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SCIENTIFIC PAPERS

BY

SIR GEORGE HOWARD DARWIN,

K.C.B., F.R.S.

FELLOW OF TRINITY COLLEGE PLUMIAN PROFESSOR IN THE UNIVERSITY OF CAMBRIDGE

VOLUME II TIDAL FRICTION

AND

COSMOGONY

CAMBRIDGE :

AT THE UNIVERSITY PRESS 1908

CTambriDge :

PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.

I

Scic Library

PREFACE

rinHE papers contained in the present volume form in effect a single -*- investigation in speculative astronomy. The tidal oscillations of the mobile parts of a planet must be subject to frictional resistance, and this simple cause gives rise to a diversity of astronomical effects worthy of examination.

The earlier portion of the investigation was undertaken with the object of explaining, if possible, the obliquity of the earth's equator to the ecliptic, and the results attained were so fruitful and promising that it seemed well to examine the whole subject with the closest attention, and to discuss the various collateral points which arose in the course of the work.

It is the experience of every investigator that he reaches his goal by a devious path, and this, at least, has been the case in the present group of papers. If then the whole field were now to be retraversed, it is almost certain that the results might be obtained more shortly. Then, again, there is another cause which precludes brevity, for when an entirely new subject is under consideration every branch road must be examined with care. By far the greater number of the forks in the road lead only to blind alleys ; and it is often impossible to foresee, at the cross roads, which will be the main highway, and which a blind alley. Clearness of view is only reached by the pioneer after much labour, and as he first passes along his path he has to grope his way in the dark without the help of any sign- post.

This may be illustrated by what actually occurred to me, for when I first found the quartic equation (p. 102) which expresses the identity between the lengths of the day and of the month, I only regarded it as giving the configuration towards which the retrospective integration was leading back. I well remember thinking that it was just as well to find the other roots of the equation, although I had no suspicion that anything of interest would be discovered thereby. As of course I ought to have foreseen, the result threw a flood of light on the whole subject, for it showed how the system must have degraded, through loss of energy, from a configuration represented by the first real root to another represented by the second. Moreover the motion in the first configuration was found to be unstable whilst that in

vi PREFACE.

the second was stable. Thus this quartic equation led to the remarkably simple and illuminating view of the theory of tidal friction contained in the fifth paper (p. 195); and yet all this arose from a point which appeared at first sight barely worth examining.

I wish now, after the lapse of more than twenty years, to avail myself of this opportunity of commenting on some portions of the work and of reviewing the theory as a whole.

The observations of Dr Hecker* and of others do not afford evidence of any considerable amount of retardation in the tidal oscillations of the solid earth, for, within the limits of error of observation, the phase of the oscilla- tion appears to be the same as if the earth were purely elastic. Then again modern researches in the lunar theory show that the secular acceleration of the moon's mean motion is so nearly explained by means of pure gravitation as to leave but a small residue to be referred to the effects of tidal friction. We are thus driven to believe that at present tidal friction is producing its inevitable effects with extreme slowness. But we need not therefore hold that the march of events was always so leisurely, and if the earth was ever wholly or in large part molten, it cannot have been the case.

In any case frictional resistance, whether it be much or little and whether applicable to the solid planet or to the superincumbent ocean, is a true cause of change, and it remains desirable that its effects should be investigated. Now for this end it was necessary to adopt some consistent theory of friction- ally resisted tides, and the hypothesis of the earth's viscosity afforded the only available theory of the kind. Thus the first paper in the present volume is devoted to the theory of the tides of a viscous spheroid. It may be that nothing material is added by solving the problem also for the case of elastico- viscosity, but it was well that that hypothesis should also be examined.

I had at a previous date endeavoured to determine the amount of modi- fication to which Lord Kelvin's theory of the tides of an elastic globe must be subject in consequence of the heterogeneity of the earth's density, and this investigation is reproduced in the second paper. Dr Herglotz has also treated the problem by means of some laborious analysis, and finds the change due to heterogeneity somewhat greater than I had done. But we both base our conclusions on assumptions which seem to be beyond the reach of verification, and the probability of correctness in the results can only be estimated by means of the plausibility of the assumptions.

The differential equations which specify the rates of change in the various elements of the motions of the moon and the earth were found to be too

" Veroffentl. d. K. Prniss. Qcodiit. Intl., Neue Folge, No. 32, Potsdam, 1907.

PREFACE. Vll

complex to admit of analytical integration, and it therefore became necessary to solve the problem numerically. It was intended to draw conclusions as to the history of the earth and moon, and accordingly the true values of the mass, size and speed of rotation of the earth were taken as the basis of computation. But the earth was necessarily treated as being homogeneous, and thus erroneous values were involved for the ellipticity, for the precessional constant and for the inequalities in the moon's motion due to the oblateness of the earth. It was not until the whole of the laborious integrations had been completed that it occurred to me that an appropriate change in the linear dimensions of the homogeneous earth might afford approximately correct values for every other element. Such a mechanically equivalent substitute for the earth is determined on p. 439, and if my integrations should ever be repeated I suggest that it would be advantageous to adopt the numerical values there specified as the foundation for the computations.

The third paper contains the investigation of the secular changes in the motions of the earth and moon, due to tidal friction, when the lunar orbit is treated as circular and coincident with the ecliptic. The differential equations are obtained by means of the disturbing forces, but the method of the disturbing function is much more elegant. The latter method is used in the sixth paper (p. 208), which is devoted especially to finding the changes in the eccentricity and the inclination of the orbit. However the analysis is so complicated that I do not regret having obtained the equations in two independent ways. As the sixth paper was intended to be supplementary to the third, the disturbing function is developed with the special object of finding the equations for the eccentricity and the inclination, but an artifice is devised whereby it may also be made to furnish the equations for the other elements. It would only need a slight amount of modification to obtain the equations for all the elements simultaneously by straightforward analysis.

This paper also contains an investigation of the motion of a satellite moving about an oblate planet by means of equations, which give simul- taneously the nutations of the planet and the corresponding inequalities in the motion of the satellite. The equations are afterwards extended so as to include the effects of tidal friction. I found this portion of the work far more arduous than anything else in the whole series of researches.

The developments and integrations in all these papers are carried out with what may perhaps be regarded as an unnecessary degree of elaboration, but it was impossible to foresee what terms might become important. It does not, however, seem worth while to comment further on minor points such as these.

Viii PREFACE.

For the astronomer who is interested in cosmogony the important point is the degree of applicability of the theory as a whole to celestial evolution. To me it seems that the theory has rather gained than lost in the esteem of men of science during the last 25 years, and I observe that several writers are disposed to accept it as an established acquisition to our knowledge of cosmogony.

Undue weight has sometimes been laid on the exact numerical values assigned for defining the primitive configuration of the earth and moon. In so speculative a matter close accuracy is unattainable, for a different theory of frictionally retarded tides would inevitably lead to a slight difference in the conclusion ; moreover such a real cause as the secular increase in the masses of the earth and moon through the accumulation of meteoric dust, and possibly other causes are left out of consideration.

The exact nature of the process by which the moon was detached from the earth must remain even more speculative. I suggested that the fission of the primitive planet may have been brought about by the synchronism of the solar tide with the period of the fundamental free oscillation of the planet, and the suggestion has received a degree of attention which I never anticipated. It may be that we shall never attain to a higher degree of certainty in these obscure questions than we now possess, but I would main- tain that we may now hold with confidence that the moon originated by a process of fission from the primitive planet, that at first she revolved in an orbit close to the present surface of the earth, and that tidal friction has been the principal agent which transformed the system to its present configuration.

The theory for a long time seemed to lie open to attack on the ground that it made too great demands on time, and this has always appeared to me the greatest difficulty in the way of its acceptance. If we were still compelled to assent to the justice of Lord Kelvin's views as to the period of time which has elapsed since the earth solidified, and as to the age of the solar system, we should also have to admit that the theory of evolution under tidal influence is inapplicable to its full extent. Lord Kelvin's contributions to cosmogony have been of the first order of importance, but his arguments on these points no longer carry conviction with them. Lord Kelvin contended that the actual distribution of land and sea proves that the planet solidified at a time when the day had nearly its present length. If this were true the effects of tidal friction relate to a period antecedent to the solidifica- tion. But I have always felt convinced that the earth would adjust its ellipticity to its existing speed of rotation with close approximation. The calculations contained in Paper 9, the plasticity of even the most refractory

PREFACE. ix

forms of matter under great stresses, and the contortions of geological strata appear to me, at least, conclusive against Lord Kelvin's view.

The researches of Mr Strutt on the radio-activity of rocks prove that we cannot regard the earth simply as a cooling globe, and therefore Lord Kelvin's argument as to the age of the earth as derived from the observed gradient of temperature must be illusory. Indeed even without regard to the initial temperature of the earth acquired by means of secular contraction, it is hard to understand why the earth is not hotter inside than it is.

It seems probable that Mr Strutt may be able to obtain a rough numerical scale of geological time by means of his measurements of the radio-activity of rocks, and although he has not yet been able to formulate such a scale with any degree of accuracy, he is already confident that the periods involved must be measured in hundreds or perhaps even thousands of millions of years*. The evidence, taken at its lowest, points to a period many times as great as was admitted by Lord Kelvin for the whole history of the solar system.

Lastly the recent discovery of the colossal internal energy resident in the atom shows that it is unsafe to calculate the age of the sun merely from mechanical energy, as did Helmholtz and Kelvin. It is true that the time has not yet arrived at which we can explain exactly the manner in which the atomic energy may be available for maintaining the sun's heat, but when the great age of the earth is firmly established the insufficiency of the supply of heat to the sun by means of purely mechanical energy will prove that atomic energy does become available in some way. On the whole then it may be maintained that deficiency of time does not, according to our present state of knowledge, form a bar to the full acceptability of the theory of terrestrial evolution under the influence of tidal friction.

It is very improbable that tidal friction has been the dominant cause of change in any of the other planetary sub-systems or in the solar system itself, yet it seems to throw light on the distribution of the satellites amongst the several planets. It explains the identity of the rotation of the moon with her orbital motion, as was long ago pointed out by Kant and Laplace, and it tends to confirm the correctness of the observations according to which Venus always presents the same face to the sun. Finally it has been held by Dr See and by others to explain some of the peculiarities of the orbits of double stars.

Lord Kelvin's determination of the strain of an elastic sphere and the solution of the corresponding problem of the tides of a viscous spheroid suggested another interesting question with respect to the earth. This problem is to find the strength of the materials of which the earth must be

* Some of Mr Strutt's preliminary computations are given in Proc. Roy. Soc. A, Vol. 81, p. 272 (1908).

X PREFACE.

built so as to prevent the continents from sinking and the sea bed from rising; this question is treated in Paper 9 (p. 459). The existence of an isostatic layer, at which the hydrostatic pressure is uniform, at no great depth below the earth's surface, is now well established. This proves that I have underestimated in my paper the strength of the superficial layers necessary to prevent subsidence and elevation. The strength of granite and of other rocks is certainly barely adequate to sustain the continents in position, and Mr Hayford* seeks to avoid the difficulty by arguing that the earth is actually 'a failing structure," and that the subsidence of the continents is only prevented by the countervailing effects of the gradually increasing weight of sedimentation on the adjoining sea-beds.

In his address to the Geological Section of the British Association at Dublin (1908) Professor Joly makes an interesting suggestion which bears on this subject. He supposes that the heat generated by the radio-active materials in sediment has exercised an important influence in bringing about the elevation of mountain ranges and of the adjoining continents.

A subsidiary outcome of this same investigation was given in Vol. I. of these papers, when I attempted to determine the elastic oscillations of the superficial layers of the earth under the varying pressures of the tides and of the atmosphere. Dr Hecker may perhaps be able to verify or disprove these theoretical calculations when he makes the final reduction of his valuable observations with horizontal pendulums at Potsdam.

When the first volume of these papers was published Lord Kelvin was still alive, and I had the pleasure of receiving from him a cordial letter of thanks for my acknowledgement of the deep debt I owe him. His name also occurs frequently in the present volume, and if I dissent from some of his views, I none the less regard him as amongst the greatest of those who have tried to guess the riddle of the history of the universe.

The chronological list of my papers is repeated in this second volume, together with a column showing in which volume they are or will be reproduced.

In conclusion I wish to thank the printers and readers of the Cambridge University Press for their marvellous accuracy and care in setting up the type and in detecting some mistakes in the complicated analysis contained in these papers.

G. H. DARWIN.

October, 1908.

* Phil. Soc. Wathington, Vol. 15 (1907), p. 57.

CONTENTS

Chronological List of Papers with References to the Volumes in

which they are or probably will be contained . ... . xii Erratum in Vol. I .......... xvi

TIDAL FRICTION AND COSMOGONY.

1. On the Bodily Tides of Viscous and Semi-elastic Spheroids, and

on the Ocean Tides upon a Yielding Nucleus ... 1

[Philosophical Transactions of the Royal Society, Part i. Vol. 170 (1879), pp. 1-35.]

2. Note on Thomson's Theory of the Tides of an Elastic Sphere . 33

[Messenger of Mathematics, vm. (1879), pp. 23 26.]

3. On the Precession of a Viscous Spheroid, and on the Remote

History of the Earth 36

[Philosophical Transactions of the Royal Society, Part n. Vol. 170 (1879), pp. 447—530.]

4. Problems connected with the Tides of a Viscous Spheroid . 140

[Philosophical Transactions of the Royal Society, Part II. Vol. 170 (1879), pp. 539—593.]

5. The Determination of the Secular Effects of Tidal Friction by

a Graphical Method ........ 195

[Proceedings of the Royal Society of London, xxix. (1879), pp. 168 181.]

6. On the Secular Changes in the Elements of the Orbit of a

Satellite revolving about a Tidally Distorted Planet . . 208

[Philosophical Transactions of the Royal Society, Vol. 171 (1880), pp. 713—891.]

7. On the Analytical Expressions which give the History of a Fluid

Planet of Small Viscosity, attended by a Single Satellite . 383 [Proceedings of the Royal Society, Vol. xxx. (1880), pp. 255—278.]

8. On the Tidal Friction of a Planet attended by Several Satellites,

and on the Evolution of the Solar System .... 406

[Philosophical Transactions of the Royal Society, Vol. 172 (1881), pp. 491—535.]

9. On the Stresses caused in the Interior of the Earth by the Weight

of Continents and Mountains ....... 459

[Philosophical Transactions of the Royal Society, Vol. 173 (1882), pp. 187— 230, with which is incorporated " Note on a previous paper," Proc. Roy. Soc. Vol. 38 (1885), pp. 322—328.]

INDEX = ..„•,. . 515

CHRONOLOGICAL LIST OF PAPERS WITH REFERENCES TO THE VOLUMES IN WHICH THEY ARE OR PROBABLY WILL BE CONTAINED.

Probable volume

in collected TITLE AND REFERENCE

1875 On two applications of Peaucellier's cells. London Math. Soc. Proc., IV 6, 1875, pp. 113, 114.

1875 On some proposed forms of slide-rule. London Math. Soc. Proc., 6, IV 1875, p. 113.

1875 The mechanical description of equipotential lines. London Math. Soc. IV Proc., 6, 1875, pp. 115—117.

1875 On a mechanical representation of the second elliptic integral. Mes- IV senger of Math., 4, 1875, pp. 113-115.

1875 On maps of the World. Phil. Mag., 50, 1875, pp. 431—444. IV

1876 On graphical interpolation and integration. Brit. Assoc. Rep., 1876, IV

p. 13.

1876 On the influence of geological changes on the Earth's axis of rotation. Ill Roy. Soc. Proc., 25, 1877, pp. 328—332 ; Phil. Trans., 167, 1877, pp. 271—312.

1876 On an oversight in the Mecaniqiie Celeste, and on the internal densities III

of the planets. Astron. Soc. Month. Not., 37, 1877, pp. 77—89.

1877 A geometrical puzzle. Messenger of Math., 6, 1877, p. 87. IV

1877 A geometrical illustration of the potential of a distant centre of force IV Messenger of Math., 6, 1877, pp. 97, 98.

1877 Note on the ellipticity of the Earth's strata. Messenger of Math., 6, III 1877, pp. 109, 110.

1877 On graphical interpolation and integration. Messenger of Math., 6, IV 1877, pp. 134—136.

1877 On a theorem in spherical harmonic analysis. Messenger of Math., 6, IV 1877, pp. 165—168.

1877 On a suggested explanation of the obliquity of planets to their orbits. Ill PhiL Mag., 3, 1877, pp. 188—192.

1877 On fallible measures of variable quantities, and on the treatment of IV meteorological observations. Phil. Mag., 4, 1877, pp. 1—14.

CHRONOLOGICAL LIST OF PAPERS.

Xlll

YEAR

1878 1878

1878 1879

1879 1879 1879 1880

1880

1880 1881

1881 1881

1882

TITLE AND REFERENCE

On Professor Haughton's estimate of geological time.

Probable volume

In collected

papers

1882

Roy. Soc. Proc., Ill 27, 1878, pp. 179—183.

On the bodily tides of viscous and semi-elastic spheroids, and on the II Ocean tides on a yielding nucleus. Roy. Soc. Proc., 27, 1878, pp. 419—424; Phil. Trans., 170, 1879, pp. 1—35.

On the precession of a viscous spheroid. Brit. Assoc. Rep., 1878, omitted pp. 482—485.

On the precession of a viscous spheroid, and on the remote history of II the Earth. Roy. Soc. Proc., 28, 1879, pp. 184—194 ; Phil. Trans., 170, 1879, pp. 447—538.

Problems connected with the tides of a viscous spheroid. Roy. Soc. II Proc., 28, 1879, pp. 194—199 ; Phil. Trans., 170, 1879, pp. 539—593.

Note on Thomson's theory of the tides of an elastic sphere. Messenger II

of Math., 8, 1879, pp. 23—26. The determination of the secular effects of tidal friction by a graphical II

method. Roy. Soc. Proc., 29, 1879, pp. 168—181.

On the secular changes in the elements of the orbit of a satellite II revolving about a tidally distorted planet. Roy. Soc. Proc., 30, 1880, pp. 1—10; Phil. Trans., 171, 1880, pp. 713—891.

On the analytical expressions which give the history of a fluid planet of II small viscosity, attended by a single satellite. Roy. Soc. Proc., 30, 1880, pp. 255—278.

On the secular effects of tidal friction. Astr. Nachr., 96, 1880, omitted col. 217—222.

On the tidal friction of a planet attended by several satellites, and on II the evolution of the solar system. Roy. Soc. Proc., 31, 1881, pp. 322—325; PhiL Trans., 172, 1881, pp. 491—535.

On the stresses caused in the interior of the Earth by the weight of II continents and mountains. Phil. Trans., 173, 1882, pp. 187 230; Amer. Journ. Sci., 24, 1882, pp. 256—269.

(Together with Horace Darwin.) On an instrument for detecting and I

measuring small changes in the direction of the force of gravity. Brit. Assoc. Rep., 1881, pp. 93—126 ; Annal. Phys. Chem., Beibl. 6, 1882, pp. 59—62.

On variations in the vertical due to elasticity of the Earth's surface. I

Brit. Assoc. Rep., 1882, pp. 106—119; Phil. Mag., 14, 1882, pp. 409—427.

On the method of harmonic analysis used in deducing the numerical omitted values of the tides of long period, and on a misprint in the Tidal Report for 1872. Brit. Assoc. Rep., 1882, pp. 319—327.

A numerical estimate of the rigidity of the Earth. Brit. Assoc. Rep., I

1882, pp. 472—474 ; § 848, Thomson and Tait's Nat. Phil, second edition.

XIV CHRONOLOGICAL LIST OF PAPERS.

Probable volume

YEAR TITLE AND REFERENCE

1883 Report on the Harmonic analysis of tidal observations. Brit. Assoc. I

Rep., 1883, pp. 49—117.

1883 On the figure of equilibrium of a planet of heterogeneous density. Ill Roy. Soc. Proc., 36, pp. 158—166.

1883 On the horizontal thrust of a mass of sand. Instit. Civ. Engin. Proc., IV

71, 1883, pp. 350—378.

1884 On the formation of ripple-mark in sand. Roy. Soc. Proc., 36, 1884, IV

pp. 18—43.

1884 Second Report of the Committee, consisting of Professors G. H. Darwin omitted

and J. C. Adams, for the harmonic analysis of tidal observations. Drawn up by Professor G. H. Darwin. Brit. Assoc. Rep., 1884, pp. 33-35.

1885 Note on a previous paper. Roy. Soc. Proc., 38, pp. 322—328. ' 1 1

1885 Results of the harmonic analysis of tidal observations. (Jointly with omitted A. W. Baird.) Roy. Soc. Proc., 39, pp. 135—207.

1885 Third Report of the Committee, consisting of Professors G. H. Darwin I

and J. C. Adams, for the harmonic analysis of tidal observations. Drawn up by Professor G. H. Darwin. Brit. Assoc. Rep., 1885, pp. 35—60.

1886 Report of the Committee, consisting of Professor G. H. Darwin, I

Sir W. Thomson, and Major Baird, for preparing instructions for the practical work of tidal observation ; and Fourth Report of the Committee, consisting of Professors G. H. Darwin and J. C. Adams, for the harmonic analysis of tidal observations. Drawn up by Professor G. H. Darwin. Brit. Assoc. Rep., 1886, pp. 40—58.

1886 Presidential Address. Section A, Mathematical and Physical Science. IV Brit. Assoc. Rep., 1886, pp. 511—518.

1886 On the correction to the equilibrium theory of tides for the continents. I

i. By G. H. Darwin. IL By H. H. Turner. Roy. Soc. Proc., 40, pp. 303—315.

1886 On Jacobi's figure of equilibrium for a rotating mass of fluid. Roy. Ill Soc. Proc., 41, pp. 319—336.

1886 On the dynamical theory of the tides of long period. Roy. Soc. Proc., I

41, pp. 337—342.

1886 Article ' Tides.' (Admiralty) Manual of Scientific Inquiry. I

1887 On figures of equilibrium of rotating masses of fluid. Roy. Soc. Proc., 42, III

pp. 359—362 ; Phil. Trans., 178A, pp. 379—428.

1887 Note on Mr Davison's Paper on the straining of the Earth's crust in IV

cooling. Phil. Trans., 178A, pp. 242—249.

1888 Article ' Tides.' Encyclopaedia Britannica. Certain sections in I 1888 On the mechanical conditions of a swarm of meteorites, and on theories IV

of cosmogony. Roy. Soc. Proc., 45, pp. 3—16; Phil. Trans., 180A, pp. 1—69.

CHRONOLOGICAL LIST OF PAPERS.

TITLE AND REFERENCE

Probable volume

in collected

papers

1889 Second series of results of the harmonic analysis of tidal observations, omitted Roy. Soc. Proc., 45, pp. 556—611.

1889 Meteorites and the history of Stellar systems. Roy. Inst. Rep., Friday, omitted

Jan. 25, 1889.

1890 On the harmonic analysis of tidal observations of high and low water. I

Roy. Soc. Proc., 48, pp. 278—340.

1891 On tidal prediction. Bakerian Lecture. Roy. Soc. Proc., 49, pp. 130— I

133; Phil. Trans., 182A, pp. 159—229.

1892 On an apparatus for facilitating the reduction of tidal observations. I

Roy. Soc. Proc., 52, pp. 345—389.

18% On periodic orbits. Brit. Assoc. Rep., 1896, pp. 708, 709. omitted

1897 Periodic orbits. Acta Mathematica, 21, pp. 101 242, also (with IV

omission of certain tables of results) Mathem. Annalen, 51,

pp. 523—583. [by S. S. Hough. On certain discontinuities connected with periodic IV

orbits. Acta Math., 24 (1901), pp. 257—288.]

1899 The theory of the figure of the Earth carried to the second order of III

small quantities. Roy. Astron. Soc. Month. Not., 60, pp. 82 124.

1900 Address delivered by the President, Professor G. H. Darwin, on IV

presenting the Gold Medal of the Society to M. H. Poincare. Roy. Astron. Soc. Month. Not., 60, pp. 406—415.

1901 Ellipsoidal harmonic analysis. Roy. Soc. Proc., 68, pp. 248—252; III

Phil. Trans., 197 A, pp. 461—557.

1901 On the pear-shaped figure of equilibrium of a rotating mass of liquid. Ill Roy. Soc. Proc., 69, pp. 147, 148; Phil. Trans., 198A, pp. 301—331.

Article ' Tides.' Encyclopaedia Britannica, supplementary volumes.

1902

Certain sections in I III

1902 The stability of the pear-shaped figure of equilibrium of a rotating mass

of liquid. Roy. Soc. Proc., 71, pp. 178—183 ; Phil. Trans., 200A, pp. 251—314.

1903 On the integrals of the squares of ellipsoidal surface harmonic functions. Ill

Roy. Soc. Proc., 72, p. 492; Phil. Trans., 203A, pp. 111—137.

1903 The approximate determination of the form of Maclaurin's spheroid. Ill

Trans. Amer. Math. Soc., 4, pp. 113 133. 1903 The Eulerian nutation of the Earth's axis. Bull. Acad. Roy. de IV

Belgique (Sciences), pp. 147—161.

1905 The analogy between Lesage's theory of gravitation and the repulsion IV of light. Roy. Soc. Proc., 76A, pp. 387—410.

1905 Address by Professor G. H. Darwin, President. Brit. Assoc. Rep., IV

1905, pp. 3—32.

1906 On the figure and stability of a liquid satellite. Roy. Soc. Proc., 77A, III

pp. 422—425 ; Phil. Trans., 206A, pp. 161—248.

CHRONOLOGICAL LIST OF PAPERS.

Probable vol

YEAR TITLE AND REFERENCE

1908 Tidal observations of the ' Discovery.' National Antarctic Expedition I

1901—4, Physical Observations, pp. 1—12.

1908 Discussion of the tidal observations of the ' Scotia.' National Antarctic omitted Expedition 1901 4, Physical Observations, p. 16.

1908 Further consideration of the figure and stability of the pear-shaped figure III of a rotating mass of liquid. Roy. Soc. Proc., 80A, pp. 166—7 ; Phil. Trans., 208A, pp. 1—19.

1908 Further note on Maclaurin's Ellipsoid. Trans. Amer. Math. Soc., 9, III pp. 34—38.

1908 (Together with S. S. Hough.) Article 'Bewegung der Hydrosphare' IV (The Tides). Encyklopadie der mathematischen Wissenschaften, VL 1, 6. 83 pp.

Unpublished Article ' Tides.' Encyclopaedia Britannica, new edition to be published hereafter (by permission of the proprietors).

Certain sections in I

ERRATUM IN VOL. I.

p. 275, equation (26), line 9 from foot of page, should read

1.

ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS*

[Philosophical Transactions of the Royal Society, Part I. Vol. 170 (1879),

pp. 1—35.]

IN a well-known investigation Sir William Thomson has discussed the problem of the bodily tides of a homogeneous elastic sphere, and has drawn therefrom very important conclusions as to the great rigidity of the earth -J-.

Now it appears improbable that the earth should be perfectly elastic ; for the contortions of geological strata show that the matter constituting the earth is somewhat plastic, at least near the surface. We know also that even the most refractory metals can be made to flow under the action of sufficiently great forces.

Although Sir W. Thomson's investigation has gone far to overthrow the old idea of a semi-fluid interior to the earth, yet geologists are so strongly impressed by the fact that enormous masses of rock are being, and have been, poured out of volcanic vents in the earth's surface, that the belief is not yet extinct that we live on a thin shell over a sea of molten lava. Under these circumstances it appears to be of interest to investigate the consequences which would arise from the supposition that the matter constituting the earth is of a viscous or imperfectly elastic nature ; for if the interior is

* [Since the date of this paper important contributions to the subject have been made by Professor Horace Lamb in his papers on " The Oscillations of a Viscous Spheroid," Proc. Lond. Math. Soc., Vol. xm. (1881-2), p. 51; "On the Vibrations of an Elastic Sphere," ibid., p. 189, and " On the Vibrations of a Spherical Shell," ibid., Vol. xiv. (1882-3), p. 50. See also a paper by T. J. Bromwich, Proc. Lond. Math. Soc., Vol. xxx. (1898-9), p. 98.]

t Sir William states that M. Lame had treated the subject at an earlier date, but in an entirely different manner. I am not aware, however, that M. Lame had fully discussed the subject in its physical aspect.

D. II. 1

2 INTRODUCTION. [1

constituted in this way, then the solid crust, unless very thick, cannot possess rigidity enough to repress the tidal surgings, and these hypotheses must give results fairly conformable to the reality. The hypothesis of imperfect elasticity will be principally interesting as showing how far Sir W. Thomson's results are modified by the supposition that the elasticity breaks down under continued stress.

In this paper, then, I follow out these hypotheses, and it will be seen that the results are fully as hostile to the idea of any great mobility of the interior of the earth as is that of Sir W. Thomson.

The only terrestrial evidence of the existence of a bodily tide in the earth would be that the ocean tides would be less in height than is indicated by theory. The subject of this paper is therefore intimately connected with the theory of the ocean tides.

In the first part the equilibrium tide-theory is applied to estimate the reduction and alteration of phase of ocean tides as due to bodily tides, but that theory is acknowledged on all hands to be quite fallacious in its explanation of tides of short period.

In the second part of this paper, therefore, I have considered the dynamical theory of tides in an equatorial canal running round a tidally-distorted nucleus, and the results are almost the same as those given by the equi- librium theory.

The first two sections of the paper are occupied with the adaptation of Sir W. Thomson's work * to the present hypotheses ; as, of course, it was impossible to reproduce the whole of his argument, I fear that the investigation will only be intelligible to those who are either already acquainted with that work, or who are willing to accept my quotations therefrom as established.

As some readers may like to know the results of this inquiry without going into the mathematics by which they are established, I have given in Part III. a summary of the whole, and have as far as possible relegated to that part of the paper the comments and conclusions to be drawn. I have tried, however, to give so much explanation in the body of the paper as will make it clear whither the argument is tending.

The case of pure viscosity is considered first, because the analysis is somewhat simpler, and because the results will afterwards admit of an easy extension to the case of elastico-viscosity.

* His paper will be found in Phil. Trant., 1863, p. 573, and §§ 733—737 and 834—846 of Thomson and Tail's Natural Philosophy.

1879]

THE EQUATIONS OF MOTION.

THE BODILY TIDES OF Viscous AND ELASTICO-VISCOUS SPHEROIDS.

1. Analogy between the flow of a viscous body and the strain of an elastic one.

The general equations of flow of a viscous fluid, when the effects of inertia are 'neglected, are

(1)

-

where x, y, z are the rectangular coordinates of a point of the fluid ; a, j3, 7 are the component velocities parallel to the axes ; p is the mean of the three pressures across planes perpendicular to the three axes respectively ; X, Y, Z are the component forces acting on the fluid, estimated per unit volume; v is the coefficient of viscosity; and V2 is the Laplacian operation

d2 d* d*

Besides these we have the equation of continuity -r- + -, h -r^ = 0.

Also if P, Q, R, S, T, U are the normal and tangential stresses estimated in the usual way across three planes perpendicular to the axes

, dy) '

Now in an elastic solid, if a, (3, 7 be the displacements, m-fyi the coefficient of dilatation, and n that of rigidity, and if B = -r- + -5 h -T- ; the equations of equilibrium are

<tt ,

' ~z "p ¥1

dy

.(3)

* Thomson and Tail's Natural Philosophy, § 698, eq. (7) and (8).

1—2

4 COMPARISON WITH THE CASE OF AN ELASTIC SOLID. [1

Also

dx ' dy ' dz

(4)

and S, T, U have the same forms as in (2), with n written instead of v.

Therefore if we put -p = $ (P + Q + R), we have p = - (m - £n) B, so that (3) may be written

- —^ * + ttV'a + X = 0, &c.. &c.

Also P = - --— " p + 2nd,a, Q = &c., R = &c.

in %n cuv

Now if we suppose the clastic solid to be incompressible, so that m is infinitely large compared to n, then it is clear that the equations of equi- librium of the incompressible elastic solid assume exactly the same form as those of flow of the viscous fluid, n merely taking the place of v.

Thus every problem in the equilibrium of an incompressible elastic solid has its counterpart in a problem touching the state of flow of an incom- pressible viscous fluid, when the effects of inertia are neglected ; and the solution of the one may be made applicable to the other by merely reading for " displacements " " velocities," and for the coefficient of " rigidity " that of " viscosity."

2. A sphere under influence of bodily force. Sir W. Thomson has solved the following problem :

To find the displacement of every point of the substance of an elastic sphere exposed to no surface traction, but deformed infinitesimally by an equilibrating system of forces acting bodily through the interior.

If for " displacement " we read velocity, and for " elastic " viscous, we have the corresponding problem with respect to a viscous fluid, and mutatis mutandis the solution is the same.

But we cannot find the tides of a viscous sphere by merely making the equilibrating system of forces equal to the tide-generating influence of the sun or moon, because the substance of the sphere must be supposed to have the power of gravitation.

For suppose that at any time the equation to the free surface of the earth

00

(as the viscous sphere may be called for brevity) is r = a + 2o-t-, where o^ is

1879] INTRODUCTION OF GRAVITATION. 5

a surface harmonic. Then the matter, positive or negative, filling the space represented by So-; exercises an attraction on every point of the interior; and this attraction, together with that of a homogeneous sphere of radius a, must be added to the tide-generating influence to form the whole force in the interior of the sphere. Also it is a spheroid, and no longer a true sphere with which we have to deal. If, however, we cut a true sphere of radius a out of the spheroid (leaving out 2cr;), then by a proper choice of surface actions, the tidal problem may be reduced to finding the state of flow in a true sphere under the action of (i) an external tide-generating influence, (ii) the attraction of the true sphere, and of the positive and negative matter filling the space So-;, but (iii) subject to certain surface forces.

Since (i) and (ii) together constitute a bodily force, the problem only differs from that of Sir W. Thomson in the fact that there are forces acting on the surface of the sphere.

Now as we are only going to consider small deviations from sphericity, these surface actions will be of small amount, and an approximation will be permissible.

It is clear that rigorously there is tangential action* between the layer of matter So-; and the true sphere, but by far the larger part of the action is normal, and is simply the weight (either positive or negative) of the matter which lies above or below any point on the surface of the true sphere.

Thus, in order to reduce the earth to sphericity, the appropriate surface action is a normal traction equal to $wS<r;, where g is gravity at the surface, and wjs_the_m.ass per unit volume of the matter constituting the earth.

In order to show what alteration this normal surface traction will make in Sir W. Thomson's solution, I must now give a short account of his method of attacking the problem.

He first shows that, where there is a potential function, the solution of the problem may be subdivided, and that the complete values of a, y3, 7 consist of the sums of two parts which are to be found in different ways. The first part consists of any values of a, /3, 7, which satisfy the equations throughout the sphere, without reference to surface conditions. As far as regards the second part, the bodily force is deemed to be non-existent and is replaced by certain surface actions, so calculated as to counteract the surface actions which correspond to the values of a, ft, 7 found in the first part of the solution. Thus the first part satisfies the condition that there is a

* I shall consider some of the effects of this tangential action in a future paper, viz.: " Problems connected with the Tides of a Viscous Spheroid," read before the Royal Society on December 19th, 1878. [Paper 4.]

6 LORD KELVIN'S SOLUTION FOR AN ELASTIC SPHERE. [1

bodily force, and the second adds the condition that the surface forces are zero. The first part of the solution is easily found, and for the second part Sir W. Thomson discusses the case of an elastic sphere under the action of any surface tractions, but without any bodily force acting on it. The component surface tractions parallel to the three axes, in this problem, are supposed to be expanded in a series of surface harmonics ; and the harmonic terms of any order are shown to have an effect on the displacements inde- pendent of those of every other order. Thus it is only necessary to consider the typical component surface tractions Af, Bf, C< of the order i.

He proves that (for an incompressible elastic solid for which m is infinite) this one surface traction A,-, B,, C,- produces a displacement throughout the sphere given by

__- + _

-1 ]2(2t» + i) dx t-

+ 2t

with symmetrical expressions for /8 and 7; where functions defined b

functions defined by

.(6)

<A"-''-) + I, <B"-'-'> + 5 <

In the case considered by Sir W. Thomson of an elastic sphere deformed by bodily stress and subject to no surface action, we have to substitute in (5) and (6) only those surface actions which are equal and opposite to the surface forces corresponding to the first part of the solution f ; but in the case which we now wish to consider, we must add to these latter the com- ponents of the normal traction