into lunar days and hours.
On each lunar day the tide for which we are
searching presents itself in the same way, so
that it is always high and low water at the same
hour of the lunar clock, with exactly two high
waters and two low waters in the lunar day.

Now the other simple tides are governed by
other scales of time, so that in a long succession
of days their high waters and low waters occur
at every hour of the lunar clock. If then we
find the average curve of rise and fall of the
water, when the time is divided into lunar days
and hours, and if we use for the average a long
\PageSep{214}
succession of days, all the other tide waves will
disappear, and we shall be left with only the
lunar semidiurnal tide, purified from all the
others which really coëxist with it.

The numerical process of averaging thus leads
to the obliteration of all but one of the ideal
satellites, and this is the foundation of the
method of harmonic analysis. The average
lunar tide curve may be looked on as the outcome
of a single day of observation, when all
but the selected satellite have been obliterated.
The height of the average wave, and the interval
after lunar noon up to high water, are the
two tidal constants for the lunar semidiurnal
tide, and they enable us to foretell that tide for
all future time.

If the tide curve were partitioned into other
days and hours of appropriate lengths, it would
be possible by a similar process of averaging to
single out another of the constituent tide waves,
and to determine its two tidal constants, which
contain the elements of prediction with respect
to it. By continued repetition of operations of
this kind, all the constituents of practical importance
can be determined, and recorded numerically
by means of their pairs of tidal constants.

The possibility of the disentanglement has
now been demonstrated, but the work of carrying
out these numerical operations would be
\PageSep{215}
\index{Schedule for reducing tidal observations}%
fearfully laborious. The tide curve would have
to be partitioned into about a dozen kinds of
days of various lengths, and the process would
entail measurements at each of the $24$~hours of
each sort of day throughout the whole series.
There are about nine thousand hours in a year,
and it would need about a hundred thousand
measurements of the curve to evaluate twelve
different partial tides; each set of measured
heights would then have to be treated separately
to find the several sorts of averages. Work of
this kind has usually to be done by paid computers,
and the magnitude of the operation
would make it financially prohibitive. It is,
however, fortunately possible to devise abridged
methods, which bring the work within manageable
limits.

In order to minimize the number of measurements,
the tide curve is only measured at each
of the $24$~exact hours of ordinary time, the
height at noon being numbered $0$~hr., and that
at midnight $12$~hrs., and so on up to $24$~hrs.
After obtaining a set of $24$~measurements for
each day, the original tide curve is of no further
use. The number of measurements involved is
still large, but not prohibitive. It would be
somewhat too technical, in a book of this kind,
to explain in detail how the measured heights of
the water at the exact hours of ordinary time
may be made to give, with fair approximation,
\PageSep{216}
\index{Schedule for reducing tidal observations}%
the heights at the exact hours of other time
scales. It may, however, be well to explain that
this approximate method is based on the fact,
that each exact hour of any one of the special
time scales must of necessity fall within half an
hour of one of the exact hours of ordinary time.
The height of the water at the nearest ordinary
hour is then accepted as giving the height at the
exact hour of the special time. The results, as
computed in this way, are subjected to a certain
small correction, which renders the convention
accurate enough for all practical purposes.

A schedule, serviceable for all time and for
all places, is prepared which shows the hour of
ordinary time lying nearest to each successive
hour of any one of the special times. The successive
$24$~hourly heights, as measured on the tide
curve, are entered in this schedule, and when
the entry is completed the heights are found to
be arranged in columns, which follow the special
time scale with a sufficiently good approximation
to accuracy. A different form of schedule is
required for each partial tide, and the entry of
the numbers therein is still enormously laborious,
although far less so than the re-partitions and
re-measurements of the tide curve would be.

The operation of sorting the numbers into
schedules has been carried out in various ways.
In the work of the Indian Survey, the numbers
\index{Indian Survey!method of reducing tidal observations}%
have been re-copied over and over again. In
\PageSep{217}
\index{Abacus for reducing tidal observations|(}%
\index{Darwin, G. H.!tidal abacus|(}%
\index{Indian Survey!method of reducing tidal observations}%
the office of the United States Coast Survey use
\index{United States Coast Survey!method of reducing tidal observations}%
is made of certain card templates pierced with
holes. These templates are laid upon the tabulation
of the measurements of the tide curve,
and the numbers themselves are visible through
the holes. On the surface of the template lines
are drawn from hole to hole, and these lines
indicate the same grouping of the numbers as
that given by the Indian schedules. Dr.~Börgen,
\index{Borgen@Börgen, method of reducing tidal observations}%
of the Imperial German Marine Observatory
\index{German method of reducing tidal observations}%
at Wilhelmshaven, has used sheets of tracing
paper to attain the same end. The Indian procedure
is unnecessarily laborious, and the American
and German plans appear to have some
disadvantage in the fact that the numbers to be
added together lie diagonally across the page.
I am assured by some professional computers
that diagonal addition is easy to perform correctly;
nevertheless this appeared to me to be
so serious a drawback, that I devised another
plan by which the numbers should be brought
into vertical columns, without the necessity of
re-copying them. In my plan each day is treated
as a unit and is shifted appropriately. It might
be thought that the results of the grouping
would be considerably less accurate than in the
former methods, but in fact there is found to be
no appreciable loss of accuracy.

I have $74$~narrow writing-tablets of xylonite,
divided by lines into $24$~compartments; the
\PageSep{218}
tablets are furnished with spikes on the under
side, so that they can be fixed temporarily in any
position on an ordinary drawing-board. The
compartments on each strip are provided for the
entry of the $24$~tidal measurements appertaining
to each day. Each strip is stamped at its end
with a number specifying the number of the day
to which it is appropriated.

The arrangement of these little tablets, so that
the numbers written on them may fall into columns,
is indicated by a sheet of paper marked
with a sort of staircase, which shows where each
tablet is to be set down, with its spikes piercing
the guide sheet. When the strips are in place,
as shown in~\fig{33}, the numbers fall into $48$~columns,
numbered $0$,~$1$,~\dots~$23$, $0$,~$1$,~\dots~$23$
twice over. The guide sheet shown in the figure~\figref{33}
is the one appropriate for the lunar semidiurnal
tide for the fourth set of $74$~days of
a year of observation. The upper half of the
tablets are in position, but the lower ones are
left unmounted, so as the better to show the
staircase of marks.

Then I say that the average of all the $74$~numbers
standing under the two~$0$'s combined
will give the average height of water at $0$~hr.\
of lunar time, and the average of the numbers
under~$1$, that at $1$~hr.\ of lunar time, and so forth.
Thus, after the strips are pegged out, the computer
has only to add the numbers in columns in
%% Plate 2
\TallFig{33}{Tidal Abacus}{jpg}
%% Facing page
%[Blank Page]
\PageSep{219}
order to find the averages. There are other
sheets of paper marked for such other rearrangements
of the strips that each new setting gives
one of the required results; thus a single writing
of the numbers serves for the whole computation.
It is usual to treat a whole year of
observations at one time, but the board being
adapted for taking only $74$~successive days, five
series of writings are required for $370$~days,
which is just over a year. The number~$74$ was
chosen for simultaneous treatment, because $74$~days
is almost exactly five semilunations, and
accordingly there will always be five spring tides
on record at once.

In order to guard the computer against the
use of the wrong paper with any set of strips,
the guide sheets for the first set of $74$~days are
red; for the second they are yellow; for the
third green; for the fourth blue; for the fifth
violet, the colors being those of the rainbow.

The preparation of these papers entailed a
great deal of calculation in the first instance, but
the tidal computer has merely to peg out the
tablets in their right places, verifying that the
numbers stamped on the ends of the strips agree
with the numbers on the paper. The addition
of the long columns of figures is certainly laborious,
but it is a necessary incident of every
method of reducing tidal observations.

The result of all the methods is that for each
\PageSep{220}
partial tide we have a set of $24$~numbers, which
represent the oscillations of the sea due to the
isolated action of one of the ideal satellites, during
the period embraced between two successive
passages of that satellite to the south of the
place of observation. The examination of each
partial tide wave gives its height, and the interval
of time which elapses after its satellite has
passed the meridian until it is high water for
that particular tide. The height and interval
are the tidal constants for that particular tide, at
the port of observation.
\index{Observation!reduction of tidal|)}%

The results of this ``reduction of the observations''
are contained in some fifteen or twenty
pairs of tidal constants, and these numbers contain
a complete record of the behavior of the sea
at the place in question.

\begin{Authorities}
G.~H. Darwin, \Title{Harmonic Analysis, \&c.}: ``Report to British Association,''
1883.

G.~H. Darwin, \Title{On an apparatus for facilitating the reduction of
\index{Darwin, G. H.!tidal abacus|)}%
tidal observations}: ``Proceedings of the Royal Society,'' vol.~lii.\
1892.
\end{Authorities}
\index{Abacus for reducing tidal observations|)}%
\index{Reduction of tidal observations|)}%
\PageSep{221}


\Chapter{XIII}
{Tide Tables}

\First{A tide table} professes to tell, at a given
\index{Tables, tide|(}%
place and on a given day, the time of high and
low water, together with the height of the rise
and the depth of the fall of the water, with
reference to some standard mark on the shore.
A perfect tide table would tell the height of the
water at every moment of the day, but such a
table would be so bulky that it is usual to predict
only the high and low waters.

There are two kinds of tide table, namely,
those which give the heights and times of high
and low water for each successive day of each
year, and those which predict the high and low
water only by reference to some conspicuous
celestial phenomenon. Both sorts of tide table
refer only to the particular harbor for which they
are prepared.

The first kind contains definite forecasts for
each day, and may be called a special tide table.
Such a table is expensive to calculate, and must
be published a full year beforehand. Special
tide tables are published by all civilized countries
for their most important harbors. I believe that
\PageSep{222}
\index{Indian Survey!tide tables}%
\index{United States Coast Survey!tide tables of}%
the most extensive publications are those of the
Indian Government for the Indian Ocean, and
of the United States Government for the coasts
of North America. The Indian tables contain
\index{America, North, tide tables for}%
predictions for about thirty-seven ports.

The second kind of table, where the tide is
given by reference to a celestial phenomenon,
may be described as a general one. It is here
necessary to refer to the Nautical Almanack for
the time of occurrence of the celestial phenomenon,
and a little simple calculation must then be
made to obtain the prediction. The phenomenon
to which the tide is usually referred is the passage
of the moon across the meridian of the place of
observation, and the table states that high and
low water will occur so many hours after the
moon's passage, and that the water will stand at
such and such a height.

The moon, at her change, is close to the sun
and crosses the meridian at noon; she would
then be visible but for the sun's brightness, and
if she did not turn her dark side towards us.
She again crosses the meridian invisibly at midnight.
At full moon she is on the meridian,
visibly at midnight, and invisibly at noon. At
waxing half moon she is visibly on the meridian
at six at night, and at waning half moon at six
in the morning. The hour of the clock at which
the moon passes the meridian is therefore in effect
a statement of her phase. Accordingly the
\PageSep{223}
relative position of the sun and moon is directly
involved in a statement of the tide as corresponding
to a definite hour of the moon's passage. A
table founded on the time of the moon's passage
must therefore involve the principal lunar and
solar semidiurnal tides.

At places where successive tides differ but little
from one another, a simple table of this kind
suffices for rough predictions. The curves marked
Portsmouth in \fig{34} show graphically the interval
after the moon's passage, and the height
of high water at that port, for all the hours of
the moon's passage. We have seen in \Ref{Chapter}{X}.\
that the tide in the North Atlantic is principally
due to a wave propagated from the Southern
Ocean. Since this wave takes a considerable
time to travel from the Cape of Good Hope to
England, the tide here depends, in great measure,
on that generated in the south at a considerable
time earlier. It has therefore been found better
to refer the high water to a transit of the moon
which occurred before the immediately preceding
one. The reader will observe that it is noted on
the upper figure that $28$~hours have been subtracted
from the Portsmouth intervals; that is
to say, the intervals on the vertical scale marked
$6$,~$7$,~$8$ hours are, for Portsmouth, to be interpreted
as meaning $34$,~$35$,~$36$ hours. These are
the hours which elapse after any transit of the
moon up to high water. The horizontal scale is
\PageSep{224}
\index{Moon and earth!tidal prediction by reference to transit|(}%
one of the times of moon's transit and of phases
of the moon; the vertical scale in the lower figure
is one of feet, and it shows the height to
which the water will rise measured from a certain
mark ashore. These Portsmouth curves do not
extend beyond 12~o'clock of moon's transit; this
is because there is hardly any diurnal inequality,
\index{Diurnal inequality!complicates prediction}%
and it is not necessary to differentiate the hours
as either diurnal or nocturnal, the statement being
equally true of either day or night. Thus
if the Portsmouth curves had been extended onward
from $12$~hours to $24$~hours of the clock time
of the moon's passage, the second halves of the
curves would have been merely the duplicates of
the first halves.\footnote
  {Before the introduction of the harmonic analysis of the tides
  described in preceding chapters, tidal observations were ``reduced''
  by the construction of such figures as these, directly from
  the tidal observations. Every high water was tabulated as appertaining
  to a particular phase of the moon, both as to its height
  and as to the interval between the moon's transit and the occurrence
  of high water. The average of a long series of observations
  may be represented in the form of curves by such figures
  as these.}

But the time of the moon's passage leaves her
angular distance from the equator and her linear
distance from the earth indeterminate; and since
the variability of both of these has its influence
on the tide, corrections are needed which add
something to or subtract something from the
tabular values of the interval and height, as dependent
solely on the time of the moon's passage.
\PageSep{225}
\index{Diurnal inequality!complicates prediction}%
\index{Interval from moon's transit to high water!at Portsmouth and at Aden}%
The sun also moves in a plane which is oblique
to the equator, and so similar allowances must be
made for the distance of the sun from the equator,
and for the variability in his distance from the
earth. In order to attain accuracy with a tide
table of this sort, eight or ten corrections are
needed, and the use of the table becomes complicated.

It is, however, possible by increasing the number
of such figures or tables to introduce into
them many of the corrections referred to; and
the use of a general tide table then becomes comparatively
simple. The sun occupies a definite
position with reference to the equator, and stands
at a definite distance from the earth on each day
of the year; also the moon's path amongst the
stars does not differ very much from the sun's.
Accordingly a tide table which states the interval
after the moon's passage to high or low water
and the height of the water on a given day of
\index{Height of tide!at Portsmouth and at Aden}%
the year will directly involve the principal inequalities
in the tides. As the sun moves slowly
amongst the stars, a table applicable to a given
day of the year is nearly correct for a short time
before and after that date. If, then, a tide table,
stating the time and height of the water by reference
to the moon's passage, be computed for
say every ten days of the year, it will be very
nearly correct for five days before and for five
days after the date for which it is calculated.
\PageSep{226}

The curves marked Aden, March and June, in
\index{Prediction of tide!example at Aden|(}%
\fig{34}, show the intervals and heights of tide,
on the 15th of those months at that port, for all
the hours of the moon's passage. The curves are
to be read in the same way as those for Portsmouth,
but it is here necessary to distinguish the
hours of the day from those of the night, and
accordingly the clock times of moon's transit are
numbered from $0$~hr.\ at noon up to $24$~hrs.\ at
the next noon. The curves for March differ so
much from those for June, that the corrections
would be very large, if the tides were treated at
Aden by a single pair of average curves as at
Portsmouth.

The law of the tides, as here shown graphically,
may also be stated numerically, and the
use of such a table is easy. The process will be
best explained by an example, which happens to
be retrospective instead of prophetic. It will involve
that part of the complete table (or series of
curves) for Aden which applies to the 15th of
March of any year. Let it be required then to
find the time and height of high water on March~17,
1889. The Nautical Almanack for that year
shows that on that day the moon passed the meridian
of Aden at eleven minutes past noon of
Aden time, or in astronomical language at $0$~hr.\
$11$~mins. Now the table, or the figure of intervals,
shows that if the moon had passed at $0$~hr.,
or exactly at noon, the interval would have been
\PageSep{227}
$8$~hrs.\ $9$~mins., and that if she had passed at $0$~hr.\
$20$~mins., or 12.20~\PM\ of the day, the interval
would have been $7$~hrs.\ $59$~mins. But on
March~17th the moon actually crossed at $0$~hr.\
\Figure{34}{Curves of Intervals and Heights at Portsmouth
and at Aden}{png}
$11$~mins., very nearly halfway between noon and
$20$~mins.\ past noon. Hence the interval was
halfway between $8$~hrs.\ $9$~mins.\ and $7$~hrs.\ $59$~mins.,
so that it was $8$~hrs.\ $4$~mins. Accordingly
it was high water $8$~hrs.\ $4$~mins.\ after the moon
\PageSep{228}
crossed the meridian. But the moon crossed at
$0$~hr.\ $11$~mins., therefore the high water occurred
at 8.15~\PM.

Again the table of heights, or the figure, shows
that on March~15th, if the moon crossed at $0$~hr.\
$0$~min.\ the high water would be $6.86$~ft.\ above
a certain mark ashore, and if she crossed at $0$~hr.\
$20$~mins.\ the height would be $6.92$~ft. But on
March~17th the moon crossed halfway between
$0$~hr.\ $0$~min.\ and $0$~hr.\ $20$~mins., and therefore
the height was halfway between $6.86$~ft.\ and
$6.92$~ft., that is to say, it was $6.89$~ft., or $6$~ft.\
$11$~in. We therefore conclude that on March~17,
1889, the sea at high water rose to $6$~ft.\
$11$~in., at 8.15~\PM. I have no information as
to the actual height and time of high water on
that day, but from the known accuracy of other
predictions at Aden we may be sure that this
agrees pretty nearly with actuality. The predictions
derived from this table are markedly improved
when a correction, either additive or subtractive,
is applied, to allow for the elliptic motion
of the moon round the earth. On this particular
occasion the moon stood rather nearer the earth
than the average, and therefore the correction to
the height is additive; the correction to the time
also happens to be additive, although it could
not be foreseen by general reasoning that this
would be the case. The corrections for March~17,
1889, are found to add about $2$~mins.\ to the
\PageSep{229}
time, bringing it to 8.17~\PM, and nearly two
inches to the height, bringing it to $7$~ft.\ $1$~in.

This sort of elaborate general tide table has
been, as yet, but little used. It is expensive to
calculate, in the first instance, and it would occupy
two or three pages of a book. The expense
is, however, incurred once for all, and the table
is available for all time, provided that the tidal
observations on which it is based have been good.
A sea captain arriving off his port of destination
would not take five minutes to calculate the two
or three tides he might require to know, and the
information would often be of the greatest value
to him.

As things stand at present, a ship sailing to
most Chinese, Pacific, or Australian ports is only
furnished with a statement, often subject to considerable
error, that the high water will occur at
so many hours after the moon's passage and will
rise so many feet. The average rise at springs and
neaps is generally stated, but the law of the variability
according to the phases of the moon is wanting.
But this is not the most serious defect in the
information, for it is frequently noted that the
tide is ``affected by diurnal inequality,'' and this
note is really a warning to the navigator that he
cannot foretell the time of high water within two
or three hours of time, or the height within several
feet.

Tables of the kind I have described would
\PageSep{230}
\index{Prediction of tide!method of computing|(}%
banish this extreme vagueness, but they are more
likely to be of service at ports of second-rate importance
than at the great centres of trade, because
at the latter it is worth while to compute
full special tide tables for each year.
\index{Tables, tide!method of calculating|(}%

It is unnecessary to comment on the use of
tables containing predictions for definite days,
since it merely entails reference to a book, as to
a railway time table. Such special tables are undoubtedly
the most convenient, but the number
of ports which can ever be deemed worthy of the
great expense incidental to their preparation
must always be very limited.

\TB

We must now consider the manner in which
tide tables are calculated. It is supposed that
careful observations have been made, and that
the tidal constants, which state the laws governing
the several partial tides, have been accurately
determined by harmonic analysis. The analysis
of tidal observations consists in the dissection of
the aggregate tide wave into its constituent partial
waves, and prediction involves the recomposition
or synthesis of those waves. In the synthetic
\index{Synthesis of partial tides for prediction|(}%
process care must be taken that the partial
waves shall be recompounded in their proper
relative positions, which are determined by the
places of the moon and sun at the moment of
time chosen for the commencement of prediction.

The synthesis of partial waves may be best
\index{Moon and earth!tidal prediction by reference to transit|)}%
\index{Prediction of tide!example at Aden|)}%
\PageSep{231}
arranged in two stages. It has been shown in
\Ref{Chapter}{XI}.\ that the partial waves fall naturally
into three groups, of which the third is practically
insignificant. The first and second are the
semidiurnal and diurnal groups. The first process
is to unite each group into a single wave.

We will first consider the semidiurnal group.
Let us now, for the moment, banish the tides
from our minds, and imagine that there are two
trains of waves traveling simultaneously along a
straight canal. If either train existed by itself
every wave would be exactly like all its brethren,
both in height, length, and period. Now suppose
that the lengths and periods of the waves
of the two coëxistent trains do not differ much
from one another, although their heights may
differ widely. Then the resultant must be a single
train of waves of lengths and periods intermediate
between those of the constituent waves,
but in one part of the canal the waves will be
high, where the two sets of crests fall in the
same place, whilst in another they will be low,
where the hollow of the smaller constituent wave
falls in with the crest of the larger. If only one
part of the canal were visible to us, a train of
waves would pass before us, whose heights would
gradually vary, whilst their periods would change
but little.

In the same way two of the semidiurnal tide
waves, when united by the addition of their separate
\PageSep{232}
displacements from the mean level, form a
single wave of variable height, with a period still
semidiurnal, although slightly variable. But
there is nothing in this process which limits the
synthesis to two waves, and we may add a third
and a fourth, finally obtaining a single semidiurnal
wave, the height of which varies according
to a very complex law.

A similar synthesis is then applied to the second
group of waves, so that we have a single
variable wave of approximately diurnal period.
The final step consists in the union of the single
semidiurnal wave with the single diurnal one into
a resultant wave. When the diurnal wave is
large, the resultant is found to undergo very
great variability both in period and height. The
principal variations in the relative positions of
the partial tide waves are determined by the
phases of the moon and by the time of year, and
there is, corresponding to each arrangement of
the partial waves, a definite form for the single
resultant wave. The task of forming a general
tide table therefore consists in the determination
of all the possible periods and heights of the resultant
wave and the tabulation of the heights
and intervals after the moon's passage of its high
and low waters.

I supposed formerly that the captain would
himself calculate the tide he required from the
general tide table, but such calculation may be
\PageSep{233}
\index{Machine, tide-predicting}%
\index{Predicting machine for tides|(}%
done beforehand for every day of a specified
year, and the result will be a special tide table.
There are about $\DPchg{1400}{1,400}$ high and low waters in
a year, so that the task is very laborious, and
has to be repeated each year.

\TB

It is, however, possible to compute a special
tide table by a different and far less laborious
method. In this plan an ingenious mechanical
device replaces the labor of the computer. The
first suggestion for instrumental prediction of
tides was made, I think, by Sir William Thomson,
now Lord Kelvin, in~1872. Mr.~Edward
\index{Kelvin, Lord!tide predicting machine}%
Roberts bore an important part in the practical
\index{Roberts, E., the tide-predicting machine}%
realization of such a machine, and a tide predicter
was constructed by Messrs.\ Légé for the
\index{Lege@Légé, constructor of tide-predicting machine}%
Indian Government under his supervision. This
is, as yet, the only complete instrument in existence.
But others are said to be now in course
of construction for the Government of the
United States and for that of France. The
Indian machine cost so much and works so well,
that it is a pity it should not be used to the full
extent of its capacity. The Indian Government
has, of course, the first claim on it, but the use
of it is allowed to others on the payment of a
small fee. I believe that, pending the construction
of their own machine, the French authorities
are obtaining the curves for certain tidal
predictions from the instrument in London.
\index{Prediction of tide!method of computing|)}%
\index{Synthesis of partial tides for prediction|)}%
\PageSep{234}

Although the principle involved in the tide
predicter is simple, yet the practical realization
of it is so complex that a picture of the whole
machine would convey no idea of how it works.
I shall therefore only illustrate it diagrammatically,
in~\fig{35}, without any pretension to scale
or proportion. The reader must at first imagine
that there are only two pulleys, namely, $A$~and~$B$,
so that the cord passes from the fixed end~$F$
under~$A$ and over~$B$, and so onward to the pencil.
The pulley~$B$ is fixed, and the pulley~$A$ can slide
vertically up and down in a slot, which is not
shown in the diagram. If $A$~moves vertically
through any distance, the pencil must clearly
move through double that distance, so that
when $A$~is highest the pencil is lowest, and vice
versa.

The pencil touches a uniformly revolving
drum, covered with paper; thus if the pulley~$A$
executes a simple vertical oscillation, the pencil
draws a simple wave on the drum. Now the
pulley is mounted on an inverted T-shaped
frame, and a pin, fixed in a crank~$C$, engages in
the slit in the horizontal arm of the T-piece.
When the crank~$C$ revolves, the pulley~$A$ executes
a simple vertical oscillation with a range depending
on the throw of the crank.\footnote
  {I now notice that the throw of the crank~$C$ is too small to
  have allowed the pencil to draw so large a wave as that shown
  on the drum. But as this is a mere diagram, I have not thought
  it worth while to redraw the whole.}
The position
\PageSep{235}
of the pin is susceptible of adjustment on the
crank, so that its throw and the range of oscillation
of the pulley can be set to any required
\Figure[0.7]{35}{Diagram of Tide-predicting Instrument}{png}
length---of course within definite limits determined
by the size of the apparatus.

The drum is connected to the crank~$C$ by a
train of wheels, so that as the crank rotates the
drum also turns at some definitely proportional
rate. If, for example, the crank revolves twice
for one turn of the drum, the pencil will draw a
simple wave, with exactly two crests in one circumference
of the drum. If one revolution of
the drum represents a day, the graphical time
scale is $24$~hours to the circumference of the
\PageSep{236}
drum. If the throw of the crank be one inch,
the pulley will oscillate with a total range of two
inches, and the pencil with a total range of four
inches. Then taking two inches lengthwise on
the drum to represent a foot of water, the curve
drawn by the pencil might be taken to represent
the principal solar semidiurnal tide, rising one
foot above and falling one foot below the mean
sea level.

I will now show how the machine is to be
adjusted so as to give predictions. We will
suppose that it is known that, at noon of the
first day for which prediction is required, the
solar tide will stand at $1$~ft.\ $9$~in.\ above mean
sea level and that the water will be rising. Then,
the semi-range of this tide being one foot, the
pin is adjusted in the crank at one inch from
the centre, so as to make the pencil rock through
a total range of $4$~inches, representing $2$~feet.
The drum is now turned so as to bring the noon-line
of its circumference under the pencil, and
the crank is turned so that the pencil shall be
$3\frac{1}{2}$~inches (representing $1$~ft.\ $9$~in.\ of water)
below the middle of the drum, and so that when
the machine starts, the pencil will begin to descend.
The curve being drawn upside-down,
the pencil is set below the middle line because
the water is to be above mean level, and it must
begin to descend because the water is to ascend.
The train of wheels connecting the crank and
\PageSep{237}
drum is then thrown into gear, and the machine
is started; it will then draw the solar tide curve,
on the scale of $2$~inches to the foot, for all
time.

If the train of wheels connecting the crank to
the drum were to make the drum revolve once
whilst the crank revolves $1.93227$~times, the
curve would represent a lunar semidiurnal tide.
The reason of this is that $1.93227$~is the ratio
of $24$~hours to $12$~h.\ $25$~m.\ $14$~s., that is to say,
of a day to a lunar half day. We suppose the
circumference of the drum still to represent an
ordinary day of $24$~hours, and therefore the
curve drawn by the pencil will have lunar semidiurnal
periodicity. In order that these curves
may give predictions of the future march of that
tide, the throw of the crank must be set to give the
correct range and its angular position must give
the proper height at the moment of time chosen
for beginning. When these adjustments are
made the curve will represent that tide for all
time.

We have now shown that, by means of appropriate
trains of wheels, the machine can be made
to predict either the solar or the lunar tide; but
we have to explain the arrangement for combining
them. If, still supposing there to be
only the two pulleys $A$,~$B$, the end~$F$ of the cord
were moved up or down, its motion would be
transmitted to the pencil, whether the crank~$C$
\PageSep{238}
and pulley~$A$ were in motion, or at rest; but if
they were in motion, the pencil would add the
motion of the end of the cord to that of the
pulley. If then there be added another fixed
pulley~$B'$, and another movable pulley~$A'$, driven
by a crank and T-piece (not shown in the diagram),
the pencil will add together the movements
of the two pulleys $A$~and~$A'$. There must
now be two trains of wheels, one connecting $A$
with the drum and the other for~$A'$. If a single
revolution of the drum causes the crank~$C$ to
turn twice, whilst it makes the crank of~$A'$ rotate
$1.93227$~times, the curve drawn will represent
the union of the principal solar and lunar semidiurnal
tides. The trains of wheels requisite for
transmitting motion from the drum to the two
cranks in the proper proportions are complicated,
but it is obviously only a matter of calculation
to determine the numbers of the teeth in the
several wheels in the trains. It is true that rigorous
accuracy is not attainable, but the mechanism
is made so nearly exact that the error in the
sum of the two tides would be barely sensible
even after $\DPchg{3000}{3,000}$~revolutions of the drum. It is
of course necessary to set the two cranks with
their proper throws and at their proper angles
so as to draw a curve which shall, from the noon
of a given day, correspond to the tide at a given
place.

It must now be clear that we may add as
\PageSep{239}
many more movable pulleys as we like. When
the motion of each pulley is governed by an
appropriate train of wheels, the movement of
the pencil, in as far as it is determined by that
pulley, corresponds to the tide due to one of our
ideal satellites. The resultant curve drawn on
the drum is then the synthesis of all the partial
tides, and corresponds with the motion of the
sea.

The instrument of the Indian Government
unites twenty-four partial tides. In order to
trace a tide curve, the throws of all the cranks
are set so as to correspond with the known
heights of the partial tides, and each crank is set
at the proper angle to correspond with the moment
of time chosen for the beginning of the tide
table. It is not very difficult to set the cranks
and pins correctly, although close attention is of
course necessary. The apparatus is then driven
by the fall of a weight, and the paper is fed
automatically on to the drum and coiled off on
to a second drum, with the tide curve drawn on
it. It is only necessary to see that the paper
runs on and off smoothly, and to write the date
from time to time on the paper as it passes, in
order to save future trouble in the identification
of the days. It takes about four hours to run
off the tides for a year.

The Indian Government sends home annually
the latest revision of the tidal constants for
\PageSep{240}
thirty-seven ports in the Indian Ocean. Mr.~Roberts
sets the machine for each port, so as to
correspond with noon of a future 1st~of January,
and then lets it run off a complete tide
curve for a whole year. The curve is subsequently
measured for the time and height of
each high and low water, and the printed tables
are sold at the moderate price of four rupees.
The publication is made sufficiently long beforehand
to render the tables available for future
voyages. These tide tables are certainly amongst
the most admirable in the world.

\TB

It is characteristic of England that the machine
is not, as I believe, used for any of the
home ports, and only for a few of the colonies.
The neglect of the English authorities is not,
however, so unreasonable as it might appear to
be. The tides at English ports are remarkably
simple, because the diurnal inequality is practically
absent. The applicability of the older
methods of prediction, by means of such curves
as that for Portsmouth in~\fig{34}, is accordingly
easy, and the various corrections are well determined.
The arithmetical processes are therefore
not very complicated, and ordinary computers
are capable of preparing the tables with but
little skilled supervision. Still it is to be regretted
that this beautiful instrument should not
be more used for the home and colonial ports.
\PageSep{241}

The excellent tide tables of the Government
of the United States have hitherto been prepared
by the aid of a machine of quite a different
character, the invention of the late Professor
Ferrel. This apparatus virtually carries out
\index{Ferrel, tide-predicting instrument}%
\index{Predicting machine for tides!Ferrel's}%
that process of compounding all the waves together
into a single one, which I have described
as being done by a computer for the formation
of a general tide table. It only registers, however,
the time and height of the maxima and
minima---the high and low waters. I do not
think it necessary to describe its principle in
detail, because it will shortly be superseded by a
machine like, but not identical with, that of the
Indian Government.

\begin{Authorities}
G.~H. Darwin, \Title{On Tidal Prediction}. ``Philosophical Transactions
of the Royal Society,'' A.~1891, pp.~159--229.

In the example of the use of a general tide table at Aden,
given in this chapter, the datum from which the height is measured
is $0.37$~ft.\ higher than that used in the Indian Tide Tables;
\index{Tables, tide|)}%
\index{Tables, tide!method of calculating|)}%
accordingly $4\frac{1}{2}$~inches must be added to the height, in order to
bring it into accordance with the official table.

Sir William Thomson, \Title{Tidal Instruments}, and the subsequent
discussion. ``Institute of Civil Engineers,'' vol.~lxv.

William Ferrel, \Title{Description of a Maxima and Minima Tide-predicting
\index{Machine, tide-predicting}%
Machine}. ``United States Coast Survey,'' 1883.
\end{Authorities}
\index{Predicting machine for tides|)}%
\PageSep{242}


\Chapter[Accuracy of Tidal Prediction]{XIV}
{The Degree of Accuracy of Tidal Prediction}

\First{The} success of tidal predictions varies much
\index{Atmospheric pressure!influence on tidal prediction}%
\index{Prediction of tide!errors in|(}%
according to the place of observation. They are
not unfrequently considerably in error in our
latitude, and throughout those regions called by
sailors ``the roaring forties.'' The utmost that
can be expected of a tide table is that it shall
be correct in calm weather and with a steady
barometer. But such conditions are practically
non-existent, and in the North Atlantic the great
variability in the meteorological elements renders
tidal prediction somewhat uncertain.

The sea generally stands higher when the
barometer is low, and lower when the barometer
is high, an inch of mercury corresponding to
rather more than a foot of water. The pressure
of the air on the sea in fact depresses it in those
places where the barometer is high, and allows it
to rise where the opposite condition prevails.

Then again a landward wind usually raises the
\index{Wind!perturbation of, in tidal prediction}%
sea level, and in estuaries the rise is sometimes
very great. There is a known instance when the
Thames at London was raised by five feet in a
strong gale. Even on the open coast the effect
\PageSep{243}
\index{Atmospheric pressure!influence on tidal prediction}%
of wind is sometimes great. A disastrous example
\index{Wind!perturbation of, in tidal prediction}%
of this was afforded on the east coast of England
in the autumn of~1897, when the conjunction
of a gale with springtide caused the sea to
do an enormous amount of damage, by breaking
embankments and flooding low-lying land.

But sometimes the wind has no apparent effect,
and we must then suppose that it had been blowing
previously elsewhere in such a way as to depress
the water at the point at which we watch it.
The gale might then only restore the water to its
normal level, and the two effects might mask one
another. The length of time during which the
wind has lasted is clearly an important factor,
because the currents generated by the wind must
be more effective in raising or depressing the sea
level the longer they have lasted.

It does not then seem possible to formulate
any certain system of allowance for barometric
pressure and wind. There are, at each harbor,
certain rules of probability, the application of
which will generally lead to improvement in the
prediction; but occasionally such empirical corrections
will be found to augment the error.

But notwithstanding these perturbations, good
tide tables are usually of surprising accuracy
even in northern latitudes; this may be seen
from the following table showing the results of
comparisons between prediction and actuality at
Portsmouth. The importance of the errors in
\index{Errors in tidal prediction|(}%
\PageSep{244}
height depends of course on the range of tide;
it is therefore well to note that the average ranges
of tide at springs and neaps are $13$~ft.\ $9$~in.\ and
$7$~ft.\ $9$~in.\ respectively.
\begin{table}[hbtp!]
\caption{Table of Errors in the Prediction of High Water at
Portsmouth in the Months of January, May, And
\index{Portsmouth, table of errors in tidal predictions}%
September, 1897.}
\[
\begin{array}{|c|c||c|c|}
\hline
\multicolumn{2}{|c||}{\ColHead{Time}} &
\multicolumn{2}{c|}{\ColHead{Height}} \\
\hline
\ColHead[1.2in]{Magnitude of error} &
\ColHead[0.5in]{Number of cases} &
\ColHead[1.2in]{Magnitude of error} &
\ColHead[0.5in]{Number of cases} \\
\hline
&& \TEntry{Inches} & \\
\Z0\mm\text{ to }\Z5\mm&  69 & \Z0\text{ to }\Z6 &  89 \\
\Z6\mm\text{ to }10\mm &  50 & \Z7\text{ to }12  &  58 \\
 11\mm\text{ to }15\mm &  25 & 13\text{ to }18   &  24 \\
 16\mm\text{ to }20\mm &  10 & 19\text{ to }24   & \Z6 \\
 21\mm\text{ to }25\mm &  11 & \Dash & \Dash \\
 26\mm\text{ to }30\mm & \Z7 & \Dash & \Dash \\
 31\mm\text{ to }35\mm & \Z4 & \Dash & \Dash \\
     52\mm             & \Z1 & \Dash & \Dash \\
\hline
\Strut\Dash            & \llap{$1$}77&\Dash & \llap{$1$}77 \\
\hline
\end{array}
\]
\end{table}
\begin{table}[hbtp!]
\caption{Errors in Height for the Year 1892,
Excepting Part of July}
\[
\begin{array}{|c|c|}
\hline
\ColHead{Magnitude of error} & \ColHead{Number of cases} \\
\hline
\TEntry{Inches} & \\
\Z0\text{ to }\Z6& 381   \\
\Z7\text{ to }12 & 228   \\
 13\text{ to }18 & \Z52  \\
 19\text{ to }24 & \Z\Z8 \\
 31              & \Z\Z1 \\
\hline
\Strut\Dash & 670 \\
\hline
\end{array}
\]
\end{table}
\PageSep{245}

\begin{Remark}
\NB---The comparison seems to indicate that these predictions
might be much improved, because the predicted height is
nearly always above the observed height, and because the diurnal
inequality has not been taken into account sufficiently, if at
all.
\end{Remark}

In tropical regions the weather is very uniform,
and in many places the ``meteorological
tides'' produced by the regularly periodic variations
of wind and barometric pressure are taken
into account in tidal predictions.

The apparent irregularity of the tides at Aden
is so great, that an officer of the Royal Engineers
has told me that, when he was stationed there
many years ago, it was commonly believed that
the strange inequalities of water level were due
to the wind at distant places. We now know
that the tide at Aden is in fact marvelously
regular, although the rule according to which it
proceeds is very complex. In almost every month
in the year there are a few successive days when
there is only one high water and one low water
in the $24$~hours; and the water often remains
almost stagnant for three or four hours at a
time. This apparent irregularity is due to the
diurnal inequality, which is very great at Aden,
whereas on the coasts of Europe it is insignificant.

I happen to have a comparison with actuality
of a few predictions of high water at Aden,
where the maximum range of the tide is about
$8$~ft.\ $6$~in. They embrace the periods from March~10
\index{Errors in tidal prediction|)}%
\PageSep{246}
\index{Aden, errors of tidal prediction at}%
\index{Tables, tide!amount of error in}%
to April~9, and again from November~12 to
December~12, 1884. In these two periods there
were $118$~high waters, but through an accident
to the tide gauge one high water was not registered.
On one occasion, when the regular semidiurnal
sequence of the tide would lead us to
expect high water, there occurred one of those
periods of stagnation to which I have referred.
Thus we are left with $116$~cases of comparison
between the predicted and actual high waters.

The results are exhibited in the following
table:---
\[
\begin{array}{|c|c||c|c|}
\hline
\multicolumn{2}{|c||}{\ColHead{Time}} &
\multicolumn{2}{c|}{\ColHead{Height}} \\
\hline
\ColHead{Magnitude of errors} &
\ColHead[0.5in]{Number of high waters} &
\ColHead{Magnitude of errors} &
\ColHead[0.5in]{Number of high waters} \\
\hline
&& \TEntry{Inches} & \\
\Z0\mm\PadTxt{ and }{to}\Z5\mm &  35 & 0 & 15 \\
\Z5\mm\PadTxt{ and }{to}10\mm &  32 & 1 & 48 \\
 10\mm\PadTxt{ and }{to}15\mm  &  19 & 2 & 28 \\
 15\mm\PadTxt{ and }{to}20\mm  &  19 & 3 & 14 \\
 20\mm\PadTxt{ and }{to}25\mm  & \Z5 & 4 & 11 \\
 26\mm\text{ and }28\mm        & \Z2 &
\multicolumn{1}{c|}{\TEntry{No high water}} & \Z1 \\
 33\mm\text{ and }36\mm        & \Z2 & \Dash & \Dash \\
 56\mm\text{ and }57\mm        & \Z2 & \Dash & \Dash \\
\multicolumn{1}{|c|}{\TEntry{No high water}} & \Z1 & \Dash & \Dash \\
\hline
\Strut & 117 && 117 \\
\hline
\end{array}
\]

It would be natural to think that when the
prediction is erroneous by as much as $57$~minutes,
it is a very bad one; but I shall show that
\PageSep{247}
\index{Tables, tide!amount of error in}%
this would be to do injustice to the table. On
several of the occasions comprised in this list
the water was very nearly stagnant. Now if the
water only rises about a foot from low to high
water in the course of four or five hours, it is
almost impossible to say with accuracy when it
was highest, and two observers might differ in
their estimate by half an hour or even by an
hour.

In the table of comparison there are $11$~cases
in which the error of time is equal to or greater
than twenty minutes, and I have examined these
cases in order to see whether the water was then
nearly stagnant. A measure of the degree of
stagnation is afforded by the amount of the rise
from low water to high water, or of the fall from
high water to low water. The following table
gives a classification of the errors of time according
to the rise or fall:---
\begin{table}[hp!]
\caption{Analysis of Errors in Time.}
\centering
\begin{tabular}{|c|c|}
\hline
\ColHead[1.5in]{Ranges from low water to high water} &
\ColHead{Errors of time} \\
\hline
\Strut
Nil                 & \Dash \\
$6$ in.\ to $8$ in. & $22$, $26$, $28$, $56$, $57$~minutes \\
$13$ in.            & $36$~minutes \\
$17$ in.            & $22$ \Ditto{minutes} \\
$19$ in.            & $33$ \Ditto{minutes} \\
$2$ ft.\ $10$ in.   & $22$ \Ditto{minutes} \\
$3$ ft.\ $\Z9$ in.  & $23$ \Ditto{minutes} \\
$3$ ft.\ $11$ in.   & $20$ \Ditto{minutes} \\
\hline
\end{tabular}
\end{table}
\PageSep{248}

There are then only three cases when the rise
of water was considerable, and in the greatest of
them it was only $3$~ft.\ $11$~in.

If we deduct all the tides in which the range
between low and high water was equal to or less
than $19$~inches, we are left with $108$~predictions,
and in these cases the greatest error in time is
$23$~mins. In $86$~cases the error is equal to or less
than a quarter of an hour. This leaves $22$~cases
where the error was greater than $15$~mins.\ made
up as follows: $18$~cases with error greater than
$15$~mins.\ and less than $20$~mins.\ and $3$~cases with
errors of $20$~mins., $22$~mins., $23$~mins. Thus in
$106$ out of~$108$ predictions the error of time was
equal to or less than $20$~minutes.

Two independent measurements of a tide
curve, for the determination of the time of high
water, lead to results which frequently differ by
five minutes, and sometimes by ten minutes. It
may therefore be claimed that these predictions
have a very high order of accuracy as regards
time.

Turning now to the heights, out of $116$~predictions
the error in the predicted height was
equal to or less than $2$~inches in $91$~cases, it
amounted to $3$~inches in $14$~cases, and in the
remaining $11$~cases it was $4$~inches. It thus appears
that, as regards the height of the tide also,
the predictions are of great accuracy. This
short series of comparisons affords a not unduly
\PageSep{249}
favorable example of the remarkable success attainable,
where tidal observation and prediction
have been thoroughly carried out at a place
subject to only slight meteorological disturbance.

If our theory of tides were incorrect, so that
we imagined that there was a partial tide wave
of a certain period, whereas in fact such a wave
has no true counterpart in physical causation,
the reduction of a year of tidal observation would
undoubtedly assign some definite small height,
and some definite retardation of the high water
after the passage of the corresponding, but
erroneous, satellite. But when a second series
of observations is reduced, the two tidal constants
would show no relationship to their previous
evaluations. If then reductions carried
out year after year assign, as they do, fairly
consistent values to the tidal constants, we may
feel confident that true physical causation is involved,
even when the heights of some of the
constituent tide waves do not exceed an inch
or two.

Prediction must inevitably fail, unless we have
lighted on the true causes of the phenomena;
success is therefore a guarantee of the truth of
the theory. When we consider that the incessant
variability of the tidal forces, the complex
outlines of our coasts, the depth of the sea and
the earth's rotation are all involved, we should
\PageSep{250}
regard good tidal prediction as one of the
greatest triumphs of the theory of universal
gravitation.

\begin{Authorities}
The Portsmouth comparisons were given to the author by the
Hydrographer of the Admiralty, Admiral Sir W.~J. Wharton.

G.~H. Darwin, \Title{On Tidal Prediction}. ``Philosophical Transactions
of the Royal Society,'' A.~1891.
\end{Authorities}
\index{Prediction of tide!errors in|)}%
\PageSep{251}


\Chapter[Rigidity of the Earth]{XV}
{Chandler's Nutation---The Rigidity of the
Earth}

\First{In} the present chapter I have to explain the
\index{Nutation!Chandler's|(}%
\index{Variation of latitude|(}%
origin of a tide of an entirely different character
from any of those considered hitherto. It may
fairly be described as a true tide, although it is
not due to the attraction of either the sun or
the moon.

We have all spun a top, and have seen it, as
boys say, go to sleep. At first it nods a little,
but gradually it settles down to perfect steadiness.
Now the earth may be likened to a top,
and it also may either have a nutational or nodding
motion, or it may spin steadily; it is only
by observation that we can decide whether it is
nodding or sound asleep.

The equator must now be defined as a plane
through the earth's centre at right angles to the
axis of rotation, and not as a plane fixed with
reference to the solid earth. The latitude of
\index{Latitude!periodic variations of|(}%
any place is the angle\footnote
  {This angle is technically called the geocentric latitude; the
  distinction between true and geocentric latitude is immaterial in
  the present discussion.}
between the equator and
\PageSep{252}
a line drawn from the centre of the earth to the
place of observation. Now when the earth
nutates, the axis of rotation shifts, and its
extremity describes a small circle round the spot
which is usually described as the pole. The
equator, being perpendicular to the axis of rotation,
of course shifts also, and therefore the
latitude of a place fixed on the solid earth varies.
During the whole course of the nutation, the
earth's axis of rotation is always directed towards
the same point in the heavens, and therefore the
angle between the celestial pole and the vertical
or plumb-line at the place of observation must
oscillate about some mean value; the period of
the oscillation is that of the earth's nutation.
This movement is called a ``free'' nutation,
because it is independent of the action of external
forces.

There are, besides, other nutations resulting
from the attractions of the moon and sun on the
protuberant matter at the equator, and from the
same cause there is a slow shift in space of the
earth's axis, called the precession. These movements
are said to be ``forced,'' because they are
due to external forces. The measurements of
the forced nutations and of the precession afford
the means of determining the period of the free
nutation, if it should exist. It has thus been
concluded that if there is any variation in the
latitude, it should be periodic in $305$~days; but
\PageSep{253}
only observation can decide whether there is
such a variation of latitude or not.

Until recently astronomers were so convinced
of the sufficiency of this reasoning, that, when
they made systematic examination of the latitudes
of many observatories, they always searched
for an inequality with a period of $305$~days.
Some thought that they had detected it, but
when the observations extended over long periods,
it always seemed to vanish, as though what
they had observed were due to the inevitable
errors of observation. At length it occurred
to Mr.~Chandler to examine the observations
\index{Chandler, free nutation of earth, and variation of latitude|(}%
of latitude without any prepossession as to the
period of the inequality. By the treatment of
enormous masses of observation, he came to the
conclusion that there is really such an inequality,
but that the period is $427$~days instead of $305$~days.
He also found other inequalities in the
motion of the axis of rotation, of somewhat
obscure origin, and of which I have no occasion
to say more.\footnote
  {They are perhaps due to the unequal melting of polar
  ice and unequal rainfall in successive years. These irregular
  variations in the latitude are such that some astronomers are
  still skeptical as to the reality of Chandler's nutation, and think
  that it will perhaps be found to lose its regularly rhythmical
  character in the future.}

The question then arises as to how the theory
can be so amended as to justify the extension of
the period of nutation. It was, I believe, Newcomb,
\PageSep{254}
\index{Newcomb, S., theoretical explanation of Chandler's nutation}%
of the United States Naval Observatory,
who first suggested that the explanation is to be
sought in the fact that the axis of rotation is an
axis of centrifugal repulsion, and that when it
shifts, the distribution of centrifugal force is
changed with reference to the solid earth, so
that the earth is put into a state of stress, to
which it must yield like any other elastic body.
The strain or yielding consequent on this stress
must be such as to produce a slight variability
in the position of the equatorial protuberance
with reference to places fixed on the earth.
Now the period of $305$~days was computed on
the hypothesis that the position of the equatorial
protuberance is absolutely invariable, but
periodic variations of the earth's figure would
operate so as to lengthen the period of the free
nutation, to an extent dependent on the average
elasticity of the whole earth.
\index{Elasticity of earth}%

Mr.~Chandler's investigation demanded the
utmost patience and skill in marshaling large
masses of the most refined astronomical observations.
His conclusions are not only of the
greatest importance to astronomy, but they also
give an indication of the amount by which the
solid earth is capable of yielding to external
forces. It would seem that the average stiffness
of the whole earth must be such that it yields a
little less than if it were made of steel.\footnote
  {Mr.~S. S. Hough, p.~338 of the paper referred to in the list
\index{Hough, S. S.!rigidity of earth}%
  of authorities at the end of the chapter.}
But
\PageSep{255}
\index{Elasticity of earth}%
the amount by which the surface yields remains
unknown, because we are unable to say what
proportion of the aggregate change is superficial
and what is deep-seated. It is, however, certain
that the movements are excessively small, because
the circle described by the extremity of
the earth's axis of rotation, about the point on
the earth which we call the pole, has a radius of
only fifteen feet.

It is easily intelligible that as the axis of
rotation shifts in the earth, the oceans will tend
to swash about, and that a sort of tide will be
generated. If the displacement of the axis were
considerable, whole continents would be drowned
by a gigantic wave, but the movement is so
small that the swaying of the ocean is very
feeble. Two investigators have endeavored to
detect an oceanic tide with a period of $427$~days;
they are Dr.~Bakhuyzen of Leyden and
\index{Bakhuyzen on tide due to variation of latitude}%
Mr.~Christie of the United States Coast Survey.
\index{Christie, A. S., tide due to variation of latitude}%
The former considered observations of sea-level
on the coasts of Holland, the latter those on the
coasts of the United States; and they both conclude
that the sea-level undergoes a minute
variability with a period of about $430$~days. A
similar investigation is now being prosecuted by
the Tidal Survey of India, and as the Indian
tidal observations are amongst the best in the
world, we may hope for the detection of this
minute tide in the Indian Ocean also.
\PageSep{256}
\index{Earth and moon!rigidity of|(}%
\index{Rigidity of earth|(}%

The inequality in water level is so slight and
extends over so long a period that its measurement
cannot yet be accepted as certain. The
mean level of the sea is subject to slight irregular
variations, which are probably due to unequal
rainfall and unequal melting of polar ice in
successive years. But whatever be the origin of
these irregularities they exceed in magnitude the
one to be measured. The arithmetical processes,
employed to eliminate the ordinary tides and the
irregular variability, will always leave behind
some residual quantities, and therefore the examination
of a tidal record will always apparently
yield an inequality of any arbitrary period whatever.
It is only when several independent determinations
yield fairly consistent values of the
magnitude of the rise and fall and of the moment
of high water, that we can feel confidence
in the result. Now although the reductions of
Bakhuyzen and Christie are fairly consistent
\index{Bakhuyzen on tide due to variation of latitude}%
\index{Christie, A. S., tide due to variation of latitude}%
with one another, and with the time and height
suggested by Chandler's nutation, yet it is by no
means impossible that accident may have led to
this agreement. The whole calculation must
therefore be repeated for several places and at
several times, before confidence can be attained
in the detection of this latitudinal tide.

\TB

The prolongation of the period of Chandler's
nutation from $305$ to $427$~days seems to indicate
\index{Latitude!periodic variations of|)}%
\index{Nutation!Chandler's|)}%
\index{Variation of latitude|)}%
\PageSep{257}
that our planet yields to external forces, and we
naturally desire to learn more on so interesting
a subject. Up to fifty years ago it was generally
held that the earth was a globe of molten
matter covered by a thin crust. The ejection of
lava from volcanoes and the great increase of
temperature in mines seemed to present evidence
in favor of this belief. But the geologists and
physicists of that time seemed not to have perceived
that the inference might be false, if great
pressure is capable of imparting rigidity to matter
at a very high temperature, because the interior
of the earth might then be solid although
very hot. Now it has been proved experimentally
that rock expands in melting, and a physical
corollary from this is that when rock is under
great pressure a higher temperature is needed to
melt it than when the pressure is removed. The
pressure inside the earth much exceeds any that
can be produced in the laboratory, and it is uncertain
up to what degree of increase of pressure
the law of the rise of the temperature of
melting would hold good; but there can be no
doubt that, in so far as experiments in the laboratory
can be deemed applicable to the conditions
prevailing in the interior of the earth, they
tend to show that the matter there is not improbably
solid.

But Lord Kelvin reinforces this argument
\index{Kelvin, Lord!rigidity of earth|(}%
from another point of view. Rock in the solid
\index{Chandler, free nutation of earth, and variation of latitude|)}%
\PageSep{258}
condition is undoubtedly heavier than when it is
molten. Now the solidified crust on the surface
of a molten planet must have been fractured
many times during the history of the planet,
and the fragments would sink through the liquid,
and thus build up a solid nucleus. It will
be observed that this argument does not repose
on the rise in the melting temperature of rock
through pressure, although it is undoubtedly
reinforced thereby.

Hopkins was, I think, the first to adduce arguments
\index{Hopkins on rigidity of earth}%
of weight in favor of the earth's solidity.
He examined the laws of the precession and
nutation of a rigid shell inclosing liquid, and
found that the motion of such a system would
differ to a marked degree from that of the earth.
From this he concluded that the interior of the
earth was not liquid.

Lord Kelvin has pointed out that although
Hopkins's investigation is by no means complete,
yet as he was the first to show that the
motion of the earth as a whole affords indications
of the condition of the interior, an important
share in the discovery of the solidity of the
earth should be assigned to him. Lord Kelvin
then resumed Hopkins's work, and showed that
if the liquid interior of the planet were inclosed
in an unyielding crust, a very slight departure
from perfect sphericity in the shell would render
the motion of the system almost identical with
\PageSep{259}
\index{Height of tide!reduced by elastic yielding of earth}%
that of a globe solid from centre to surface,
although this would not be the case with the
more rapid nutations. A yet more important
deficiency in Hopkins's investigation is that he
\index{Hopkins on rigidity of earth}%
did not consider that, unless the crust were more
rigid than the stiffest steel, it would yield to the
surging of the imprisoned liquid as freely as
india-rubber; and, besides, that if the crust
yielded freely, the precession and nutations of
the whole mass would hardly be distinguishable
from those of a solid globe. Hopkins's argument,
as thus amended by Lord Kelvin, leads
to one of two alternatives: either the globe is
solid throughout, or else the crust yields with
nearly the same freedom to external forces as
though it were liquid.

We have now to show that the latter hypothesis
is negatived by other considerations. The
oceanic tides, as we perceive them, consist in a
motion of the water relatively to the land. Now
if the solid earth were to yield to the tidal forces
with the same freedom as the superjacent sea,
the cause for the relative movement of the sea
would disappear. And if the solid yielded to
some extent, the apparent oceanic tide would be
proportionately diminished. The very existence
of tides in the sea, therefore, proves at least that
the land does not yield with perfect freedom.

Lord Kelvin has shown that the oceanic tides,
on a globe of the same rigidity as that of glass,
\PageSep{260}
would only have an apparent range of two fifths
of those on a perfectly rigid globe; whilst, if
the rigidity was equal to that of steel, the fraction
of diminution would be two thirds. I have
myself extended his argument to the hypothesis
that the earth may be composed of a viscous
material, which yields slowly under the application
of continuous forces, and also to the hypothesis
of a material which shares the properties
of viscosity and rigidity, and have been led to
\index{Rigidity of earth|)}%
analogous conclusions.

The difficulty of the problem of oceanic tides
is so great that we cannot say how high the tides
would be if the earth were absolutely rigid, but
Lord Kelvin is of opinion that they certainly
\index{Kelvin, Lord!rigidity of earth|)}%
would not be twice as great as they are, and he
concludes that the earth possesses a greater average
stiffness than that of glass, although perhaps
not greater than that of steel. It is proper to
add that the validity of this argument depends
principally on the observed height of an inequality
of sea level with a period of a fortnight. This
is one of the partial tides of the third kind, which
I described in \Ref{Chapter}{XI}.\ as practically unimportant,
and did not discuss in detail. The value
of this inequality in the present argument is due
to the fact that it is possible to form a much
closer estimate of its magnitude on a rigid earth
\index{Earth and moon!rigidity of|)}%
than in the case of the semidiurnal and diurnal
tides.
\PageSep{261}
\index{Darwin, G. H.!rigidity of earth}%
\index{Earthquakes!shock perceptible at great distance}%

It may ultimately be possible to derive further
indications concerning the physical condition of
the inside of the earth from the science of seismology.
The tremor of an earthquake has frequently
been observed instrumentally at an enormous
distance from its origin; as, for example,
when the shock of a Japanese earthquake is
perceived in England.

The vibrations which are transmitted through
the earth are of two kinds. The first sort of wave
is one in which the matter through which it passes
is alternately compressed and dilated; it may be
described as a wave of compression. In the
second sort the shape of each minute portion of
the solid is distorted, but the volume remains
unchanged, and it may be called a wave of distortion.
These two vibrations travel at different
speeds, and the compressional wave outpaces
the distortional one. Now the first sign of a
distant earthquake is that the instrumental record
shows a succession of minute tremors.
These are supposed to be due to waves of compression,
and they are succeeded by a much
more strongly marked disturbance, which, however,
lasts only a short time. This second phase
in the instrumental record is supposed to be due
to the wave of distortion.

If the natures of these two disturbances are
correctly ascribed to their respective sources, it
is certain that the matter through which the vibration
\PageSep{262}
\index{Darwin, G. H.!rigidity of earth}%
has passed was solid. For, although a
compressional wave might be transmitted without
much loss of intensity, from a solid to a
liquid and back again to a solid, as would have
to be the case if the interior of the earth is molten,
yet this cannot be true of the distortional
wave. It has been supposed that vibrations due
to earthquakes pass in a straight line through
the earth; if then this could be proved, we
should know with certainty that the earth is
solid, at least far down towards its centre.

Although there are still some---principally
amongst the geologists---who believe in the existence
of liquid matter immediately under the
solid crust of the earth,\footnote
  {See the Rev.~Osmond Fisher's \Title{Physics of the Earth's Crust}.}
\index{Fisher, Osmond, on molten interior of earth}%
yet the arguments which
I have sketched appear to most men of science
conclusive against such belief.

\begin{Authorities}
Mr.~S.~C. Chandler's investigations are published in the ``Astronomical
Journal,'' vol.~11 and following volumes. A summary
is contained in ``Science,'' May~3, 1895.

R.~S. Woodward, \Title{Mechanical Interpretation of the Variations of
\index{Woodward on variation of latitude}%
Latitude}, ``Ast.\ Journ.'' vol.~15, May,~1895.

Simon Newcomb, \Title{On the Dynamics of the Earth's Rotation},
``Monthly Notices of the R.~Astron.\ Soc.,'' vol.~52 (1892),
p.~336.

S.~S. Hough, \Title{The Rotation of an Elastic Spheroid}, ``Philosoph.\
\index{Hough, S. S.!Chandler's nutation}%
Trans.\ of the Royal Society,'' A.~1896, p.~319. He indicates a
slight oversight on the part of Newcomb.

H.~G. van~de Sande Bakhuyzen, \Title{Ueber die Aenderung der Polhoehe},
``Astron.\ Nachrichten,'' No.~3261.
\PageSep{263}

A.~S. Christie, \Title{The Latitude-variation Tide}, ``Phil.\ Soc.\ of
Washington, Bulletin,'' vol.~12 (1895), p.~103.

Lord Kelvin, in Thomson and Tait's ``Natural Philosophy,''
on the Rigidity of the Earth; and ``Popular Lectures,'' vol.~3.

G.~H. Darwin, \Title{Bodily Tides of Viscous and Semi-elastic Spheroids,~\&c.},
``Philosoph.\ Trans.\ of the Royal Society,'' Part.~I.
1879.
\end{Authorities}
\PageSep{264}


%[** TN: Footnote mark handled by \Chapter logic]
\Chapter{XVI}
{Tidal Friction}
\footnotetext{A considerable portion of this and of the succeeding chapter
  appeared as an article in \Title{The Atlantic Monthly} for April,~1898.}

\First{The} fact that the earth, the moon, and the
\index{Friction of tides|(}%
planets are all nearly spherical proves that in
early times they were molten and plastic, and
assumed their present round shape under the
influence of gravitation. When the material of
which any planet is formed was semi-liquid
through heat, its satellites, or at any rate the
sun, must have produced tidal oscillations in the
molten rock, just as the sun and moon now produce
the tides in our oceans.

Molten rock and molten iron are rather sticky
or viscous substances, and any movement which
agitates them must be subject to much friction.
Even water, which is a very good lubricant, is
not entirely free from friction, and so our present
oceanic tides must be influenced by fluid
friction, although to a far less extent than the
molten solid just referred to. Now, all moving
systems which are subject to friction gradually
come to rest. A train will run a long way when
the steam is turned off, but it stops at last, and
\PageSep{265}
a fly-wheel will continue to spin for only a limited
time. This general law renders it certain that
the friction of the tide, whether it consists in the
swaying of molten lava or of an ocean, must be
retarding the rotation of the planet, or at any
rate retarding the motion of the system in some
way.

It is the friction upon its bearings which brings
a fly-wheel to rest; but as the earth has no bearings,
it is not easy to see how the friction of the
tidal wave, whether corporeal or oceanic, can
tend to stop its rate of rotation. The result
must clearly be brought about, in some way, by
the interaction between the moon and the earth.
Action and reaction must be equal and opposite,
and if we are correct in supposing that the friction
of the tides is retarding the earth's rotation,
there must be a reaction upon the moon which
must tend to hurry her onwards. To give a
homely illustration of the effects of reaction, I
may recall to mind how a man riding a high
bicycle, on applying the brake too suddenly, was
thrown over the handles. The desired action
was to stop the front wheel, but this could not
be done without the reaction on the rider, which
sometimes led to unpleasant consequences.

The general conclusion as to the action and
reaction due to tidal friction is of so vague a
character that it is desirable to consider in detail
how they operate.
\PageSep{266}

The circle in \fig{36} is supposed to represent
the undisturbed shape of the planet, which rotates
in the direction of the curved arrow. A portion
of the orbit of the satellite is indicated by part
\Figure{36}{Frictionally retarded Tide}{png}
of a circle, and the direction of its motion is
shown by an arrow. I will first suppose that the
water lying on the planet, or the molten rock of
which it is formed, is a perfect lubricant devoid
of friction, and that at the moment represented
in the figure the satellite is at~$M'$. The fluid will
then be distorted by the tidal force until it assumes
the egg-like shape marked by the ellipse,
projecting on both sides beyond the circle. It
will, however, be well to observe that if this figure
represents an ocean, it must be a very deep
one, far deeper than those which actually exist
on the earth; for we have seen that it is only in
deep oceans that the high water stands underneath
and opposite to the moon; whereas in
shallow water it is low water where we should
\PageSep{267}
naturally expect high water. Accepting the hypothesis
that the high tide is opposite to the
moon, and supposing that the liquid is devoid of
friction, the long axis of the egg is always directed
straight towards the satellite~$M'$, and the
liquid maintains a continuous rhythmical movement,
so that as the planet rotates and the satellite
revolves, it always maintains the same shape
and attitude towards the satellite.

But when, as in reality, the liquid is subject to
friction, it gets belated in its rhythmical rise and
fall, and the protuberance is carried onward by
the rotation of the planet beyond its proper
place. In order to make the same figure serve
for this condition, I set the satellite backward to~$M$;
for this amounts to just the same thing, and
is less confusing than redrawing the protuberance
in its more advanced position. The planet
then constantly maintains this shape and attitude
with regard to the satellite, and the interaction
between the two will be the same as though the
planet were solid, but continually altering its
shape.

We have now to examine what effects must
follow from the attraction of the satellite on an
egg-shaped planet, when the two constantly
maintain the same attitude relatively to each
other. It will make the matter somewhat easier
of comprehension if we replace the tidal protuberances
by two particles of equal masses, one at~$P$,
\PageSep{268}
\index{Earth and moon!rotation retarded by tidal friction}%
\index{Retardation of earth's rotation}%
and the other at~$P'$. If the masses of these
particles be properly chosen, so as to represent
the amount of matter in the protuberances, the
proposed change will make no material difference
in the action.

The gravitational attraction of the satellite is
greater on bodies which are near than on those
which are far, and accordingly it attracts the
particle~$P$ more strongly than the particle~$P'$. It
is obvious from the figure that the attraction on~$P$
must tend to stop the planet's rotation, whilst
\index{Rotation!retarded by tidal friction}%
that on~$P'$ must tend to accelerate it. If a man
pushes equally on the two pedals of a bicycle,
the crank has no tendency to turn, and besides
there are dead points in the revolution where
pushing and pulling have no effect. So also in
the astronomical problem, if the two attractions
were exactly equal, or if the protuberances were
at a dead point, there would be no resultant effect
on the rotation of the planet. But it is
obvious that here the retarding pull is stronger
than the accelerating one, and that the set of the
protuberances is such that we have passed the
dead point. It follows from this that the primary
effect of fluid friction is to throw the tidal
protuberance forward, and the secondary effect
is to retard the planet's rotation.

It has been already remarked that this figure is
drawn so as to apply only to the case of corporeal
tides or to those of a very deep ocean. If
\PageSep{269}
\index{Moon and earth!retardation of motion by tidal friction}%
the ocean were shallow and frictionless, it would
be low water under and opposite to the satellite.
If then the effect of friction were still to throw
the protuberances forward, the rotation of the
planet would be accelerated instead of retarded.
But in fact the effect of fluid friction in a shallow
ocean is to throw the protuberances backward,
and a similar figure, drawn to illustrate such a
displacement of the tide, would at once make it
clear that here also tidal friction will lead to the
retardation of the planet's rotation. Henceforth
then I shall confine myself to the case illustrated
by~\fig{36}.

Action and reaction are equal and opposite,
and if the satellite pulls at the protuberances,
they pull in return on the satellite. The figure
shows that the attraction of the protuberance~$P$
tends in some measure to hurry the satellite onward
in its orbit, whilst that of~$P'$ tends to retard
it. But the attraction of~$P$ is stronger than that
of~$P'$, and therefore the resultant of the two is a
force tending to carry the satellite forward in the
direction of the arrow.

If a stone be whirled at the end of an elastic
string, a retarding force, such as the friction of
the air, will cause the string to shorten, and an
accelerating force will make it lengthen. In the
same way the satellite, being as it were tied to
the planet by the attraction of gravitation, when
subjected to an onward force, recedes from the
\PageSep{270}
\index{Moon and earth!retardation of motion by tidal friction}%
planet, and moves in a spiral curve at ever increasing
distances. The time occupied by the
satellite in making a circuit round the planet is
prolonged, and this lengthening of the periodic
time is not merely due to the lengthening of the
arc described by it, but also to an actual retardation
of its velocity. It appears paradoxical that
the effect of an accelerating force should be a
retardation, but a consideration of the mode in
which the force operates will remove the paradox.
The effect of the tangential accelerating
force on the satellite is to make it describe an
increasing spiral curve. Now if the reader will
draw an exaggerated figure to illustrate part of
such a spiral orbit, he will perceive that the central
force, acting directly towards the planet,
must operate in some measure to retard the velocity
of the satellite. The central force is very
great compared with the tangential force due to
the tidal friction, and therefore a very small
fraction of the central force may be greater than
the tangential force. Although in a very slowly
increasing spiral the fraction of the central force
productive of retardation is very small, yet it is
found to be greater than the tangential accelerating
force, and thus the resultant effect is a
retardation of the satellite's velocity.

The converse case where a retarding force results
in increase of velocity will perhaps be more
intelligible, as being more familiar. A meteorite,
\PageSep{271}
rushing through the earth's atmosphere, moves
faster and faster, because it gains more speed
from the attraction of gravity than it loses by the
friction of the air.

Now let us apply these ideas to the case of the
earth and the moon. A man standing on the
planet, as it rotates, is carried past places where
the fluid is deeper and shallower alternately; at
the deep places he says that it is high tide, and
at the shallow places that it is low tide. In \fig{36}
it is high tide when the observer is carried
past~$P$. Now it was pointed out that when there
is no fluid friction we must put the moon at~$M'$,
but when there is friction she must be at~$M$.
Accordingly, if there is no friction it is high tide
when the moon is over the observer's head, but
when there is friction the moon has passed his
zenith before he reaches high tide. Hence he
would remark that fluid friction retards the time
of high tide.

A day is the name for the time in which the
earth rotates once, and a month for the time in
which the moon revolves once. Then since tidal
friction retards the earth's rotation and the
moon's revolution, we may state that both the
day and the month are being lengthened, and
that these results follow from the retardation of
the time of high tide.

It must also be noted that the spiral in which
the moon moves is an increasing one, so that her
\PageSep{272}
\index{Assyrian records of eclipses}%
distance from the earth also increases. These
are absolutely certain and inevitable results of
the mechanical interaction of the two bodies.

At the present time the rates of increase of
the day and month are excessively small, so that
it has not been found possible to determine them
with any approach to accuracy. It may be well
to notice in passing that if the rate of either increase
of element were determinable, that of the
other would be deducible by calculation.

The extreme slowness of the changes within
historical times is established by the early records
in Greek and Assyrian history of eclipses of the
\index{Eclipses, ancient, and earth's rotation}%
\index{Greek!records of ancient eclipses}%
sun, which occurred on certain days and in certain
places. Notwithstanding the changes in the
calendar, it is possible to identify the day according
to our modern reckoning, and the identification
of the place presents no difficulty.
Astronomy affords the means of calculating the
exact time and place of the occurrence of an
eclipse even three thousand years ago, on the
supposition that the earth spun at the same rate
then as now, and that the complex laws governing
the moon's motion are unchanged.

The particular eclipse referred to in history is
known, but any considerable change in the
earth's rotation and in the moon's position would
have shifted the position of visibility on the
earth from the situation to which modern computation
would assign it. Most astronomical
\PageSep{273}
observations would be worthless if the exact time
of the occurrence were uncertain, but in the
case of eclipses the place of observation affords
\index{Eclipses, ancient, and earth's rotation}%
just that element of precision which is otherwise
wanting. As, then, the situations of the ancient
eclipses agree fairly well with modern computations,
we are sure that there has been no great
change within the last three thousand years,
either in the earth's rotation or in the moon's
motion. There is, however, a small outstanding
discrepancy which indicates that there has been
some change. But the exact amount of change
involves elements of uncertainty, because our
knowledge of the laws of the moon's motion is
not yet quite accurate enough for the absolutely
perfect calculation of eclipses which occurred
many centuries ago. In this way, it is known
that within historical times the retardation of the
earth's rotation and the recession of the moon
have been at any rate very slow.

It does not, however, follow from this that
the changes have always been equally slow; indeed,
it may be shown that the efficiency of tidal
friction increases with great rapidity as we bring
the tide-generating satellite nearer to the planet.

It has been shown in \Ref{Chapter}{V}.\ that the intensity
of tide-generating force varies as the inverse
cube of the distance between the moon and
the earth, so that if the moon's distance were
reduced successively to $\frac{1}{2}$,~$\frac{1}{3}$,~$\frac{1}{4}$, of its original distance,
\PageSep{274}
the force and the tide generated by it
would be multiplied $8$,~$27$,~$64$ times. But the
efficiency of tidal friction increases far more rapidly
than this, because not only is the tide itself
augmented, but also the attraction of the moon.
In order to see how these two factors will coöperate,
let us begin by supposing that the
height of the tide remains unaffected by the approach
or retrogression of the moon. Then the
same line of argument, which led to the conclusion
that tide-generating force varies inversely as
the cube of the distance, shows that the action
of the moon on protuberances of definite magnitude
must also vary inversely as the cube of the
distance. But the height of the tide is not in
fact a fixed quantity, but varies inversely as the
cube of the distance, so that when account is
taken both of the augmentation of the tide and
of the increased attraction of the moon, it follows
that the tidal retardation of the earth's rotation
must vary as the inverse sixth power of
the distance. Now since the sixth power of~$2$ is~$64$,
the lunar tidal friction, with the moon at
half her present distance, would be $64$~times as
efficient as at present. Similarly, if her distance
were diminished to a third and a quarter of what
it is, the tidal friction would act with $729$ and
$4,096$~times its present strength. Thus, although
the action may be insensibly slow now, it must
have gone on with much greater rapidity when
the moon was nearer to us.
\PageSep{275}

There are many problems in which it would
be very difficult to follow the changes according
to the times of their occurrence, but where it is
possible to banish time from consideration, and
to trace the changes themselves, in due order,
without reference to time. In the sphere of
common life, we know the succession of stations
which a train must pass between London and
Edinburgh, although we may have no time-table.
This is the case with our astronomical
problem; for although we have no time-table,
yet the sequence of the changes in the system
can be traced accurately.

Let us then banish time, and look forward to
the ultimate outcome of the tidal interaction of
the moon and earth. The day and the month
\index{Day, change in length of, under tidal friction}%
\index{Month, change in, under tidal friction|(}%
are lengthening at relative rates which are calculable,
although the absolute rates in time are
unknown. It will suffice for a general comprehension
of the problem to know that the present
rate of increase of the day is much more rapid
than that of the month, and that this will hold
good in the future. Thus, the number of rotations
of the earth in the interval comprised in
one revolution of the moon diminishes; or, in
other words, the number of days in the month
diminishes, although the month itself is longer
than at present. For example, when the day
shall be equal in length to two of our actual
days, the month may be as long as thirty-seven
\PageSep{276}
of our days, and then the earth will spin round
only about eighteen times in the month.

This gradual change in the day and month
\index{Day, change in length of, under tidal friction}%
proceeds continuously until the duration of a
rotation of the earth is prolonged to fifty-five of
our present days. At the same time the month,
or the time of revolution of the moon round the
earth, will also occupy fifty-five of our days.
Since the month here means the period of the
return of the moon to the same place among the
stars, and since the day is to be estimated in
the same way, the moon must then always face
the same part of the earth's surface, and the
two bodies must move as though they were
united by a bar. The outcome of the lunar
tidal friction will therefore be that the moon
and the earth go round as though locked together,
in a period of fifty-five of our present
days, with the day and the month identical in
length.

Now looking backward in time, we find the
day and the month shortening, but the day
changing more rapidly than the month. The
earth was therefore able to complete more revolutions
in the month, although that month was
itself shorter than it is now. We get back in
fact to a time when there were $29$~rotations of
the earth in a month instead of~$27\frac{1}{3}$, as at present.
This epoch is a sort of crisis in the history
of the moon and the earth, for it may be proved
\PageSep{277}
that there never could have been more than $29$~days
in the month. Earlier than this epoch, the
days were fewer than~$29$, and later fewer also.
Although measured in years, this epoch in the
earth's history must be very remote, yet when we
contemplate the whole series of changes it must
be considered as a comparatively recent event.
In a sense, indeed, we may be said to have passed
recently through the middle stage of our history.

Now, pursuing the series of changes further
back than the epoch when there was the maximum
number of days in the month, we find the
earth still rotating faster and faster, and the
moon drawing nearer and nearer to the earth,
and revolving in shorter and shorter periods.
But a change has now supervened, so that the
rate at which the month is shortening is more
rapid than the rate of change in the day. Consequently,
the moon now gains, as it were, on
the earth, which cannot get round so frequently
in the month as it did before. In other words,
the number of days in the month declines from
the maximum of~$29$, and is finally reduced to
one. When there is only one day in the month,
\index{Month, change in, under tidal friction|)}%
the earth and the moon go round at the same
rate, so that the moon always looks at the same
side of the earth, and so far as concerns the
motion they might be fastened together by a
rigid bar.

This is the same conclusion at which we arrived
\PageSep{278}
with respect to the remote future. But
the two cases differ widely; for whereas in the
future the period of the common rotation will
be $55$~of our present days, in the past we find
the two bodies going round each other in between
three and five of our present hours. A
satellite revolving round the earth in so short a
period must almost touch the earth's surface.
The system is therefore traced until the moon
nearly touches the earth, and the two go round
each other like a single solid body in about three
to five hours.

The series of changes has been traced forward
and backward from the present time, but it will
make the whole process more intelligible, and
the opportunity will be afforded for certain further
considerations, if I sketch the history again
\index{History!of earth and moon|(}%
in the form of a continuous narrative.

Let us imagine a planet attended by a satellite
which revolves so as nearly to touch its surface,
and continuously to face the same side of the
planet's surface. If now, for some reason, the
satellite's month comes to differ very slightly
from the planet's day, the satellite will no longer
continuously face the same side of the planet,
but will pass over every part of the planet's
equator in turn. This is the condition necessary
for the generation of tidal oscillations in the
planet, and as the molten lava, of which we
suppose it to be formed, is a sticky or viscous
\PageSep{279}
fluid, the tidal oscillations must be subject to
friction. Tidal friction will then begin to do its
work, but the result will be very different according
as the satellite revolves a little faster or
a little slower than the planet. If it revolves a
little faster, so that the month is shorter than
the day, we have a condition not contemplated
in~\fig{36}; it is easy to see, however, that as
the satellite is always leaving the planet behind
it, the apex of the trial protuberance must be
directed to a point behind the satellite in its
orbit. In this case the rotation of the planet
must be \DPtypo{acclerated}{accelerated} by the tidal friction, and the
satellite will be drawn inward towards the planet,
into which it must ultimately fall. In the application
of this theory to the earth and moon, it
is obvious that the very existence of the moon
negatives the hypothesis that the initial month
was even infinitesimally shorter than the day.
We must then suppose that the moon revolved
a little more slowly than the earth rotated. In
this case the tidal friction would retard the
earth's rotation, and force the moon to recede
from the earth, and so perform her orbit more
slowly. Accordingly, the primitive day and the
primitive month lengthen, but the month increases
much more rapidly than the day, so that
the number of days in a month increases. This
proceeds until that number reaches a maximum,
which in the case of our planet is about~$29$.
\PageSep{280}
\index{Instability!nature of dynamical, and initial of moon's motion|(}%
\index{Stability!nature of dynamical}%

After the epoch of the maximum number of
days in the month, the rate of change in the
length of the day becomes less rapid than that
in the length of the month; and although both
periods increase, the number of days in the
month begins to diminish. The series of
changes then proceeds until the two periods
come again to an identity, when we have the
earth and the moon as they were at the beginning,
revolving in the same period, with the
moon always facing the same side of the earth.
But in her final condition the moon will be a
long way off the earth instead of being quite
close to it.

Although the initial and final states resemble
each other, yet they differ in one respect which
is of much importance, for in the initial condition
the motion is unstable, whilst finally it is
stable. The meaning of this is, that if the
moon were even infinitesimally disturbed from
the initial mode of motion, she would necessarily
either fall into the planet, or recede therefrom,
and it would be impossible for her to continue
to move in that neighborhood. She is unstable
in the same sense in which an egg when balanced
on its point is unstable; the smallest mote
of dust will upset it, and practically it cannot
stay in that position. But the final condition
resembles the case of the egg lying on its side,
which only rocks a little when we disturb it.
\PageSep{281}
\index{Stability!nature of dynamical}%
So if the moon were slightly disturbed from her
final condition, she would continue to describe
very nearly the same path round the earth, and
would not assume some entirely new form of
orbit.

It is by methods of rigorous argument that
the moon is traced back to the initial unstable
condition when she revolved close to the earth.
But the argument here breaks down, and calculation
is incompetent to tell us what occurred
before, and how she attained that unstable mode
of motion. If we were to find a pendulum
swinging in a room, where we knew that it had
been undisturbed for a long time, we might, by
observing its velocity and allowing for the resistance
of the air, conclude that at some previous
moment it had just been upside down, but
calculation could never tell us how it had
reached that position. We should of course
feel confident that some one had started it.
Now a similar hiatus must occur in the history
of the moon, but it is not so easy to supply the
missing episode. It is indeed only possible to
speculate as to the preceding history.

But there is some basis for our speculation;
for I say that if a planet, such as the earth,
made each rotation in three hours, it would very
nearly fly to pieces. The attraction of gravity
would be barely strong enough to hold it together,
just as the cohesive strength of iron is
\PageSep{282}
\index{Forced oscillation!due to solar tide, possibly related to birth of moon|(}%
\index{Moon and earth!origin of}%
insufficient to hold a fly-wheel together if it is
spun too fast. There is, of course, an important
distinction between the case of the ruptured
fly-wheel and the supposed break-up of the
earth; for when a fly-wheel breaks, the pieces
are hurled apart as soon as the force of cohesion
fails, whereas when a planet breaks up through
too rapid rotation, gravity must continue to
hold the pieces together after they have ceased
to form parts of a single body.

Hence we have grounds for conjecturing that
the moon is composed of fragments of the primitive
planet which we now call the earth, which
detached themselves when the planet spun very
swiftly, and afterwards became consolidated. It
surpasses the power of mathematical calculation
to trace the details of the process of this rupture
and subsequent consolidation, but we can hardly
doubt that the system would pass through a
period of turbulence, before order was reëstablished
in the formation of a satellite.

I have said above that rapid rotation was probably
the cause of the birth of the moon, but it
may perhaps not have been brought about by
this cause alone. There are certain considerations
which make it difficult to ascertain the
initial common period of revolution of the moon
and the earth with accuracy; it may lie between
three and five hours. Now I think that such
a speed might not quite suffice to cause the
\index{Instability!nature of dynamical, and initial of moon's motion|)}%
\PageSep{283}
\index{Moon and earth!origin of}%
primitive planet to break up. In \Ref{Chapter}{XVIII}.\
we shall consider in greater detail the conditions
under which a rotating mass of liquid would
rupture, but for the present it may suffice to say
that, where the rotating body is heterogeneous in
density, like the earth, the exact determination
of the limiting speed of rotation is not possible.
Is there, then, any other cause which might coöperate
with rapid rotation in producing rupture?
I think there is such a cause, and, although
we are here dealing with guesswork, I
will hazard the suggestion.

The primitive planet, before the birth of the
moon, was rotating rapidly with reference to the
sun, and it must therefore have been agitated by
solar tides. In \Ref{Chapter}{IX}.\ it was pointed out
that there is a general dynamical law which enables
us to foresee the magnitude of the oscillations
of a system under the action of external
forces. That law depended on the natural or
free period of the oscillation of the system when
disturbed and left to itself, free from the intervention
of external forces. We saw that the
more nearly the periodic forces were timed to
agree with the free period, the greater was the
amplitude of the oscillations of the system. Now
it is easy to calculate the natural or free period
of the oscillation of a homogeneous liquid globe
of the same density as the earth, namely, five
and a half times as heavy as water; the period
\PageSep{284}
\index{Sun!possible influence of, in assisting birth of moon}%
is found to be $1$~hour $34$~minutes. The heterogeneity
of the earth introduces a complication of
which we cannot take account, but it seems likely
that the period would be from $1\frac{1}{2}$ to $2$~hours.
The period of the solar semidiurnal tide is half a
\index{Solar!possible effect of tide in assisting birth of moon}%
day, and if the day were from $3$ to $4$ of our present
hours the forced period of the tide would
be in close agreement with the free period of
oscillation.

May we not then conjecture that as the rotation
of the primitive earth was gradually reduced
by solar tidal friction, the period of the solar tide
was brought into closer and closer agreement
with the free period, and that consequently the
solar tide increased more and more in height?
In this case the oscillation might at length become
so violent that, in coöperation with the
rapid rotation, it shook the planet to pieces, and
that huge fragments were detached which ultimately
became our moon.

There is nothing to tell us whether this theory
affords the true explanation of the birth of the
moon, and I say that it is only a wild speculation,
incapable of verification.

But the truth or falsity of this speculation
does not militate against the acceptance of the
general theory of tidal friction, which, standing
on the firm basis of mechanical necessity, throws
much light on the history of the earth and the
moon, and correlates the lengths of our present
day and month.
\index{Forced oscillation!due to solar tide, possibly related to birth of moon|)}%
\PageSep{285}
\index{Sun!possible influence of, in assisting birth of moon}%

I have said above that the sequence of events
has been stated without reference to the scale of
time. It is, however, of the utmost importance
\index{Time!requisite for evolution of moon}%
to gain some idea of the time requisite for all the
changes in the system. If millions of millions
of years were necessary, the theory would have
to be rejected, because it is known from other
lines of argument that there is not an unlimited
bank of time on which to draw. The uncertainty
as to the duration of the solar system is
\index{Solar!possible effect of tide in assisting birth of moon}%
wide, yet we are sure that it has not existed for
an almost infinite past.

Now, although the actual time scale is indeterminate,
it is possible to find the minimum time
adequate for the transformation of the moon's
orbit from its supposed initial condition to its
present shape. It may be proved, in fact, that
if tidal friction always operated under the conditions
most favorable for producing rapid change,
the sequence of events from the beginning until
to-day would have occupied a period of between
$50$ and $60$~millions of years. The actual period,
of course, must have been much greater. Various
lines of argument as to the age of the solar
system have led to results which differ widely
among themselves, yet I cannot think that the
applicability of the theory is negatived by the
magnitude of the period demanded. It may be
that science will have to reject the theory in its
full extent, but it seems unlikely that the ultimate
\PageSep{286}
\index{Moon and earth!rotation annulled by tidal friction and present libration}%
verdict will be adverse to the preponderating
influence of the tide in the evolution of our
planet.

\TB

If this history be true of the earth and moon,
\index{History!of earth and moon|)}%
it should throw light on many peculiarities of the
solar system. In the first place, a corresponding
series of changes must have taken place in the
moon herself. Once on a time the moon must
have been molten, and the great extinct volcanoes
revealed by the telescope are evidences of
her primitive heat. The molten mass must have
been semi-fluid, and the earth must have raised
in it enormous tides of molten lava. Doubtless
the moon once rotated rapidly on her axis, and
the frictional resistance to her tides must have
impeded her rotation. This cause must have
\index{Rotation!of moon annulled by tidal friction}%
added to the moon's recession from the earth,
but as the moon's mass is only an eightieth part
of that of the earth, the effect on the moon's
orbit must have been small. The only point to
which we need now pay attention is that the
rate of her rotation was reduced. She rotated
then more and more slowly until the tide solidified,
and thenceforward and to the present day
she has shown the same face to the earth. Kant
\index{Kant!rotation of moon}%
and Laplace in the last century, and Helmholtz
\index{Helmholtz!on rotation of the moon}%
\index{Laplace!on rotation of moon}%
in recent times, have adduced this as the explanation
of the fact that the moon always shows
us the same face. Our theory, then, receives a
\PageSep{287}
striking confirmation from the moon; for, having
ceased to rotate relatively to us, she has actually
advanced to that condition which may be
foreseen as the fate of the earth.

The earth tide in the moon is now solidified
so that the moon's equator is not quite circular,
and the longer axis is directed towards the earth.
Laplace has considered the action of the earth
\index{Laplace!on rotation of moon}%
on this solidified tide, and has shown that the
moon must rock a little as she moves round the
earth. In consequence of this rocking motion or
libration of the moon, and also of the fact that
her orbit is elliptic, we are able to see just a little
more than half of the moon's surface.

\TB

Thus far I have referred in only one passage
to the influence of solar tides, but these are of
considerable importance, being large enough to
cause the conspicuous phenomena of spring and
neap tides. Now, whilst the moon is retarding
the earth's rotation, the sun is doing so also.
But these solar tides react only on the earth's
motion round the sun, leaving the moon's motion
round the earth unaffected. It might perhaps
be expected that parallel changes in the
earth's orbit would have proceeded step by step,
and that the earth might be traced to an origin
close to the sun. The earth's mass is less than $\frac{1}{300,000}$~part
of the sun's, and the reactive effect on the
earth's orbit round the sun is altogether negligible.
\PageSep{288}
It is improbable, in fact, that the year is,
from this cause at any rate, longer by more than
a few seconds than it was at the very birth of
the solar system.

Although the solar tides cannot have had any
perceptible influence upon the earth's movement
in its orbit, they will have affected the rotation
of the earth to a considerable extent. Let us
imagine ourselves transported to the indefinite
future, when the moon's orbital period and the
earth's diurnal period shall both be prolonged to
$55$~of our present days. The lunar tide in the
earth will then be unchanging, just as the earth
tide in the moon is now fixed; but the earth will
be rotating with reference to the sun, and, if
there are still oceans on the earth, her rotation
will be subject to retardation in consequence of
the solar tidal friction. The day will then become
longer than the month, whilst the moon
will at first continue to revolve round the earth
in $55$~days. Lunar tides will now be again generated,
but as the motion of the earth will be
very slow relatively to the moon, the oscillations
will also be very slow, and subject to little friction.
But that friction will act in opposition to
the solar tides, and the earth's rotation will to
some slight extent be assisted by the moon.
The moon herself will slowly approach the earth,
moving with a shorter period, and must ultimately
fall back into the earth. We know that
\PageSep{289}
there are neither oceans nor atmosphere on the
moon, but if there were such, the moon would
have been subject to solar tidal friction, and
would now be rotating slower than she revolves.

%[** TN: Not hyperlinking chapter reference]
\begin{Authorities}
See the end of Chapter~XVII.
\end{Authorities}
\PageSep{290}


\Chapter[Tidal Friction]{XVII}
{Tidal Friction (Continued)}

\First{It} has been shown in the last chapter that the
prolongation of the day and of the month under
the influence of tidal friction takes place in such
a manner that the month will ultimately become
longer than the day. Until recent times no case
had been observed in the solar system in which
a satellite revolved more rapidly than its planet
rotated, and this might have been plausibly adduced
as a reason for rejecting the actual efficiency
of solar tidal friction in the process of
celestial evolution. At length however, in~1877,
Professor Asaph Hall discovered in the system
\index{Hall, Asaph, discovery of Martian satellites|(}%
of the planet Mars a case of the kind of motion
\index{Mars!discovery of satellites|(}%
which we foresee as the future fate of the moon
and earth, for he found that the planet was attended
by two satellites, the nearer of which has
\index{Satellites!discovery of those of Mars|(}%
a month shorter than the planet's day. He gives
an interesting account of what had been conjectured,
partly in jest and partly in earnest, as to
the existence of satellites attending that planet.
This foreshadowing of future discoveries is so
curious that I quote the following passage from
Professor Hall's paper. He writes:---
\PageSep{291}

``Since the discovery of the satellites of Mars,
the remarkable statements of Dean Swift and
Voltaire concerning the satellites of this planet,
and the arguments of Dr.~Thomas Dick and
others for the existence of such bodies, have attracted
so much attention, that a brief account
of the writings on this subject may be interesting.

``The following letter of Kepler was written
\index{Kepler!argument respecting Martian satellites}%
to one of his friends soon after the discovery by
Galileo in~1610 of the four satellites of Jupiter,
\index{Galileo!discovery of Jupiter's satellites}%
and when doubts had been expressed as to the
reality of this discovery. The news of the discovery
was communicated to him by his friend
Wachenfels; and Kepler says:---

``\,`Such a fit of wonder seized me at a report
which seemed to be so very absurd, and I was
thrown into such agitation at seeing an old dispute
between us decided in this way, that between
his joy, my coloring, and the laughter of
both, confounded as we were by such a novelty,
we were hardly capable, he of speaking, or I of
listening. On our parting, I immediately began
to think how there could be any addition to the
number of the planets without overturning my
``Cosmographic Mystery,'' according to which
Euclid's five regular solids do not allow more
than six planets round the sun\dots. I am so
far from disbelieving the existence of the four
circumjovial planets, that I long for a telescope,
to anticipate you, if possible, in discovering \emph{two}
\PageSep{292}
\index{Kepler!argument respecting Martian satellites}%
round Mars, as the proportion seems to require,
\emph{six} or \emph{eight} round Saturn, and perhaps \emph{one} each
round Mercury and Venus.'

``Dean Swift's statement concerning the satellites
\index{Swift, satire on mathematicians|(}%
of Mars is in his famous satire, `The
Travels of Mr.~Lemuel Gulliver.' After describing
\index{Gulliver@\Title{Gulliver's Travels}, satire on mathematics|(}%
his arrival in Laputa, and the devotion
of the Laputians to mathematics and music,
Gulliver says:---

``\,`The knowledge I had in mathematics gave
me great assistance in acquiring their phraseology,
which depended much upon that science,
and music; and in the latter I was not unskilled.
Their ideas were perpetually conversant in lines
and figures. If they would, for example, praise
the beauty of a woman, or of any other animal,
they describe it by rhombs, circles, parallelograms,
ellipses, and other geometrical terms, or
by words of art drawn from music, needless here
to repeat\dots. And although they are dexterous
enough upon a piece of paper, in the management
of the rule, the pencil, and the divider,
yet in the common actions and the behavior of
life, I have not seen a more clumsy, awkward,
and unhandy people, nor so slow and perplexed
in their conceptions upon all subjects, except
those of mathematics and music. They are very
bad reasoners, and vehemently given to opposition,
unless when they happen to be of the right
opinion, which is seldom their case\dots. These
\PageSep{293}
people are under continual disquietudes, never
enjoying a minute's peace of mind; and their
disturbances proceed from causes which very
little affect the rest of mortals. Their apprehensions
arise from several changes they dread
in the celestial bodies. For instance, that the
earth, by the continual approaches of the sun
towards it, must, in the course of time, be absorbed,
or swallowed up. That the face of the
sun will, by degrees, be encrusted with its own
effluvia, and give no more light to the world.
That the earth very narrowly escaped a brush
from the tail of the last comet, which would
have infallibly reduced it to ashes; and that the
next, which they have calculated for one-and-thirty
years hence, will probably destroy us.
For if, in its perihelion, it should approach
within a certain degree of the sun (as by their
calculations they have reason to dread,) it will
receive a degree of heat ten thousand times
more intense than that of red-hot glowing iron;
and, in its absence from the sun, carry a blazing
tail ten hundred thousand and fourteen miles
long; through which, if the earth should pass
at the distance of one hundred thousand miles
from the nucleus, or main body of the comet, it
must, in its passage, be set on fire, and reduced
to ashes. That the sun, daily spending its rays,
without any nutriment to supply them, will at
last be wholly consumed and annihilated; which
\PageSep{294}
must be attended with the destruction of this
earth, and of all the planets that receive their
light from it.

``\,`They are so perpetually alarmed with the
apprehension of these, and the like impending
dangers, that they can neither sleep quietly in
their beds, nor have any relish for the common
pleasures and amusements of life. When they
meet an acquaintance in the morning, the first
question is about the sun's health, how he looked
at his setting and rising, and what hopes they had
to avoid the stroke of the approaching comet\dots.
They spend the greatest part of their lives
in observing the celestial bodies, which they do
by the assistance of glasses, far excelling ours in
goodness. For although their largest telescopes
do not exceed three feet, they magnify much
more than those of a hundred with us, and show
the stars with greater clearness. This advantage
has enabled them to extend their discoveries
much further than our astronomers in Europe;
for they have made a catalogue of ten thousand
fixed stars, whereas the largest of ours do not
contain above one-third of that number\dots.
They have likewise discovered two lesser stars,
or satellites, which revolve about Mars; whereof
the innermost is distant from the centre of the
primary planet exactly three of his diameters,
and the outermost, five; the former revolves in
the space of ten hours, and the latter in twenty-one
\PageSep{295}
and a half; so that the squares of their
periodical times are very near in the same proportion
with the cubes of their distance from
the centre of Mars; which evidently shows them
to be governed by the same law of gravitation
that influences the other heavenly bodies.'

``The reference which Voltaire makes to the
\index{Voltaire, satire on mathematicians, and Martian satellites}%
moons of Mars is in his `Micromegas, Histoire
Philosophique.' Micromegas was an inhabitant
of Sirius, who, having written a book which a
suspicious old man thought smelt of heresy, left
Sirius and visited our solar system. Voltaire
says:---

``\,`Mais revenons à nos voyageurs. En sortant
de Jupiter, ils traversèrent un espace d'environ
cent millions de lieues, et ils côtoyèrent
la planète de Mars, qui, comme on sait, est cinq
fois plus petite que noire petit globe; ils virent
deux lunes qui servent à cette planète, et qui ont
échappé aux regards de nos astronomes. Je sais
bien que le père \emph{Castel} écrira, et même plaisamment,
\index{Castel, Father, ridiculed by Voltaire}%
contre l'existence de ces deux lunes; mais
je m'en rapporte à ceux qui raisonnent par analogie.
Ces bons philosophes-là savent combien il
serait difficile que Mars, qui est si loin du soleil,
se passât à moins de deux lunes.'

``The argument by analogy for the existence
of a satellite of Mars was revived by writers like
Dr.~Thomas Dick, Dr.~Lardner, and others. In
\index{Dick, argument as to Martian satellites}%
\index{Lardner, possibility of Martian satellites}%
addition to what may be called the analogies of
\index{Gulliver@\Title{Gulliver's Travels}, satire on mathematics|)}%
\index{Swift, satire on mathematicians|)}%
\PageSep{296}
astronomy, these writers appear to rest on the
idea that a beneficent Creator would not place
a planet so far from the sun as Mars without
giving it a satellite. This kind of argument has
passed into some of our handbooks of astronomy,
and is stated as follows by Mr.~Chambers
\index{Chambers on possible existence of Martian satellites}%
in his excellent book on `Descriptive Astronomy,'
2d~edition, p.~89, published in~1867:---

``\,`As far as we know, Mars possesses no satellite,
though analogy does not forbid, but rather,
on the contrary, infers the existence of one; and
its never having been seen, in this case at least,
proves nothing. The second satellite of Jupiter
is only $\frac{1}{43}$~of the diameter of the primary, and
a satellite $\frac{1}{43}$~of the diameter of Mars would
be less than $100$~miles in diameter, and therefore
of a size barely within the reach of our largest
telescopes, allowing nothing for its possibly close
proximity to the planet. The fact that one of
the satellites of Saturn was only discovered a
few years ago renders the discovery of a satellite
of Mars by no means so great an improbability
as might be imagined.'

``Swift seems to have had a hearty contempt
for mathematicians and astronomers, which he
has expressed in his description of the inhabitants
of Laputa. Voltaire shared this contempt,
\index{Voltaire, satire on mathematicians, and Martian satellites}%
and delighted in making fun of the philosophers
whom Frederick the Great collected at Berlin.
The `père Castel' may have been le~père Louis
\index{Castel, Father, ridiculed by Voltaire}%
\PageSep{297}
Castel, who published books on physics and
mathematics at Paris in 1743 and~1758. The
probable origin of these speculations about the
moons of Mars was, I think, Kepler's analogies.
Astronomers failing to verify these, an opportunity
was afforded to satirists like Swift and
Voltaire to ridicule such arguments.''\footnote
  {\Title{Observations and Orbits of the Satellites of Mars}, by Asaph
  Hall. Washington, Government Printing Office, 1878.}

As I have already said, these prognostications
were at length verified by Professor Asaph Hall
in the discovery of two satellites, which he named
Phobos and Deimos---Fear and Panic, the dogs
\index{Deimos, a satellite of Mars}%
\index{Phobos, a satellite of Mars}%
of war. The period of Deimos is about $30$~hours,
and that of Phobos somewhat less than $8$~hours,
whilst the Martian day is of nearly the same
length as our own. The month of the inner
minute satellite is thus less than a third of the
planet's day; it rises to the Martians in the west,
and passes through all its phases in a few hours;
sometimes it must even rise twice in a single
Martian night. As we here find an illustration
of the condition foreseen for the earth and moon,
it seems legitimate to suppose that solar tidal
friction has retarded the planet's rotation until it
has become slower than the revolution of one of
the satellites. It would seem as if the ultimate
fate of Phobos will be absorption in the planet.

Several of the satellites of Jupiter and of Saturn
present faint inequalities of coloring, and
\PageSep{298}
\index{Jupiter!satellites constantly face planet}%
\index{Saturn!satellites always face the planet}%
telescopic examination has led astronomers to believe
that they always present the same face to
their planets. The theory of tidal friction would
\index{Planets!rotation of some, annulled by tidal friction}%
certainly lead us to expect that these enormous
planets should work out the same result for their
relatively small satellites that the earth has produced
\index{Satellites!discovery of those of Mars|)}%
\index{Satellites!rotation of those of Jupiter and Saturn annulled}%
in the moon.

The proximity of the planets Mercury and
\index{Mercury, rotation of}%
Venus to the sun should obviously render solar
\index{Venus, rotation of}%
tidal friction far more effective than with us.
The determination of the periods of rotation of
\index{Rotation!of Mercury, Venus, and satellites of Jupiter and Saturn annulled by tidal friction}%
these planets thus becomes a matter of much interest.
But the markings on their disks are so
obscure that the rates of their rotations have remained
under discussion for many years. Until
recently the prevailing opinion was that in both
cases the day was of nearly the same length as
ours; but a few years ago Schiaparelli of Milan,
\index{Schiaparelli on rotation of Venus and Mercury}%
an observer endowed with extraordinary acuteness
of vision, announced as the result of his observations
that both Mercury and Venus rotate
only once in their respective years, and that
each of them constantly presents the same face
to the sun. These conclusions have recently been
confirmed by Mr.~Percival Lowell from observations
\index{Lowell, P., on rotations of Venus and Mercury}%
made in Arizona. Although on reading
the papers of these astronomers it is not easy
to see how they can be mistaken, yet it should
be noted that others have failed to detect the
markings on the planet's disks, although they
\index{Hall, Asaph, discovery of Martian satellites|)}%
\index{Mars!discovery of satellites|)}%
\PageSep{299}
\index{Lowell, P., on rotations of Venus and Mercury}%
apparently enjoyed equal advantages for observation.\footnote
  {Dr.~See, a member of the staff of the Flagstaff Observatory,
  Arizona, tells me that he has occasionally looked at these planets
  through the telescope, although he took no part in the systematic
  observation. In his opinion it would be impossible for any one
  at Flagstaff to doubt the reality of the markings. There are,
  however, many astronomers of eminence who suspend their
  judgment, and await confirmation by other observers at other
  stations.}

If, as I am disposed to do, we accept these observations
as sound, we find that evidence favorable
to the theory of tidal friction is furnished
by the planets Mercury and Venus, and by the
\index{Mercury, rotation of}%
\index{Venus, rotation of}%
satellites of the earth, Jupiter and Saturn, whilst
\index{Earth and moon!figure of}%
\index{Earth and moon!adjustment of figure to suit change of rotation|(}%
the Martian system is yet more striking as an
instance of an advanced stage in evolution.

\TB

It is well known that the figure of the earth
is flattened by the diurnal rotation, so that the
polar axis is shorter than any equatorial diameter.
At the present time the excess of the equatorial
radius over the polar radius is $\frac{1}{290}$~part of
either of them. Now in tracing the history of
the earth and moon, we found that the earth's
rotation had been retarded, so that the day is
now longer than it was. If then the solid earth
has always been absolutely unyielding, and if an
ocean formerly covered the planet to a uniform
depth, the sea must have gradually retreated
towards the poles, leaving the dry land exposed
at the equator. If on the other hand the solid
\PageSep{300}
\index{Geological evidence of earth's plasticity}%
\index{Plasticity of earth under change of rotation|(}%
earth had formerly its present shape, there must
then have been polar continents and a deep equatorial
sea.

But any considerable change in the speed of
the earth's rotation would, through the action of
gravity, bring enormous forces to bear on the
solid earth. These forces are such as would, if
they acted on a plastic material, tend to restore
the planet's figure to the form appropriate to its
changed rotation. It has been shown experimentally
by M.~Tresca and others that even very
\index{Tresca on flow of solids}%
rigid and elastic substances lose their rigidity
and their elasticity, and become plastic under the
action of sufficiently great forces. It appears to
me, therefore, legitimate to hold to the belief in
the temporary rigidity of the earth's mass, as explained
in \Ref{Chapter}{XV}., whilst contending that
under a change of rotational velocity the earth
may have become plastic, and so have maintained
a figure adapted to its speed. Geological observation
shows that rocks have been freely twisted
and bent near the earth's surface, and it is impossible
to doubt that under altered rotation the
deeper portions of the earth would have been
subjected to very great stress. I conjecture that
the internal layers might adapt themselves by
continuous flow, whilst the superficial portion
might yield impulsively. Earthquakes are probably
due to unequal shrinkage of the planet in
cooling, and each shock would tend to bring the
\PageSep{301}
strata into their position of rest; thus the earth's
surface would avail itself of the opportunity afforded
by earthquakes of acquiring its proper
shape. The deposit in the sea of sediment, derived
from the denudation of continents, affords
another means of adjustment of the figure of the
planet. I believe then that the earth has always
maintained a shape nearly appropriate to its rotation.
The existence of the continents proves
that the adjustment has not been perfect, and we
shall see reason to believe that there has been
also a similar absence of complete adjustment in
the interior.

But the opinion here maintained is not shared
by the most eminent of living authorities, Lord
Kelvin; for he holds that the fact that the average
\index{Kelvin, Lord!denies adjustment of earth's figure to changed rotation}%
figure of the earth corresponds with the
actual length of the day proves that the planet
was consolidated at a time when the rotation was
but little more rapid than it is now. The difference
between us is, however, only one of degree,
for he considers that the power of adjustment is
slight, whilst I hold that it would be sufficient
to bring about a considerable change of shape
within the period comprised in geological history.

If the adjustment of the planet's figure were
perfect, the continents would sink below the
ocean, which would then be of uniform depth.
But there is no superficial sign, other than the
dry land, of absence of adaptation to the present
\PageSep{302}
\index{Moon and earth!inequality in motion indicates internal density of earth}%
rotation---unless indeed the deep polar sea discovered
by Nansen be such. Yet, as I have
hinted above, some tokens still exist in the earth
\index{Earth and moon!internal density}%
of the shorter day of the past. The detection of
this evidence depends however on arguments of
so technical a character that I cannot hope in
such a work as this to do more than indicate the
nature of the proof.

The earth is denser towards the centre than
outside, and the layers of equal density are concentric.
\index{Density!of earth, law of internal}%
If then the materials were perfectly
plastic throughout, not only the surface, but
also each of these layers would be flattened to a
definite extent, which depends on the rate of rotation
and on the law governing the internal
density of the earth. Although the rate at
which the earth gets denser is unknown, yet it is
possible to assign limits to the density at various
depths. Thus it can be proved that at any internal
point the density must lie between two
values which depend on the position of the point
in question. So also, the degree of flattening at
any internal point is found to lie between two
extreme limits, provided that all the internal layers
are arranged as they would be if the whole
mass were plastic.

Now variations in the law of internal density
and in the internal flattening would betray themselves
to our observation in several ways. In
the first place, gravity on the earth's surface
\index{Earth and moon!adjustment of figure to suit change of rotation|)}%
\index{Gravity, variation according to latitude}%
\index{Plasticity of earth under change of rotation|)}%
\PageSep{303}
\index{Meteorological!conditions dependent on earth's rotation}%
\index{Moon and earth!inequality in motion indicates internal density of earth}%
\index{Nutation!value of, indicates internal density of earth}%
would be changed. The force of gravity at the
\index{Gravity, variation according to latitude}%
poles is greater than at the equator, and the law
of its variation according to latitude is known.
In the second place the amount of the flattening
of the earth's surface would be altered, and the
present figure of the earth is known with considerable
exactness. Thirdly the figure and law of
density of the earth govern a certain irregularity
or inequality in the moon's motion, which has
been carefully evaluated by astronomers. Lastly
the precessional and nutational motion of the
earth is determined by the same causes, and these
motions also are accurately known. These four
facts of observation---gravity, the ellipticity of
\index{Ellipticity of earth's strata in excess for present rotation}%
the earth, the lunar inequality, and the precessional
and nutational motion of the earth---are
so intimately intertwined that one of them cannot
be touched without affecting the others.

Now Édouard Roche, a French mathematician,
\index{Roche, E.!ellipticity of internal strata of earth}%
has shown that if the earth is perfectly plastic,
so that each layer is exactly of the proper shape
for the existing rotation, it is not possible to adjust
the unknown law of internal density so as
\index{Precession, value of, indicates internal density of earth}%
to make the values of all these elements accord
with observation. If the density be assumed
such as to fit one of the data, it will produce a
disagreement with observation in others. If,
however, the hypothesis be abandoned that the
internal strata all have the proper shapes, and if
it be granted that they are a little more flattened
\PageSep{304}
\index{Ellipticity of earth's strata in excess for present rotation}%
\index{Geological evidence of earth's plasticity!as to retardation of earth's rotation|(}%
than is due to the present rate of rotation, the
data are harmonized together; and this is just
what would be expected according to the theory
of tidal friction. But it would not be right to
attach great weight to this argument, for the
absence of harmony is so minute that it might
be plausibly explained by errors in the numerical
data of observation. I notice, however, that the
most competent judges of this intricate subject
are disposed to regard the discrepancy as a
reality.

\DPchg{}{\TB}

We have seen in the preceding chapter that
the length of day has changed but little within
historical times. But the period comprised in
written history is almost as nothing compared
with the whole geological history of the earth.
We ought then to consider whether geology furnishes
any evidence bearing on the theory of
tidal friction. The meteorological conditions on
the earth are dependent to a considerable extent
on the diurnal rotation of the planet, and therefore
those conditions must have differed in the
past. Our storms are of the nature of aerial eddies,
and they derive their rotation from that of
the earth. Accordingly storms were probably
more intense when the earth spun more rapidly.
The trunks of trees should be stronger than they
are now to withstand more violent storms. But
I cannot learn that there is any direct geological
evidence on this head, for deciduous trees with
\PageSep{305}
\index{Ripple mark in sand preserved in geological strata}%
stiff trunks seem to have been a modern product
of geological time, whilst the earlier trees more
nearly resembled bamboos, which yield to the
wind instead of standing up to it. It seems possible
that trees and plants would not be exterminated,
even if they suffered far more wreckage
than they do now. If trees with stiff trunks
could only withstand the struggle for existence
when storms became moderate in intensity, their
absence from earlier geological formations would
be directly due to the greater rapidity of the
earth's rotation in those times.

According to our theory the tides on the seacoast
must certainly have had a much wider
range, and river floods must probably have been
more severe. The question then arises whether
these agencies should have produced sedimentary
deposits of coarser grain than at present. Although
I am no geologist, I venture to express a
doubt whether it is possible to tell, within very
wide limits, the speed of the current or the range
of the tide that has brought down and distributed
any sedimentary deposit. I doubt whether any
geologist would assert that floods might not have
been twice or thrice as frequent, or that the tide
might not have had a very much greater range
than at present.

In some geological strata ripple-marks have
been preserved which exactly resemble modern
ones. This has, I believe, been adduced as an
\PageSep{306}
argument against the existence of tides of great
range. Ripples are, however, never produced
by a violent scour of water, but only by gentle
currents or by moderate waves. The turn of
the tide must be gentle to whatever height it
rises, and so the formation of ripple-mark should
have no relationship to the range of tide.

It appears then that whilst geology affords no
direct confirmation of the theory, yet it does not
present any evidence inconsistent with it. Increased
activity in the factors of change is important
to geologists, since it renders intelligible
a diminution in the time occupied by the history
of the earth; and thus brings the views of the
\index{Earth and moon!probably once molten}%
geologist and of the physicist into better harmony.

Although in this discussion I have maintained
the possibility that a considerable portion of the
changes due to tidal friction may have occurred
within geological history, yet it seems to me
probable that the greater part must be referred
back to pre-geological times, when the planet
was partially or entirely molten.

\TB

The action of the moon and sun on a plastic
and viscous planet would have an effect of which
some remains may perhaps still be traceable.
The relative positions of the moon and of the
frictionally retarded tide were illustrated in the
last chapter by~\fig{36}. That figure shows that
\index{Geological evidence of earth's plasticity!as to retardation of earth's rotation|)}%
\PageSep{307}
the earth's rotation is retarded by forces acting
\index{Earth and moon!distortion under primeval tidal friction}%
on the tidal protuberances in a direction adverse
to the planet's rotation. As the plastic substance,
of which we now suppose the planet to
be formed, rises and falls rhythmically with the
tide, the protuberant portions are continually
subject to this retarding force. Meanwhile the
internal portions are urged onward by the
inertia due to their velocity. Accordingly there
must be a slow motion of the more superficial
portions with reference to the interior. From
the same causes, under present conditions, the
whole ocean must have a slow westerly drift, although
it has not been detected by observation.

Returning however to our plastic planet, the
equatorial portion is subjected to greater force
than the polar regions, and if meridians were
painted on its surface, as on a map, they would
gradually become distorted. In the equatorial
belt the original meridional lines would still run
north and south, but in the northern hemisphere
they would trend towards the northeast, and in
the southern hemisphere towards the southeast.
This distortion of the surface would cause the
surface to wrinkle, and the wrinkles should be
warped in the directions just ascribed to the
meridional lines. If the material yielded very
easily I imagine that the wrinkles would be
small, but if it were so stiff as only to yield with
difficulty they might be large.
\PageSep{308}

There can be no doubt as to the correctness
\index{History!of earth and moon|(}%
of this conclusion as to a stiff yet viscous planet,
but the application of these ideas to the earth is
hazardous and highly speculative. We do, however,
observe that the continents, in fact, run
\index{Continents, trend of, possibly due to primeval tidal friction}%
roughly north and south. It may appear fanciful
to note, also, that the northeastern coast of
America, the northern coast of China, and the
southern extremity of South America have the
proper theoretical trends. But the northwestern
coast of America follows a line directly adverse
to the theory, and the other features of the globe
are by no means sufficiently regular to inspire
much confidence in the justice of the conjecture.\footnote
  {See, also, W. Prinz, \Title{Torsion apparente des planètes}, ``Annuaire
  de l'Obs.~R. de~Bruxelles,'' 1891.}

\TB

We must now revert to the astronomical aspects
of our problem. It is natural to inquire
whether the theory of tidal friction is competent
to explain any peculiarities of the motion of the
moon and earth other than those already considered.
It has been supposed thus far that the
moon moves over the earth's equator in a circular
orbit, and that the equator coincides with the
plane in which the earth moves in its orbit. But
the moon actually moves in a plane different
from that in which the earth revolves round the
sun, her orbit is not circular but elliptic, and the
\PageSep{309}
earth's equator is oblique to the orbit. We must
consider, then, how tidal friction will affect these
three factors.

Let us begin by considering the obliquity of
the equator to the ecliptic, which produces the
seasonal changes of winter and summer. The
problem involved in the disturbance of the motion
of a rotating body by any external force is
too complex for treatment by general reasoning,
and I shall not attempt to explain in detail the
interaction of the moon and earth in this respect.

The attractions of the moon and sun on the
equatorial protuberance of the earth causes the
earth's axis to move slowly and continuously
with reference to the fixed stars. At present,
the axis points to the pole-star, but $13,000$~years
hence the present pole-star will be $47°$~distant
from the pole, and in another $13,000$~years it
will again be the pole-star. Throughout this
precessional movement the obliquity of the equator
to the ecliptic remains constant, so that winter
and summer remain as at present. There is
also, superposed on the precession, the nutational
or nodding motion of the pole to which I referred
in \Ref{Chapter}{XV}. In the absence of tidal
friction the attractions of the moon and sun on
the tidal protuberance would slightly augment
the precession due to the solid equatorial protuberance,
and would add certain very minute
nutations of the earth's axis; the amount of
\PageSep{310}
these tidal effects, is, however, quite insignificant.
But under the influence of tidal friction,
the matter assumes a different aspect, for the
earth's axis will not return at the end of each
nutation to exactly the same position it would
have had in the absence of friction, and there is
a minute residual effect which always tends in
the same direction. A motion of the pole may
be insignificant when it is perfectly periodic, but
it becomes important in a very long period of
time when the path described is not absolutely
reëntrant. Now this is the case with regard to
the motion of the earth's axis under the influence
of frictionally retarded tides, for it is found
to be subject to a gradual drift in one direction.

In tracing the history of the earth and moon
backwards in time we found the day and month
growing shorter, but at such relative speeds that
the number of days in the month diminished until
the day and month became equal. This conclusion
remains correct when the earth is oblique
to its orbit, but the effect on the obliquity is
\index{Ecliptic, obliquity of, due to tidal friction|(}%
\index{Obliquity of ecliptic, effects of tidal friction on|(}%
found to depend in a remarkable manner upon
the number of days in the month. At present
and for a long time in the past the obliquity
is increasing, so that it was smaller long ago.
But on going back to the time when the day
was six and the month twelve of our present
hours we find that the tendency for the obliquity
to increase vanishes. In other words, if
\PageSep{311}
there are more than two days in a month the
obliquity will increase, if less than two it will
diminish.

Whatever may be the number of days in the
month, the rate of increase or diminution of
obliquity varies as the obliquity which exists at
the moment under consideration. If, then, a
planet be spinning about an axis absolutely perpendicular
to the plane of its satellite's orbit, the
obliquity remains invariable. But if we impart
infinitesimal obliquity to a planet whose day is
less than half a month, that infinitesimal obliquity
will increase; whilst, if the day is more
than half a month, the infinitesimal obliquity
will diminish. Accordingly, the motion of a
planet spinning upright is stable, if there are
less than two days in a month, and unstable if
there are more than two.

It is not legitimate to ascribe the whole of
the present obliquity of~$23\frac{1}{2}°$ to the influence of
tidal friction, because it appears that when there
were only two days in the month, the obliquity
was still as much as~$11°$. It is, moreover, impossible
to explain the considerable obliquity of the
other planets to their orbits by this cause. It
must, therefore, be granted that there was some
unknown cause which started the planets in rotation
about axes oblique to their orbits. It remains,
however, certain that a planet, rotating primitively
without obliquity, would gradually become
\PageSep{312}
inclined to its orbit, although probably not to so
great an extent as we find in the case of the
earth.

The next subject to be considered is the fact
that the moon's orbit is not circular but eccentric.
Here, again, it is found that if the tides
were not subject to friction, there would be no
sensible effect on the shape of the moon's path,
but tidal friction produces a reaction on the
moon tending to change the degree of eccentricity.
In this case, it is possible to indicate by
general reasoning the manner in which this reaction
operates. We have seen that tidal reaction
tends to increase the moon's distance from the
earth. Now, when the moon is nearest, in perigee,
the reaction is stronger than when she is
furthest, in apogee. The effect of the forces in
perigee is such that the moon's distance at the
next succeeding apogee is greater than it was at
the next preceding apogee; so, also, the effect
of the forces in apogee is an increase in the perigeal
distance. But the perigeal effect is stronger
than the apogeal, and, therefore, the apogeal distances
increase more rapidly than the perigeal
ones. It follows, therefore, that, whilst the orbit
as a whole expands, it becomes at the same time
more eccentric.
\index{Ecliptic, obliquity of, due to tidal friction|)}%
\index{Obliquity of ecliptic, effects of tidal friction on|)}%

The lunar orbit is then becoming more eccentric,
and numerical calculation shows that in
very early times it must have been nearly circular.
\PageSep{313}
\index{Eccentricity of orbit!due to tidal friction}%
\index{Moon and earth!eccentricity of orbit increased by tidal friction}%
\index{Saint@St.\ Vénant on flow of solids}%
\index{See, T. J. J.!eccentricity of orbits of double stars}%
But mathematical analysis indicates that in
this case, as with the obliquity, the rate of
increase depends in a remarkable manner upon
the number of days in the month. I find in
fact that if eighteen days are less than eleven
months the eccentricity will increase, but in the
converse case it will diminish; in other words
the critical stage at which the eccentricity is
stationary is when $1\frac{7}{11}$~days is equal to the
month. It follows from this that the circular
orbit of the satellite is dynamically stable or
\index{Orbit!of double stars, very eccentric}%
unstable according as $1\frac{7}{11}$~days is less or greater
than the month.

The effect of tidal friction on the eccentricity
has been made the basis of extensive astronomical
speculations by Dr.~See. I shall revert to
this subject in \Ref{Chapter}{XIX}., and will here
merely remark that systems of double stars are
\index{Stars!double, eccentricity of orbits}%
found to revolve about one another in orbits of
great eccentricity, and that Dr.~See supposes
that the eccentricity has arisen from the tidal
action of each star on the other.

The last effect of tidal friction to which I
have to refer is that on the plane of the moon's
orbit. The lunar orbit is inclined to that of the
earth round the sun at an angle of~$5°$, and the
problem to be solved is as to the nature of the
effect of tidal friction on that inclination. The
nature of the relation of the moon's orbit to the
ecliptic is however so complex that it appears
\index{History!of earth and moon|)}%
\PageSep{314}
\index{Eccentricity of orbit!due to tidal friction}%
\index{Moon and earth!eccentricity of orbit increased by tidal friction}%
hopeless to explain the effects of tidal action
without the use of mathematical language, and
I must frankly give up the attempt. I may,
however, state that when the moon was near the
earth she must have moved nearly in the plane
of the earth's equator, but that the motion gradually
changed so that she has ultimately come to
move nearly in the plane of the ecliptic. These
two extreme cases are easily intelligible, but the
transition from one case to the other is very
complicated. It may suffice for this general
account of the subject to know that the effects
of tidal friction are quite consistent with the
present condition of the moon's motion, and
with the rest of the history which has been
traced.

This discussion of the effects of tidal friction
may be summed up thus:---

If a planet consisted partly or wholly of molten
lava or of other fluid, and rotated rapidly about
an axis perpendicular to the plane of its orbit,
and if that planet was attended by a single satellite,
revolving with its month a little longer than
the planet's day, then a system would necessarily
be developed which would have a strong resemblance
to that of the earth and moon.

A theory reposing on \textit{veræ causæ} which brings
into quantitative correlation the lengths of the
present day and month, the obliquity of the
ecliptic, the eccentricity and the inclination of
\PageSep{315}
the lunar orbit, should have strong claims to
acceptance.

\begin{Authorities}
G.~H. Darwin. A series of papers in the ``Phil.\ Trans.\ Roy.\
\index{Darwin, G. H.!papers on tidal friction}%
Soc.'' pt.~i.\ 1879, pt.~ii.\ 1879, pt.~ii.\ 1880, pt.~ii.\ 1881, pt.~i.\ 1882,
and abstracts (containing general reasoning) in the corresponding
Proceedings; also ``Proc.\ Roy.\ Soc.''\ vol.~29, 1879, p.~168 (in
part republished in Thomson and Tait's \Title{Natural Philosophy}),
and vol.~30, 1880, p.~255.

Lord Kelvin, \Title{On Geological Time}, ``Popular Lectures and
\index{Kelvin, Lord!on geological time}%
Addresses,'' vol.~iii. Macmillan, 1894.

Roche. The investigations of Roche and of others are given
in Tisserand's \Title{Mécanique Céleste}, vol.~ii. Gauthier-Villars, 1891.
\index{Tisserand, Roche's investigations as to earth's figure}%

Tresca and St.~Vénant, \Title{Sur l'écoulement des Corps Solides},
``Mémoires des Savants Étrangers,'' Académie des Sciences de
Paris, vols.\ 18~and~20.

Schiaparelli, \Title{Considerazioni sul moto rotatorio del pianeta
\index{Schiaparelli on rotation of Venus and Mercury}%
Venere}. Five notes in the ``Rendiconti del R.~Istituto Lombardo,''
vol.~23, and \Title{Sulla rotazione di Mercurio}, ``Ast.\ Nach.,''
No.~2944. An abstract is given in ``Report of Council of R.
Ast.\ Soc.,'' Feb.~1891.

Lowell, Mercury, ``Ast.\ Nach.,'' No.~3417. \Title{Mercury and Determination
\index{Lowell, P., on rotations of Venus and Mercury}%
of Rotation Period~\dots\ of Venus}, ``Monthly Notices
R. Ast.\ Soc.,'' vol.~57, 1897, p.~148. \Title{Further proof},~\&c., \textit{ibid}.\
p.~402.

Douglass, \Title{Jupiter's third Satellite}, ``Ast.\ Nach.,'' No.~3432.
\index{Douglass, rotation of Jupiter's satellites}%
\Title{Rotation des IV~Jupitersmondes}, ``Ast.\ Nach.,'' No.~3427, confirming
Engelmann, \Title{Ueber~\dots\ Jupiterstrabanten}, Leipzig, 1871.

Barnard, \Title{The third and fourth Satellites of Jupiter}, ``Ast.\
\index{Barnard, rotation of Jupiter's satellites}%
Nach.,'' No.~3453.
\end{Authorities}
\index{Friction of tides|)}%
\PageSep{316}


\Chapter[Figures of Equilibrium]{XVIII}
{The Figures of Equilibrium of a Rotating
Mass of Liquid}

\First{The} theory of the tides involves the determination
\index{Equilibrium, figures of, of rotating liquid|(}%
\index{Figure of equilibrium!of rotating liquid|(}%
\index{Rotating liquid, figures of equilibrium|(}%
of the form assumed by the ocean under
the attraction of a distant body, and it now
remains to discuss the figure which a rotating
mass of liquid may assume when it is removed
from all external influences. The forces which
act upon the liquid are the mutual gravitation
of its particles, and the centrifugal force due to
its rotation. If the mass be of the appropriate
shape, these two opposing forces will balance
one another, and the shape will be permanent.
The problem in hand is, then, to determine
what shapes of this kind are possible.

In 1842 a distinguished Belgian physicist, M.~Plateau,\footnote
  {He is justly celebrated not only for his discoveries, but also
  for his splendid perseverance in continuing his researches after
  he had become totally blind.}
devised an experiment which affords
\index{Capillarity of liquids, and Plateau's experiment|(}%
\index{Plateau, experiment on figure of rotating globule|(}%
a beautiful illustration of the present subject.
The experiment needs very nice adjustment in
several respects, but I refer the reader to
Plateau's paper for an account of the necessary
\PageSep{317}
\index{Surface tension of liquids}%
precautions. Alcohol and water may be so
mixed as to have the same density as olive oil.
If the adjustment of density is sufficiently exact,
a mass of oil will float in the mixture, in the
form of a spherical globule, without any tendency
to rise or fall. The oil is thus virtually
relieved from the effect of gravity. A straight
wire, carrying a small circular disk at right
angles to itself, is then introduced from the top
of the vessel. When the disk reaches the
globule, the oil automatically congregates itself
round the disk in a spherical form, symmetrical
with the wire.

The disk is then rotated slowly and uniformly,
and carries with it the oil, but leaves the surrounding
mixture at rest. The globule is then
seen to become flattened like an orange, and as
the rotation quickens it dimples at the centre,
and finally detaches itself from the disk in the
form of a perfect ring. This latter form is only
transient; for the oil usually closes in again
round the disk, or sometimes, with slightly different
manipulation, the ring may break into
drops which revolve round the centre, rotating
round their axes as they go.

The force which holds a drop of water, or
this globule of oil, together is called ``surface
tension'' or ``capillarity.'' It is due to a certain
molecular attraction, quite distinct from
that of gravitation, and it produces the same
\PageSep{318}
effect as if the surface of the liquid were enclosed
in an elastic skin. There is of course no
actual skin, and yet when the liquid is stirred
the superficial particles attract their temporary
neighbors so as to restore the superficial elasticity,
continuously and immediately. The intensity
of surface tension depends on the nature
\index{Surface tension of liquids}%
of the material with which the liquid is in contact;
thus there is a definite degree of tension
in the skin of olive oil in contact with spirits
and water.

A globule at rest necessarily assumes the form
of a sphere under the action of surface tension,
but when it rotates it is distorted by centrifugal
force. The polar regions become less curved,
and the equatorial region becomes more curved,
until the excess of the retaining power at the
equator over that at the poles is sufficient to
restrain the centrifugal force. Accordingly the
struggle between surface tension and centrifugal
force results in the assumption by the globule
of an orange-like shape, or, with greater speed
of rotation, of the other figures of equilibrium.
\index{Capillarity of liquids, and Plateau's experiment|)}%

In very nearly the same way a large mass of
gravitating and rotating liquid will naturally
assume certain definite forms. The simplest
case of the kind is when the fluid is at rest in
space, without any rotation. Then mutual gravitation
is the only force which acts on the system.
The water will obviously crowd together
\PageSep{319}
into the smallest possible space, so that every
particle may get as near to the centre as its
neighbors will let it. I suppose the water to be
incompressible, so that the central portion, although
pressed by that which lies outside of it,
does not become more dense; and so the water
does not weigh more per cubic foot near the
centre than towards the outside. Since there
is no upwards and downwards, or right and
left about the system, it must be symmetrical in
every direction; and the only figure which possesses
this quality of universal symmetry is the
sphere. A sphere is then said to be a figure of
equilibrium of a mass of fluid at rest.

If such a sphere of water were to be slightly
deformed, and then released, it would oscillate
to and fro, but would always maintain a nearly
spherical shape. The speed of the oscillation
depends on the nature of the deformation impressed
upon it. If the water were flattened to
the shape of an orange and released, it would
spring back towards the spherical form, but
would overshoot the mark, and pass on to a
lemon shape, as much elongated as the orange
was flattened. It would then return to the
orange shape, and so on backwards and forwards,
passing through the spherical form at
each oscillation. This is the simplest kind of
oscillation which the system can undergo, but
there is an infinite number of other modes of
\index{Plateau, experiment on figure of rotating globule|)}%
\PageSep{320}
any degree of complexity. The mathematician
can easily prove that a liquid globe, of the same
density as the earth, would take an hour and a
half to pass from the orange shape to the lemon
shape, and back to the orange shape. At present,
the exact period of the oscillation is not
the important point, but it is to be noted that if
the body be set oscillating in any way whatever,
it will continue to oscillate and will always remain
nearly spherical. We say then that the
sphere is a stable form of equilibrium of a mass
of fluid. The distinction between stability and
instability has been already illustrated in \Ref{Chapter}{XVI}.\
by the cases of an egg lying on its
side and balanced on its end, and there is a
similar distinction between stable and unstable
modes of motion.

Let us now suppose the mass of water to rotate
slowly, all in one piece as if it were solid.
We may by analogy with the earth describe the
axis of rotation as polar, and the central plane,
at right angles to the axis, as equatorial. The
equatorial region tends to move outwards in consequence
of the centrifugal force of the rotation,
and this tendency is resisted by gravitation which
tends to draw the water together towards the
centre. As the rotation is supposed to be very
slow, centrifugal force is weak, and its effects are
small thus the globe is very slightly flattened at
the poles, like an orange or like the earth itself.
\PageSep{321}
Such a body resembles the sphere in its behavior
when disturbed; it will oscillate, and its average
figure in the course of its swing is the orange
shape. It is therefore stable.

But it has been discovered that the liquid may
also assume two other alternative forms. One
of these is extremely flattened and resembles a
flat cheese with rounded edges. As the disk of
liquid is very wide, the centrifugal force at the
equator is very great, although the rotation is
very slow. In the case of the orange-shaped figure,
the slower the rotation the less is the equatorial
centrifugal force, because it diminishes
both with diminution of radius and fall of speed.
But in the cheese shape the equatorial centrifugal
force gains more by the increase of equatorial
radius than it loses by diminution of rotation.
Therefore the slower the rotation the broader the
disk, and, if the rotation were infinitely slow, the
liquid would be an infinitely thin, flat, circular
disk.

The cheese-like form differs in an important
respect from the orange-like form. If it were
slightly disturbed, it would break up, probably
into a number of detached pieces. The nature
of the break-up would depend on the disturbance
from which it started, but it is impossible to trace
the details of the rupture in any case. We say
then that the cheese shape is an unstable figure
of equilibrium of a rotating mass of liquid.
\PageSep{322}
\index{Stability!of figures of equilibrium}%

The third form is strikingly different from
either of the preceding ones. We must now imagine
the liquid to be shaped like a long cigar,
and to be rotating about a central axis perpendicular
to its length. Here again the ends of
the cigar are so distant from the axis of rotation
that the centrifugal force is great, and with infinitely
slow rotation the figure becomes infinitely
long and thin. Now this form resembles the
cheese in being unstable. It is remarkable that
these three forms are independent of the scale on
which they are constructed, for they are perfectly
similar whether they contain a few pounds of
water or millions of tons.\footnote
  {It is supposed that they are more than a fraction of an inch
  across, otherwise surface tension would be called into play.}
If the period of rotation
and the density of the liquid are given,
the shapes are absolutely determinable.

The first of the three figures resembles the
earth and may be called the planetary figure, and
\index{Planetary figure of equilibrium of rotating liquid}%
I may continue to refer to the other two as the
cheese shape and the cigar shape. The planetary
and cheese shape are sometimes called the spheroids
of Maclaurin, after their discoverer, and
\index{Maclaurin!figure of equilibrium of rotating liquid|(}%
the cigar shape is generally named after Jacobi,
\index{Jacobi, figure of equilibrium of rotating liquid|(}%
the great German mathematician. For slow rotations
the planetary form is stable, and the
cheese and cigar are unstable. There are probably
other possible forms of equilibrium, such as
a ring, or several rings, or two detached masses
\PageSep{323}
revolving about one another like a planet and
satellite, but for the present I only consider these
three forms.

Now imagine three equal masses of liquid, infinitely
distant from one another, and each rotating
\Figure[0.85]{37}{}{png}
at the same slow speed, and let one of them
have the planetary shape, the second the cheese
shape, and the third the cigar shape. When the
rotations are simultaneously and equally augmented,
we find the planetary form becoming
flatter, the cheese form shrinking in diameter
and thickening, and the cigar form shortening
and becoming fatter. There is as yet no change
in the stability, the first remaining stable and
\index{Stability!of figures of equilibrium}%
the second and third unstable. The three figures
are illustrated in~\fig{37}, but the cigar shape
is hardly recognizable by that name, since it has
already become quite short and its girth is
considerable.
\PageSep{324}

Now it has been proved that as the cigar shape
shortens, its tendency to break up becomes less
marked, or in other words its degree of instability
diminishes. At a certain stage, not as yet
exactly determined, but which probably occurs
when the cigar is about twice as long as broad,
the instability disappears and the cigar form just
becomes stable. I shall have to return to the
consideration of this phase later. The condition
of the three figures is now as follows: The planetary
form of Maclaurin has become much flattened,
but is still stable; the cigar form of Jacobi
has become short and thick, and is just stable;
and the cheese form of Maclaurin is still unstable,
\index{Maclaurin!figure of equilibrium of rotating liquid|)}%
but its diameter has shrunk so much that the
figure might be better described as a very flat
orange.

On further augmenting the rotation the form
of Jacobi still shrinks in length and increases in
girth, until its length becomes equal to its
greater breadth. Throughout the transformation
the axis of rotation has always remained the
shortest of the three, so that when the length
becomes equal to the shorter equatorial diameter,
the shape is not spherical, but resembles that of
a much flattened orange. In fact, at this stage
Jacobi's figure of equilibrium has degenerated to
\index{Jacobi, figure of equilibrium of rotating liquid|)}%
identity with the planetary shape. One of the
upper ovals in \fig{38} represents the section of
the form in which the planetary figure and the
\PageSep{325}
cigar figure coalesce, the former by continuous
flattening, the latter by continuous shortening.
The other upper figure represents the form to
which the cheese-like figure of Maclaurin has
\Figure[0.85]{38}{}{png}
\index{Poincaré!figure of rotating liquid}%
been reduced; it will be observed that it presents
some resemblance to the coalescent form.

When the rotation is further augmented, there
is no longer the possibility of an elongated Jacobian
figure, and there remain only the two
spheroids of Maclaurin. But an important change
has now supervened, for both these are now unstable,
and indeed no stable form consisting of a
single mass of liquid has yet been discovered.

Still quickening the rotation, the two remaining
forms, both unstable, grow in resemblance to
one another, until at length they become identical
in shape. This limiting form of Maclaurin's
spheroids is shown in the lower part of~\fig{38}.
If the liquid were water, it must rotate in $2$~hours
\PageSep{326}
$25$~minutes to attain this figure, but it would be
unstable.

A figure for yet more rapid rotation has not
been determined, but it seems probable that
dimples would be formed on the axis, that the
dimples would deepen until they met, and that
the shape would then be annular. The actual
existence of such figures in Plateau's experiment
is confirmatory of this conjecture.

We must now revert to the consideration of
the cigar-shaped figure of Jacobi, at the stage
when it has just become stable. The whole of
this argument depends on the fact that any figure
of equilibrium is a member of a continuous
series of figures of the same class, which gradually
transforms itself as the rotation varies. Now
M.~Poincaré has proved that, when we follow a
\index{Poincaré!law of interchange of stability}%
given series of figures and find a change from instability
to stability, we are, as it were, served with
a notice that there exists another series of figures
coalescent with the first at that stage. We have already
seen an example of this law, for the planetary
figure of Maclaurin changed from stability
to instability at the moment of its coalescence
with the figure of Jacobi. Now I said that when
the cigar form of Jacobi was very long it was
unstable, but that when its length had shrunk to
about twice its breadth it became stable; hence
we have notice that at the moment of change
another series of forms was coalescent with the
\PageSep{327}
cigar. It follows also from Poincaré's investigation
\index{Poincaré!law of interchange of stability}%
\index{Poincaré!figure of rotating liquid}%
that the other series of forms must have
been stable before the coalescence.

Let us imagine then a mass of liquid in the
form of Jacobi's cigar-shaped body rotating at
the speed which just admits of stability, and let
us pursue the series of changes backwards by
making it rotate a little slower. We know that
this retardation of rotation lengthens Jacobi's
figure, and induces instability, but Poincaré has
not only proved the existence and stability of the
other series, but has shown that the shape is
something like a pear.

Poincaré's figure is represented approximately
in~\fig{38}, but the mathematical difficulty of the
problem has been too great to admit of an absolutely
exact drawing. The further development
of the pear shape is unknown, when the rotation
slackens still more. There can, however, be
hardly any doubt that the pear becomes more
constricted in the waist, and begins to resemble
an hour-glass; that the neck of the hour-glass
becomes thinner, and that ultimately the body
separates into two parts. It is of course likewise
unknown up to what stage in these changes
Poincaré's figure retains its stability.

I have myself attacked this problem from an
entirely different point of view, and my conclusions
throw an interesting light on the subject,
although they are very imperfect in comparison
\PageSep{328}
\index{Darwin, G. H.!hour-glass figure of rotating liquid|(}%
with Poincaré's masterly work. To understand
this new point of view, we must consider a new
series of figures, namely that of a liquid planet
attended by a liquid satellite. The two bodies
are supposed to move in a circle round one another,
and each is also to revolve on its axis at
such a speed as always to exhibit the same face
to its neighbor. Such a system, although divided
into two parts, may be described as a figure of
equilibrium. If the earth were to turn round
once in twenty-seven days, it would always show
to the moon the same side, and the moon actually
does present the same side to us. In this
case the earth and the moon would form such a
system as that I am describing. Both the planet
and the satellite are slightly flattened by their
rotations, and each of them exercises a tidal influence
on the other, whereby they are elongated
towards the other.

The system then consists of a liquid planet
and liquid satellite revolving round one another,
so as always to exhibit the same face to one another,
and each tidally distorting the other. It
is certain that if the two bodies are sufficiently
far apart the system is a stable one, for if any
slight disturbance be given, the whole system will
not break up. But little is known as yet as to
the limiting proximity of the planet and satellite,
which will insure stability.

Now if the rotations and revolutions of the
\PageSep{329}
bodies be accelerated, the two masses must be
brought nearer together in order that the greater
attraction may counterbalance the centrifugal
force. But as the two are brought nearer the
tide-generating force increases in intensity with
great rapidity, and accordingly the tidal elongation
of the two bodies is much augmented.

A time will at length come when the ends of
the two bodies will just touch, and we then have
a form shaped like an hour-glass with a very
\Figure[0.7]{39}{Hour-glass Figure of Equilibrium}{png}
\PageLabel[pg]{329}% [** TN: Used by reference on p. 356 of the original]
thin neck. The form is clearly Poincaré's figure,
at an advanced stage of its evolution.

The figure~\figref{39} shows the form of one possible
\PageSep{330}
figure of this class; it arises from the coalescence
of two equal masses of liquid, and the
shape shown was determined by calculation.
But there are any number of different sorts of
hour-glass shapes, according to the relative sizes
of the planet and satellite which coalesce; and
in order to form a continuous series with Poincaré's
pear, it would be necessary to start with
a planet and satellite of some definitely proportionate
sizes. Unfortunately I do not know
what the proportion may be. There are, however,
certain indications which may ultimately
lead to a complete knowledge of the series of
figures from Jacobi's cigar shape down to the
planet and satellite. It may be shown---and I
shall have in \Ref{Chapter}{XX}.\ to consider the point
more in detail---that if our liquid satellite had
only, say, a thousandth of the mass of the planet,
and if the two bodies were brought nearer one
another, at a certain calculable distance the tidal
action of the big planet on the very small satellite
would become so intense that it would tear
it to pieces. Accordingly the contact and coalescence
of a very small satellite with a large
planet is impossible. It is, however, certain that
a large enough satellite---say of half the mass
of the planet---could be brought up to contact
with the planet, without the tidal action of the
planet on the satellite becoming too intense to
admit of the existence of the latter. There
\PageSep{331}
must then be some mass of the satellite, which
will just allow the two to touch at the same
moment that the tidal action of the larger on
the smaller body is on the point of disrupting
it. Now I suspect, although I do not know,
that the series of figures which we should find in
this case is in fact Poincaré's series. This discussion
shows that the subject still affords an
interesting field for future mathematicians.

These investigations as to the form of rotating
masses of liquid are of a very abstract character,
and seem at first sight remote from practical
conclusions, yet they have some very interesting
applications.

The planetary body of Maclaurin is flattened
at the poles like the actual planets, and the
degree of its flattening is exactly appropriate to
the rapidity of its rotation. Although the planets
are, at least in large part, composed of solid
matter, yet that matter is now, or was once,
sufficiently plastic to permit it to yield to the
enormous forces called into play by rotation and
gravitation. Hence it follows that the theory
of Maclaurin's figure is the foundation of that
of the figures of planets, and of the variation of
gravity at the various parts of their surfaces.
In the liquid considered hitherto, every particle
attracted every other particle, the fluid was
equally dense throughout, and the figure assumed
was the resultant of the battle between
\PageSep{332}
\index{Figure of planets and their density}%
\index{Gravity, variation according to latitude}%
\index{Saturn!law of density and figure}%
the centrifugal force and gravitation. At every
part of the liquid the resultant attraction was
directed nearly, but not quite, towards the
centre of the shape. But if the attraction had
everywhere been directed exactly to the centre,
the degree of flattening would have been
diminished. We may see that this must be so,
because if the rotation were annulled, the mass
would be exactly spherical, and if the rotation
were not annulled, yet the forces would be such
as to make the fluid pack closer, and so assume
a more nearly spherical form than when the
forces were not absolutely directed to the centre.
It may be shown in fact that the flattening is
$2\frac{1}{2}$~times greater in the case of Maclaurin's
body than it is when the seat of gravitation is
exactly central.

In the case of actual planets the denser matter
\index{Planets!figures and internal densities}%
must lie in the centre and the less dense outside.
If the central matter were enormously
denser than superficial rock, the attraction would
be directed towards the centre. There are then
two extreme cases in which the degree of flattening
can be determined,---one in which the density
\index{Density!of planets determinable from their figures}%
of the planet is the same all through, giving
Maclaurin's figure; the other when the density
is enormously greater at the centre. The flattening
in the former is $2\frac{1}{2}$~times as great as in
the latter. The actual condition of a real planet
must lie between these two extremes. The
\index{Darwin, G. H.!hour-glass figure of rotating liquid|)}%
\PageSep{333}
\index{Figure of planets and their density}%
knowledge of the rate of rotation of a planet
and of the degree of its flattening furnishes us
with some insight into the law of its internal
density. If it is very much less flat than Maclaurin's
\index{Density!of planets determinable from their figures}%
figure, we conclude that it is very dense
in its central portion. In this way it is known
with certainty that the central portions of the
planets Jupiter and Saturn are much denser,
\index{Jupiter!figure and law of internal density}%
\index{Planets!figures and internal densities}%
compared with their superficial portions, than is
the case with the earth.

I do not propose to pursue this subject into
the consideration of the law of the variation of
gravity on the surface of a planet; but enough
has been said to show that these abstract investigations
have most important practical applications.

\begin{Authorities}
Plateau, ``Mémoires de l'Académie Royale de~Belgique,''
vol.~xvi. 1843.

Thomson and Tait's \Title{Natural Philosophy} or other works on
hydrodynamics give an account of figures of equilibrium.

Poincaré, \Title{Sur l'équilibre d'une masse fluide animée d'un mouvement
de rotation}, ``Acta Mathematica,'' vol.~7, 1885.

An easier and different presentation of the subject is contained
in an inaugural dissertation by Schwarzschild (Annals of Munich
\index{Schwarzschild!exposition of Poincaré's theory}%
Observatory, vol.~iii. 1896). He considers that Poincaré's
proof of the stability of his figure is not absolutely conclusive.

G.~H. Darwin, \Title{Figures of Equilibrium of Rotating Masses of
\index{Darwin, G. H.!Jacobi's ellipsoid}%
Fluid}, ``Transactions of Royal Society,'' vol.~178, 1887.

G.~H. Darwin, \Title{Jacobi's Figure of Equilibrium},~\&c., ``Proceedings
\index{Equilibrium, figures of, of rotating liquid|)}%
\index{Figure of equilibrium!of rotating liquid|)}%
Roy.\ Soc.,'' vol.~41, 1886, p.~319.

S.~Krüger, \Title{Ellipsoidale Evenwichtsvormen},~\&c., Leeuwen, Leiden,
\index{Krüger, figures of equilibrium of liquid}%
1896; \Title{Sur l'ellipsoïde de Jacobi}, ``Nieuw Archief voor Wiskunde,''
2d~series, 3d~part, 1898. The author shows that G.~H.
Darwin had been forestalled in much of his work on Jacobi's
figure, and he corrects certain mistakes.
\end{Authorities}
\index{Rotating liquid, figures of equilibrium|)}%
\PageSep{334}


\Chapter{XIX}
{The Evolution of Celestial Systems}

\First{Men} will always aspire to peer into the remote
\index{Evolution of celestial systems|(}%
\index{Solar!system, nebular hypothesis as to origin of|(}%
past to the utmost of their power, and the fact
that their success or failure cannot appreciably
influence their life on the earth will never deter
them from such endeavors. From this point
of view the investigations explained in the last
chapter acquire much interest, since they form
the basis of the theories of cosmogony which
seem most probable by the light of our present
knowledge.

We have seen that an annular figure of equilibrium
\index{Kant!nebular hypothesis|(}%
\index{Nebular hypothesis|(}%
actually exists in Plateau's experiment,
and it is almost certainly a possible form amongst
celestial bodies. Plateau's ring has however
only a transient existence, and tends to break up
into globules, spinning on their axes and revolving
round the centre. In this result we saw a
close analogy with the origin of the planets, and
regarded his experiment as confirmatory of the
Nebular Hypothesis, of which I shall now give a
short account.\footnote
  {My knowledge of the history of the Nebular Hypothesis is
  entirely derived from an interesting paper by Mr.~G.~F. Becker,
\index{Becker, G. F., on Nebular Hypothesis}%
  on ``Kant as a Natural Philosopher,'' \Title{American Journal of Science},
  vol.~v. Feb.~1898.}
\PageSep{335}

The first germs of this theory are to be found
in Descartes' ``Principles of Philosophy,'' published
\index{Descartes, vortical theory of cosmogony}%
in~1644. According to him the sun and
planets were represented by eddies or vortices in
a primitive chaos of matter, which afterwards
formed the centres for the accretion of matter.
As the theory of universal gravitation was propounded
for the first time half a century later
than the date of Descartes' book, it does not
seem worth while to follow his speculations
further. Swedenborg formulated another vortical
cosmogony in~1734, and Thomas Wright of
\index{Wright, Thomas, on a theory of cosmogony}%
Durham published in 1750 a book of preternatural
dullness on the same subject. It might not
have been worth while to mention Wright, but
that Kant acknowledges his obligation to him.

The Nebular Hypothesis has been commonly
associated with the name of Laplace, and he undoubtedly
\index{Laplace!nebular hypothesis|(}%
avoided certain errors into which his
precursors had fallen. I shall therefore explain
Laplace's theory, and afterwards show how he
was, in most respects, really forestalled by the
great German philosopher Kant.

Laplace supposed that the matter now forming
the solar system once existed in the form of a
lens-shaped nebula of highly rarefied gas, that it
rotated slowly about an axis perpendicular to the
present orbits of the planets, and that the nebula
extended beyond the present orbit of the furthest
planet. The gas was at first expanded by heat,
\PageSep{336}
and as the surface cooled the central portion
condensed and its temperature rose. The speed
of rotation increased in consequence of the contraction,
according to a well known law of mechanics
called ``the conservation of moment of
momentum;''\footnote
  {Kant fell into error through ignorance of the generality of
  this law, for he imagined that rotation could be generated from
  rest.}
the edges of the lenticular mass
of gas then ceased to be continuous with the
more central portion, and a ring of matter was
detached, in much the same way as in Plateau's
experiment. Further cooling led to further contraction
and consequently to increased rotation,
until a second ring was shed, and so on successively.
The rings then ruptured and aggregated
themselves into planets whilst the central nucleus
formed the sun.

Virtually the same theory had been propounded
by Kant many years previously, but I am not
aware that there is any reason to suppose that
Laplace had ever read Kant's works. In a paper,
to which I have referred above, Mr.~G.~F.
Becker makes the following excellent summary
\index{Becker, G. F., on Nebular Hypothesis|(}%
of the relative merits of Kant and Laplace; he
writes:---

``Kant seems to have anticipated Laplace almost
completely in the more essential portions
of the nebular hypothesis. The great Frenchman
was a child when Kant's theory was issued,
\PageSep{337}
and the `Système du Monde,' which closes with
the nebular hypothesis, did not appear until
1796. Laplace, like Kant, infers unity of origin
for the members of the solar system from the
similarity of their movements, the small obliquity
and small eccentricity of the orbits of either
planets or satellites.\footnote
  {``The retrograde satellites of Uranus were discovered by
  Herschel in~1787, but Laplace in his hypothesis does not refer to
  them.''}
Only a fluid extending
throughout the solar system could have produced
such a result. He is led to conclude that the
atmosphere of the sun, in virtue of excessive
heat, originally extended beyond the solar system
and gradually shrank to its present limits. This
nebula was endowed with moment of momentum
which Kant tried to develop by collisions. Planets
formed from zones of vapor, which on breaking
agglomerated\dots. The main points of
comparison between Kant and Laplace seem to
be these. Kant begins with a cold, stationary
nebula which, however, becomes hot by compression
and at its first regenesis would be in a state
of rotation. It is with a hot, rotating nebula
that Laplace starts, without any attempt to account
for the heat. Kant supposes annular
zones of freely revolving nebulous matter to
gather together by attraction during condensation
of the nebula. Laplace supposes rings left
\index{Laplace!nebular hypothesis|)}%
behind by the cooling of the nebula to agglomerate
\PageSep{338}
in the same way as Kant had done. While
both appeal to the rings of Saturn as an example
of the hypothesis, neither explains satisfactorily
why the planetary rings are not as stable
as those of Saturn. Both assert that the positive
rotation of the planets is a necessary
consequence of agglomeration, but neither is
sufficiently explicit. The genesis of satellites is
for each of them a repetition on a small scale of
the formation of the system\dots. While Laplace
assigns no cause for the heat which he ascribes
to his nebula, Lord Kelvin goes further
back and supposes a cold nebula consisting of
separate atoms or of meteoric stones, initially
possessed of a resultant moment of momentum
equal or superior to that of the solar system.
Collision at the centre will reduce them to a
vapor which then expanding far beyond Neptune's
orbit will give a nebula such as Laplace
postulates.\footnote
  {\Title{Popular Lectures}, vol.~i.\ p.~421.}
Thus Kelvin goes back to the same
initial condition as Kant, excepting that Kant
endeavored (of course vainly) to develop a moment
of momentum for his system from collisions.''\footnote
  {Becker, \Title{Amer.\ Journ.\ Science}, vol.~v. 1898, pp.~107,~108.}
\index{Becker, G. F., on Nebular Hypothesis|)}%

There is good reason for believing that the
Nebular Hypothesis presents a true statement in
outline of the origin of the solar system, and of
the planetary subsystems, because photographs
%% Plate 3
\TallFig[0.85]{40}{Nebula in Andromeda}{jpg}
%% Facing page
%[Blank Page]
\PageSep{339}
of nebulæ have been taken recently in which we
can almost see the process in action. \Fig{40} is
a reproduction of a remarkable photograph by
Dr.~Isaac Roberts of the great nebula in the constellation
\index{Roberts, I., photograph of nebula in Andromeda}%
of Andromeda. In it we may see the
\index{Andromeda, nebula in}%
\index{Nebula in Andromeda}%
lenticular nebula with its central condensation,
the annulation of the outer portions, and even
the condensations in the rings which will doubtless
at some time form planets. This system is
built on a colossal scale, compared with which
our solar system is utterly insignificant. Other
\index{Solar!system, nebular hypothesis as to origin of|)}%
\index{Solar!system, distribution of satellites in|(}%
nebulæ show the same thing, and although they
are less striking we derive from them good
grounds for accepting this theory of evolution
as substantially true.

\TB

I explained in \Ref{Chapter}{XVI}.\ how the theory
of tidal friction showed that the moon took her
origin very near to the present surface of the
earth. But it was also pointed out that the same
theory cannot be invoked to explain an origin
for the planets at a point close to the sun. They
must in fact have always moved at nearly their
present distances. In the same way the dimensions
of the orbits of the satellites of Mars, Jupiter,
\index{Satellites!distribution of, in solar system|(}%
Saturn, and Neptune cannot have been
largely augmented, whatever other effects tidal
friction may have had. We must therefore still
rely on the Nebular Hypothesis for the explanation
\index{Nebular hypothesis|)}%
of the main features of the system as a
whole.
\index{Kant!nebular hypothesis|)}%
\PageSep{340}

It may, at first sight, appear illogical to maintain
that an action, predominant in its influence
on our satellite, should have been insignificant
in regulating the orbits of all the other bodies
of the system. But this is not so, for whilst the
earth is only $80$~times as heavy as the moon, Saturn
weighs about $4,600$~times as much as its
satellite Titan, which is by far the largest satellite
in the solar system; and all the other satellites
are almost infinitesimal in comparison with their
primaries. Since, then, the relationship of the
moon to the earth is unique, it may be fairly contended
that a factor of evolution, which has been
predominant in our own history, has been relatively
insignificant elsewhere.

There is indeed a reason explanatory of this
singularity in the moon and earth; it lies in the
fact that the earth is nearer to the sun than any
other planet attended by a satellite. To explain
the bearing of this fact on the origin of satellites
and on their sizes, I must now show how tidal
friction has probably operated as a perturbing
influence in the sequence of events, which would
be normal according to the Nebular Hypothesis.

We have seen that rings should be shed from
the central nucleus, when the contraction of the
nebula has induced a certain degree of augmentation
of rotation. Now if the rotation were
retarded by some external cause, the genesis of
a ring would be retarded, or might be entirely
prevented.
\PageSep{341}

The friction of the solar tides in a planetary
nebula furnishes such an external cause, and accordingly
the rotation of a planetary nebula near
to the sun might be so much retarded that a ring
would never be detached from it, and no satellite
would be generated. From this point of view
it is noteworthy that Mercury and Venus have
no satellites; that Mars has two, Jupiter five,
and that all the exterior planets have several
satellites. I suggest then that the solar tidal
friction of the terrestrial nebula was sufficient to
retard the birth of a satellite, but not to prevent
it, and that the planetary mass had contracted
to nearly the present dimensions of the earth
and had partially condensed into the solid and
liquid forms, before the rotation had augmented
sufficiently to permit the birth of a satellite.
When satellites arise under conditions which are
widely different, it is reasonable to suppose that
their masses will also differ much. Hence we can
understand how it has come about that the relationship
between the moon and the earth is so
unlike that between other satellites and their
planets. In \Ref{Chapter}{XVII}.\ I showed that there
are reasons for believing that solar tidal friction
\index{Solar!system, distribution of satellites in|)}%
has really been an efficient cause of change, and
this makes it legitimate to invoke its aid in explaining
the birth and distribution of satellites.
\index{Satellites!distribution of, in solar system|)}%

\TB

In speaking of the origin of the moon I have
\PageSep{342}
\index{Eccentricity of orbit!theory of, in case of double stars}%
\index{See, T. J. J.!theory of evolution of double stars|(}%
been careful not to imply that the matter of
which she is formed was necessarily first arranged
in the form of a ring. Indeed, the genesis of
the hour-glass figure of equilibrium from Jacobi's
form and its fission into two parts indicate the
possibility of an entirely different sequence of
events. It may perhaps be conjectured that the
moon was detached from the primitive earth in
this way, possibly with the help of tidal oscillations
due to the solar action. Even if this suggestion
is only a guess, it is interesting to make
such speculations, when they have some basis of
reason.

In recent years astronomers have been trying,
principally by aid of the spectroscope, to determine
the orbits of pairs of double stars around
\index{Stars!theory of evolution|(}%
one another. It has been observed that, in the
majority of these systems, the masses of the two
component stars do not differ from one another
extremely; and Dr.~See, who has specially devoted
himself to this research, has drawn attention
to the great contrast between these systems
and that of the sun, attended by a retinue of
infinitesimal planets. He maintains, with justice,
that the paths of evolution pursued in the two
cases have probably also been strikingly different.

It is hardly credible that two stars should
have gained their present companionship by an
accidental approach from infinite space. They
cannot always have moved as they do now, and
\PageSep{343}
so we are driven to reflect on the changes which
might supervene in such a system under the
action of known forces.

The only efficient interaction between a pair
of celestial bodies, which is known hitherto, is
a tidal one, and the friction of the oscillations
introduces a cause of change in the system.
Tidal friction tends to increase the eccentricity
of the orbit in which two bodies revolve about
one another, and its efficiency is much increased
when the pair are not very unequal in mass and
when each is perturbed by the tides due to the
other. The fact that the orbits of the majority
of the known pairs are very eccentric affords a
reason for accepting the tidal explanation. The
only adverse reason, that I know of, is that the
eccentricities are frequently so great that we
may perhaps be putting too severe a strain on
the supposed cause.

But the principal effect of tidal friction is the
repulsion of the two bodies from one another,
so that when their history is traced backwards
we ultimately find them close together. If then
this cause has been as potent as Dr.~See believes
it to have been, the two components of a binary
system must once have been close together.
From this stage it is but a step to picture to
ourselves the rupture of a nebula, in the form
of an hour-glass, into two detached masses.

The theory embraces all the facts of the case,
\PageSep{344}
and as such is worthy of at least a provisional
acceptance. But we must not disguise from
ourselves that out of the thousands, and perhaps
millions of double stars which may be visible
from the earth, we only as yet know the orbits
and masses of a dozen.

Many years ago Sir John Herschel drew a
\index{Herschel, observations of twin nebulæ}%
number of twin nebulæ as they appear through
a powerful telescope. The drawings probably
possess the highest degree of accuracy attainable
by this method of delineation, and the shapes
present evidence confirmatory of the theory of
the fission of nebulæ adopted by Dr.~See. But
since Herschel's time it has been discovered that
many details, to which our eyes must remain forever
blind, are revealed by celestial photography.
The photographic film is, in fact, sensitive to
those ``actinic'' rays which we may call invisible
light, and many nebulæ are now found to be
hardly recognizable, when photographs of them
are compared with drawings. A conspicuous
example of this is furnished by the great nebula
in Andromeda, illustrated above in~\fig{40}.

Photographs, however, do not always aid interpretation,
for there are some which serve only
to increase the chaos visible with the telescope.
We may suspect, indeed, that the complete system
of a nebula often contains masses of cold
and photographically invisible gas, and in such
cases it would seem that the true nature of the
whole will always be concealed from us.
\PageSep{345}

Another group of strange celestial objects is
that of the spiral nebulæ, whose forms irresistibly
\index{Nebulae@Nebulæ, description of various}%
suggest violent whirlpools of incandescent
gas. Although in all probability the motion of
the gas is very rapid, yet no change of form has
been detected. We are here reminded of a
rapid stream rushing past a post, where the form
of the surface remains constant whilst the water
itself is in rapid movement; and it seems reasonable
to suppose that in these nebulæ it is
only the lines of the flow of the gas which are
visible. Again, there are other cases in which
the telescopic view may be almost deceptive in
its physical suggestions. Thus the Dumb-Bell
\index{Dumb-bell nebula, description of photograph of}%
nebula (27~Messier Vulpeculæ), as seen telescopically,
might be taken as a good illustration of a
nebula almost ready to split into two stars. If
this were so, the rotation would be about an
axis at right angles to the length of the nebula.
But a photograph of this object shows that the
system really consists of a luminous globe surrounded
by a thick and less luminous ring, and
that the opacity of the sides of the ring takes a
bite, as it were, out of each side of the disk, and
so gives it the apparent form of a dumb-bell.
In this case the rotation must be about an axis
at right angles to the ring, and therefore along
the length of the dumb-bell. It is proper to
add that Dr.~See is well aware of this, and does
not refer to this nebula as a case of incipient
fission.
\PageSep{346}

I have made these remarks in order to show
that every theory of stellar evolution must be
full of difficulty and uncertainty. According to
our present knowledge Dr.~See's theory appears
to have much in its favor, but we must await its
confirmation or refutation from the results of
future researches with the photographic plate,
the spectroscope, and the telescope.

\begin{Authorities}
Mr.~G.~F. Becker (\Title{Amer.\ Jour.\ Science}, vol.~v. 1898, art.~xv.)\
gives the following references to Kant's work: \Title{Sämmtliche
Werke}, ed.~Hartenstein, 1868 (Tidal Friction and the Aging of
the Earth), vol.~i.\ pp.~179--206; (Nebular Hypothesis), vol.~i.\
pp.~207--345.

Laplace, \Title{Système du Monde}, last appendix; the tidal retardation
of the moon's rotation is only mentioned in the later
editions.

T.~J.~J. See, \Title{Die Entwickelung der Doppelstern-systeme}, ``Inaugural
Dissertation,'' 1892. Schade, Berlin.

T.~J.~J. See, \Title{Evolution of the Stellar Systems}, vol.~i.\ 1896.
Nichols Press, Lynn, Massachusetts. Also a popular article,
\Title{The Atlantic Monthly}, October, 1897.

G.~H. Darwin, \Title{Tidal Friction~\dots\ and Evolution}, ``Phil.\ Trans.\
\index{Darwin, G. H.!evolution of satellites}%
Roy.\ Soc.,'' part~ii.\ 1881, p.~525.
\end{Authorities}
\index{Evolution of celestial systems|)}%
\index{See, T. J. J.!theory of evolution of double stars|)}%
\index{Stars!theory of evolution|)}%
\PageSep{347}


\Chapter[Saturn's Rings]{XX}
{Saturn's Rings\protect\footnotemark}

\footnotetext{Part of this chapter appeared as an article in \Title{Harper's
  Magazine} for June,~1889.}

\First{To} the naked eye Saturn appears as a brilliant
\index{Saturn!description and picture|(}%
star, which shines, without twinkling, with a
yellowish light. It is always to be found very
nearly in the ecliptic, moving slowly amongst
the fixed stars at the rate of only thirteen degrees
per annum. It is the second largest
planet of the solar system, being only exceeded
in size by the giant Jupiter. It weighs $91$~times
as much as our earth, but, being as light as cork,
occupies $690$~times the volume, and is nine times
as great in circumference. Notwithstanding its
great size it rotates around its axis far more
rapidly than does the earth, its day being only
$10\frac{1}{2}$~of our hours. It is ten times as far from
the sun as we are, and its year, or time of revolution
round the sun, is equal to thirty of our
years. It was deemed by the early astronomers
to be the planet furthest from the sun, but that
was before the discovery by Herschel, at the
end of the last century, of the further planet
Uranus, and that of the still more distant Neptune
by Adams and Leverrier in the year~1846.
\PageSep{348}

The telescope has shown that Saturn is attended
by a retinue of satellites almost as numerous
as, and closely analogous to, the planets
circling round the sun. These moons are eight
in number, are of the most various sizes, the
largest as great as the planet Mars, and the
smallest very small, and are equally diverse in
respect of their distances from the planet. But
besides its eight moons Saturn has another attendant
absolutely unique in the heavens; it is
girdled with a flat ring, which, like the planet
itself, is only rendered visible to us by the
illumination of sunlight. \Fig{41}, to which
further reference is made below, shows the general
appearance of the planet and of its ring.
The theory of the physical constitution of that
ring forms the subject of the present chapter.

A system so rich in details, so diversified and
so extraordinary, would afford, and doubtless
has afforded, the subject for many descriptive
essays; but description is not my present object.

The existence of the ring of Saturn seems
now a very commonplace piece of knowledge,
and yet it is not $300$~years since the moons of
Jupiter and Saturn were first detected, and since
suspicion was first aroused that there was something
altogether peculiar about the Saturnian
system. These discoveries, indeed, depended
entirely on the invention of the telescope. It
may assist the reader to realize how necessary
\PageSep{349}
%[** TN: Oriented vertically in the original]
\Figure{41}{The Planet Saturn}{jpg}
\PageSep{350}
the aid of that instrument was when I say that
Saturn, when at his nearest to us, is the same in
size as a sixpenny piece held up at a distance of
$210$~yards.

It was the celebrated Galileo who first invented
\index{Galileo!Saturn's ring}%
a combination of lenses such as is still
used in our present opera-glasses, for the purpose
of magnifying distant objects.

In July of~1610 he began to examine Saturn
with his telescope. His most powerful instrument
only magnified $32$~times, and although
such an enlargement should have amply sufficed
to enable him to make out the ring, yet he persuaded
himself that what he saw was a large
bright disk, with two smaller ones touching it,
one on each side. His lenses were doubtless
imperfect, but the principal cause of his error
must have been the extreme improbability of the
existence of a ring girdling the planet. He
wrote an account of what he had seen to the
Grand Duke of Tuscany, Giuliano de'~Medici,
and to others; he also published to the world an
anagram which, when the letters were properly
arranged, read as follows: ``Altissimum planetam
tergeminum observavi'' (I have seen the
furthest planet as triple), for it must be remembered
that Saturn was then the furthest known
planet.

In 1612 Galileo again examined Saturn, and
was utterly perplexed and discouraged to find
\PageSep{351}
his triple star replaced by a single disk. He
writes, ``Is it possible that some mocking demon
has deceived me?'' And here it may be well to
remark that there are several positions in which
Saturn's rings vanish from sight, or so nearly
vanish as to be only visible with the most powerful
modern telescopes. When the plane of the
ring passes through the sun, only its very thin
edge is illuminated; this was the case in~1612,
when Galileo lost it; secondly, if the plane of
the ring passes through the earth, we have only
a very thin edge to look at; and thirdly, when
the sun and the earth are on opposite sides of
the ring, the face of the ring which is presented
to us is in shadow, and therefore invisible.

Some time afterwards Galileo's perplexity was
increased by seeing that the planet had then a
pair of arms, but he never succeeded in unraveling
the mystery, and blindness closed his career
as an astronomer in~1626.

About thirty years after this, the great Dutch
astronomer Huyghens, having invented a new
\index{Huyghens, discovery of Saturn's ring}%
sort of telescope (on the principle of our present
powerful refractors), began to examine the planet
and saw that it was furnished with two loops or
handles. Soon after the ring disappeared; but
when, in~1659, it came into view again, he at
last recognized its true character, and announced
that the planet was attended by a broad, flat
ring.
\PageSep{352}

A few years later it was perceived that there
were two rings, concentric with one another.
The division, which may be easily seen in drawings
of the planet, is still named after Cassini,
\index{Cassini, discovery of division in Saturn's rings}%
one of its discoverers. Subsequent observers
have detected other less marked divisions.

Nearly two centuries later, namely, in~1850,
Bond in America and Dawes in England, independently
\index{Bond, discovery of inner ring of Saturn}%
\index{Dawes, discovery of inner ring of Saturn}%
and within a fortnight of the same
time, observed that inside of the well-known
bright rings there is another very faint dark
ring, which is so transparent that the edge of
the planet is visible through it. There is some
reason to believe that this ring has really become
more conspicuous within the last $200$~years,
so that it would not be right to attribute the
lateness of its detection entirely to the imperfection
of earlier observations.

It was already discovered in the last century
that the ring is not quite of the same thickness
at all points of its circumference, that it is not
strictly concentric with the planet, and that it
revolves round its centre. Herschel, with his
magnificent reflecting telescope, detected little
beads on the outer ring, and by watching these
he concluded that the ring completes its revolution
in $10\frac{1}{2}$~hours.

This sketch of the discovery and observation
of Saturn's rings has been necessarily very incomplete,
but we have perhaps already occupied
too much space with it.
\PageSep{353}

\Fig{41} exhibits the appearance of Saturn and
his ring. The drawing is by Bond of Harvard
University, and is considered an excellent one.

It is usual to represent the planets as they are
seen through an astronomical telescope, that is
\Figure{42}{Diagram of Saturn and his Rings}{png}
to say, reversed. Thus in \fig{41} the south
pole of the planet is at the top of the plate, and
unless the telescope were being driven by clockwork,
the planet would appear to move across
the field of view from right to left.

The plane of the ring is coincident with the
equator of the planet, and both ring and equator
are inclined to the plane of the planet's orbit at
an angle of $27$~degrees.

A whole essay might be devoted to the discussion
of this and of other pictures, but we must
confine ourselves to drawing attention to the
well-marked split, called Cassini's division, and
\PageSep{354}
to the faint internal ring, through which the
edge of the planet is visible.

The scale on which the whole system is constructed
is best seen in a diagram of concentric
circles, showing the limits of the planet's body
and of the successive rings. Such a diagram,
with explanatory notes, is given in~\fig{42}.

An explanation of the outermost circle, called
\emph{Roche's limit}, will be given later. The following
are the dimensions of the system:---
\begin{center}
\begin{tabular}{l>{\qquad}r}
Equatorial diameter of planet     &  $73,000$ miles \\
Interior diameter of dark ring    &  $93,000$ \Ditto{miles} \\
Interior diameter of bright rings & $111,000$ \Ditto{miles} \\
Exterior diameter of bright rings & $169,000$ \Ditto{miles}
\end{tabular}
\end{center}
We may also remark that the radius of the
limit of the rings is $2.38$~times the mean radius
of the planet, whilst Roche's limit is $2.44$~such
radii. The greatest thickness of the ring is uncertain,
but it seems probable that it does not
exceed $200$~or $300$~miles.

The pictorial interest, as we may call it, of all
this wonderful combination is obvious, but our
curiosity is further stimulated when we reflect on
the difficulty of reconciling the existence of this
strange satellite with what we know of our own
planet and of other celestial bodies.

It may be admitted that no disturbance to our
ordinary way of life would take place if Saturn's
\index{Saturn!description and picture|)}%
rings were annihilated, but, as Clerk-Maxwell
has remarked, ``from a purely scientific point of
\PageSep{355}
view, they become the most remarkable bodies in
the heavens, except, perhaps, those still less \emph{useful}
bodies---the spiral nebulæ. When we have
actually seen that great arch swung over the
equator of the planet without any visible connection,
we cannot bring our minds to rest. We
cannot simply admit that such is the case, and
describe it as one of the observed facts of nature,
not admitting or requiring explanation. We
must either explain its motion on the principles
of mechanics, or admit that, in Saturnian realms,
there can be motion regulated by laws which we
are unable to explain.''

I must now revert to the subject of \Ref{Chapter}{XVIII}.\
and show how the investigations, there
explained, bear on the system of the planet. We
then imagined a liquid satellite revolving in a
circular orbit about a liquid planet, and supposed
that each of these two masses moved so as always
to present the same face to the other. It was
pointed out that each body must be somewhat
flattened by its rotation round an axis at right
angles to the plane of the orbit, and that the
tidal attraction of each must deform the other.
In the application of this theory to the system of
Saturn it is not necessary to consider further the
tidal action of the satellite on the planet, and we
must concentrate our attention on the action of
the planet on the satellite. We have found reason
to suppose that the earth once raised enormous
\PageSep{356}
\index{Saturn!theory of ring|(}%
tides in the moon, when her body was
molten, and any planet must act in the same way
on its satellite. When, as we now suppose, the
satellite moves so as always to present the same
face to the planet, the tide is fixed and degenerates
into a permanent distortion of the equator
of the satellite into an elliptic shape. If the
satellite is very small compared with its planet,
and if it is gradually brought closer and closer
to the planet, the tide-generating force, which
varies inversely as the cube of the distance, increases
with great rapidity, and we shall find the
satellite to assume a more and more elongated
shape. When the satellite is not excessively
small, the two bodies may be brought together
until they actually touch, and form the hour-glass
figure exhibited in \fig{39}, \PageRef{p.}{329}.

The general question of the limiting proximity
of a liquid planet and satellite which just insures
stability is as yet unsolved. But it has been
proved that there is one case in which instability
sets in. Édouard Roche has shown that this approach
\index{Roche, E.!theory of limit and Saturn's ring|(}%
up to contact is not possible when the
satellite is very small, for at a certain distance
the tidal distortion of a small satellite becomes
so extreme that it can no longer subsist as a
single mass of fluid. He also calculated the
form of the satellite when it is elongated as much
as possible. \Fig{43} represents the satellite in
its limiting form. We must suppose the planet
\PageSep{357}
about which it revolves to be a large globe, with
its centre lying on the prolongation of the longest
axis of the egg-like body in the direction
of~$E$. As it revolves, the longest axis of the satellite
always points straight towards its planet.
The egg, though not strictly circular in girth, is
\Figure[0.7]{43}{Roche's Figure of a Satellite when elongated
to the utmost}{png}
very nearly so. Thus another section at right
angles to this one would be of nearly the same
shape. One diameter of the girth is in fact only
longer than the other by a seventeenth part.
The shortest of the three axes of the slightly flattened
egg is at right angles to the plane of the
orbit in which the satellite revolves. The longest
axis of the body is nearly twice as long as
either of the two shorter ones; for if we take
the longest as~$\DPchg{1000}{1,000}$, the other two would be $496$
and~$469$. \Fig{43} represents a section through
the two axes equal respectively to~$\DPchg{1000}{1,000}$ and to~$469$,
so that we are here supposed to be looking
at the satellite's orbit edgewise.
\PageSep{358}

But, as I have said, Roche determined not
\index{Earth and moon!Roche's limit for}%
only the shape of the satellite when thus elongated
to the utmost possible extent, but also in
its nearness to the planet, and he proved that if
the planet and satellite be formed of matter of
the same density, the centre of such a satellite
must be at a distance from the planet's centre of
$2\frac{11}{25}$~of the planet's radius. This distance of $2\frac{11}{25}$
or $2.44$~of a planet's radius I call Roche's limit
for that planet. The meaning of this is that inside
of a circle drawn around a planet at a distance
so proportionate to its radius no small
satellite can circulate; the reason being that if
a lump of matter were started to revolve about
the planet inside of that circle, it would be torn
to pieces under the action of the forces we have
been considering. It is true that if the lump of
matter were so small as to be more properly described
as a stone than as a satellite, then the
cohesive force of stone might be strong enough
to resist the disruptive force. But the size for
which cohesion is sufficient to hold a mass of
matter together is small compared with the
smallest satellite.

I have said that Roche's limit as evaluated at
$2.44$~radii is dependent on the assumption of
equal densities in the satellite and planet. If
the planet be denser than the satellite, Roche's
limit is a larger multiple of the planet's radius,
and if it be less dense the multiple is smaller.
\PageSep{359}
But the variation of distance is not great for
considerable variations in the relative densities
of the two bodies, the law being that the~$2.44$
must be multiplied by the cube root of the ratio
of the density of the planet to that of the satellite.
If for example the planet be on the average
of its whole volume twice as dense as the
satellite, the limit is only augmented from $2.44$
to $3$~times the planet's radius; and if it be half
as dense, the $2.44$ is depressed to~$1.94$. Thus
the variation of density of the planet from a
half to twice that of the planet---that is to
say, the multiplication of the smaller density by
four---only changes Roche's limit from $2$ to $3$~radii.
It follows from this that, within pretty
wide limits of variation of relative densities,
Roche's limit changes but little.

The only relative density of planet and satellite
that we know with accuracy is that of the
earth and moon. Now the earth is more dense
than the moon in the proportion of $8$~to~$5$; hence
Roche's limit for the earth is the cube root of~$\frac{8}{5}$
multiplied by~$2.44$, that is to say, it is $2.86$~times
the earth's radius. It follows that if the moon
were to revolve at a distance of less than $2.86$~radii,
or $11,000$~miles, she would be torn to pieces
by the earth's tidal force.

If this result be compared with the conclusions
drawn from the theory of tidal friction, it follows
that at the earliest stage to which the moon was
\PageSep{360}
\index{Saturn!Roche's limit for}%
traced, she could not have existed in her present
form, but the matter which is now consolidated
in the form of a satellite must then have been a
mere swarm of loose fragments. Such fragments,
if concentrated in one part of the orbit, would
be nearly as efficient in generating tides in the
planet as though they were agglomerated in the
form of a satellite. Accordingly the action of
tidal friction does not necessitate the agglomeration
of the satellite. The origin and earliest history
of the moon must always remain highly
speculative, and it seems fruitless to formulate
exact theories on the subject.\footnote
  {Mr.~Nolan has criticised the theory of tidal friction from
\index{Nolan, criticism of tidal theory of moon's origin}%
  this point of view (\Title{Genesis of the Moon}, Melbourne, 1885; also
  \Title{Nature}, Feb.~18 and July~29, 1886).}

When we apply this reasoning to the other
planets, exact data are wanting. The planet
Mars resembles the earth in so many respects
\index{Mars!Roche's limit}%
that it is reasonable to suppose that there is much
the same relationship between the densities of
the planet and satellites as with us. As with the
case of the earth and moon, this would bring
Roche's limit to $2.86$~times the planet's radius.
The satellite Phobos, however, revolves at a
distance of $2.75$~radii of Mars; hence we are
bound to suppose that the density of Phobos is
a very little more nearly equal to that of Mars
than in the case of the moon and earth; if
it were not so, Phobos would be disrupted by
\PageSep{361}
tidal action. How interesting it will be if future
generations shall cease to see the satellite Phobos,
for they will then conclude that Phobos has been
drawn within the charmed circle, and has been
broken to pieces.

In considering the planets Jupiter and Saturn,
\index{Jupiter!Roche's limit for}%
we are deprived of the indications which are useful
in the case of Mars. The satellites are probably
solid, and these planets are known to have
a low mean density. Hence it is probable that
Roche's limit is a somewhat smaller multiple than
$2.44$~of the radii of Jupiter and Saturn. The
only satellite which is in danger is the innermost
and recently discovered satellite of Jupiter, which
revolves at $2.6$~times the planet's mean radius,
for with the same ratio of densities as obtains
here the satellite would be broken up. This confirms
the conclusion that the mean density of
Jupiter is at least not greater than that of the
satellite.

We are also ignorant of the relative densities
of Saturn and its satellites, and so in the figure
Roche's limit is placed at $2.44$~times the planet's
radius, corresponding to equal densities. But
the density of the planet is very small, and therefore
the limit is almost certainly slightly nearer
to the planet than is shown.

This system affords the only known instance
where matter is clearly visible circulating round
an attractive centre at a distance certainly less
\PageSep{362}
than the theoretical limit, and the belief seems
justified that Saturn's rings consist of dust and
fragments.

Although Roche himself dismissed this matter
in one or two sentences, he saw the full bearing
of his remarks, and to do him justice we should
date from~1848 the proof that Saturn's rings
consist of meteoric stones.

The theoretical limit lies just outside the limit
of the rings, but we may suspect that the relative
densities of the planet and satellite are such that
the limit should be displaced to a distance just
inside of the outer edge of the ring, because any
solid satellite would almost necessarily have a
mean density greater than that of the planet.

Although Roche's paper was published about
fifty years ago, it has only recently been mentioned
in text-books and general treatises. Indeed,
it has been stated that Bond was the first
in modern times to suggest the meteoric constitution
of the rings. His suggestion, based on
telescopic evidence, was however made in~1851.

\TB

And now to explain how a Cambridge mathematician
to whom reference was made above, in
ignorance of Roche's work of nine years before,
\index{Roche, E.!theory of limit and Saturn's ring|)}%
arrived at the same conclusion. In~1857, Clerk-Maxwell,
one of the most brilliant men of science
who have taught in the University of Cambridge,
and whose early death we still deplore, attacked
\PageSep{363}
\index{Instability!of Saturn's ring}%
the problem of Saturn's rings in a celebrated
essay, which gained for him what is called the
Adams prize. Laplace had early in the century
considered the theory that the ring is solid, and
Maxwell first took up the question of the motion
\index{Maxwell on Saturn's ring|(}%
of such a solid ring at the point where it had
been left. He determined what amount of
weighting at one point of a solid uniform ring is
necessary to insure its steady motion round the
planet. He found that there must be a mass
attached to the circumference of the ring weighing
$4\frac{1}{2}$~times as much as the ring itself. In fact,
the system becomes a satellite with a light ring
attached to it.

``As there is no appearance,'' he says, ``about
the rings justifying a belief in so great an irregularity,
the theory of the solidity of the rings
becomes very improbable. When we come to
consider the additional difficulty of the tendency
of the fluid or loose parts of the ring to accumulate
at the thicker parts, and thus to destroy that
nice adjustment of the load on which the stability
depends, we have another powerful argument
against solidity. And when we consider the immense
size of the rings and their comparative
thinness, the absurdity of treating them as rigid
bodies becomes self-evident. An iron ring of
such a size would be not only plastic, but semi-fluid,
under the forces which it would experience,
and we have no reason to believe these rings to
\PageSep{364}
\index{Instability!of Saturn's ring}%
be artificially strengthened with any material
unknown on this earth.''

The hypothesis of solidity being condemned,
Maxwell proceeds to suppose that the ring is
composed of a number of equal small satellites.
This is a step towards the hypothesis of an indefinite
number of meteorites of all sizes. The
consideration of the motion of these equal satellites
affords a problem of immense difficulty, for
each satellite is attracted by all the others and
by the planet, and they are all in motion.

If they were arranged in a circle round the
planet at equal distances, they might continue to
revolve round the planet, provided that each
satellite remained in its place with mathematical
exactness. Let us consider that the proper place
of each satellite is at the ends of the spokes of
a revolving wheel, and then let us suppose that
none of them is exactly in its place, some being
a little too far advanced, some a little behind,
some too near and some too far from the centre
of the wheel---that is to say, from the planet---then
we want to know whether they will swing
to and fro in the neighborhood of their places,
or will get further and further from their places,
and whether the ring will end in confusion.

Maxwell treated this problem with consummate
skill, and showed that if the satellites were
not too large, confusion would not ensue, but
each satellite would oscillate about its proper
place.
\PageSep{365}
\index{Stability!of Saturn's ring}%

At any moment there are places where the
satellites are crowded and others where they are
spaced out, and he showed that the places of
crowding and of spacing out will travel round
the ring at a different speed from that with
which the ring as a whole revolves. In other
words, waves of condensation and of rarefaction
are propagated round the ring as it rotates.

He constructed a model, now in the laboratory
at Cambridge, to exhibit these movements; it is
pretty to observe the changes of the shape of the
ring and of the crowding of the model satellites
as they revolve.

I cannot sum up the general conclusions at
which Maxwell arrived better than by quoting
his own words.

In the summary of his paper he says:---

``If the satellites are unequal, the propagation
of waves will no longer be regular, but the disturbances
of the ring will in this, as in the
former case, produce only waves, and not growing
confusion. Supposing the ring to consist,
not of a single row of large-satellites, but of a
cloud of evenly distributed unconnected particles,
we found that such a cloud must have a
very small density in order to be permanent, and
that this is inconsistent with its outer and inner
parts moving with the same angular velocity.
Supposing the ring to be fluid and continuous,
we found that it will necessarily be broken up
into small portions.
\PageSep{366}
\index{Stability!of Saturn's ring}%

``We conclude, therefore, that the rings must
consist of disconnected particles; these may be
either solid or liquid, but they must be independent.
The entire system of rings must therefore
consist either of a series of many concentric
rings, each moving with its own velocity, and
having its own system of waves, or else of a confused
multitude of revolving particles, not arranged
in rings, and continually coming into
collision with each other.

``Taking the first case, we found that in an
indefinite number of possible cases the mutual
perturbation of two rings, stable in themselves,
might mount up in time to a destructive magnitude,
and that such cases must continually occur
in an extensive system like that of Saturn, the
only retarding cause being the possible irregularity
of the rings.

``The result of long-continued disturbance
was found to be the spreading out of the rings
in breadth, the outer rings pressing outward,
while the inner rings press inward.

``The final result, therefore, of the mechanical
theory is, that the only system of rings which
can exist is one composed of an indefinite number
of unconnected particles, revolving round the
planet with different velocities according to their
respective distances. These particles may be
arranged in a series of narrow rings, or they may
move through each other irregularly. In the
\PageSep{367}
\index{Keeler, spectroscopic examination of Saturn's ring|(}%
first case the destruction of the system will be
very slow, in the second case it will be more
rapid, but there may be a tendency towards an
arrangement in narrow rings, which may retard
the process.

``We are not able to ascertain by observation
the constitution of the two outer divisions of the
system of rings, but the inner ring is certainly
transparent, for the limb (\ie~edge) of Saturn
has been observed through it. It is also certain,
that though the space occupied by the ring is
transparent, it is not through the material particles
of it that Saturn was seen, for his limb was
observed without distortion; which shows that
there was no refraction, and therefore that the
rays did not pass through a medium at all, but
between the solid or liquid particles of which the
ring is composed. Here then we have an optical
argument in favor of the theory of independent
particles as the material of the rings. The
two outer rings may be of the same nature, but
not so exceedingly rare that a ray of light can
pass through their whole thickness without encountering
one of the particles.''
\index{Maxwell on Saturn's ring|)}%

\TB

The last link in the chain of evidence has been
furnished by recent observations made in America.
If it can be proved that every part of the
apparently solid ring moves round the planet's
centre at a different rate, and that the speed at
\PageSep{368}
\index{Meteoric constitution of Saturn's ring}%
\index{Spectroscopic proof of rotation of Saturn's ring}%
each part is appropriate at its distance from the
centre, the conclusion is inevitable that the ring
consists of scattered fragments.

Every one must have noticed that when a
train passes at full speed with the whistle blowing,
there is an abrupt fall in the pitch of the
note. This change of note is only apparent to
the stationary listener, and is caused by the
crowding together of the waves of sound as the
train approaches, and by their spacing out as it
recedes. The same thing is true of light-waves,
and if we could imagine a colored light to pass
us at an almost inconceivable velocity it would
change in tint as it passed.\footnote
  {This statement is strictly correct only of monochromatic
  light. I might, in the subsequent argument, have introduced
  the limitation that the moving body shall emit only monochromatic
  light. The qualification would, however, only complicate
  the statement, and thus render the displacement of the lines of
  the spectrum less easily intelligible.}
Now there are certain
lines in the spectrum of sunlight, and the
shifting of their positions affords an excessively
delicate measure of a change which, when magnified
enormously, would produce a change of
tint. For example, the sun is a rotating body,
and when we look at its disk one edge is approaching
us and the other is receding. The
two edges are infinitesimally of different colors,
and the change of tint is measurable by the displacement
of the lines I have mentioned. In
the same way Saturn's ring is illuminated by
sunlight, and if different portions are moving at
\PageSep{369}
\index{Spectroscopic proof of rotation of Saturn's ring}%
different velocities, those portions are infinitesimally
of different colors. Now Professor Keeler,
the present director of the Lick Observatory, has
actually observed the reflected sunlight from the
several parts of Saturn's ring, and he finds that
the lines in the spectrum of the several parts
are differently displaced. From measurement of
these displacements he has concluded that every
part of the ring moves at the same pace as if it
were an independent satellite. The proof of the
meteoric constitution of the ring is therefore
\index{Meteoric constitution of Saturn's ring}%
complete.

It would be hard to find in science a more
beautiful instance of arguments of the most
diverse natures concentrating themselves on a
definite and final conclusion.

\begin{Authorities}
Édouard Roche, \Title{La figure d'une masse fluide soumise à l'attraction
\index{Roche, E.!stability of ellipsoid of}%
d'un point éloigné}, ``Mém.\ Acad.\ de~Montpelier,'' vol.~i.\
(Sciences), 1847--50.

Maxwell, \Title{Stability of Saturn's Rings}, Macmillan, 1859.

Keeler, \Title{Spectroscopic Proof of the Meteoric Constitution of
Saturn's Rings}, ``Astrophysical Journal,'' May, 1895; see also
\index{Keeler, spectroscopic examination of Saturn's ring|)}%
\index{Saturn!theory of ring|)}%
the same for June, 1895.

Schwarzschild, \Title{Die Poincarésche Theorie des Gleichgewichts},
\index{Schwarzschild!stability of Roche's ellipsoid}%
``Annals of Munich Observatory,'' vol.~iii.\ 1896. He considers
the stability of Roche's ellipsoid.
\end{Authorities}
\PageSep{370}
%[Blank Page]
\PageSep{371}
\BackMatter
\printindex
\iffalse
INDEX

Abacus for reducing tidal observations#abacus, 217-220.

Abbadie, tidal deflection of vertical#Abbadie, 143, 144.

Aden, errors of tidal prediction at#Aden, 246.

Adriatic, tide in#Adriatic, 186.

Airy, Sir G. B.#Airy,
  tides in rivers, 75;
  attack on Laplace, 181;
  cotidal chart, 188;
  \Title{Tides and Waves}, 192.

America, North, tide tables for#America, 222.

Analysis, harmonic, of tide#analysis, 193-210.

Andromeda, nebula in#Andromeda, 339.

Annual and semi-annual tides#annual, 206.

Arabian theories of tide#Arab, 77-79.

Aristotle on tides#Aristotle, 81.

Assyrian records of eclipses, 272.

Atlantic, tide in#Atlantic, 186-188.

Atmospheric pressure,
  cause of seiches, 40;
  distortion of soil by, 145, 146;
  influence on tidal prediction, 242, 243.

Atmospheric waves, Helmholtz on#Helmholtz, 48-51.

Attraction,
  of weight resting on elastic slab proportional to slope, 136, 137;
  of tide calculated, 143.

Baird, \Title{Manual for Tidal Observation}#Baird, 16.

Bakhuyzen on tide due to variation of latitude#Bakhuyzen, 255, 256.

Barnard, rotation of Jupiter's satellites#Barnard, 315.

Barometric pressure. |see{Atmospheric pressure}. 0

Becker, G. F., on Nebular Hypothesis#Becker, 334, 336-338.

Bernoulli, Daniel, essay on tides#Bernoulli, 86, 88.

Bertelli on Italian seismology#Bertelli, 126, 127.

Bifilar. |see{Pendulum}. 0

Borgen@Börgen, method of reducing tidal observations#Börgen, 217.

Bond, discovery of inner ring of Saturn#Bond, 352.

Bore,
  definition, 59;
  bore-shelter, 63;
  diagram of rise in Tsien-Tang, 66;
  pictures, 67;
  rivers where found, 71;
  causes, 72;
  Chinese superstition, 68-70.

Browne, E. G., Arabian theories of tide#Browne, 77-79.

Cambridge, experiments with bifilar pendulum at#bifilar, 115-125.

Canal,
  theory of tide wave in, 165-167;
  critical depth, 163-165;
  tides in ocean partitioned into canals, 175;
  canal in high latitude, 174-176.

Capillarity of liquids, and Plateau's experiment#Plateau, 316-318.

Cassini, discovery of division in Saturn's rings#Cassini, 352.

Castel, Father, ridiculed by Voltaire#Castel, 295, 296.

Cavalleri, essay on tides#Cavalleri, 86.

Centripetal and centrifugal forces#Centripetal, 91-93.

Chambers on possible existence of Martian satellites#Chambers, 296.
\PageSep{372}

Chandler, free nutation of earth, and variation of latitude#Chandler, 253-257.

Chinese
  superstition as to bore, 68-70;
  theories of tide, 76, 77.

Christie, A. S., tide due to variation of latitude#Christie, 255, 256.

Constants, tidal, explained#constants, 195.

Continents, trend of, possibly due to primeval tidal friction#continents, 308.

Cotidal chart, 188;
  for diurnal tide hitherto undetermined, 191, 192.

Currents, tidal, in rivers#tidal current, 56.

Curve, tide#tide curve,
  irregularities in, 10-16;
  at Bombay, 12;
  partitioned into lunar time, 213.

D'Abbadie. |see{Abbadie}. 0

Darwin, G. H.#Darwin,
  bifilar pendulum, 115-125;
  harmonic analysis, 210;
  tidal abacus, 217-220;
  distortion of earth's surface by varying loads, 134-148;
  rigidity of earth, 261, 262;
  papers on tidal friction, 315;
  hour-glass figure of rotating liquid, 328-332;
  Jacobi's ellipsoid, 333;
  evolution of satellites, 346.

Darwin, Horace, bifilar pendulum#Horace, 115-125.

Davis, method of presenting tide-generating force#Davis, 96, 97.

Davison, history of bifilar and horizontal pendulums#Davison, 133.

Dawes, discovery of inner ring of Saturn#Dawes, 352.

Dawson coöperates in investigation of seiches#Dawson, 48.

Day, change in length of, under tidal friction#day, 275, 276.

Deflection of the vertical, 109-133;
  experiments to measure, 115-125;
  due to tide, 134-143.

Deimos, a satellite of Mars#Deimos, 297.

Denison, F. Napier, vibrations and seiches on lakes#Denison, 48-53.

Density
  of earth, law of internal, 302;
  of planets determinable from their figures, 332, 333.

Descartes, vortical theory of cosmogony#Descartes, 335.

Dick, argument as to Martian satellites#Dick, 295.

Dimple,
  in soil, due to weight, 123;
  form of, in elastic slab, 135.

Distortion of soil
  by weight, 123;
  by varying loads, 134-148.

Diurnal inequality
  observed by Seleucus, 84, 85;
  according to equilibrium theory, 156;
  in Laplace's solution, 179;
  in Atlantic, Pacific, and Indian Oceans, 180;
  not shown in cotidal chart, 191;
  in harmonic method, 205;
  complicates prediction, 224, 225.

Douglass, rotation of Jupiter's satellites#Douglass, 315.

Dumb-bell nebula, description of photograph of#dumb-bell, 345.

Dynamical theory of tide-wave, 163-181.

Earth and moon#Earth,
  diagram, 93;
  rotation of, effects on tides, 177;
  rigidity of, 256-260;
  rotation retarded by tidal friction, 268;
  figure of, 299;
  adjustment of figure to suit change of rotation, 299-302;
  internal density, 302;
  probably once molten, 306;
  distortion under primeval tidal friction, 307;
  Roche's limit for, 358.

Earthquakes,
  a cause of seiches, 39;
  microsisms and earth tremors, 125-127;
  shock perceptible at great distance, 261.

Ebb and flow defined#ebb, 56.

Eccentricity of orbit
  due to tidal friction, 313, 314;
  theory of, in case of double stars, 342.
\PageSep{373}

Eclipses, ancient, and earth's rotation#eclipses, 272, 273.

Ecliptic, obliquity of, due to tidal friction#ecliptic, 308-312.

Eddies, tidal oscillation involves#eddies, 177.

Ehlert, observation with horizontal pendulum#Ehlert, 132.

Elastic distortion#elastic
  of soil by weight, 123;
  of earth by varying loads, 134-148;
  calculation and illustration, 138-140;
  by atmospheric pressure, 145-147.

Elasticity of earth#elasticity, 254, 255.

Elliptic tide, lunar#elliptic tide, 204.

Ellipticity of earth's strata in excess for present rotation#ellipticity, 303, 304.

Energy, tidal, utilization of#tidal energy, 73, 74.

Equatorial canal, tide wave in#canal, 173.

Equilibrium, figures of, of rotating liquid#equilibrium, 316-333.

Equilibrium theory of tides#equilibrium, 149-162;
  chart and law of tide, 151-153;
  defects of, 160.

Errors in tidal prediction#errors, 243-245.

Establishment of port,
  definition, 161, 162;
  zero in equilibrium theory, 161;
  shown in cotidal chart, 189.

Estuary, annual meteorological tide in#estuary, 207, 208.

Euler, essay on tides#Euler, 86.

Europe, tides on coasts of#Europe, 188.

Evolution of celestial systems, 334-346.

Ferrel, tide-predicting instrument#Ferrel, 241.

Figure of equilibrium
  of ocean under tidal forces, 151-153;
  of rotating liquid, 316-333.

Figure of planets and their density, 332, 333.

Fisher, Osmond, on molten interior of earth#Fisher, 262.

Flow and ebb defined#ebb, 56.

Forced oscillation,
  principle of, 169, 170;
  due to solar tide, possibly related to birth of moon, 282-284.

Forced wave, explanation and contrast with free wave#forced wave, 164.

Forces,
  centripetal and centrifugal, 91-93;
  tide-generating, 93-108;
  numerical estimate, 109-111;
  deflection of vertical by, 109-133;
  figure of equilibrium under tidal, 151-153;
  those of sun and moon compared, 156-158.

Forel
  on seiches, 17-38;
  list of papers, 53, 54.

Free oscillation contrasted with forced#free oscillation, 169, 170.

Free wave, explanation and contrast with forced#free wave, 164.

Friction of tides, 264-315.

Galileo,
  blames Kepler for his tidal theory, 85;
  discovery of Jupiter's satellites, 291;
  Saturn's ring, 350.

Gauge, tide#tide gauge,
  description of, 6-11;
  site for, 14.

Geneva,
  seiches in lake, 17-28;
  model of lake, 28.

Geological evidence of earth's plasticity#plasticity, 300;
  as to retardation of earth's rotation, 304-306.

German method of reducing tidal observations#German, 217.

Giles on Chinese theories of the tide#Giles, 76, 77.

Gravity, variation according to latitude#gravity, 302, 303, 332.

Greek
  theory and description of tides, 81-85;
  records of ancient eclipses, 272.

Gulliver@\Title{Gulliver's Travels}, satire on mathematics#Gulliver, 292-295.

Hall, Asaph, discovery of Martian satellites#Hall, 290-298.

Hangchow, the bore at#Hangchow, 60-70.
\PageSep{374}

Harmonic analysis
  initiated by Lord Kelvin, 87;
  account of, 193-210.

Height of tide#height
  due to ideal satellite, 198;
  at Portsmouth and at Aden, 225;
  reduced by elastic yielding of earth, 259.

Helmholtz
  on atmospheric waves, 48-51;
  on rotation of the moon, 286.

Herschel, observations of twin nebulæ#Herschel, 344.

High water
  under moon in equilibrium theory, 160;
  position in shallow and deep canals in dynamical theory, 171, 172.

History
  of tidal theories, 76-88;
  of earth and moon, 278-286, 308-313.

Hopkins on rigidity of earth#Hopkins, 258, 259.

Horizontal tide-generating force, 107.

Horizontal tide-generating force |see{also Pendulum}. 0

Hough, S. S.#Hough,
  frictional extinction of waves, 47;
  dynamical solution of tidal problem, 181;
  rigidity of earth, 254;
  Chandler's nutation, 262.

Hugli, bore on the#Hugli, 71.

Huyghens, discovery of Saturn's ring#Huyghens, 351.

Icelandic theory of tides, 79, 80.

Indian Survey,
  method of reducing tidal observations, 216, 217;
  tide tables, 222.

Instability,
  nature of dynamical, and initial of moon's motion, 280-282;
  of Saturn's ring, 363, 364.

Interval from moon's transit to high water
  in case of ideal satellite, 198;
  at Portsmouth and at Aden, 225.

Italian investigations in seismology, 125-130.

Jacobi, figure of equilibrium of rotating liquid#Jacobi, 322-324.

Japan, frequency of earthquakes#Japan, 130, 131.

Jupiter,
  satellites constantly face planet, 298;
  figure and law of internal density, 333;
  Roche's limit for, 361.

Kant,
  rotation of moon, 286;
  nebular hypothesis, 334-339.

Keeler, spectroscopic examination of Saturn's ring#Keeler, 367-369.

Kelvin, Lord#Kelvin,
  initiates harmonic analysis, 87, 199;
  calculation of tidal attraction, 143;
  tide predicting machine, 233;
  rigidity of earth, 257-260;
  denies adjustment of earth's figure to changed rotation, 301;
  on geological time, 315.

Kepler,
  ideas concerning tides, 85, 86;
  argument respecting Martian satellites, 291, 292.

Krüger, figures of equilibrium of liquid#Krüger, 333.

Lakes,
  seiches in, 17-54;
  mode of rocking in seiches, 24, 25;
  vibrations, 41-53;
  tides in, 182-185.

Lamb, H., presentation of Laplace's theory#Lamb, 181.

Laplace,
  theory of tides, 86-88, 177-180;
  on rotation of moon, 286, 287;
  nebular hypothesis, 335-337.

Lardner, possibility of Martian satellites#Lardner, 295.

Latitude,
  tidal wave in canal in high, 174-176;
  periodic variations of, 251-256.

Lege@Légé, constructor of tide-predicting machine#Légé, 233.

Level of sea affected by atmospheric pressure#atmospheric pressure, 146.

Limnimeter, a form of tide gauge#limnimeter, 24.

Lowell, P., on rotations of Venus and Mercury#Lowell, 298, 299, 315.

Low water. |see{High water}.
\PageSep{375}

Lubbock, Sir J., senior, on tides#Lubbock, 87.

Lunar
  tide-generating force compared with solar, 156-158;
  tide, principal, 201;
  elliptic tide, 204;
  time, 213.

Machine, tide-predicting#tide-predicting, 233, 241.

Mackerel sky, evidence of air-waves#mackerel, 49.

Maclaurin,
  essay on tides, 86;
  figure of equilibrium of rotating liquid, 322-324.

Magnússon on Icelandic theories of tides#Magnússon, 79, 80.

Marco Polo, resident of Hangchow#Marco, 70.

Mars,
  discovery of satellites, 290-298;
  Roche's limit, 360.

Maxwell on Saturn's ring#Maxwell, 363-367.

Mediterranean Sea, tides in#Mediterranean, 185, 186.

Mercury, rotation of#Mercury, 298, 299.

Meteoric constitution of Saturn's ring#meteoric, 368, 369.

Meteorological
  tides, 206, 207;
  conditions dependent on earth's rotation, 303.

Microphone as a seismological instrument#microphone, 128-130.

Microsisms, minute earthquakes#microsisms, 125-127.

Mills worked by the tide#mills, 74, 75.

Milne on seismology#Milne, 125, 130.

Month, change in, under tidal friction#month, 275-277.

Moon and earth,
  diagram, 93;
  tide-generating force compared with sun's, 156-158;
  tide due to ideal, moving in equator, 193, 194;
  ideal satellites replacing actual, 199, 200;
  tidal prediction by reference to transit, 224-230;
  retardation of motion by tidal friction, 269, 270;
  origin of, 282, 283;
  rotation annulled by tidal friction and present libration, 286;
  inequality in motion indicates internal density of earth, 302, 303;
  eccentricity of orbit increased by tidal friction, 313, 314.

Moore, Captain#Moore,
  illustrations of bore, 67;
  survey of Tsien-Tang-Kiang, 60-70.

Neap and spring tides#neap
  in equilibrium theory, 159;
  represented by principal lunar and solar tides, 204.

Nebula in Andromeda#Andromeda, 339.

Nebulae@Nebulæ, description of various#Nebulæ, 345.

Nebular hypothesis, 334-339.

Newcomb, S., theoretical explanation of Chandler's nutation#Newcomb, 254.

Newton,
  founder of tidal theory, 86;
  theory of tide in equatorial canal, 172.

Nolan, criticism of tidal theory of moon's origin#Nolan, 360.

Nutation,
  value of, indicates internal density of earth, 303;
  Chandler's, 251-256.

Obliquity of ecliptic, effects of tidal friction on#obliquity, 310-312.

Observation,
  methods of tidal, 6-14;
  reduction of tidal, 211-220.

Orbit
  of moon and earth, 93-95;
  of double stars, very eccentric, 313.

Pacific Ocean, tide in, affects Atlantic#Pacific, 186, 187.

Partial tides in harmonic method#partial tides, 199.

Paschwitz, von Rebeur#Paschwitz,
  on horizontal pendulum, 130-132;
  tidal deflection of vertical at Wilhelmshaven, 144.

Pendulum,
  curves traced by, under tidal force, 111, 112;
  bifilar, 115-125;
  as seismological instrument, 126, 127;
  horizontal, 130-132.
\PageSep{376}

Petitcodiac, bore in the#Petitcodiac, 71.

Phobos, a satellite of Mars#Phobos, 297.

Planetary figure of equilibrium of rotating liquid#planetary figure, 322.

Planets,
  rotation of some, annulled by tidal friction, 298;
  figures and internal densities, 332, 333.

Plasticity of earth under change of rotation#plasticity, 300-302.

Plateau, experiment on figure of rotating globule#Plateau, 316-319.

Plemyrameter, observation of seiches with#plemyrameter, 19-22.

Poincaré,
  law of interchange of stability, 326, 327;
  figure of rotating liquid, 325, 327.

Polibius on tides at Cadiz#Polibus, 83.

Portsmouth, table of errors in tidal predictions#Portsmouth, 244.

Posidonius on tides#Posidonius, 81-84.

Precession, value of, indicates internal density of earth#internal density, 303.

Predicting machine for tides, 233-241;
  Ferrel's, 241.

Prediction of tide,
  due to ideal satellite, 200;
  example at Aden, 226-230;
  method of computing, 230-233;
  errors in, 242-250.

Pressure of atmosphere, elastic distortion of soil by#distortion, 145, 146.

Principle of forced oscillations#forced oscillations, 169, 170.

Rebeur. |see{Paschwitz}. 0

Reduction of tidal observations, 211-220.

Retardation of earth's rotation, 268.

Rigidity of earth#rigidity, 256-260.

Ripple mark in sand preserved in geological strata#ripple mark, 305.

Rivers,
  tide wave in, 55-59;
  Airy on tide in, 75;
  annual meteorological tide in, 206.

Roberts, E., the tide-predicting machine#Roberts, 233.

Roberts, I., photograph of nebula in Andromeda#Roberts, 339.

Roche, E.#Roche,
  ellipticity of internal strata of earth, 303;
  theory of limit and Saturn's ring, 356-362;
  stability of ellipsoid of, 369.

Roman description of tides#Roman, 81-85.

Rossi on Italian seismology#Rossi, 128-130.

Rotating liquid, figures of equilibrium#rotating liquid, 316-333.

Rotation
  of earth involved in tidal problem, 177;
  retarded by tidal friction, 268;
  of moon annulled by tidal friction, 286;
  of Mercury, Venus, and satellites of Jupiter and Saturn annulled by tidal friction, 298.

Russell, observation of seiches in New South Wales#Russell, 47.

Saint@St.\ Vénant on flow of solids#Vénant, 313.

Satellites,
  tide due to single equatorial, 195, 196;
  ideal replacing sun and moon in harmonic analysis, 199, 200;
  discovery of those of Mars, 290-298;
  rotation of those of Jupiter and Saturn annulled, 298;
  distribution of, in solar system, 339-341.

Saturn,
  satellites always face the planet, 298;
  law of density and figure, 332;
  description and picture, 347-354;
  theory of ring, 356-369;
  Roche's limit for, 360.

Schedule for reducing tidal observations, 215, 216.

Schiaparelli on rotation of Venus and Mercury#Schiaparelli, 298, 315.

Schwarzschild,
  exposition of Poincaré's theory, 333;
  stability of Roche's ellipsoid, 369.

Sea,
  vibrations of, 44, 45;
  level affected by atmospheric pressure, 146.
\PageSep{377}

See, T. J. J.,
  eccentricity of orbits of double stars, 313;
  theory of evolution of double stars, 342-346.

Seiches,
  definition, 17;
  records of, 21;
  longitudinal and transverse, 25-27;
  periods of, 27;
  causes of, 39, 40.

Seine, bore in the#Seine, 71.

Seismology, 133.

Seleucus, observation of tides of Indian Ocean#Seleucus, 84, 85.

Semidiurnal tide
  in equilibrium theory, 153-156;
  in harmonic method, 201-204.

Severn, bore in the#Severn, 71.

Slope of soil
  due to elastic distortion, 136;
  calculation and illustration of, 138-140.

Solar
  tide-generating force compared with lunar, 156-158;
  principal tide, 202;
  possible effect of tide in assisting birth of moon, 284, 285;
  system, nebular hypothesis as to origin of, 334-339;
  system, distribution of satellites in, 339-341.

Spectroscopic proof of rotation of Saturn's ring#spectroscop, 368, 369.

Spring and neap tides
  in equilibrium theory, 159;
  represented by principal lunar and solar tides, 203.

Stability,
  nature of dynamical, 280, 281;
  of figures of equilibrium, 322, 323;
  of Saturn's ring, 365, 366.

Stars,
  double, eccentricity of orbits, 313;
  theory of evolution, 342-346.

Storms a cause of seiches#storms, 39, 40.

Strabo on tides#Strabo, 81-85.

Stupart coöperates in investigation of seiches#Stupart, 48.

Sun,
  tide-generating force of, compared with that of moon, 156-158;
  ideal, replacing real sun in harmonic analysis, 201;
  possible influence of, in assisting birth of moon, 284, 285.

Surface tension of liquids#surface tension, 317, 318.

Swift, satire on mathematicians#Swift, 292-295.

Synthesis of partial tides for prediction#synthesis, 230-233.

Tables, tide#tide tables, 221-241;
  method of calculating, 230-241;
  amount of error in, 246, 247.

Thomson, Sir W. |see{Kelvin}. 0

Tidal problem. |see{Laplace, Harmonic Analysis, etc.} 0

Tide,
  definition, 1-3;
  general description, 4-6.

Tide, |see{also other headings; \eg\ for tide-generating force, |see{Force}}. 0

Time,
  lunar, 213;
  requisite for evolution of moon, 285.

Tisserand, Roche's investigations as to earth's figure#Tisserand, 315.

Tremors, earth#tremors, 125.

Tresca on flow of solids#Tresca, 300.

Tromometer, a seismological instrument#tromometer, 126, 127.

Tsien-Tang-Kiang, the bore in#Tsien, 60-70.

United States Coast Survey,
  method of reducing tidal observations, 217;
  tide tables of, 222.

Variation of latitude, 251-256.

Vaucher, record of a great seiche at Geneva#Vaucher, 17.

Venus, rotation of#Venus, 298, 299.

Vertical. |see{Deflection}. 0

Vibration of lakes, 41-53.

Voltaire, satire on mathematicians, and Martian satellites#Voltaire, 295, 296.

Vortical motion in oceanic tides#vorticial, 177, 178.
\PageSep{378}

Waves
  in deep and shallow water, 29;
  speed of, 31;
  composition of, 33-37;
  in atmosphere, 48-50;
  forced and free, 164;
  of tide in equatorial canal, 173;
  in canal in high latitude, 174-176;
  propagated northward in Atlantic, 186-188.

Wharton, Sir W. J., illustration of bore#Wharton, 69.

Whewell
  on tides, 87;
  empirical construction of tide tables, 87-90;
  on cotidal charts, 188, 189.

Wind,
  a cause of seiches, 39;
  vibrations of lakes due to, 41, 42;
  a cause of meteorological tides, 206;
  perturbation of, in tidal prediction, 242, 243.

Woodward on variation of latitude#Woodward, 262.

Wright, Thomas, on a theory of cosmogony#Wright, 335.

Wye, bore in the#Wye, 71.
\fi
\PageSep{379}
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