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SCIENTIFIC PAPERS 



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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 V 
SUPPLEMENTARY VOLUME 

CONTAINING 

BIOGRAPHICAL MEMOIRS BY SIR FRANCIS DARWIN 

AND PROFESSOR E. W. BROWN, 
LECTURES ON HILL'S LUNAR THEORY, ETC. 



EDITED BY 
F. J. M. STRATTON, M.A., AND J. JACKSON, M.A., B.Sc. 



Cambridge : 

at the University Press 
1916 



Cambringr : 

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



PREFACE 

T)EFORE his death Sir George Darwin expressed the view that his 
lectures on Hill's Lunar Theory should be published. He made no 
claim to any originality in them, but he believed that a simple presentation 
of Hill's method, in which the analysis was cut short while the fundamental 
principles of the method were shewn, might be acceptable to students of 
astronomy. In this belief we heartily agree. The lectures might also 
with advantage engage the attention of other students of mathematics 
who have not the time to enter into a completely elaborated lunar theory. 
They explain the essential peculiarities of Hill's work and the method of 
approximation used by him in the discussion of an actual problem of 
nature of great interest. It is hoped that sufficient detail has been given 
to reveal completely the underlying principles, and at the same time not 
be too tedious for verification by the reader. 

During the later years of his life Sir George Darwin collected his 
principal works into four volumes. It has been considered desirable to 
publish these lectures together with a few miscellaneous articles in a fifth 
volume of his works. Only one series of lectures is here given, although 
he lectured on a great variety of subjects connected with Dynamics, Cos- 
mogony, Geodesy, Tides, Theories of Gravitation, etc. The substance of 
many of these is to be found in his scientific papers published in the four 
earlier volumes. The way in which in his lectures he attacked problems 
of great complexity by means of simple analytical methods is well illustrated 
in the series chosen for publication. 

Two addresses are included in this volume. The one gives a view of 
the mathematical school at Cambridge about 1880, the other deals with 
the mathematical outlook of 1912. 



vi PREFACE 

The previous volumes contain all the scientific papers by Sir George 
Darwin published before 1910 which he wished to see reproduced. They 
do not include a large number of scientific reports on geodesy, the tides and 
other subjects which had involved a great deal of labour. Although the 
reports were of great value for the advancement and encouragement of 
science, he did not think it desirable to reprint them. We have not 
ventured to depart from his own considered decision ; the collected lists 
at the beginning of these volumes give the necessary references for such 
papers as have been omitted. We are indebted to the Royal Astronomical 
Society for permission to complete Sir George Darwin's work on Periodic 
Orbits by reproducing his last published paper. 

The opportunity has been taken of securing biographical memoirs of 
Darwin from two different points of view. His brother, Sir Francis Darwin, 
writes of his life apart from his scientific work, while Professor E. W. Brown, 
of Yale University, writes of Darwin the astronomer, mathematician and 
teacher. 

F. J. M. S. 
J. J. 

GREENWICH, 

6 December 1915. 



CONTENTS 

PORTRAIT OP SIR GEORGE DARWIN ...... Frontispiece 

PAGE 

MEMOIR OF SIR GEORGE DARWIN BY HIS BROTHER SIR FRANCIS DARWIN ix 

THE SCIENTIFIC WORK OP SIR GEORGE DARWIN BY PROFESSOR E. W. 

BROWN ............ xxxiv 

INAUGURAL LECTURE (DELIVERED AT CAMBRIDGE, IN 1883, ON ELECTION TO 

THE PLUMIAN PROFESSORSHIP) ... ..... 1 

INTRODUCTION TO DYNAMICAL ASTRONOMY ....... 9 

LECTURES ON HILL'S LUNAR THEORY ....... 16 

1. Introduction ........... 16 

2. Differential Equations of Motion and Jacobi's Integral . 17 

3. The Variational Curve ........ 22 

4. Differentia] Equations for Small Displacements from the 

Variational Curve . . . . . . . . 26 

5. Transformation of the Equations in 4 . . . . 29 

6. Integration of an important type of Differential Equation . 36 

7. Integration of the Equation for Sp ..... 39 

8. Introduction of the Third Coordinate ..... 43 

9. Results obtained ......... 45 

10. General Equations of Motion and their solution ... 46 

11. Compilation of Results ........ 52 

Note 1. On the Infinite Determinant of 5 ..... 53 

Note 2. On the periodicity of the integrals of the equation 



where = + ! cos 2r + 2 cos 4r + . . . . . . . 55 

ON LIBRATING PLANETS AND ON A NEW FAMILY OP PERIODIC ORBITS . 59 
[Monthly Notices of the Royal Astronomical Society, Vol. 72 (1912), pp. 642 

658.] 

ADDRESS TO THE INTERNATIONAL CONGRESS OF MATHEMATICIANS AT 

CAMBRIDGE IN 1912 .......... 76 

INDEX ........ 80 



MEMOIR OF SIR GEORGE DARWIN 

BY 
HIS BROTHER SIR FRANCIS DARWIN 

George Howard, the fifth 1 child of Charles and Emma Darwin, was 
born at Down July 9th, 1845. Why he was christened 2 George, I cannot 
say. It was one of the facts on which we founded a theory that our parents 
lost their presence of mind at the font and gave us names for which there 
was neither the excuse of tradition nor of preference on their own part. 
His second name, however, commemorates his great-grandmother, Mary 
Howard, the first wife of Erasmus Darwin. It seems possible that George's 
ill-health and that of his father were inherited from the Howards. This at 
any rate was Francis Galton's view, who held that his own excellent health 
was a heritage from Erasmus Darwin's second wife. George's second name, 
Howard, has a certain appropriateness in his case for he was the genea- 
logist and herald of our family, and it is through Mary Howard that the 
Darwins can, by an excessively devious route, claim descent from certain 
eminent people, e.g. John of Gaunt. This is shown in the pedigrees which 
George wrote out, and in the elaborate genealogical tree published in Pro- 
fessor Pearson's Life of Francis Galton. George's parents had moved to 
Down in September 1842, and he was born to those quiet surroundings of 
which Charles Darwin wrote "My life goes on like clock-work and I am 
fixed on the spot where I shall end it 3 ." It would have been difficult to 
find a more retired place so near London. In 1842 a coach drive of some 
twenty miles was the only means of access to Down ; and even now that 
railways have crept closer to it, it is singularly out of the world, with little 
to suggest the neighbourhood of London, unless it be the dull haze of smoke 
that sometimes clouds the sky. In 1842 such a village, communicating with 
the main lines of traffic only by stony tortuous lanes, may well have been 
enabled to retain something of its primitive character. Nor is it hard to 
believe in the smugglers and their strings of pack-horses making their way 
up from the lawless old villages of the Weald, of which the memory then 
still lingered. 

1 The third of those who survived childhood. 

2 At Maer, the Staffordshire home of his mother. 

3 Life and Letters of Charles Darwin, vol. i. p. 318. 



X MEMOIR OF SIR GEORGE DARWIN 

George retained throughout life his deep love for Down. For the lawn 
with its bright strip of flowers; and for the row of big lime trees that 
bordered it. For the two yew trees between which we children had our 
swing, and for many another characteristic which had become as dear and 
as familiar to him as a human face. He retained his youthful love of 
the "Sand-walk," a little wood far enough from the house to have for us 
a romantic character of its own. It was here that our father took his daily 
exercise, and it has ever been haunted for us by the sound of his heavy 
walking stick striking the ground as he walked. 

George loved the country round l)own, and all its dry chalky valleys 
of ploughed land with "shaws," i.e. broad straggling hedges on their 
crests, bordered by strips of flowery turf. The country is traversed by 
many foot-paths, these George knew well and used skilfully in our walks, 
in which he was generally the leader. His love for the house and the 
neighbourhood was I think entangled with his deepest feelings. In later 
years, his children came with their parents to Down, and they vividly 
remember his excited happiness, and how he enjoyed showing them his 
ancient haunts. 

In this retired region we lived, as children, a singularly quiet life 
practically without friends and dependent on our brothers and sisters for 
companionship. George's earliest recollection was of drumming with his 
.spoon and fork on the nursery table because dinner was late, while a 
barrel-organ played outside. Other memories were less personal, for instance 
the firing of guns when Sebastopol was supposed to have been taken. His 
diary of 1852 shows a characteristic interest in current events and in the 
picturesqueness of Natural History: 

The Duke is dead. Dodos are out of the world. 

He perhaps carried rather far the good habit of re-reading one's favourite 
authors. He told his children that for a year or so he read through every 
day the story of Jack the Giant Killer, in a little chap-book with coloured 
pictures. He early showed signs of the energy which marked his character 
in later life. I am glad to remember that I became his companion and 
willing slave. There was much playing at soldiers, and I have a clear 
remembrance of our marching with toy guns and knapsacks across the 
field to the Sand-walk. There we made our bivouac with gingerbread, 
and milk, warmed (and generally smoked) over a " touch-wood " fire. I was 
a private while George was a sergeant, and it was part of my duty to stand 
.sentry at the far end of the kitchen-garden until released by a bugle-call 
from the lawn. I have a vague remembrance of presenting my fixed bayonet 
at my father to ward off a kiss which seemed to me inconsistent with my 
military duties. Our imaginary names and heights were written up on the 
wall of the cloak-room. .George, with romantic exactitude, made a small 



BY SIR FRANCIS DARWIN XI 

foot rule of such a size that he could conscientiously record his height as 
6 feet and mine as slightly less, in accordance with my age and station. 

Under my father's instruction George made spears with leaded heads 
which he hurled with remarkable skill by means of an Australian throwing 
stick. I used to skulk behind the big lime trees on the lawn in the character 
of victim, and I still remember the look of the spears flying through the air 
with a certain venomous waggle. Indoors, too, we threw at each other lead- 
weighted javelins which we received on beautiful shields made by the village 
carpenter and decorated with coats of arms. 

Heraldry was a serious pursuit of his for many years, and the London 
Library copies of Guillim and Edmonson 1 were generally at Down. He 
retained a love of the science through life, and his copy of Percy's Reliques 
is decorated with coats of arms admirably drawn and painted. In later life 
he showed a power of neat and accurate draughtsmanship, and some of the 
illustrations in his father's books, e.g. in Climbing Plants, are by his hand. 

His early education was given by governesses : but the boys of the family 
used to ride twice or thrice a week to be instructed in Latin by Mr Reed, the 
Rector of Hayes the kindest of teachers. For myself, I chiefly remember 
the cake we used to have at 11 o'clock and the occasional diversion of looking 
at the pictures in the great Dutch bible. George must have impressed his 
parents with his solidity and self-reliance, since he was more than once 
allowed to undertake alone the 20 mile ride to the hpuse of a relative at 
Hartfield in Sussex. For a boy of ten to bait his pony and order his 
luncheon at the Edenbridge inn was probably more alarming than the 
rest of the adventure. There is indeed a touch of David Copperfield in 
his recollections, as preserved in family tradition. " The waiter always said, 
' What will you have for lunch, Sir ? ' to which he replied, ' What is there ? ' 
and the waiter said, ' Eggs and bacon ' ; and, though he hated bacon more 
than anything else in the world, he felt obliged to have it." 

On August 16th, 1856, George was sent to school. Our elder brother, 
William, was at Rugby, and his parents felt his long absences from home 
such an evil that they fixed on the Clapham Grammar School for their 
younger sons. Besides its nearness to Down, Clapham had the merit of 
giving more mathematics and science than could then be found in public 
schools. It was kept by the Rev. Charles Pritchard 2 , a man of strong 
character and with a gift for teaching mathematics by which George un- 
doubtedly profited. In (I think) 1861 Pritchard left Clapham and was 
succeeded by the Rev. Alfred Wrigley, a man of kindly mood but without 
the force or vigour of Pritchard. As a mathematical instructor I imagine 

1 Guillim, John, A display of heraldry, 6th ed., folio 1724. Edmonsou, J., A complete body 
of heraldry, folio 1780. 

- Afterwards Savilian Professor of Astronomy at Oxford. Born 1808, died 1893. 

b 2 



Xll MEMOIR OF SIR GEORGE DARWIN 

Wrigley was a good drill-master rather than an inspiring teacher. Under 
him the place degenerated to some extent; it no longer sent so many boys 
to the Universities, and became more like a " crammer's " and less like a public 
school. My own recollections of George at Clapham are coloured by an abiding 
gratitude for his kindly protection of me as a shrinking and very unhappy 
" new boy " in 1860. 

George records in his diary that in 1863 he tried in vain for a Minor 
Scholarship at St John's College, Cambridge, and again failed to get one at 
Trinity in 1864, though he became a Foundation Scholar in 1866. These 
facts suggested to me that his capacity as a mathematician was the result of 
slow growth. I accordingly applied to Lord Moulton, who was kind enough 
to give me his impressions : 

My memories of your brother during his undergraduate career 
correspond closely to your suggestion that his mathematical power 
developed somewhat slowly and late. Throughout most if not the 
whole of his undergraduate years he was in the same class as myself 
and Christie, the ex-Astronomer Royal, at Routh's 1 . We all recognised 
him as one who was certain of being high in the Tripos, but he did not 
display any of that colossal power of work and taking infinite trouble 
that characterised him afterwards. On the contrary, he treated his 
work rather jauntily. At that time his health was excellent and he 
took his studies lightly so that they did not interfere with his enjoy- 
ment of other things 2 . I remember that as the time of the examination 
came near I used to tell him that he was unfairly handicapped in being 
in such robust health and such excellent spirits. 

Even when he had taken his degree I do not think he realised his 
innate mathematical power.... It has been a standing wonder to me that 
he developed the patience for making the laborious numerical calcu- 
lations on which so much of his most original work was necessarily 
based. He certainly showed no tendency in that direction during his 
undergraduate years. Indeed he told me more than once in later life 
that he detested Arithmetic and that these calculations were as tedious 
and painful to him as they would have been to any other man, but that 
he realised that they must be done and that it was impossible to train 
anyone else to do them. 

As a Freshman he " kept " (i.e. lived) in A 6, the staircase at the N.W. 
corner of the New Court, afterwards moving to F 3 in the Old Court, 
pleasant rooms entered by a spiral staircase on the right of the Great Gate. 
Below him, in the ground floor room, now used as the College offices, lived 
Mr Colvill, who remained a faithful but rarely seen friend as long as George 
lived. 

Lord Moulton, who, as we have seen, was a fellow pupil of George's at 
Routh's, was held even as a Freshman to be an assured Senior Wrangler, 

1 The late Mr Routh was the most celebrated Mathematical " Coach " of his day. 

2 Compare Charles Darwin's words: "George has not slaved himself, which makes his 
success the more satisfactory" (More Letters of C. Darwin, vol. n. p. 287). 



BY SIR FRANCIS DARWIN Xlll 

a prophecy that he easily made good. The second place was held by George, 
and was a much more glorious position than he had dared to hope for. In 
those days the examiners read out the list in the Senate House at an early 
hour, 8 a.m. I think. George remained in bed and sent me to bring the 
news. I remember charging out through the crowd the moment the magni- 
ficent " Darwin of Trinity " had followed the expected " Moulton of St John's." 
I have a general impression of a cheerful crowd sitting on George's bed and 
literally almost smothering him with congratulations. He received the 
following characteristic letter from his father 1 : 

DOWN, Jan. 24th [1868]. 
My dear old fellow, 

I am so pleased. I congratulate you with all my heart and 
soul. I always said from your early days that such energy, per- 
severance and talent as yours would be sure to succeed : but I never 
expected such brilliant success as this. Again and again I congratulate 
you. But you have made my hand tremble so I can hardly write. The 
telegram came here at eleven. We have written to W. and the boys. 
God bless you, my dear old fellow may your life so continue. 

Your affectionate Father, 

CH. DARWIN. 

In those days the Tripos examination was held in the winter, and the 
successful candidates got their degrees early in the Lent Term : George 
records in his diary that he took his B.A. on January 25th, 1868 : also 
that he won the second of the two Smith's Prizes, the first being the 
natural heritage of the Senior Wrangler. There is little to record in this 
year. He had a pleasant time in the summer coaching Clement Bunbury, 
the nephew of Sir Charles, at his beautiful place Barton Hall in Suffolk. 
In the autumn he was elected a Fellow of Trinity, as he records, " with 
Galabin, young Niven, Clifford, [Sir Frederick] Pollock, and [Sir Sidney] 
Colvin." W. K. Clifford was the well-known brilliant mathematician who 
died comparatively early. 

Chief among his Cambridge friends were the brothers Arthur, Gerald 
and Frank Balfour. The last-named was killed, aged 31, in a climbing 
accident in 1882 on the Aiguille Blanche near Courmayeur. He was 
remarkable both for his scientific work and for his striking and most lovable 
personality. George's affection for him never faded. Madame Raverat remem- 
bers her father (not long before his death) saying with emotion, " I dreamed 
Frank Balfour was alive." I imagine that tennis was the means of bringing 
George into contact with Mr Arthur Balfour. What began in this chance 
way grew into an enduring friendship, and George's diary shows how much 
kindness and hospitality he received from Mr Balfour. George had also the 

1 Emma Darwin, A Century of Family Letters, vol. n. p. 186. 



XIV MEMOIR OF SIR GEORGE DARWIN 

advantage of knowing Lord Rayleigh at Cambridge, and retained his friend- 
ship through his life. 

In the spring of 1869 he was in Paris for two months working at French. 
His teacher used to make him write original compositions, and George gained 
a reputation for humour by giving French versions of all the old Joe Millers 
and ancient stories he could remember. 

It was his intention to make the Bar his profession 1 , and in October 1869 
we find him reading with Mr Tatham, in 1870 and 1872 with the late 
Mr Montague Crackenthorpe (then Cookson). Again, in November 1871, -he 
was a pupil of Mr W. G. Harrison. The most valued result of his legal work 
was the friendship of Mr and Mrs Crackenthorpe, which he retained throughout 
his life. During these years we find the first indications of the circumstances 
which forced him to give up a legal career namely, his failing health and 
his growing inclination towards science 2 . Thus in the summer of 1869, when 
we were all at Caerdeon in the Barmouth valley, he writes that he " fell ill " : 
and again in the winter of 1871. His health deteriorated markedly during 
1872 and 1873. In the former year he went to Malvern and to Homburg 
without deriving any advantage. I have an impression that he did not 
expect to survive these attacks ; but I cannot say at what date he made this 
forecast of an early death. In January 1873 he tried Cannes: and "came 
back very ill." It was in the spring of this year that he first consulted Dr 
(afterwards Sir Andrew) Clark, from whom he received the kindest care. 
George suffered from digestive troubles, sickness and general discomfort and 
weakness. Dr Clark's care probably did what was possible to make life more 
bearable, and as time went on his health gradually improved. In 1894 he 
consulted the late Dr Eccles, and by means of the rest-cure, then something 
of a novelty, his weight increased from 9 stone to 9 stone 11 pounds. I gain 
the impression that this treatment produced a permanent improvement, 
although his health remained a serious handicap throughout his life. 

Meanwhile he had determined on giving up the Bar, and settled, in 
October 1873, when he was 28 years old, at Trinity in Nevile's Court next 
the Library (G 4). His diary continues to contain records of ill-health and 
of various holidays in search of improvement. Thus in 1873 we read " Very 
bad during January. Went to Cannes and stayed till the end of April." Again 
in 187 4, "February to July very ill." In spite of unwelmess he began in 1872 3 
to write on various subjects. He sent to Macmillaris Magazine 3 an enter- 
taining article, "Development in Dress," where the various survivals in modern 

1 He was called in 1874 but did not practise. 

2 As a boy he had energetically collected Lepidoptera during the years 1858 61, but the first 
vague indications of a leaning towards physical science may perhaps be found in his joining the 
Sicilian eclipse expedition, Dec. 1870 Jan. 1871. It appears from Nature, Dec. 1, 1870, that 
George was told off to make sketches of the Corona. 

3 Macmillan's Magazine, 1872, vol. xxvi. pp. 410 416. 



BY SIR FRANCIS DARWIN XV 

costume were recorded and discussed from the standpoint of evolution. In 
1873 he wrote " On beneficial restriction to liberty of marriage 1 ," a eugenic 
article for which he was attacked with gross unfairness and bitterness by the 
late St George Mivart. He was defended by Huxley, and Charles Darwin 
formally ceased all intercourse with Mivart. We find mention of a " Globe 
Paper for the British Association " in 1873. And in the following year he 
read a contribution on "Probable Error" to the Mathematical Society 2 on 
which he writes in his diary, " found it was old." Besides another paper in the 
Messenger of Mathematics, he reviewed " Whitney on Language 3 ," and wrote 
a " defence of Jevons " which I have not been able to trace. In 1875 he 
was at work on the " flow of pitch," on an " equipotential tracer," on slide 
rules, and sent a paper on " Cousin Marriages " to the Statistical Society 4 . It 
is not my province to deal with these papers ; they are here of interest as 
showing his activity of mind and his varied interests, features in character 
which were notable throughout his life. 

The most interesting entry in his diary for 1875 is " Paper on Equi- 
potentials much approved by Sir W. Thomson." This is the first notice of an 
association of primary importance in George's scientific career. Then came 
his memoir " On the influence of geological changes in the earth's axis of 
rotation." Lord Kelvin was one of the referees appointed by the Council of 
the Royal Society to report on this paper, which was published in the Philo- 
sophical Transactions in 1877. 

In his diary, November 1878, George records "paper on tides ordered to 
be printed." This refers to his work " On the bodily tides of viscous and 
semi-elastic spheroids, etc.," published in the Phil. Trails, in 1879. It was in 
regard to this paper that his father wrote to George on October 29th, 1878 5 : 

My dear old George, 

I have been quite delighted with your letter and read it all 
with eagerness. You were very good to write it. All of us are 
delighted, for considering what a man Sir William Thomson is, it is 
most grand that you should have staggered him so quickly, and that he 
should speak of your 'discovery, etc.'... Hurrah for the bowels of the 
earth and their viscosity and for the moon and for the Heavenly bodies 
and for my son George (F.R.S. very soon)... 6 . 

The bond of pupil and master between George Darwin and Lord Kelvin, 
originating in the years 1877 8, was to be a permanent one, and developed 

1 Contemporary Review, 1873, vol. xxu. pp. 412 426. 

2 Not published. 

3 Contemporary Review, 1874, vol. xxiv. pp. 894904. 

* Journal of the Statistical Society, 1875, vol. xxxvni. pt 2, pp. 153 182, also pp. 183 184, 
and pp. 344348. 

5 Probably he heard informally at the end of October what was not formally determined till 
November. 

6 Emma Darwin, A Century of Family Letters, 1915, vol. n. p. 233. 



XVI MEMOIR OF SIR GEORGE DARWIN 

not merely into scientific co-operation but into a close friendship. Sir Joseph 
Larmor has recorded 1 that George's "tribute to Lord Kelvin, to whom he 
dedicated volume I of his Collected Papers 2 ... gave lively pleasure to his 
master and colleague." His words were: 

Early in my scientific career it was my good fortune to be brought 
into close personal relationship with Lord Kelvin. Many visits to Glas- 
gow and to Largs have brought me to look up to him as my master, and 
I cannot find words to express how much I owe to his friendship and to 
his inspiration. 

During these years there is evidence that he continued to enjoy the 
friendship of Lord Rayleigh and of Mr Balfour. We find in his diary 
records of visits to Terling and to Whittingeharne, or of luncheons at 
Mr Balfour's house in Carlton Gardens for which George's scientific com- 
mittee work in London gave frequent opportunity. In the same way we 
find many records of visits to Francis Galton, with whom he was united alike 
by kinship and affection. 

Few people indeed can have taken more pains to cultivate friendship 
than did George. This trait was the product of his affectionate and emi- 
nently sociable nature and of the energy and activity which were his chief 
characteristics. In earlier life he travelled a good deal in search of health 3 , 
and in after years he attended numerous congresses as a representative 
of scientific bodies. He thus had unusual opportunities of making the 
acquaintance of men. of other nationalities, and some of his warmest friend- 
ships were with foreigners. In passing through Paris he rarely failed to visit 
M. and Mme d'Estournelles and " the d'Abbadies." It was in Algiers in 1878 
and 1879 that he cemented his friendship with the late J. F. MacLennan, 
author of Primitive Marriage ; and in 1880 he was at Davos with the same 
friends. In 1881 he went to Madeira, where he received much kindness from 
the Blandy family doubtless through the recommendation of Lady Kelvin. 

Cambridge. 

We have seen that George was elected a Fellow of Trinity in October 
1868, and that five years later (Oct. 1873) he began his second lease of 
a Cambridge existence. There is at first little to record : he held at this 
time no official position, and when his Fellowship expired he continued to 
live in College busy with his research work and laying down the earlier tiers 

1 Nature, Dec. 12, 1912. 

2 It was in 1907 that the Syndics of the Cambridge University Press asked George to prepare 
a reprint of his scientific papers, which the present volume brings to an end. George was 
deeply gratified at an honour that placed him in the same class as Lord Kelvin, Stokes, Cayley, 
Adams, Clerk Maxwell, Lord Eayleigh and other men of distinction. 

3 Thus in 1872 he was in Homburg, 1873 in Cannes, 1874 in Holland, Belgium, Switzerland 
and Malta, 1876 in Italy and Sicily. 



BY SIR FRANCIS DARWIN xvii 

of the monumental series of papers in the present volumes. This soon led to 
his being proposed (in Nov. 1877) for the Royal Society, and elected in June 
1879. The principal event in this stage of his Cambridge life was his 
election 1 in 1883 as Plumian Professor of Astronomy and Experimental 
Philosophy. His predecessor in the Chair was Professor Challis, who had 
held office since 1836, and is now chiefly remembered in connection with 
Adams and the planet Neptune. The professorship is not necessarily con- 
nected with the Observatory, and practical astronomy formed no part of 
George's duties. His lectures being on advanced mathematics usually 
attracted but few students; in the Long Vacation however, when he 
habitually gave one of his courses, there was often a fairly large class. 

George's relations with his class have been sympathetically treated by 
Professor E. W. Brown, than whom no one can speak with more authority, 
since he was one of my brother's favourite pupils. 

In the late '70's George began to be appointed to various University 
Boards and Syndicates. Thus from 1878 82 he was on the Museums and 
Lecture Rooms Syndicate. In 1879 he was placed on the Observatory 
Syndicate, of which he became an official member in 1883 on his election 
to the Plumian Professorship. In the same way he was on the Special Board 
for Mathematics. He was on the Financial Board from 19001 to 19034 
and on the Council of the Senate in 19056 and 19089. But he never 
became a professional syndic one of those virtuous persons who spend their 
lives in University affairs. In his obituary of George (Nature, Dec. 12, 1912), 
Sir Joseph Larmor writes : 

In the affairs of the University of which he was an ornament, 
Sir George Darwin made a substantial mark, though it cannot be said 
that he possessed the patience in discussion that is sometimes a 
necessary condition to taking a share in its administration. But his wide 
acquaintance and friendships among the statesmen and men of affairs of 
the time, dating often from undergraduate days, gave him openings for 
usefulness on a wider plane. Thus, at a time when residents were 
bewailing even more than usual the inadequacy of the resources of the 
University for the great expansion which the scientific progress of the 
age demanded, it was largely on his initiative that, by a departure from 
all precedent, an unofficial body was constituted in 1899 under the name 

1 The voting at University elections is in theory strictly confidential, but in practice this is 
unfortunately not always the case. George records in his diary the names of the five who voted 
for him and of the four who supported another candidate. None of the electors are now living. 
The election occurred in January, and in June he had the great pleasure and honour of being 
re-elected to a Trinity Fellowship. His daughter, Madame Baverat, writes: "Once, when I was 
walking with my father on the road to Madingley village, he told me how he had walked there, 
on the first Sunday he ever was at Cambridge, with two or three other freshmen ; and how, when 
they were about opposite the old chalk pit, one of them betted him 20 that he (my father) 
would never be a professor of Cambridge University : and said my father, with great indignation, 
' He never paid me.' " 



xviii MEMOIR OF SIR GEORGE DARWIN 

of the Cambridge University Association, to promote the further endow- 
ment of the University by interesting its graduates throughout the 
Empire in its progress and its more pressing needs. This important 
body, which was organised under the strong lead of the late Duke of 
Devonshire, then Chancellor, comprises as active members most of the 
public men who owe allegiance to Cambridge, and has already by its 
interest and help powerfully stimulated the expansion of the University 
into new fields of national work ; though it has not yet achieved 
financial support on anything like the scale to which American seats 
of learning are accustomed. 

The Master of Christ's writes : 

May 3lst, 1915. 

My impression is that George did not take very much interest in 
the petty details which are so beloved by a certain type of University 
authority. ' Comma hunting ' and such things were not to his taste, 
and at Meetings he was often rather distrait : but when anything of 
real importance came up he was of extraordinary use. He was especially 
good at drafting letters, and over anything he thought promoted the 
advancement of the University along the right lines he would take 
endless trouble writing and re-writing reports and letters till he got 
them to his taste. The sort of movements which interested him most 
were those which connected Cambridge with the outside world. He 
was especially interested in the Appointments Board. A good many of 
us constantly sought his advice and nearly always took it : but, as I say, 
I do not think he cared much about the ' parish pump,' and was usually 
worried at long Meetings. 

Professor Newall has also been good enough to give me his impressions : 

His weight in the Committees on which I have had personal 
experience of his influence seems to me to have depended in large 
measure on his realising very clearly the distinction between the 
importance of ends to be aimed at and the difficulty of harmonising 
the personal characteristics of the men who might be involved in the 
work needed to attain the ends. The ends he always took seriously; 
the crotchets he often took humorously, to the great easement of many 
situations that are liable to arise on a Committee. I can imagine that 
to those who had corns his direct progress may at times have seemed 
unsympathetic and hasty. He was ready to take much trouble in for- 
mulating statements of business with great precision a result doubtless 
of his early legal experiences. I recall how he would say. ' If a thing has 
to be done, the minute should if possible make some individual respon- 
sible for doing it.' He would ask, ' Who is going to do the work ? If a 
man has to take the responsibility, we must do what we can to help him 
and not hamper him by unnecessary restrictions and criticisms.' His 
helpfulness came from his quickness in seizing the important point and 
his readiness to take endless trouble in the important work of looking 
into details before and after the meetings. The amount of work that he 
did in response to the requirements of various Committees was very 
great, and it was curious to realise in how many cases he seemed to 
have diffidence as to the value of his contributions. 



BY SIR FRANCIS DARWIN XIX 

But on the whole the work which, in spite of ill-health, he was able to 
carry out in addition to professional duties and research, was given to matters 
unconnected with the University, but of a more general importance. To 
these we shall return. 

In 1884 he became engaged to Miss Maud Du Puy of Philadelphia. 
She came of an old Huguenot stock, descending from Dr John Du Puy 
who was born in France in 1679 and settled in New York in 1713. They 
were married on July 22nd, 1884, and this event happily coloured the 
remainder of George's life. As time went on and existence became fuller 
and busier, she was able by her never-failing devotion to spare him much 
arrangement and to shield him from fatigue and anxiety. In this way he 
was helped and protected in the various semi-public functions in which he 
took a principal part. Nor was her help valued only on these occasions, for 
indeed the comfort and happiness of every day was in her charge. There is 
a charming letter 1 from George's mother, dated April 15th, 1884 : 

Maud had to put on her wedding-dress in order to say at the 
Custom-house in America that she had worn it, so we asked her to 
come down and show it to us. She came down with great simplicity 
and quietness... only really pleased at its being admired and at looking 
pretty herself, which was strikingly the case. She was a little shy at 
coming in, and sent in Mrs Jebb to ask George to come out and see it 
first and bring her in. It was handsome and simple. I like seeing 
George so frivolous, so deeply interested in which diamond trinket 
should be my present, and in her new Paris morning dress, in which he 
felt quite unfit to walk with her. 

Later, probably in June, George's mother wrote 2 to Miss Du Puy, " Your 
visit here was a great happiness to me, as something in you (I don't know 
what) made me feel sure you would always be sweet and kind to George 
when he is ill and uncomfortable." These simple and touching words may 
be taken as a forecast of his happy married life. 

In March 1885 George acquired by purchase the house Newnham 
Grange 3 , Avhich remained his home to the end of his life. It stands at the 
southern end of the Backs, within a few yards of the river where it bends 
eastward in flowing from the upper to the lower of the two Newnham water- 
mills. I remember forebodings as to dampness, but they proved wrong 
even the cellars being remarkably dry. The house is built of faded 
yellowish bricks with old tiles on the roof, and has a pleasant home-like air. 

1 Emma Darwin, A Century of Family Letters. Privately printed, 1904, vol. 11. p. 350. 

2 Emma Darwin, A Century of Family Letters, 1912, vol. n. p. 266. 

3 At that time it was known simply as Newnham, but as this is the name of the College and 
was also in use for a growing region of houses, the Darwins christened it Newnham Grange. The 
name Newnham is now officially applied to the region extending from Silver Street Bridge to the 
Barton Road. 



XX MEMOIR OF SIR GEORGE DARWIN 

It was formerly the house of the Beales family 1 , one of the old merchant 
stocks of Cambridge. This fact accounts for the great barn-like granaries 
which occupied much of the plot near the high road. These buildings were 
in part pulled down, thus making room for a lawn tennis court, while what 
was not demolished made a gallery looking on the court as well as play-room 
for the children. At the eastern end of the property a cottage and part of 
the granaries were converted into a small house of an attractively individual 
character, for which I think tenants have hitherto been easily found among 
personal friends. It is at present inhabited by Lady Corbett. One of the 
most pleasant features of the Grange was the flower-garden and rockery 
on the other side of the river, reached by a wooden bridge and called " the 
Little Island 2 ." The house is conveniently close to the town, yet has a most 
pleasant outlook, to the north over the Backs while there is the river and the 
Fen to the south. The children had a den or house in the branches of a 
large copper beech tree, overhanging the river. They were allowed to use 
the boat, which was known as the Griffin from the family crest with which 
it was adorned. None of them were drowned, though accidents were not 
unknown ; in one of these an eminent lady and well-known writer, who was 
inveigled on to the river by the children, had to wade to shore near Silver 
Street bridge owing to the boat running aground. 

The Darwins had five children, of whom one died an infant : of the others, 
Charles Galton Darwin has inherited much of his father's mathematical 
ability, and has been elected to a Mathematical Lectureship at Christ's 
College. He is now in the railway service of the Army in France. The 
younger son, William, has a commission in the 18th Battalion of the Durham 
Light Infantry. George's elder daughter is married to Monsieur Jacques 
Raverat. Her skill as an artist has perhaps its hereditary root in her 
father's draughtsmanship. The younger daughter Margaret lives with her 
mother. 

George's relations with his family were most happy. His diary never 
fails to record the dates on which the children came home, or the black days 
which took them back to school. There are constantly recurring entries in 
his diary of visits to the boys at Marlborough or Winchester. Or of the 

1 The following account of Newnham Grange is taken from C. H. Cooper's Memorials of 
Cambridge, 1866, vol. in. p. 262 (note) : ' ' The site of the hermitage was leased by the Corpo- 
ration to Oliver Grene, 20 Sep., 31 Eliz. [1589]. It was in 1790 leased for a long term to 
Patrick Beales, from whom it came to his brother S. P. Beales, Esq., who erected thereon a 
substantial mansion and mercantile premises now occupied by his son Patrick Beales, Esq., 
alderman, who purchased the reversion from the Corporation in 1839." Silver Street was formerly 
known as Little Bridges Street, and the bridges which gave it this name were in charge of a 
hermit, hence the above reference to the hermitage. 

2 This was to distinguish it from the " Big Island," both being leased from the town. Later 
George acquired in the same way the small oblong kitchen garden on the river bank, and bought 
the freehold of the Lammas land on the opposite bank of the river. 



BY SIR FRANCIS DARWIN Xxi 

journeys to arrange for the schooling of the girls in England or abroad. 
The parents took pains that their children should have opportunities of 
learning conversational French and German. 

George's characteristic energy showed itself not only in these ways but 
also in devising bicycling expeditions and informal picnics, for the whole 
family, to the Fleam Dyke, to Whittlesford, or other pleasant spots near home 
and these excursions he enjoyed as much as anyone of the party. As he 
always wished to have his children with him, one or more generally accom- 
panied him and his wife when they attended congresses or other scientific 
gatherings abroad. 

His house was the scene of many Christmas dinners, the first of which 
I find any record being in 1886. These meetings were often made an 
occasion for plays acted by the children ; of these the most celebrated was 
a Cambridge version of Romeo and Juliet, in which the hero and heroine 
were scions of the rival factions of Trinity and St John's. 

Games and Pastimes. 

As an undergraduate George played tennis not the modern out-door 
game, but that regal pursuit which is sometimes known as the game of 
kings and otherwise as the king of games. When George came up as' an 
undergraduate there were two tennis courts in Cambridge, one in the East 
Road, the other being the ancient one that gave its name to Tennis Court 
Road and was pulled down to make room for the new buildings of Pembroke. 
In this way was destroyed the last of the College tennis courts of which we 
read in Mr Clark's History. I think George must have had pleasure in the 
obvious development of the tennis court from some primaeval court-yard in 
which the pent-house was the roof of a shed, and the grille a real window 
or half-door. To one brought up on evolution there is also a satisfaction 
about the French terminology which survives in e.g. the Tambour and 
the Dedans. George put much thought into acquiring a correct style of 
play for in tennis there is a religion of attitude corresponding to that which 
painfully regulates the life of the golfer. He became a good tennis player as 
an undergraduate, and was in the running for a place in the inter-University 
match. The marker at the Pembroke court was Henry Harradine, whom we 
all sincerely liked and respected, but he was not a good teacher, and it was 
only when George came under Henry's sons, John and Jim Harradine, at the 
Trinity and Clare courts, that his game began to improve. He continued to 
play tennis for some years, and only gave it up after a blow from a tennis 
ball in January 1895 had almost destroyed the sight of his left eye. 

In 1910 he took up archery, and zealously set himself to acquire the 
correct mode of standing, the position of the head and hands, etc. He kept 
an archery diary in which each day's shooting is carefully analysed and the 



xxii MEMOIR OF SIR GEORGE DARWIN 

results given in percentages. In 1911 he shot on 131 days: the last occasion 
on which he took out his bow was September 13, 1912. 

I am indebted to Mr H. Sherlock, who often shot with him at Cambridge, 
for his impressions. He writes : " I shot a good deal with your brother the 
year before his death ; he was very keen on the sport, methodical and pains- 
taking, and paid great attention to style, and as he had a good natural 
' loose,' which is very difficult to acquire, there is little doubt (notwithstanding 
that he came to Archery rather late in life) that had he lived he would have 
been above the average of the men who shoot fairly regularly at the public 
Meetings." After my brother's death, Mr Sherlock was good enough to look 
at George's archery note-book. "I then saw," he writes, "that he had 
analysed them in a way which, so far as I am aware, had never been done 
before." Mr Sherlock has given examples of the method in a sympathetic 
obituary published (p. 273) in The Archer s Register^. George's point was 
that the traditional method of scoring is not fair in regard to the areas of the 
coloured rings of the target. Mr Sherlock records in his Notice that George 
joined the Royal Toxophilite Society in 1912, and occasionally shot in the 
Regent's Park. He won the Norton Cup and Medal (144 arrows at 120 
yards) in 1912. 

There was a billiard table at Down, and George learned to play fairly 
well though he had no pretension to real proficiency. He used to play at 
the Athenaeum, and in 1911 we find him playing there in the Billiard 
Handicap, but a week later he records in his diary that he was "knocked 
out." 



Scientific Committees. 

George served for many years on the Solar Physics Committee and on 
the Meteorological Council. With regard to the latter, Sir Napier Shaw 
has at my request supplied the following note : 

It was in February 1885 upon the retirement of Warren De la Rue 
1 that your brother George, by appointment of the Royal Society, joined 
the governing body of the Meteorological Office, at that time the 
Meteorological Council. He remained a member until the end of the 
Council in 1905 and thereafter, until his death, he was one of the two 
nominees of the Royal Society upon the Meteorological Committee, the 
new body which was appointed by the Treasury to take over the control 
of the administration of the Office. 

It will be best to devote a few lines to recapitulating the salient 
features of the history of the official meteorological organisation because, 
otherwise, it will be difficult for anyone to appreciate the position in 
which Darwin was placed. 

1 The Archer's Register for 19121913, by H. Walrond. London, The Field Office, 1913. 



BY SIR FRANCIS DARWIN XXlli 

In 1854 a department of the Board of Trade was constituted under 
Admiral R. FitzRoy to collect and discuss meteorological information 
from ships, and in 1860, impressed by the loss of the ' Royal Charter,' 
FitzRoy began to collect meteorological observations by telegraph from 
land stations and chart them. Looking at a synchronous chart and 
conscious that he could gather from it a much better notion of coming 
weather than anyone who had only his own visible sky and barometer 
to rely upon, he formulated 'forecasts' which were published in the 
newspapers and ' storm warnings ' which were telegraphed to the ports. 

This mode of procedure, however tempting it might be to the 
practical man with the map before him, was criticised as not complying 
with the recognised canons of scientific research, and on FitzRoy's 
untimely death in 1865 the Admiralty, the Board of Trade and the 
Royal Society elaborated a scheme for an office for the study of weather 
in due form under a Director and Committee, appointed by the Royal 
Society, and they obtained a grant in aid of 10,000 for this purpose. 
In this transformation it was Galton, I believe, who took a leading part 
and to him was probably due the initiation of the new method of study 
which was to bring the daily experience, as represented by the map, 
into relation with the continuous records of the meteorological elements 
obtained at eight observatories of the Kew type, seven of which were 
immediately set on foot, and Galton devoted an immense amount of 
time and skill to the reproduction of the original curves so that the 
whole sequence of phenomena at the seven observatories could be taken 
in at a glance. Meanwhile the study of maps was continued and a good 
deal of progress was made in our knowledge of the laws of weather. 

But in spite of the wealth of information the generalisations were 
empirical and it was felt that something more than the careful examina- 
tion of records was required to bring the phenomena of weather within 
the rule of mathematics and physics, so in 1876 the constitution of the 
Office was changed and the direction of its work was placed in Commis- 
sion with an increased grant. The Commissioners, collectively known 
as the Meteorological Council, were a remarkably distinguished body of 
fellows of the Royal Society, and when Darwin took the place of 
De la Rue, the members were men subsequently famous, as Sir Richard 
Strachey, Sir William Wharton, Sir George Stokes, Sir Francis Galton, 
Sir George Darwin, with E. J. Stone, a former Astronomer Royal for 
the Cape. 

It was understood that the attack had to be made by new methods 
and was to be entrusted partly to members of the Council themselves, 
with the staff of the Office behind them, and partly to others outside 
who should undertake researches on special points. Sir Andrew Noble, 
Sir William Abney, Dr W. J. Russell, Mr W. H. Dines, your brother 
Horace and myself came into connection with the Council in this way. 

Two important lines of attack were opened up within the Council 
itself. The first was an attempt, under the influence of Lord Kelvin, 
to base an explanation of the sequence of weather upon harmonic 
analysis. As the phenomena of tides at any port could be synthesized 
by the combinations of waves of suitable period and amplitude, so the 
sequence of weather could be analysed into constituent oscillations the 
general relations of which would be recognisable although the original 



MEMOIR OF SIR GEORGE DARWIN 

composite result was intractable on direct inspection. It was while this 
enterprise was in progress that Darwin was appointed to the Council. 
His experience with tides and tidal analysis was in a way his title 
to admission. He and Stokes were the mathematicians of the Council 
and were looked to for expert guidance in the undertaking. At first 
the individual curves were submitted to analysis in a harmonic analyser 
specially built for the purpose, the like of which Darwin had himself 
used or was using for his work on tides ; but afterwards it was decided 
to work arithmetically with the numbers derived from the tabulation of 
the curves; and the identity of the individual curves was merged in 
' five-day means.' The features of the automatic records from which so 
much was hoped in 1865, after twelve years of publication in facsimile, 
were practically never seen outside the room in the Office in which they 
were tabulated. 

It is difficult at this time to point to any general advances in 
meteorology which can be attributed to the harmonic analyser or its 
arithmetical equivalent as a process of discussion, though it still remains 
a powerful method of analysis. It has, no doubt, helped towards the 
recognition of the ubiquity and simultaneity of the twelve-hour term in 
the diurnal change of pressure which has taken its place among funda- 
mental generalisations of meteorology and the curious double diurnal 
change in the wind at any station belongs to the same category; but 
neither appears to have much to do with the control of weather. 
Probably the real explanation of the comparative fruitlessness of the 
effort lies in the fact that its application was necessarily restricted to 
the small area of the British Isles instead of being extended, in some 
way or other, to the globe. 

It is not within my recollection that Darwin was particularly 
enthusiastic about the application of harmonic analysis. When I was 
appointed to the Council in 1897, the active pursuit of the enterprise 
had ceased. Strachey who had taken an active part in the discussion 
of the results and contributed a paper on them to the Philosophical 
Transactions, was still hopeful of basing important conclusions upon the 
seasonal peculiarities of the third component, but the interest of other 
members of the Council was at best languid. 

The other line of attack was in connection with synoptic charts. For 
the year from August 1892 to August 1893 there was an international 
scheme for circumpolar observations in the Northern Hemisphere, and 
in connection therewith the Council undertook the preparation of daily 
synoptic charts of the Atlantic and adjacent land areas. A magnificent 
series of charts was produced and published from which great results 
were anticipated. But again the conclusions drawn from cursory inspec- 
tion were disappointing. At that time the suggestion that weather 
travelled across the Atlantic in so orderly a manner that our weather 
could be notified four or five days in advance from New York had a 
considerable vogue and the facts disclosed by the charts put an end to 
any hope of the practical development of that suggestion. Darwin was 
very active in endeavouring to obtain the help of an expert in physics 
for the discussion of the charts from a new point of view, but he was 
unsuccessful. 

Observations at High Level Stations were also included in the 



BY SIR FRANCIS DARWIN XXV 

Council's programme. A station was maintained at Hawes Junction 
for some years, and the Observatories on Ben Nevis received their 
support. But when I joined the Council in 1897 there was a pervading 
sense of discouragement. The forecasting had been restored as the result 
of the empirical generalisations based on the work of the years 1867 to 
1878, but the study had no attractions for the powerful analytical minds 
of the Council ; and the work of the Office had settled down into the 
assiduous compilation of observations from sea and land and the regular 
issue of forecasts and warnings in the accustomed form. The only part 
which I can find assigned to Darwin with regard to forecasting is an 
endeavour to get the forecast worded so as not to suggest more assurance 
than was felt. 

I do not think that Darwin addressed himself spontaneously to 
meteorological problems, but he was always ready to help. He was 
very regular in his attendance at Council and the Minutes show that 
after Stokes retired all questions involving physical measurement or 
mathematical reasoning were referred to him. There is a short and 
very characteristic report from him on the work of the harmonic 
analyser and a considerable number upon researches by Mr Dines or 
Sir G. Stokes on anemometers. It is hardly possible to exaggerate 
his aptitude for work of that kind. He could take a real interest in 
things that were not his own. He was full of sympathy and appreciation 
for efforts of all kinds, especially those of young men, and at the same 
time, using his wide experience, he was perfectly frank and fearless not 
only in his judgment but also in the expression of it. He gave one the 
impression of just protecting himself from boredom by habitual loyalty 
and a finely tempered sense of duty. My earliest recollection of him on 
the Council is the thrilling production of a new version of the Annual 
Report of the Council which he had written because the original had 
become more completely ' scissors and paste ' than he could endure. 

After the Office came into my charge in 1900, so long as he lived, 
I never thought of taking any serious step without first consulting him 
and he was always willing to help by his advice, by his personal influ- 
ence and by his special knowledge. For the first six years of the time 
I held a college fellowship with the peculiar condition of four public 
lectures in the University each year and no emolument. One year, 
when I was rather overdone, Darwin took the course for me and devoted 
the lectures to Dynamical Meteorology. I believe he got it up for the 
occasion, for he professed the utmost diffidence about it, but the progress 
which we have made in recent years in that subject dates from those 
lectures and the correspondence which arose upon them. 

In Council it was the established practice to proceed by agreement 
and not by voting ; he had a wonderful way of bringing a discussion to 
a head by courageously ' voicing ' the conclusion to which it led and 
frankly expressing the general opinion without hurting anybody's 
feelings. 

This letter has, I fear, run to a great length, but it is not easy 
to give expression to the powerful influence which he exercised upon 
all departments of official meteorology without making formal contribu- 
tions to meteorological literature. He gave me a note on a curious 
point in the evaluation of the velocity equivalents of the Beaufort Scale 



XXVI MEMOIR OF SIR GEORGE DARWIN 

which is published in the Office Memoirs No. 180, and that is all I have 
to show in print, but he was in and behind everything that was done 
and personally, I need hardly add, I owe to him much more than this or 
any other letter can fully express. 

On May 6, 1904, he was elected President of the British Association 
the South African meeting. 

On July 29, 1905, he embarked with his wife and his son Charles and 
arrived on August 15 at the Cape, where he gave the first part of his 
Presidential Address. Here he had the pleasure of finding as Governor 
Sir Walter Hely-Hutchinson, whom he had known as a Trinity undergraduate. 
He was the guest of the late Sir David Gill, who remained a close friend for 
the rest of his life. George's diary gives his itinerary which shows the 
trying amount of travel that he went through. A sample may be quoted : 
August 19 Embark, 

22 Arrive at Durban, 
23 Mount Edgecombe, 
24 Pietermaritzburg, 

26 Colenso, 
27 Lady smith, 
28 Johannesburg. 

At Johannesburg he gave the second half of his Address. Then on by 
Bloemfontein, Kimberley, Bulawayo, to the Victoria Falls, where a bridge had 
to be opened. Then to Portuguese Africa on September 16, 17, where he 
made speeches in French and English. Finally he arrived at Suez on 
October 4 and got home October 18. 

It was generally agreed that his Presidentship was a conspicuous success. 
The following appreciation is from the obituary notice in The Observatory, 
Jan. 1913, p. 58: 

The Association visited a dozen towns, and at each halt its President 
addressed an audience partly new, and partly composed of people who 
had been travelling with him for many weeks. At each place this 
latter section heard with admiration a treatment of his subject wholly 
fresh and exactly adapted to the locality. 

Such duties are always trying and it should not be forgotten that tact was 
necessary in a country which only two years before was still in the throes 
of war. 

In the autumn he received the honour of being made a K.C.B. The 
distinction was doubly valued as being announced to him by his friend 
Mr Balfour, then Prime Minister. 

From 1899 to 1900 he was President of the Royal Astronomical Society. 
One of his last Presidential acts was the presentation of the Society's Medal 
to his friend M. Poincare. 



BY SIR FRANCIS DARWIN XXvii 

He had the unusual distinction of serving twice as President of the 
Cambridge Philosophical Society, once in 1890 92 and again 1911 12. 

In 1891 he gave the Bakerian Lecture 1 of the Royal Society, his subject 
being "Tidal Prediction." This annual prselection dates from 1775 and the 
list of lecturers is a distinguished roll of names. 

In 1897 he lectured at the Lowell Institute at Boston, and this was 
the origin of his book on Tides, published in the following year. Of this 
Sir Joseph Larmor says 2 that "it has taken rank with the semi-popular 
writings of Helmholtz and Kelvin as a model of what is possible in the 
exposition of a scientific subject." It has passed through three English 
editions, and has been translated into many foreign languages. 

International Associations. 

During the last ten or fifteen years of his life George was much occupied 
with various International bodies, e.g. the International Geodetic Association, 
the International Association of Academies, the International Congress of 
Mathematicians and the Seismological Congress. 

With regard to the last named it was in consequence of George's report 
to the Royal Society that the British Government joined the Congress. It 
was however with the Geodetic Association that he was principally connected. 

Sir Joseph Larmor (Nature, December 12, 1912) gives the following 
account of the origin of the Association: 

The earliest of topographic surveys, the model which other national 
surveys adopted and improved upon, was the Ordnance Survey of the 
United Kingdom. But the great trigonometrical survey of India, started 
nearly a century ago, and steadily carried on since that time by officers 
of the Royal Engineers, is still the most important contribution to the 
science of the figure of the earth, though the vast geodetic operations in 
the United States are now following it closely. The gravitational and 
other complexities incident on surveying among the great mountain 
masses of the Himalayas early demanded the highest mathematical 
assistance. The problems originally attacked in India by Archdeacon 
Pratt were afterwards virtually taken over by the Royal Society, and its 
secretary, Sir George Stokes, of Cambridge, became from 1864 onwards 
the adviser and referee of the survey as regards its scientific enterprises. 
On the retirement of Sir George Stokes, this position fell very largely to 
Sir George Darwin, whose relations with the India Office on this and 
other affairs remained close, and very highly appreciated, throughout 
the rest of his life. 

The results of the Indian survey have been of the highest importance 
for the general science of geodesy.... It came to be felt that closer 
cooperation between different countries was essential to practical 
progress and to coordination of the work of overlapping surveys. 

1 See Prof. Brown's Memoir, p. xlix. 

2 Nature, 1912. See also Prof. Brown's Memoir, p. 1. 



XXViii MEMOIR OF SIR GEORGE DARWIN 

The further history of George's connection with the Association is told in 
the words of its Secretary, Dr van d. Sande Bakhuyzen, to whom I am greatly 
indebted. 

On the proposal of the Royal Society, the British Government, after 
having consulted the Director of the Ordnance Survey, in 1898, resolved 
upon the adhesion of Great Britain to the International Geodetic Asso- 
ciation, and appointed as its delegate, G. H. Darwin. By his former 
researches and by his high scientific character, he, more than any other, 
was entitled to this position, which would afford him an excellent 
opportunity of furthering, by his recommendations, the study of theo- 
retical geodesy. 

The meeting at Stuttgart in 1898 was the first which he attended, 
and at that and the following conferences, Paris 1900, Copenhagen 1903, 
Budapest 1906, London- Cambridge 1909, he presented reports on the 
geodetic work in the British Empire. To Sir David Gill's report on the 
geodetic work in South Africa, which he delivered at Budapest, Darwin 
added an appendix in which he relates that the British South Africa 
Company, which had met all the heavy expense of the part of the survey 
along the 30th meridian through Rhodesia, found it necessary to make 
various economies, so that it was probably necessary to suspend the 
survey for a time. This interruption would be most unfortunate for the 
operations relating to the great triangulation from the Southern part of 
Cape Colony to Egypt, but, happily, by the cooperation of different 
authorities, all obstacles had been overcome and the necessary money 
found, so that the triangulation could be continued. So much for 
Sir George Darwin's communication ; it is correct but incomplete, as it 
does not mention that it was principally by Darwin's exertions and by 
his personal offer of financial help that the question was solved and the 
continuation of this great enterprise secured. 

To the different researches which enter into the scope of the Geodetic 
Association belong the researches on the tides, and it is natural that 
Darwin should be chosen as general reporter on that subject; two 
elaborate reports were presented by him at the conferences of Copen- 
hagen and London. 

In Copenhagen he was a member of the financial committee, and at 
the request of this body he presented a report on the proposal to determine 
gravity at sea, in which he strongly recommended charging Dr Hecker 
with that determination using the method of Prof. Mohn (boiling 
temperature of water and barometer readings). At the meeting of 
1906 an interesting report was read by him on a question raised by 
the Geological Congress : the cooperation of the Geodetic Association 
in geological researches by means of the anomalies in the intensity 
of gravitation. 

By these reports and recommendations Darwin exercised a useful 
influence on the activity of the Association, but his influence was to be 
still increased. In 1907 the Vice-president of the Association, General 
Zacharias, died, and the permanent committee, whose duty it was to 
nominate his provisional successor, chose unanimously Sir George 
Darwin, and this choice was confirmed by the next General Conference 
in London. 



BY SIR FRANCIS DARWIN XXIX 

We cannot relate in detail his valuable cooperation as a member of 
the council in the various transactions of the Association, for instance on 
the junction of the Russian and Indian triangulations through Pamir, 
but we must gratefully remember his great service to the Association 
when, at his invitation, the delegates met in 1909 for the 16th General 
Conference in London and Cambridge. 

With the utmost care he prepared everything to render the Con- 
ference as interesting and agreeable as possible, and he fully succeeded. 
Through his courtesy the foreign delegates had the opportunity of making 
the personal acquaintance of several members of the Geodetic staff of 
England and its colonies, and of other scientific men, who were invited 
to take part in the conference ; and when after four meetings in London 
the delegates went to Cambridge to continue their work, they enjoyed 
the most cordial hospitality from Sir George and Lady Darwin, who, 
with her husband, procured them in Newnham Grange happy leisure 
hours between their scientific labours. 

At this conference Darwin delivered various reports, and at the 
discussion on Hecker's determination of the variation of the vertical by 
the attraction of the moon and sun, he gave an interesting account of 
the researches on the same subject made by him and his brother Horace 
more than 20 years ago, which unfortunately failed from the bad conditions 
of the places of observation. 

In 1912 Sir George, though already over- fatigued by the preparations 
for the mathematical congress in Cambridge, and the exertions entailed 
by it, nevertheless prepared the different reports on the geodetic work 
in the British Empire, but alas his illness prevented him from assisting 
at the conference at Hamburg, where they were presented by other 
British delegates. The conference thanked him and sent him its best 
wishes, but at the end of the year the Association had to deplore the loss 
of the man who in theoretical geodesy as well as in other branches of 
mathematics and astronomy stood in the first rank, and who for his 
noble character was respected and beloved by all his colleagues in the 
International Geodetic Association. 

Sir Joseph Larmor writes 1 : 

Sir George Darwin's last public appearance was as president of the 
fifth International Congress of Mathematicians, which met at Cambridge 
on August 22 28, 1912. The time for England to receive the congress 
having obviously arrived, a movement was initiated at Cambridge, with 
the concurrence of Oxford mathematicians, to send an invitation to the 
fourth congress held at Rome in 1908. The proposal was cordially 
accepted, and Sir George Darwin, as doyen of the mathematical school 
at Cambridge, became chairman of the organising committee, and was 
subsequently elected by the congress to be their president. Though 
obviously unwell during part of the meeting, he managed to discharge 
the delicate duties of the chair with conspicuous success, and guided 
with great verve the deliberations of the final assembly of what turned 
out to be a most successful meeting of that important body. 

1 Nature, Dec. 12, 1912. 



XXX MEMOIR OF SIR GEORGE DARWIN 

Personal Characteristics. 
His daughter, Madame Raverat, writes : 

I think most people might not realise that the sense of adventure 
and romance was the most important thing in my father's life, except his 
love of work. He thought about all life romantically and his own life 
in particular; one could feel it in the quality of everything he said 
about himself. Everything in the world was interesting and wonderful 
to him and he had the power of making other people feel it. 

He had a passion for going everywhere and seeing everything; 
learning every language, knowing the technicalities of every trade ; and 
all this emphatically not from the scientific or collector's point of view, but 
from a deep sense of the romance and interest of everything. It was 
splendid to travel with him ; he always learned as much as possible of 
the language, and talked to everyone ; we had to see simply everything 
there was to be seen, and it was all interesting like an adventure. For 
instance at Vienna I remember being taken to a most improper music hall; 
and at Schonbrunn hearing from an old forester the whole secret history of 
the old Emperor's son. My father would tell us the stories of the places 
we went to with an incomparable conviction, and sense of the reality 
and dramaticness of the events. It is absurd of course, but in that 
respect he always seemed to me a little like Sir Walter Scott 1 . 

The books he used to read to us when we were quite small, 
and which we adored, were Percy's Reliques and the Prologue to the 
Canterbury Tales. He used often to read Shakespeare to himself, 
I think generally the historical plays, Chaucer, Don Quixote in Spanish, 
and all kind of books like Joinville's Life of St Louis in the old French. 

I remember the story of the death of Gordon told so that we all 
cried, I think; and Gladstone could hardly be mentioned in consequence. 
All kinds of wars and battles interested him, and I think he liked archery 
more because it was romantic than because it was a game. 

During his last illness his interest in the Balkan war never failed. 
Three weeks before his death he was so ill that the doctor thought him 
dying. Suddenly he rallied from the half-unconscious state in which he 
had been lying for many hours and the first words he spoke on opening 
his eyes were : " Have they got to Constantinople yet ? " This was very 
characteristic. I often wish he was alive now, because his under- 
standing and appreciation of the glory and tragedy of this war would 
be like no one else's. 

His daughter Margaret Darwin writes : 

He was absolutely unselfconscious and it never seemed to occur to 
him to wonder what impression he was making on others. I think it 
was this simplicity which made him so good with children. He seemed 
to understand their point of view and to enjoy with them in a way that 

1 Compare Mr Chesterton's Twelve Types, 1903, p. 190. He speaks of Scott's critic in the 
Edinburgh Review : "The only thing to be said about that critic is that he had never been 
a little boy. He foolishly imagined that Scott valued the plume and dagger of Marmion for 
Marmion's sake. Not being himself romantic, he could not understand that Scott valued 
the plume because it was a plume and the dagger because it was a dagger." 



BY SIR FRANCIS DARWIN XXxi 

is not common with grown-up people. I shall never forget how when 
our dog had to be killed he seemed to feel the horror of it just as I did, 
and how this sense of his really sharing my grief made him able to 
comfort me as nobody else could. 

He took a transparent pleasure in the honours that came to him, 
especially in his membership of foreign Academies, in which he and 
Sir David Gill had a friendly rivalry or "race," as they called it. I think 
this simplicity was one of his chief characteristics, though most im- 
portant of all was the great warmth and width of his affections. He 
would take endless trouble about his friends, especially in going to see 
them if they were lonely or ill; and he was absolutely faithful and 
generous in his love. 

After his mother came to live in Cambridge, I believe he hardly ever 
missed a day in going to see her even though he might only be able to stay 
a few minutes. She lived at some distance off and he was often both busy 
and tired. This constancy was very characteristic. It was shown once more 
in his many visits to Jim Harradine, the marker at the tennis court, on what 
proved to be his death-bed. 

His energy and his kindness of heart were shown in many cases of distress. 
For instance, a guard on the Great Northern Railway was robbed of his savings 
by an absconding solicitor, and George succeeded in collecting some 300 
for him. In later years, when his friend the guard became bedridden, George 
often went to see him. Another man whom he befriended was a one-legged 
man at Balsham whom he happened to notice in bicycling past. He took the 
trouble to see the village authorities and succeeded in sending the man to 
London to be fitted with an artificial leg. 

In these and similar cases there was always the touch of personal 
sympathy. For instance he pensioned the widow of his gardener, and he 
often made the payment of her weekly allowance the excuse for a visit. 

In another sort of charity he was equally kind-hearted, viz. in answering 
the people who wrote foolish letters to him on scientific subjects and here 
as in many points he resembled his father. 

His sister, Mrs Litchfield, has truly said 1 of George that he inherited his 
father's power of work and much of his " cordiality and warmth of nature 
with a characteristic power of helping others." He resembled his father in 
another quality, that of modesty. His friend and pupil E. W. Brown writes : 

He was always modest about the importance of his researches. 
He would often wonder whether the results were worth the labour they 
had cost him and whether he would have been better employed in some 
other way. 

His nephew Bernard, speaking of George's way of taking pains to be 
friendly and forthcoming to anyone with whom he came in contact, says : 

1 Emma Darwin, A Century of Family Letters, 1915, vol. n. p. 146. 



xxxii MEMOIR OF SIR GEORGE DARWIN 

He was ready to take other people's pleasantness and politeness at 
its apparent value and not to discount it. If they seemed glad to see him, 
he believed that they were glad. If he liked somebody, he believed 
that the somebody liked him, and did not worry himself by wondering 
whether they really did like him. 

Of his energy we have evidence in the amount of work contained in 
these volumes. There was nothing dilatory about him, and here he again 
resembled his father who had markedly the power of doing things at the 
right moment, and thus avoiding waste of time and discomfort to others. 
George had none of a characteristic which was denned in the case of Henry 
Bradshaw, as "always doing something else." After an interruption he could 
instantly reabsorb himself in his work, so that his study was not kept as a 
place sacred to peace and quiet. 

His wife is my authority for saying that although he got so much done, 
it was not by working long hours. Moreover the days that he was away 
from home made large gaps in his opportunities for steady application. His 
diaries show in another way that his researches by no means took all his 
time. He made a note of the books he read and these make a considerable 
record. Although he read much good literature with honest enjoyment, he 
had not a delicate or subtle literary judgment. Nor did he care for music. 
He was interested in travels, history, and biography, and as he could remember 
what he read or heard, his knowledge was wide in many directions. His 
linguistic power was characteristic. He read many European languages. 
I remember his translating a long Swedish paper for my father. And he 
took pleasure in the Platt Deutsch stories of Fritz Reuter. 

The discomfort from which he suffered during the meeting at Cambridge 
of the International Congress of Mathematicians in August 1912, was in fact 
the beginning of his last illness. An exploratory operation showed that he 
was suffering from malignant disease. Happily he was spared the pain that 
gives its terror to this malady. His nature was, as we have seen, simple and 
direct with a pleasant residue of the innocence and eagerness of childhood. 
In the manner of his death these qualities were ennobled by an admirable 
and most unselfish courage. As his vitality ebbed away his affection only 
showed the stronger. He wished to live, and he felt that his power of work 
and his enjoyment of life were as strong as ever, but his resignation to the 
sudden end was complete and beautiful. He died on Dec. 7, 1912, and was 
buried at Trumpington. 



BY SIR FRANCIS DARWIN 

HONOURS, MEDALS, DEGREES, SOCIETIES, ETC. 

Order. K.C.B. 1905. 
Medals 1 . 

1883. Telford Medal of the Institution of Civil Engineers. 

1884. Royal Medal 2 . 

1892. Royal Astronomical Society's Medal. 

1911. Copley Medal of the Royal Society. 

1912. Royal Geographical Society's Medal. 



Fellow of Trinity College, Cambridge, and Plumian Professor in the 
University. 

Vice-President of the International Geodetic Association, Lowell Lecturer 
at Boston U.S. (1897). 

Member of the Meteorological and Solar Physics Committees. 

Past President of the Cambridge Philosophical Society 3 , Royal Astro- 
nomical Society, British Association. 

Doctorates, etc. of Universities. 

Oxford, Dublin, Glasgow, Pennsylvania, Padua (Socio onorario), Gottingen, 
Christiania, Cape of Good Hope, Moscow (honorary member). 

Foreign or Honorary Membership of Academies, etc. 

Amsterdam (Netherlands Academy), Boston (American Academy), 
Brussels (Royal Society), Calcutta (Math. Soc.), Dublin (Royal Irish 
Academy), Edinburgh (Royal Society); Halle (K. Leop.-Carol. Acad.), 
Kharkov (Math. Soc.), Mexico (Soc. "Antonio Alzate"), Moscow (Imperial 
Society of the Friends of Science), New York, Padua, Philadelphia (Philo- 
sophical Society), Rome (Lincei), Stockholm (Swedish Academy), Toronto 
(Physical Society), Washington (National Academy), Wellington (New 
Zealand Inst.). 

Correspondent of Academies, etc. at 

Acireale (Zelanti), Berlin (Prussian Academy), Buda Pest (Hungarian 
Academy), Frankfort (Senckenberg. Natur. Gesell.), Gottingen (Royal Society), 
Paris, St Petersburg, Turin, Istuto Veneto, Vienna 4 . 

1 Sir George's medals are deposited in the Library of Trinity College, Cambridge. 

2 Given by the Sovereign on the nomination of the Royal Society. 

3 Re-elected in 1912. 

4 The above list is principally taken from that compiled by Sir George for the Year-Book of 
the Royal Society, 1912, and may not be quite complete. 

It should be added that he especially valued the honour conferred on him in the publication 
of his collected papers by the Syndics of the University Press. 



THE SCIENTIFIC WORK OF SIR GEORGE DARWIN 

BY 
PROFESSOR E. W. BROWN 

The scientific work of Darwin possesses two characteristics which cannot 
fail to strike the reader who glances over the titles of the eighty odd papers 
which are gathered together in the four volumes which contain most of his 
publications. The first of these characteristics is the homogeneous nature 
of his investigations. After some early brief notes, on a variety of subjects, 
he seems to have set himself definitely to the task of applying the tests of 
mathematics to theories of cosmogony, and to have only departed from it 
when pressed to undertake the solution of practical problems for which there 
was an immediate need. His various papers on viscous spheroids concluding 
with the effects of tidal friction, the series on rotating masses of fluids, even 
those on periodic orbits, all have the idea, generally in the foreground, of 
developing the consequences of old and new assumptions concerning the past 
history of planetary and satellite systems. That he achieved so much, in 
spite of indifferent health which did not permit long hours of work at his 
desk, must have been largely due to this single aim. 

The second characteristic is the absence of investigations undertaken for 
their mathematical interest alone ; he was an applied mathematician in the 
strict and older sense of the word. In the last few decades another school of 
applied mathematicians, founded mainly by Poincare, has arisen, but it differs 
essentially from the older school. Its votaries have less interest in the 
phenomena than in the mathematical processes which are used by the student 
of the phenomena. They do not expect to examine or predict physical 
events but rather to take up the special classes of functions, differential 
equations or series which have been used by astronomers or physicists, to 
examine their properties, the validity of the arguments and the limitations 
which must be placed on the results. Occasionally theorems of great physical 
importance will emerge, but from the primary point of view of the investigations 
these are subsidiary results. Darwin belonged essentially to the school which 
studies the phenomena by the most convenient mathematical methods. Strict 
logic in the modern sense is not applied nor is it necessary, being replaced in 
most cases by intuition which guides the investigator through the dangerous 
places. That the new school has done great service to both pure and applied 
mathematics can hardly be doubted, but the two points of view of the subject 



SCIENTIFIC WORK XXXV 

will but rarely be united in the same man if much progress in either direction 
is to be made. Hence we do not find and do not expect to find in Darwin's 
work developments from the newer point of view. 

At the same time, he never seems to have been affected by the problem- 
solving habits which were prevalent in Cambridge during his undergraduate 
days and for some time later. There was then a large number of mathema- 
ticians brought up in the Cambridge school whose chief delight was the 
discovery of a problem which admitted of a neat mathematical solution. 
The chief leaders were, of course, never very seriously affected by this 
attitude ; they had larger objects in view, but the temptation to work out 
a problem, even one of little physical importance, when it would yield to 
known mathematical processes, was always present. Darwin kept his aim 
fixed. If the problem would not yield to algebra he has recourse to 
arithmetic; in either case he never seemed to hesitate to embark on the 
most complicated computations if he saw a chance of attaining his end. 
The papers on ellipsoidal harmonic analysis and periodic orbits are instructive 
examples of the labour which he would undertake to obtain a knowledge of 
physical phenomena. 

One cannot read any of his papers without also seeing another feature, 
his preference for quantitative rather than qualitative results. If he saw 
any possibility of obtaining a numerical estimate, even in his most specu- 
lative work, he always made the necessary calculations. His conclusions 
thus have sometimes an appearance of greater precision than is warranted 
by the degree of accuracy of the data. But Darwin himself was never 
misled by his numerical conclusions, and he is always careful to warn his 
readers against laying too great a stress on the numbers he obtains. 

In devising processes to solve his problems, Darwin generally adopted 
those which would lead in a straightforward manner to the end he had 
in view. Few " short cuts " are to be found in his memoirs. He seems to 
have felt that the longer processes often brought out details and points 
of view which would otherwise have been concealed or neglected. This is 
particularly evident in the papers on Periodic Orbits. In the absence of 
general methods for the discovery and location of the curves, his arithmetic 
showed classes of orbits which would have been difficult to find by analysis, 
and it had a further advantage in indicating clearly the various changes 
which the members of any class undergo when the parameter varies. Yet, 
in spite of the large amount of numerical work which is involved in many 
of his papers, he never seemed to have any special liking for either algebraic 
or numerical computation; it was something which "had to be done." Unlike 
J. C. Adams and G. W. Hill, who would often carry their results to a large 
number of places of decimals, Darwin would find out how high a degree of 
accuracy was necessary and limit himself to it. 



XXXVI THE SCIENTIFIC WORK OF 

The influence which Darwin exerted has been felt in many directions. 
The exhibition of the necessity for quantitative and thorough analysis of the 
problems of cosmogony and celestial mechanics has been perhaps one of his 
chief contributions. It has extended far beyond the work of the pupils who 
were directly inspired by him. While speculations and the framing of new 
hypotheses must continue, but little weight is now attached to those which 
are defended by general reasoning alone. Conviction fails, possibly because 
it is recognised that the human mind cannot reason accurately in these 
questions without the aids furnished by mathematical symbols, and in any 
case language often fails to carry fully the argument of the writer as against 
the exact implications of mathematics. If for no other reason, Darwin's work 
marks an epoch in this respect. 

To the pupils who owed their first inspiration to him, he was a constant 
friend. First meeting them at his courses on some geophysical or astro- 
nomical subject, he soon dropped the formality of the lecture-room, and they 
found themselves before long going to see him continually in the study at 
Newnham Grange. Who amongst those who knew him will fail to remember 
the sight of him seated in an armchair with a writing board and papers 
strewn about the table and floor, while through the window were seen 
glimpses of the garden filled in summer time with flowers ? While his 
lectures in the class-room were always interesting and suggestive, the chief 
incentive, at least to the writer who is proud to have been numbered amongst 
his pupils and friends, was conveyed through his personality. To have spent 
an hour or two with him, whether in discussion on "shop" or in general 
conversation, was always a lasting inspiration. And the personal attachment 
of his friends was strong ; the gap caused by his death was felt to be far 
more than a loss to scientific progress. Not only the solid achievements 
contained in his published papers, but the spirit of his work and the example 
of his life will live as an enduring memorial of him. 



Darwin's first five papers, all published in 1875, are of some interest as 
showing the mechanical turn of his mind and the desire, which he never lost, 
for concrete illustrations of whatever problem might be interesting him. 
A Peaucellier's cell is shown to be of use for changing a constant force into 
one varying inversely as the square of the distance, and it is applied to the 
description of equipotential lines. A method for describing graphically the 
second elliptic integral and one for map projection on the face of a polyhedron 
are also given. There are also a few other short papers of the same kind but 
of no special importance, and Darwin says that he only included them in his 
collected works for the sake of completeness. 

His first important contributions obviously arose through the study 
of the works of his predecessors, and though of the nature of corrections to 



SIR GEORGE DARWIN XXXvii 

previously accepted or erroneous ideas, they form definite additions to the 
subject of cosmogony. The opening paragraph of the memoir "On the 
influence of geological changes in the earth's axis of rotation " describes the 
situation which prompted the work. " The subject of the fixity or mobility 
of the earth's axis of rotation in that body, and the possibility of variations 
in the obliquity of the ecliptic, have from time to time attracted the notice 
of mathematicians and geologists. The latter look anxiously for some grand 
cause capable of producing such an enormous effect as the glacial period. 
Impressed by the magnitude of the phenomenon, several geologists have 
postulated a change of many degrees in the obliquity of the ecliptic and 
a wide variability in the position of the poles on the earth ; and this, again, 
they have sought to refer back to the upheaval and subsidence of continents." 
He therefore subjects the hypothesis to mathematical examination under 
various assumptions which have either been put forward by geologists or 
which he considers a, priori probable. The conclusion, now well known to 
astronomers, but frequently forgotten by geologists even at the present time, 
is against any extensive wanderings of the pole during geological times. 
" Geologists and biologists," writes Professor Barrell J , " may array facts 
which suggest such hypotheses, but the testing of their possibility is really 
a problem of mathematics, as much as are the movements of precession, 
and orbital perturbations. Notwithstanding this, a number of hypotheses 
concerning polar migration have been ingeniously elaborated and widely 
promulgated without their authors submitting them to these final tests, or 
in most cases even perceiving that an accordance with the known laws of 
mechanics was necessary.... A reexamination of these assumptions in the 
light of forty added years of geological progress suggests that the actual 
changes have been much less and more likely to be limited to a fraction 
of the maximum limits set by Darwin. His paper seems to have checked 
further speculation upon this subject in England, but, apparently unaware 
of its strictures, a number of continental geologists and biologists have 
carried forward these ideas of polar wandering to the present day. The 
hypotheses have grown, each creator selecting facts and building up from 
his particular assortment a fanciful hypothesis of polar migration unre- 
strained even by the devious paths worked out by others." The methods 
used by Darwin are familiar to those who investigate problems connected 
with the figure of the earth, but the whole paper is characteristic of his style 
in the careful arrangement of the assumptions, the conclusions deduced 
therefrom, the frequent reduction to numbers and the summary giving the 
main results. 

It is otherwise interesting because it was the means of bringing Darwin 
into close connection with Lord Kelvin, then Sir William Thomson. The 

1 Science, Sept. 4, 1914, p. 333. 



XXXviii THE SCIENTIFIC WORK OF 

latter was one of the referees appointed by the Royal Society to report on it, 
and, as Darwin says, " He seemed to find that on these occasions the quickest 
way of coming to a decision was to talk over the subject with the author 
himself at least this was frequently so as regards myself." Through his 
whole life Darwin, like many others, prized highly this association, and he 
considered that his whole work on cosmogony " may be regarded as the 
scientific outcome of our conversation of the year 1877; but," he adds, "for 
me at least science in this case takes the second place." 

Darwin at this time was thirty-two years old. In the three years since 
he started publication fourteen memoirs and short notes, besides two statis- 
tical papers on marriage between first cousins, form the evidence of his 
activity. He seems to have reached maturity in his mathematical power 
and insight into the problems which he attacked without the apprenticeship 
which is necessary for most investigators. Probably the comparatively late 
age at which he began to show his capacity in print may have something to 
do with this. Henceforth development is rather in the direction of the full 
working out of his ideas than growth of his powers. It seems better there- 
fore to describe his further scientific work in the manner in which he arranged 
it himself, by subject instead of in chronological order. And here we have 
the great advantage of his own comments, made towards the end of his 
life when he scarcely hoped to undertake any new large piece of work. 
Frequent quotation will be made from these remarks which occur in the 
prefaces to the volumes, in footnotes and in his occasional addresses, 

The following account of the Earth-Moon series of papers is taken bodily 
from the Notice in the Proceedings of the Royal Society l by Mr S. S. Hough, 
who was himself one of Darwin's pupils. 

" The conclusions arrived at in the paper referred to above were based on 
the assumption that throughout geological history, apart from slow geological 
changes, the Earth would rotate sensibly as if it were rigid. It is shown that 
a departure from this hypothesis might possibly account for considerable 
excursions of the axis of rotation within the Earth itself, though these would 
be improbable, unless, indeed, geologists were prepared to abandon the view 
' that where the continents now stand they have always stood ' ; but no such 
effect is possible with respect to the direction of the Earth's axis in space. 
Thus the present condition of obliquity of the Earth's equator could in no 
way be accounted for as a result of geological change, and a further cause 
had to be sought. Darwin foresaw a possibility of obtaining an explanation 
in the frictional resistance to which the tidal oscillations of the mobile parts 
of a planet must be subject. The investigation of this hypothesis gave rise 
to a remarkable series of papers of far-reaching consequence in theories of 
cosmogony and of the present constitution of the Earth. 

1 Vol. 89 A, p. i. 



SIR GEORGE DARWIN XXXIX 

" In the first of these papers, which is of preparatory character, ' On the 
Bodily Tides of Viscous and Semi-elastic Spheroids, and on the Ocean Tides 
on a Yielding Nucleus' (Phil Trans., 1879, vol. 170), he adapts the analysis 
of Sir William Thomson, relating to the tidal deformations of an elastic 
sphere, to the case of a sphere composed of a viscous liquid or, more generally, 
of a material which partakes of the character either of a solid or a fluid 
according to the nature of the strain to which it is subjected. For momentary 
deformations it is assumed to be elastic in character, but the elasticity is 
considered as breaking down with continuation of the strain in such a manner 
that under very slow variations of the deforming forces it will behave sensibly 
as if it were a viscous liquid. The exact law assumed by Darwin was dictated 
rather by mathematical exigencies than by any experimental justification, but 
the evidence afforded by the flow of rocks under continuous stress indicates 
that it represents, at least in a rough manner, the mechanical properties 
which characterise the solid parts of the Earth. 

" The chief practical result of this paper is summed up by Darwin himself 
by saying that it is strongly confirmatory of the view already maintained by 
Kelvin that the existence of ocean tides, which would otherwise be largely 
masked by the yielding of the ocean bed to tidal deformation, points to 
a high effective rigidity of the Earth as a whole. Its value, however, 
lies further in the mathematical expressions derived for the reduction in 
amplitude and retardation in phase of the tides resulting from viscosity 
which form the starting-point for the further investigations to which the 
author proceeded. 

"The retardation in phase or 'lag' of the tide due to the viscosity 
implies that a spheroid as tidally distorted will no longer present a 
symmetrical aspect as if no such cause were operative. The attractive forces 
on the nearer and more distant parts will consequently form a non-equi- 
librating system with resultant couples tending to modify the state of 
rotation of the spheroid about its centre of gravity. The action of these 
couples, though exceedingly small, will be cumulative with lapse of time, 
and it is their cumulative effects over long intervals which form the subject 
of the next paper, 'On the Precession of a Viscous Spheroid and on the 
Remote History of the Earth ' (Phil. Trans., 1879, vol. 170, Part II, pp. 447 
530). The case of a single disturbing body (the Moon) is first considered, 
but it is shown that if there are two such bodies raising tidal disturbances 
(the Sun and Moon) the conditions will be materially modified from the 
superposed results of the two disturbances considered separately. Under 
certain conditions of viscosity and obliquity the obliquity of the ecliptic 
will increase, and under others it will diminish, but the analysis further 
yields ' some remarkable results as to the dynamical stability or instability 
of the system... for moderate degrees of viscosity, the position of zero 



xl THE SCIENTIFIC WORK OF 

obliquity is unstable, but there is a position of stability at a high obliquity. 
For large viscosities the position of zero obliquity becomes stable, and 
(except for a very close approximation to rigidity) there is an unstable 
position at a larger obliquity, and again a stable one at a still larger one.' 

" The reactions of the tidal disturbing force on the motion of the Moon 
are next considered, and a relation derived connecting that portion of the 
apparent secular acceleration of the Moon's mean motion, which cannot be 
otherwise accounted for by theory, with the heights and retardations of the 
several bodily tides in the Earth. Various hypotheses are discussed, but with 
the conclusion that insufficient evidence is available to form 'any estimate 
having any pretension to accuracy... as to the present rate of change due to 
tidal friction.' 

"But though the time scale involved must remain uncertain, the nature 
of the physical changes that are taking place at the present time is practi- 
cally free from obscurity. These involve a gradual increase in the length 
of the day, of the month, and of the obliquity of the ecliptic, with a gradual 
recession of the Moon from the Earth. The most striking result is that 
these changes can be traced backwards in time until a state is reached when 
the Moon's centre would be at a distance of only about 6000 miles from the 
Earth's surface, while the day and month would be of equal duration, 
estimated at 5 hours 36 minutes. The minimum time which can have 
elapsed since this condition obtained is further estimated at about 54 million 
years. This leads to the inevitable conclusion that the Moon and Earth at 
one time formed parts of a common mass and raises the question of how and 
why the planet broke up. The most probable hypothesis appeared to be 
that, in accordance with Laplace's nebular hypothesis, the planet, being 
partly or wholly fluid, contracted, and thus rotated faster and faster, until the 
ellipticity became so great that the equilibrium was unstable. 

" The tentative theory put forward by Darwin, however, differs from the 
nebular hypothesis of Laplace in the suggestion that instability might set 
in by the rupture of the body into two parts rather than by casting off a 
ring of matter, somewhat analogous to the rings of Saturn, to be afterwards 
consolidated into the form of a satellite. 

"The mathematical investigation of this hypothesis forms a subject to 
which Darwin frequently reverted later, but for the time he devoted himself 
to following up more minutely the motions which would ensue after the 
supposed planet, which originally consisted of the existing Earth and Moon 
in combination, had become detached into two separate masses. In the 
final section of a paper 'On the Secular Changes in the Elements of the 
Orbit of a Satellite revolving about a Tidally Distorted Planet' (Phil. 
Trans., 1880, vol. 171), Darwin summarises the results derived in his 
different memoirs. Various factors ignored in the earlier investigations, 



SIR GEORGE DARWIN xli 

such as the eccentricity and inclination of the lunar orbit, the distribution 
of the heat generated by tidal friction and the effects of inertia, were duly 
considered and a complete history traced of the evolution resulting from 
tidal friction of a system originating as two detached masses nearly in 
contact with one another and rotating nearly as though they were parts 
of one rigid body. Starting with the numerical data suggested by the 
Earth-Moon System, ' it is only necessary to postulate a sufficient lapse of 
time, and that there is not enough matter diffused through space to resist 
materially the motions of the Moon and Earth,' when 'a system would 
necessarily be developed which would bear a strong resemblance to our own.' 
'A theory, reposing on verae causae, which brings into quantitative correlation 
the lengths of the present day and month, the obliquity of the ecliptic, 
and the inclination and eccentricity of the lunar orbit, must, I think, have 
strong claims to acceptance.' 

" Confirmation of the theory is sought and found, in part at least, in the 
case of other members of the Solar System which are found to represent 
various stages in the process of evolution indicated by the analysis. 

" The application of the theory of tidal friction to the evolution of the 
Solar System and of planetary sub-systems other than the Earth-Moon 
System is, however, reconsidered later, ' On the Tidal Friction of a Planet 
attended by Several Satellites, and on the Evolution of the Solar System ' 
(Phil. Trans., 1882, vol. 172). The conclusions drawn in this paper are 
that the Earth-Moon System forms a unique example within the Solar 
System of its particular mode of evolution. While tidal friction may 
perhaps be invoked to throw light on the distribution of the satellites 
among the several planets, it is very improbable that it has figured as the 
dominant cause of change of the other planetary systems or in the Solar 
System itself." 

For some years after this series of papers Darwin was busy with practical 
tidal problems but he returned later " to the problems arising in connection 
with the genesis of the Moon, in accordance with the indications previously 
arrived at from the theory of tidal friction. It appeared to be of interest to 
trace back the changes which would result in the figures of the Earth and 
Moon, owing to their mutual attraction, as they approached one another. 
The analysis is confined to the consideration of two bodies supposed con- 
stituted of homogeneous liquid. At considerable distances the solution of the 
problem thus presented is that of the equilibrium theory of the tides, but, 
as the masses are brought nearer and nearer together, the approximations 
available for the latter problem cease to be sufficient. Here, as elsewhere, 
when the methods of analysis could no longer yield algebraic results, Darwin 
boldly proceeds to replace his symbols by numerical quantities, and thereby 
succeeds in tracing, with considerable approximation, the forms which such 



xlii THE SCIENTIFIC WORK OF 

figures would assume when the two masses are nearly in contact. He even 
carries the investigation farther, to a stage when the two masses in part 
overlap. The forms obtained in this case can only be regarded as satisfying 
the analytical, and not the true physical conditions of the problem, as, of 
course, two different portions of matter cannot occupy the same space. 
They, however, suggest that, by a very slight modification of conditions, 
a new form could be found, which would fulfil all the conditions, in which 
the two detached masses are united into a single mass, whose shape has been 
variously described as resembling that of an hour-glass, a dumb-bell, or a pear. 
This confirms the suggestion previously made that the origin of the Moon was 
to be sought in the rupture of the parent planet into two parts, but the theory 
was destined to receive a still more striking confirmation from another source. 

"While Darwin was still at work on the subject, there appeared the great 
memoir by M. Poincare, 'Sur 1'equilibre d'une masse fiuide animee d'un 
mouvement de rotation ' (Acta Math., vol. 7). 

"The figures of equilibrium known as Maclaurin's spheroid and Jacobi's 
ellipsoid were already familiar to mathematicians, though the conditions of 
stability, at least of the latter form, were not established. By means of 
analysis of a masterly character, Poincare succeeded in enunciating and 
applying to this problem the principle of exchange of stabilities. This prin- 
ciple maybe briefly indicated as follows: Imagine a dynamical system such as 
a rotating liquid planet to be undergoing evolutionary change such as would 
result from a gradual condensation of its mass through cooling. Whatever 
be the varying element to which the evolutionary changes may be referred, 
it may be possible to define certain relatively simple modes of motion, the 
features associated with which will, however, undergo continuous evolution. 
If the existence of such modes has been established, M. Poincare shows that 
the investigation of their persistence or 'stability' may be made to depend 
on the evaluation of certain related quantities which he defines as coeffi- 
cients of stability. The latter quantities will be subject to evolutionary 
change, and it may happen that in the course of such change one or more 
of them assumes a zero value. Poincare shows that such an occurrence 
indicates that the particular mode of motion under consideration coalesces 
at this stage with one other mode which likewise has a vanishing coefficient 
of stability. Either mode will, as a rule, be possible before the change, but 
whereas one will be stable the other will be unstable. The same will be 
true after the change, but there will be an interchange of stabilities, whereby 
that which was previously stable will become unstable, and vice versd. 
An illustration of this principle was found in the case of the spheroids of 
Maclaurin and the ellipsoids of Jacobi. The former in the earlier stages of 
evolution will represent a stable condition, but as the ellipticity of surface 
increases a stage is reached where it ceases to be stable and becomes unstable. 



SIR GEORGE DARWIN xliii 

At this stage it is found to coalesce with Jacobi's form which involves in its 
further development an ellipsoid with three unequal axes. Poincare shows 
that the latter form possesses in its earlier stages the requisite elements of 
stability, but that these in their turn disappear in the later developments. 
In accordance with the principle of exchange of stabilities laid down by 
him, the loss of stability will occur at a stage where there is coalescence 
with another form of figure, to which the stability will be transferred, and 
this form he shows at its origin resembles the pear which had already been 
indicated by Darwin's investigation. The supposed pear-shaped figure was 
thus arrived at by two entirely different methods of research, that of Poincare 
tracing the processes of evolution forwards and that of Darwin proceeding 
backwards in time. 

" The chain of evidence was all but complete ; it remained, however, to 
consider whether the pear-shaped figure indicated by Poincare, stable in its 
earlier forms, could retain its stability throughout the sequence of changes 
necessary to fill the gap between these forms and the forms found by Darwin. 

" In later years Darwin devoted much time to the consideration of this 
problem. Undeterred by the formidable analysis which had to be faced, he 
proceeded to adapt the intricate theory of Ellipsoidal Harmonics to a form in 
which it would admit of numerical application, and his paper ' Ellipsoid 
Harmonic Analysis' (Phil. Trans., A, 1901, vol. 197), apart from the appli- 
cation for which it was designed, in itself forms a valuable contribution 
to this particular branch of analysis. With the aid of these preliminary 
investigations he succeeded in tracing with greater accuracy the form of the 
pear-shaped figure as established by Poincare, ' On the Pear-shaped Figure of 
Equilibrium of a Rotating Mass of Liquid' (Phil. Trans., A, 1901, vol. 198), 
and, as he considered, in establishing its stability, at least in its earlier forms. 
Some doubt, however, is expressed as to the conclusiveness of the argument 
employed, as simultaneous investigations by M. Liapounoff pointed to an 
opposite conclusion. Darwin again reverts to this point in a further paper 
'On the Figure and Stability of a Liquid Satellite' (Phil. Trans., A, 1906, 
vol. 206), in which is considered the stability of two isolated liquid masses in 
the stage at which they are in close proximity, i.e., the condition which would 
obtain, in the Earth-Moon System, shortly after the Moon had been severed 
from the Earth. The ellipsoidal harmonic analysis previously developed is 
then applied to the determination of the approximately ellipsoidal forms 
which had been indicated by Roche. The conclusions arrived at seem to 
point, though not conclusively, to instability at the stage of incipient rupture, 
but in contradistinction to this are quoted the results obtained by Jeans, who 
considered the analogous problems of the equilibrium and rotation of infinite 
rotating cylinders of liquid. This problem is the two-dimensional analogue 
of the problems considered by Darwin and Poincare', but involves far greater 

d2 



xliv THE SCIENTIFIC WORK OF 

simplicity of the conditions. Jeans finds solutions of his problem strictly 
analogous to the spheroids of Maclaurin, the ellipsoids of JacobL and the 
pear of Poincare, and is able to follow the development of the latter until the 
neck joining the two parts has become quite thin. He is able to establish 
conclusively that the pear is stable in its early stages, while there is no 
evidence of any break in the stability up to the stage when it divides itself 
into two parts." 

Darwin's own final comments on this work next find a place here. 
He is writing the preface to the second volume of his Collected Works in 
1908, after which time nothing new on the subject came from his pen. 
"The observations of Dr Hecker," he says, "and of others do not afford 
evidence of any considerable amount of retardation in the tidal oscilla- 
tions of the solid earth, for, within the limits of error of observation, the 
phase of the oscillation 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.... 

" 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 configurations 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 dif- 
ference 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 



SIR GEORGE DARWIN xlv 

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 
maintain 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 the theory of evolution under 
tidal influence as 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 con- 
tended 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 
solidification. But I have always felt convinced that the earth would adjust 
its ellipticity to its existing speed of rotation with close approximation." 

After some remarks concerning the effects of the discovery of radio- 
activity and the energy resident in the atom on estimates of geological time, 
he continues, " 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." 

Since this was written much information bearing on the point has been 
gathered from the stellar universe. The curious curves of light-changes in 
certain classes of spectroscopic binaries have been well explained on the 
assumption that the two stars are close together and under strong tidal 
distortion. Some of these, investigated on the same hypothesis, even seem 
to be in actual contact. In chap, xx of the third edition (1910) of his book 
on the Tides, Darwin gives a popular summary of this evidence which had 



xlvi THE SCIENTIFIC WORK OF 

in the interval been greatly extended by the discovery and application of 
the hypothesis to many other similar systems. In discussing the question 
Darwin sets forth a warning. He points out that most of the densities 
which result from the application of the tidal theory are very small compared 
with that of the sun, and he concludes that these stars are neither homo- 
geneous nor incompressible. Hence the figures calculated for homogeneous 
liquid can only be taken to afford a general indication of the kind of figure 
which we might expect to find in the stellar universe. 

Perhaps Darwin's greatest service to cosmogony was the successful effort 
which he made to put hypotheses to the test of actual calculation. Even 
though the mathematical difficulties of the subject compel the placing of 
many limitations which can scarcely exist in nature, yet the solution of even 
these limited problems places the speculator on a height which he cannot 
hope to attain by doubtful processes of general reasoning. If the time 
devoted to the framing and setting forth of cosmogonic hypotheses by various 
writers had been devoted to the accurate solution of some few problems, the 
newspapers and popular scientific magazines might have been less interesting 
to their readers, but we should have had more certain knowledge of our 
universe. Darwin himself engaged but little in speculations which were 
not based on observations or precise conclusions from definitely stated 
assumptions, and then only as suggestions for further problems to be 
undertaken by himself or others. And this view of progress he communi- 
cated to his pupils, one of whom, Mr J. H. Jeans, as mentioned above, is 
continuing with success to solve these gravitational problems on similar 
lines. 

The nebular hypothesis of Kant and Laplace has long held the field as 
the most probable mode of development of our solar system from a nebula. 
At the present time it is difficult to say what are its chief features. Much 
criticism has been directed towards every part of it, one writer changing 
a detail here, another there, and still giving to it the name of the best known 
exponent. The only salient point which seems to be left is the main hypo- 
thesis that the sun, planets and satellites were somehow formed during the 
process of contraction of a widely diffused mass of matter to the system as 
we now see it. Some writers, including Darwin himself, regard a gaseous 
nebula contracting under gravitation as the essence of Laplace's hypotheses, 
distinguishing this condition from that which originates in the accretion 
of small masses. Others believe that both kinds of matter may be present. 
After all it is only a question of a name, but it is necessary in a discussion to 
know what the name means. 

Darwin's paper, "The mechanical conditions of a swarm of meteorites," 
is an attempt to show that, with reasonable hypotheses, the nebula and the 
small masses under contraction by collisions may have led to the same result. 



SIR GEORGE DARWIN xlvii 

In his preface to volume IV he says with respect to this paper : " Cosraogonists 
are of course compelled to begin their survey of the solar system at some 
arbitrary stage of its history, and they do not, in general, seek to explain 
how the solar nebula, whether gaseous or meteoritic, came to exist. My 
investigation starts from the meteoritic point of view, and I assume the 
meteorites to be moving indiscriminately in all directions. But the doubt 
naturally arises as to whether at any stage a purely chaotic motion of the 
individual meteorites could have existed, and whether the assumed initial 
condition ought not rather to have been an aggregate of flocks of meteorites 
moving about some central condensation in orbits which intersect one another 
at all sorts of angles. If this were so the chaos would not be one consisting 
of individual stones which generate a quasi-gas by their collisions, but it 
would be a chaos of orbits. But it is not very easy to form an exact picture 
of this supposed initial condition, and the problem thus seems to elude 
mathematical treatment. Then again have I succeeded in showing that a 
pair of meteorites in collision will be endowed with an effective elasticity ? 
If it is held that the chaotic motion and the effective elasticity are quite 
imaginary, the theory collapses. It should however be remarked that an 
infinite gradation is possible between a chaos of individuals and a chaos 
of orbits, and it cannot be doubted that in most impacts the colliding stones 
would glance from one another. It seems to me possible, therefore, that my 
two fundamental assumptions may possess such a rough resemblance to truth 
as to produce some degree of similitude between the life-histories of gaseous 
and meteoritic nebulae. If this be so the Planetesimal Hypothesis of 
Chamberlain and Moulton is nearer akin to the Nebular Hypothesis than 
the authors of the former seem disposed to admit. 

" Even if the whole of the theory could be condemned as futile, yet the 
paper contains an independent solution of the problem of Lane and Ritter ; 
and besides the attempt to discuss the boundary of an atmosphere, where 
the collisions have become of vanishing rarity, may still perhaps be worth 
something." 

In writing concerning the planetesimal hypothesis, Darwin seems to have 
forgotten that one of its central assumptions is the close approach of two 
stars which by violent tidal action drew off matter in spiral curves which 
became condensed into the attendants of each. This is, in fact, one of the 
most debatable parts of the hypothesis, but one on which it is possible to 
get evidence from the distribution of such systems in the stellar system. 
Controversy on the main issue is likely to exist for many years to come. 

Quite early in his career Darwin was drawn into practical tidal problems 
by being appointed on a Committee of the British Association with Adams, 
to coordinate and revise previous reports drawn up by Lord Kelvin. He 
evidently felt that the whole subject of practical analysis of tidal observations 



xlviii THE SCIENTIFIC WORK OF 

needed to be set forth in full and made clear. His first report consequently 
contains a development of the equilibrium theory of the Tides, and later, 
after a careful analysis of each harmonic component, it proceeds to outline in 
detail the methods which should be adopted to obtain the constants of each 
component from theory or observation, as the case needed. Schedules and 
forms of reduction are given with examples to illustrate their use. 

There are in reality two principal practical problems to be considered. 
The one is the case of a port with much traffic, where it is possible to obtain 
tide heights at frequent intervals and extending over a long period. While 
the accuracy needed usually corresponds to the number of observations, it is 
always assumed that the ordinary methods of harmonic analysis by which all 
other terms but that considered are practically eliminated can be applied ; 
the corrections when this is not the case are investigated and applied. The 
other problem is that of a port infrequently visited, so that we have only 
a short series of observations from which to obtain the data for the compu- 
tation of future tides. The possible accuracy here is of course lower than in 
the former case but may be quite sufficient when the traffic is light. In his 
third report Darwin takes up this question. The main difficulty is the 
separation of tides which have nearly the same period and which could not 
be disentangled by harmonic analysis of observations extending over a very 
few weeks. Theory must therefore be used, not only to obtain the periods, 
but also to give some information about the amplitudes and phases if this 
separation is to be effected. The magnitude of the tide-generating force is 
used for the purpose. Theoretically this should give correct results, but it is 
often vitiated by the form of the coast line and other circumstances depending 
on the irregular shape of the water boundary. Darwin shows however that 
fair prediction can generally be obtained ; the amount of numerical work is 
of course much smaller than in the analysis of a year's observations. This 
report was expanded by Darwin into an article on the Tides for the Admiralty 
Scientific Manual. 

Still another problem is the arrangement of the analysis when times and 
heights of high and low water alone are obtainable ; in the previous papers 
the observations were supposed to be hourly or obtained from an automatic- 
ally recording tide-gauge. The methods to be used in this case are of course 
well known from the mathematical side ; the chief problem is to reduce the 
arithmetical work and to put the instructions into such a form that the 
ordinary computer may use them mechanically. The problem was worked 
out by Darwin in 1890, and forms the subject of a long paper in the 
Proceedings of the Royal Society. 

A little later he published the description of his now well known abacus, 
designed to avoid the frequent rewriting of the numbers when the harmonic 
analysis for many different periods is needed. Much care was taken to obtain 



SIR GEORGE DARWIN xlix 

the right materials. The real objection to this, and indeed to nearly all the 
methods devised for the purpose, is that the arrangement and care of the 
mechanism takes much longer time than the actual addition of the numbers 
after the arrangement has been made. In this description however there 
are more important computing devices which reduce the time of computa- 
tion to something like one-fifth of that required by the previous methods. 
The principal of these is the one in which it is shown how a single set 
of summations of 9000 hourly values can be made to give a good many 
terms, by dividing the sums into proper groups and suitably treating 
them. 

Another practical problem was solved in his Bakerian Lecture " On Tidal 
Prediction." In a previous paper, referred to above, Darwin had shown how 
the tidal constants of a port might be obtained with comparatively little 
expense from a short series of high and low water observations. These, 
however, are of little value unless the port can furnish the funds necessary 
to predict the future times and heights of the tides. Little frequented ports 
can scarcely afford this, and therefore the problem of replacing such pre- 
dictions by some other method is necessary for a complete solution. " The 
object then," says Darwin, " of the present paper, is to show how a general 
tide-table, applicable for all time, may be given in such a form that anyone, 
with an elementary knowledge of the Nautical Almanac, may, in a few 
minutes, compute two or three tides for the days on which they are required. 
The tables will also be such that a special tide-table for any year may be 
computed with comparatively little trouble." 

This, with the exception of a short paper dealing with the Tides in the 
Antarctic as shown by observations made on the Discovery, concludes Darwin's 
published work on practical tidal problems. But he was constantly in corre- 
spondence about the subject, and devoted a good deal of time to government 
work and to those who wrote for information. 

In connection with these investigations it was natural that he should 
turn aside at times to questions of more scientific interest. Of these the 
fortnightly tide is important because by it some estimate may be reached as 
to the earth's rigidity. The equilibrium theory while effective in giving the 
periods only for the short-period tides is much more nearly true for those of 
long period. Hence, by a comparison of theory and observation, it is possible 
to see how much the earth yields to distortion produced by the moon's 
attraction. Two papers deal with this question. In the first an attempt is 
made to evaluate the corrections to the equilibrium theory caused by the 
continents; this involves an approximate division of the land and sea 
surfaces into blocks to which calculation may be applied. In the second 
tidal observations from various parts of the earth are gathered together for 
comparison with the theoretical values. As a result, Darwin obtains the 



1 THE SCIENTIFIC WORK OF 

oft-quoted expression for the rigidity of the earth's mass, namely, that it is 
effectively about that of steel. An attempt made by George and Horace 
Darwin to measure the lunar disturbance of gravity by means of the 
pendulum is in reality another approach to the solution of the same problem. 
The attempt failed mainly on account of the local tremors which were pro- 
duced by traffic and other causes. Nevertheless the two reports contain 
much that is still interesting, and their value is enhanced by a historical 
account of previous attempts on the same lines. Darwin had the satisfaction 
of knowing that this method was later successful in the hands of Dr Hecker 
whose results confirmed his first estimate. Since his death the remarkable 
experiment of Michelson 1 with a pipe partly filled with water has given 
a precision to the determination of this constant which much exceeds that 
of the older methods ; he concludes that the rigidity and viscosity are at least 
equal to and perhaps exceed those of steel. 

It is here proper to refer to Darwin's more popular expositions of the 
work of himself and others. He wrote several articles on Tides, notably for 
the Encyclopaedia Britannica and for the Encyclopaedic der Mathematischen 
Wissenschaften, but he will be best remembered in this connection for his 
volume The Tides which reached its third edition not long before his 
death. The origin of it was a course of lectures in 1897 before the Lowell 
Institute of Boston, Massachusetts. An attempt to explain the foundations 
and general developments of tidal theory is its main theme. It naturally 
leads on to the subject of tidal friction and the origin of the moon, and 
therewith are discussed numerous questions of cosmogony. From the point 
of view of the mathematician, it is not only clear and accurate but gives the 
impression, in one way, of a tour de force. Although Darwin rarely has to 
ask the reader to accept his conclusions without some description of the 
nature of the argument by which they are reached, there is not a single 
algebraic symbol in the whole volume, except in one short footnote where, on 
a minor detail, a little algebra is used. The achievement of this, together 
with a clear exposition, was no light task, and there are few examples to be 
found in the history of mathematics since the first and most remarkable of all, 
Newton's translation of the effects of gravitation into geometrical reasoning. 
The Tides has been translated into German (two editions), Hungarian, 
Italian and Spanish. 

In 1877 the two classical memoirs of G. W. Hill on the motion of the 
moon were published. The first of these, Researches in the Lunar Tlieory, 
contains so much of a pioneer character that in writing of any later work on 
celestial mechanics it is impossible to dismiss it with a mere notice. One 
portion is directly concerned with a possible mode of development of the 
lunar theory and the completion of the first step, in the process. The usual 

1 Astrophysical Journal, March, 1914. 



SIR GEORGE DARWIN 11 

method of procedure has been to consider the problem of three bodies as an 
extension of the case of two bodies in which the motion of one round the 
other is elliptic. Hill, following a suggestion of Euler which had been 
worked out by the latter in some detail, starts to treat the problem as a 
very special particular case of the problem of three bodies. One of them, 
the earth, is of finite mass ; the second, the sun, is of infinite mass and at 
an infinite distance but is revolving round the former with a finite and 
constant angular velocity. The third, the moon, is of infinitesimal mass, but 
moves at a finite distance from the earth. Stated in this way, the problem 
of the moon's motion appears to bear no resemblance to reality. It is, 
however, nothing but a limiting case where certain constants, which are 
small in the case of the actual motion, have zero values. The sun is 
actually of very great mass compared with the earth, it is very distant as 
compared with the distance of the moon, its orbit round the earth (or vice 
versd) is nearly circular, and the moon's mass is small compared with that 
of the earth. The differential equations which express the motion of 
the moon under these limitations are fairly simple and admit of many 
transformations. 

Hill simplifies the equations still further, first by supposing the moon 
so started that it always remains in the same fixed plane with the earth 
and the sun (its actual motion outside this plane is small). He then uses 
moving rectangular axes one of which always points in the direction of the 
sun. Even with all these limitations, the differential equations possess many 
classes of solutions, for there will be four arbitrary constants in the most 
general values of the coordinates which are to be derived in the form of a 
doubly infinite series of harmonic terms. His final simplification is the 
choice of one of these classes obtained by giving a zero value to one of 
the arbitrary constants ; in the moon's motion this constant is small. The 
orbit thus obtained is of a simple character but it possesses one important 
property; relative to the moving axes it is closed and the body following 
it will always return to the same point of it (relative to the moving axis) 
after the lapse of a definite interval. In other words, the relative motion 
is periodic. 

Hill develops this solution literally and numerically for the case of our 
satellite with high accuracy. This accuracy is useful because the form of 
the orbit depends solely on the ratio of the mean rates of motion of the sun 
and moon round the earth, and these rates, determined from centuries of 
observation, are not affected by the various limitations imposed at the outset. 
The curve does not differ much from a circle to the eye but it includes the 
principal part of one of the chief differences of the motion from that in a 
circle with uniform velocity, namely, the inequality long known as the 
"variation "; hence the name since given to it, " the Variational Orbit." Hill, 



Ill THE SCIENTIFIC WORK OF 

however, saw that it was of more general interest than its particular appli- 
cation to our satellite. He proceeds to determine its form for other values 
of the mean rates of motion of the two bodies. This gives a family of 
periodic orbits whose form gradually varies as the ratio is changed; the 
greater the ratio, the more the curve differs from a circle. 

It is this idea of Hill's that has so profoundly changed the whole outlook 
of celestial mechanics. Poincare took it up as the basis of his celebrated 
prize essay of 1887 on the problem of three bodies and afterwards expanded 
his work into the three volumes ; Les methodes nouvelles de la Mecanique 
Celeste. His treatment throughout is highly theoretical. He shows that 
there must be many families of periodic orbits even for specialised problems 
in the case of three bodies, certain general properties are found, and much 
information concerning them which is fundamental for future investigation 
is obtained. 

It is doubtful if Darwin had paid any special attention to Hill's work 
on the moon for at least ten" years after its appearance. All this time he 
was busy with the origin of the moon and with tidal work. Adams had 
published a brief reswne of his own work on lines similar to those of Hill 
immediately after the memoirs of the latter appeared, but nothing further 
on the subject came from his pen. The medal of the Royal Astronomical 
Society was awarded to Hill in 1888, and Dr Glaisher's address on his work 
contains an illuminating analysis of the methods employed and the ideas 
which are put forward. Probably both Darwin and Adams had a con- 
siderable share in making the recommendation. Darwin often spoke of his 
difficulties in assimilating the work of others off his own beat and possibly 
this address started him thinking about the subject, for it was at his recom- 
mendation in the summer of 1888 that the writer took up the study of Hill's 
papers. "They seem to be veiy good," he said, "but scarcely anyone knows 
much about them." 

He lectured on Hill's work for the first time in the Michaelmas Term 
of 1893, and writes of his difficulties in following parts of them, more 
particularly that on the Moon's Perigee which contains the development of 
the infinite determinant. He concludes, "I can't get on with my own work 
until these lectures are over but Hill's papers are splendid." One of his 
pupils on this occasion was Dr P. H. Cowell, now Director of the Nautical 
Almanac office. The first paper of the latter was a direct result of these 
lectures and it was followed later by a valuable series of memoirs in which 
the constants of the lunar orbit and the coefficients of many of the periodic 
terms were obtained with great precision. Soon after these lectures Darwin 
started his own investigations on the subject. But they took a different 
line. The applications to the motion of the moon were provided for and 
Poincare had gone to the foundations. Darwin felt, however, that the work of 



SIR GEORGE DARWIN liii 

the latter was far too abstract to satisfy those who, like himself, frequently 
needed more concrete results, either for application or for their own mental 
satisfaction. In discussing periodic orbits he set himself the task of tracing 
numbers of them in order, as far as possible, to get a more exact knowledge 
of the various families which Poincare's work had shown must exist. Some 
of Hill's original limitations are dropped. Instead of taking a sun of infinite 
mass and at an infinite distance, he took a mass ten times that of the 
planet and at a finite distance from that body. The orbit of each round 
the other is circular and of uniform motion, the third body being still of 
infinitesimal mass. Any periodic orbit which may exist is grist to his mill 
whether it circulate about one body or both or neither. 

Darwin saw little hope of getting any extensive results by solutions of 
the differential equations in harmonic series. It was obvious that the slow- 
ness of convergence or the divergence would render the work far too doubtful. 
He adopted therefore the tedious process of mechanical quadratures, starting 
at an arbitrary position on the #-axis with an arbitrary speed in a direction 
parallel to the y-axis. Tracing the orbit step-by-step, he again reaches the 
#-axis. If the final velocity there is perpendicular to the axis, the orbit is 
periodic. If not, he starts again with a different speed and traces another 
orbit. The process is continued, each new attempt being judged by the 
results of the previous orbits, until one is obtained which is periodic. The 
amount of labour involved is very great since the actual discovery of a 
periodic orbit generally involved the tracing of from three to five or even 
more non-periodic paths. Concerning one of the orbits he traced for his last 
paper on the subject, he writes: "You may judge of the work when I tell 
you that I determined 75 positions and each averaged f hr. (allowing for 
correction of small mistakes which sometimes is tedious). You will see 
that it is far from periodic....! have now got six orbits of this kind." And all 
this to try and find only one periodic orbit belonging to a class of whose 
existence he was quite doubtful. 

Darwin's previous work on figures of equilibrium of rotating fluids made 
the question of the stability of the motion in these orbits a prominent factor 
in his mind. He considered it an essential part in their classification. To 
determine this property it was necessary, after a periodic orbit had been 
obtained, to find the effect of a small variation of the conditions. For this 
purpose, Hill's second paper of 1877, on the Perigee of the Moon, is used. 
After finding the variation orbit in his first paper, Hill makes a start 
towards a complete solution of his limited differential equations by finding 
an orbit, not periodic and differing slightly from the periodic orbit already 
obtained. The new portion, the difference between the two, when expressed 
as a sum of harmonic terms, contains an angle whose uniform rate of change, 
c, depends only on the constants of the periodic orbit. The principal 



Hv THE SCIENTIFIC WORK OF 

portion of Hill's paper is devoted to the determination of c with great 
precision. For this purpose, the infinite determinant is introduced and 
expanded into infinite series, the principal parts of which are expressed by 
a finite number of well known functions; the operations Hill devised to 
achieve this have always called forth a tribute to his skill. Darwin uses 
this constant c in a different way. If it is real, the orbit is stable, if 
imaginary, unstable. In the latter case, it may be a pure imaginary or a 
complex number ; hence the necessity for the two kinds of unstability. 

In order to use Hill's method, Darwin is obliged to analyse a certain 
function of the coordinates in the periodic orbit into a Fourier series, and to 
obtain the desired accuracy a large number of terms must be included. 
For the discovery of c from the infinite determinant, he adopts a mode of 
expansion of his own better suited to the purpose in hand. But in any case 
the calculation is laborious. In a later paper, he investigates the stability 
by a different method because Hill's method fails when the orbit has 
sharp flexures. 

For the classification into families, Darwin follows the changes according 
to variations in the constant of relative energy, C. The differential equations 
referred to the moving axes admit a Jacobian integral, the constant of 
which is C. One property of this integral Hill had already developed, 
namely, that the curve obtained by making the kinetic energy zero is one 
which the body cannot cross. Darwin draws the curves for different values 
of C with care. He is able to show in several cases the origin of the 
families he has found and much use is made of Poincare's proposition, that 
all such families originate in pairs, for following the changes. But even 
his material is sometimes insufficient, especially where two quite different 
pairs of families originate near the same point on the #-axis, and some later 
corrections of the classification partly by himself and partly by Mr S. S. Hough 
were necessary. In volume iv of his collected works these corrections are 
fully explained. 

The long first memoir was published in 1896. Nothing further on the 
subject appeared from his hand until 1909 when a shorter paper containing 
a number of new orbits was printed in the Monthly Notices of the Royal 
Astronomical Society. Besides some additions and corrections to his older 
families he considers orbits of ejection and retrograde orbits. During the 
interval others had been at work on similar lines while Darwin with 
increasing duties thrust upon him only found occasional opportunities to 
keep his calculations going. A final paper which appears in the present 
volume was the outcome of a request by the writer that a trial should be 
made to find a member of a librating class of orbits for the mass ratio 1:10 
which had been shown to exist and had been traced for the mass ratio 1 : 1048. 
The latter arose in an attempt to consider the orbits of the Trojan group of 



SIR GEORGE DARWIN Iv 

asteroids. He failed to find one but in the course of his work discovered 
another class of great interest, which shows the satellite ultimately falling 
into the planet. He concludes, "My attention was first drawn to periodic 
orbits by the desire to discover how a Laplacian ring could coalesce into 
a planet. With this object in view I tried to discover how a large planet 
could affect the mean motion of a small one moving in a circular orbit at 
the same mean distance. After various failures the investigation drifted 
towards the work of Hill and Poincare, so that the original point of view 
was quite lost and it is not even mentioned in my paper on 'Periodic Orbits.' 
It is of interest, to me at least, to find that the original aspect of the problem 
has emerged again." It is of even greater interest to one of his pupils to 
find that after more than twenty years of work on different lines in celestial 
mechanics, Darwin's last paper should be on the same part of the subject to 
which both had been drawn from quite different points of view. 

Thus Darwin's work on what appeared to be a problem in celestial 
mechanics of a somewhat unpractical nature sprang after all from and 
finally tended towards the question which had occupied his thoughts nearly 
all his life, the genesis and evolution of the solar system. 



INAUGUEAL LECTUEE 

(DELIVERED AT CAMBRIDGE, IN 1883, ON ELECTION TO THE 
PLUMIAN PROFESSORSHIP) 

I PROPOSE to take advantage of the circumstance that this is the first of 
the lectures which I am to give, to say a few words on the Mathematical 
School of this University, and especially of the position of a professor in 
regard to teaching at the present time. 

There are here a number of branches of scientific study to which there 
are attached laboratories, directed by professors, or by men who occupy the 
position and do the duties of professors, but do not receive their pay from, 
nor full recognition by, the University. Of these branches of science I have 
comparatively little to say. 

You are of course aware of the enormous impulse which has been given 
to experimental science in Cambridge during the last ten years. It would 
indeed have been strange if the presence of such men as now stand at the 
head of those departments had not created important Schools of Science. 
And yet when we consider the strange constitution of our University, it 
may be wondered that they have been able to accomplish this. I suspect 
that there may be a considerable number of men who go through their 
University course, whose acquaintance with the scientific activity of the place 
is limited by the knowledge that there is a large building erected for some 
obscure purpose in the neighbourhood of the Corn Exchange. Is it possible 
that any student of Berlin should be heard to exclaim, " Helmholtz, who is 
Helmholtz ?" And yet some years ago I happened to mention the name of 
one of the greatest living mathematicians, a professor in this University, 
in the presence of a first class man and fellow of his College, and he made 
just such an exclamation. 

This general state of apathy to the very existence of science here has 

now almost vanished, but I do not think I have exaggerated what it was 

some years ago. Is not there a feeling of admiration called for for those, who 

by their energy and ability have raised up all the activity which we now see? 

D.L. 1 



2 INAUGURAL LECTURE 

For example, Foster arrived here, a stranger to the University, without 
University post or laboratory. I believe that during his first term Balfour 
and one other formed his whole class. And yet holding only that position 
of a College lecturer which he holds at this minute, he has come to make 
Cambridge the first Physiological School of Great Britain, and the range of 
buildings which the University has put at his disposal has already proved 
too small for his requirements*. His pupil Balfour had perhaps a less 
uphill game to play, for the germs of the School of Natural Science were 
already laid when he began his work as a teacher. But he did not merely 
aid in the further developments of what he found, for he struck out in a 
new line that line of study which his own original work has gone, I 
believe, a very long way to transform and even create. He did not live 
to see the full development of the important school and laboratory which 
he had founded. But thanks to his impulse it is now flourishing, and will 
doubtless prosper under the able hands into which the direction has fallen. 
His name ought surely to live amongst us for what he did ; for those who 
had the fortune to be his friends the remembrance of him cannot die, for 
what he was. 

I should be going too far astray were I to continue to expatiate on the 
work of Rayleigh, Stuart, and the others who are carrying on the develop- 
ment of practical work in various branches within these buildings. It must 
suffice to say that each school has had its difficulties, and that those diffi- 
culties have been overcome by the zeal of those concerned in the management. 

But now let us turn to the case of the scientific professors who have no 
laboratories to direct, and I speak now of the mathematical professors. In 
comparison with the prosperity of which I have been speaking, I think 
it is not too much to say that there is no vitality. I belong to this class of 
professors, and I am far from flattering myself that I can do much to impart 
life to the system. But if I shall not succeed I may perhaps be pardoned 
if I comfort myself by the reflection, that it may not be entirely my own fault. 

The University has however just entered on a new phase; I have the 
honour to be the first professor elected under the new Statutes now in force. 
A new scheme for the examinations in Mathematics is in operation, and it 
may be that such an opportunity will now be afforded as has hitherto been 
wanting. We can but try to avail ourselves of the chance. 

To what causes are we to assign the fact that our most eminent 
teachers of mathematics have hitherto been very frequently almost without 
classes ? It surely cannot be that Stokes, Adams and Cayley have nothing 
to say worth hearing by students of mathematics. Granting the possibility 

* Sir Michael Foster was elected the first Professor of Physiology a few weeks after the 
delivery of this lecture. 



INAUGURAL LECTURE 3 

that a distinguished man may lack the power of exposition, yet it is inad- 
missible that they are all deficient in that respect. No, the cause is not far 
to seek, it lies in the Mathematical Tripos. How far it is desirable that the 
system should be so changed, that it will be advisable for students in their 
own interest to attend professorial lectures, I am not certain ; but it can 
scarcely be doubted that if there were no Tripos, the attendance at such 
lectures would be larger. 

In hearing the remarks which I am about to make on the Mathematical 
Tripos, you must bear in mind that I have hitherto taken no part in mathe- 
matical teaching of any kind, and therefore must necessarily be a bad judge 
of the possibilities of mathematical training, and of its effects on most minds. 
A year and a half ago I took part as Additional Examiner in the Mathe- 
matical Tripos, and I must confess that I was a good deal discouraged by what 
I saw. Now do not imagine that I flatter myself I was one jot better in all 
these respects than others, when I went through the mill. I too felt the 
pressure of time, and scribbled down all I could in my three hours, and 
doubtless presented to my examiners some very pretty muddles. I can only 
congratulate myself that the men I examined were not my competitors. 

In order to determine whether anything can be done to improve this 
state of things, let us consider the merits and demerits of our Mathematical 
School. One of the most prominent evils is that our system of examination 
has a strong tendency to make men regard the subjects more as a series of 
isolated propositions than as a whole ; and much attention has to be paid to a 
point, which is really important for the examination, viz. where to begin and 
where to leave off in answering a question. The coup d'oeil of the whole 
subject is much impaired ; but this is to some extent inherent in any system 
of examination. This result is, however, principally due to our custom of 
setting the examinees to reproduce certain portions of the books which they 
have studied ; that is to say this evil arises from the " bookwork " questions. 
I have a strong feeling that such questions should be largely curtailed, and 
that the examinees should by preference be asked for transformations and 
modifications of the results obtained in the books. I suppose a certain amount 
of bookwork must be retained in order to permit patient workers, who are 
not favoured by any mathematical ability, to exhibit to the examiners that 
they have done their best. But for men with any mathematical power 
there can be no doubt that such questions as I suggest would give a far 
more searching test, and their knowledge of the subject would not have 
to be acquired in short patches. 

I should myself like to see an examination in which the examinees were 
allowed to take in with them any books they required, so that they need not 
load their memories with formulae, which no original worker thinks of trying 

12 



4 INAUGURAL LECTURE 

to remember. A first step in this direction has been taken by the intro- 
duction of logarithm tables into the Senate House; and I fancy that a 
terrible amount of incompetence was exhibited in the result. I may remark 
by the way that the art of computation is utterly untaught here, and that 
readiness with figures is very useful in ordinary life. I have done a good 
deal of such work myself, but I had to learn it by practice and from a few 
useful hints from others who had mastered it. 

It is to be regretted that questions should be set in examinations which 
are in fact mere conjuring tricks with symbols, a kind of double acrostic; 
another objectionable class of question is the so-called physical question which 
has no relation to actual physics. This kind of question was parodied once 
by reference to "a very small elephant, whose weight may be neglected, etc." 
Examiners have often hard work to find good questions, and their difficulties 
are evidenced by such problems as I refer to. I think, however, that of late 
this kind of exercise is much less frequent than formerly. 

I am afraid the impression is produced in the minds of many, that if 
a problem cannot be solved in a few hours, it cannot be solved at all. At any 
rate there seems to be no adequate realisation of the process by which most 
original work is done, when a man keeps a problem before him for weeks, 
months, years and gnaws away from time to time when any new light may 
strike him. 

I think some of our text books are to blame in this ; they impress the 
student in the same way that a high road must appear to a horse with 
blinkers. The road stretches before him all finished and macadamised, 
having existed for all he knows from all eternity, and he sees nothing of 
by-ways and foot-paths. Now it is the fact that scarcely any subject is so 
way worn that there are not numerous unexplored by-paths, which may lead 
across to undiscovered countries. I do not advocate that the student should 
be led along and made to examine all the cul-de-sacs and blind alleys, as he 
goes ; he would never get on if he did so, but I do protest against that tone 
which I notice in many text books that mathematics is a spontaneously 
growing fruit of the tree of knowledge, and that all the fruits along that 
road have been gathered years ago. Rather let him see that the whole 
grand work is the result of the labours of an army of men, each exploring 
his little bit, and that there are acres of untouched ground, where he too may 
gather fruit : true, if he begins on original work, he may think that he has 
discovered something new and may very likely find that someone has been 
before him; but at least he too will have had the enormous pleasure of 
discovery. 

There is another fault in the system of examinations, but I hardly know 
whether it can be appreciably improved. It is this : the system gives very 



INAUGURAL LECTURE 5 

little training in the really important problem both of practical life and of 
mathematics, viz. the determination of the exact nature of the question 
which is to be attacked, the making up of your mind as to what you will do. 
Everyone who has done original work knows that at first the subject gene- 
rally presents itself as a chaos of possible problems, and careful analysis 
is necessary before that chaos is disentangled. The process is exactly that 
of a barrister with his brief. A pile of papers is set before him, and from 
that pile he has to extract the precise question of law or fact on which 
the whole turns. When he has mastered the story and the precise point, 
he has generally done the more difficult part of his work. In most cases, 
it is exactly the same in mathematical work; and when the question has 
been pared down until its characteristics are those of a Tripos question, of 
however portentous a size, the battle is half won. It only remains to the 
investigator then to avail himself of all the "morbid aptitude for the 
manipulation of symbols" which he may happen to possess. 

In examination, however, the whole of this preparatory part of the work 
is done by the examiner, and every examiner must call to mind the weary 
threshing of the air which he has gone through in trying "to get a question" 
out of a general idea. Now the limitation of time in an examination makes 
this evil to a large extent irremediable; but it seems to me that some good 
may be done by requesting men to write essays on particular topics, 
because in this case their minds are not guided by a pair of rails carefully 
prepared by an examiner. 

In the report on the Tripos for 1882, I spoke of the slovenliness of style 
which characterised most of the answers. It appears to me that this is really 
much more than a mere question of untidiness and annoyance to examiners. 
The training here seems to be that form and style are matters of no moment, 
and answers are accordingly sent up in examination which are little more 
than rough notes of solutions. But I insist that a mathematical writer 
should attend to style as much as a literary man. 

Some of our Cambridge writers on mathematics seem never to have 
recovered from the ill effects of their early training, even when they devote 
the rest of their life to original work. I wish some of you would look at the 
artistic mode of presentation practised by Gauss, and compare it with the 
standard of excellence which passes muster here. Such a comparison will 
not prove gratifying to our national pride. 

Where there is slovenliness of style it is, I think, almost certain that 
there will be wanting that minute attention to form on which the successful, 
or at least easy, marshalling of a complex analytical development depends. 
The art of carrying out such work has to be learnt by trial and error by 
the men trained in our school, and yet the inculcation of a few maxims 



6 -INAUGURAL LECTURE 

would generally be of great service to students, provided they are made to 
attend to them in their work. The following maxims contain the pith of 
the matter, although they might be amplified with advantage if I were to 
detain you over this point for some time. 

1st. Choose the notation with great care, and where possible use a 
standard notation. 

2nd. Break up the analysis into a series of subsections, each of which 
may be attended to in detail. 

3rd. Never attempt too many transformations in one operation. 

4th. Write neatly and not quickly, so that in passing from step to step 
there may be no mistakes of copying. 

A man who undertakes any piece of work, and does not attend to some 
such rules as these, doubles his chances of mistake ; even to short pieces 
of work such as examination questions the same applies, and I have little 
doubt that many a score of questions have been wrongly worked out from 
want of attention to these points. 

It is true that great mathematicians have done their work in very 
various styles, but we may be sure that those who worked untidily gave 
themselves much unnecessary trouble. Within my own knowledge I may 
say that Thomson [Lord Kelvin] works in a copy-book, which is produced at 
Railway Stations and other conveniently quiet places for studious pursuits ; 
Maxwell worked in part on the backs of envelopes and loose sheets of paper 
crumpled up in his pocket*; Adams' manuscript is as much a model of 
neatness in mathematical writing as Person's of Greek writing. There is, of 
course, no infallibility in good writing, but believe me that untidiness surely 
has its reward in mistakes. I have spoken only on the evils of slovenliness 
in its bearing on the men as mathematicians I cannot doubt that as a 
matter of general education it is deleterious. 

I have dwelt long on the demerits of our scheme, because there is hope 
of amending some of them, but of the merits there is less to be said because 
they are already present. The great merit of our plan seems to me to be 
reaped only by the very ablest men in the year. It is that the student is 
enabled to get a wide view over a great extent of mathematical country, 
and if he has not assimilated all his knowledge thoroughly, yet he knows 
that it is so, and he has a fair introduction to many subjects. This 
advantage he would have lost had he become a pure specialist and original 
investigator very early in his career. But this advantage is all a matter 
of degree, and even the ablest man cannot cover an indefinitely long course 

* I think that he must have been only saved from error by his almost miraculous physical 
insight, and by a knowledge of the time when work must be done neatly. But his Electricity 
was crowded with errata, which have now been weeded out one by one. 



INAUGURAL LECTURE 7 

in his three years. Year by year new subjects were being added to the 
curriculum, and the limit seemed to have been exceeded ; whilst the 
disastrous effects on the weaker brethren were becoming more prominent. 
I cannot but think that the new plan, by which a man shall be induced to 
become a partial specialist, gives us better prospects. 

Another advantage we gain by our strict competition is that a man must 
be bright and quick; he must not sit mooning over his papers; he is quickly 
brought to the test, either he can or he cannot do a definite problem in 
a finite time if he cannot he is found out. Then if our scheme checks 
original investigation, it at least spares us a good many of those pests of 
science, the man who churns out page after page of x, y, z, and thinks he 
has done something in producing a mass of froth. That sort of man is 
quickly found out here, both for his own good and the good of the world 
at large. Lastly this place has the advantage of having been the training 
school of nearly all the English mathematicians of eminence, and of having 
always attracted as it continues to attract whatever of mathematical 
ability is to be found in the country. These are great merits, and in the 
endeavour to remove blemishes, we must see that we do not destroy them. 

A discussion of the Mathematical Tripos naturally brings us face to face 
with a much abused word, namely "Cram." 

The word connotes bad teaching, and accordingly teaching with reference 
to examinations has been supposed to be bad because it has been called 
cram. The whole system of private tuition commonly called coaching has 
been nick-named cram, and condemned accordingly. I can only say for 
myself that I went to a private tutor whose name is familiar to everyone 
in Cambridge, and found the most excellent and thorough teaching; far 
be it from me to pretend that I shall prove his equal as a teacher. What- 
ever fault is to be found, it is not with the teaching, but it lies in the 
system. It is obviously necessary that when a vast number of new subjects 
are to be mastered the most rigorous economy in the partition of the student's 
time must be practised, and he is on no account to be allowed to spend 
more than the requisite minimum on any one subject, even if it proves 
attractive to him. The private tutor must clearly, under the old regime, 
act as director of studies for his pupils strictly in accordance with exami- 
nation requirements; for place in the Tripos meant pounds, shillings, and 
pence to the pupil. The system is now a good deal changed, and we may 
hope that it will be possible henceforth to keep the examination less 
incessantly before the student, who may thus become a student of a subject, 
instead of a student for a Tripos. 

And now. I think you must see the peculiar difficulties of a professor of 
mathematics ; his vice has been that he tried to teach a subject only, and 



8 INAUGURAL LECTURE 

private tutors felt, and felt justly, that they could not, in justice to their 
pupils' prospects, conscientiously recommend the attendance at more than 
a very small number of professorial lectures. But we are now at the begin- 
ning of a new regime and it may be that now the professors have their 
chance. But I think it depends much more on the examiners than on the 
professors. If examiners can and will conduct the examinations in such 
a manner that it shall "pay" better to master something thoroughly, than 
to have a smattering of much, we shall see a change in the manner of 
learning. Otherwise there will not be much change. I do not know how 
it will turn out, but I do know that it is the duty of professors to take such 
a chance if it exists. 

My purpose is to try my best to lecture in such a way as will impart an 
interest to the subject itself and to help those who wish to learn, so that 
they may reap advantage in examinations provided the examinations are 
conducted wisely. 



INTRODUCTION TO DYNAMICAL ASTRONOMY 

THE field of dynamical astronomy is a wide one and it is obvious that 
it will be impossible to consider even in the most elementary manner 
all branches of it ; for it embraces all those effects in the heavens which may 
be attributed to the effects of gravitation. In the most extended sense of 
the term it may be held to include theories of gravitation itself. Whether 
or not gravitation is an ultimate fact beyond which we shall never penetrate 
is as yet unknown, but Newton, whose insight into physical causation was 
almost preternatural, regarded it as certain that some further explanation 
was ultimately attainable. At any rate from the time of Newton down to 
to-day men have always been striving towards such explanation it must be 
admitted without much success. The earliest theory of the kind was that 
of Lesage, promulgated some 170 years ago. He conceived all space to be 
filled with what he called ultramundane corpuscles, moving with very great 
velocities in all directions. They were so minute and so sparsely distributed 
that their mutual collisions were of extreme rarity, whilst they bombarded 
the grosser molecules of ordinary matter. Each molecule formed a partial 
shield to its neighbours, and this shielding action was held to furnish an 
explanation of the mutual attraction according to the law of the inverse 
square of the distance, and the product of the areas of the sections of the 
two molecules. Unfortunately for this theory it is necessary to assume that 
there is a loss of energy at each collision, and accordingly there must be 
a perpetual creation of kinetic energy of the motion of the ultramundane 
corpuscles at infinity. The theory is further complicated by the fact that 
the energy lost by the corpuscle at each collision must have been communi- 
cated to the molecule of matter, and this must occur at such a rate as to 
vaporize all matter in a small fraction of a second. Lord Kelvin has, how- 
ever, pointed out that there is a way out of this fundamental difficulty, for 
if at each collision the ultramundane corpuscle should suffer no loss of total 
kinetic energy but only a transformation of energy of translation into energy 
of internal vibration, the system becomes conservative of energy and the 
eternal creation of energy becomes unnecessary. On the other hand, gravi- 
tation will not be transmitted to infinity, but only to a limited distance. 



10 INTRODUCTION TO DYNAMICAL ASTRONOMY 

I will not refer further to this conception save to say that I believe that no 
man of science is disposed to accept it as affording the true road. 

It may be proved that if space were an absolute plenum of incompressible 
fluid, and that if in that fluid there were points towards which the fluid 
streams from all sides and disappears, those points would be urged towards 
one another with a force varying inversely as the square of the distance 
and directly as the product of the intensities of the two inward streams. 
Such points are called sinks and the converse, namely points from whence 
the fluid streams, are called sources. Now two sources also attract one 
another according to the same law; on the other hand a source and a sink 
repel one another. If we could conceive matter to be all sources or all sinks 
we should have a mechanical theory of gravitation, but no one has as yet 
suggested any means by which this can be realised. Bjerknes of Christiania 
has, however, suggested a mechanical means whereby something of the kind 
may be realised. Imagine an elastic ball immersed in water to swell and 
contract rhythmically, then whilst it is contracting the motion of the sur- 
rounding water is the same as that due to a sink at its centre, and whilst 
it is expanding the motion is that due to a source. Hence two balls which 
expand and contract in exactly the same phase will attract according to the 
law of gravitation on taking the average over a period of oscillation. If, 
however, the pulsations are in opposite phases the resulting force is one of 
repulsion. If then all matter should resemble in some way the pulsating 
balls we should have an explanation, but the absolute synchronism of the 
pulsations throughout all space imports a condition which does not commend 
itself to physicists. I may mention that Bjerknes has actually realised these 
conclusions by experiment. Although it is somewhat outside our subject 
I may say that if a ball of invariable volume should execute a small 
rectilinear oscillation, its advancing half gives rise to a source and the 
receding half to a sink, so that the result is what is called a doublet. Two 
oscillating balls will then exercise on one another forces analogous to that 
of magnetic particles, but the forces of magnetism are curiously inverted. 
This quasi-magnetism of oscillating balls has also been treated experi- 
mentally by Bjerknes. However curious and interesting these speculations 
and experiments may be, I do not think they can afford a working hypothesis 
of gravitation. 

A new theory of gravitation which appears to be one of extraordinary 
ingenuity has lately been suggested by a man of great power, viz. Osborne 
Reynolds, but I do not understand it sufficiently to do more than point 
out the direction towards which he tends. He postulates a molecular ether. 
I conceive that the molecules of ether are all in oscillation describing orbits 
in the neighbourhood of a given place. If the region of each molecule be 
replaced by a sphere those spheres may be packed in a hexagonal arrangement 



INTRODUCTION TO DYNAMICAL ASTRONOMY 11 

completely filling all space. We may, however, come to places where the 
symmetrical piling is interrupted, and Reynolds calls this a region of misfit. 

Then, according to this theory, matter consists of misfit, so that matter is 
the deficiency of molecules of ether. Reynolds claims to show that whilst 
the particular molecules which don't fit are continually changing the amount 
of misfit is indestructible, and that two misfits attract one another. The 
theory is also said to explain electricity. Notwithstanding that Reynolds 
is not a good exponent of his own views, his great achievements in science 
are such that the theory must demand the closest scrutiny. 

The newer theories of electricity with which the name of Prof. J. J. 
Thomson is associated indicate the possibility that mass is merely an electro- 
dynamic phenomenon. This view will perhaps necessitate a revision of all 
our accepted laws of dynamics. At any rate it will be singular if we shall 
have to regard electrodynamics as the fundamental science, and subsequently 
descend from it to the ordinary laws of motion. How much these notions 
are in the air is shown by the fact that at a congress of astronomers, held in 
1902 at Gottingen, the greater part of one day's discussion was devoted 
to the astronomical results which would follow from the new theory of 
electrons. 

I have perhaps said too much about the theories of gravitation, but it 
should be of interest to you to learn how it teems with possibilities and how 
great is the present obscurity. 

Another important subject which has an intimate relationship with 
Dynamical Astronomy is that of abstract dynamics. This includes the 
general principles involved in systems in motion under the action of con- 
servative forces and the laws which govern the stability of systems. Perhaps 
the most important investigators in this field are Lagrange and Hamilton, 
and in more recent times Lord Kelvin and Poincare. 

Two leading divisions of dynamical astronomy are the planetary theory 
and the theory of the motion of the moon and of other satellites. A first 
approximation in all these cases is afforded by the case of simple elliptic 
motion, and if we are to consider the case of comets we must include 
parabolic and hyperbolic motion round a centre. Such a first approximation 
is, however, insufficient for the prediction of the positions of any of the bodies 
in our solar system for any great length of time, and it becomes necessary 
to include the effects of the disturbing action of one or more other bodies. 
The problem of disturbed revolution may be regarded as a single problem 
in all its cases, but the defects of our analysis are such that in effect its 
several branches become very distinct from one another. It is usual to 
speak of the problem of disturbed revolution as the problem of three bodies, 
for if it were possible to solve the case where there are three bodies we 



12 INTRODUCTION TO DYNAMICAL ASTRONOMY 

should already have gone a long way towards the solution of that more 
complex case where there are any number of bodies. 

Owing to the defects of our analysis it is at present only possible to 
obtain accurate results of a general character by means of tedious expansions. 
All the planets and all the satellites have their motions represented with 
more or less accuracy by ellipses, but this first approximation ceases to be 
satisfactory for satellites much more rapidly than is the case for planets. 
The eccentricities of the ellipses and the inclinations of the orbits are in most 
cases inconsiderable. It is assumed then that it is possible to effect the 
requisite expansions in powers of the eccentricities and of suitable functions 
of the inclinations. Further than this it is found necessary to expand in 
powers of the ratios of the mean distances of the disturbed and disturbing 
bodies from the centre. It is at this point that the first marked separation 
of the lunar and planetary theories takes place. In the lunar theory the 
distance of the sun (disturber) from the earth is very great compared with 
that of the moon, and we naturally expand in this ratio in order to start 
with as few terms as possible. In the planetary theory the ratio of the 
distances of the disturbed and disturbing bodies two planets from the sun 
may be a large fraction. For example, the mean distances of Venus and the 
earth are approximately in the ratio 7 : 10, and in order to secure sufficient 
accuracy a large number of terms is needed. In the case of the planetary 
theory the expansion is delayed as long as possible. 

Again, in the lunar theory the mass of the disturbing body is very 
great compared with that of the primary, a ratio on which it is evident that 
the amount of perturbation greatly depends. On the other hand, in the 
planetary theory the disturbing body has a very small mass compared with 
that of the primary, the sun. From these facts we are led to expect that 
large terms will be present in the expressions for the motion of the moon 
due to the action of the sun, and that the later terms in the expansion will 
rapidly decrease; and in the planetary theory we expect large numbers of 
terms all of about equal magnitude and none of them very great. This 
expectation is, however, largely modified by some further remarks to be made. 

You know that a dynamical system may have various modes of free 
oscillation of various periods. If then a disturbing force with a period differ- 
ing but little from that of one of the modes of free oscillation acts on the 
system for a long time it will generate an oscillation of large amplitude. 

A familiar instance of this is in the roll of a ship at sea. If the incidence 
of the waves on the ship is such that the succession of impulses is very 
nearly identical in period with the natural period of the ship, the roll becomes 
large. In analysis this physical fact is associated with a division by a small 
divisor on integration. 



INTRODUCTION TO DYNAMICAL ASTRONOMY 13 

As an illustration of the simplest kind suppose that the equation of motion 
of a system under no forces were 



Then we know that the solution is 

x = A cos nt + B sin nt, 

that is to say the free period is . Suppose then such a system be acted on 
by a perturbing force F cos (n e) t, where e is small; the equation of motion is 



and the solution corresponding to such a disturbing force is 



If e is small the amplitude becomes great, and this arises, as has been said, by 
a division by a small divisor. 

Now in both lunar and planetary theories the coefficients of the periodic 
terms become frequently much greater than might have been expected 
a priori. In the lunar theory before this can happen in such a way as to 
cause much trouble the coefficients have previously become so small that it 
is not necessary to consider them. But suppose in the planetary theory n, n' 
are the mean motions of two planets round the primary. Then coefficients 
will continually be having multipliers of the forms 



n 



and 



Y 



in i'n' \in i'n'J 

where i, i' are small positive integers. In general the larger i, i' the smaller is 
the coefficient to begin with, but owing to the fact that the ratio n : n may 
very nearly approach that of two small integers a coefficient may become very 
great; e.g. 5 Jovian years nearly equal 2 of Saturn, while the ratio of 
the mean distances is 6 : 11. The result is a large long inequality with a 
period of 913 years in the motions of those two planets. The periods of the 
principal terms in the moon's motion are generally short, but some have 
large coefficients, so that the deviation from elliptic motion is well marked. 

The general problem of three bodies is in its infancy, and as yet but little 
is known as to the possibilities in the way of orbits and as to their stabilities. 

Another branch of our subject is afforded by the precession and nutation 
of the earth, or any other planet, under the influence of the attractions of 
disturbing bodies. This is the problem of disturbed rotation and it presents 
a strong analogy with the problem of disturbed elliptic motion. When a top 



14 INTRODUCTION TO DYNAMICAL ASTRONOMY 

spins with absolute steadiness we say that it is asleep. Now the earth in its 
rotation may be asleep or it may not be so there is nothing but observation 
which is capable of deciding whether it is so or not. This is equally true 
whether the rotation takes place under external perturbation or not. If the 
earth is asleep its motion presents a perfect analogy with circular orbital 
motion; if it wobbles the analogy is with elliptic motion. The analogy is 
such that the magnitude of the wobble corresponds with the eccentricity of 
orbit and the position of greatest departure with the longitude of pericentre. 
Until the last 20 years it has always been supposed that the earth is asleep 
in its rotation, but the extreme accuracy of modern observation, when sub- 
jected to the most searching analysis by Chandler and others, has shewn 
that there is actually a small wobble. This is such that the earth's axis of 
rotation describes a small circle about the pole of figure. The theory of 
precession indicated that this circle should be described in a period of 
305 days, and all the earlier astronomers scrutinised the observations with 
the view of detecting such an inequality. It was this preconception, appa- 
rently well founded, which prevented the detection of the small inequality 
in question. It was Chandler who first searched for an inequality of un- 
known period and found a clearly marked one with a period of 428 days. 
He found also other smaller inequalities with a period of a year. This 
wandering of the pole betrays itself most easily to the observer by changes 
in the latitude of the place of observation. 

The leading period in the inequality of latitude is then one of 428 days. 
Tha theoretical period of 305 days was, as I have said, apparently well 
established, but after the actual period was found to be 428 days Newcomb 
pointed out that if the earth is not absolutely rigid, but slightly changes 
its shape as the axis of rotation wanders, such a prolongation of period 
would result. Thus these purely astronomical observations end by affording 
a measure of the effective rigidity of the earth's mass. 

The theory of the earth's figure and the variation of gravity as we vary 
our position on the surface or the law of variation of gravity as we descend 
into mines are to be classified as branches of dynamical astronomy, although 
in these cases the velocities happen to be zero. This theory is intimately 
connected with that of precession, for it is from this that we conclude that 
the free wobble of the perfectly rigid earth should have a period of 305 days. 
The ellipticity of the earth's figure also has an important influence on the 
motion of the moon, and the determination of a certain inequality in the 
moon's motion affords the means of finding the amount of ellipticity of the 
earth's figure with perhaps as great an accuracy as by any other means. 
Indeed in the case of Jupiter, Saturn, Mars, Uranus and Neptune the 
ellipticity is most accurately determined in this way. The masses also of the 
planets may be best determined by the periods of their satellites. 



INTRODUCTION TO DYNAMICAL ASTRONOMY 15 

The theory of Saturn's rings is another branch. The older and now 
obsolete views that the rings are solid or liquid gave the subject various 
curious and difficult mathematical investigations. The modern view now 
well established that they consist of an indefinite number of meteorites 
which collide together from time to time presents a number of problems of 
great difficulty. These were ably treated by Maxwell, and there does not 
seem any immediate prospect of further extension in this direction. 

Then the theory of the tides is linked to astronomy through the fact that 
it is the moon and sun which cause the tides, so that any inequality in their 
motions is reflected in the ocean. 

On the fringe of our subject lies the whole theory of figures of equi- 
librium of rotating liquids with the discussion of the stability of the various 
possible forms and the theory of the equilibrium of gaseous planets. In this 
field there is yet much to discover. 

This subject leads on immediately to theories of the origin of planetary 
systems and to cosmogony. Tidal theory, on the hypothesis that the tides 
are resisted by friction, leads to a whole series of investigations in speculative 
astronomy whose applications to cosmogony are of great interest. 

Up to a recent date there was little evidence that gravitation held good 
outside the solar system, but recent investigations, carried out largely by 
means of the spectroscopic determinations of velocities of stars in the line of 
sight, have shewn that there are many other systems, differing very widely 
from our own, where the motions seem to be susceptible of perfect ex- 
planation by the theory of gravitation. These new extensions of gravitation 
outside our system are leading to many new problems of great difficulty 
and we may hope in time to acquire wider views as to the possibilities of 
motion in the heavens. 

This hurried sketch of our subject will shew how vast it is, and I cannot 
hope in these lectures to do more than touch on some of the leading topics. 



HILL'S LUNAR THEORY 



1. Introduction*. 

AN account of Hill's Lunar Theory can best be prefaced by a few 
quotations from Hill's original papers. These will indicate the peculiarities 
which mark off his treatment from that of earlier writers and also, to some 
extent, the reasons for the changes he introduced. Referring to the well- 
known expressions which give, for undisturbed elliptic motion, the rectangular 
coordinates as explicit functions of the time expressions involving nothing 
more complicated than Bessel's functions of integral order Hill writes : 

" Here the law of series is manifest, and the approximation can easily be 
carried as far as we wish. But the longitude and latitude, variables employed 
by nearly all lunar theorists, are far from having such simple expressions; in 
fact their coefficients cannot be finitely expressed in terms of Besselian 
functions. And if this is true in the elliptic theory how much more likely is 
a similar thing to be true when the complexity of the problem is increased 
by the consideration of disturbing forces?... There is also another advantage 
in employing coordinates of the former kind (rectangular) : the differential 
equations are expressed in purely algebraic functions, while with the latter 
(polar) circular functions immediately present themselves." 

In connection with the parameters to be used in the expansions Hill 
argues thus: 

" Again as to parameters all those who have given literal developments, 
Laplace setting the example, have used the parameter m, the ratio of the 
sidereal month to the sidereal year. But a slight examination, even of the 
results obtained, ought to convince anyone that this is a most unfortunate 
selection in regard to convergence. Yet nothing seems to render the 
parameter desirable, indeed the ratio of the synodic month to the sidereal 
year would appear to be more naturally suggested as a parameter." 

* The references in this section are to Hill's "Researches in the Lunar Theory" first published 
(1878) in the American Journal of Mathematics, vol. i. pp. 5 26, 129 147 and reprinted in 
Collected Mathematical Works, vol. i. pp. 284 335. Hill's other paper connected with these 
lectures is entitled "On the Part of the Motion of the Lunar Perigee which is a function of the 
Mean Motions of the Sun and Moon," published separately in 1877 by John Wilson and Son, 
Cambridge, Mass., and reprinted in Acta Mathematica, vol. vin. pp. 136, 1886 and in Collected 
Mathematical Works, vol. i. pp. 243270. 



COORDINATES 17 

When considering the order of the differential equations and the method 
of integration, Hill wrote: 

"Again the method of integration by undetermined coefficients is most 
likely to give us the nearest approach to the law of series; and in this 
method it is as easy to integrate a differential equation of the second order 
as one of the first, while the labour is increased by augmenting the number 
of variables and equations. But Delaunay's method doubles the number of 
variables in order that the differential equations may be all of the first order. 
Hence in this disquisition I have preferred to use the equations expressed in 
terms of the coordinates rather than those in terms of the elements ; and, in 
general, always to diminish the number of unknown quantities and equations 
by augmenting the order of the latter. In this way the labour of making a 
preliminary development of R in terms of the elliptic elements is avoided." 

We may therefore note the characteristics of Hill's method as follows: 

(1) Use of rectangular coordinates. 

(2) Expansion of series in powers of the ratio of the synodic month to 
the sidereal year. 

(3) Use of differential equations of the second order which are solved by 
assuming series of a definite type and equating coefficients. 

In these lectures we shall obtain only the first approximation to the 
solution of Hill's differential equations. The method here followed is not 
that given by Hill, although it is based on the same principles as his method. 
Our work only involves simple algebra, and probably will be more easily 
understood than Hill's. If followed in detail to further approximations, it 
would prove rather tedious, but it leads to the results we require without too 
much labour. If it is desired to follow out the method further, reference 
should be made to Hill's own writings. 



2. Differential Equations of Motion and Jacobis Integral. 

Let E, M, m denote the masses or positions of the earth, moon, and sun, 
and let G be the centre of inertia of E and M. Let x, y, z be the rect- 
angular coordinates of M with E as origin, and let x, y', z' be the coordinates 
of m referred to parallel axes through G. The coordinates of M relative to 

P 1 ff F 

the axes through G are clearly - ^x, " ^y, j^ + j^ z ' those of E are 

The distances EM, Em', Mm are denoted 



.x 



E+M E+M y ' E-+M 



18 



HILLS LUNAR THEORY 



by r, r lt A respectively. It is assumed that G describes a Keplerian ellipse 
round mf so that x, y ', z! are known functions of the time. The accelerations 
of M relative to E are shewn in the diagram. 




We have 



Fig. 1. 

a? + f + ^ 2 , ' 



E 



Hence 



MX 



M 



E 



, 



Ex 
E+M 



. . the direction cosines of EM are -=- , ^- , ^, 

dx dy dz 



, E + M/dr, dr, 3rA 

Em are ^ I *"*, x , -^ , 
M \dx dy dz/' 



dy 

E+M/dk 3A 3A\ 

,, Mm are ^ -^ , -^ , ^ ) . 

E \dx dy dz / 

If X, Y, Z denote the components of acceleration of M relative to axes 
through E, 



FORCE FUNCTION 



19 



m' E + M 8r x ' 
A 2 E 3x rf M Hx 



where 



M \ m> E + M m/ E+ M 



r A E n M 

Similarly, F=|^, Z-^. 
' 



(1). 



then 



Let r be the distance between G and m', and let be the angle m'OM ; 



and 



/ 
rr 



Since r is very small compared with r', 



E + M I 



Hence F = ^~ + 

But the second term does not involve x, y, z, and may be dropped. 

.*. F = -| TJ- (| cos 2 6 |) (2), 

r 3 
neglecting terms in -^ . 

We will now find an approximate expression for F, paying attention to 
the magnitude of the various terms in the actual earth-moon-sun system. 
As a first rough approximation, r' is a constant a', and Gm' rotates with 
uniform angular velocity ri. This neglects the effect on the sun of the earth 
and moon not being collected at G (this effect is very small), and it neglects 
the eccentricity of the solar orbit. In order that the coordinates of the sun 
relative to the earth might be nearly constant, we introduce axes x, y 

22 



20 HILL'S LUNAR THEORY 

rotating with angular velocity n' in the plane of the sun's orbit round the 
earth; the #-axis being so chosen that it passes through the sun. When 
required, a -axis is taken perpendicular to the plane of x, y. As before, let 
x, y, z be the coordinates of the moon ; the sun's coordinates will be approxi- 
mately a, 0, 0. In this approximation r cos 6 = x and 
E+M ' _ ^ 
r a 3 a 3 

This suggests the following general form for F, instead of that given in 
equation (2) : 

E + M , m' ,/r 2 cos 2 



For the sake of future developments, we now introduce a new notation. 

Let v be the moon's synodic mean motion and put m = = --- ,*. In the 

v n-ri 

case of our moon, m is approximately ^ : this is a small quantity in 
powers of which our expressions will be obtained. If we neglect E and M 

compared with m', we have m' = w' 2 a' 3 , whence r . = n' 3 = i/ 2 m 2 . Let us also 

a 3 

write E + M = xv 2 , and then we get 



For convenience we write 
/d /s 

and then F + $ n' 2 (x 2 + y 2 ) = v 2 P + m 2 (3# 2 - z>) + n"| . 

The equations of motion for uniformly rotating axesf are 
d*x . ,dn 



d*y ,dx dF 

j-% + 2n ^7 - n y = ^- 

dt* dt dy 

d*z = aP 

dt 2 ~ dz 

* In the lunar theory n' is supposed to be a known constant, while n (or m) is one of the 
constants of integration the value of which is not yet determined and can only be determined 
from the observations. So far n (or m) is quite arbitrary. 

t See any standard treatise on Dynamics. 



JACOBI'S INTEGRAL 



21 



which give 



We might write T= vt, and on dividing the equations by i/ 2 use T hence- 
forth as equivalent to time; or we might choose a special unit of time such 
that i' is unity. In either case our equations become 



d*x dy KX , an 

-T-J - 2m -/- + - 3m 2 a; = 
dr 2 dr 1* ex 



d*y 

d*z 
d? 



dx 



icy 
+ 



a_n 
^ 
a_n 

dz 



If we multiply these equations respectively by 2 
them, we have 



.(3). 



, 2 - and add 



_ /an ^ an dy a_n ^ 
+ ' 



The whole of the left-hand side is a complete differential ; the right-hand 

side needs the addition of the term 2 -^-. 

OT 

Let us put for brevity 



Then 



If the earth moved round the sun with uniform angular velocity n', the 
axis of x would always pass through the sun, and therefore we should have 



r 



y' = z' = 



and 



,, xx + yy + zz 
r cos 8 = , - = x, 
r 



22 



HILL'S LUNAR THEORY 



giving 



- r 2 cos 2 6 a? = 0. 



In this case fl would vanish. It follows that fl must involve as a factor 
the eccentricity of the solar orbit. 

It is proposed as a first approximation to neglect that eccentricity, and 
this being the case, our equations become 



^r* - 2m ^ + 7? - 3m2a; = 



.(5). 



Of these equations one integral is known, viz. Jacobi's integral, 



F 2 = 2 



- m 2 ^ 2 + C. 



3. The Variational Curve. 

In ordinary theories the position of a satellite is determined by the 
departure from a simple ellipse fixed or moving. The moving ellipse is 
preferred to the fixed one, because it is found that the departures of the 
actual body from the moving ellipse are almost of a periodic nature. But 
the moving ellipse is not the solution of any of the equations of motion 
occurring in the theory. Instead of referring the true orbit to an ellipse, 
Hill introduced as the orbit of reference, or intermediate orbit, a curve 
suggested by his differential equations, called the "variational curve." 

We have already neglected the eccentricity of the solar orbit, and will 
now go one step further and neglect the inclination of the lunar orbit to the 
ecliptic, so that g disappears. If the path of a body whose motion satisfies 



.(6) 



intersects the a?-axis at right angles, the circumstances of the motion before 
and after intersection are identical, but in reverse order. Thus, if time 
be counted from the intersection, x = /(r 2 ), y = T/(r 2 ) ; for if in the dif- 
ferential equations the signs of y and T are reversed, but x left unchanged, 
the equations are unchanged. 

A similar result holds if the path intersects y at right angles, for if 
x and T have signs changed, but y is unaltered, the equations are unaltered. 



THE VARIATIONAL CURVE 23 

Now it is evident that the body may start from a given point on the 
#-axis, and at right angles to it, with different velocities, and that within 
certain limits it may reach the axis of y and cross it at correspondingly 
different angles. If the right angle lie between some of these, we judge 
from the principle of continuity that there is some intermediate velocity with 
which the body would arrive at and cross the t/-axis at right angles. 

If the body move from one axis to the other, crossing both at right 
angles, it is plain that the orbit is a closed curve symmetrical to both axes. 
Thus is obtained a particular solution of the differential equations. This 
solution is the " variational curve." While the general integrals involve four 
arbitrary constants, the variational curve has but two, which may be taken to 
be the distance from the origin at the x crossing and the time of crossing. 

For the sake of brevity, we may measure time from the instant of 
crossing x. 

Then since x is an even function of T and y an odd one, both of 
period 2?r, it must be possible to expand x and y by Fourier Series thus 

x = A cos T + A l cos 3r + A 2 cos 5r + ...... , 

y = B sin T + B l sin 3r + B 2 sin or + ....... 

When T is a multiple of tr, y = 0; and when it is an odd multiple 

of J, # = 0: also in the first case -5- = and in the second -/- = 0. Thus 
2 dr dr 

these conditions give us the kind of curve we want. It will be noted that 
there are no terms with even multiples of r ; such terms have to be omitted 

if x. ~r~ are to vanish at T = Tr/2, etc. 
dr 

We do not propose to follow Hill throughout the arduous analysis by 
which he determines the nature of this curve with the highest degree of 
accuracy, but will obtain only the first rough approximation to its form 
thereby merely illustrating the principles involved. 

Accordingly we shall neglect all terms higher than those in 3r. It is 
also convenient to change the constants into another form. Thus we write 



We have one constant less than before, but it will be seen that this is 
sufficient, for in fact A l and B^ only differ by terms of an order which we 
are going to neglect. We assume a,, a_! to be small quantities. 
Hence x = (a + a_^ cos T + a l cos 3r, 

y = (a - a_ x ) sin r + ^ sin 3r. 



24 HILL'S LUNAR THEORY 

Since cos 3r = 4 cos 3 r 3 cos r = cos r (1 - 4 sin 2 T), 
sin 3r = - 4 sin 3 T + 3 sin T = sin r (1 4 cos 2 T), 

we have a? = a cos T 



f 1 + ^ + a ~* - --' sin 2 rl ) 
a, a J ^ 



Neglecting powers of a,, a_j higher than the first, we deduce 



1 If a a + a_j 



a a 

ffix 

-^ = - [(ao + a_j) cos T + 9aj cos 3r] = cos T [a,, + 9aj + a_ x 36aj sin 2 T], 

d 2 y 

= - [(a - a_j) sin T + 9a 1 sin 3r] = - sin T [a,, 9aj - a_j -f 36aj cos 2 T]. 



With the required accuracy 
2m ^ = 2ma cos r, 2m ^ 
Substituting these results in the differential equations, (6), we get 



2m = 2ma cos r, 2m - = - 2ma sin r, and 3m 2 ; = 3m 2 ao cos r. 



a 



_ 1 _ I + _ lsinlT 3 = 

a 3 V a,, ao 



sin T - 1 + - _ . coss T _ 2ra 



THE VARIATIONAL CURVE 25 

Equating to zero the coefficients of cos r, cos r sin 2 T, sin T, sin T cos 2 T, 
we get 






ao a 3 

_ i + fetg=; _ 2m + - 3 ( 1 + "" "" ~-' ) =0 

Q Q 3 \ a I 



(7). 



As there are only three equations for the determination of , , - 

a 3 ' a a 

our assumption that A 1 = B l = aj is justified to the order of small quantities 
considered. 

Half the sum and difference of the first two give 
-l-2m-fm 2 + 3 =0, 




Therefore 3 = 1 + 2m + f m 2 , 

JLaj 3a_i = _ to ^ Qrder of accuracV) 
^ a 

also ^ + ^=i = 0, from the third equation ; 

a,, a 



.(8). 



= l + 2m + fm 2 
a*' 

Hence x = a [(1 |f m 2 ) cos T + ^m 2 cos 3r], 

y = ao [(1 4- |fm 2 ) sin T + T 3 ff m 2 sin 3r], 
or perhaps more conveniently for future work 

x ao cos T [1 m 2 f m 2 sin 2 T] \ , . 

y = ao sin T [1 + m 2 -f f m 2 cos 2 T] j 

It will be seen that these are the equations to an oval curve, the semi- 
axes of which are a (l m 2 ), a (l+m 2 ) along and perpendicular to the line 
joining the earth and sun. If r, be the polar coordinates of a point on the 
curve, 

r 2 = ^ [1 - 2m 2 cos 2r], 

giving r = a [1 m 2 cos 2r] 



26 HILL'S LUNAR THEORY 

Also tan = y - = tan r [1 + 2m 2 + f m 2 ] 



giving 6 = T + J^-m 2 sin 2r ........................... (11). 

If a be the mean distance corresponding to a mean motion n in an 
undisturbed orbit, Kepler's third law gives 

KV Z ........................... (12). 



t, , n n n' + n' 

But -= - -r- =l + m. 

v n n 

Hence (1 + m) 2 a? = K = a^ (1 + 2m + f m 2 ), 

V_ 1 + 2m + m 2 
a 8 m 1 + 2m + m 2 + |m 2 ' 
and a = a(l-m 2 ) .............................. (13). 

This is a relation between a and the undisturbed mean distance. 



4. Differential Equations 
for Small Displacements from the Variational Curve. 

If the solar perturbations were to vanish, m would be zero and we should 
have x = a cos r, y = a sin T so that the orbit would be a circle. We may 
therefore consider the orbit already found as a circular orbit distorted by solar 
influence. [We have indeed put ft = 0, but the terms neglected are small 
and need not be considered at present.] As the circular orbit is only a 
special solution of the problem of two bodies, we should not expect the 
variational curve to give the actual motion of the moon. In fact it is known 
that the moon moves rather in an ellipse of eccentricity ^ than in a circle or 
variational curve. The latter therefore will only serve as an approximation 
to the real orbit in the same way as a circle serves as an approximation to an 
ellipse. An ellipse of small eccentricity can be obtained by " free oscillations " 
about a circle, and what we proceed to do is to determine free oscillations 
about the variational curve. We thus introduce two new arbitrary constants 
determining the amplitude and phase of the oscillations and so get the 
general solution of our differential equations (6). The procedure is exactly 
similar to that used in dynamics for the discussion of small oscillations about 
a steady state, i.e., the moon is initially supposed to lie near the variational 
curve, and its subsequent motion is determined relatively to this curve. At 
first only first powers of the small quantities will be used an approximation 



SMALL DISPLACEMENTS 



27 



which corresponds to the first powers of the eccentricity in the elliptic theory. 
If required, further approximations can be made. 

Suppose then that x, y are the coordinates of a point on the variational 
curve which we have found to satisfy the differential equations of motion and 
that x + Bx, y + By are the coordinates of the moon in her actual orbit, then 
since x, y satisfy the equations it is clear that the equations to be satisfied 
by Bx, By are 



.(14). 



Now it is not convenient to proceed im- 
mediately from these equations as you may see by 
considering how you would proceed if the orbit of 
reference were a simple undisturbed circle. The 
obvious course is to replace Bx, By by normal 
and tangential displacements Bp, Bs. 

Suppose then that </> denotes the inclination 
of the outward normal of the variational curve to 
the #-axis. Then we have 



...... (.-*-",) 




Fi g- 



Bx = Bp cos <f> Bs sin < 
By = Bp sin <f> + Bs cos <f> 

Multiply the first differential equation (14) by cos < and the second by 
sin < and add ; and again multiply the first by sin <f> and the second by cos </> 
and subtract. We have 

d* Bx . . , d?By _ 



d*Bx _ 

+ cos< /> 



r dr 
+ K cos <f>B f ^ ) + K sin <f>B ( 

,72$,,, 

+ 2m 



dSx~] 
-j- 

dr ] 



3m 2 cos <]>Sx = 
d8x~ 



...(16). 



- K sin <f>B (^\ + K cos <f>B l^J + 3m 2 sin $Bx = , 

Now we have from (15) 

Bp = Bx cos <f> + By sin </>, Ss = 8 sin (f> + By cos <. 
Therefore 

-y^ = cos < -i h sin ^> -r^ + ( Bx sin </> + By cos </>) -5- 



28 



HILL'S LUNAR THEORY 



Hence the two expressions which occur in the second group of terms of 
(16) are 

, dSy . , dox dSs . d<f> 
cos 6 -- - sin 6 -j- = -j h op -f- , 
r dr r dr dr r dr 

^dSy ,dSx dSp s d(f> 

sin <b ~- + cos <t> p- = ^- OS-Y-. 
r dr r dr dr dr 

When we differentiate these again, we obtain the first group of terms in 
(16). Inverting the order of the equations we have 



dr dr 



r. \AJ\JU . . \AJ\J*AJ \ \AJ\Lf 

cos <f> -^=- sin <f> -7 -r- 
dr Y dr J dr 



dS 



dr 2 dr dr 

Substituting in (16), we have as our equations 



+ * cos 4>8 + K sin (f>8 - 3m 2 cos 



dr 2 



dd> 

+ m + 







...(17). 



,, n 

+ 2m -^ + 2 
r/ dr 



- sin <f>8 - + K cos <f>S *\ + 3m 2 sin <j>Sx = 



Variation of the Jacobian integral 



gives 

Now 



= = 

rj \drj r 



dx dSx dy d&y K ~ 
-=- -=- + -f -jSL =--Sr 
dr dr dr dr r 2 



* We could introduce a term 5C, but the variation of the orbit which we are introducing 
18 one for which C is unaltered. 



TRANSFORMATIONS 29 



d8x , d&p . , d6 . , d8s . dd> 

and j = cos 9 ~ os cos d> ~- sin d> -^ -- sin 68p^r- , 

dr ^ dr ^ dr r dr r dr 



Hence 

Also 

= - ^ (^Sa; + t/%) = ^ (gp cos 9 8s sin 9) ^ (8p sin < + Ss cos </>) 

= - [Sp (x cos 9 + 2/ sin 9) + 8s ( a; sin 9 + T/ cos 0)]. 

Thus, retaining the term 3m 2 #&e in its original form, the varied Jacobian 
integral becomes 



= -[Sp (x cos<f> + y sin <) + 8s ( a; sin 9 + y cos <f>)] + 3m 2 8a?. . .(18). 

Before we can solve the differential equations (17) for 8p, 8s we require to 
express all the other variables occurring in them in terms of T by means of 
the equations obtained in 3. 

5. Transformation of the equations in 4. 

We desire to transform the differential equations (17) so that the only 
variables involved will be Sp, Bs, r. We shall then be in a position to solve 
for Sp, &s in terms of T. 

We have 

rSr = ac8x + ySy = (x cos </> + y sin 9) Sp + ( x sin <J> + y cos <f>) 8s. 

Hence 



cos 9 S (-) + sm<t>8(^ 

1 3 

= (8x cos </> + 8y sin 9) - (x cos $ + y sin <) rSr 

8 3 f 
= (x 2 cos 2 <f) +y 2 sin 2 < + 27/ sin <f> cos </>) op 

+ ( x 2 sin $ cos <f>+xy cos 2 (f> xy sin 2 <f> + y z sin < cos 9) 8s 



( i (* 2 - 2/ 2 ) s i n 2< /> + ^ cos 2 01 



30 HILL'S LUNAR THEORY 

-sin *() + 008*8 () 

1 o 

= (- 8x sin (j> + 8y cos 0) - - (- x sin + y cos 0) r Sr 

of" 

= ( a 2 sin cos $ xy sin 2 <f>+xy cos 2 <b + y z sin cos 

r 3 r 5 |_ 

+ (x 2 sin 2 + y 2 cos 2 2xy sin $ cos 0) 8s 
i ( - 2/ 2 ) ^n 20 + *y cos 20} Sp 

+ (i (tf 2 + y 2 ) - H* 2 - /) cos 2 - ^ sin 20} Sal 



We shall consider the terms 3m 2 8ac . d> later (p. 33). 

sm r 

The next step is to substitute throughout the differential equations (17) 
the values of x, y and which correspond to the undisturbed orbit. For 
simplicity in writing we drop the linear factor ao. It can be easily 
introduced when required. 

We have already found, in (9), 

x = cos T (1 |f m 2 ) + -j^m 2 cos 3r = cos r (1 - m 2 | m 2 sin 2 T), 
y = sin T (1 + jf m 2 ) + fk m2 sin 3r = sin T (1 + m 2 + f m 2 cos 2 T). 

Then 

-j- = sin T (1 \ m 2 + f m 2 cos 2 T) = sin T (1 + m 2 f m 2 sin 2 T), 

-jp = cosr(l +fm 2 fm 2 sin 2 T)= cos r(l - m 2 + f m 2 cos 2 r). 
Whence 
V 2 = (-T-J + f -jt j = sin 2 r (1 + m 2 f m 2 sin 2 r) + cos 2 r (1 - m' 2 + f m 2 cos 2 T) 

= 1 m 2 cos 2r + f m 2 cos 2r = 1 + f m 2 cos 2r 
= 1 + |m 2 - 7m 2 sin 2 T = 1 - |m 2 + 7m 2 cos 2 r. 
Therefore 

^=1 + \ m 2 - f m 2 cos 2 T = 1 - f m 2 + f m 2 sin 2 r = 1 - fm 2 cos 2r. 



TRANSFORMATIONS 



31 



-*= 

Therefore 

sin </> = sin T (1 + m 2 - f m 2 sin 2 r - m 2 + f m 2 sin 2 T) 

= sin r (1 f m 2 + f m 2 sin 2 T) = sin r (1 - f m 2 cos 2 T), 
cos < = cos T (1 m 2 + f m 2 cos 2 T + |m 2 - f m 2 cos 2 T) 

= cos T (1 + f m 2 - f m 2 cos 2 T) = cos T (1 + f m 2 sin 2 T) ; 
sin 2< = sin 2r (1 |m 2 cos 2r), 
cos 2<f> = cos 2r + f m 2 sin 2 2r ; 



sin T (1+ |m 2 J^m 2 cos 2 T). 
Summing the squares of these, 

= cos 2 r (1 - f m 2 + J^m 2 sin 2 T) + sin 2 T (1 + f m 2 - ^m 2 cos 2 T) 
= 1 - f m 2 cos 2r, 

and thence d = ^ fm 2 cos2T ........................... (21). 

Differentiating again ~~ = |m 2 sin 2r. 

We are now in a position to evaluate all the earlier terms in the 
differential equations (17). 

Thus 

_ 



-T-r- 
dr 2 



d<b\ 2 dd>^ ~dSp fdd> 

-r 1 - + 2m -i- + 2 , -=*- 
dr/ drj dr \dr 



+ m 



+ |m 2 sin 2 



32 HILL'S LUNAR THEORY 

We now have to evaluate the several terms involvings and ?/ in (18), (19), (20). 
x cos + y sin = cos 2 T (1 m 2 f m 2 sin 2 r + f m 2 sin 2 T) 

+ sin 2 T (1 + m 2 + f m 2 cos 2 T f m 2 cos 2 T) 
= 1 m 2 cos 2r, 
- x sin <j) + y cos = sin T cos T (1 m 2 f m 2 sin 2 r f m 2 cos 2 T) 

+ sin T cos T (1 + m 2 + f m 2 cos 2 T + f m 2 sin 2 r) 
= 2m 2 sin 2r ; 
r 2 = # 2 + 2/ 2 = 1 2m 2 cos 2r, 

a? - ?/ 2 = cos 2 T (1 - 2m 2 - f m 2 sin 2 r) 

sin 2 T (1 + 2m 2 + f m 2 cos 2 T) 
= cos 2r 2m 2 f m 2 sin 2 2r, 
xy=\ sin 2r (1 + f m 2 cos 2r) ; 
(a? - f) cos 20 = cos 2 2r - 2m 2 cos 2r - f m 2 sin 2 2r cos 2r 

+ f m 2 sin 2 2r cos 2r 
= cos 2r (cos 2r - 2m 2 + \ m 2 sin 2 2r), 
(a 2 - y 2 ) sin 20 = sin 2r (cos 2r - 2m 2 - f m 2 sin 2 2r - f m 2 cos 2 2r) 

= sin 2r (cos 2r - JJ- m 2 - ^ m 2 cos 2 2r) ; 
xy cos 20 = J sin 2r (cos 2r + f m 2 sin 2 2r + f m 2 cos 2 2r) 

= sin 2r (cos 2r + |m 2 - m 2 cos 2 2r), 
t/ sin 20 = \ sin 2 2r (1 f m 2 cos 2r + 1 m 2 cos 2r) 

= sin 2 2r (1 - m 2 cos 2r). 
Therefore 

- y 2 ) cos 20 + ?/ sin 20 
= \ cos 2 2r m 2 cos 2r + ^m 2 sin 2 2r cos 2r + ^ sin 2 2r ^m 2 sin 2 2r cos 2r 

= 1 - 2m2 cos 2r = r2 ' 



These are the coefficients of -^ in the expression (19) for 
cos 08 ( - ) +sin 08 



and of -f in the expression (20) for sin 08 (^ ) + cos 08 ( J) . 



TRANSFORMATIONS 33 

Again 

~ i (^ ~ 2/ 2 ) sm 2<#> + xy cos 20 

= - \ sin 2r (cos 2r - -U- m 2 - \ m 2 cos 2 2r) 

+ \ sin 2r (cos 2r + \ m 2 | m 2 cos 2 2r) 
= 2m 2 sin 2r. 

Then since to the order zero, r 2 = 1, we have 

3 (^ cos 20 - 5^ sin 2</>) = 6m 2 sin 2r. 

This is the coefficient of - ^ in cos (f>8 (^} + sin 08 (^) and of - ^ in 



Hence we have 



3m 2 cos 2r) - 6m 2 Ss sin 2r 



- sin 08 + cos 08 = - . 6m 2 sin 2r + - 
3 3 7 13 r 3 



= - 6m 2 8^ sin 2r + 8s (1 + 3m 2 cos 2r) , 



...(23). 



These two expressions are to be multiplied by K in the differential 
equations (17). 

The other terms which occur in the differential equations are 3m 2 cos <f>8x 
and + 3m 2 sin <f>8x. 

Since m 2 occurs in the coefficient we need only go to the order zero of 
small quantities in cos (/>&*; and sin </>&e. 

Thus 

3m 2 8# cos = 3m 2 (Bp cos r 8s sin T) cos r = f m 2 Sp (1 + cos 2r) - f m 2 Ss sin 2r, 
3m 2 &c sin = 3m 2 (8p cos r 8s sin T) sin r = f m 2 8p sin 2r - f m 2 8s (1 cos 2r). 

Now /c = 1 + 2m + f m 2 , and hence 



* cos <f>8 + K sin </>S \ - 3rn 2 S# cos </> 

= - 28p (1 -f 3m 2 cos 2r + 2m + f m 2 ) - 6m 2 8s sin 2r 
- f m 2 8p (1 + cos 2r) + f m 2 8s sin 2r 

= - 28p [1 + 2m + f m 2 + J^m 2 cos 2r] - f m 2 &? sin 2r, 

3 



34 HILL'S LUNAR THEORY 

- K sin 08 (^j + cos </>8 fQ + 3m 2 Ban sin < 

= - 6m 2 Bp sin 2r + 8s (1 + 2m + f m 2 + 3m 2 cos 2r) 
4- f m 2 8p sin 2r 8s (|m 2 f m 2 cos 2r) 

= -fm 2 8psin2T + 8s(l + 2m + f m 2 cos 2r). 
Hence 



+ K cos <f>8 -J + K sin <f>8 ( ) - 3m 2 cos <b&x = 
\rv Vr 3 ^ 

becomes 

- gp [1 + 2m - |m 2 cos 2r] - 2 -p (1 + m - |m 2 cos 2r) - f m 2 &? sin 2r 

- 28p [1 + 2m + |m 2 + Y-m 2 cos 2r] - |m 2 8s sin 2r = 
or ? - 8^ [3 + 6m + |m 2 + 5m 2 cos 2r] - 2 (1 + m - -f m 2 cos 2r) 



This is the first of our equations transformed. 
Again the second equation is 

dd> 

-- 



- sin <S + K cos <f>S - + 3m 2 sin <j>8a; = 0, 
and it becomes 
^ + g s (_ 1 _ 2m + |m 2 cos 2r) + 2 ^ (1 + m - f m 2 cos 2r) + fm 2 Sp sin 2r 

- f m 2 Sp sin 2r -f Bs(l + 2m + |m 2 cos 2r) = 0. 
Whence 

^ 4- 7m 2 8s cos 2r + 2 ^ (1 + m - |m 2 cos 2r) - 2m 2 8p sin 2r = 0. . .(25). 

This is the second of our equations transformed. 
The Jacob ian integral gives 
dSs *, dd> 



rc> , . <s 

y -- y^ [P ( x cos ^ + y sm ^) + . 

3m 2 cos r (8p cos r - 8s sin T) - (1 + 2m + f m 2 - f m 2 cos 2r 
+ 3m 2 cos 2r) [8p (1 - m 2 cos 2r) + 2m 2 8s sin 2r] 



TRANSFORMATIONS 35 



- 8p (1 + 2m + f m 2 + f m 2 cos 2r - m 2 cos 2r) - 2m* 8s sin 2r 
= - Sp (1 + 2m - f m 2 cos 2r) - f m 2 &? sin 2r. 

Substituting for -~ its value from (21) 

(IT 

= - Sp (1 + 2m - f m 2 cos 2r) - Sp (1 - f m 2 cos 2r) - f m 2 &? sin 2r 



= - 4Sp (1 + m f m 2 cos 2r) 7m 2 8s sin 2r 
g- (1 + m - f m 2 cos 2r)= - 48p (1 + 2rn + m 2 - f m 2 cos 2-r) - 7m a & sin 2 r 



-^ (1 + m - f m 2 cos 2r)+ 7m 2 5s sin 2r = - 48p (1 + 2m + rn 2 f m 2 cos 2r) 

......... (26). 

This expression occurs in (24), and therefore can be used to eliminate 

dSs . .. 
- T - from it. 
dr 

Substituting we get 

? + Sp [- 3 - 6m - f m 2 - 5m 2 cos 2r + 4 + 8m + 4m 2 - 10m 2 cos 2r] = 0, 



+ 2m |m* 15m 2 cos 2r] = 0. 

u/ J 

And -j = 2S/> (1 4- m f m 2 cos 2r) - f m 2 Ss sin 2r 



If we differentiate the second of these equations, which it is to be 
remembered was derived from Jacobi's integral and therefore involves our 
second differential equation, we get 



,. . . dSs 
and eliminating -j 



-j- + 7m 2 8s cos 2r - 7m a 8p sin 2r + 2 (1 + m - f m 2 cos 2r) 



32 



36 HILL'S LUNAR THEORY 

or ^ + 7 m 2g s cos 2r + 2 (1 + m - 4 m 2 eos 2r) ^ - 2m s 8p sin 2r = 0, 
a,T dr 

and this is as might be expected our second differential equation which was 
found above. Hence we only require to consider the equations (2"7). 

6. Integration of an important type of Differential Equation. 

The differential equation for Sp belongs to a type of great importance 
in mathematical physics. We may write the typical equation in the form 

d?x 

-77 + (o + 2@j cos 2t + 2 2 cos 4 +...)# = 0, 

where , x, 2 , are constants depending on increasing powers of a small 
quantity m. It is required to find a solution such that x remains small for 
all values of t. 

Let us attempt the apparently obvious process of solution by successive 
approximations. 

Neglecting 1? 2 , > we g e ^ as a first approximation 
x = A cos (t Vo + e). 

Using this value for x in the term multiplied by j, and neglecting @ 2 > 
@ 3 , ..., we get 

~ -f # + 4, {cos [* (V 4- 2) + e] + cos [t (V o - 2) + e]} = 0. 

Solving this by the usual rules we get the second approximation 

2) + e] 



. = A jcos P V@ + e] + ^ -s ^V3 

( 4(Ve. + i 



Again using this we have the differential equation 

^ + @ tf + 4! {cos [* (V6 + 2) + e] + cos [t <Y@ - 2) - e]} 

J CV 2 

= ~^ ^ cos ^ (Ve + 4 > + e l + cos (* ^o + f)} 

/o +1) 

2 

1 1} l cos (* ^^ + e > + cos t* (^^o - 4) + e]} 

{cos [t (Vo + 4) + e] + cos [< (V" - 4) + e]} = 0. 
Now this equation involves terms _of the form B cos (t Vi + e) ; on 



_ 

integration terms of the form Gt sin (t VB + e) will arise. But these terms 
are not periodic and do not remain small when t increases, x will therefore 
not remain small and the argument will fail. The assumption on which these 
approximations have been made is that the period of the principal term of 
x can be determined from HQ alone and is independent of n 2 , ____ But the 



HILL'S DIFFERENTIAL EQUATION 37 

appearance of secular terras leads us to revise this assumption and to take as 
a first approximation 

x = A cos (ct V^ + e), 

where c is nearly equal to 1 and will be determined, if possible, to prevent 
secular terms arising. 

It will, however, be more convenient to write as a first approximation 

# = A cos (ct + e), 
where c is nearly equal to V<>- 

Using this value of x in the term involving t , our equation becomes 

/2/ 

-jp+<>v + A%i {cos [(c + 2) t + e] + cos [(c - 2) t + e]} = 0, 

and the second approximation is 

A 

x = A cos (ct + e) + - -^ yl e cos t( c + 2 ) i + <0 

A 



( e - gye. 

Proceeding to another approximation with this value of a\ we get 

d~x 

jp + # + 40, [cos [(c + 2) t + e] + cos [(c - 2) * + e]] 

2 

l cos K c + 4) + e] + cos (c* + e)} 



+ A z (cos [(c + 4) t + e] + cos [(c -4)t + e]} = 0. 

We might now proceed to further approximations but just as a term in 
cos (ct + e) generates in the solution terms in 

cos [(c 2) t + e] and cos [(c 4) t + e], 
terms in cos [(c 2) t + e] and cos [(c 4)t+ e] 

will generate new terms in cos (c + e), i.e. terms of exactly the same nature 
as the term initially assumed. Hence to get our result it will be best to 
begin by assuming a series containing all the terms which will arise. 

Various writers have found it convenient to introduce exponential instead 
of trigonometric functions. Following their example we shall therefore write 
the differential equation in the form 

+ 



O ..................... (28), 

at- _oo 

where _,: = i, 

* This is not a solution of the previous equation, unless we actually put c = v GO in the 
first term. 



38 HILL'S LUNAR THEORY 

and the solution is assumed to be 



where the ratios of all the coefficients Aj, and c, are to be determined by 
equating coefficients of different powers of e fv -1 . 

Substituting this expression for x in the differential equation, we get 



and equating to zero the coefficient of e( c + 2 ^) fx - 1 

- (C + 2j) 2 ^ + 4j + Aj_, 0, + ^-_ 2 @ 2 + 

+ 4 j+1 _, + A j+2 _ 2 + -4j+,0_ 3 + = 0. 

Hence the succession of equations is 



o- ( c - 4) 2 ]^_ 2 + ^-j + _ 2 A + _ 3 A + -,A z + ... = 0, 

-4_ 2 + [0 -(c - 2) 2 ] ^_j + @_ 3 ^ + _ 2 4j + @_ 3 j[ 2 + ... = o, 

(6 - c 2 ) 4 + O-^! + @_ 2 .<4 2 + ... =0, 
[o - (c + 2) 2 ] ^l a + _ 1 A a +...=0, 

2 + . . . = o. 



We clearly have an infinite determinantal equation for c. 
If we take only three columns and rows, we get 
[@ - (c - 2)'] [ - c 2 ] [0 - (c + 2) 2 ] - e x [o - (c - 2) 2 ] - , 2 [ - (c + 2) 2 ] 

- 2 2 (o - c 2 ) + 2^, = 0, 

[(o - c 2 - 4) 2 - 16c 2 ] [@ - c 2 ] - 2^ ( - c 2 - 4) - 2 2 (o - c 2 ) + 2^, = 0. 
If we neglect ( c 2 ) 3 which is certainly small 
[- 8 (0 - c 2 ) + 16 + 16 (0 - c 2 ) - 16 J [o - c 2 ] 

- (0 - c 2 ) [20J 2 + 2 2 ] + S! 2 + 2 1 2 @ 2 = 0, 

8 (o - c 2 ) 2 + (o - c 2 ) (16 - 16 - 2! 2 - 2 2 ) + 80J 2 + 2^, = 0, 
(o - c 2 ) 2 + 2 ( - c 2 ) (1 - o - i! 2 - T V 2 2 ) + ! 2 + i! 2 , = 0. 

Since j 2 , 2 2 are small compared with 1 - , and 2 compared with 1, we 
have as a rougher approximation 

(c 2 - ) 2 + 2 (6, - 1) (c 2 - ) = - A 



DETERMINATION OF Sp 39 

whence c 2 - @ = - (0 - 1) \/(@ - 1) 2 -B 1 2 , 



c 2 - 1 
Now c 2 = o when fy = 0. Hence we take the positive sign and get 

c = N/l+V(@ -l)2-@7 ........................ (29), 

which is wonderfully nearly correct. 

For farther discussion of the equation for c, see Notes 1, 2, pp. 53, 55. 

7. Integration of the Equation for Sp. 

We now return to the Lunar Theory and consider the solution of our 
differential equation. Assume it to be 

Bp = A_, cos [(c - 2) T + e] 4- A cos (cr + e) + A l cos [(c + 2) r + e]. 
On substitution in (27) we get 

A_, [(I + 2m - |m 2 - 15m 2 cos 2r) - (c - 2) 2 ] cos [(c - 2) T -f e] 
+ A [(1 + 2m - m 2 - 15m 2 cos 2r) - c 2 ] cos (cr + e) 
+ A! [(1 + 2m - m 2 - 15m 2 cos 2r) - (c + 2) 2 ] cos [(c + 2) T + e] = 0. 
Then we equate to zero the coefficients of the several cosines. 
1st cos (cr + e) gives 

- J mM., + A, (1 + 2m - & m 2 - c 2 ) - J^m 3 ^ = 0. 
2nd cos [(c - 2) r + e] gives 

A_i [1 + 2m - m 2 - (c - 2) 2 ] - ^ m 2 ^ = 0. 
3rd cos [(c + 2) t + e] gives 

- i5mM + A l [1 + 2m - m 2 - (c + 2) 2 ] = 0. 

If we neglect terms in m 2 the first equation gives us c 2 = 1 + 2m, and 
therefore c=l+m, c 2 = -(l-m), c + 2 = 3 + m. 

The second and third equations then reduce to 



From this it follows that A_^ is at least of order m and A l at least of 
order m 2 . 

Then since we are neglecting higher powers than m 2 , the first equation 
reduces to 

A (1 + 2m - m 2 - c 2 ) = 0, 

so that c 2 = 1+ 2m m 2 or c = 1 + m - f m 2 . 

Thus (c - 2) 2 = (1 - m + f m 2 ) 2 = 1 - 2m + f m 2 , 

and 1 + 2m - m 2 - (c - 2) 2 = 4m - 3m 2 . 



40 HILL'S LUNAR THEORY 

Hence the second equation becomes 

A_, (4m - 3m 2 ) = J^mM ; 

and since A^ is of order m, the term 3m 2 J._! is of order m 3 and therefore 
negligible. Hence 

4m-4_] = J^-m 2 ^l or A__ l = imA , 

and we cannot obtain A^ to an order higher than the first. 
The third equation is 



We have seen that A^ can only be obtained to the first order; so it is 
useless to retain terms of a higher order in A^ Hence our solution is 



Hence 8p = A, [cos (CT + e) + J^m cos [(c - 2) T + e] j ............ (30). 

In order that the solution may agree with the more ordinary notation we 
write A = a e, and obtain 

8p = a^e cos (CT 4- e) - Jg 5 -ma e cos [(c 2) T + e]| , 

where c = 1 + m f m 2 j 

To the first order of small quantities the equation (27) for 8s was 



= 2(1 + m) aoe cos (cr + e) + ^ma e cos [(c - 2) T + e]. 
If we integrate and note that c = 1 + m so that c 2= (1 m). we have 
8s = 2a e sin (cr + e) - -^ma^e sin [(c - 2) r + e] ...... (32). 

We take the constant of integration zero because e = will then corre- 
spond to no displacement along the variational curve. 

In order to understand the physical meaning of the results let us consider 
the solution when m = 0, i.e. when the solar perturbation vanishes. 

Then 8p = a ecos (CT+ e), 8s = 2a e sin (CT + e). 

In the undisturbed orbit 

x a cos T, y = a sin r so that <b = r, 
and 8x = 8p cos <f> 8s sin (f), 

8y = 8p sin <f> + 8s cos <f> ; 

8x = a e cos (CT + e) cos T 2a e sin (CT -f- e) sin r, 
8y = a, e cos (CT + e) sin T + 2a e sin (CT + e) cos T. 



MOTION OF PERIGEE 41 

Therefore writing X = x + Bos, Y = y + By, X = R cos 0, Y = R sin , 
X = a [cos r e cos (CT + e) cos T 2e sin (CT + e) sin r], 
T = a [sin T e cos (CT + e) sin T + 2e sin (CT + e) cos T]. 

Therefore -R 2 = a, 2 [1 - 2e cos (CT + ?)] 

JJ.^[l-.co.( e r + .)]. r+ -^L__ J (33). 

Again cos = cos T - 2e sin (CT + e) sin T. 
sin = sin T + 2e sin (CT + e) cos T. 
Hence sin ( T) = 2e sin (CT + e), 
giving = T + 2esin(cT + e) (34). 

It will be noted that the equations for R, are of the same form as the 
first approximation to the radius vector and true longitude in undisturbed 
elliptic motion. When we neglect the solar perturbation by putting m = 
we see that e is to be identified with the eccentricity and CT + e with the 
mean anomaly. 

* We can interpret c in terms of the symbols of the ordinary lunar theories. 
When no perturbations are considered the moon moves in an ellipse. The 
perturbations cause the moon to deviate from this simple path. If a fixed 
ellipse is taken, these deviations increase with the time. It is found, 
however, that if we consider the ellipse to be fixed in shape and size but with 
the line of apses moving with uniform angular velocity, the actual motion of 
the moon differs from this modified elliptic motion only by small periodic 
quantities. If n denote as before the mean sidereal motion of the moon and 

-T- the mean motion of the line of apses, the argument entering into the 
d/t 

elliptic inequalities is ( n -7- j t + e. This must be the same as CT + e, i.e. as 

c (n ri)t + e. 

drs . 
Hence n r- = c (n - n ), 

d*ts n n 

giving ' 'ji 1 ~~ c 

ndt n 



1 +m 



A determination of c is therefore equivalent to a determination of the rate 
of change of perigee ; the value of c we have already obtained gives 



T. 
ndt 




f m 2 . 



* From here till the foot of this page a slight knowledge of ordinary lunar theory is 
supposed. The results given are not required for the further development of Hill's theory. 



42 HILL'S LUNAR THEORY 

Returning to our solution, and for simplicity again dropping the factor 
a , we have from (31), (32) 

Bp = - l me cos [(c 2) r + e] ecos(cr+e), 

8s = ifme sin [(c 2) r + e] + 2e sin (CT + e). 

Also cos (f> = cos T, sin < = sin r to the first order of small quantities, and 

8a = Bp cos <f> Bs sin <j>, By = 8p sin <f> + Bs cos <. 
Therefore 

&r = -^me cos [(c 2) r + e] cos T e cos (CT + e) cos r 

+ Aj 5 -me sin [(c 2) T + e] sin r *2e sin (CT + e) sin T, 
By = -^me cos [(c 2) r + e] sin r e cos (CT + e) sin T 

-^-me sin [(c 2) T + e] cos T + 2e sin (CT + e) cos T. 

Now let X x + Bx, Y = y + By and we have by means of the values of x, 
y in the variational curve 

X = cos T [1 m 2 f m 2 sin 2 T i/-me cos {(c 2) T 4- e} e cos (cr + e)] 
+ sin T [J^me sin {(c 2) T 4- e} 2e sin (CT 4- e)], 

F= sin T [1 + m 2 + f m 2 cos 2 T ip-me cos {(c 2) T + e} e cos (CT + e)] 
cos T [J^-me sin {(c 2) T + e} 2e sin (CT + e)]. 

Writing R 2 = X 2 + F 2 , we obtain to the requisite degree of approximation 
E 2 = cos 2 T [1 - 2m 2 - f m 2 sin 2 T - J^me cos {(c - 2) T + e} - 2e cos (CT 4- e)] 
+ sin 2 T [1 + 2m 2 + f m 2 cos 2 T me cos {(c 2) T + e} - 2e cos (CT + e)] 
+ sin 2r [-^-me sin {(c - 2) T + e} - 2e sin (CT + e)] 
- sin 2T [J^-me sin {(c - 2) T + e} - 2e sin (CT + e)], 
.R 2 = 1 - 2m 2 cos 2T - J^-me cos {(c - 2) T + e} - 2e cos (CT + e). 
Hence reintroducing the factor a which was omitted for the sake of brevity 

R = a [1 - e cos (CT + e) - igme cos {(c - 2) T + e} - m 2 cos 2r]. . .(35). 
This gives the radius vector ; it remains to find the longitude. 
We multiply the expressions for X, Y by l/R, i.e. by 

1 + e cos (CT + e) + J^me cos [(c 2) T + e] + m 2 cos 2r, 
and remembering that 

m 2 cos 2T = m 2 2m 2 sin 2 T = 2m 2 cos 2 T m 2 , 
we get 

cos = cos T [1 -^-m 2 sin 2 T] + sin T [^f-me sin {(c 2) T + e} 2e sin (CT -f e)], 
sin = sin T [1 + J^-m 8 cos 2 T] - cos T [j>-me sin {(c - 2) T + e} - 2e sin (cr 4- e)]. 
Whence 

sin (0 - T) = -V-m 2 sin 2T - J^me sin {(c - 2) T + e} + 2e sin (CT + e), 



THE THIRD COORDINATE 43 

or to our degree of approximation 

= T + J^-m 2 sin 2r - J^-me sin {(c - 2) T + ej + 2e sin (cr + e). . .(36). 
We now transform these results into the ordinary notation. 

* Let /, v be the moon's mean and true longitudes, and /' the sun's mean 
longitude. Then being the moon's true longitude relatively to the moving 
axes, we have 

v = @ + I'. 

Also T + I' = (n ri) t + n't = I, 

/. r = l-l'. 
We have seen that cr + e is the moon's mean anomaly, or I' &, 



Then substituting these values in the expressions for R and and 
adding I' to the latter we have on noting that a = a (1 m 2 ) 

R = a [1 m 2 e cos (I or) -^-me cos (I 21' + OT) m 2 cos 2(1 I'} 

equation of centre evection variation 

v = I + 2e sin (I - or) + if-me sin (I 21' + or) + -^-m 2 sin 2(1 I') 
equation of centre evection variation 

(37). 

The names of the inequalities in radius vector and longitude are written 
below, and the values of course agree with those found in ordinary lunar 
theories. 

8. Introduction of the Third Coordinate. 
Still keeping II = 0, consider the differential equation for z in (5) 
d?z KZ _ 
~dr* + r* +m ~ Z= ' 

From (8) a = 1 + 2m + f m 2 , 

a 

and from (10) ^ = 1 + 3m 2 cos 2r. 

The equation may therefore be written 

d-z 

^~ + z (1 + 2m + fm 2 + 3m 2 cos 2r) = 0. 

This is an equation of the type considered in 6 and therefore we 
assume 

z = #_! cos {(g - 2) r + } + B cos (gr + ) + B, cos {(y + 2)r+ ]. 
* From here till the end of this paragraph is not a part of Hill's theory, it is merely a 
comparison with ordinary lunar theories. 



44 HILL'S LUNAR THEORY 

On substitution we get 

#_! [- (g - 2) 2 + 1 + 2m + f m 2 + 3m* cos 2r] cos \_(g - 2) r + ] 
+ [- # 2 + 1 + 2m + f m 2 4- 3m 2 cos 2r] cos (#T + ) 
+ B l [- (g + 2) 2 + 1 + 2m + f m 2 + 3m 2 cos 2r] cos [(0 + 2) T + ] = 0. 

The coefficients of cos (<?T + ), cos [(g 2) T + ], cos [(# + 2) T + ] give 
respectively 

fm 2 ^ + B [-g n - + I + 2m +fm 2 ] +fm 2 B, = \ 
_ 1 [-(#-2) 2 + l + 2m + fm 2 ] + im 2 =0 I ......... (38). 

fm 2 B + B, [- (g + 2) 2 + 1 + 2m + fm 2 ] - } 

As a first approximation drop the terms in m 2 . The first of these equa- 

D 

tions then gives g z = 1 + 2m. The third equation then shews that -^ is of 
order m 2 . But a factor m can be removed from the second equation shewing 

D 

that -^ is of order m and can only be determined to this order. Hence 
A> 

B l can be dropped. [Cf. pp. 39, 40.] 

Considering terms in m 2 we now get from the first equation 

g- = 1 + 2m + f m 2 . 
Therefore g = I + m -f f m 2 m 2 = 1 + m + f m 2 , 

(g 2) 2 = (1 m) 2 = 1 2m, neglecting terms in m 2 . 

The second equation then gives 

B_ l = -mB , 
and the solution is 

] ............ (39). 



We shall now interpret this equation geometrically. To do so we neglect 
the solar perturbation and we get 

........................... (40). 



Now consider the moon to move in a plane orbit inclined at angle i to 
the ecliptic and let H be the longitude of the lunar 
node, I the longitude of the moon, /3 the latitude. 

The right-angled spherical triangle gives ^/ 

tan ft = tan i sin (I O) 

and therefore T^o. 

z = r tan /& = r tan i sin (I - O). Fig. 3. 




RESULTS OBTAINED 45 

As we are only dealing with a first approximation we may put r = s^ and 
so we interpret 

.Bo = 



* We can easily find the significance of g, for differentiating this equation 
with respect to the time we get 

dtl 

g(n- n ) = n - t 

. dtt = l g(n-n') 
ndt n 

= 1- (J 



1 +m 

= |m 2 to our approximation. 

Thus we find that the node has a retrograde motion. 
We have gr + =1- H -far, 

( (J _ 2) T + = I - II - TT - 2 (I - I'} 



If we write s = tan /8, k = tan i, we find 

6- = fcsin(Z-n) + fm&sm<7-2r + n) ............... (41). 

The lasL term in this equation is called the evection in latitude. 

9. Results obtained. 

We shall now shortly consider the progress we have made towards the 
actual solution of the moon's motion. We obtained first of all a special 
solution of the differential equations assuming the motion to be in the ecliptic 
and neglecting certain terms in the force function denoted by flf. This gave 
us a disturbed circular orbit in the plane of the ecliptic. We have since 
introduced the first approximation to two free oscillations about this motion, 
the one corresponding to eccentricity of the orbit, the other to an inclination 
of the orbit to the ecliptic. 

It is found to be convenient to refer the motion of the moon to the pro- 
jection on the ecliptic. We will denote by r^ the curtate radius vector, so 
that r a 2 = a? + y 2 , r 2 = rf + z 2 - ; the x, y axes rotating as before with angular 
velocity n' in the plane of the ecliptic. In determining the variational curve, 
3, we put H = 0, r = rj. It will appear therefore that in finding the actual 
motion of the moon we shall require to consider not only O but new terms in 
z 1 . In the next section we shall discuss the actual motion of the moon, making 
use of the approximations we have already obtained. 

* From here till end of paragraph is a comparison with ordinary lunar theories. 
t The of p. 20, not that of the preceding paragraph. 



46 HILL'S LUNAR THEORY 

10. General Equations of Motion and their solution. 
We have r^ = x' 2 + y' 2 and r 2 = r^ 2 + 2 2 . 

1 1 /., 3 2 \ .1 1 /., 1 ^ 2 \ 

Hence _ = _ (l _ _ _ ) ; and -_- (l _ s _), 

to our order of accuracy. 

The original equations (3) may now be written 

d?x dy KX an 3 KZ^X 

-j 2 - 2m -^ H ; 3m 2 x = ~- + - - 

d*y 9 dx Ky _ an 3 Kz z y , 

dr^ + m dr + ^. = dy~ + 2~rS > ' 

d?z_ K z , _ an 3 K2 s 

dr 2 ^s "^g 2 ?* 5 

If we multiply by 2 ~ , 2~, 2 ~ and add, we find that the Jacobian 
dr dr dr 

integral becomes 



where F 2 = 1^-f (^f= f^V + 



Ntfw H = |m 2 (, r 2 cos 2 6-x 



n srx' + yy' + zz xx' -f yy' 

and cos 6 = -JUL _ = -- -2^. smce ^ 

rr ?T 

Hence 
n = f m 2 3 (xx' + yyj - 



When we neglected n and z, we found the solution 

x = a [(1 - if m 2 ) cos T + T ^m 2 cos 3r], 
y = a [(1 + jfm 2 ) sin r + T 3 s m 2 sin 3r]. 

We now require to determine the effect of the terms introduced on the 
right, and for brevity we write 

an 3 KZ-X an 3 Kz*- 



When we refer to 4 and consider how the differential equations for 
$p, 8s were formed from those for 80;, By, we see that the new terms X, Y on 
the right-hand sides of the differential equations for 8x, By will lead to new 
terms X cos </> + Y sin <f>, X sin < -f F cos </> on the right-hand sides of those 
for Bp, Bs. 



SECOND APPROXIMATION 47 

Hence taking the equations (24) and (25) for 8p and 8s and introducing 
these new terms, we find 



- 7m 2 8s sin 2r = X cos </> + Fsin <, 
^ + 7m 2 8s cos 2r + 2 ^ (1 + m - f m 2 cos 2r) - 2m 2 8p sin 2r 

= X sin < + Fcos $. 

In this analysis we shall include all terms to the order mJk 2 , where A; is the 
small quantity in the expression for z. Terms involving m 2 ^ 2 will therefore 

be neglected. In the variation of the Jacobian integral the term -j- - can 

dr dr 

obviously be neglected. The variation of the Jacobian integral therefore 
gives (cf. pp. 29, 35) 

~ = -28p(l + m- f m 2 cos 2r) - f m 2 8s sin 2r 

i r panda; , an d y t an dz\,-. L r ^ /^\ 2 11 

+ TT K--T- + j +^^ :r T + 2 ^C -- ; T- r ...... (44), 

FiL/oV^dr dy rfr d2 dr/ 2 [ rf \dTj ) J 

where 8G will be chosen as is found most convenient. [In the previous work 
we chose 8C = 0.] 

By means- of this equation we can eliminate 8s from the differential 
equation for Sp. For 

2 ^- ( 1+ m - |m 2 cos 2r) + 7m 2 8s sin 2r 
dr 

= - 4>8p (1 + 2m + m 2 - fm 2 cos 2r) 

2 /n , r T T /an dx an cfy an ^ 

+ - TT (1 + m - f m 2 cos 2r) ( ^- T- + ^- -f + -^- ^- 

FI ^^LJoVa^^r a^/ rfr a^ rfr 



and therefore 

!*L*P. + 8p(l+2m- ^m 2 - 15m 2 cos 2r) = X cos </> + F sin 

2 ., , 

+ T7 - (1 + m - f m 2 cos 2r) 

v 



We first neglect n, and consider X, Y as arising only from terms 
in z 2 , i.e. 

_ 3 KZ Z X v _ 3 /c^y 

-27^' ~2 r^' 

3 /t^ 2 
.-. X cos ^> + F sin (/> = ^ s (* cos<f> + y sin ^>). 



48 HILL'S LUNAR THEORY 

To the required order of accuracy, 

z = K cos (gr + ), - 3 = 1 + 2m, 
a 

TJ = a , </> = T, # = a cos T, t/ = a sin T. 
.-. JT cos < 4- Fsin < = f (1 + 2m) k 2 ^ [1 + cos 2 (gr + f )]. 
Also to order m 



= (1 + 2m) k*a, 2 cos 2 (gr + ) + g 2 & 2 a 2 sin* (gr + ) 

= (l + 2m)& 2 a 2 , 
since g* = 1 + 2m. 

The equation for 8p becomes therefore, as far as regards the new terms 
now introduced, 

2m) = f(l+ 2m) ^a [1 + cos 2 (gr + )] 

+ -' [80 - (1 + 2m) #V]. 

a 

Hence Bp - f & 2 a - - - [8C - (I + 2m) & 2 a 2 ] 



but /7 2 = 1 + 2m, and therefore 1 + 2m - 4(/ 2 = - 3 (1 + 2m), 



Again the varied Jacobian integral is 



-L [8C- (1 + 2m) & 



2V 
= - f (1 + m) k"a - Jj- [80 - (T + 2m) #a a ] -4- ^ (1 + m) & 2 a cos 2 

In order that 8s may not increase with the time we choose 8(7 so that the 
constant term is zero, 

.*. 8(7 = mA^ao, 

and -T^- = \ (1 + m) &% cos 2 (gr + ), 

giving 8s = !& 2 a sin2 (gr + ) (46), 

as there is no need to introduce a new constantf. Using the value of 8(7 just 

found we get . . 

8p = ^ k-a | A,- 2 ao cos 2 (gr + ) (47). 

Having obtained 8p and 8s, we now require 8x, 8y. These are 

8x = 8p cos ^ 8s sin <f>, 
8y = 8p sin < + 8s cos </>. 

* It is of course onlv the special integral we require. The general integral when the right- 
hand side is zero has already been dealt with, 7. 
t Cf. same point in connection with equation (32). 



THE REDUCTION 49 

In this case with sufficient accuracy < = T, 

cos r la,ok 2 cos r cos 2 (gr + ) - ^a ^ sin r sin 2 (gr + f), 
sin r - &J<? sin r cos 2 ( gr + ) + 0^ cos r sin 2 (gr + ) 

Dropping the recent use of X, Y in connection with the forces and using 
as before X = x + 8x, T = y + 8y we have 

X = a cos T (1 - p 2 ) - ^aA.- 2 cos r cos 2 (#r + - ia A; 2 sin r sin 

F = a sin T (1 - |& 2 ) - la^ 2 sin T cos 2 (#T + ) + ^a^'cos T sin 2 

R* = X 2 + F 2 = ao 2 (1 - P 2 ) - lao^ 2 cos 2 (#T + ^), 

^=ao[l-p 2 -^0082(^+0] .......................................... (48). 

We thus get corrected result in radius vector as projected on to the ecliptic. 

jy 
Again cos <R) = -~ = cos r \k 2 sin T sin 2 (^T -f ^), 

7 

sin = ^ = sin T + ^& 2 cos r sin 2 ( #T + ), 

.................. (49). 



Hence we have as a term in the moon's longitude ^& 2 sin 2 ( #T + ). Terms 
of this type are called the reduction ; they result from referring the moon's 
orbit to the ecliptic. 

We have now only to consider the terms depending on H. We have seen 
that fl vanishes when the solar eccentricity, e, is put equal to zero. We shall 
only develop O as far as first power of e', 

The radius vector r', and the true longitude v', of the sun are given to the 
required approximation by 

r' = a (1 e' cos (n't nr')}, 
v' = n't 4- 2e' sin (n't -or'). 
Hence x = r' cos (v' - n't) = r' = a' (1 e' cos (n't *r')}, 

y' = r' sin (v f n't) = 2aV sin (n't /). 
And n't = mr ; 

/ . XX + yy =x-e'x cos (mr - VF') + ley sin (mr is'), 
a 



_ 

V a / 

a' 5 

.= 1 + 5e' cos (mr CT'), 

r 6 



50 HILL'S LUNAR THEORY 

I 6 (xx + yy')"* a? \ = ~- e'a? cos (mr -57') + Qrrfe'xy sin (mr -or'), 
m 2 (a? + y z + #} ( 1 - ^) = - f m 2 (a? + f + * 2 ) e' cos (mr - -57'), 



n = mV [3# 2 cos (mr TO-') + Qxy sin (mr or') f y 2 cos (mr /)], 
for we neglect m 2 ^ 2 when multiplied by e', 

^ = 6mV [a; cos (mr OT') + y sin (mr r')], 
^ = 6mV [a; sin (mr nr') ^y cos (mr or')]. 

It is sufficiently accurate for us to take 

x = a ft cos r, y = a sin T, 
</> = r; 

/ . -^- cos <f> 4- v sin d> = 6m 2 e'a [cos 2 T cos (mr -cr') + cos T sin T sin (mr CT' 

oa; dy 

+ cos T sin T sin (mr -or') ^ sin 2 r cos (mr tzr')] 
= 3m 2 e'a [cos (mr -ar') + cos 2r cos (mr is') + 2 sin 2r sin (mr -or')] 
- \ cos (mr -sr') + \ cos 2r cos (mr ') 
= 3mVa [^ cos (mr w') + f cos {(2 + m) r -c/} + f cos {(2 m) T + CT' 

+ cos {(2 - m) T + w'} - cos {(2 + m) T - *'}] 
= fmVao [cos (mr w') | cos {(2 + m) T - r'} + f cos {(2 - m) r + w'j 

Again 

T: f- + ^ r^ = 6m 2 e'a [ sin T cos T cos (mr -or') sin 2 T sin (mr -or') 
9ic dr 3y dr 

+ cos 2 T sin (HIT or') ^ sin r cos T cos (mr tsr')] 
= 3mVa [ f sin 2r cos (mr -cr') + 2 cos 2r sin (mr -ST')] 
= |mVa [- f sin {(2 + m) r r'} f sin {(2 - m) T 4- w'} 

+ 2 sin {(2 + m) r - r'} - 2 sin {(2 - m) T + r'}] 
= fmVa [ sin {(2 + m) T - r'] - f sin {(2 - m) r + or'}], 



an an . _ r. an rf an 

_,o.^ + em* + 2j(^ 3; + 

- cos {(2 + m) T - -OT'} + 7 cos {(2 m) T + w'}]. 



THE ANNUAL EQUATION 51 

Hence to the order required 

+ (1 + 2m) &p = |mVa [cos (mr r') - cos {(2 + m) T - w'} 

4- 7 cos {(2 - m) T + tsr'}], 
)T--sr'} 7cos{(2-m) 



-(4-4m)-fl+2m 
= |mVa [cos (mr - tsr') + ^ cos {(2 + m) T w'} f cos {(2 - m) T + '}] 

...... (50), 



-3mVao[cos(mT ') + i cos {(2 + m) T r'} -|cos {(2- 

- fmV [^ cos {(2 + m) r - r'} - \ cos {(2 - m) T + w' 
ao [cos (mr c/) + -|| cos {(2 + m) T r'} 



/. 8s = Sme'ao sin (mr OT') 3mVa [-^ sin {(2 -f m) r r'} 

-lfsin{(2-m)r+ OT '}] ...... (51). 

Hence to order me', to which order only our result is correct, 
Sp = 0, Ss 3me'a sin (mr IT'). 

And following our usual method for obtaining new terms in radius vector 
and longitude 

Bx = 8p cos <f> 8s sin 0, 8y = Sp sin <j> + 8s cos </>, 
Sx = Bs sin T, Sy = Ss cos r, 

Z = ao [cos T + 3me' sin T sin (mr - ')], 
F = a [sin T 3me' cos T sin (mr -sr')], 

U 2 = a 2 [1 + 3me' sin 2r sin (mr - or') - 3me' sin 2r sin (mr - -ar')] = a 2 

...... (52), 

and to the order required there is no term in radius vector 
cos = cos T + 3me' sin r sin (mr nr'), 
sin = sin T 3me' cos T sin (mr w'), 
sin ( T) = 3me' sin (mr -or'), 

= T 3me' sin (mr ts') ........................... (53). 

The new term in the longitude is - 3me' sin (V - '). This term is called 
the annual equation. 

42 



52 HILL'S LUNAR THEORY 



11. Compilation of Results. 

Let v be the longitude, s the tangent of the latitude (or to our order 
simply the latitude). When we collect our results we find 

v = I + 2e sin (I - r) + J^me sin (I - 21' + ) + ^-m 2 sin 2 (Z - O 

(mean equation to evection variation 

longitude the centre 

= nt + t) 

- \k* sin 2 (I - n) - 3me' sin (7 - r'), 
reduction annual equation 

s = k sin (I - ft) + |mfe sin (Z - 2Z' + fl). 
evection in latitude 

For R, the projection of the radius vector on the ecliptic, we get 
R = a [1 - m 2 - \k* - e cos (I - r) - Jg 5 -me cos (I - 21' + r) - m 2 cos 2 (Z - I'} 

equation to the evection variation 

centre 

+ ^cos2(Z-n>] (54). 

reduction 

To get the actual radius vector we require to multiply by sec /3, i.e. by 
1 + P 2 sin 2 (I - n) or 1 + k 2 - {k* cos 2(1- fl). 

This amounts to removing the terms \k* + \k* cos 2(1 fl). The radius 
vector then is 

a [1 - m 2 - e cos (I - is} - ^me cos (I - 21' + &)- m 2 cos 2(1- I')]. 

This is independent of k, but k will enter into product terms of higher 
order than we have considered. The perturbations are excluded by putting 
m = and the value of the radius vector is then independent of k as it 
should be. The quantity of practical importance is not the radius vector but 
its reciprocal. To our degree of approximation it is 

- [1 + m 2 + e cos (I -sr) + ^me cos (I 21' + vr) + m 2 cos 2(1- I')]. 

It may be noted in conclusion that the terms involving only e in the 
coefficient, and designated the equation to the centre, are not perturbations 
but the ordinary elliptic inequalities. There are terms in e 2 but these have 
not been included in our work. 



NOTE ON THE INFINITE DETERMINANT 58 

NOTE 1. On the Infinite Determinant of 5. 

We assume (as has been justified by Poincare) that we may treat the 
infinite determinant as though it were a finite one. 

For every row corresponding to + i there is another corresponding to - i, 
and there is one for i = 0. 

If we write c for c the determinant is simply turned upside down. 
Hence the roots occur in pairs and if c is a root c is also a root. 

If for c we write c 2j, where j is an integer, we simply shift the centre 
of the determinant. 

Hence if c is a root, c 2j are also roots. 
But these are the roots of cos TTC = cos TTC O . 
Therefore the determinant must be equal to 
k (cos TTC cos TTC O ). 
If all the roots have been enumerated, k is independent of c. 

Now the number of roots cannot be affected by the values assigned to 
the 's. Let usput @ 1 = @ 2 = 3 = ... = 0. 

The determinant then becomes equal to the product of the diagonal terms 
and the equation is 



CD = Vo is one pair of roots, and all the others are given by c + 2i. 
Hence there are no more roots and k is independent of c. 

The determinant which we have obtained is inconvenient because the 
diagonal elements increase as we pass away from the centre while the non- 
diagonal elements are of the same order of magnitude for all the rows. But 
the roots of the determinant are not affected if the rows are multiplied by 
numerical constants and we can therefore introduce such numerical multi- 
pliers as we may find convenient. 

The following considerations indicate what multipliers may prove useful. 
If we take a finite determinant from the centre of the infinite one it can be 
completely expanded by the ordinary processes. Each of the terms in the 
expansion will only involve c through elements from the principal diagonal 
and the term obtained by multiplying all the elements of this diagonal will 
contain the highest power of c. When the determinant has (2i +1) rows 
and columns, the highest power of c will be cpW+V. We wish to associate 
the infinite determinant with cos TTC. Now 



54 HILL'S LUNAR THEORY 

The first 2i + 1 terms of this product may be written 



and the highest power of c in this product is 

4c 2 4c 2 4c 2 



Hence we multiply the ith row below or above the central row by /x r ^7i 



The ith diagonal term below the central term will now be . . 

v* 1 ) ~ * 

and will be denoted by [i\. It clearly tends to unity as i tends to infinity by 
positive or negative values. The ith row below the central row will now 
read 

-4O 2 -4! , -4@! -46 2 



- 1 ' (4i) 2 - 1 ' > (4f) 2 - 1 ' (4t) 2 - 1 ' ' 

The new determinant which we will denote by V (c) has the same roots 
as the original one and so we may write 

V (c) = k' {COS 7TC - COS TTCo}, 

where k' is a new numerical constant. But it is easy to see that k' = l. 
This was the object of introducing the multipliers and that it is true is easily 
proved by taking the case of (^ = 8 8 = ... = and = , in which case the 
determinant reduces to cos TTC. We thus have the equation 

V (c) = COS 7TC COS 7TC , 

which can be considered as an identity in c. 
Putting c = we get 

V(0)=1-COS7TC . 

V (0) depends only on the @'s; written so as to shew the principal elements 
it is 



4 , 



If 6 lf 2 , etc. vanish, the solution of the differential equation is cos(V +e) 
or c = \/o- But in this case the determinant has only diagonal terms and 
the product of the diagonal terms of V (0) is 1 cos TT \/ or 2 sin 2 \ir \/o- 



NOTE ON HILL'S DIFFERENTIAL EQUATION 



55 



Hence we may divide each row by its diagonal member and put 
2 sin 2 ^TT \/0o outside. 
If therefore 



A(0) = 



@4 



i6-0 ' 



@> 1 ? /C^ ' 1 ? /uV ' " * ' 

lo Vz lo Oo 



01 - 01 02 03 

" 4-0 ' ' 4^0 ' 4^0 ' ~4^o' - 



00 
03 



0- 



I 



2 

0o 



^3 02 01 ^ 01 

4-0 ' 4-0o ' 4-0 ' ' "4^0^ ' 



V (0) = 2 sin 2 |ir V0o A (0). 

Now since cos TTC O = 1 - V (0) = 1 - 2 sin 2 fa \/0 A (0), 



an equation to be solved for c (or c). 

Clearly for stability A(0) must be positive and A (0) < cosec 2 \TT \/0 - 
Hill gives other transformations. 



NOTE 2*. On the periodicity of the integrals of the equation 



where 



= + 0j cos 2r + 0.^ cos 4r + .... 



Since the equation remains unchanged when r becomes r + TT, it follows 
that if 8p = F(T) is a solution F(r + TT) is also a solution. 

Let <f>(r) be a solution subject to the conditions that when 

T = 0, 8p = l, ^ = 0; i.e. (0) = 1, f(0) = 0. 

Let -^(T) be a second solution subject to the conditions that when 
T = 0, 8p = 0, ^ = 1; i-e. >(0) = 0, ^'(0) = 1. 

* This treatment of the subject was pointed out to Sir George Darwin by Mr S. S. Hough. 



56 HILL'S LUNAR THEORY 

It is clear that <f> (T) is an even function of r, and ^ (T) an odd one, so 
that 



Then the general solution of the equation is 



where A and B are two arbitrary constants. 

Since <f> (r + TT), -fy (r + TT) are also solutions of the equation, it follows 
that 



, (55), 

) \JT } T tcy \T ) ) 

where a, $, 7, 8 are definite constants. 
If possible let A : B be so chosen that 



where v is a numerical constant. 

When we substitute for F its values in terms of <f> and i/r, we obtain 

A(f> (T + 7r) + B^(r + 7r) = v[A<j> (T) + Bty (T)]. 
Further, substituting for < (r + TT), i/r (r + TT) their values, we have 

A K (r) + W (r)] + B [ 7 </> (T) + Sf (T)] = v \A$ (r) + B^r (T)], 
whence [4 (a - i/) + 5 7 ] <^> (r) + [4/3 + # (S - i/)] ^ (r) = 0. 
Since this is satisfied for all values of T, 

A (a - v) + By = 0, 

A/3 + B (8 - v) = 0, 

/. (a - v) (8 - i/) - #y = 0, 

i. e . !/> _ ( + S) v + 08 - /3y = 0, 

an equation for v in terms of the constants a, /3, 7, S. This equation can be 
simplified. 

Since ^ + ^ = 0, ^ + @^ = , 

^d 2 ^ . d 2 d> 
we have ^^-^^ = - 

On integration of which 

<-vJr' tyfi = const. 

But 0(0)=1, ^ 7 (0) = 1, ^(0) = 0, ^(0)=0. 

Therefore the constant is unity ; and 

d>(T)^'(T)-^( T )f (r)=l (56). 



NOTE ON HILL'S DIFFERENTIAL EQUATION 57 

But putting T = in the equations (55), and in the equations obtained by 
differentiating them, 

</> (TT) = a< (0) + ^ (0) = a, 



Therefore by (56), a8 - 7 = 1. 

Accordingly our equation for i/ is 

i/ 2 - (a + 8) y + 1 = 



If now we put T = ^TT in (55) and the equations obtained by 
differentiating them, 



- 7 f (|TT 
+ 1 7 



__ 

a-1 7 ' </>'( 

( W + 1 g + 1 



But since <^> (TT) -ty-' (^TT) ^>' (TT) -/r (|TT) = 1 we have 



Hence the equation for v may be written in five different forms, viz. 

\ (v + 1) = (TT) = ^' (TT) = (^TT) ^r* (^r) + 0' (^TT) + (fr) 

= 1 + 2f (|TT) ^ (ITT) = 2<j> (|TT) ^' (fir) - 1 ............ (57). 

It remains to determine the meaning of v in terms of the c introduced in 
the solution by means of the infinite determinant. 

The former solution was 

Sp = 2 {Aj cos (c + 2j) T + Bj sin (c + 2j) T}, 

- 00 

where Aj : Bj as cos e : sin 6. 

In the solution </> (T) we have <(0)=1, </>'(0) = 0, and </> (T) is an even 
function of T. Hence to get </> (T) from 8p we require to put *S,Aj = 1, and 
BJ = for all values of j. 



58 HILL'S LUNAR THEORY 

This gives <f> (TT) = 2 {A 5 cos (c + 2j) TT} 

= COS 7TC 2, Aj = COS 7TC. 

Similarly we may shew that ty' (TT) = cos TTC. 
It follows from equations (57) that 

cos TTC = (J> (TT) = ifr' (TT), 
cos 2 TTC = < (iir) t' (W ; S^ |TTC = - ^ ( 

We found on p. 55 that sin 2 ^ < 7rc = sin 2 ^7rV A (0), where A(0) is a 
certain determinant. Hence the last solution being of this form, we have 
the value of the determinant A (0) in terms of <f> and ty, viz. 



From this new way of looking at the matter it appears that the value of 
c may be found by means of the two special solutions (f> and i/r. 



ON LIBRA TING PLANETS AND ON A NEW FAMILY 
OF PERIODIC ORBITS 

1. Librating Planets. 

IN Professor Ernest Brown's interesting paper on "A New Family of 
Periodic Orbits " (M.N., R.A.S., vol. LXXL, 1911, p. 438) he shews how to 
obtain the orbit of a planet which makes large oscillations about the vertex 
of the Lagrangian equilateral triangle. In discussing this paper I shall 
depart slightly from his notation, and use that of my own paper on " Periodic 
Orbits" (Scientific Papers, vol. iv., or Acta Math., vol. LI.). "Jove," J, of 
mass 1, revolves at distance 1 about the " Sun," S, of mass v, and the orbital 
angular velocity is n, where n? = v + 1. 

The axes of reference revolve with SJ as axis of x, and the heliocentric 
and jovicentric rectangular coordinates of the third body are x, y and 
x 1, y respectively. The heliocentric and jovicentric polar co-ordinates are 
respectively r, 6 and p, ^. The potential function for relative energy is H. 

The equations of motion and Jacobian integral, from which Brown 
proceeds, are 

&r de (de rt \ an 



an 

.(i). 

2&-C 

where 2n = v (r 2 + -) + (p* + -J 

The following are rigorous transformations derived from these equations, 
virtually given by Brown in approximate forms in equation (13), and at the 
foot of p. 443 : 



60 ON LIBEATING PLANETS 



,drdB . fdd\* ^.dr r c 

. dr d 3 r d*r 



where 

- d -=?- l -(- 

p z 



cos (tf - 1^) [3 - 5 cos' ( - 1 )], 



an 



A great diversity of forms might be given to these functions, but the fore- 
going seemed to be as convenient for computation as I could devise. 

It is known that when v is less than 24'9599* the vertex of the equi- 
lateral triangle is an unstable solution of the problem, and if the body is 
displaced from the vertex it will move away in a spiral orbit. Hence for 
small values of v there are no small closed periodic orbits of the kind 
considered by Brown. But certain considerations led him to conjecture that 

* "Periodic Orbits," Scientific Papers, vol. iv., p. 73. 



AND ON A NEW FAMILY OF PERIODIC ORBITS 61 

there might still exist large oscillations of this kind. The verification of 
such a conjecture would be interesting, and in my attempt to test his idea 
I took v equal to 10. This value was chosen because the results will thus 
form a contribution towards that survey of periodic orbits which I have made 
in previous papers for v equal to 10. 

Brown's system of approximation, which he justifies for large values of v, 
may be described, as far as it is material for my present object, as follows: 

We begin the operation at any given point r, 0, such that p is greater 
than unity. 

d?r d?r 

Then in (2) and (3) -^r and -j are neglected, and we thence find 
ctt ctt 

dr dff 

dt' dt' 

d 3 r 

By means of these values of the first differentials, and neglecting -j 

dt 

d*r d z r 

and -7^ in (4), we find -^ from (4). 



Returning to (2) and (3) and using this value of -, , we re-determine the 



first differentials, and repeat the process until the final values of -=- and -j- 

dt dt 

remain unchanged. We thus obtain the velocity at this point r, 6 on the 

d 3 r d*r 
supposition that -^ , -, are negligible, and on substitution in the last of (1) 

we obtain the value of C corresponding to the orbit which passes through the 
chosen point. 

Brown then shews how the remainder of the orbit may be traced with all 
desirable accuracy in the case where v is large. It does not concern me to 
follow him here, since his process could scarcely be applicable for small values 
of v. But if his scheme should still lead to the required result, the remainder 
of the orbit might be traced by quadratures, and this is the plan which 
I have adopted. If the orbit as so determined proves to be clearly non- 
periodic, it seems safe to conclude that no widely librating planets can exist 
for small values of v. 

I had already become fairly confident from a number of trials, which will 
be referred to hereafter, that such orbits do not exist ; but it seemed worth 
while to make one more attempt by Brown's procedure, and the result appears 
to be of sufficient interest to be worthy of record. 

For certain reasons I chose as my starting-point 
a? = - '36200, ye = '93441, 
which give r =l'00205, . /B O = 1-65173. 



62 ON LIBRATING PLANETS 

The successive approximations to C were found to be 

33-6977, 33-7285, 33'7237, 33-7246, 337243. 

I therefore took the last value as that of C, and found also that the direction 
of motion was given by < = 2 21'. These values of x , y , < , and C then 
furnish the values from which to begin the quadratures. 

Fig. I shews the result, the starting-point being at B. The curve was 
traced backwards to A and onwards to C, and the computed positions are 
shewn by dots connected into a sweeping curve by dashes. 




-.4 -3 -2 -1 



w^ 



1-2 

1-J 

~~ Curve 







^ 







-2-3 -4 -5 -6 -7 -8 



r 
r 
r 



9 J 1-1 1-2 1-3 




Fig. 1. Results derived from Professor Brown's Method. 

From A back to perijove and from C on to J the orbit was computed as 
undisturbed by the Sun*. Within the limits of accuracy adopted the body 
collides with J. 

* When the body has been traced to the neighbourhood of J, let it be required to determine 
its future position on the supposition that the solar perturbation is negligible. Since the axes 
of reference are rotating, the solution needs care, and it may save the reader some trouble if I set 
down how it may be done conveniently. 

Let the coordinates, direction of motion, and velocity, at the moment ( = when solar 
perturbation is to be neglected, be given by x , y (or TO, 6 0i and p , \f/ ), 0o, F ; and generally 



AND ON A NEW FAMILY OF PERIODIC ORBITS 63 

Since the curve comes down on to the negative side of the line of syzygy 
SJ it differs much from Brown's orbits, and it is clear that it is not periodic. 
Thus his method fails, and there is good reason to believe that his conjecture 
is unfounded. 

After this work had been done Professor Brown pointed out to me in 
a letter that if his process be translated into rectangular coordinates, the 
approximate expressions for dx/dt and dy/dt will have as a divisor the 
function 

Q 



The method will then fail if Q vanishes or is small. 

let the suffix to any symbol denote its value at this epoch. Then the mean distance a, mean 
motion p, and eccentricity e are found from 



a (1 - e 2 ) = [ F Po cos (0 - >Ao) + wpo 2 ] 2 - 
Let t = r be the time of passage of perijove, so that when T is positive perijove is later than the 
epoch t = 0. 

At any time t let p, v, E be radius vector, true and eccentric anomalies ; then 

p = a (1-ecos.E), 
p% cos $v = ai (1 - e$ cos %E, 



On putting t = 0, E and r may be computed from these formulae, and it must be noted that 
when r is positive EQ and TO are to be taken as negative. 

The position of the body as it passes perijove is clearly given by 

x - 1 = a (1 - e) cos (^ - v - nr), y = a (1 - e) sin (\f/ -VQ- TIT). 

Any other position is to be found by assuming a value for E, computing p, v, t, \f/, and using the 
formulae 

x-l=pcoB\f/, y = psin\f/. 

In order to find V and we require the formulae 

1 dp _ ae sin E /ta dv _ [a (1 - e 2 )]^ a$ /j.a 
p dt~ p ' p ' dt~ p ' p ' p ' 
and 



The value of V as computed from these should be compared with that derived from 



and if the two agree pretty closely, the assumption as to the insignificance of solar perturbation 
is justified. 

If the orbit is retrograde about J, care has to be taken to use the signs correctly, for v and E 
will be measured in a retrograde direction, whereas ^ will be measured in the positive direction. 

A similar investigation is applicable, mutatis mutandis, when the body passes very close to S. 



64 ON LIBRATING PLANETS 

I find that if we write F = + , the divisor may be written in the 
r* p A 

form 

Q = (3n 2 + T) (3n 2 - 2F) + ^- s sin 6 sin ^. 

Now, Mr T. H. Brown, Professor Brown's pupil, has traced one portion of 
the curve Q = 0, corresponding to i> = 10, and he finds that it passes rather 
near to the orbit I have traced. This confirms the failure of the method 
which I had concluded otherwise. 



2. Variation of an Orbit. 

A great difficulty in determining the orbits of librating planets by 
quadratures arises from the fact that these orbits do not cut the line of 
syzygies at right angles, and therefore the direction of motion is quite inde- 
terminate at every point. I endeavoured to meet this difficulty by a method 
of variation which is certainly feasible, but, unfortunately, very laborious. 
In my earlier attempts I had drawn certain orbits, and I attempted to utilise 
the work by the method which will now be described. 

The stability of a periodic orbit is determined by varying the orbit. The 
form of the differential equation which the variation must satisfy does not 
depend on the fact that the orbit is periodic, and thus the investigation in 
8, 9 of my paper on " Periodic Orbits " remains equally true when the 
varied orbit is not periodic. 

Suppose, then, that the body is displaced from a given point of a non- 
periodic orbit through small distances $q V ~ - along the outward normal and 
8s along the positive tangent, then we must have 



where 
and 

Also 



AND ON A NEW FAMILY OF PERIODIC ORBITS 65 

Since it is supposed that the coordinates, direction of motion, and radius 
of curvature R have been found at a number of points equally distributed 
along the orbit, it is clear that may be computed for each of those 
points. 

At the point chosen as the starting-point the variation may be of two 
kinds : 



(1) 8q = a, ^ = 0, where a is a constant, 

(2) 8q = 0, -y^ = 6, where 6 is a constant. 

Each of these will give rise to an independent solution, and if in either of 
them a or b is multiplied by any factor, that factor will multiply all the 
succeeding results. It follows, therefore, that we need not concern ourselves 
with the exact numerical values of a or b, but the two solutions will give us 
all the variations possible. In the first solution we start parallel with the 
original curve at the chosen point on either side of it, and at any arbitrarily 
chosen small distance. In the second we start from the chosen point, but at 
any arbitrary small inclination on either side of the original tangent. 

The solution of the equations for 8q and 8s have to be carried out step by 
step along the curve, and it may be worth while to indicate how the work 
may be arranged. 

The length of arc from point to point of the unvaried orbit may be 
denoted by As, and we may take four successive values of "V, say *P n -i> 
, "^rn-i, "^n+2, as affording a sufficient representation of the march 
of the function throughout the arc As between the points indicated by 
n to n + 1. 

If the differential equation for 8q be multiplied by (As) 2 , and if we 
introduce a new independent variable z such that dz = ds/ks, and write 
X = V (As) 2 , the equation becomes 



dz a 

and z increases by unity as the arc increases by As. 

Suppose that the integration has been carried as far as the point n, and 
that 8q , d8qjdz are the values at that point; then it is required to find 8q l} 
d8q 1 /dz at the point n + 1 . 

If the four adjacent values of X are Z_j, X , X lt X 2 , and if 

8 1 = X l - Z , 8, = i [(Z s - 2X, + Z ) + (X, - 2X + Z-0], 
Bessel's formula for the function X is 






66 ON LIBRATING PLANETS 

We now assume that throughout the arc n to n + 1, 

* + Q^ + 



where Q 2) Q 3 , Q 4 have to be determined so as to satisfy the differential 
equation. 

On forming the product X&q, integrating, and equating coefficients, we 
find Q 2 = %X 8q Q , and the values of Q 3 , Q t are easily found. In carrying out 
this work I neglect all terms of the second order except X 2 . 

The result may be arranged as follows : 
Let A=I-X - 



then, on putting z = 1, we find 



- . 

dz dz 

When the SP's have been computed, the X's and A, B, C, D are easily 
found at each point of the unvaried orbit. We then begin the two solutions 
from the chosen starting-point, and thus trace Sq and d8q/dz from point to 
point both backwards and forwards. The necessary change of procedure when 
As changes in magnitude is obvious. 

The procedure is tedious although easy, but the work is enormously 
increased when we pass on further to obtain an intelligible result from the 
integration. When 8q and d&qfdz have been found at each point, a further 
integration has to be made to determine 8s, and this has, of course, to be done 
for each of the solutions. Next, we have to find the normal displacement Sp 
(equal to SqV~-), and, finally, 8p, s have to be converted into rectangular 
displacements 8x, By. 

The whole process is certainly very laborious; but when the result is 
attained it does furnish a great deal of information as to the character of the 
orbits adjacent to the orbit chosen for variation. I only carried the work 
through in one case, because I had gained enough information by this single 
instance. However, it does not seem worth while to record the numerical 
results in that case. 

In the variation which has been described, C is maintained unchanged, 



AND ON A NEW FAMILY OF PERIODIC ORBITS 67 

but it is also possible to vary G. If C becomes C+8C, it will be found that 
the equations assume the form 



But this kind of variation cannot be used with much advantage, for 
although it is possible to evaluate 8q and &s for specific initial values of BC, 
&q, dSq/ds at a specific initial point, only one single varied orbit is so deducible. 
In the previous case we may assign any arbitrary values, either positive or 
negative, to the constants denoted by a and b, and thus find a group of varied 
orbits. 



3. A New Family of Periodic Orbits. 

In attempting to discover an example of an orbit of the kind suspected 
by Brown, I traced a number of orbits. Amongst these was that one which 
was varied as explained in 2, although when the variation was effected I did 
not suspect it to be in reality periodic in a new way. It was clear that it 
could not be one of Brown's orbits, and I therefore put the work aside and 
made a fresh attempt, as explained in 1. Finally, for my own satisfaction, 
I completed the circuit of this discarded orbit, and found to my surprise that 
it belonged to a new and unsuspected class of periodic. The orbit in question 
is that marked 33 '5 in fig. 3, where only the half of it is drawn which lies on 
the positive side of SJ. 

It will be convenient to use certain terms to indicate the various parts 
of the orbits under discussion, and these will now be explained. Periodic 
orbits have in reality neither beginning nor end ; but, as it will be convenient 
to follow them in the direction traversed from an orthogonal crossing of the 
line of syzygies, I shall describe the first crossing as the " beginning " and the 
second orthogonal crossing of SJ as the " end." I shall call the large curve 
surrounding the apex of the Lagrangian equilateral triangle the " loop," and 
this is always described in the clockwise or negative direction. The portions 
of the orbit near J will be called the " circuit," or the " half-" or " quarter- 
circuit," as the case may be. The "half-circuits" about J are described 
counter-clockwise or positively, but where there is a complete " circuit " it is 
clockwise or negative. For example, in fig. 3 the orbit 33'5 " begins " with 
a positive quarter-circuit, passes on to a negative " loop," and " ends " in a 
positive quarter-circuit. Since the initial and final quarter-circuits both cut 
SJ at right angles, the orbit is periodic, and would be completed by a similar 
curve on the negative side of S J. In the completed orbit positively described 

52 



68 



ON LIBRATING PLANETS 



half-circuits are interposed between negative loops described alternately on 
the positive and negative sides of SJ. 

Having found this orbit almost by accident, it was desirable to find other 
orbits of this kind ; but the work was too heavy to obtain as many as is 
desirable. There seems at present no way of proceeding except by conjecture, 
and bad luck attended the attempts to draw the curve when C is 33'25. The 
various curves are shewn in fig. 2, from which this orbit may be constructed 
with substantial accuracy. 

In fig. 2 the firm line of the external loop was computed backwards, 
starting at right angles to SJ from x = '95, y = 0, the point to which 480 is 
attached. After the completion of the loop, the curve failed to come down 




6 --5 4 --3 ~2 - 



449 



s 

Fig. 2. Orbits computed for the Case of (7=33-25. 

close to J as was hoped, but came to the points marked 10 and 0. The 
" beginnings " of two positively described quarter-circuits about J are shewn 
as dotted lines, and an orbit of ejection, also dotted, is carried somewhat 
further. Then there is an orbit, shewn in firm line, "beginning" with a 
negative half-circuit about J, and when this orbit had been traced half-way 
through its loop it appeared that the body was drawing too near to the curve 
of zero velocity, from which it would rebound, as one may say. This orbit is 
continued in a sense by a detached portion starting from a horizontal tangent 
at x = '2, 2/=l'3. It became clear ultimately that the horizontal tangent 
ought to have been chosen with a somewhat larger value for y. From these 



AND ON A NEW FAMILY OF PERIODIC ORBITS 



69 



attempts it may be concluded that the periodic orbit must resemble the 
broken line marked as conjectural, and as such it is transferred to fig. 3 and 
shewn there as a dotted curve. I shall return hereafter to the explanation 
of the degrees written along these curves. 

Much better fortune attended the construction of the orbit 33 <I 75 shewn 
in fig. 3, for, although the final perijove does not fall quite on the line of 
syzygies, yet the true periodic orbit can differ but little from that shewn. 
It will be noticed that in this case the orbit " ends " with a negative half- 
circuit, and it is thus clear that if we were to watch the march of these 



2 -3 -4 -5 -6 -7 -3 -9 1-1 1-2 1-3 




s - 1 



Fig. 3. Three Periodic Orbits. 

orbits as C falls from 33'75 to 33'5 we should see the negative half-circuit 
shrink, pass through the ejectional stage, and emerge as a positive quarter- 
circuit when C is 33'5. 

The three orbits shewn in fig. 3 are the only members of this family that 
I have traced. It will be noticed that they do not exhibit that regular 
progress from member to member which might have been expected from the 
fact that the values of C are equidistant from one another. It might be 
suspected that they are really members of different families presenting similar 
characteristics, but I do not think this furnishes the explanation. 



70 ON LIB RATING PLANETS 

In describing the loop throughout most of its course the body moves 
roughly parallel to the curve of zero velocity. For the values of C involved 
here that curve is half of the broken horse-shoe described in my paper on 
"Periodic Orbits" (Scientific Papers, vol. iv., p. 11, or Acta Math,, vol. xxi. 
(1897)). Now, for v = 10 the horse-shoe breaks when C has fallen to 34'91, 
and below that value each half of the broken horse-shoe, which delimits the 
forbidden space, shrinks. Now, since the orbits follow the contour of the 
horse-shoe, it might be supposed that the orbits would also shrink as C falls 
in magnitude. On the other hand, as G falls from 33'5 to 33'25, our figures 
shew that the loop undoubtedly increases in size. This latter consideration 
would lead us to conjecture that the loop for 33'75 should be smaller than 
that for 33'5. Thus, looking at the matter from one point of view, we should 
expect the orbits to shrink, and from another to swell as C falls in value. 
It thus becomes intelligible that neither conjecture can be wholly correct, 
and we may thus find an explanation of the interlacing of the orbits as shewn 
in my fig. 3. 

It is certain from general considerations that families of orbits must 
originate in pairs, and we must therefore examine the origin of these orbits, 
and consider the fate of the other member of the pair. 

It may be that for values of C greater than 33'75 the initial positive 
quarter-circuit about J is replaced by a negative half-circuit; but it is 
unnecessary for the present discussion to determine whether this is so or not, 
and it will suffice to assume that when C is greater than 33'75 the "beginning " 
is as shewn in my figure. The " end " of 3375 is a clearly marked negative 
half-circuit, and this shews that the family originates from a coalescent pair of 
orbits " ending " in such a negative half-circuit, with identical final orthogonal 
crossing of SJ in which the body passes from the negative to the positive 
side of S J. 

This coalescence must occur for some critical value of C between 34'91 
and 33*75, and it is clear that as C falls below that critical value one 
of the "final" orthogonal intersections will move towards S and the other 
towards J. 

In that one of the pair for which the intersection moves towards S the 
negative circuit increases in size ; in the other in which it moves towards J 
the circuit diminishes in size, and these are clearly the orbits which have 
been traced. We next see that the negative circuit vanishes, the orbit 
becomes ejectional, and the motion about J both at "beginning" and "end" 
has become positive. 

It may be suspected that when C falls below 33'25 the half-circuits 
round J increase in magnitude, and that the orbit tends to assume the 
form of a sort of asymmetrical double figure-of-8, something like the figure 



AND ON A NEW FAMILY OF PERIODIC ORBITS 71 

which Lord Kelvin drew as an illustration of his graphical method of curve- 
tracing*. 

In the neighbourhood of Jove the motion of the body is rapid, but the 
loops are described very slowly. The number of degrees written along the 
curves in fig. 2 represent the angles turned through by Jove about the Sun 
since the moment corresponding to the position marked 0. Thus the firm 
line which lies externally throughout most of the loop terminates with 480. 
Since this orbit cuts SJ orthogonally, it may be continued symmetrically on 
the negative side of SJ, and therefore while the body moves from the point 
to a symmetrical one on the negative side Jove has turned through 960 round 
the Sun, that is to say, through 2f revolutions. 

Again, in the case of the orbit beginning with a negative half-circuit, 
shewn as a firm line, Jove has revolved through 280 by the time the point 
so marked is reached. We may regard this as continued in a sense by the 
detached portion of an orbit marked with 0, 113, 203; and since 280+ 203 
is equal to 483, we again see that the period of the periodic orbit must be 
about 960, or perhaps a little more. 

In the cases of the other orbits more precise values may be assigned. For 
0=33-5, the angle nT (where T is the period) is 1115 or 31 revolutions of 
Jove ; and for C= 33'75, nT is 1235 or 3'4 revolutions. 

It did not seem practicable to investigate the stability of these orbits, but 
we may suspect them to be unstable. 

The numerical values for drawing the orbits (7=33-5 and 33'75 are given 
in an appendix, but those for the various orbits from which the conjectural 
orbit C= 33-25 is constructed are omitted. I estimate that it is as laborious 
to trace one of these orbits as to determine fully half a dozen of the simpler 
orbits shewn in my earlier paper. 

Although the present contribution to our knowledge is very imperfect, 
yet it may be hoped that it will furnish the mathematician with an 
intimation worth having as to the orbits towards which his researches must 
lead him. 

The librating planets were first recognised as small oscillations about the 
triangular positions of Lagrange, and they have now received a very remark- 
able extension at the hands of Professor Brown. It appears to me that the 
family of orbits here investigated possesses an interesting relationship to 
these librating planets, for there must be orbits describing double, triple, 
and multiple loops in the intervals between successive half- circuits about 
Jove. Now, a body which describes its loop an infinite number of times, 

* Popular Lectures, vol. i., 2nd ed., p. 81; Phil. Mag., vol. xxxiv., 1892, p. 443. 



72 ON LIBRATING PLANETS 

before it ceases to circulate round the triangular point, is in fact a libr'ating 
planet. It may be conjectured that when the Sun's mass v is yet smaller 
than 10, no such orbit as those traced is possible. When v has increased 
to 10, probably only a single loop is possible ; for a larger value a double loop 
may be described, and then successively more frequently described multiple 
loops will be reached. When v has reached 24'9599 a loop described an 
infinite number of times must have become possible, since this is the smallest 
value of v which permits oscillation about the triangular point. If this idea 
is correct, and if N denotes the number expressing the multiplicity of the 
loop, then as v increases dN/dv must tend to infinity ; and I do not see why 
this should not be the case. 

These orbits throw some light on cosmogony, for we see how small planets 
with the same mean motion as Jove in the course of their vicissitudes tend 
to pass close to Jove, ultimately to be absorbed into its mass. We thus see 
something of the machinery whereby a large planet generates for itself a clear 
space in which to circulate about the Sun. 

My attention was first drawn to periodic orbits by the desire to discover 
how a Laplacian ring could coalesce into a planet. With that object in view 
I tried to discover how a large planet would affect the motion of a small one 
moving in a circular orbit at the same mean distance. After various failures 
the investigation drifted towards the work of Hill and Poincare, so that the 
original point of view was quite lost and it is not even mentioned in my paper 
on "Periodic Orbits." It is of interest, to me at least, to find that the original 
aspect of the problem has emerged again. 



APPENDIX. 

Numerical results of Quadratures. 
C = 33-5. 

Perijove # = 1-0171, y =-'0034, taken as zero. 

Time from perijove up to = -2'1 is given by nt=9 25'. 

* x y <i> 2n/F 

-2-1 + -8282 + -0980 +66 10 2-408 

2-0 -7409 -1467 55 53 2 -829 

1-9 -6625 -2084 48 36 2 -876 

1-8 -5894 -2766 46 3 2 '768 

1-7 -5171 '3457 46 55 2'655 

1-6 -4425 '4124 49 46 2'584 

1-5 '3641 '4744 53 39 2-568 

-1-4 + -2814 + -5306 +57 56 2 "613 



AND ON A NEW FAMILY OF PERIODIC ORBITS 



73 



2n/F 



-1-3 


+ -1948 


+ -5805 


+ 62 8 


2-728 


1-2 


1049 


6243 


65 51 


2-930 


1-1 


+ -0126 


6628 


68 38 


3-251 


1-0 


- -0810 


6979 


69 46 


3-760 


9 


1747 


7330 


68 7 


4-598 


85 


2207 


7526 


65 13 


5-240 


8 


2653 


7754 


60 1 


6-133 


75 


3068 


8035 


50 51 


7-377 


725 


3252 


8203 


44 2 


8-139 


7 


3412 


8395 


35 17 


8-944 


675 


3537 


8611 


24 33 


9-664 


65 


3617 ' 


8848 


12 27 


10-129 


625 


3644 


9096 


+ 13 


10-224 


6 


3620 


9344 


-10 56 


10-009 


575 


3552 


9584 


20 31 


9-655 


55 


3448 


9811 


28 30 


9-205 


5 


3161 


1 -0220 


40 48 


8-448 


45 


2806 


1-0571 


49 38 


7-872 


4 


2405 


1-0869 


56 51 


7-460 


3 


1518 


1-1326 


68 4 


6-961 


2 


- -0565 


1626 


76 47 


6-730 


- -1 


+ -0421 


1791 


83 58 


6-647 


o 


1419 


1842 


-90 


6-633 


+ '05 


1919 


1830 


180 + 87 21 


6-630 


1 


2418 


1797 


84 54 


6-626 


15 


2915 


1-1742 


82 38 


6-609 


2 


3410 


T1669 


80 31 


6-572 


3 


4389 


1-1470 


76 31 


6-432 


4 


5353 


1-1203 


72 33 


6-201 


5 


6295 


1 -0869 


68 16 


5-912 


6 


7208 


1 -0461 


63 29 


5-605 


7 


8081 


9974 


58 8 


5-313 


8 


8902 


9404 


52 12 


5-055 


9 


9656 


8748 


45 39 


4-842 


1-0 


1-0326 


8006 


38 22 


4-671 


1-1 


1-0889 


7181 


30 11 


4-540 


1-2 


1-1321 


6280 


20 46 


4-435 


1-3 


1-1585 


5318 


9 38 


4-326 


1-35 


1-1642 


4821 


180+ 3 16 


4-250 


1-4 


1-1641 


4322 


180- 3 40 


4-141 


1-45 


1-1577 


3826 


11 5 


3-983 


1-5 


1 -1448 


3343 


18 44 


3-758 


1-55 


1-1257 


2881 


26 8 


3-460 


T6 


1-1011 


2446 


32 39 


3-100 


1-65 


1-0723 


2038 


37 33 


2-701 


1-7 


1 -0408 


1650 


40 4 


2-291 


+ 1-75 


+ 1-0087 


+ '1267 


180 -39 12 


1-893 




Time from s = 


1-75 to perijove given 


by w<=5 58'. 






Coordinates of 


perijove #='9501, y= 


- -0029. 





5-5 



74 



ON LIBRATING PLANETS 



The following additional positions were calculated backwards from a perijove at 
95, y=0, 0=180. 



+ -9500 

9512 

9647 

9756 

9874 

1-0128 

1-0390 

1-0649 

1-0893 

1-1114 

1-1463 

+ 1-1661 



+ 0000 
0531 
0797 



1127 
1436 
1738 
2043 
2360 



3412 
4186 



180+ 
180 -22 30 

30 52 

34 48 
37 37 
40 37 
40 56 
39 12 

35 51 

31 16 
20 10 

180- 8 40 



This supplementary orbit becomes indistinguishable in a figure of moderate size from 
the preceding orbit, which is therefore accepted as being periodic. The period is given by 
n7 T =1115-4 = 3-l revolutions of Jove. 



(7 = 33-75. 

This orbit was computed from a conjectural starting-point which seemed likely to lead 
to the desired result ; the computation was finally carried backwards from the starting- 
point. The coordinates of perijove were found to be x = 1-0106, y ='0006, which may be 
taken as virtually on the line of syzygies. The motion from perijove is direct. 
s x y $ 2n/V 



perijove 

- -35 

- -3 

- -25 



+ 1-0106 
9652 
9184 
8713 
8251 



1 


7391 


o-o 


6625 


1 


5911 


2 


5202 


3 


4465 


4 


3685 


5 


2858 


6 


1987 


7 


1081 


8 


+ -0147 


9 


- -0805 


1-0 


1764 


1 


2713 


15 


3173 


2 


3601 


225 


3791 


25 


- -3951 



+ 0006 
0403 
0578 
0744 
0936 
1444 
2084 
2785 
3490 
4165 
4791 
5352 
5844 
6265 
6622 
6929 
7213 
7525 
7721 
7977 
8140 

+ -8332 



very 


nearly 


66 38 


1-140 


71 6 


1-635 


69 27 


2-075 


65 3 


2-447 


54 15 


2-882 


47 


2-946 


44 44 


2-850 


46 


2-749 


49 13 


2-686 


53 29 


2-675 


58 10 


2-723 


62 52 


2-838 


67 13 


3-036 


70 49 


3-348 


73 11 


3-834 


73 25 


4-631 


69 17 


6-090 


63 50 


7-333 


53 25 


9-236 


45 6 


10-360 


33 54 


11-840 



AND ON A NEW FAMILY OF PERIODIC ORBITS 75 

x y $ 2nlV 



1-275 


- -4064 


+ -8553 


+ 19 53 


12-955 


1-3 


4118 


8796 


+ 4 42 


13-412 


1-325 


4108 


9046 


- 9 14 


13-174 


1-35 


4043 


9287 


20 35 


12-599 


1-375 


3936 


9513 


29 25 


11-945 


1-4 


3800 


9723 


36 21 


11-364 


1-45 


3466 


1-0096 


46 23 


10-471 


1-5 


3082 


1 -0416 


53 25 


9-849 


1-6 


2227 


1-0940 


62 21 


9-034 


1-7 


1317 


1-1356 


67 59 


8-347 


1-8 


- -0377 


1 -1696 


72 2 


7-618 


2-0 


+ -1563 


1-2184 


79 17 


6-140 


2-2 


3547 


1-2407 


-88 13 


4-966 


2-4 


5541 


1-2300 


180 + 81 54 


4-182 


2-6 


7487 


1-1845 


71 49 


3-665 


2-8 


9322 


1-1057 


61 40 


3-305 


3-0 


1-0989 


9956 


51 24 


3-052 


3-2 


1 -2429 


8573 


40 54 


2-873 


3-4 


1-3588 


6946 


29 55 


2-751 


3-6 


1-4402 


5123 


18 1 


2-682 


3-8 


1-4797 


3168 


180+ 4 28 


2-670 


4-0 


1-4674 


1181 


180 -12 14 


2-733 


4-1 


1 -4377 


+ -0227 


23 43 


2-806 


4-2 


1-3894 


- -0646 


35 38 


2-910 


4-3 


1 -3208 


1366 


52 23 


3-027 


4-35 


1-2787 


1635 


62 47 


3-068 


4-4 


1 -2322 


1817 


74 47 


3-063 


4-45 


1-1829 


1892 


180 -88 15 


2-983 


4-5 


1-1332 


1845 


+ 77 25 


2-780 


4-55 


1 -0863 


1676 


63 8 


2-477 


4-6 


1-0448 


1399 


49 32 


2-101 


4-65 


1-0108 


1034 


36 18 


1-683 


4-7 


9867 


- -0598 


21 1 


1-234 


perijove 


+ -990 


+ -on 


about 49 





The orbit is not rigorously periodic, but an extremely small change at the beginning 
would make it so. The period is given by n T= 1234 -6 = 3-43 revolutions of Jove. 



ADDRESS 

(DELIVERED BEFORE THE INTERNATIONAL CONGRESS OF MATHEMATICIANS 
AT CAMBRIDGE IN 1912) 

FOUR years ago at our Conference at Rome the Cambridge Philosophical 
Society did itself the honour of inviting the International Congress of 
Mathematicians to hold its next meeting at Cambridge. And now I, as 
President of the Society, have the pleasure of making you welcome here. 
I shall leave it to the Vice-Chancellor, who will speak after me, to express 
the feeling of the University as a whole on this occasion, and I shall 
confine myself to my proper duty as the representative of our Scientific 
Society. 

The Science of Mathematics is now so wide and is already so much 
specialised that it may be doubted whether there exists to-day any man 
fully competent to understand mathematical research in all its many diverse 
branches. I, at least, feel how profoundly ill-equipped I am to represent 
our Society as regards all that vast field of knowledge which we classify as 
pure mathematics. I must tell you frankly that when I gaze on some of the 
papers written by men in this room I feel myself much in the same position 
as if they were written in Sanskrit. 

But if there is any place in the world in which so one-sided a President 
of the body which has the honour to bid you welcome is not wholly out of 
place it is perhaps Cambridge. It is true that there have been in the past 
at Cambridge great pure mathematicians such as Cayley and Sylvester, but 
we surely may claim without undue boasting that our University has played 
a conspicuous part in the advance of applied mathematics. Newton was 
a glory to all mankind, yet we Cambridge men are proud that fate ordained 
that he should have been Lucasian Professor here. But as regards the part 
played by Cambridge I refer rather to the men of the last hundred years, 
such as Airy, Adams, Maxwell, Stokes, Kelvin, and other lesser lights, who 
have marked out the lines of research in applied mathematics as studied in 
this University. Then too there are others such as our Chancellor, Lord 
Rayleigh, who are happily still with us. 



ADDRESS 77 

Up to a few weeks ago there was one man who alone of all mathematicians 
might have occupied the place which I hold without misgivings as to his 
fitness ; I mean Henri Poincar6. It was at Rome just four years ago that 
the first dark shadow fell on us of that illness which has now terminated so 
fatally. You all remember the dismay which fell on us when the word passed 
from man to man " Poincare is ill." We had hoped that we might again 
have heard from his mouth some such luminous address as that which he 
gave at Rome ; but it was not to be, and the loss of France in his death 
affects the whole world. 

It was in 1900 that, as president of the Royal Astronomical Society, 
I had the privilege of handing to Poincare the medal of the Society, and 
I then attempted to give an appreciation of his work on the theory of the 
tides, on figures of equilibrium of rotating fluid and on the problem of the 
three bodies. Again in the preface to the third volume of my collected 
papers I ventured to describe him as my patron Saint as regards the papers 
contained in that volume. It brings vividly home to me how great a man 
he was when I reflect that to one incompetent to appreciate fully one half of 
his work yet he appears as a star of the first magnitude. 

It affords an interesting study to attempt to analyze the difference in the 
textures of the minds of pure and applied mathematicians. I think that 
I shall not be doing wrong to the reputation of the psychologists of half 
a century ago when I say that they thought that when they had successfully 
analyzed the way in which their own minds work they had solved the problem 
before them. But it was Sir Francis Galton who shewed that such a view is 
erroneous. He pointed out that for many men visual images form the most 
potent apparatus of thought, but that for others this is not the case. Such 
visual images are often quaint and illogical, being probably often founded on 
infantile impressions, but they form the wheels of the clockwork of many 
minds. The pure geometrician must be a man who is endowed with great 
powers of visualisation, and this view is confirmed by my recollection of the 
difficulty of attaining to clear conceptions of the geometry of space until 
practice in the art of visualisation had enabled one to picture clearly the 
relationship of lines and surfaces to one another. The pure analyst probably 
relies far less on visual images, or at least his pictures are not of a geometrical 
character. I suspect that the mathematician will drift naturally to one branch 
or another of our science according to the texture of his mind and the nature 
of the mechanism by which he works. 

I wish Galton, who died but recently, could have been here to collect 
from the great mathematicians now assembled an introspective account 
of the way in which their minds work. One would like to know whether 
students of the theory of groups picture to themselves little groups of dots ; 
or are they sheep grazing in a field? Do those who work at the theory 



78 ADDRESS 

of numbers associate colour, or good or bad characters with the lower 
ordinal numbers, and what are the shapes of the curves in which the 
successive numbers are arranged ? What I have just said will appear pure 
nonsense to some in this room, others will be recalling what they see, and 
perhaps some will now for the first time be conscious of their own visual 
images. 

The minds of pure and applied mathematicians probably also tend to 
differ from one another in the sense of aesthetic beauty. Poincare has well 
remarked in his Science et Methode (p. 57) : 

" On peut s'etonner de voir invoquer la sensibilite a propos de demon- 
strations mathe'matiques qui, semble-t-il, ne peuvent interesser que 1'in- 
telligence. Ce serait oublier le sentiment de la beaute mathematique, de 
1'harmonie des nombres et des formes, de 1'elegance geometrique. C'est un 
vrai sentiment esthetique que tous les vrais mathematiciens connaissent. 
Et c'est bien la de la sensibilite." 

And again he writes : 

"Les combinaisons utiles, ce sont pr6cisement les plus belles, je veux dire 
celles qui peuvent le mieux charmer cette sensibilite speciale que tous les 
mathematiciens connaissent, mais que les profanes ignorent au point qu'ils 
sont souvent tentes d'en sourire." 

Of course there is every gradation from one class of mind to the other, 
and in some the aesthetic sense is dominant and in others subordinate. 

In this connection I would remark on the extraordinary psychological 
interest of Poincare's account, in the chapter from which I have already 
quoted, of the manner in which he proceeded in attacking a mathematical 
problem. He describes the unconscious working of the mind, so that his 
conclusions appeared to his conscious self as revelations from another world. 
I suspect that we have all been aware of something of the same sort, and 
like Poincare have also found that the revelations were not always to be 
trusted. 

Both the pure and the applied mathematician are in search of truth, but 
the former seeks truth in itself and the latter truths about the universe in 
which we live. To some men abstract truth has the greater charm, to others 
the interest in our universe is dominant. In both fields there is room for 
indefinite advance ; but while in pure mathematics every new discovery 
is a gain, in applied mathematics it is not always easy to find the direction 
in which progress can be made, because the selection of the conditions 
essential to the problem presents a preliminary task, and afterwards there 
arise the purely mathematical difficulties. Thus it appears to me at least, 
that it is easier to find a field for advantageous research in pure than in 



ADDRESS 79 

applied mathematics. Of course if we regard an investigation in applied 
mathematics as an exercise in analysis, the correct selection of the essential 
conditions is immaterial ; but if the choice has been wrong the results lose 
almost all their interest. I may illustrate what I mean by reference to 
Lord Kelvin's celebrated investigation as to the cooling of the earth. He 
Avas not and could not be aware of the radio-activity of the materials of which 
the earth is formed, and I think it is now generally acknowledged that the 
conclusions which he deduced as to the age of the earth cannot be maintained ; 
yet the mathematical investigation remains intact. 

The appropriate formulation of the problem to be solved is one of the 
greatest difficulties which beset the applied mathematician, and when he 
has attained to a true insight but too often there remains the fact that 
his problem is beyond the reach of mathematical solution. To the layman 
the problem of the three bodies seems so simple that he is surprised to learn 
that it cannot be solved completely, and yet we know what prodigies of 
mathematical skill have been bestowed on it. My own work on the subject 
cannot be said to involve any such skill at all, unless indeed you describe as 
skill the procedure of a housebreaker who blows in a safe-door with dynamite 
instead of picking the lock. It is thus by brute force that this tantalising 
problem has been compelled to give up some few of its secrets, and great as 
has been the labour involved I think it has been worth while. Perhaps this 
work too has done something to encourage others such as Stormer* to similar 
tasks as in the computation of the orbits of electrons in the neighbourhood 
of the earth, thus affording an explanation of some of the phenomena of the 
aurora borealis. To put at their lowest the claims of this clumsy method, 
which may almost excite the derision of the pure mathematician, it 
has served to throw light on the celebrated generalisations of Hill and 
Poincare. 

I appeal then for mercy to the applied mathematician and would ask 
you to consider in a kindly spirit the difficulties under which he labours. 
If our methods are often wanting in elegance and do but little to satisfy that 
aesthetic sense of which I spoke before, yet they are honest attempts to 
unravel the secrets of the universe in which we live. 

We are met here to consider mathematical science in all its branches. 
Specialisation has become a necessity of modern work and the intercourse 
which will take place between us in the course of this week will serve to 
promote some measure of comprehension of the work which is being carried 
on in other fields than our own. The papers and lectures which you will 
hear will serve towards this end, but perhaps the personal conversations 
outside the regular meetings may prove even more useful. 

* Videnskabs Selskab, Christiania, 1904. 



INDEX TO VOLUME V 



Abacus, xlviii 

Address to the International Congress of 

Mathematicians in Cambridge, 1912, 76 
Annual Equation, 51 
Apse, motion of, 41 

B 

Bakerian lecture, xlix 
Bakhuyzen, Dr Van d. Sande, Sir George 

Darwin's connection with the International 

Geodetic Association, xxviii 
Barrell, Prof., Cosmogony as related to Geology 

and Biology, xxxvii 
British Association, South African Meeting, 

1905, xxvi 
Brown, Prof. E. W., Sir George Darwin's 

Scientific Work, xxxiv ; new family of 

periodic orbits, 59 



his first papers, xxxvi; characteristics of 
his work, xxxiv; his influence on cosmo- 
gony, xxxvi; his relationship with his 
pupils, xxxvi; on his own work, 79 

Darwin, Margaret, on Sir George Darwin's 
personal characteristics, xxx 

Differential Equation, Hill's, 36; periodicity 
of integrals of, 55 

Differential Equations of Motion, 17 

Dynamical Astronomy, introduction to, 9 

E 

Earth-Moon theory of Darwin, described by 

Mr S. S. Hough, xxxviii 
Earth's figure, theory of, 14 
Ellipsoidal harmonics, xliii 
Equation, annual, 51; of the centre, 43 
Equations of motion, 17, 46 
Equilibrium of a rotating fluid, xlii 
Evection, 43; in latitude, 45 



Cambridge School of Mathematics, 1, 76 
Chamberlain and Moulton, Planetesimal Hypo- 
thesis, xlvii 

Committees, Sir George Darwin on, xxii 
Congress, International, of Mathematicians at 
Cambridge, 1912, 76; note by Sir Joseph 
Larmor, xxix 

Cosmogony, Sir George Darwin's influence on, 
xxxvi; as related to Geology and Biology, 
by Prof. Barrell, xxxvii 



Darwin, Charles, ix; letters of, xiii, xv 
Darwin, Sir Francis, Memoir of Sir George 

Darwin by, ix 

Darwin, Sir George, genealogy, ix; boyhood, 
x; interested in heraldry, xi; education, xi; 
at Cambridge, xii, xvi; friendships, xiii, xvi ; 
ill health, xiv; marriage, xix; children, 
xx ; house at Cambridge, xix; games and 
pastimes, xxi; personal characteristics, xxx; 
energy, xxxii; honours, xxxiii; university 
work, described by Sir Joseph Larmor, the 
Master of Christ's, and Prof. Newall, xvii, 
xviii; work on scientific committees, xxii; 
association with Lord Kelvin, xv, xxxvii; 
scientific work, by Prof. E. W. Brown, xxxiv ; 



Galton, Sir Francis, ix ; analysis of difference 

in texture of different minds, 77 
Geodetic Association, International, xxvii, 

xxviii 
Glaisher, Dr J. W. L., address on presenting 

the gold medal of the E.A.S. to G. W. 

Hill, lii 
Gravitation, theory of, 9; universal, 15 

H 

Harmonics, ellipsoidal, xliii 

Hecker's observations on retardation of tidal 
oscillations in the solid earth, xliv, 1 

Hill, G. W., Lunar Theory, 1; awarded gold 
medal of B.A.S., lii; lectures by Darwin 
on Lunar Theory, lii, 16; characteristics 
of his Lunar Theory, 16; Special Differ- 
ential Equation, 36; periodicity of integrals 
of, 55; infinite determinant, 38, 53 

Hough, S. S., Darwin's work on Earth-Moon 
Theory, xxxviii; Periodic Orbits, liv 

I 

Inaugural lecture, 1 
Infinite determinant, Hill's, 38, 53 
Introduction to Dynamical Astronomy, 9 



INDEX TO VOLUME V 



SI 



Jacobi's ellipsoid, xlii; integral, 21 
Jeans, J. H., on rotating liquids, xliii 

K 

Kant, Nebular Hypothesis, xlvi 
Kelvin, associated with Sir George Darwin, xv, 
xxxvii; cooling of earth, xlv, 79 



Laplace, Nebular Hypothesis, xlvi 

Larmor, Sir Joseph, Sir George Darwin's 
work on university committees, xvii ; Inter- 
national Geodetic Association, xxvii; Inter- 
national Congress of Mathematicians at 
Cambridge 1912, xxix 

Latitude of the moon, 43 

Latitude, variation of, 14 

Lecture, inaugural, 1 

Liapounoff's work on rotating liquids, xliii 

Librating planets, 59 

Lunar and planetary theories compared, 11 

Lunar Theory, lecture on, 16 

M 

Maclaurin's spheroid, xlii 

Master of Christ's, Sir George Darwin's work 
on university committees, xviii 

Mathematical School at Cambridge, 1, 76; 
text-books, 4; Tripos, 3 

Mathematicians, International Congress of, 
Cambridge, 1912, xxix, 76 

Mechanical condition of a swarm of meteor- 
ites, xlvi 

Meteorological Council, by Sir Napier Shaw, 
xxii 

Michelson's experiment on rigidity of earth, 1 

Moulton, Chamberlain and, Planetesimal 
Hypothesis, xlvii 

N 
Newall, Prof., Sir George Darwin's work on 

university committees, xviii 
Numerical work on cosmogony, xlvi 
Numerical work, great labour of, liii 



Orbit, variation of an, 64 
Orbits, periodic, see Periodic 



Pear-shaped figure of equilibrium, xliii 



Perigee, motion of, 41 

Periodic orbits, Darwin begins papers on, liii; 
great numerical difficulties of, liii ; stability 
of, liii ; classification of, by Jacobi's integral, 
liv; new family of, 59, 67 

Periodicity of integrals of Hill's Differential 
Equation, 55 

Planetary and lunar theories compared, 11 

Planetesimal Hypothesis of Chamberlain and 
Moulton, xlvii 

Poincare, reference to, by Sir George Darwin, 
77 ; on equilibrium of fluid mass in rotation, 
xlii; Les Methodes Nouvelles de la Mecanique 
Celeste, lii; Science et Methode, quoted, 78 

Pupils, Darwin's relationship with his, xxxvi 

R 

Eaverat, Madame, on Sir George Darwin's 

personal characteristics, xxx 
Eeduction, the, 49 
Kigidity of earth, from fortnightly tides, xlix; 

Michelson's experiment, 1 
Roche's ellipsoid, xliii 
Rotating fluid, equilibrium of, xlii 

S 

Saturn's rings, 15 

Shaw, Sir Napier, Meteorological Council, xxii 
Small displacements from variational curve, 26 
South African Meeting of the British Asso- 
ciation, 1905, xxvi 
Specialisation in Mathematics, 76 

T 

Text-books, mathematical, 4 

Third coordinate introduced, 43 

Tidal friction as a true cause of change, xliv 

Tidal problems, practical, xlvii 

Tide, fortnightly, xlix 

Tides, The, xxvii, 1 

Tides, articles on, 1 

Tripos, Mathematical, 3 

U 

University committees, Sir George Darwin on, 
by Sir Joseph Larmor, the Master of Christ's, 
and Prof. Newall, xvii, xviii 



Variation, the, 43 ; of an orbit, 64 ; of lati- 
tude, 14 

Variational ctirve, defined, 22; determined, 
23; small displacements from, 26 



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