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Full text of "The tides and kindred phenomena in the solar system"

THE TIDES 

AND KINDRED PHENOMENA IN THE 
SOLAR SYSTEM 

THE SUBSTANCE OF LECTURES DELIVERED 

IN 1897 AT THE LOWELL INSTITUTE, 

BOSTON, MASSACHUSETTS 

BY 

GEORGE HOWARD DARWIN 

PLUMIAN PBOFES80R AND FELLOW OF TRINITY COLLEGE IN THJ6 
UNIVERSITY OF CAMBRIDGE 




BOSTON AND NEW YORK 
HOUGHTON, MIFFLIN AND COMPANY 

($$e Htoers'tOe press, CamfanD0e 
1899 




COPYRIGHT, 1898, BY GEORGE HOWARD DARWIN 
ALL RIGHTS RESERVED 



PREFACE 



IN 1897 I delivered a course of lectures on 
i;he Tides at the Lowell Institute in Boston, 
Massachusetts, and this book contains the sub- 
stance 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 
tlie occasion for making that effort. 



vi PREFACE 

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 vol- 
ume ; 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 prac- 
tical 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 in- 
volved 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 



PREFACE 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 sub- 
jects 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 Cam- 
bridge, but I have to acknowledge the courtesy 
of the proprietors of Harper's, the Century, 
and the Atlantic Monthly magazines, in supply- 
ing me with some important illustrations. 

A considerable portion of Chapter III. on the 
" Bore " is to appear as an article in the Century 
Magazine for October, 1898, and the reproduc- 
tions of Captain Moore's photographs of the 
66 Bore " in the Tsien-Tang-Kiang have been 
prepared for that article. The 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 num- 
ber for October, 1890. The greater portion of 
the text and the whole of the illustrations of 
Chapter XX. were originally published in Har- 
per's Magazine for June, 1889. Lastly, por- 
tions of Chapters XV. and XVI. appeared in 
the Atlantic Monthly for April, 1898, published 



viii PREFACE 

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 cor- 
diality of their reception, and my many friends 
across the Atlantic for their abundant hospital- 
ity and kindness. 

G. H. DARWIN. 
CAMBRIDGE, August, 1898. 



CONTENTS 



CHAPTER I 

PAGES 
TIDES AND METHODS OF OBSERVATION 

Definition of tide ....... 1-3 

Oceanic tides 4 5 

Methods of observation 

Tide-gauge .... 7-12 
Tide-curve ...... 

Site for tide-gauge 

Irregularities in tide-curve 14, 15 

Authorities ...... 16 

CHAPTER II 

SEICHES IN LAKES 

Meaning of seiche 17 

Uses of scientific apparatus ...... 18 

Forel's plemyrameter . . . - . 19,20 

Records of the level of the lake 

Interpretation of record 21-23 

Limnimeter . 

Mode of oscillation in seiches 25-28 

Wave motion in deep and in shallow water . . . 29-32 

Composition of waves ...... 32-36 

Periods of seiches 37, 38 

Causes of seiches 39, 40 

Vibrations due to wind and to steamers . . . 41-47 

Aerial waves and their action on lakes and on the sea 48-53 

Authorities 53, 54 



x CONTENTS 

CHAPTER III 

TIDES IX RIVERS TIDE MILLS 

Definition of ebb and flow 56 

Tidal currents in rivers 56-58 

Progressive change of wave in shallow water . . 58, 59 

The bore 59 

Captain Moore's survey of the Tsien-Tang-Kiang . 60-64 

Diagram of water-levels during the flow . . . 64, 65 

Chinese superstition ....... 68-71 

Pictures of the bore ....... 69 

Other cases of bores ....... 71 

Causes of the bore 72 

Tidal energy 73, 74 

Tide mills 74, 75 

Authorities . 75 



CHAPTER IV 

HISTORICAL SKETCH 

Theories of the Chinese 76, 77 

Theories of the Arabs 77-79 

Theories of the Norsemen 79, 80 

Writings of Posidonius and Strabo .... 80-84 

Seleucus the Babylonian on the diurnal inequality . 84, 85 

Galileo and Kepler 85 

Newton and his successors 86-88 

Empirical method of tidal prediction .... 88-90 

Authorities 90 



CHAPTER V 

TIDE-GENERATING FORCE 

Inertia and centrifugal force 91-93 

Orbital motion of earth and moon .... 93-95 

Tide-generating force 96--100 

Law of its dependence on the moon's distance . . 101-103 

Earth's rotation . . . 103, 104 



CONTENTS xi 

Second explanation of tide-generating force . . . 104, 105 

Horizontal tide-generating force .... 105, 106 

Successive changes of force in the course of a day . 107, 108 

'Authorities 108 



CHAPTER VI 

DEFLECTION OF THE VERTICAL 

Deflection of a pendulum by horizontal tidal force . 109-111 

Path pursued by a pendulum under tidal force . . 111-113 

Object of measuring the deflection of a pendulum . 113-115 

Attempt to measure deflection by bifilar pendulum . 115-125 

Microsisms 125-127 

The microphone as a seismological instrument . . 127-130 

Paschwitz's work with the horizontal pendulum . 131, 132 
Supposed measurement of the lunar deflection of gravity 132 

Authorities . 133 



CHAPTER VII . 

THE ELASTIC DISTORTION OF THE EARTH'S SURFACE BY 
VARYING LOADS 

Distortion of an elastic surface by superincumbent load . 134-137 
Application of the theory to the earth . . . 137, 138 

Effects of tidal load 138-140 

Probable deflections at various distances from the coast 140-143 
Deflections observed by M. d'Abbadie and by Dr. Pasch- 

witz 143-145 

Effects of atmospheric pressure on the earth's surface . 145-147 
Authorities . 148 



CHAPTER VIII 

EQUILIBRIUM THEORY OF TIDES 

Explanation of the figure of equilibrium . 
Map of equilibrium tide .... 
Tides according to the equilibrium theory 
Solar tidal force compared with lunar . 



149-151 
151-153 
153-156 
156-158 



xii 



CONTENTS 



Composition of lunar and solar tides .... 158, 159 
Points of disagreement between theory and fact . . 159-162 
Authorities . . . . . . . . 162 

| 

CHAPTER IX 

DYNAMICAL THEORY OF THE TIDE WAVE 

Free and forced waves in an equatorial canal . . 163-165 

Critical depth of canal 165-167 

General principle as to free and forced oscillations . 167-174 

Inverted and direct oscillation 172, 173 

Canal in latitude 60 174, 175 

Tides where the planet is partitioned into canals . . 175 

Removal of partitions ; vortical motion of the water 176, 177 

Critical latitude where the rise and fall vanish . . 177, 178 

Diurnal inequality ....... 178-180 

Authorities . 181 



CHAPTER X 



TIDES IN LAKES COTIDAL CHART 



The tide in a lake 
The Mediterranean Sea . 
Derived tide of the Atlantic . 
Cotidal chart . 
Authorities . 



. 182-185 
185, 186 

. 186-188 

188-192 

192 



CHAPTER XI 

HARMONIC ANALYSIS OF THE TIDE 

Tide in actual oceans due to single equatorial satellite 193-196 
Substitution of ideal satellites for the moon . . 197-199 
Partial tide due to each ideal satellite . . . 199-201 
Three groups of partial tides . . . . . 201 
Semidiurnal group ....... 201204 

Diurnal group 204-206 

Meteorological tides 206, 207 

Shallow water tides 207-210 

Authorities . . 210 



CONTENTS xiii 

CHAPTER XII 

REDUCTION OF TIDAL OBSERVATIONS 

Method of singling out a single partial tide . . 211-214 

Variety of plans adopted 214-217 

Tidal abacus 217-220 

Authorities 220 

CHAPTER XIII 

TIDE TABLES 

Definition of special and general tables . . . 221 
Reference to moon's transit ...... 222, 223 

Examples at Portsmouth and at Aden .... 223-228 

General inadequacy of tidal information . . . 229, 230 

Method of calculating tide tables 230-233 

Tide-predicting machine 233-241 

Authorities 241 

CHAPTER XIV 

THE DEGREE OF ACCURACY OF TIDAL PREDICTION 

Effects of wind and barometric pressure . . . 242, 243 
Errors at Portsmouth ...... 243, 244 

Errors at Aden 246-249 

Authorities 250 

CHAPTER XV 
CHANDLER'S NUTATION THE RIGIDITY OF THE EARTH 

Nutation of the earth and variation of latitude . . 251-254 

Elasticity of the earth 254, 255 

Tide due to variation of latitude . . . . 255,256 

Rigidity of the earth 256-260 

Transmission of earthquake shocks .... 261, 262 
Authorities 262, 263 



xiv 



CONTENTS 
CHAPTER XVI 

TIDAL FRICTION 



Friction retards the tide ..... 
Retardation of planetary rotation .... 
Reaction on the satellite ..... 

Ancient eclipses of the sun 

Law of variation of tidal friction with moon's distance 
Transformations of the month and of the day . 
Initial and final conditions of motion 

Genesis of the moon 

Minimum time requisite for the evolution 

Rotation of the moon 

The month ultimately to be shorter than the day . 



. 264-267 

267-269 

. 269-272 

272, 273 
. 273-275 

275-280 
. 280, 281 

281-285 
. 285, 286 

286, 287 
. 287-289 



CHAPTER XVII 
TIDAL FRICTION (continued) 

Discovery of the Martian satellites .... 290-298 
Rotation of Mercury, of Venus, and of the Jovian 

satellites 298, 299 

Adaptation of the earth's figure to changed rotation . 299-302 
Ellipticity of the internal strata of the earth . . 302-304 

Geological evidence 304-306 

Distortion of a plastic planet and trend of continents . 306-308 

Obliquity of the ecliptic 308-312 

Eccentricity of lunar orbit 312, 313 

Eccentricity of the orbits of double stars . . . 313 

Plane of the lunar orbit 313, 314 

Short summary 314, 315 

Authorities 315 



CHAPTER XVIII 

THE FIGURES OF EQUILIBRIUM OF A ROTATING MASS OF 
LIQUID 

Plateau's experiment 316-319 

Stability of a celestial sphere of liquid .... 319-321 
The two ellipsoids of Maclaurin and that of Jacobi . 321-323 



CONTENTS xv 

Transitions with change of rotation .... 323, 324 
Coalescence of Jacobi's with Maclaurin's ellipsoid . 324-326 
Pomca^'s law of stability and coalescence . . .326,327 
Poincare's pear-shaped figure ..... 327, 328 
Hour-glass figure of equilibrium ..... 328-332 

Figures of planets 332, 333 

Authorities , 333 



CHAPTER XIX 

THE EVOLUTION OF CELESTIAL SYSTEMS 

The Nebular Hypothesis 334-338 

Nebula in Andromeda 338, 339 

Distribution of satellites in the solar system . . 339-341 
Genesis of celestial bodies by fission .... 342 

Dr. See's speculations as to systems of double stars . 342-344 
Diversity of celestial bodies . . . ' . . 344-346 
Authorities 346 

CHAPTER XX 

SATURN'S RINGS 

Description 347, 348 

Discovery of Saturn's rings ..... 348-352 

Diagram of the rings . 353-356 

Roche's investigation ... . 356, 357 

Roche's limit 358-360 

The limit for the several planets .... 360, 361 

Meteoric constitution of Saturn's rings .... 361, 362 

Maxwell's investigations . . . . . . 362-367 

Spectroscopic examination of the rings .... 367-369 

Authorities 369 



LIST OF ILLUSTRATIONS 



FULL-PAGE 
no. 

23. HORIZONTAL TIDE-GENERATING FORCE . to face p. 106 

33. TIDAL ABACUS . . . . . . " 218 

40. NEBULA IN ANDROMEDA " 339 

IN TEXT 

PAGE 

1. WELL FOR TIDE-GAUGE 7 

2. PIPE OF TIDE-GAUGE 9 

3. INDIAN TIDE-GAUGE . . . . . .10 

4. L^G^'S TIDE-GAUGE 11 

5. BOMBAY TIDE-CURVE FROM NOON, APRIL 22, TO 

NOON, APRIL 30, 1884 14 

6. SITES FOR A TIDE-GAUGE 15 

7. PLEMYRAMETER . 20 

8, 9. KECORDS OF SEICHES AT ]VIAN . . . 23 

10. MAP OF LAKE OF GENEVA ... . . .26 

11. WAVE IN DEEP WATER 30 

12. WAVE IN SHALLOW WATER 31 

13. SIMPLE WAVE 33 

COMPOSITION OF Two EQUAL AND OPPOSITE WAVES 34 

14. VIBRATIONS DUE TO STEAMERS .... 45 

15. PROGRESSIVE CHANGE OF A WAVE IN SHALLOW 

WATER 59 

16. CHART OF THE ESTUARY OF THE TSIEN-TANG-KIANG 61 

17. BORE-SHELTER ON THE TSIEN-TANG-KIANG ' . 64 

18. DIAGRAM OF THE FLOW OF THE TIDE ON THE TSIEN- 

TANG-KIANG 66 

19. PICTURES OF THE BORE ON THE TSIEN-TANG-KIANG 69 

20. EARTH AND MOON 93 



xviii LIST OF ILLUSTRATIONS 

21. REVOLUTION OF A BODY WITHOUT ROTATION . 98 

22. TIDE-GENERATING FORCE 100 

24. DEFLECTION OF A PENDULUM ; THE MOON AND OB- 

SERVER ON THE EQUATOR Ill 

25. DEFLECTION OF A PENDULUM ; THE MOON IN N. DE- 

CLINATION 15, THE OBSERVER IN N. LATITUDE 

30 112 

26. BIFILAR PENDULUM 115 

27. FORM OF DIMPLE IN AN ELASTIC SURFACE . 135 

28. DISTORTION OF LAND AND SEA-BED BY TIDAL LOAD 139 

29. CHART OF EQUILIBRIUM TIDES . . . .152 

30. FORCED OSCILLATIONS OF A PENDULUM . . 171 

31. THE TIDE IN A LAKE 184 

32. CHART OF COTIDAL LINES 190 

34. CURVES OF INTERVALS AND HEIGHTS AT PORTS- 

MOUTH AND AT ADEN 227 

35. DIAGRAM OF TIDE-PREDICTING INSTRUMENT . 235 

36. FRICTIONALLY RETARDED TIDE .... 266 

37. MACLAURIN'S AND JACOBI'S ELLIPSOIDS OF EQUILI- 

BRIUM 323 

38. FIGURES OF EQUILIBRIUM 325 

39. HOUR-GLASS FIGURE OF EQUILIBRIUM . . . 329 

41. THE PLANET SATURN 349 

42. DIAGRAM OF SATURN AND HIS RINGS . . . 353 

43. ROCHE'S FIGURE OF A SATELLITE WHEN ELONGATED 

TO THE UTMOST 357 



THE TIDES 



CHAPTER I 

TIDES AND METHODS OF OBSERVATION 

THE great wave caused by an earthquake is 
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 per- 
sistent 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 



2 TIDES AND METHODS OF OBSERVATION 

alternately raise and depress the sea-level, and 
thus produce a sort of tide. Then in the Trop- 
ics 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 wiU 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 



DEFINITION OF "TIDE" 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 ad- 
jective " 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 cer- 
tainly 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" 



4 TIDES AND METHODS OF OBSERVATION 

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 
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 par- 
lance 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 periodi- 
cally, there are certain moments when it is high- 
est 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. 



TIDAL PHENOMENA 5 

The most elementary observations would show 
that the time of high water has an intimate re- 
lationship to the moon's position. The moon, in 
fact, passes the meridian on the average fifty 
ninutes later on each. succeeding day, so that if 
ligh 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 
i;he moon was south. This rule is far from being 
exact, for it would be found that the interval 
j'rom the moon's passage to high water differs 
considerably according to the age of the moon. 
I shall not, however, attempt to explain at pre- 
sent how this rough rule as to the time of high 
water must be qualified, so as to convert it into 
un 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 



6 TIDES AND METHODS OF OBSERVATION 

least. The greater tides are called " springs," 
and the smaller " neaps." 

The currents produced in the sea by tides are 
often very complicated where the open sea is 
broken by islands and headlands, and the know- 
ledge 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 
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 obser- 
vation. I shall, therefore, explain how tides are 
observed scientifically, and how the facts are col- 
lected upon which the scientific treatment of the 
tides is based. 

The rise and fall of the sea may, of course, be 



TIDE-GAUGE 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 con- 
tinually disturbed by waves, observations of this 
kind are not susceptible of accuracy, and for 




FIG. 1. WELL FOB TIDE-GAUGE 

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 sensi- 
bly disturbing the level of the water. This re- 



8 TIDES AND METHODS OF OBSERVATION. 

suit is obtained in a considerable variety of ways, 
but one of them may be described as typical of 
aU. 

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 fas- 
tened 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 for- 
wards in a straight line. The mechanism is, 



TIDE-GAUGE 



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, 



Upper buoy 





Lower nun buoy 
Rose 



FIG. 2. PIPE OF TIDE-GAUGE 

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 pen- 
cil 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 



10 TIDES AND METHODS OF OBSERVATION 

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 




FIG. 3. INDIAN TIDE-GAUGE 

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 re- 
turned to the same part of the paper from which 



TIDE-GAUGE 



11 




FIG. 4. LEGO'S TIDE-GAUGE 



12 TIDES AND METHODS OF OBSERVATION 

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 y ester day , 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 
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, 



TIDE-GAUGE 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 oc- 
curred at 5.50 P.M., and the water fell to 5 ft. 
2 in., whereas the next low water, occurring at 
5.45 A.M. 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 irregu- 
larity 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 gen- 
erally 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. 



14 TIDES AND METHODS OF OBSERVATION 

These irregularities in the tide-curve are not 
due to tides, and as the object of the observa- 
tion is to determine the nature of the tides it is 




FIG- 5. BOMBAY TIDE-CURVE FROM NOON, APRIL 22, 
TO NOON, APRIL 30, 1884 

desirable to choose a site for the gauge where 
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 sea- 



TIDE-GAUGE 15 

shore, 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 sub- 
merge it. This little observation simply points 




B 

FIG- 6. SITES FOB A TIDE-GAUGE 

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 insig- 
nificant facts as this. An interesting light is 
indeed thrown on the origin of these notches on 
tide-curves by an investigation, not very directly 



16 TIDES AND METHODS OF OBSERVATION 

connected with tidal observation, on which I shall 
make a digression in the following chapter. 

AUTHORITIES. 

Baird's Manual for Tidal Observations (Taylor & Francis, 
1886). Price 7s. 6d. Figs. 1, 2, 3, 6 are reproduced from this 
work. 

The second form of tide-gauge shown in fig. 4 is made by 
Messrs. Le'ge, and is reproduced from a woodcut kindly pro- 
vided by them. 

Sir William Thomson's (Lord Kelvin's) Popular Lectures and 
Addresses, vol. iii. (Macmillau, 1891). 



CHAPTER II 

C SEICHES IN LAKES 

IT has been known for nearly three centuries 
that the water of the Lake of Geneva is apt to 
rise and fall by a few inches, sometimes irregu- 
larly and sometimes with more or less regularity ; 
and the same sort of oscillation has been ob- 
served 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. 1 

Observations were made by Vaucher in the 
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 oscil- 
lated through three or four feet ; but hardly any 
systematic observation had been undertaken when 
Professor Forel, of Lausanne, attacked the sub- 
-ject, and it is his very interesting observations 
which I propose to describe. 

Doctor Forel is not a mathematician, but is 

1 The word " seiche " is a purely local one. It has been al- 
leged to be derived from " seche," but I can see no reason for 
associating dryness with the phenomenon. 



18 SEICHES IN LAKES 

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 work- 
ings 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 nar- 
rative, 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 prin- 
cipal reason for elaborate appliances in the labo- 
ratories 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 help- 
less, 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 in- 
vestigators seems to remain almost unaffected by 
them. But I must not convey the impression 



FOREL'S SCIENTIFIC APPARATUS 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 investiga- 
tions in which every possible refinement of ap- 
paratus is necessary. But I do say that the 
number of great investigators is but little in- 
creased 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 inves- 
tigations was both simple and delicate. Its prin- 
ciple was founded on casual observation at the port 
of Merges, 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 cur- 
rent 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 in- 
strument, illustrated in fig. 7, and called by him 
a plemyrameter, for noting currents of even the 
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 



20 



SEICHES IN LAKES 



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 




FIG. 7. PLEMYRAMETEB 

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 re- 
mained 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 uni- 
formly. 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 



PLEMYRAMETER 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 princi- 
pally interested. 

In order to understand the record of observa- 
tions, 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 por- 
tions 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., 



22 SEICHES IN LAKES 

when it rose again. Further explanation seems 
unnecessary, as it should now be easy to read 
this diagram, and that shown in fig. 9. 

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 Switzer- 
land. In order to render intelligible the method 
by which For el 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 signifi- 
cance. 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 



RECORDS OF PLEMYRAMETER 



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 



60m. 2h. 10.m. 20m 30m. 40m. 50m. 3h. 10m. 20m. 30m 40m. 50m. 4h. 



8 



\ 



u_ 




JljU 




lOh. I Om. 20m 30m. 40m. 50m llh 10m. 20m 30m. 40m. 50m Ob 10m. 20m. 30m. 40m 50m 





a 




30 Sept 74 



FIGS. 8, 9. RECORDS OF SEICHES AT EVIAN 

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 ob- 
tained by Forel, and yet even here the oscilla- 
tions 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 plemyra- 
meter for some time, Forel used another form of 



24 SEICHES IN LAKES 

apparatus, by which he could observe the ampli- 
tude 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 
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 compara- 
tively 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 
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." 
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 
1 Deuxieme IZtude, p. 544. 



OSCILLATION OF THE WHOLE LAKE 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 
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 man- 
ifestation of a far more important phenomenon. 
It is the whole water of the lake which is rock- 
ing. It is a gigantic impulse which moves the 
whole liquid mass of Leman throughout its 
length, breadth, and depth. ... 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 
grandest oscillatory movement which man can 
study on the face of our globe." 

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 practi- 
cally straight. 

ForeFs analysis of seiches led him to conclude 
that the oscillations were of two kinds, the longi- 
1 Les Seiches, Vagues d' Oscillation, p. 11. 



26 SEICHES IN LAKES 

tudinal 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 




ViLLENEUVE 



GENEVA 

FIG. 10. MAP OF LAKE OF GENEVA 

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 longitu- 
dinal 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 longitudi- 
nal seiche. In this mode the water at the mid- 



MULTINODAL SEICHES 27 

die 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, 
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 hori- 
zontal 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 min- 
utes, which he conjectured to be multinodal. 

The second group of seiches were transverse, 
being observable at Morges and Evian. It was 



28 SEICHES IN LAKES 

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 Evian, 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 explana- 
tion. 

The great difficulty of applying deductive 
reasoning to the oscillations of a sheet of water 
of irregular outline and depth led Forel to con- 
struct 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 confirma- 
tion of his theories, although the periods of 
oscillation in the model of course differed enor- 
mously 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 



WAVE MOTION 29 

swinging at the rates observed. I must, there- 
fore, 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 never- 
theless graduate into one another. The distinc- 
tion 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 shal- 
low, 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 ex- 
tends quite to the bottom, and in water which is 
neither deep nor shallow, the condition of affairs 
is intermediate. The two figures, 11 and 12, 
show the nature of the movement in the two 
classes of waves. In both cases the dotted lines 
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 



30 



SEICHES IN LAKES 



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 




FIG. 11. WAVE IN DEEP WATER 

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 



SPEED OF WAVES 



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 shal- 
low when it is less. Thus the same water may 
be shallow for long waves and deep for short 
ones. For example, the sea is very shallow for 




FIG. 12. WAVE IN SHALLOW WATEB 

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 rela- 
tive to wave-length. 

The rate at which a wave moves can be ex- 
actly calculated from mathematical formula, 
from which it appears that in the deep sea a 
wave 63 metres in length travels at 36 kilome- 
tres per hour, or, in British measure, a wave of 
68 yards in length travels 22| 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 quar- 
ter 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 



32 SEICHES IN LAKES 

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^ metres deep (quarter of 10 metres), it 
would travel 18 kilometres (half of 36 kilome- 
tres) 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 de- 
pends only on depth. The average depth of the 
Lake of Geneva may be taken as about 150 me- 
tres, 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 

&c. 100 71 71 100 71 71 100 &c. 

and plot out, at equal distances, a figure of 



COMPOSITION OF WAVES 33 

heights proportional to these numbers, setting 
off the positive numbers above and the negative 
numbers below a horizontal line, I get the sim- 
ple wave line shown in fig. 13. Now, if this 
>vave is traveling to the right, the same series of 
] lumbers will represent the wave at a later time, 
\vhen they are all displaced towards the right, as 
in the dotted line. 

Now turn to the following schedule of num- 
bers, and consider those which are written in the 
t op 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 
1 vere plotted out as a wave in fig. 13 ; in the top 

100 71 71 100 71 71 100 



FIG. 13. SIMPLE WAVE 

row of the second group they are the same, but 
moved one space to the right, so that they repre- 
sent 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 
i Iways displaced one more space to the right ; 
they thus represent the successive positions of a 



34 SEICHES IN LAKES 



-100 -71 
-100 -71 


A 7t 


100 
100 


71 
71 


\ 

/V 1 


.,-100 ^ 
- 1 OH 


-WO -143 / \42 


0<9 


/4# / (7 -\2 


-too 


-71 -100 / 
-7, / 


-71 

71 


100\ 


71 

71 


m , 
/ 


^ 71 

-71 - 


\ 

-71 ^ 

100\ -71 


- 142 -/wo 





too 


Si 


flOO 


- 


100 


-Y^ 


< 


71 


100 


-71 
71 


' 71 

/V 1 


100 71 
-100 -71 


. S 

L 





\0 








f 


\ ^ 








/ o 


71 
71 


100\ 


-71 

71 


-100 / -71 

/ " 


\ " 

-100\ -71 







' 71 
71 


!Jt2 


100 


\o 


-/TOO 


142 


-100 


\ / 


/? 


11$ 


100 
100 


71 

71 


y, 

A-" - 


-100 
100 


-71 
-71 


\/ 

A 


71 

71 


100 
100 


200 


142/0 


-V^ - 2(70 


-/g 


/ o \ 


^ 


200 



t \ / \ 

71 100 / 71 -71 -100 ' -71 71 

71. o -71 -100\-7l 71 100\ 71 

142 /wo o -wo -\42 rfioo o loo \4$ 

\ V 

100 71 / -71 -100 -71 3 



/ 71 71 / -71 

0\-71 -100 -71 0\ 71 



100 71 



\o o o / o \o 



-71 71 100/71 -71 -100 / -71 

-71 -100\-71 71 100\ 71 -71 

-11$ -100 \0 /WO 11$ 100 \0 -flOO -1J2 

-100 71 Q / n 10 71 / ~ 11 " 10 

-100 -71 (T 71 100 71 O -71 -100 

-200 -142 1L2 WO H2 -11$ -200 

COMPOSITION OF Two EQUAL AND OPPOSITE WAVES 



COMPOSITION OF WAVES 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 coexist, 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 result- 
ant 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 



36 SEICHES IN LAKES 

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 repre- 
sent 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 bino- 
dal, 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, travel- 
ing in opposite directions, when compounded to- 
gether 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 : 



200 


142 





142 


200 


142 


100 





100 


142 

















142 


100 





100 


142 


200 


142 





142 


200 


142 


100 





100 


142 

















142 


100 





100 


142 


200 


142 





142 


200 



Since the middle column consists of zero 
throughout, the water neither rises nor falls 



SEICHES IN LAKE OF GENEVA 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 rock- 
ing 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 -}| of an hour, or 70 minutes. 
Porel, in fact, found the period to be 73 min- 
utes. He expresses this result by saying that 



38 SEICHES IN LAKES 

a uninodal longitudinal seiche in the Lake of 
Geneva has a period of 73 minutes. His obser- 
vations 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 cer- 
tain 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 ob- 
servation 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. 



CAUSES OF SEICHES 39 

The height of the waves called " seiches " is 
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 instru- 
ments 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 Porel, 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 pro- 
duce 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. 
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 
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. 
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 



40 . SEICHES IN LAKES 

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 
to the consideration of an interesting specula- 
tion 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. 
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 meteorolo- 
gists. 

Although, then, it is possible to indicate causes 
competent to produce seiches, yet we cannot as 
yet point out the particular cause for any indi- 
vidual seiche. The complication of causes is so 



VIBRATIONS OF LAKES 41 

great that this degree of uncertainty will prob- 
ably 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 re- 
sembled 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 
from certain quarters, vibrations arise which are 
quite distinct from the ordinary visible wave 
motion. The period of the visible waves on the 



42 SEICHES IN LAKES 

Lake of Geneva is from 4 to 5 seconds, 1 whereas 
vibrations have periods ranging from 45 seconds 
to 4 minutes. Thus there is a clear line sepa- 
rating 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 ob- 
served, 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 
only visible effect is short waves, can be respons- 
ible 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 Merges, the 

1 I observed when it was blowing half a gale on Ullswater, in 
Cumberland, that the waves had a period of about a second. 



VIBRATIONS OF LAKES 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 rang- 
ing 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 vibra- 
tions persist for a long time after any renewal 
of them by the boat has ceased. These vibra- 
tions are called by Forel " the subsequent steamer 
vibrations." 

That the agitation of the water should con- 
tinue for more than two hours is very remark- 
able, 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 



44 SEICHES IN LAKES 

noticeable when the steamer rounds the mole of 
Ouchy, when she is still at a distance of 10 kilo- 
metres. 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. 1 

In this figure the line 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 obser- 
vations were taken, but the record begins again 
at eight o'clock. The portion marked b b f shows 
weak vibrations, probably due to steamers pass- 
ing 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 c c'. The point 
d shows the arrival of this boat at Morges, and 
d f shows the effect of another boat coming from 
Geneva! The portion marked 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 oc- 
cur in the sea, for he says : " What are these 
1 From Les Seiches, V agues d' Oscillation fixe des Lacs, 1876. 



VIBRATIONS OF THE SEA 

oscillations with periods of 
5, 10, 20, or 100 minutes, 
which are sometimes irregu- 
lar ? Are they analogous to 
our seiches ? Not if we de- 
fine seiches as uninodal os- 
cillations, for it is clear that 
if, in a closed basin of 70 
kilometres in length, unino- 
dal 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 vibra- 
tions, and, provisionally, I 
shall call them by the name 
of ' vibrations of the sea.' I 
venture to invite men of sci- 
ence who live on the sea- 
coast to follow this study. 
It presents a fine subject for 
research, either in the inter- 
pretation of the phenomenon 
or in the establishment of 
the relations between these 
movements and meteorologi- 
cal conditions." * 



45 



Seiches et Vibrations des Lacs et de la Mer, 1879, p. 5. 



46 SEICHES IN LAKES 

These vibrations are obviously due to the wind 
or to steamers, but it is a matter of no little sur- 
prise that such insignificant causes should pro- 
duce 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 instan- 
taneously 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 im- 
pulse, but the waves of enormous length and 
those of very small length are of no practical 
importance, and we need only consider the mod- 
erate 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 gen- 
eral 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 iso- 
lated disturbance will generate long waves and 



CAUSES OF VIBRATIONS 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 dis- 
turbance due to a steamer or to the wind as con- 
sisting 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 rapid- 
ity of decrease of short waves than long ones. 1 
It must then be left to future investigators to 
elucidate these points. 

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 govern- 
ment 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 

1 See, however, S. S. Hough, Proc. Lond. Math. Soc., xxviii. 
p. 276. 



48 SEICHES IN LAKES 

comparatively small basin of Geneva. This idea 
appears to have struck Mr. Napier Denison of 
Toronto, and he has been so fortunate as to en- 
list the interest of Mr. Bell Dawson, the chief of 
the Canadian Tidal Survey, and of Mr. Stupart, 
the director of the Meteorological Department. 
Mr. Denison's attention has been, in the first in- 
stance, 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 shal- 
low 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 be- 
low. 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 pass- 
ing into an upper current, which blows from a 
different direction. This abrupt change of tem- 
perature 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 



ATMOSPHERIC WAVES 49 

of fluid cannot slide over another, without gen- 
erating waves at the surface of separation. We 
are familiar with this fact in the case of sea- 
waves generated by wind. A mackerel sky 
proves also the applicability to currents of air of 
Helmholtz's observation. In this case the moist- 
ure 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 vis- 
ible ripple-mark. A mackerel sky is not seen in 
stormy weather, for it affords proof of the exist- 
ence of an upper layer of air sliding with only 
moderate velocity over a lower layer. The dis- 
tance 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 dura- 
tion 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 mag- 
nitude. 

Although the problem involved is not a sol- 
uble one, yet Helmholtz has used the analogy of 



50 SEICHES IN LAKES 

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 illus- 
trated 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 con- 
clusion. 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 a 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 pre- 
dicted in another. Now oceanic waves gener- 
ated by wind afford the known case, and Helm- 
holtz has thence determined by analogy the 



WAVE-LENGTH BY ANALOGY 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 ar- 
rangement 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. 1 

1 A gust of wind will cause the barometer to vary, without a 
corresponding change in the density of the air. It is not there- 
fore safe to interpret the oscillations of the barometer as being 
due entirely to true changes of pressure. If, however, the in- 
termittent 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. 



52 SEICHES IN LAKES 

Mr. Denison's application of this theory con- 
sists 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 en- 
gaged in a systematic comparison of the simul- 
taneous excursions of the water and of the ba- 
rometer 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 

o 

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 fore- 
casting of the weather. 



AUTHORITIES 53 

I must, however, emphasize that these conclu- 
sions 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. 

AUTHORITIES. 

Papers by Dr. Forel on Seiches. 

" Bibliotheque Universelle, Archives des Sciences physiques 
et naturelles," Geneva : 
Formule des Seiches, 1876. 
Limnimetre Enregistreur, 1876. 
Essai monographique, 1877. 
Causes des Seiches, Sept. 15, 1878. 
Limnographe, 15 Ddc., 1878. 
Seiche du 20 Fevrier, 1879, 15 Avril, 1879. 
Seiches dicrotes, 15 Jan., 1880. 
Formules des Seiches, 15 Sept., 1885. 

" Bulletin de la Soc. Vaudoise des Sciences naturelles : " 
Premiere tftude, 1873. 
Deuxieme tftude, 1875. 

Limnimetrie du Lac Lemon. I re Se'rie. Bull. xiv. 1877. 
IP Se'rie. Bull. xv. IIP Se'rie. Bull. xv. 1879. 

" Actes de la Soc. helv. Anderinatt : " 
Les Seiches, Vagues d" Oscillation, 1875. 

" Association Franchise pour 1'avancement," etc. : 
Seiches et Vibrations, Congres de Montpelier, 1879. 

" Annales de Chimie et de Physique : " 
Les Seiches, Vagues d' 'Oscillation, 1876. 
Un Limnimetre Enregistreur, 1876. 

Helmholtz, Sitzungsberichte der Preuss. Akad. der Wissen- 
schaft, July 25, 1889 ; transl. by Abbe in Smithsonian Reports. 



54 SEICHES IN LAKES 

F. Napier Denison : 

Secondary Undulations . . . found in Tide-Gauges. "Proc. 

Canadian Institute," Jan. 16, 1897. 
The Great Lakes as a Sensitive Barometer. " Proc. Canadian 

Institute," Feb. 6, 1897. 
Same title, but different paper, " Canadian Engineer," Oct. 

and Nov., 1897. 



CHAPTER III 1 

TIDES IN RIVERS TIDE MILLS 

SINCE most important towns are situated on 
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 rhyth- 
mical rise and fall of the sea generate waves 
which would travel up the river, whatever were 
the cause of the oscillation of the sea. Accord- 
ingly, 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 Chapter II. that long waves 
progress in shallow water at a speed which de- 
pends only on the depth of the water, and that 

1 The account of the bore in this chapter appeared as an 
article in the Century Magazine for August, 1898. The illus- 
trations then used are now reproduced, through the courtesy of 
the proprietors. 



56 TIDES IN RIVERS TIDE MILLS 

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 there- 
fore 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 
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 synony- 
mously ; but we shall see that this usage is in- 
correct. 

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 con- 
sidered 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, 



TIDAL CURRENTS IN RIVERS 57 

very simple. Whenever the water stands above 
the mean level the current is up-stream and pro- 
gresses 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 cur- 
rents 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 super- 



58 TIDES IN RIVERS TIDE MILLS 

posed 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 accord- 

o 

ing 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 
faUs, 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 shal- 
low water as calculated theoretically. If the 
steepening of the advancing slope of a wave 



CHANGE OF FORM IN SHALLOW WAVE 59 

were carried to an extreme, the wave would pre- 
sent 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 



FIG. 15. PROGRESSIVE CHANGE OF A TRAIN 
OF WAVES IN SHALLOW WATER 

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 cooper- 
ates with the natural change in 'the shape of a 
wave as it runs into shallow water, so as to exag- 
gerate the steepness of the advancing slope of 
the wave. 

There are in the estuaries of many rivers 
broad flats of mud or sand which are nearly dry 
at low water, and in such situations the tide not 
unf requently 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 
French mascare. 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 



60 TIDES IN RIVERS TIDE MILLS 

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 
advantage of supplementing verbal description 
by photographs. 

The estuary on which the observations were 
made is that of the Tsien-Tang-Kiang, a consid- 
erable river which flows into the China Sea about 
sixty miles south of the great Yang-Tse-Kiang. 
At most places the bore occurs only intermit- 
tently, 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 
nothing more than a mere mention of it is to be 
found in any previous publication. 

In 1888 Captain Moore, K. 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 re- 
ports drawn up after his two visits. The an- 
nexed 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 



SURVEY OF TSIEN-TANG-KIANG 



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 Pan- 
dora and Gulnare, towing the sailing cutter 



LCANO I. 




FIG. 16. CHART OF THE ESTUARY OF THE TSIEN-TANG-KIANG 

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, 



62 TIDES IN RIVERS TIDE MILLS 

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 en- 
gines 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 P. M. 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 A. M. a current of ex- 
treme 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 Bruns- 
wick 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 A. M. they had been struck 
by a violent rush of water in a succession of big 
ripples. In a few moments they were afloat in 



DANGERS OF THE BORE C3 

an eight-knot current ; in ten minutes the wate.r 
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 pol- 
ished 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 warn- 
ing, 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. Im- 
mediately after the passing of the bore the 
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 pas* 
sage of the bore, the water rises on the platform 
very rapidly, but the junks are just able to float 



64 TIDES IN RIVERS TIDE MILLS 

in safety. Captain Moore gives a graphic ac- 
count 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 




FIG- 17. BORE-SHELTER ON THE TSIEN-TANG-KIANG 

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 na- 
ture of the bore with admirable clearness. The 
observations of water-level were taken simul- 
taneously 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 



SIMULTANEOUS OBSERVATIONS 65 

the lines marked Rambler and Volcano -repre- 
sents 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 simul- 
taneously. 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 Vol- 
cano 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 Ram- 
bler 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 



66 



TIDES IN RIVERS TIDE MILLS 



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 Kamhler 
Island, outside, and Haining, in the river. Al- 
most exactly at midnight the strain broke down 
and the bore started somewhere between Ram- 
bler 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 

PROJECTION OP BORE AND AFTER RUSH. 
HAINING. RAMBLER. I. .^ -. p M lftRa W.VOLCANO I 

FEET FEET <" SEPT. P. M. 1888. ,. EIT 

U RAMBLER i. TO HAINING LJ fo. VOLCANO I. TO RAMBLER I. 51 MILES. 
26 MILES. '- -" "* 




FIG- 18. DIAGRAM OP THE FLOW OF THE TIDE ON 

THE TsiEN-TANG-KlANG 

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, 



DESCRIPTION OF THE BORE 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 suf- 
fice to say that there is no bore down-stream, 
although there is at one time a very violent 
current. 

In 1892 Captain Moore succeeded, with con- 
siderable difficulty, in obtaining photographs of 
the bore as it passed Haining. They tell more 
of the violence of the wave than could be con- 
veyed by any amount of description. The pho- 
tographs, 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 dis- 
appears 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 photo- 
graph, 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 



68 TIDES IN RIVERS TIDE MILLS 

October 9, 1892. The first of these photo- 
graphs was taken at 1.29 p. M., and the second 
represents the view only one minute later. 

The Chinese regard the bore with superstitious 
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 assassi- 
nated 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 Hang- 
chow, 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, how- 
ever, did not have the desired effect, as the high 
tide came in as before ; and it was at last deter- 



PICTURES OF THE BORE 




FIG- 19. PICTURES OF THE BORE ON THE TSIEN-TANG-KIANG 



70 TIDES IN RIVERS TIDE MILLS 

mined 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 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 exist- 
ence 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 cen- 
turies ago ; for Marco Polo, in the thirteenth 
century, stayed about a year and a half at 
Hangchow, and gives so faithful and minute 
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 inter- 
mittent. I have also learned from Captain 
Moore himself that at the time of the great 
Taiping rebellion, the suppression of which was 
principally due to " Chinese " Gordon, the in ten- 



OTHER CASES OF BORES 71 

sity of the bore was far less than it is to-day. 
This shows that the bore is liable to great vari- 
ability, 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 obser- 
vations some five or six thousand people as- 
sembled 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 advan- 
cing 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 
in England, on the Seine in France, on the Petit- 
codiac in Canada, on the Hugli in India, and 
doubtless in many other places. In general, 
however, it is only at spring tides and with cer- 



72 TIDES IN RIVERS TIDE MILLS 

tain 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, 
and a rapid rise of water-level. 

I have shown, at the beginning of this chap- 
ter, 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 com- 
bine 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 ab- 
solutely sudden one. I think that it would be 
impossible, from the mere inspection of an estu- 
ary, 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 irreg- 
ularity in the depth and form of an estuary ren- 
ders 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 comprehen- 
sion of the causes which produce the observed 
result. 



UTILIZATION OF TIDAL ENERGY 73 

The manner in which the Chinese avail them- 
selves of the after-rush for ascending the river 
affords an illustration of the utilization by man- 
kind of tidal energy. In going up-stream, a 
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 con- 
sideration 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 



74 TIDES IN RIVERS TIDE MILLS 

serviceable power. If we picture to ourselves the 
immense weight of a large ship, we may be de- 
luded 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 in- 
stallation would be far better bestowed on water- 
wheels in rivers or on wind-mills. I am glad to 
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 im- 
practicability of his plan. 

We may, then, fairly conclude that, with ex- 
isting mechanical appliances, the attempt to util- 
ize 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 



AUTHORITIES 75 

adaptable configuration of shoals in an estuary. 
There are, no doubt, many such mills in the 
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 auto- 
matically 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 bound- 
less energy of the tide, rivers and wind and fuel 
are likely for all time to be incomparably more 
important for the use of mankind. 

AUTHORITIES. 

On waves in rivers see Airy's article on Tides and Waves in 
the " Encyclopaedia Metropolitaiia." Some of his results will 
also be found in the article Tides in the " Encyclopaedia Britan- 
nica." 

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

Further Report, &c., by the same author and publisher, 1893. 



CHAPTER IV 

HISTORICAL SKETCH 

I CANNOT claim to have made extensive inves- 
tigations as to the ideas of mankind at different 
periods on the subject of the tides, but I pro- 
pose 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, pro- 
fessing 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, 
Icelandic, and classical literatures of such pas- 
sages as they were able to discover. 

I learn from Professor Giles that Chinese 
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 



CHINESE THEORIES 77 

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 sum- 
mer 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 na- 
ture 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. 

Mr. E. G. Browne has translated for me the 
following passage from the " Wonders of Crea- 
tion " of Zakariyya ibn Muhammad ibn Mah- 
mud al Qazvim, who died in A. D. 1283. 1 

" Section treating of certain wonderful condi- 
tions 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. 

1 Wustenfeld's edition, pp. 103, 104. 



78 HISTORICAL SKETCH 

66 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 . . . 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 cur- 
rents 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 



AKABIC AND ICELANDIC THEORIES 79 

begins again until she reaches the eastern hori- 
zon. This is the account of the flow and ebb 
of the sea. 

" The agitation of the sea resembles the agi- 
tation 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 poeti- 
cal 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.' ' 

Mr. Magnusson has kindly searched the old 
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 blow- 
ing 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 



80 HISTORICAL SKETCH 

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 liter- 
ature for grotesque theories of the tides. In 
1722 E. Barlow, gentleman, in " An Exact Sur- 
vey of the Tide," 1 attributes it to the pressure 
of the moon on the atmosphere. And theories 
not less absurd have been promulgated during 
the last twenty years. 

The Greeks and Komans, living on the shores 
of the Mediterranean, had not much occasion to 
learn about the tide, and the passages in classi- 

1 " The Second Edition, with Curious Maps." (London: John 
Hooke, 1722.) 



OBSERVATIONS OF POSIDONIUS 81 

cal 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 
detail is Posidonius, and we have to rely for our 
knowledge of his work entirely on quotations 
from him by Strabo. 1 

Posidonius says that Aristotle attributed the 
flow and ebb of the sea at Cadiz to the moun- 
tainous 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 : 2 

" 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 (east- 
ern) horizon by the distance of one sign of the 
zodiac (i. e. 30) the sea begins to flow, and en- 

1 My attention was drawn to Strabo by a passage in Sir W. 
Thomson's (Lord Kelvin's) Popular Lectures, 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 Posi- 
donius (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. 

2 Teubner's Strabo, i. p. 236. 



82 HISTORICAL SKETCH 

croaches 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 (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 re- 
treats 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 



OBSERVATIONS OF POSIDONIUS 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, arid greater until the summer 
solstice." 

This is an excellent account of the tides at 
Cadiz, but I doubt whether there is any founda- 
tion for that part which was derived from hearsay. 
Lord Kelvin remarks, however, that it is inter- 
esting 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." 



84 HISTORICAL SKETCH 

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 evi- 
dence as to an annual inequality in the tides. 

There is another very interesting passage in 
Strabo, the meaning of which was obviously un- 
known 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 pheno- 
mena in remoter parts of the world. Strabo 
writes : 

" Anyhow Posidonius says that Seleucus of 
the Ked Sea [also called the Babylonian] de- 
clares that there is a certain irregularity and reg- 
ularity 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 irregu- 
larity 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 sol- 
stices 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 



SELEUCUS THE BABYLONIAN 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 
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 chap- 
ter) 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 
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. 

Many centuries elapsed after the classical 
period before any scientific thought was be- 
stowed on the tides. Kepler recognized the 
tendency of the water on the earth to move 
towards the sun and the moon, but he was un- 
able to submit his theory to calculation. Gal- 
ileo expresses his regret that so acute a man as 
Kepler should have produced a theory, which 
appeared to him to reintroduce the occult quali- 
ties of the ancient philosophers. His own expla- 
nation referred the phenomenon to the rotation 
of the earth, and he considered that it afforded 
a principal proof of the Copernican system. 



86 HISTORICAL SKETCH 

The theory of tide-generating force which will 
be set forth in Chapter V. is due to Newton, 
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, 
and Cavalleri. The first three adopted, not only 
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 Mac- 
laurin contained remarkable advances in mathe- 
matical knowledge, but did not add greatly to 
the theory of the tides. The Jesuit priest 
Cavalleri adopted the theory of vortices to ex- 
plain the tides, and it is not worth while to 
follow him in his erroneous and obsolete specu- 
lations. 

Nothing of importance was added to our 
knowledge until the great French mathematician 
Laplace took up the subject in 1774. It was he 
who for the first time fully recognized the diffi- 
culty 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 



HARMONIC ANALYSIS 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 Lub- 
bock senior, Whewell, Airy, and Lord Kelvin. 
The work of Lubbock and Whewell is chiefly 
remarkable for the coordination 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 al- 
ready 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 mo- 
tion 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 
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. 



88 HISTORICAL SKETCH 

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 here- 
after of the theory of oceanic tides is based on 
the work of Newton, Bernoulli, Laplace, and 
Kelvin, in proportions of which it would be 
difficult to assign the relative importance. 

The connection between the moon and the 
tide is so obvious that long before the formula- 
tion of a satisfactory theory fairly accurate pre- 
dictions of the tides were made and published. 
On this head Whewell 1 has the following inter- 
esting passage : 

"The course which analogy would have recom- 
mended for the cultivation of our knowledge of 
tides would have been to ascertain by an analy- 
sis 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 

1 History of the Inductive Sciences, 1837, vol. ii. p. 248 et seq. 



EMPIRICAL METHOD OF PREDICTION 89 

the application of knowledge to the useful pur- 
poses of life, being thus separated from the 
promotion of the theory, was naturally treated 
as a gainful property, and preserved by secrecy. 
. . . Liverpool, London, and other places, had 
their tide tables, constructed by un divulged 
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 re- 
sented as an infringement of the rights of pro- 
perty. 

" 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 busi- 
ness ; and that they were called upon, as the 
guardians of the established theory of the uni- 
verse, to compare it in the greatest possible 



90 HISTORICAL SKETCH 

detail with the facts. Mr. Lubbock was the 
first mathematician who undertook the extensive 
labors which such a conviction suggested. Find- 
ing that regular tide observations had been made 
at the London docks from 1795, he took nine- 
teen 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 un- 
important 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 sci- 
entific process were more exact than those which 
resulted from any of the secrets ; and thus prac- 
tice was brought into its proper subordination to 
theory." 

AUTHORITIES. 

The history from Galileo to Laplace is to be found in the 
Mecanique Celeste of Laplace, book xiii. chapter i. 
The other authorities are quoted in the text or in footnotes. 



CHAPTER V 

TIDE-GENERATING FORCE 

IT would need mathematical reasoning to fully 
explain how the attractions of the sun and moon 
give rise to tide-generating forces. But as this 
book is not intended for the mathematician, I 
must endeavor to dispense with technical lan- 
guage. 

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 mo- 
tion 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 per- 
missible to abstract our attention from the mo- 
tion 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 con- 
tinuously deflected from its straight path by the 



92 TIDE-GENERATING FORCE 

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 cen- 
tre when we consider the thing whirled. A sim- 
ilar 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 mechan- 
ism by which the moon is kept revolving round 
the earth. There is not of course any visible 
connection between the two bodies, but an invis- 
ible bond is provided by the attraction of grav- 
ity, which replaces the string which unites the 
stone to the hand. The moon, then, whirls 



THE MOON'S ORBITAL MOTION 93 

round the earth at just such a rate and at just 
such a distance, that her resistance to circular 
motion, called centrifugal force, is counterbal- 
anced by the centripetal tendency of gravity. If 
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 



Axis 




) o 

/ 



Earth 240,000 nules Moon 

FIG. 20 EARTH AND MOON 

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 
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 



94 TIDE-GENERATING FORCE 

solar tide will become obvious. But the anal- 
ogy 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 grav- 
ity 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 propor- 
tions. The upper figure is more convenient for 
our present argument, and the planet and satel- 
lite may be described as the earth and the moon, 
notwithstanding . the exaggeration of their rela- 
tive 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 3000 miles in the oppo- 
site direction. The 3000 and 237,000 miles 
together make up the 240,000 miles which sepa- 
rate the centres of the two bodies. 



THE MOON'S ORBITAL MOTION 95 

A system may now be devised so as to resem- 
ble 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 con- 
sequence 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. Fix- 
ing 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 act- 
ing on the earth is equal and opposite to the 
attraction of the moon on it. In this experi- 
ment 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 



96 TIDE-GENERATING FORCE 

acted on by these forces, and accordingly a 
closer analysis is necessary. 

It will now simplify matters if we make a sup- 
position which departs from actuality, introdu- 
cing 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 3000 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 revo- 
lution 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 sep- 
arated 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 de- 
scribed, and in this case the radii of the circles 



DIURNAL ROTATION ANNULLED 97 

described by any two particles in the arrow are 
always parallel. The parallelism of the centri- 
fugal forces at the two ends of the arrow is 
indicated in the figure. Then again, the centri- 
fugal 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 cen- 
trifugal forces. The same must therefore be 
true of the earth when deprived of diurnal rota- 
tion. Accordingly every particle of the ideal- 
ized non-rotating earth is continuously subject to 
equal and parallel centrifugal forces, in conse- 
quence of the revolution of the earth's centre 
in its monthly orbit with a radius of 3000 
miles. 1 

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 attraction al 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 per- 
fect balance on the whole, there is not that uni- 

1 I owe the suggestion of this method of presenting the ori- 
gin of tide-generating force to Professor Davis of Harvard 
University. 



98 



TIDE-GENERATING FORCE 



formity which would render the balance perfect 
at every particle. 

As far as concerns the totality of the attrac- 
tion 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 




FIG. 21. REVOLUTION OF A BODY WITHOUT ROTATION 

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 



DIURNAL ROTATION ANNULLED 99 

and sixty-one radii respectively. Hence the at- 
tractions 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 cen- 
trifugal 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 at- 
traction is stronger than the average it overbal- 
ances 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 ex- 
hibits 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 



100 



TIDE-GENERATING FORCE 



in the contest between centrifugal and centri- 
petal tendencies. The point v in the figure is 
the middle of the hemisphere, which at the mo- 
ment portrayed faces full towards the moon. It 




FIG. 22. TIDE-GENERATING FORCE 

is the middle of the round disk which the man in 
the moon looks at. The middle of the face in- 
visible 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 indi- 
cated 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 overbal- 
anced, as indicated by the arrow pointing away 
from the moon. 

I shall have to refer hereafter to the intensi- 



LAW OF VARIATION 101 

ties 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 ^ or ^ as a 
measure of the intensity of the moon's attraction 
at the earth's centre. The particle which occu- 
pies the centre of the earth is also that particle 
which is at the average distance of all the parti- 
cles constituting the earth's mass. Hence 6 -p or 
3^ 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 ai or 3^. 

The attraction therefore at v exceeds the aver- 
age by i ^ 2 , or 3^3300- It will be well to 
express these results in decimals ; now ^j is 
.000,287,27, and ^ is .000,277,78, so that the 
difference is .000,009,49. It is important to 
notice that ~ or ^m is equal to .000,009,26; 
so that the difference is nearly equal to 6 -p. 

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 ^ or ^. The at- 
traction at i therefore falls below the average by 
6P 6P, or ai si; that is, by .000,277,78 



102 TIDE-GENERATING FORCE 

.000,268,75, which is equal to .000,009,03. 
This again does not differ much from ~. 

These calculations show that the excess of the 
actual attraction at v above the average attrac- 
tion 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 ^ on the adopted scale of measure- 
ment. 

The use of any particular scale of measure- 
ment 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. 1 

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 

1 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\ and 
a;+l radii. Now (a; I) 2 is nearly equal to x 2 2x or to 

a; 2 (l |), and therefore (a j; 1)a is nearly equal to a;2(1 _2), which is 

nearly equal to ^(1+J). Hence ( ^^ j* is nearly equal to 
ji. By a similar argument (a;+l) 2 is nearly equal to # 2 (l + f), 
and ^- 2 is nearly equal to ^ (1--|); so that &-$+& is nearly 
equal to Jj. 



LAW OF VARIATION 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 in- 
wards 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 direc- 
tion. 

The only point in which the system consid- 
ered 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, 



104 TIDE-GENERATING FORCE 

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 shaU 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 dis- 
placed 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 paral- 
lelism 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 



LAW OF VARIATION 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 inten- 
sity 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 inhab- 
itant 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 



106 TIDE-GENERATING FORCE 

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 hori- 
zontally 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 down- 
wards away from or towards some point in the 
sky other than the zenith, it may be decomposed 
into two forces, one vertically upwards or down- 
wards, and another along the horizontal sur- 
face. 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 



FIG. 23. HORIZONTAL TIDE-GENERATING FORCE 



HORIZONTAL FORCE 107 

with a system of forces which may be represented 
by the arrows in the perspective picture of hori- 
zontal 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 verti- 
cally 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 horizon- 
tal force where the moon is over his head or un- 
der 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 pic- 
ture at A I? and is gradually carried along to A^ 
by the earth's rotation, and so onwards. Just be- 
fore 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 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 



108 TIDE-GENERATING FORCE 

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, ac- 
cording 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 formulae. I shall in the follow- 
ing chapter consider the general nature of the 
changes which the forces undergo at any point 
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 coexists 
with the lunar force. I shall hereafter show 
how the relative importance of these two influ- 
ences is to be determined. 

AUTHORITIES. 

Any mathematical work on the theory of the tides; for exam- 
ple, Thomson and Tait's Natural Philosophy, Lamb's Hydrody- 
namics, Bassett's Hydrodynamics, article Tides, " Encycl. Britan.," 
Laplace's Mecanique Celeste, &c. 



CHAPTER VI 

DEFLECTION OF THE VERTICAL 

THE intensity of tide-generating force is to be 
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 be- 
tween 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 simi- 
larly show the effect of a horizontal force by the 
displacement of the bubble. When dealing with 
tidal forces the displacements of either the pen- 
dulum 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 



110 DEFLECTION OF THE VERTICAL 

moon may be shown by a calculation, which in- 
volves the mass and distance of the moon, to 
have an intensity of 11>66 1 0tU(M) of gravity. 1 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 deflec- 
tion of the weight would be 11<66 1 0000 of 10 metres, 

1 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. 

The moon weighs -gL- of the earth, and has a radius ^ as large. 
Hence lunar gravity on the moon's surface is ^x4 2 , or 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 ^X^Q 2 of terrestrial gravity, and lunar gravity at the 
point V is lx -j^g 2 of the same. Therefore the tidal force at V 
is x -^-$1 |x ^9-2 of terrestrial gravity. On multiplying the 
squares of 236 and of 240 by 5, we find that this difference is 
imfrBTF TTff/oTo ^ 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 ^,-f^,-Q-Q^ ^ 
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 | of the vertical force 
at V or i. To find f of the above fraction we must augment the 
denominator by one third part. Hence the maximum horizontal 
force is yT.-^V.'oFo f gravity. This number does not agree ex- 
actly 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 



NUMERICAL ESTIMATE 



111 



or j-j^g of a millimetre. In English measure this 
is 29^00 f 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 
as much in the opposite direction. Hence the 




FIG. 24. DEFLECTION OF A PENDULUM ; THE MOON AND 
OBSERVER ON THE EQUATOR 

excursion to and fro of the pendulum under the 
lunar influence will be j^ of an inch. With a 
pendulum one metre, or 3 ft. 3 in. in length, 
the range of motion of the pendulum bob is 
i4poo of an inch. For any pendulum of manage- 
able 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. 

The curves which are traced out by a pendu- 
lum present an infinite variety of forms, corre- 



112 DEFLECTION OF THE VERTICAL 

spending 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 celes- 
tial 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 
which the bob occupies successive positions are 
marked on the curve. 

If the larger ellipse be taken to show the dis- 
placement of a pendulum when the sun and 




FIG. 25. DEFLECTION OF A PENDULUM; THE MOON IN N. 
DECLINATION 15, THE OBSERVER IN N. LATITUDE 30 

moon cooperate 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 



THE PATH OF A PENDULUM 113 

of the sun's force. Here also the hours are 
marked at the successive positions of the pen- 
dulum, 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 actu- 
ality, 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-gen- 
erating 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 formulae 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 sev- 
eral attempts have been made by more or less 



114 DEFLECTION OF THE VERTICAL 

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 perpendicu- 
larly 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 attrac- 
tion, 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 know- 
ledge 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 



BIFILAR PENDULUM 



115 



the moon, and the comparison of the observed 
oscillation, with that computed on the supposi- 
tion 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 interest- 
ing as giving a measure of the 
stiffness of the earth as a whole. 

I must pass over the various 
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 displace- 
ments of a pendulum with refer- 
ence to the ground on which it 
stood. The principle of the ap- 
paratus used for this purpose is 
due to Lord Kelvin ; it is very 
simple, although the practical 
application of it was not easy. 

Fig. 26 shows diagrammatically, and not drawn 
to scale, a pendulum 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 



ID 

Mirror 

FIG. 26. BIFILAB 
PENDUIAJM 



116 DEFLECTION OF THE VERTICAL 

B 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 ver- 
tical lines passing through the two points of sus- 
pension are ^ 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 im- 
age of the light danced and wandered inces- 
santly. 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 



BIFILAR PENDULUM 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 pro- 
jected 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 pro- 
jected, 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 vis- 
cosity or stickiness than either pure spirits 01 
pure water. 

The copper tube, with the pendulum and mir- 
ror-box, was supported on three legs resting on 



118 DEFLECTION OF THE VERTICAL 

a block of stone weighing a .ton, and this stood 
on the native gravel in a north room in the lab- 
oratory at Cambridge. The whole instrument 
was immersed in a water-jacket, which was fur- 
nished 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 pre- 
liminary 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 obser- 
vation fixed. In our instrument the image of 
the movable gas jet was observed by a fixed tel- 
escope 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. 



SENSITIVENESS OF THE INSTRUMENT 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 im- 
age 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 un- 
til 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 precau- 
tions were shown by experience to be necessary, 
and were possibly even insufficient to guard the 
instrument from the effects of changes of temper- 
ature. 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 ap- 
proximate 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 appli- 
ances 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 



120 DEFLECTION OF THE VERTICAL 

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 ipoo 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 act- 
ually 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 station- 
ary gas jet were made without the use of the tel- 
escope. The scale on which the reflected spot 
of light fell was laid on the ground at about 



WARPING OF THE SOIL 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 watch- 
ing 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 incred- 
ible 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 pres- 
sure 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 im- 
age 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 re- 
sult, for we found that the readiest way of bring- 
ing 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 



122 DEFLECTION OF THE VERTICAL 

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 instru- 
ment 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 dis- 
placement 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 be- 
low 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 Ibs. on 



WARPING OF THE SOIL 123 

the stone paved floor must have been pretty 
small, and the slope of the sides of that dimple 
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 observa- 
tion would be rendered impossible by the street 
traffic and by the ordinary work of the labora- 
tory. 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 pedes- 
tal smart blows with a stick, without producing 
any sensible effect. But it was not quite easy to 
try this experiment, because there was a consid- 
erable 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 



124 DEFLECTION OF THE VERTICAL 

many times the amount of the lunar effect which 
we sought to measure. 

An example will explain how this was : A se- 
ries 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 in- 
terrupted 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. In- 
deed, 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 be- 
tween 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 to- 
wards which it had been creeping for four days 
previously. 

Notwithstanding all our precautions the pen- 
dulum was never at rest, and the image of the 
flame was always trembling and dancing, or wav- 
ing 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 



INSTABILITY OF THE PENDULUM 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 move- 
ments is still to some extent uncertain. Quite 
recent investigations by Professor Milne seem to 
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 perturba- 
tions, it remains certain that a large proportion 
of these mysterious movements are due to minute 
earthquakes. 

Some part of the displacements of our pen- 
dulum 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. 1 

The minute earthquakes of which I have 
spoken are called by Italian observers micro- 
sisms, and this name has been very generally 
adopted. The literature on the subject of seis- 
mology is now very extensive, and it would be 
out of place to attempt to summarize here the 

1 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. 



126 DEFLECTION OF THE VERTICAL 

conclusions which have been drawn from obser- 
vation. I may, however, permit myself to add a 
few words to indicate the general lines of the re- 
search, 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 pen- 
dulums 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 
may be summarized as follows : 

The oscillation of the pendulum is generally 
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, ac- 
cording 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 mi- 
croseismic movements are especiaUy apt to be in 
a vertical direction. They are always so when 



ITALIAN SEISMOLOGY 127 

the earthquake is local, but the vertical move- 
ments are sometimes absent when the shock 
occurs elsewhere. Sometimes there is no move- 
ment 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 
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 sum- 
mer solstice. 

At Florence a period of earthquakes is pre- 
saged by the magnitude and frequency of oscil- 
latory movements in a vertical direction. These 
movements are observable at intervals and dur- 
ing several hours after each shock. 

Some very curious observations on microsisms 
have also been made in Italy with the micro- 
phone, by which very slight movements of the 
soil are rendered audible. 

Cavaliere de Rossi, of Kome, 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 



128 DEFLECTION OF THE VERTICAL 

roads, he placed his microphone at a depth of 20 
metres 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 
voUeys, and metallic or bell-like sounds " (fre- 
miti, 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 caUs " rumbling (?) 
or crackling " (rullo o crepito). The roaring 
(fremito) was the only noise which he could re- 
produce 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 earth- 
quake." 

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. 



THE MICROPHONE 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. 

Eossi 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 (i colpi di 
moschetteria), and the undulatory shocks gave 
the roarings (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 [pattito 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 cir- 
cuit ; but that this was not the case, and that 
"ticking" is merely a mode of representing a 



130 DEFLECTION OF THE VERTICAL 

natural noise is proved by the fact that he sub- 
sequently says that he considers the ticking to 
be " a telluric phenomenon." 

Kossi then took the microphone to the Sol- 
fatara of Pozzuoli, and here, although no sensi- 
ble 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 Sol- 
fatara 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 
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 dis- 
tinguished investigator in seismology is Professor 
Milne, formerly of the Imperial College of Engi- 
neering at Tokyo. His residence in Japan gave 
him peculiar opportunities of studying earth- 
quakes, for there is, in that country, at least one 
earthquake per diem of sufficient intensity to 



THE HORIZONTAL PENDULUM 131 

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 
Kebeur-Paschwitz, whose early death deprived 
the world of a skillful and enthusiastic investi- 
gator. 

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

The horizontal pendulum is like a door in its 
mode of suspension. If a doorpost be abso- 
lutely 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 posi- 
tion 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 pendu- 
lum at Wilhelmshaven, Potsdam, Strassburg, and 
Orotava. He almost convinced himself at one 
time that he could detect, amidst the wanderings 



132 DEFLECTION OF THE VERTICAL 

of the curves of record, a periodicity correspond- 
ing 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 obser- 
vations at Strassburg have been continued by 
M. Ehlert. His results show an excellent con- 
sistency with those of Paschwitz, and are there- 
fore 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 indi- 
rect manner an effect on the motion of a pen- 
dulum 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. 



AUTHORITIES 133 



AUTHORITIES. 

G. H. Darwin and Horace Darwin, "Reports to the British 
Association for the Advancement of Science : " 

Measurement of the Lunar Disturbance of Gravity. York 

meeting, 1881, pp. 93-126. 
Second Report on the same, with appendix. Southampton 

meeting, 1882, pp. 95-119. 
E. von Rebeur-Paschwitz, Das Horizontalpendel. 

Nova Acta Leop. Carol. Akad.," 1892, vol. Ix. no. 1, p. 213; 

also " Brit. Assoc. Reports," 1893. 

E. von Rebeur-Paschwitz, Ueber Horizontalpendel-Beobach- 
tungen in Wilhelmshaven, Potsdam und Puerto Orotava auf Ten- 
erifa. 

" Astron. Nachrichten," vol. cxxx. pp. 194-215. 
R. Ehlert, Horizontalpendel-Beobachtungen. 

" Beitrage zur Geophysik," vol. iii. Part I., 1896. 
C. Davison, History of the Horizontal and Bifilar Pendulums. 
" Appendix to Brit. Assoc. Report on Earth Tremors." Ips- 
wich meeting, 1895, pp. 184-192. 
" British Association Reports of Committees." 

On Earth Tremors, 1891-95 (the first being purely formal). 
On Seismological Investigation, 1896. 

The literature on Seismology is very extensive, and would 
need a considerable index ; the reader may refer to Earthquakes 
and to Seismology by John Milne. Both works form volumes in 
the International Scientific Series, published by Kegan Paul, 
Trench, Triibner & Co. 



CHAPTER VII 

THE ELASTIC DISTORTION OF THE EARTH'S SUR- 
FACE BY VARYING LOADS 

WHEN the tide rises and falls on the seacoast, 
many millions of tons of water are brought alter- 
nately nearer and further from the land. Ac- 
cordingly 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 vary- 
ing tidal load. 

We saw in Chapter VI. how a weight resting 
on the floor in the neighborhood of our pendu- 
lum produced a dimple by which the massive 
stone pedestal of our instrument was tilted over. 
Now as low tide changes to high tide the posi- 
tion 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 



FORM OF DIMPLE 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 elas- 
tic 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 




a 

FIG. 27. FORM OF DIMPLE IN AN ELASTIC SURFACE 

of the original distance the slope will be aug- 
mented 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 



136 DISTORTION OF THE EARTH'S SURFACE 

will be as shown, but the centre will not be infi- 
nitely 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 pro- 
duced by the pressure of a weight, that weight, 
being endowed with gravitation, attracts any 
other body with a f or$e varying inversely as the 
square of the distance. It follows, therefore, 
that the slope of the dimple is everywhere ex- 
actly proportional to the gravitational attraction 
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 su- 
perposition 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 pro- 
portional 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 pro- 
duced by the pressure of the weights. It may 



SLOPE PROPORTIONAL TO ATTRACTION 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 
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. Calcula- 
tion indicates that, with the known average elas- 
ticity of rock, the tilting of the soil is likely to 



138 DISTORTION OF THE EARTH'S SURFACE 

be far greater than the other two effects com- 
bined. 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 im- 
portant 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 distin- 
guish the direct effects of the moon's tidal at- 
traction from the indirect effects, but I am not 
very hopeful of success in this respect. It was 
pointed out in 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 obtain- 
able 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 sim- 



WARPING OF SOIL 139 

plifies 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 




FIG. 28. DISTORTION OF LAND AND SEA-BED BY TIDAL LOAD 

vastly simpler than in actuality, yet it will suffice 
to give interesting indications of what must 
occur in nature. 

The figure 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 



140 DISTORTION OF THE EARTH'S SURFACE 

water at the left. The figure would be re- 
versed 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 be- 
comes 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 dis- 
torted 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 augmenta- 
tion 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 mod- 
erate, 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 



WARPING THE LAND 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 usu- 
ally 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 stiff est 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 3J miles, and ac- 
cordingly a hundredth of a second of arc, usu- 
ally 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 : - 

Distance from high- Slope of the land measured in 

water mark hundredths of seconds of arc 



10 metres 

100 metres 

1 kilometre 
10 kilometres 
20 kilometres 

100 kilometres 



10 
8 
6 
4 

34 
2 



142 DISTORTION OF THE EARTH'S SURFACE 

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 stiff- 
ness of rock these angles of slope must be aug- 
mented 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 : 

Distance from high- Apparent range of deflection 

water mark of the vertical 



10 metres . 

100 metres . 
1 kilometre 
10 kilometres 
20 kilometres 

100 kilometres 



0".25 

0".20 

0".15 

O'MO 

0".084 

0".050 



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 



WARPING OF THE LAND 143 

results may be taken as fairly representative, al- 
though perhaps somewhat underestimated. Lord 
Kelvin has made an entirely independent esti- 
mate of the probable deflection of a pendulum 
by the direct attraction of the sea at high tide. 
He supposes the tide to have a range of 10 feet 
from low water to high water, and he then esti- 
mates 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 gen- 
eral 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 re- 
sults present a satisfactory agreement. 

This speculative investigation receives confir- 
mation from observation. The late M. d' Abba- 
die established an observatory at his chateau of 



144 DISTORTION OF THE EARTH'S SURFACE 

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 grav- 
ity. 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 mer- 
cury. 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 1 and 73 low waters, and he also 
found that in about two thirds of the observa- 
tions 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 Kebeur-Paschwitz also noted deflec- 
tions due to the tide at Wilhelmshaven in Ger- 

1 Presumably the observation at one high water was defective. 



THEORY CONFIRMED 145 

many. 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 observa- 
tory were built on rock. 

This investigation has another interesting ap- 
plication, 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 Ibs. 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 be- 
fore, 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 baro- 
metric gradients on the earth's surface makes it 



146 DISTORTION OF THE EARTH'S SURFACE 

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 J 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 
pressure leads also to other curious conclusions. 
I have remarked before that the sea must re- 
spond 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 cen- 
timetres under the very high barometer as com- 
pared 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 baro- 
metric inequalities do not linger quite long 



EFFECTS OF ATMOSPHERIC PRESSURE 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 com- 
pression 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 eleva- 
tion of the sea under high and low barometer. 

If delicate observations are made with some 
form of pendulum, the air waves and the conse- 
quent distortions of the soil should have a sensi- 
ble 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 ^ of a second of arc ; this would 
be augmented to ^ 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 
^ 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 unyield- 
ing earth. Accordingly we have yet another 
reason why the direct measurement of the tidal 
force presents a problem of the extremest diffi- 
culty. 



148 DISTORTION OF THE EARTH'S SURFACE 

AUTHORITIES. 

G. H. Darwin, Appendix to the Second Report on Lunar Disturb- 
ance of Gravity. " Brit. Assoc. Reports." Southampton, 1882. 
Reprint of the same in the " Philosophical Magazine." 

d'Abbadie, Recherches sur la verticale. " Ann. de la Soc. Scient. 
de Bruxelles," 1881. 

von Rebeur-Paschwitz, Das Horizontalpendel. "Nova A eta 
K. Leop. Car. Akad.," Band 60, No. 1, 1892. 



CHAPTER VIII 

EQUILIBRIUM THEORY OF TIDES 

IT is clearly necessary to proceed step by step 
towards the actual conditions of the tidal prob- 
lem, and I shall begin by supposing that the 
oceans cover the whole earth, leaving no dry 
land. It has been shown in Chapter V. that the 
tidal force is the resultant of opposing centrifu- 
gal and centripetal forces. The motion of the 
system is therefore one of its most essential fea- 
tures. 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 con- 
sidered 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 



150 EQUILIBRIUM THEORY OF TIDES 

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 cir- 
cle, 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 fig- 
ure 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 revolu- 
tion " " prolate " because it is elongated, and 
" of revolution " because it is symmetrical with 
respect to the line pointing at the moon. Ac- 
cordingly 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 



EQUILIBRIUM TIDE 151 

enough to bring the surface to its form of equi- 
librium, 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 spin- 
ning at its actual rate, and with the moon revolv- 
ing 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 hemi- 
spheres, 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 lati- 
tude, in the middle of the other hemisphere. 
These are the two points v and i of figs. 22 and 
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 sur- 
face of the globe. It will be convenient to mea- 
sure 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 con- 
centric with the two spots, but on the actual 



152 EQUILIBRIUM THEORY OF TIDES 




TERRESTRIAL OBSERVER 153 

globe they would be so. These circles show 
where, on the adopted scale of measurement, the 
elevation of height is successively 1J, 1, i . The 
fourth circle, marked in chain dot, shows where 
there is no elevation or depression above the ori- 
ginal surface. The next succeeding and dotted 
circle shows where there is a depression of | , and 
the last dotted line is the circle of lowest water 
where the depression is 1 ; it is the circle D D of 
fig. 22, and the circle of the shadow in fig. 23. 

The elevation above the original spherical sur- 
face 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 per- 
spective 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 lati- 
tude. If, for example, his post of observation is 
in latitude 30 N., he will pass along the second 



154 EQUILIBRIUM THEORY OF TIDES 

parallel to the north of the equator. This par- 
allel cuts several of the circles which indicate the 
elevation and depression of the water, and there- 
fore 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 fall- 
ing 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-|- 
hours to pass from low water to high water. At 
low water he was depressed by 1 below the ori- 
ginal level, and at high water he is raised by 
i above that level, so that the range from low 
water to high water is represented by 1J. After 
the passage across the middle of the hemisphere, 
the water level falls, and after about 5^ hours 
more the water is again lowest, and the depres- 
sion 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 



DIURNAL INEQUALITY 155 

height represented by about If. In this half of 
his daily course the range of tide is from 1 below 
to If above, and is therefore 2f, whereas before 
it was only 1|. The fact that the range of two 
successive tides is not the same is of great im- 
portance in tidal theory ; it is called the diurnal 
inequality of the tide. 

It will have been noticed that in the left hemi- 
sphere the range of fall below the original spher- 
ical 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 ori- 
ginally 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 oscilla- 
tions 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 



156 EQUILIBRIUM THEORY OF TIDES 

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 dis- 
placement 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 hemi- 
sphere is in fact the same as the left upside down. 
It is this inversion which causes the two succes- 
sive tides to be unlike one another, or, in other 
words, gives rise to the diurnal inequality. But 
there is one case where inversion makes no differ- 
ence ; this is when the central spot is on the 
equator in the left hemisphere, for its inversion 
then makes the right hemisphere an exact repro- 
duction of the left one. In this case therefore 
the two successive tides are exactly alike, and 
there is no diurnal inequality. Hence the diur- 
nal 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 accord- 
ing to similar laws, although with less intensity. 



SOLAR TIDAL FORCE 157 

We must now consider how the relative strengths 
of the actions of the two bodies are to be de- 
termined. It was indicated in Chapter V. that 
tide-generating force varies inversely as the cube 
of the distance from the earth of the tide-gen- 
erating body. The force of gravity varies in- 
versely 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 real- 
ity, 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 com- 
pare 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 25J 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. 



158 EQUILIBRIUM THEORY OF TIDES 

We have, then, a force which is 251 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 1 59ths, 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 distor- 
tion 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 dis- 
tortion of the surface due to the sun is super- 
posed 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 dia- 
metrically 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 
one another, the two distortions are at right 
angles, and the low water of the solar tide con- 
spires 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 
of the lunar tide, is deducted from the lunar 
tide. Since one and a half is three times a half, 



SOLAR TIDAL FORCE 159 

it follows that when the moon and sun act to- 
gether 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. 

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, how- 
ever, 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 



160 EQUILIBRIUM THEORY OF TIDES 

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 mid- 
night. 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 



ROUGH RULE FOR PREDICTION 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 
"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 mari- 
ner the establishment of all ports should be 
zero. But observation shows that the establish- 
ment 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 mid- 
night ; 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 equi- 
librium 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 



162 EQUILIBRIUM THEORY OF TIDES 

this result can possibly be consistent with the 
theory of universal gravitation. 

The existence of continental barriers across 
the oceans must obviously exercise great influ- 
ence 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 gravita- 
tion. 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. 

AUTHORITIES . 

An exposition of the equilibrium theory will be found in any 
mathematical work on the subject, or in the article Tides in the 
" Encyclopedia Britannica." 



CHAPTER IX 

DYNAMICAL THEORY OF THE TIDE WAVE 

THE most serious difficulties in the complete 
tidal problem do not arise in a certain special 
case which was considered by Newton. His sup- 
position was that the sea is confined to a canal 
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 Chapter 
II., and the earth's rotation and the moon's at- 
traction 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 the- 
ory of the tide, namely, where the canal is 13| 
miles deep. In this case the wave travels 1,042 
miles an hour, so that it would complete the 



164 DYNAMICAL THEORY OF TIDE WAVE 

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 f 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 
when once produced it travels free from the ac- 
tion of external forces, and would persist forever, 
were it not for the friction to which water is 
necessarily subject. But the leading character- 
istic 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 
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 de- 
gree 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 



FORCED AND FREE WAVES 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 min- 
utes 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 dif- 
ferences are not material to the present argu- 
ment. The earth turns with respect to the sun 
in exactly one day, or as we may more conven- 
iently 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| miles ; accordingly 
in such a canal any long free wave just keeps 
pace with the sun. 

We have seen in Chapter V. that the solar 
tide-generating force 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 
of 13f 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 



166 DYNAMICAL THEORY OF TIDE WAVE 

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 there- 
fore, 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| 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 ex- 
act depth, the tides would be of very great 
height, and the theory would be even more com- 
plex than it is. It is, however, hardly necessary 
to consider this special case in further detail. 

We conclude then that for the depth of 13f 
miles, the wave becomes infinite in height, in 
the qualified sense of infinity which I have de- 
scribed. We may feel sure that the existence 
of the quasi-infinite tide betokens that the be- 
havior of the water in a canal shallower than 
13f 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 



CRITICAL DEPTH OF CANAL 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 
12,500 miles apart, and therefore antipodal to 
one another. The period of a wave is the time 



168 DYNAMICAL THEORY OF TIDE WAVE 

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 be- 
tween 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 deep- 
ened to 13| miles, the free wave travels at the 
same pace as the forced wave, and therefore the 
two periods are coincident. For depths greater 
than 13f 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| miles in 
depth. 



GENERAL DYNAMICAL PRINCIPLE 169 

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 de- 
pends 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. 1 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 move- 
ment of my hand is the forced period, and the 
free period is two seconds, being the time occu- 
pied 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 pendu- 
lum bob grow greater and greater without limit, 

1 A pendulum of one metre in length is commonly called a 
seconds-pendulum, although its complete period is two seconds. 



170 DYNAMICAL THEORY OF TIDE WAVE 

because the successive impulses are stored up in 
the pendulum, which swings further and fur- 
ther with each successive impulse. This case is 
exactly analogous with the quasi-infinite tides 
which would arise in a canal 13f 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 excur- 
sion 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 pen- 
dulum 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 



GENERAL DYNAMICAL PRINCIPLE 



171 



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 




Slow 
FIG. 30. FORCED OSCILLATIONS OP A PENDULUM 

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 13f miles deep, the period of the sun's dis- 
turbing force is 12 hours and is greater than the 



172 DYNAMICAL THEORY OF TIDE WAVE 

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 
shown that these forces are tending to make high 
water immediately under the sun and at its anti- 
podes, 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 13f 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 



CRITICAL DEPTH FOR LUNAR TIDE 173 

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, 
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. 1 

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 pre- 
diction of the equilibrium theory is so wide from 
the truth. Notwithstanding, however, this con- 
demnation 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. 

When the sea is confined to an equatorial 
canal, the tidal problem is much simpler than 

1 It is worthy of remark that if the canal had a depth of be- 
tween 13| and 13 miles, the solar tides would be inverted, and 
the lunar tides would be direct. We should then, at the equa- 
tor, have springtide 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| miles the solar tide might be 
greater than the lunar. But these exceptional cases have only 
a theoretical interest. 



174 DYNAMICAL THEORY OF TIDE WAVE 

when the ocean covers the whole planet, and 
this is much simpler than when the sea is inter- 
rupted 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 par- 
allels of latitude. 

The circumference of a canal in latitude 60 
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 dis- 
tance 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 equato- 
rial waves. The period of a free wave in lati- 
tude 60 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 



EFFECT OF THE EARTH'S ROTATION 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 
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 modi- 
fied 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. 



176 DYNAMICAL THEORY OF TIDE WAVE 

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 re- 
moval 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, car- 
ries 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 be- 
hind by the earth. Since the earth spins from 
west to east, a southerly current acquires a west- 
ward trend. Conversely, when water is carried 
northward of its proper latitude, it leaves the 
earth behind and is carried eastward. Hence 
the water cannot oscillate northward and south- 
ward, without at the same time oscillating east- 
ward and westward. Since in an ocean not 
partitioned into canals, the water must necessa- 
rily move not only east and west but also north 



DIRECT AND INVERTED TIDES 177 

and south, it follows that tidal movements in the 
ocean must result in eddies or vortices. The 
eddying motion of the water must exist every- 
where, but it would be impossible, without math- 
ematical 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 
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 indi- 
cated that somewhere in between higher latitudes 
and the equator, the tide would be of an unde- 
fined character, with an enormous range of rise 
and fall. The complete solution of the prob- 
lem shows, however, that this indication of the 
canal theory is wrong, and that the tidal varia- 
tion 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 posi- 
tion depends on the depth of the sea, where 
there is neither rise nor fall of tide. 



178 DYNAMICAL THEORY OF TIDE WAVE 

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 de- 
press 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 oscil- 
lation 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. 1 

Now let us take one more step towards actu- 
ality, and suppose the earth's equator to be 

1 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. 



DIURNAL INEQUALITIES OF TIDE 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 mathemati- 
cal solution of the problem shows that this con- 
clusion 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 equa- 
tor, it is found that the high water which the 
equilibrium theory would make the larger is act- 
ually 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 
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 mathema- 
tician Laplace. 



180 DYNAMICAL THEORY OF TIDE WAVE 

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 ex- 
planation had been found for the absence of 
that inequality in Europe. But since the days 
of Laplace much has been learnt about the tides 
in the Pacific and Indian oceans, and we now 
know that a large diurnal inequality is almost 
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 expla- 
nations 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 chap- 
ter I shall show how a further approximation to 
the truth is attainable. 



AUTHORITIES 181 

AUTHORITIES. 

The canal theory in its elementary form is treated in many 
works on Hydrodynamics, and in Tides, "Encyclopaedia Bri- 
tannica." 

An elaborate treatment of the subject is contained in Airy's 
Tides and Waves, " Encyclopaedia 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 Mecanique Celeste, but it 
is better studied in more recent works. 

A full presentment of this theory is contained in Professor 
Horace Lamb's Hydrodynamics, Camb. Univ. Press, 1895, chap- 
ter viii. 

Important papers, extending Laplace's work, by Mr. S. S. 
Hough, are contained in the Philosophical Transactions of the 
Royal Society, A. 1897, pp. 201-258, and A. 1898, pp. 139- 
185. 



CHAPTER X 

TIDES IN LAKES COTIDAL CHART 

IF the conditions of the tidal problem are to 
agree with reality, an ocean must be considered 
which is interrupted by continental barriers of 
land. The case of a sea or lake entirely sur- 
rounded 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 Chap- 
ter 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 ac- 
tion 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 instanta- 



NUMERICAL ESTIMATE 183 

neously 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 be- 
fore 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 Chapter V. that the maximum 
horizontal force due to the moon has an inten- 
sity equal to noo P ar * f g ray ity? and that 
therefore a pendulum 10 metres long is deflected 
through ii^ooo of 10 metres, or through ^ of 
a millimetre. Now suppose our lake, 200 kilo- 
metres 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 c D represent the level of the lake as 
undisturbed, and A B an exaggerated pendulum. 
When the tide-generating force displaces the 
pendulum to A B', the surface of the lake must 
assume the position c' D'. Now A B being 10 
metres, B B' may range as far as ^ of a milli- 
metre ; and it is obvious that c c' must bear the 



184 TIDES IN LAKES COTIDAL CHART 

same relation to c B that B B' does to A B. 
Hence c c' at its greatest may be u ^m ^ na ^ 
the length of the lake. The lake is supposed 
to be twice 100 kilometres in length, and 100 
kilometres is 10 million centimetres; thus c c' 
is f^ centimetre, or -^ 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 If centimetres or f 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 2J centimetres. 




FIG. 31. THE TIDE IN A LAKE 

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 



NUMERICAL ESTIMATE 185 

inches. 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 as- 
serted 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. 

The Straits of Gibraltar are so narrow, that 
the amount of water which can flow through 
them in the six hours which elapse between 
high and low water in the Atlantic is inconsid- 



186 TIDES IN LAKES COTIDAL CHART 

erable. Hence the Mediterranean Sea is virtu- 
ally 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 in- 
conspicuous 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 
acts as a resonator for the tidal oscillation, in the 
same way that a hollow vessel, tuned to a partic- 
ular 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 
forces have almost uninterrupted sway, but the pro- 
montories of Africa and of South America must 
profoundly affect the progress of the tide wave 
from east to west. The Atlantic Ocean forms a 
great bay in this vaster tract of water. If this 
inlet were closed by a barrier from the Cape of 



ATLANTIC TIDES 187 

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, al- 
though 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 be- 
tween 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 de- 
pendent 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 aug- 
mented in that region. Then the energy is 
weakened by spreading, where the sea broadens 
again, and it is again reconcentrated by the pro- 
jection of the North American coast line towards 
Europe. Hence, even in this case, ideally simpli- 
fied 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 



188 TIDES IN LAKES 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 what- 
ever 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 it- 
self. It may be conjectured that on the coast of 
Europe the latter is of less importance than the 
former. It is interesting to reflect that our tides 
to-day depend even more on what occurred yes- 
terday 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 con- 
jecture, for the problem is one of insoluble com- 
plexity. 

Some sixty years ago Whewell, and after him 
Airy, drew charts illustrative of what has just 
been described. A map showing the march 
of the tide wave is reproduced from Airy's 



MARCH OF THE TIDE WAVE 189 

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 
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 at- 
tempt 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 in- 
terest. 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 know- 
ledge 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 Green- 
wich times of high water at full and change of 
moon. They thus purport to represent the suc- 
cessive 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 hun- 



190 TIDES IN LAKES COTIDAL CHART 




COTIDAL CHART 191 

dreds 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 graph- 
ical representation of the retardation of the 
wave, as it runs into shallower water. 

But even if this chart were perfectly trust- 
worthy, 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 suc- 
cessive 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. 

This sketch of the difficulties in the solution 



192 TIDES IN LAKES COTIDAL CHART 

of the full tidal problem might well lead to de- 
spair of the possibility of tidal prediction on our 
coasts. I shall, however, show in the next chap- 
ter how such prediction is possible. 

AUTHORITIES. 

For cotidal charts see Whewell, Phil. Trans. Roy. Soc. 1833, 
or Airy's Tides and Waves, " Encyclopaedia Metropolitana." 



CHAPTER XI 

HARMONIC ANALYSIS OF THE TIDE 

IT is not probable that it will ever be possible 
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 suf- 
fice for accurate prediction of the tide at any 
place. 

Although the equilibrium theory is totally 
false as regards its prediction of the time of pas- 
sage 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 state- 
ment 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 



194 HARMONIC ANALYSIS OF THE TIDE 

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-mor- 
row. 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 be- 
low 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 



TIDE DUE TO EQUATORIAL SUN 195 

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 geo- 
graphically, 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 
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, expres- 
sive of height and interval, would virtually con- 
tain tidal prediction for the solar tide for all 
time. 

Theory here gives us some power of foresee- 
ing 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 



196 HARMONIC ANALYSIS OF THE TIDE 

actions of the sun and moon makes it safe to con- 
clude 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 proportion- 
ality 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 coexist, 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. 



IDEAL SATELLITES 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 con- 
siders the moon alone and replaces it by a num- 
ber of satellites of various masses, which move 
in various planes. It is a matter of indifference 
that such a system of satellites could not main- 
tain the orbits assigned to them if they were al- 
lowed 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 verti- 
cally overhead in latitude 45 ; others repel in- 
stead 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. 



198 HARMONIC ANALYSIS OF THE TIDE 

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 charac- 
ter, 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 
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 coexist, 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 suf- 
fice to know that if we could at will annul all 
the ideal satellites except one, and observe its 



IDEAL SUNS 199 

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 satel- 
lites as subjects of observation, and the materials 
for a lunar tide table for all time would be ob- 
tained. 

The motion of the sun round the earth is ana- 
logous 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 abso- 
lutely 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 num- 
ber 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. 

In this method the aggregate tide wave is con- 
sidered as the sum of a number of simple waves 
following one another at exactly equal intervals 
of time, and always presenting a constant rise 



200 HARMONIC ANALYSIS OF THE TIDE 

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 sec- 
ond. The instant at which any one of the sat- 
ellites 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 coexistent waves 
of sensible magnitude, a long series of observa- 
tions 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 elim- 
inated by the taking of averages. It may be 
well to reiterate that each harbor has to be con- 
sidered 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 moder- 
ate accuracy some eight partial waves suffice, but 
if high accuracy is to be attained, we have to 



SUMMATION OF SIMPLE WAVES 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 
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 prin- 
cipal lunar semidiurnal tide." It is the tide 
raised by an ideal satellite, which moves in a cir- 
cle round the earth's equator. I began my ex- 
planation 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 14i seconds. The average 



202 HARMONIC ANALYSIS OF THE TIDE 

interval of time between the successive visible 
transits of the moon over the meridian of the 
place of observation is 24 hours 50 minutes 28J 
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 
same relationship to the real sun that the princi- 
pal 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^ 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| min- 
utes 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 
50J 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. Accord- 
ingly, when the lunar tide has fallen 6 hours 



SEMIDIURNAL TIDES 203 

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 ; sup- 
pose 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 14J 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 



204 HARMONIC ANALYSIS OF THE TIDE 

the crests follow one another at intervals of 12 
hours 39 minutes 30 seconds. Now the interval 
between the successive principal lunar tides was 
12 hours 25 minutes 14 seconds ; hence, this 
new tide falls behind the principal lunar tide by 
14| minutes in each half day. If this tide starts 
so that its high water agrees with that of the 
principal lunar tide, then after 13f days from 
the start, its hollow falls in with the crest of the 
former, and in 27^ 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| 
days it will be furthest, and in 27J days it will 
be nearest again. The moon must clearly ex- 
ercise a stronger tidal force and create higher 
tides when she is near than when she is far; 
hence every 27| 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 27J days, and, 
accordingly, the joint wave is larger every 27 J 
days and smaller in between. Thus this lunar 
elliptic tide represents the principal effect of the 
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. 

Now turning to the waves of the second kind, 



DIURNAL TIDES 205 

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 9J 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 obvi- 
ous that the oscillation resulting from three co- 
existent waves will be very complicated. 

All the semidiurnal tides result from waves of 
essentially similar character, although some fol- 
low 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 semidi- 
urnal waves would afford but little insight into 
the behavior of the diurnal waves. At some 



206 HARMONIC ANALYSIS OF THE TIDE 

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 con- 
struction of such a chart feasible. 

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 pre- 
valent at one season than at another, and moun- 
tain 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 con- 
siderable 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 
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 
the water in-shore as long as the wind blows in- 
land, and to lower it when the wind blows off- 
shore. Hence there results a diurnal inequality 



TIDE WAVE IN SHALLOW WATER 207 

of sea-level, which is taken into account in tidal 
prediction by means of a " solar diurnal meteoro- 
logical tide." Although these inequalities de- 
pend entirely on meteorological influences and 
have no astronomical counterpart, yet it is neces- 
sary to take them into account in tidal predic- 
tion. 

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 ex- 
aggeration 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, 



208 HARMONIC ANALYSIS OF THE TIDE 

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 con- 
sisting 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. 
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 sec- 
ond 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 min- 
utes. In estuaries, the first over-tide of the 
lunar semidiurnal tide is often of great impor- 
tance, and even the second is considerable; the 
third is usually very small, and the fourth and 



OVER-TIDES 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 semi- 
diurnal tide. But it is not usually found neces- 
sary to consider them in the cases of the less 
important partial tides. The octave, the twelfth, 
and the upper octave may be legitimately de- 
scribed as tides, because they are due to the 
attractions of the moon and of the sun, although 
they arise indirectly through the distorting influ- 
ence of the shallowness of the water. 

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 evapora- 
tion 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 repre- 
sentation 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 



210 HARMONIC ANALYSIS OF THE TIDE 

dozen simple waves than by two dozen in a tidal 
river. 

The method of analyzing a tide into its con- 
stituent 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 
Chapter XIII. 

AUTHORITY. 

G. H. Darwin, Harmonic Analysis of Tidal Observations : 
"Report to British Association." Southport, 1883. 

An outline of the method is also contained in Tides, " Ency- 
clopaedia Iji'itannica." 



CHAPTER XII 

REDUCTION OF TIDAL OBSERVATIONS 

I HAVE now to explain the process by which 
the several partial tides may be disentangled 
from one another. 

The tide gauge furnishes a complete tidal re- 
cord, 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 
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. Accord- 
ingly, 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, mutatis mutandis. 



212 REDUCTION OF TIDAL OBSERVATIONS 

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 phenome- 
non 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 reduc- 
tion 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 coexist, 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 



LUNAR TIME 213 

semidiurnal tide, of which the crests follow one 
another at intervals of 12 hours 25 minutes 14J 
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 
to make them exactly semidiurnal. Accordingly 
we describe 24 hours 50 minutes 28^ seconds as 
a lunar day, so that there are exactly two waves 
following one another in the lunar day. 

The tide curve furnishes the height of the 
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 inter- 
vals of 24 hours 50 minutes 28J 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 



214 REDUCTION OF TIDAL OBSERVATIONS 

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 coexist 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 out- 
come of a single day of observation, when all 
but the selected satellite have been obliterated. 
The height of the average wave, and the inter- 
val 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 im- 
portance can be determined, and recorded nu- 
merically by means of their pairs of tidal 
constants. 

The possibility of the disentanglement has 
now been demonstrated, but the work of carry- 
ing out these numerical operations would be 



ABRIDGED METHODS 215 

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 com- 
puters, 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 manage- 
able limits. 

In order to minimize the number of measure- 
ments, the tide curve is only measured at each 
of the 24 exact hours of ordinary time, the 
height at noon being numbered 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, 



216 REDUCTION OF 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 suc- 
cessive 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 
have been re-copied over and over again. In 



TIDAL ABACUS 217 

the office of the United States Coast Survey use 
is made of certain card templates pierced with 
holes. These templates are laid upon the tabu- 
lation 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. Bor- 
gen, of the Imperial German Marine Observatory 
at Wilhelmshaven, has used sheets of tracing 
paper to attain the same end. The Indian pro- 
cedure is unnecessarily laborious, and the Ameri- 
can 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 cor- 
rectly ; 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 



218 REDUCTION OF TIDAL OBSERVATIONS 

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 col- 
umns, 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 fah 1 into 48 
columns, numbered 0, 1, ... 23, 0, 1, ... 23 
twice over. The guide sheet shown in the fig- 
ure 33 is the one appropriate for the lunar semi- 
diurnal 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 O's combined 
will give the average height of water at 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 com- 
puter has only to add the numbers in columns in 



a?? 




TIDAL ABACUS 219 

order to find the averages. There are other 
sheets of paper marked for such other rearrange- 
ments of the strips that each new setting gives 
one of the required results ; thus a single writ- 
ing of the numbers serves for the whole com- 
putation. 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 labo- 
rious, but it is a necessary incident of every 
method of reducing tidal observations. 

The result of all the methods is that for each 



220 REDUCTION OF TIDAL OBSERVATIONS 

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, dur- 
ing 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 inter- 
val 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. 

The results of this " reduction of the observa- 
tions " are contained in some fifteen or twenty 
pairs of tidal constants, and these numbers con- 
tain a complete record of the behavior of the sea 
at the place in question. 

AUTHORITIES. 

G. H. Darwin, Harmonic Analysis, fyc. : " Report to British As- 
sociation," 1883. 

G. H. Darwin, On an apparatus for facilitating the reduction of 
tidal observations : " Proceedings of the Royal Society," vol. Hi. 
1892. 



CHAPTER XIII 

TIDE TABLES 

A TIDE TABLE professes to tell, at a given 
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 pre- 
dict 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 



222 TIDE TABLES 

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 
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 phenome- 
non, 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 mid- 
night. 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 ef- 
fect a statement of her phase. Accordingly the 



GENERAL TIDE TABLES 223 

relative position of the sun and moon is directly 
involved in a statement of the tide as correspond- 
ing 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 lit- 
tle from one another, a simple table of this kind 
suffices for rough predictions. The curves marked 
Portsmouth in fig. 34 show graphicaUy the in- 
terval 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 Chapter 
X. that the tide in the North Atlantic is princi- 
pally due to a wave propagated from the South- 
ern 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 sub- 
tracted from the Portsmouth intervals ; that is 
to say, the intervals on the vertical scale marked 
6, 7, 8 hours are, for Portsmouth, to be inter- 
preted 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 



224 TIDE TABLES 

one of the times of moon's transit and of phases 
of the moon ; the vertical scale in the lower fig- 
ure 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, 
and it is not necessary to differentiate the hours 
as either diurnal or nocturnal, the statement be- 
ing equally true of either day or night. Thus 
if the Portsmouth curves had been extended on- 
ward 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. 1 

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 de- 
pendent solely on the time of the moon's passage. 

1 Before the introduction of the harmonic analysis of the tides 
described in preceding chapters, tidal observations were "re- 
duced " by the construction of such figures as these, directly from 
the tidal observations. Every high water was tabulated as ap- 
pertaining to a particular phase of the moon, both as to its height 
and as to the interval between the moon's transit and the occur- 
rence of high water. The average of a long series of observa- 
tions may be represented in the form of curves by such figures 
as these. 



MULTIPLICITY OF CORRECTIONS 225 

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 com- 
plicated. 

It is, however, possible by increasing the num- 
ber 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 com- 
paratively 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 
the year will directly involve the principal in- 
equalities 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 re- 
ference 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. 



226 TIDE TABLES 

The curves marked Aden, March and June, in 
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 Ports- 
mouth, 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 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 graphi- 
cally, 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 in- 
volve 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 me- 
ridian of Aden at eleven minutes past noon of 
Aden time, or in astronomical language at hr. 
11 mins. Now the table, or the figure of inter- 
vals, shows that if the moon had passed at hr., 
or exactly at noon, the interval would have been 



USE OF GENERAL TIDE TABLE 



227 



8 hrs. 9 mins., and that if she had passed at 
hr. 20 mins., or 12.20 p. M. of the day, the inter- 
val would have been 7 hrs. 59 mins. But on 
March 17th the moon actually crossed at hr. 



INTERVALS 

28 Hours subtracted 

from 
Portsmouth intervals 




, \fi 6 / 8 9 10 11 12 13 
TIMES \J/ OF O MOON'S TRANSITS 

HEIGHTS 




TIMES 

FIG. 34. CURVES OF INTERVALS AND HEIGHTS AT PORTSMOUTH 
AND AT ADEN 

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 



228 TIDE TABLES 

crossed the meridian. But the moon crossed at 
hr. 11 mins., therefore the high water occurred 
at 8.15 p. M. 

Again the table of heights, or the figure, shows 
that on March 15th, if the moon crossed at hr. 
min. the high water would be 6.86 ft. above 
a certain mark ashore, and if she crossed at hr. 
20 mins. the height would be 6.92 ft. But on 
March 17th the moon crossed halfway between 
hr. min. and 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 p. M. 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 predic- 
tions derived from this table are markedly im- 
proved when a correction, either additive or sub- 
tractive, 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 



DEFICIENCY OF TIDAL INFORMATION 229 

time, bringing it to 8.17 P. M., 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 oc- 
cupy 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 con- 
siderable 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 varia- 
bility according to the phases of the moon is want- 
ing. 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 sev- 
eral feet. 

Tables of the kind I have described would 



230 TIDE TABLES 

banish this extreme vagueness, but they are more 
likely to be of service at ports of second-rate im- 
portance than at the great centres of trade, be- 
cause at the latter it is worth while to compute 
full special tide tables for each year. 

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 un- 
doubtedly 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. 

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 govern- 
ing 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 par- 
tial waves, and prediction involves the recompo- 
sition or synthesis of those waves. In the syn- 
thetic 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 



SYNTHESIS OF PARTIAL WAVES 231 

arranged in two stages. It has been shown in 

O O 

Chapter XI. that the partial waves fall naturally 
into three groups, of which the third is practi- 
cally insignificant. The first and second are the 
semidiurnal and diurnal groups. The first pro- 
cess 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 sup- 
pose that the lengths and periods of the waves 
of the two coexistent trains do not differ much 
from one another, although their heights may 
differ widely. Then the resultant must be a sin- 
gle train of waves of lengths and periods inter- 
mediate 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 sep- 



232 TIDE TABLES 

arate 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 semidiur- 
nal wave, the height of which varies according 
to a very complex law. 

A similar synthesis is then applied to the sec- 
ond 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 re- 
sultant 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 



MECHANICAL PREDICTION OF TIDES 233 

done beforehand for every day of a specified 
year, and the result will be a special tide table. 
There are about 1400 high and low waters in 
a year, so that the task is very laborious, and 
has to be repeated each year. 

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 Thom- 
son, now Lord Kelvin, in 1872. Mr. Edward 
Roberts bore an important part in the practical 
realization of such a machine, and a tide pre- 
dicter was constructed by Messrs. Lege for the 
Indian Government under his supervision. This 
is, as yet, the only complete instrument in ex- 
istence. 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 construc- 
tion of their own machine, the French authori- 
ties are obtaining the curves for certain tidal 
predictions from the instrument in London. 



234 TIDE TABLES 

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 diagrammati- 
cally, 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 depend- 
ing on the throw of the crank. 1 The position 

1 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. 



MECHANICAL PREDICTION OF TIDES 235 

of the pin is susceptible of adjustment on the 
crank, so that its throw and the range of oscilla- 
tion of the pulley can be set to any required 



B 




FIG. 35. DIAGRAM OF TIDE-PREDICTING INSTRUMENT 

length of course within definite limits deter- 
mined 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 cir- 
cumference of the drum. If one revolution of 
the drum represents a day, the graphical time 
scale is 24 hours to the circumference of the 



236 TIDE TABLES 

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 
3J 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 de- 
scend. 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 



CURVE FOR SINGLE TIDE 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 semi- 
diurnal 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 appro- 
priate 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 com- 
bining 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 



238 TIDE TABLES 

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 dia- 
gram), the pencil will add together the move- 
ments 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 semi- 
diurnal 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 rig- 
orous accuracy is not attainable, but the mechan- 
ism is made so nearly exact that the error in the 
sum of the two tides would be barely sensible 
even after 3000 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 



COMPOSITION OF TIDES 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 mo- 
ment 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 



240 TIDE TABLES 

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 Janu- 
ary, and then lets it run off a complete tide 
curve for a whole year. The curve is subse- 
quently 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 before- 
hand to render the tables available for future 
voyages. These tide tables are certainly amongst 
the most admirable in the world. 

It is characteristic of England that the ma- 
chine 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 prac- 
tically 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 deter- 
mined. 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 re- 
gretted that this beautiful instrument should not 
be more used for the home and colonial ports. 



INDIAN TIDE PREDICTER 241 

The excellent tide tables of the Government 
of the United States have hitherto been pre- 
pared by the aid of a machine of quite a differ- 
ent character, the invention of the late Professor 
Ferrel. This apparatus virtually carries out 
that process of compounding all the waves to- 
gether 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, how- 
ever, 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. 

AUTHORITIES. 

G. H. Darwin, On Tidal Prediction. " Philosophical Transac- 
tions 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 mea- 
sured is 0.37 ft. higher than that used in the Indian Tide Tables; 
accordingly 4 inches must be added to the height, in order to 
bring it into accordance with the official table. 

Sir William Thomson, Tidal Instruments, and the subsequent 
discussion. " Institute of Civil Engineers," vol. Ixv. 

William Ferrel, Description of a Maxima and Minima Tide- 
predicting Machine. " United States Coast Survey," 1883. 



CHAPTER XIV 

THE DEGREE OF ACCURACY OF TIDAL PREDICTION 

THE success of tidal predictions varies much 
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 
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 



THE EFFECT OF THE WIND 243 

of wind is sometimes great. A disastrous exam- 
ple of this was afforded on the east coast of Eng- 
land in the autumn of 1897, when the conjunc- 
tion 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 blow- 
ing previously elsewhere in such a way as to de- 
press 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 cor- 
rections 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 



244 



ACCURACY OF TIDAL PREDICTION 



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. 

TABLE OF ERRORS IN THE PREDICTION OF HIGH WATER AT 

PORTSMOUTH IN THE MONTHS OF JANUARY, MAY, AND 

SEPTEMBER, 1897. 



Time 


Height 


Magnitude of error 


Number of cases 


Magnitude of error 


Number of cases 


O m to 5 m 


69 


Inches 

to 6 


89 


6 m to i<r 


50 


7 to 12 


58 


ll m to 15 m 


25 


13 to 18 


24 


16 m to 20 m 


10 


19 to 24 


6 


21 m to 25 m 


11 








26 m to 30 m 


7 








31 m to 35 m 


4 








52 m 


1 











177 





177 



ERRORS IN HEIGHT FOR THE YEAR 1892, 
EXCEPTING PART OF JULY 



Magnitude of error 


Number of cases 


Inches 




Oto 6 


381 


7 to 12 


228 


13 to 18 


52 


19 to 24 


8 


31 


1 





670 



AT PORTSMOUTH AND AT ADEN 245 

N. B. The comparison seems to indicate that these predic- 
tions might be much improved, because the predicted height is 
nearly always above the observed height, and because the diur- 
nal inequality has not been taken into account sufficiently, if at 
all. 

In tropical regions the weather is very uni- 
form, and in many places the " meteorological 
tides " produced by the regularly periodic varia- 
tions 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 insignifi- 
cant. 

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 



246 



ACCURACY OF TIDAL PREDICTION 



10 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 regis- 
tered. On one occasion, when the regular semi- 
diurnal 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 : 



Time 


Height 


Magnitude of 


Number of 


Magnitude of 


Number of 


errors 


high waters 


errors 


high waters 






Inches 




O m to 5 m 


35 





15 


5 m to 10 m 


32 


1 


48 


10 m to 15 m 


19 


2 


28 


15 m to 20 m 


19 


3 


14 


20 m to 25 m 


5 


4 


11 


26 m and28 m 


2 


No high water 


1 


33 m and 36 m 


2 








56 m and57 m 


2 








No high water 


1 


~~~ 







117 




117 



It would be natural to think that when the 
prediction is erroneous by as much as 57 min- 
utes, it is a very bad one ; but I shall show that 



WATER APPROXIMATELY STAGNANT 247 

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 accord- 
ing to the rise or fall : 

ANALYSIS OF ERRORS IN TIME. 



Ranges from low water 
to high water 


Errors of time 


Nil 






6 in. to 8 in. 


22, 26, 28, 56, 57 minutes 


13 in. 


36 minutes 




17 in. 


22 " 




19 in. 


33 




2 ft. 10 in. 


22 " 




3ft. 9 in. 


23 " 




3 ft. 11 in. 


20 





248 ACCURACY OF TIDAL PREDICTION 

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 pre- 
dictions 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 ap- 
pears that, as regards the height of the tide also, 
the predictions are of great accuracy. This 
short series of comparisons affords a not unduly 



DEGREE OF SUCCESS 249 

favorable example of the remarkable success at- 
tainable, where tidal observation and predic- 
tion have been thoroughly carried out at a place 
subject to only slight meteorological disturb- 
ance. 

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 con- 
stants would show no relationship to their pre- 
vious 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 in- 
volved, 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 inces- 
sant 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 



250 ACCURACY OF TIDAL PREDICTION 

regard good tidal prediction as one of the 
greatest triumphs of the theory of universal 
gravitation. 

AUTHORITIES. 

The Portsmouth comparisons were given to the author by the 
Hydrographer of the Admiralty, Admiral Sir W. J. Wharton. 

G. H. Darwin, On Tidal Prediction. " Philosophical Trans- 
actions of the Royal Society," A. 1891. 



CHAPTER XV 



CHANDLER'S NUTATION THE RIGIDITY OF THE 
EARTH 



IN the present chapter I have to explain the 
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 steadi- 
ness. Now the earth may be likened to a top, 
and it also may either have a nutational or nod- 
ding 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 
any place is the angle 1 between the equator and 

1 This angle is technically called the geocentric latitude ; the 
distinction between true and geocentric latitude is immaterial in 
the present discussion. 



252 CHANDLER'S NUTATION 

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 rota- 
tion, 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 ex- 
ternal 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 move- 
ments 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 



FORCED NUTATION AND PRECESSION 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 lati- 
tudes 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 peri- 
ods, 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 
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. 1 

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, New- 

1 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. 



254 CHANDLER'S NUTATION 

comb, 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 equa- 
torial 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. 

Mr. Chandler's investigation demanded the 
utmost patience and skill in marshaling large 
masses of the most refined astronomical observa- 
tions. 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. 1 But 

1 Mr. S. S. Hough, p. 338 of the paper referred to in the list 
of authorities at the end of the chapter. 



TIDE DUE TO FREE NUTATION 255 

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, be- 
cause 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 
Mr. Christie of the United States Coast Survey. 
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 con- 
clude 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. 



256 CHANDLER'S NUTATION 

The inequality in water level is so slight and 
extends over so long a period that its measure- 
ment 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 exam- 
ination of a tidal record will always apparently 
yield an inequality of any arbitrary period what- 
ever. It is only when several independent deter- 
minations yield fairly consistent values of the 
magnitude of the rise and fall and of the mo- 
ment of high water, that we can feel confidence 
in the result. Now although the reductions of 
Bakhuyzen and Christie are fairly consistent 
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. 

The prolongation of the period of Chandler's 
nutation from 305 to 427 days seems to indicate 



CONDENSATION OF ROCK 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 gener- 
ally 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 per- 
ceived that the inference might be false, if great 
pressure is capable of imparting rigidity to mat- 
ter at a very high temperature, because the inte- 
rior of the earth might then be solid although 
very hot. Now it has been proved experimen- 
tally 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 un- 
certain up to what degree of increase of pres- 
sure 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 labo- 
ratory can be deemed applicable to the condi- 
tions prevailing in the interior of the earth, they 
tend to show that the matter there is not im- 
probably solid. 

But Lord Kelvin reinforces this argument 
from another point of view. Rock in the solid 



258 RIGIDITY OF THE EARTH 

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 liq- 
uid, 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 argu- 
ments 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 com- 
plete, yet as he was the first to show that the 
motion of the earth as a whole affords indica- 
tions of the condition of the interior, an impor- 
tant 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 



ARGUMENT FROM OCEANIC TIDES 259 

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 
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 argu 
ment, 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 hypo- 
thesis 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 super jacent 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, 



260 RIGIDITY OF THE EARTH 

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 frac- 
tion 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 applica- 
tion of continuous forces, and also to the hy- 
pothesis of a material which shares the properties 
of viscosity and rigidity, and have been led to 
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 
would not be twice as great as they are, and he 
concludes that the earth possesses a greater aver- 
age 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 Chapter XI. as practically unim- 
portant, 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 
than in the case of the semidiurnal and diurnal 
tides. 



VIBRATIONS DUE TO EARTHQUAKES 261 

It may ultimately be possible to derive further 
indications concerning the physical condition of 
the inside of the earth from the science of seis- 
mology. The tremor of an earthquake has fre- 
quently been observed instrumentally at an enor- 
mous 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 dis- 
tortion. These two vibrations travel at differ- 
ent speeds, and the compressional wave outpaces 
the distortional one. Now the first sign of a 
distant earthquake is that the instrumental re- 
cord shows a succession of minute tremors. 
These are supposed to be due to waves of com- 
pression, and they are succeeded by a much 
more strongly marked disturbance, which, how- 
ever, 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 vi- 



262 RIGIDITY OF THE EARTH 

bration has passed was solid. For, although a 
compressional wave might be transmitted with- 
out 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 mol- 
ten, 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 ex- 
istence of liquid matter immediately under the 
solid crust of the earth, 1 yet the arguments which 
I have sketched appear to most men of science 
conclusive against such belief. 

AUTHORITIES. 

Mr. S. C. Chandler's investigations are published in the " As- 
tronomical Journal," vol. 11 and following volumes. A summary 
is contained in " Science," May 3, 1895. 

R. S. Woodward, Mechanical Interpretation of the Variations of 
Latitude, Ast. Journ." vol. 15, May, 1895. 

Simon Newcomb, On the Dynamics of the Earth's Rotation, 
"Monthly Notices of the R. Astron. Soc.," vol. 52 (1892), 
p. 336. 

S. S. Hough, The Rotation of an Elastic Spheroid, " Philosoph. 
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, Ueber die Aenderung der Pol- 
hoehe, " Astron. Nachrichten," No. 3261. 

1 See the Rev. Osmond Fisher's Physics of the Earth's Crust. 






AUTHORITIES 263 

A. S. Christie, 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, Bodily Tides of Viscous and Semi-elastic Sphe- 
roids, &c., "Philosoph. Trans, of the Royal Society," Part. I. 
1879. * 



CHAPTER XVI 1 

TIDAL FRICTION 

THE fact that the earth, the moon, and the 
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 pro- 
duce 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 pre- 
sent 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 

1 A considerable portion of this and of the succeeding chapter 
appeared as an article in The Atlantic Monthly for April, 1898. 



RETARDATION OF MOTION 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 bear- 
ings, 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 fric- 
tion 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. 



TIDAL FRICTION 



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 




FIG. 36. FRICTIONALLY RETARDED TIDE 

of a circle, and the direction of its motion is 
shown Tfy 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 as- 
sumes 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 fig- 
ure 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 under- 
neath and opposite to the moon ; whereas in 
shallow water it is low water where we should 



ACTION OF FRICTION 267 

naturally expect high water. Accepting the hy- 
pothesis 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 di- 
rected straight towards the satellite M', and the 
liquid maintains a continuous rhythmical move- 
ment, so that as the planet rotates and the satel- 
lite 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 protuber- 
ance 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 continuaUy 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 protu- 
berances by two particles of equal masses, one at 



268 TIDAL FRICTION 

p, 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 
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 ef- 
fect 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 pri- 
mary 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 corpo- 
real tides or to those of a very deep ocean. If 



RETARDATION OF EARTH'S ROTATION 269 

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 on- 
ward 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 



270 TIDAL FRICTION 

planet, and moves in a spiral curve at ever in- 
creasing 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 retard- 
ation 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 para- 
dox. 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 cen- 
tral force, acting directly towards the planet, 
must operate in some measure to retard the ve- 
locity 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 acceler- 
ating force, and thus the resultant effect is a 
retardation of the satellite's velocity. 

The converse case where a retarding force re- 
sults in increase of velocity will perhaps be more 
intelligible, as being more familiar. A meteorite, 



DAY AND MONTH PROLONGED 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 

o 

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 



272 TIDAL FRICTION 

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 in- 
crease 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 
sun, which occurred on certain days and in cer- 
tain places. Notwithstanding the changes in the 
calendar, it is possible to identify the day ac- 
cording to our modern reckoning, and the iden- 
tification 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 govern- 
ing 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 com- 
putation would assign it. Most astronomical 



VARIATION WITH MOON'S DISTANCE 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 
just that element of precision which is otherwise 
wanting. As, then, the situations of the ancient 
eclipses agree fairly well with modern computa- 
tions, 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 ; in- 
deed, 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 Chapter V. that the in- 
tensity of tide-generating force varies as the in- 
verse cube of the distance between the moon and 
the earth, so that if the moon's distance were 
reduced successively to |, J, |, of its original dis- 



274 TIDAL FRICTION 

tance, the force and the tide generated by it 
would be multiplied 8, 27, 64 times. But the 
efficiency of tidal friction increases far more rap- 
idly 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 co- 
operate, let us begin by supposing that the 
height of the tide remains unaffected by the ap- 
proach or retrogression of the moon. Then the 
same line of argument, which led to the conclu- 
sion that tide-generating force varies inversely as 
the cube of the distance, shows that the action 
of the moon on protuberances of definite magni- 
tude 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 fol- 
lows that the tidal retardation of the earth's ro- 
tation 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. 



SEQUENCE OF EVENTS 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 
are lengthening at relative rates which are cal- 
culable, although the absolute rates in time are 
unknown. It will suffice for a general compre- 
hension 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 rota- 
tions of the earth in the interval comprised in 
one revolution of the moon diminishes; or, in 
other wordsj 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 



276 TIDAL FRICTION 

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 
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 to- 
gether, 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 revo- 
lutions 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 27J, as at pre- 
sent. This epoch is a sort of crisis in the history 
of the moon and the earth, for it may be proved 



SEQUENCE OF EVENTS 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 maxi- 
mum 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. Con- 
sequently, 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, 
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 ar- 



278 TIDAL FRICTION 

rived 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 be- 
tween 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 fur- 
ther considerations, if I sketch the history again 
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 



INITIAL CONDITION 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 ac- 
cording 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 acclerated by the tidal friction, and the 
satellite will be drawn inward towards the planet, 
into which it must ultimately fall. In the appli- 
cation 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 in- 
creases 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. 



280 TIDAL FRICTION 

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 begin- 
ning, 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 condi- 
tion 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 bal- 
anced on its point is unstable ; the smallest moto 
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. 



INITIAL INSTABILITY OF MOON 281 

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 cal- 
culation 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 re- 
sistance of the air, conclude that at some previ- 
ous 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 meces. The attraction of gravity 
would be barely strong enough to hold it to- 
gether, just as the cohesive strength of iron is 



282 TIDAL FRICTION 

insufficient to hold a fly-wheel together if it is 
spun too fast. There is, of course, an impor- 
tant 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 primi- 
tive 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 reestab- 
lished in the formation of a satellite. 

I have said above that rapid rotation was prob- 
ably the cause of the birth of the moon, but it 
may perhaps not have been brought about by 
this cause alone. There are certain considera- 
tions 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 



GENESIS OF MOON 283 

primitive planet to break up. In 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 co- 
operate with rapid rotation in producing rup- 
ture ? I think there is such a cause, and, al- 
though 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 Chapter IX. it was pointed out 
that there is a general dynamical law which en- 
ables us to foresee the magnitude of the oscilla- 
tions 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 inter- 
vention 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 



284 TIDAL FRICTION 

is found to be 1 hour 34 minutes. The hetero- 
geneity of the earth introduces a complication of 
which we cannot take account, but it seems likely 
that the period would be from 1| to 2 hours. 
The period of the solar semidiurnal tide is half a 
day, and if the day were from 3 to 4 of our pre- 
sent 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 rota- 
tion 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 be- 
come so violent that, in cooperation with the 
rapid rotation, it shook the planet to pieces, and 
that huge fragments were detached which ulti- 
mately 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. 



MINIMUM TIME REQUISITE 285 

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 
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 uncer- 
tainty as to the duration of the solar system is 
wide, yet we are sure that it has not existed for 
an almost infinite past. 

Now, although the actual time scale is indeter- 
minate, 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 condi- 
tions 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. Va- 
rious 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 ulti- 



286 TIDAL FRICTION 

mate verdict will be adverse to the preponderat- 
ing influence of the tide in the evolution of our 
planet. 

If this history be true of the 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 volca- 
noes 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 
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 solidi- 
fied, and thenceforward and to the present day 
she has shown the same face to the earth. Kant 
and Laplace in the last century, and Helmholtz 
in recent times, have adduced this as the expla- 
nation of the fact that the moon always shows 
us the same face. Our theory, then, receives a 



ROTATION OF THE MOON 287 

striking confirmation from the moon ; for, hav- 
ing ceased to rotate relatively to us, she has actu- 
ally 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 
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. 

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 mo- 
tion round the earth unaffected. It might per- 
haps 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 3^ 
part of the sun's, and the reactive effect on the 
earth's orbit round the sun is altogether negligi- 



288 TIDAL FRICTION 

ble. 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 be- 
come 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 gen- 
erated, 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 fric- 
tion. 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 ulti- 
mately fall back into the earth. We know that 



SOLAR TIDES 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. 

AUTHORITIES. 
See the end of Chapter XVII. 



CHAPTER XVII 

TIDAL FRICTION (CONTINUED) 

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 ad- 
duced as a reason for rejecting the actual effi- 
ciency of solar tidal friction in the process of 
celestial evolution. At length however, in 1877, 
Professor Asaph Hall discovered in the system 
of the planet Mars a case of the kind of motion 
which we foresee as the future fate of the moon 
and earth, for he found that the planet was at- 
tended by two satellites, the nearer of which has 
a month shorter than the planet's day. He gives 
an interesting account of what had been conjec- 
tured, 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 : 



SATELLITES OF MARS 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 at- 
tracted so much attention, that a brief account 
of the writings on this subject may be interesting. 

" The following letter of Kepler was written 
to one of his friends soon after the discovery by 
Galileo in 1610 of the four satellites of Jupiter, 
and when doubts had been expressed as to the 
reality of this discovery. The news of the dis- 
covery 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 dis- 
pute between us decided in this way, that be- 
tween 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. ... 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 two 



292 TIDAL FRICTION 

round Mars, as the proportion seems to require, 
six or eight round Saturn, and perhaps one each 
round Mercury and Venus.' 

" Dean Swift's statement concerning the satel- 
lites of Mars is in his famous satire, ' The 
Travels of Mr. Lemuel Gulliver.' After de- 
scribing 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 phrase- 
ology, 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, parallelo- 
grams, ellipses, and other geometrical terms, or 
by words of art drawn from music, needless here 
to repeat. . . . And although they are dexter- 
ous enough upon a piece of paper, in the man- 
agement 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 opposi- 
tion, unless when they happen to be of the right 
opinion, which is seldom their case. . . . These 



DEAN SWIFT'S SATIRE 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 appre- 
hensions 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 ab- 
sorbed, 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 



294 TIDAL FRICTION 

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. 
. . . 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. . . . 
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- 



VOLTAIRE ON MARTIAN SATELLITES 295 

one and a half; so that the squares of their 
periodical times are very near in the same pro- 
portion 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 
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 a nos voyageurs. En sor- 
tant de Jupiter, ils traverserent un espace d' en- 
viron cent millions de lieues, et ils cotoyerent 
la planete de Mars, qui, comme on sait, est cinq 
f ois plus petite que notre petit globe ; ils virent 
deux lunes qui servent a cette planete, et qui ont 
echappe aux regards de nos astronomes. Je sais 
bien que le pere Castel ecrira, et meme plaisam- 
ment, centre 1' existence de ces deux lunes ; mais 
je m'en rapporte a ceux qui raisonnent par ana- 
logie. Ces bons philosophes-la savent combien il 
serait difficile que Mars, qui est si loin du soleil, 
se passat a 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 
addition to what may be called the analogies of 



296 TIDAL FRICTION 

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 astro- 
nomy, and is stated as follows by Mr. Chambers 
in his excellent book on ' Descriptive Astro- 
nomy,' 2d edition, p. 89, published in 1867 : 

" ' As far as we know, Mars possesses no satel- 
lite, 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 ^V of the diameter of the primary, and 
a satellite ^V f 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 inhab- 
itants of Laputa. Voltaire shared this contempt, 
and delighted in making fun of the philosophers 
whom Frederick the Great collected at Berlin. 
The ( pere Castel ' may have been le pere Louis 



SATELLITES OF MARS DISCOVERED 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 oppor- 
tunity was afforded to satirists like Swift and 
Voltaire to ridicule such arguments." 

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 
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 arid 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 Sat- 
urn present faint inequalities of coloring, and 

1 Observations and Orbits of the Satellites of Mars, by Asaph 
Hall. Washington, Government Printing Office, 1878. 



298 TIDAL FRICTION 

telescopic examination has led astronomers to be- 
lieve that they always present the same face to 
their planets. The theory of tidal friction would 
certainly lead us to expect that these enormous 
planets should work out the same result for their 
relatively small satellites that the earth has pro- 
duced in the moon. 

The proximity of the planets Mercury and 
Venus to the sun should obviously render solar 
tidal friction far more effective than with us. 
The determination of the periods of rotation of 
these planets thus becomes a matter of much in- 
terest. But the markings on their disks are so 
obscure that the rates of their rotations have re- 
mained 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, 
an observer endowed with extraordinary acute- 
ness of vision, announced as the result of his ob- 
servations 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 observa- 
tions 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 



ROTATION OF MERCURY AND VENUS 299 

apparently enjoyed equal advantages for obser- 
vation. 1 

If, as I am disposed to do, we accept these ob- 
servations as sound, we find that evidence favor- 
able to the theory of tidal friction is furnished 
by the planets Mercury and Venus, and by the 
satellites of the earth, Jupiter and Saturn, whilst 
the Martian system is yet more striking as an 
instance of an advanced stage in evolution. 

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 equato- 
rial radius over the polar radius is 2^0 P ar ^ ^ 
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 

1 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. 



300 TIDAL FRICTION 

earth had formerly its present shape, there must 
then have been polar continents and a deep equa- 
torial 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 experi- 
mentally by M. Tresca and others that even very 
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 ex- 
plained in 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 obser- 
vation shows that rocks have been freely twisted 
and bent near the earth's surface, and it is im- 
possible 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 prob- 
ably due to unequal shrinkage of the planet in 
cooling, and each shock would tend to bring the 



ADAPTATION OF EARTH'S FIGURE 301 

strata into their position of rest; thus the earth's 
surface would avail itself of the opportunity af- 
forded by earthquakes of acquiring its proper 
shape. The deposit in the sea of sediment, de- 
rived 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 ro- 
tation. 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 aver- 
age 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 differ- 
ence 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 



302 TIDAL FRICTION 

rotation unless indeed the deep polar sea dis- 
covered by Nansen be such. Yet, as I have 
hinted above, some tokens still exist in the earth 
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 con- 
centric. 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 ro- 
tation 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 in- 
ternal point the density must He 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 lay- 
ers 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 them- 
selves to our observation in several ways. In 
the first place, gravity on the earth's surface 



ELLIPTICITY OF INTERNAL STRATA 303 

would be changed. The force of gravity at the 
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 consid- 
erable 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 
the earth, the lunar inequality, and the preces- 
sional and nutational motion of the earth are 
so intimately intertwined that one of them can- 
not be touched without affecting the others. 

Now Edouard Roche, a French mathematician, 
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 ad- 
just the unknown law of internal density so as 
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 



304 TIDAL FRICTION 

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. 

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 fur- 
nishes 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 there- 
fore those conditions must have differed in the 
past. Our storms are of the nature of aerial ed- 
dies, 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 



GEOLOGICAL EVIDENCE 305 

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 pos- 
sible that trees and plants would not be exter- 
minated, 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 sea- 
coast 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. Al- 
though 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 



306 TIDAL FRICTION 

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. In- 
creased activity in the factors of change is im- 
portant 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 
geologist and of the physicist into better har- 
mony. 

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. 

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 



CHANGES IN A PLASTIC PLANET 307 

the earth's rotation is retarded by forces acting 
on the tidal protuberances in a direction adverse 
to the planet's rotation. As the plastic sub- 
stance, 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, al- 
though 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. 



308 TIDAL FRICTION 

There can be no doubt as to the correctness 
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, how- 
ever, observe that the continents, in fact, run 
roughly north and south. It may appear fanci- 
ful 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 conjec- 
ture. 1 

We must now revert to the astronomical as- 
pects 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 consid- 
ered. It has been supposed thus far that the 
moon moves over the earth's equator in a circu- 
lar 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 

1 See, also, W. Prinz, Torsion apparente des planetes, "Annuaire 
de 1'Obs. R. de Bruxelles," 1891. 



OBLIQUITY OF THE ECLIPTIC 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 mo- 
tion 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 equa- 
tor to the ecliptic remains constant, so that win- 
ter and summer remain as at present. There is 
also, superposed on the precession, the nutational 
or nodding motion of the pole to which I re- 
ferred in 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 protu- 
berance, and would add certain very minute 
nutations of the earth's axis ; the amount of 



310 TIDAL FRICTION 

these tidal effects, is, however, quite insignifi- 
cant. 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 
reentrant. Now this is the case with regard to 
the motion of the earth's axis under the influ- 
ence 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 un- 
til the day and month became equal. This con- 
clusion remains correct when the earth is oblique 
to its orbit, but the effect on the obliquity is 
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 obli- 
quity to increase vanishes. In other words, if 



OBLIQUITY OF THE ECLIPTIC 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 per- 
pendicular 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 ob- 
liquity 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 1 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, impos- 
sible 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 rota- 
tion about axes oblique to their orbits. It remains, 
however, certain that a planet, rotating primi- 
tively without obliquity, would gradually become 



312 TIDAL FRICTION 

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 eccen- 
tric. 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 eccen- 
tricity. In this case, it is possible to indicate by 
general reasoning the manner in which this reac- 
tion operates. We have seen that tidal reaction 
tends to increase the moon's distance from the 
earth. Now, when the moon is nearest, in peri- 
gee, 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 peri- 
geal distance. But the perigeal effect is stronger 
than the apogeal, and, therefore, the apogeal dis- 
tances 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. 

The lunar orbit is then becoming more eccen- 
tric, and numerical calculation shows that in 
very early times it must have been nearly circu- 



ECCENTRICITY OF LUNAR ORBIT 313 

lar. 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 T 7 T days is equal to the 
month. It follows from this that the circular 
orbit of the satellite is dynamically stable or 
unstable according as lyr days is less or greater 
than the month. 

The effect of tidal friction on the eccentricity 
has been made the basis of extensive astronom- 
ical speculations by Dr. See. I shall revert to 
this subject in Chapter XIX., and will here 
merely remark that systems of double stars are 
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 



314 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 grad- 
ually 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 satel- 
lite, revolving with its month a little longer than 
the planet's day, then a system would necessarily 
be developed which would have a strong resem- 
blance to that of the earth and moon. 

A theory reposing on verce causce which brings 
into quantitative correlation the lengths of the 
present day and month, the obliquity of the 
ecliptic, the eccentricity and the inclination of 



SUMMARY 315 

the lunar orbit, should have strong claims to 
acceptance. 

AUTHORITIES. 

G. H. Darwin. A series of papers in the " Phil. Trans. Roy. 
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 Natural Philosophy), 
and vol. 30, 1880, p. 255. 

Lord Kelvin, On Geological Time, "Popular Lectures and 
Addresses," vol. iii. Macmillan, 1894. 

Roche. The investigations of Roche and of others are given 
in Tisserand's Mecanique Celeste, vol. ii. Gauthier-Villars, 1891. 

Tresca and St. Ve'nant, Sur Vecoulement des Corps Solides, 
" Mdinoires des Savants Etrangers," Acade'rnie des Sciences de 
Paris, vols. 18 and 20. 

Schiaparelli, Consider azioni sul moto rotatorio del pianeta 
Venere. Five notes in the "Rendiconti del R. Istituto Lom- 
bardo," vol. 23, and 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. Mercury and De- 
termination of Rotation Period . . . of Venus, " Monthly Notices 
R. Ast. Soc.," vol. 57, 1897, p. 148. Further proof, &c., ibid. 
p. 402. 

Douglass, Jupiter's third Satellite, "Ast. Nach.," No. 3432. 
Rotation des IV Jupitersmondes, "Ast. Nach.," No. 3427, confirm- 
ing Engelmann, Ueber . . . Jupiterstrabanten, Leipzig, 1871. 

Barnard, The third and fourth Satellites of Jupiter, "Ast. 
Nach.," No. 3453. 



CHAPTER XVIII 

THE FIGURES OF EQUILIBRIUM OF A ROTATING 
MASS OF LIQUID 

THE theory of the tides involves the determi- 
nation 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, 1 devised an experiment which affords 
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 

1 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. 



CAPILLARITY 317 

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 tend- 
ency 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 sur- 
rounding 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 dif- 
ferent 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 cer- 
tain molecular attraction, quite distinct from 
that of gravitation, and it produces the same 



318 FIGURES OF EQUILIBRIUM 

effect as if the surface of the liquid were en- 
closed 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 elasti- 
city, continuously and immediately. The in- 
tensity of surface tension depends on the nature 
of the material with which the liquid is in con- 
tact ; 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. 

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 gravi- 
tation is the only force which acts on the sys- 
tem. The water will obviously crowd together 



FLUID AT REST 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, al- 
though 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 pos- 
sesses 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 im- 
pressed 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 for- 
wards, 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 



320 FIGURES OF EQUILIBRIUM 

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 pre- 
sent, 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 re- 
main 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 Chap- 
ter 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 ro- 
tate 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 con- 
sequence 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. 



STABILITY AND INSTABILITY 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 fig- 
ure, the slower the rotation the less is the equa- 
torial centrifugal force, because it diminishes 
both with diminution of radius and fall of speed. 
But in the cheese shape the equatorial centrifu- 
gal 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. 



322 FIGURES OF EQUILIBRIUM 

The third form is strikingly different from 
either of the preceding ones. We must now im- 
agine the liquid to be shaped like a long cigar, 
and to be rotating about a central axis perpen- 
dicular 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 in- 
finitely slow rotation the figure becomes infinitely 
long and thin. Now this form resembles the 

O 

cheese in being unstable. It is remarkable that 
these three forms are independent of the scale on 
which they are constructed, for tney are perfectly 
similar whether they contain a few pounds of 
water or millions of tons. 1 If the period of ro- 
tation 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 
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 sphe- 
roids of Maclaurin, after their discoverer, and 
the cigar shape is generally named after Jacobi, 
the great German mathematician. For slow ro- 
tations the planetary form is stable, and the 
cheese and cigar are unstable. There are prob- 
ably other possible forms of equilibrium, such as 
a ring, or several rings, or two detached masses 

1 It is supposed that they are more than a fraction of an inch 
across, otherwise surface tension would be called into play. 



MACLAURIN'S AND JACOBI'S FIGURES 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, in- 
finitely distant from one another, and each rotat- 




Maclaurin's Spheroids 





Sections of Jacobi's Ellipsoid 
FIG. 37 

ing 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 aug- 
mented, 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 
the second and third unstable. The three fig- 
ures 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. 



324 FIGURES OF EQUILIBRIUM 

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 instabil- 
ity 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 plan- 
etary form of Maclaurin has become much flat- 
tened, 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, 
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 transforma- 
tion 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 
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 



COALESCENCE OF TWO FORMS 



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 



Planetary form coalescent with 
elongated form, just stable 



Flat unstable form 





Poincar^'s figure 




FIG. 38 

been reduced ; it will be observed that it pre- 
sents some resemblance to the coalescent form. 

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

Still quickening the rotation, the two remain- 
ing forms, both unstable, grow in resemblance to 
one another, until at length they become identi- 
cal 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 



326 FIGURES OF EQUILIBRIUM 

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 fig- 
ure of equilibrium is a member of a continuous 
series of figures of the same class, which gradu- 
ally transforms itself as the rotation varies. Now 
M. Poincare has proved that, when we follow a 
given series of figures and find a change from in- 
stability 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 al- 
ready seen an example of this law, for the planet- 
ary 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 



POINCARE'S FIGURE 327 

cigar. It follows also from Poincare's investiga- 
tion 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 Poincare has 
not only proved the existence and stability of the 
other series, but has shown that the shape is 
something like a pear. 

Poin care's figure is represented approximately 
in fig. 38, but the mathematical difficulty of the 
problem has been too great to admit of an abso- 
lutely 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 
Poincare' s figure retains its stability. 

I have myself attacked this problem from an 
entirely different point of view, and my conclu- 
sions throw an interesting light on the subject, 
although they are very imperfect in comparison 



328 FIGURES OF EQUILIBRIUM 

with Poineare'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 an- 
other, 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 actu- 
ally 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 in- 
fluence 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 an- 
other, 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 



HOUR-GLASS FIGURE 



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 elonga- 
tion of the two bodies is much augmented. 

A time will at length come when the ends of 

o 

the two bodies will just touch, and we then have 
a form shaped like an hour-glass with a very 




FIG. 39. HOUR-GLASS FIGURE OF EQUILIBRIUM 

thin neck. The form is clearly Poincare's fig- 
ure, at an advanced stage of its evolution. 

The figure 39 shows the form of one possible 



330 FIGURES OF EQUILIBRIUM 

figure of this class ; it arises from the coales- 
cence 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 Poin- 
care's pear, it would be necessary to start with 
a planet and satellite of some definitely propor- 
tionate sizes. Unfortunately I do not know 
what the proportion may be. There are, how- 
ever, certain indications which may ultimately 
lead to a complete knowledge of the series of 
figures from Jacobfs cigar shape down to the 
planet and satellite. It may be shown and I 
shall have in 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 satel- 
lite would become so intense that it would tear 
it to pieces. Accordingly the contact and co- 
alescence 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 



VARIATION OF GRAVITY 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 Poincare's series. This dis- 
cussion 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 plan- 
ets 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 as- 
sumed was the resultant of the battle between 



332 FIGURES OF EQUILIBRIUM 

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 cen- 
tre, 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 
2J 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 mat- 
ter must lie in the centre and the less dense out- 
side. 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 flatten- 
ing can be determined, one in which the den- 
sity of the planet is the same all through, giving 
Maclaurin's figure ; the other when the density 
is enormously greater at the centre. The flat- 
tening in the former is 2J times as great as in 
the latter. The actual condition of a real planet 
must lie between these two extremes. The 



VARIATION OF GRAVITY 333 

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 Mac- 
laurin's 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, 
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 investi- 
gations have most important practical applica- 
tions. 

AUTHORITIES. 

Plateau, "Me'moires de I'Acade'mie Royale de Belgique," 
vol. xvi. 1843. 

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

Poincare', Sur Vequilibre d'une masse Jluide animee d'un mouve- 
ment 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 Mu- 
nich Observatory, vol. iii. 1896). He considers that Poincare"s 
proof of the stability of his figure is not absolutely conclusive. 

G. H. Darwin, Figures of Equilibrium of Rotating Masses of 
Fluid, Transactions of Royal Society," vol. 178, 1887. 

G. H. Darwin, Jacobi's Figure of Equilibrium, &c., " Proceed- 
ings Roy. Soc.," vol. 41, 1886, p. 319. 

S. Kriiger, Ellipsoidale Evenwichtsvormen, &c., Leeuwen, Lei- 
den, 1896; Sur Vellipsolde de Jacobi, " Nieuw Archief voor Wis- 
kunde," 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. 



CHAPTER XIX 

THE EVOLUTION OF CELESTIAL SYSTEMS 

MEN will always aspire to peer into the remote 
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 de- 
ter 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 equi- 
librium 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 revolv- 
ing 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. 1 

1 My knowledge of the history of the Nebular Hypothesis is 
entirely derived from an interesting paper by Mr. G. F. Becker, 
on "Kant as a Natural Philosopher," American Journal of Sci- 
ence, vol. v. Feb. 1898. 



THE NEBULAR HYPOTHESIS 335 

The first germs of this theory are to be found 
in Descartes' " Principles of Philosophy," pub- 
lished 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 pro- 
pounded 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 vorti- 
cal cosmogony in 1734, and Thomas Wright of 
Durham published in 1750 a book of preternat- 
ural 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 un- 
doubtedly 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, 



336 EVOLUTION OF CELESTIAL SYSTEMS 

and as the surface cooled the central portion 
condensed and its temperature rose. The speed 
of rotation increased in consequence of the con- 
traction, according to a well known law of me- 
chanics called " the conservation of moment of 
momentum ; " 1 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 con- 
traction and consequently to increased rotation, 
until a second ring was shed, and so on succes- 
sively. 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 pa- 
per, to which I have referred above, Mr. G. F. 
Becker makes the following excellent summary 
of the relative merits of Kant and Laplace ; he 
writes : 

" Kant seems to have anticipated Laplace al- 
most completely in the more essential portions 
of the nebular hypothesis. The great French- 
man was a child when Kant's theory was issued, 

1 Kant fell into error through ignorance of the generality of 
this law, for he imagined that rotation could be generated from 
rest. 



KANT AND LAPLACE 337 

and the ' Systeme 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. 1 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. Plan- 
ets formed from zones of vapor, which on break- 
ing agglomerated. . . . 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 compres- 
sion and at its first regenesis would be in a state 
of rotation. It is with a hot, rotating nebula 

' O 

that Laplace starts, without any attempt to ac- 
count for the heat. Kant supposes annular 
zones of freely revolving nebulous matter to 
gather together by attraction during condensa- 
tion of the nebula. Laplace supposes rings left 
behind by the cooling of the nebula to agglom- 

1 " The retrograde satellites of Uranus were discovered by 
fJerschel in 1787, but Laplace in his hypothesis does not refer to 
them." 



338 EVOLUTION OF CELESTIAL SYSTEMS 

erate in the same way as Kant had done. While 
both appeal to the rings of Saturn as an exam- 
ple of the hypothesis, neither explains satisfac- 
torily why the planetary rings are not as stable 
as those of Saturn. Both assert that the posi- 
tive 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. . . . While La- 
place assigns no cause for the heat which he as- 
cribes 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 Nep- 
tune's orbit will give a nebula such as Laplace 
postulates. 1 Thus Kelvin goes back to the same 
initial condition as Kant, excepting that Kant 
endeavored (of course vainly) to develop a mo- 
ment of momentum for his system from colli- 
sions." 2 

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 

1 Popular Lectures, vol. i. p. 421. 

2 Becker, Amer. Journ. Science, vol. v. 1898, pp. 107, 108. 




FIG. 40. NEBULA IN ANDROMEDA 



DISTRIBUTION OF SATELLITES 339 

of nebulae 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 con- 
stellation of Andromeda. In it we may see the 
lenticular nebula with its central condensation, 
the annulation of the outer portions, and even 
the condensations in the rings which will doubt- 
less at some time form planets. This system is 
built on a colossal scale, compared with which 
our solar system is utterly insignificant. Other 
nebulae 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. 

I explained in 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 dimen- 
sions of the orbits of the satellites of Mars, Ju- 
piter, 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 explana- 
tion of the main features of the system as a 
whole. 



340 EVOLUTION OF CELESTIAL SYSTEMS 

It may, at first sight, appear illogical to main- 
tain 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, Sat- 
urn 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 con- 
tended that a factor of evolution, which has been 
predominant in our own history, has been rela- 
tively 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 augmen- 
tation 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. 



DISTRIBUTION OF SATELLITES 341 

The friction of the solar tides in a planetary 
nebula furnishes such an external cause, and ac- 
cordingly 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 re- 
lationship between the moon and the earth is so 
unlike that between other satellites and their 
planets. In Chapter XVII. I showed that there 
are reasons for believing that solar tidal friction 
has really been an efficient cause of change, and 
this makes it legitimate to invoke its aid in ex- 
plaining the birth and distribution of satellites. 

In speaking of the origin of the moon I have 



342 EVOLUTION OF CELESTIAL SYSTEMS 

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 oscilla- 
tions due to the solar action. Even if this sug- 

o 

gestion 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 deter- 
mine the orbits of pairs of double stars around 
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 de- 
voted himself to this research, has drawn atten- 
tion 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 



ECCENTRICITY OF DOUBLE STARS 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, 



344 EVOLUTION OF CELESTIAL SYSTEMS 

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 
number of twin nebulae 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 nebula adopted by Dr. See. But 
since Herschel's time it has been discovered that 
many details, to which our eyes must remain for- 
ever 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 nebulae 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 in- 
terpretation, for there are some which serve only 
to increase the chaos visible with the telescope. 
We may suspect, indeed, that the complete sys- 
tem 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. 



DIVERSITY OF NEBULAE 345 

Another group of strange celestial objects is 
that of the spiral nebulae, whose forms irresisti- 
bly 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 rea- 
sonable to suppose that in these nebulae 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 
nebula (27 Messier Vulpeculae), as seen telescopi- 
cally, 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 sur- 
rounded 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. 



346 EVOLUTION OF CELESTIAL SYSTEMS 

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. 

AUTHORITIES. 

Mr. G. F. Becker (Amer. Jour. Science, vol. v. 1898, art. xv.) 
gives the following references to Kant's work : Sdmmtliche 
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, Systeme du Monde, last appendix ; the tidal retarda- 
tion of the moon's rotation is only mentioned in the later 
editions. 

T. J. J. See, Die Entwickelung der Doppelstern-systeme, " In- 
augural Dissertation," 1892. Schade, Berlin. 

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

G. H. Darwin, Tidal Friction . . . and Evolution, " Phil. Trans. 
Roy. Soc.," part ii. 1881, p. 525. 



CHAPTER XX 

SATURN'S RINGS l 

To the naked eye Saturn appears as a brilliant 
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 de- 
grees 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| of our hours. It is ten times as far from 
the sun as we are, and its year, or time of revo- 
lution 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 Nep- 
tune by Adams and Leverrier in the year 1846. 

1 Part of this chapter appeared as an article in Harper's 
Magazine for June, 1889. 



348 SATURN'S RINGS 

The telescope has shown that Saturn is at- 
tended by a retinue of satellites almost as numer- 
ous 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 at- 
tendant 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 gen- 
eral 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 some- 
thing 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 



THE PLANET SATURN 



349 




f-i 

I 



350 SATURN'S RINGS 

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 in- 
vented a combination of lenses such as is still 
used in our present opera-glasses, for the pur- 
pose of magnifying distant objects. 

In July of 1610 he began to examine Saturn 
with his telescope. His most powerful instru- 
ment only magnified 32 times, and although 
such an enlargement should have amply sufficed 
to enable him to make out the ring, yet he per- 
suaded 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 plane- 
tarn tergeminum observavi " (I have seen the 
furthest planet as triple), for it must be remem- 
bered that Saturn was then the furthest known 
planet. 

In 1612 Galileo again examined Saturn, and 
was utterly perplexed and discouraged to find 



OBSERVATION BY GALILEO 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 power- 
ful 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 unravel- 
ing 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 
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. 



352 SATURN'S RINGS 

A few years later it was perceived that there 
were two rings, concentric with one another. 
The division, which may be easily seen in draw- 
ings of the planet, is still named after Cassini, 
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, inde- 
pendently 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 be- 
come more conspicuous within the last 200 years, 
so that it would not be right to attribute the 
lateness of its detection entirely to the imperfec- 
tion 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 revo- 
lution in 10J hours. 

This sketch of the discovery and observation 
of Saturn's rings has been necessarily very in- 
complete, but we have perhaps already occupied 
too much space with it. 



BOND'S OBSERVATION 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 



Roche's Limit 

Outer Ring 
Cassini's Division 




PIG. 42. DIAGRAM OP SATURN AND ms RINGS 

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 clock- 
work, 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 discus- 
sion of this and of other pictures, but we must 
confine ourselves to drawing attention to the 
well-marked split, called Cassini's division, and 



354 SATURN'S RINGS 

to the faint internal ring, through which the 
edge of the planet is visible. 

The scale on which the whole system is con- 
structed 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 
Roche's limit, will be given later. The follow- 
ing are the dimensions of the system : 

Equatorial diameter of planet . . 73,000 miles 

Interior diameter of dark ring . . 93,000 " 

Interior diameter of bright rings . . 111,000 " 

Exterior diameter of bright rings . . 169,000 " 

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 un- 
certain, 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 
rings were annihilated, but, as Clerk-Maxwell 
has remarked, " from a purely scientific point of 



THE SATURNIAN SYSTEM 355 

view, they become the most remarkable bodies in 
the heavens, except, perhaps, those still less use- 
ful bodies the spiral nebulae. When we have 
actually seen that great arch swung over the 
equator of the planet without any visible connec- 
tion, 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 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 rea- 
son to suppose that the earth once raised enor- 



356 SATURN'S RINGS 

mous 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 degener- 
ates 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, in- 
creases 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, 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. Edouard Koche has shown that, this ap- 
proach 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 



TIDAL FORCE ON SMALL SATELLITE 357 

about which it revolves to be a large globe, with 
its centre lying on the prolongation of the long- 
est axis of the egg-like body in the direction 
of E. As it revolves, the longest axis of the sat- 
ellite always points straight towards its planet. 
The egg, though not strictly circular in girth, is 




FIG. 43. ROCHE'S FIGURE OF A SATELLITE WHEN ELONGATED 
TO THE UTMOST 



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 flat- 
tened egg is at right angles to the plane of the 
orbit in which the satellite revolves. The long- 
est axis of the body is nearly twice as long as 
either of the two shorter ones ; for if we take 
the longest as 1000, the other two would be 496 
and 469. Fig. 43 represents a section through 
the two axes equal respectively to 1000 and to 
469, so that we are here supposed to be looking 
at the satellite's orbit edgewise. 



358 SATURN'S RINGS 

But, as I have said, Roche determined not 
only the shape of the satellite when thus elon- 
gated 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^| of the planet's radius. This distance of 2|J 
or 2.44 of a planet's radius I call Roche's limit 
for that planet. The meaning of this is that in- 
side of a circle drawn around a planet at a dis- 
tance 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 de- 
scribed 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. 



ROCHE'S LIMIT 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 satel- 
lite. If for example the planet be on the aver- 
age 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, 
Koche's limit changes but little. 

The only relative density of planet and satel- 
lite 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 | 
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 



360 SATURN'S RINGS 

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 agglomera- 
tion of the satellite. The origin and earliest his- 
tory of the moon must always remain highly 
speculative, and it seems fruitless to formulate 
exact theories on the subject. 1 

When we apply this reasoning to the other 
planets, exact data are wanting. The planet 
Mars resembles the earth in so many respects 
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 

1 Mr. Nolan has criticised the theory of tidal friction from 
this point of view (Genesis of the Moon, Melbourne, 1885; also 
Nature, Feb. 18 and July 29, 1886). 



ROCHE'S LIMIT 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, 
we are deprived of the indications which are use- 
ful in the case of Mars. The satellites are prob- 
ably 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 con- 
firms 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 there- 
fore 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 



362 SATURN'S RINGS 

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 men- 
tioned in text-books and general treatises. In- 
deed, it has been stated that Bond was the first 
in modern times to suggest the meteoric consti- 
tution of the rings. His suggestion, based on 
telescopic evidence, was however made in 1851. 

And now to explain how a Cambridge mathe- 
matician to whom reference was made above, in 
ignorance of Roche's work of nine years before, 
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 



MAXWELL'S INVESTIGATION 363 

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 
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 weigh- 
ing 44 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 irreg- 
ularity, 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 accumu- 
late at the thicker parts, and thus to destroy that 
nice adjustment of the load on which the stabil- 
ity depends, we have another powerful argument 
against solidity. And when we consider the im- 
mense 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 



364 SATURN'S RINGS 

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 in- 
definite number of meteorites of all sizes. The 
consideration of the motion of these equal satel- 
lites 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 consum- 
mate skill, and showed that if the satellites were 
not too large, confusion would not ensue, but 
each satellite would oscillate about its proper 
place. 



MAXWELL'S CONCLUSIONS 365 

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 dis- 
turbances of the ring will in this, as in the 
former case, produce only waves, and not grow- 
ing confusion. Supposing the ring to consist, 
not of a single row of large satellites, but of a 
cloud of evenly distributed unconnected parti- 
cles, 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. 



366 SATURN'S RINGS 

" We conclude, therefore, that the rings must 
consist of disconnected particles ; these may be 
either solid or liquid, but they must be independ- 
ent. 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 con- 
fused multitude of revolving particles, not ar- 
ranged 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 magni- 
tude, and that such cases must continually occur 
in an extensive system like that of Saturn, the 
only retarding cause being the possible irregu- 
larity 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 num- 
ber 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 



MAXWELL'S CONCLUSIONS 367 

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 (i. e. 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 parti- 
cles 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 opti- 
cal argument in favor of the theory of independ- 
ent 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 en- 
countering one of the particles." 

The last link in the chain of evidence has been 
furnished by recent observations made in Amer- 
ica. 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 



368 SATURN'S RINGS 

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 blow- 
ing, 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. 1 Now there are cer- 
tain lines in the spectrum of sunlight, and the 
shifting of their positions affords an excessively 
delicate measure of a change which, when mag- 
nified 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 ap- 
proaching us and the other is receding. The 
two edges are infinitesimally of different colors, 
and the change of tint is measurable by the dis- 
placement of the lines I have mentioned. In 
the same way Saturn's ring is illuminated by 
sunlight, and if different portions are moving at 

1 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 monochro- 
matic light. The qualification would, however, only complicate 
the statement, and thus render the displacement of the lines of 
the spectrum less easily intelligible 



KEELER'S OBSERVATION 369 

different velocities, those portions are infinitesi- 
mally 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 
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. 

AUTHORITIES. 

Edouard Roche, La figure d'une masse fluide soumise a V attrac- 
tion d'un point eloigne, " Me'm. Acad. de Montpelier," vol. i. 
(Sciences), 1847-50. 

Maxwell, Stability of Saturn's Rings, Macmillan, 1859. 

Keeler, Spectroscopic Proof of the Meteoric Constitution of 
Saturn's Rings, " Astrophysical Journal," May, 1895 ; see also 
the same for June, 1895. 

Schwarzschild, Die Poincaresche Theorie des Gleichgewichts, 
"Annals of Munich Observatory," vol. iii. 1896. He considers 
the stability of Roche's ellipsoid. 



INDEX 



ABACUS for reducing tidal obser- 
vations, 217-220. 

Abbadie, tidal deflection of verti- 
cal, 143, 144. 

Aden, errors of tidal prediction 
at, 246. 

Adriatic, tide in, 186. 

Airy, Sir G. B., tides in rivers, 
75 ; attack on Laplace, 181 ; co- 
tidal chart, 188; Tides and 
Waves, 192. 

America, North, tide tables for, 
222. 

Analysis, harmonic, of tide, 193- 
210. 

Andromeda, nebula in, 339. 

Annual and semi-annual tides, 
206. 

Arabian theories of tide, 77-79. 

Aristotle on tides, 81. 

Assyrian records of eclipses, 272. 

Atlantic, tide in, 186-188. 

Atmospheric pressure, cause of 
seiches, 40 ; distortion of soil 
by, 145, 146 ; influence on tidal 
prediction, 242, 243. 

Atmospheric waves, Helmholtz 
on, 48-51. 

Attraction, of weight resting on 
elastic slab proportional to 
slope, 136, 137; of tide calcu- 
lated, 143. 

Baird, Manual far Tidal Observa- 
tion, 16. 

Bakhuyzen on tide due to varia- 
tion of latitude, 255, 256. 

Barnard, rotation of Jupiter's 
satellites, 315. 



Barometric pressure. See Atmo- 
spheric pressure. 

Becker, G. F., on Nebular Hypo- 
thesis, 334, 336-338. 

Bernoulli, Daniel, essay on tides, 
86, 88. 

Bertelli on Italian seismology, 
126, 127. 

Bifilar. See Pendulum. 

Borgen, method of reducing tidal 
observations, 217. 

Bond, discovery of inner ring of 
Saturn, 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, 77-79. 

Cambridge, experiments with bifi- 
lar pendulum at, 115-125. 

Canal, theory of tide wave in, 
165-167 ; critical depth, 163-165 ; 
tides in ocean partitioned into 
canals, 175; canal in high lati- 
tude, 174-176. 

Capillarity of liquids, and Pla- 
teau's experiment, 316-318. 

Cassini, discovery of division in 
Saturn's rings, 352. 

Castel, Father, ridiculed by Vol- 
taire, 295, 296. 

Cavalleri, essay on tides, 86. 

Centripetal and centrifugal forces, 
91-93. 

Chambers on possible existence of 
Martian satellites, 296. 



372 



INDEX 



Chandler, free nutation of earth, 
and variation of latitude, 253- 
257. 

Chinese superstition as to bore, 
68-70 ; theories of tide, 76, 77. 

Christie, A. S., tide due to varia- 
tion of latitude, 255, 256. 

Constants, tidal, explained, 195. 

Continents, trend of, possibly due 
to primeval tidal friction, 308. 

Cotidal chart, 188; for diurnal 
tide hitherto undetermined, 191, 
192. 

Currents, tidal, in rivers, 56. 

Curve, tide, irregularities in, 10- 
16 ; at Bombay, 12 ; partitioned 
into lunar time, 213. 

D'Abbadie. See Abbadie. 

Darwin, G. H., 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 rotat- 
ing liquid, 328-332 ; Jacobi's el- 
lipsoid, 333 ; evolution of satel- 
lites, 346. 

Darwin, Horace, bifilar pendu- 
lum, 115-125. 

Davis, method of presenting tide- 
generating force, 96, 97. 

Davison, history of bifilar and 
horizontal pendulums, 133. 

Dawes, discovery of inner ring of 
Saturn, 352. 

Dawson cooperates in investiga- 
tion of seiches, 48. 

Day, change in length of, under 
tidal friction, 275, 276. 

Deflection of the vertical, 109- 
133 ; experiments to measure, 
115-125 ; due to tide, 134-143. 

Deimos, a satellite of Mars, 297. 

Denison, F. Napier, vibrations 
on lakes, 48-53. 



Density of earth, law of internal, 
302; of planets determinable 
from their figures, 332, 333. 

Descartes, vortical theory of 
cosmogony, 335. 

Dick, argument as to Martian 
satellites, 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 pre- 
diction, 224, 225. 

Douglass, rotation of Jupiter's 
satellites, 315. 

Dumb-bell nebula, description of 
photograph of, 345. 

Dynamical theory of tide- wave, 
163-181. 

Earth and moon, diagram, 93 ; 
rotation of, effects on tides, 
177 ; rigidity of, 256-260 ; rota- 
tion retarded by tidal friction, 
268; figure of, 299; adjust- 
ment 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 tre- 
mors, 125-127 ; shock percepti- 
ble at great distance, 261." 

Ebb and flow defined, 56. 

Eccentricity of orbit due to tidal 
friction, 313, 314 ; theory of, in 
case of double stars, 342. 



INDEX 



373 



Eclipses, ancient, and earth's ro- 
tation, 272, 273. 

Ecliptic, obliquity of, due to tidal 
friction, 308-312. 

Eddies, tidal oscillation involves, 
177. 

Ehlert, observation with horizon- 
tal pendulum, 132. 

Elastic distortion of soil by weight, 
123 ; of earth by varying loads, 
134-148 ; calculation and illus- 
tration, 138-140 ; by atmospheric 
pressure, 145-147. 

Elasticity of earth, 254, 255. 

Elliptic tide, lunar, 204. 

Ellipticity of earth's strata in ex- 
cess for present rotation, 303, 
304. 

Energy, tidal, utilization of, 73, 74. 

Equatorial canal, tide wave in, 173. 

Equilibrium, figures of, of rotat- 
ing liquid, 316-333. 

Equilibrium theory of tides, 149- 
162 ; chart and law of tide, 151- 
153 ; defects of, 160. 

Errors in tidal prediction, 243-245. 

Establishment of port, definition, 
161, 162 ; zero in equilibrium 
theory, 161 ; shown in cotidal 
chart, 189. 

Estuary, annual meteorological 
tide in, 207, 208. 

Euler, essay on tides, 86. 

Europe, tides on coasts of, 188. 

Evolution of celestial systems, 
334-346. 

Ferrel, tide-predicting instrument, 
241. 

Figure of equilibrium of ocean un- 
der tidal forces, 151-153 ; of ro- 
tating liquid, 316-333. 

Figure of planets and their density, 
332, 333. 

Fisher, Osmond, on molten inte- 
rior of earth, 262. 

Flow and ebb defined, 56. 



Forced oscillation, principle of, 
169, 170 ; due to solar tide, pos- 
sibly related to birth of moon, 
282-284. 

Forced wave, explanation and con- 
trast with free wave, 164. 

Forces, centripetal and centrif- 
ugal, 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, 169, 170. 

Free wave, explanation and con- 
trast with forced, 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, description of, 6-11 ; 
site for, 14. 

Geneva, seiches in lake, 17-28 ; 
model of lake, 28. 

Geological evidence of earth's 
plasticity, 300 ; as to retardation 
of earth's rotation, 304-306. 

German method of reducing tidal 
observations, 217. 

Giles on Chinese theories of the 
tide, 76, 77. 

Gravity, variation according to 
latitude, 302, 303, 332. 

Greek theory and description of 
tides, 81-85; records of ancient 
eclipses, 272. 

Gulliver's Travels, satire on math- 
ematics, 292-295. 

Hall, Asaph, discovery of Martian 

satellites, 290-298. 
Hangchow, the bore at, 60-70. 
Harmonic analysis initiated by 



374 



INDEX 



Lord Kelvin, 87; account of, 
193-210. 

Height of tide due o ideal satel- 
lite, 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 
nebulae, 344. 

High water under moon in equi- 
librium theory, 160; position in 
shallow and deep canals in dy- 
namical theory, 171, 172. 

History of tidal theories, 76-88 ; of 
earth and moon, 278-286, 308-313. 

Hopkins on rigidity of earth, 
258, 259. 

Horizontal tide-generating force, 
107. See also Pendulum. 

Hough, S. S., frictional extinction 
of waves, 47 ; dynamical solution 
of tidal problem, 181 ; rigidity of 
earth, 254 ; Chandler's nutation, 
262. 

Hugli, bore on the, 71, 

Huyghens, discovery of Saturn's 
ring, 351. 

Icelandic theory of tides, 79, 80. 

Indian Survey, method of redu- 
cing 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 satel- 
lite, 198 ; at Portsmouth and at 
Aden, 225. 

Italian investigations in seismol- 
ogy, 125-130. 

Jacobi, figure of equilibrium of 
rotating liquid, 322-324. 



Japan, frequency of earthquakes, 

.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, 367-369. 

Kelvin, Lord, initiates harmonic 
analysis, 87, 199 ; calculation of 
tidal attraction, 143 ; tide pre- 
dicting machine, 233 ; rigidity 
of earth, 257-260; denies ad- 
justment of earth's figure to 
changed rotation, 301 ; on geolo- 
gical time, 315. 

Kepler, ideas concerning tides, 85, 
86 ; argument respecting Martian 
sateUites, 291, 292. 

Kriiger, figures of equilibrium of 
liquid, 333. 

Lakes, seiches in, 17-54 ; mode of 
rocking in seiches, 24, 25 ; vibra- 
tions, 41-53 ; tides in, 182-185. 

Lamb, H., presentation of La- 
place's theory, 181. 

Laplace, theory of tides, 86-88, 177- 
180; on rotation of moon, 286, 
287; nebular hypothesis, 335-337. 

Lardner, possibility of Martian 
satellites, 295. 

Latitude, tidal wave in canal in 
high, 174-176 ; periodic variations 
of, 251-256. 

Le'ge', constructor of tide-predict- 
ing machine, 233. 

Level of sea affected by atmo- 
spheric pressure, 146. 

Limnimeter, a form of tide gauge, 
24. 

Lowell, P. v on rotations of Venus 
and Mercury, 298, 299, 315. 

Low water. See High water. 



INDEX 



375 



Lubbock, Sir J., senior, on tides, 
87. 

Lunar tide-generating force com- 
pared with solar, 156-158 ; tide, 
principal, 201 ; elliptic tide, 204 ; 
time, 213. 

Machine, tide-predicting, 233, 241. 

Mackerel sky, evidence of air- 
waves, 49. 

Maclaurin, essay on tides, 86 ; 
figure of equilibrium of rotating 
liquid, 322-324. 

Magmisson on Icelandic theories 
of tides, 79, 80. 

Marco Polo, resident of Hang- 
chow, 70. 

Mars, discovery of satellites, 290- 
298 ; Roche's limit, 360. 

Maxwell on Saturn's ring, 363- 
367. 

Mediterranean Sea, tides in, 185, 
186. 

Mercury, rotation of, 298, 299. 

Meteoric constitution of Saturn's 
ring, 368, 369. 

Meteorological tides, 206, 207; con- 
ditions dependent on earth's ro- 
tation, 303. 

Microphone as a seismological in- 
strument, 128-130. 

Microsisms, minute earthquakes, 
125-127. 

Mills worked by the tide, 74, 75. 

Milne on seismology, 125, 130. 

Month, change in, under tidal fric- 
tion, 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 ac- 
tual, 199, 200; tidal prediction by 
reference to transit, 224-230 ; re- 
tardation of motion by tidal fric- 
tion, 269, 270 ; origin of, 282, 283 ; 
rotation annulled by tidal fric- 



tion and present libration, 286 ; 
inequality in motion indicates 
internal density of earth, 302, 
303 ; eccentricity of orbit in- 
creased by tidal friction, 313, 
314. 

Moore, Captain, illustrations of 
bore, 67 ; survey of Tsien-Tang- 
Kiang, 60-70. 

Neap and spring tides in equili- 
brium theory, 159 ; represented 
by principal lunar and solar 
tides, 204. 

Nebula in Andromeda, 339. 

Nebula, description of various, 
345. 

Nebular hypothesis, 334-339. 

Newcomb, S., theoretical explana- 
tion of Chandler's nutation, 254. 

Newton, founder of tidal theory, 
86 ; theory of tide in equatorial 
canal, 172. 

Nolan, criticism of tidal theory of 
moon's origin, 360. 

Nutation, value of, indicates inter- 
nal density of earth, 303 ; Chan- 
dler's, 251-256. 

Obliquity of ecliptic, effects of 
tidal friction on, 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, 186, 187. 

Partial tides in harmonic method, 
199. 

Paschwitz, von Rebeur, on hori- 
zontal pendulum, 130-132 ; tidal 
deflection of vertical at Wil- 
helmshaven, 144. 

Pendulum, curves traced by, un- 
der tidal force, 111, 112 ; bifilar, 
115-125 ; as seismological instru- 



376 



INDEX 



ment, 126, 127 ; horizontal, 130- 
132. 

Petitcodiac, bore in the, 71. 

Phobos, a satellite of Mars, 297. 

Planetary figure of equilibrium of 
rotating liquid, 322. 

Planets, rotation of some, an- 
nulled by tidal friction, 298 ; 
figures and internal densities, 
332, 333. 

Plasticity of earth under change 
of rotation, 300-302. 

Plateau, experiment on figure of 
rotating globule, 316-319. 

Plemyrameter, observation of 
seiches with, 19-22. 

Poincare 1 , law of interchange of 
stability, 326, 327 ; figure of ro- 
tating liquid, 325, 327. 

Polibius on tides at Cadiz, 83. 

Portsmouth, table of errors in 
tidal predictions, 244. 

Posidonius on tides, 81-84. 

Precession, value of, indicates in- 
ternal density of earth, 303. 

Predicting machine for tides, 233- 
241 ; Fen-el'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, 145, 146. 

Principle of forced oscillations, 
169, 170. 

Rebeur. See Paschwitz. 

Reduction of tidal observations, 
211-220. 

Retardation of earth's rotation, 
268. 

Rigidity of earth, 256-260. 

Ripple mark in sand preserved in 
geological strata, 305. 

Rivers, tide wave in, 55-59 ; Airy 
on tide in, 75 ; annual meteoro- 
logical tide in, 206. 



Roberts, E., the tide-predicting 
machine, 233. 

Roberts, I., photograph of nebula 
in Andromeda, 339. 

Roche, E., 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, 81- 
85. 

Rossi on Italian seismology, 128- 
130. 

Rotating liquid, figures of equili- 
brium, 316-333. 

Rotation of earth involved in tidal 
problem, 177 ; retarded by tidal 
friction, 268 ; of moon annulled 
by tidal friction, 286 ; of Mer- 
cury, Venus, and satellites of 
Jupiter and Saturn annulled by 
tidal friction, 298. 

Russell, observation of seiches in 
New South Wales, 47. 

St. Ve"nant on flow of solids, 313. 

Satellites, tide due to single equato- 
rial, 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 pic- 
ture, 347-354; theory of ring, 
356-369 ; Roche's limit for, 
360. 

Schedule for reducing tidal obser- 
vations, 215, 216. 

Schiaparelli on rotation of Venus 
and Mercury, 298, 315. 

Schwarzschild, exposition of Poin- 
care' 's theory, 333; stability of 
Roche's ellipsoid, 369. 

Sea, vibrations of, 44, 45; level af- 



INDEX 



377 



fected by atmospheric pressure, 
146. 

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, 71. 

Seismology, 133. 

Seleucus, observation of tides of 
Indian Ocean, 84, 85. 

Semidiurnal tide in equilibrium 
theory, 153-156 ; in harmonic 
method, 201-204. 

Severn, bore in the, 71. 

Slope of soil due to elastic distor- 
tion, 136 ; calculation and illus- 
tration of, 138-140. 

Solar tide-generating force com- 
pared with lunar, 156-158 ; prin- 
cipal tide, 202 ; possible effect of 
tide in assisting birth of moon, 
284, 285; system, nebular hy- 
pothesis as to origin of, 334-339 ; 
system, distribution of satellites 
in, 339-341. 

Spectroscopic proof of rotation of 
Saturn's ring, 368, 369. 

Spring and neap tides in equili- 
brium 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, 39, 40. 

Strabo on tides, 81-85. 

Stupart cooperates in investigation 
of seiches, 48. 

Sun, tide-generating force of, com- 
pared with that of moon, 156-158 ; 



ideal, replacing real sun in har- 
monic analysis, 201 ; possible in- 
fluence of, in assisting birth of 
moon, 284, 285. 

Surface tension of liquids, 317, 
318. 

Swift, satire on mathematicians, 
292-295. 

Synthesis of partial tides for pre- 
diction, 230-233. 

Tables, tide, 221-241 ; method of 
calculating, 230-241 ; amount of 
error in, 246, 247. 

Thomson, Sir W. See Kelvin. 

Tidal problem. See Laplace, Har- 
monic Analysis, etc. 

Tide, definition, 1-3; general de- 
scription, 4-6. See also other 
headings ; e. g. for tide-genera- 
ting force, see Force. 

Time, lunar, 213 ; requisite for 
evolution of moon, 285. 

Tisserand, Roche's investigations 
as to earth's figure, 315. 

Tremors, earth, 125. 

Tresca on flow of solids, 300. 

Tromometer, a seismological in- 
strument, 126, 127. 

Tsien-Tang-Kiang, the bore in, 
60-70. 

United States Coast Survey, meth- 
od of reducing tidal observa- 
tions, 217 ; tide tables of, 222. 

Variation of latitude, 251-256. 

Vaucher, record of a great seiche 
at Geneva, 17. 

Venus, rotation of, 298, 299. 

Vertical. See Deflection. 

Vibration of lakes, 41-53. 

Voltaire, satire on mathemati- 
cians, and Martian satellites, 
295, 296. 

Vortical motion in oceanic tides, 
177, 178. 



378 



INDEX 



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; pro- 
pagated northward in Atlantic, 
186-188. 

Wharton, Sir W. J., illustration 
of bore, 69. 

Whewell on tides, 87; empirical 
construction of tide tables, 87- 



90; on cotidal charts, 188, 
189. 

Wind, a cause of seiches, 39 ; vi- 
brations of lakes due to, 41, 42 ; 
a cause of meteorological tides, 
206 ; perturbation of, in tidal 
prediction, 242, 243. 

Woodward on variation of lati- 
tude, 262. 

Wright, Thomas, on a theory of 
cosmogony, 335. 

Wye, bore in the, 71. 



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