% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% The Project Gutenberg EBook of Scientific Papers by Sir George Howard   %
% Darwin, by George Darwin                                                %
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
% with this eBook or online at book.klll.cc                          %
%                                                                         %
%                                                                         %
% Title: Scientific Papers by Sir George Howard Darwin                    %
%        Volume V. Supplementary Volume                                   %
%                                                                         %
% Author: George Darwin                                                   %
%                                                                         %
% Commentator: Francis Darwin                                             %
%              E. W. Brown                                                %
%                                                                         %
% Editor: F. J. M. Stratton                                               %
%         J. Jackson                                                      %
%                                                                         %
% Release Date: March 16, 2011 [EBook #35588]                             %
% Most recently updated: June 11, 2021                         %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: UTF-8                                           %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK SCIENTIFIC PAPERS ***         %
%                                                                         %
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The Project Gutenberg EBook of Scientific Papers by Sir George Howard
Darwin, by George Darwin

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


Title: Scientific Papers by Sir George Howard Darwin
       Volume V. Supplementary Volume

Author: George Darwin

Commentator: Francis Darwin
             E. W. Brown

Editor: F. J. M. Stratton
        J. Jackson

Release Date: March 16, 2011 [EBook #35588]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK SCIENTIFIC PAPERS ***
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\LARGE\textbf{SCIENTIFIC PAPERS}
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CAMBRIDGE UNIVERSITY PRESS \\[\TmpLen]
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  }{%
    \includegraphics[width=0.875\textwidth]{./images/frontis.jpg}
  }
\iffalse
[Hand-written note: From a water-colour drawing
by his daughter
Mrs Jacques Raverat
G. H. Darwin]
\fi
\end{figure}
\vfill
\clearpage
\DPPageSep{005}{iii}
\begin{center}
\setlength{\TmpLen}{12pt}%
\textbf{\Huge SCIENTIFIC PAPERS}
\vfil
\footnotesize%
BY \\[\TmpLen]
{\normalsize SIR GEORGE HOWARD DARWIN} \\
{\scriptsize K.C.B., F.R.S. \\
FELLOW OF TRINITY COLLEGE \\
PLUMIAN PROFESSOR IN THE UNIVERSITY OF CAMBRIDGE}
\vfil
VOLUME V \\
SUPPLEMENTARY VOLUME \\[\TmpLen]

{\scriptsize CONTAINING} \\

BIOGRAPHICAL MEMOIRS BY SIR FRANCIS DARWIN \\[2pt]
AND PROFESSOR E. W. BROWN, \\[2pt]
LECTURES ON HILL'S LUNAR THEORY, \textsc{etc.}
\vfil
EDITED BY \\
F. J. M. STRATTON, M.A., \textsc{and} J. JACKSON, M.A., \textsc{B.Sc.}
\vfil\vfil
\normalsize
Cambridge: \\
at the University Press \\
1916
\end{center}
\newpage
\DPPageSep{006}{iv}
\begin{center}
\null\vfill\scriptsize
\textgoth{Cambridge}: \\
PRINTED BY JOHN CLAY, M.A. \\
AT THE UNIVERSITY PRESS
\vfill
\end{center}
\newpage
\DPPageSep{007}{v}


\Chapter{Preface}

\First{Before} 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, Cosmogony,
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.
\DPPageSep{008}{vi}

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.

\footnotesize
\settowidth{\TmpLen}{F. J. M. S.\quad}%
\null\hfill\parbox{\TmpLen}{F. J. M. S.\\ J. J.}

\scriptsize
\textsc{Greenwich,} \\
\indent\indent6 \textit{December} 1915.

\normalsize
\newpage
\DPPageSep{009}{vii}
%[** TN: Table of Contents]


\Chapter{Contents}
\enlargethispage{36pt}
\ToCFrontis{Portrait of Sir George Darwin}%{Frontispiece}

\ToCPAGE

\ToCChap{Memoir of Sir George Darwin by his brother Sir Francis Darwin}
{chapter:3}%{ix}

\ToCChap{The Scientific Work of Sir George Darwin by Professor E. W.
Brown}{chapter:4}%{xxxiv}

\ToCChap{Inaugural lecture (Delivered at Cambridge, in 1883, on Election to
the Plumian Professorship)}{chapter:5}%{1}

\ToCChap{Introduction to Dynamical Astronomy}{chapter:6}%{9}

\ToCChap{Lectures on Hill's Lunar Theory}{chapter:7}%{16}

\ToCSec{§ 1.}{Introduction}{1}%{16}

\ToCSec{§ 2.}{Differential Equations of Motion and Jacobi's Integral}
{2}%{17}

\ToCSec{§ 3.}{The Variational Curve}{3}%{22}

\ToCSec{§ 4.}{Differential Equations for Small Displacements from the
Variational Curve}{4}%{26}

\ToCSec{§ 5.}{Transformation of the Equations in § 4}{5}%{29}

\ToCSec{§ 6.}{Integration of an important type of Differential Equation}
{6}%{36}

\ToCSec{§ 7.}{Integration of the Equation for~$\delta p$}{7}%{39}

\ToCSec{§ 8.}{Introduction of the Third Coordinate}{8}%{43}

\ToCSec{§ 9.}{Results obtained}{9}%{45}

\ToCSec{§ 10.}{General Equations of Motion and their solution}
{10}%{46}

\ToCSec{§ 11.}{Compilation of Results}{11}%{52}

\ToCNote{Note 1.}{On the Infinite Determinant of § 5}{note:1}%{53}

\ToCNote{Note 2.}{On the periodicity of the integrals of the equation
\[
\frac{d^{2}\, \delta p}{d\tau^{2}} + \Theta\, \delta p = 0,
\]
where $\Theta = \Theta_{0} + \Theta_{1} \cos 2\tau
  + \Theta_{2} \cos 4\tau + \dots$.}{note:2}%{55}

\ToCChap{On Librating Planets and on a New Family of Periodic Orbits}
{chapter:8}%{59}

\ToCMisc{[\textit{Monthly Notices of the Royal Astronomical Society}, Vol.~72 (1912), pp.~642--658.]}

\ToCChap{Address to the International Congress of Mathematicians at
Cambridge in 1912}{chapter:9}%{76}

\ToCChap{Index}{indexpage}%{80}
\DPPageSep{010}{viii}
% [Blank Page]
\DPPageSep{011}{ix}

\cleardoublepage
\phantomsection
\pdfbookmark[-1]{Main Matter}{Main Matter}


\Chapter{Memoir of Sir George Darwin}
\BY{His Brother Sir Francis Darwin}
\SetRunningHeads{Memoir of Sir George Darwin}{By Sir Francis Darwin}
\index{Darwin, Sir Francis, Memoir of Sir George Darwin by}%
\index{Darwin, Sir George, genealogy}%
\index{Galton, Sir Francis}%

George Howard, the fifth\footnoteN
  {The third of those who survived childhood.}
child of Charles and Emma Darwin, was
born at Down July~9th, 1845. Why he was christened\footnoteN
  {At Maer, the Staffordshire home of his mother.}
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 genealogist
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 Professor
Pearson's \textit{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\DPnote{[** TN: Hyphenated in original]} and I am
fixed on the spot where I shall end it.\footnotemarkN'' It would have been difficult to
\footnotetextN{\textit{Life and Letters of Charles Darwin}, vol.~\Vol{I.} p.~318.}%
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.
\DPPageSep{012}{x}

George retained throughout life his deep love for Down. For the lawn
\index{Darwin, Sir George, genealogy!boyhood}%
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 Down,---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:
\begin{Quote}
\centering
The Duke is dead. Dodos are out of the world.
\end{Quote}
He perhaps carried rather far the good habit of re-reading one's\DPnote{[** TN: [sic]]} 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
\DPPageSep{013}{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 loaded 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
\index{Darwin, Sir George, genealogy!interested in heraldry}%
Library copies of Guillim and Edmonson\footnoteN
  {Guillim, John, \textit{A display of heraldry}, 6th~ed., folio~1724. Edmonson,~J., \textit{A complete body
  of heraldry}, folio~1780.}
were generally at Down. He
retained a love of the science through life, and his copy of Percy's \textit{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 \textit{Climbing Plants}, are by his hand.

His early education was given by governesses: but the boys of the family
\index{Darwin, Sir George, genealogy!education}%
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 house 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 them be found in public
schools. It was kept by the Rev.~Charles Pritchard\footnotemarkN, a man of strong
\footnotetextN{Afterwards Savilian Professor of Astronomy at Oxford. Born~1808, died~1893.}%
character and with a gift for teaching mathematics by which George undoubtedly
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
\DPPageSep{014}{xii}
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
\index{Darwin, Sir George, genealogy!at Cambridge}%
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:
\begin{Quote}
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\footnotemarkN. We all recognised
\footnotetextN{The late Mr~Routh was the most celebrated Mathematical ``Coach'' of his
day.}%
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 enjoyment
of other things\footnotemarkN. I remember that as the time of the examination
\footnotetextN{Compare Charles Darwin's words: ``George has not slaved himself, which makes his
  success the more satisfactory.'' (\textit{More Letters of C.~Darwin}, vol.~\Vol{II.} p.~287)}%
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\ldots. It has been a standing wonder to me that
he developed the patience for making the laborious numerical calculations
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.
\end{Quote}

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,
\DPPageSep{015}{xiii}
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 magnificent
``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\footnotemarkN:
\footnotetextN{\textit{Emma Darwin, A Century of Family Letters}, vol.~\Vol{II.} p.~186.}%
\index{Darwin, Charles, ix; letters of}%
\begin{Letter}
  {\textsc{Down}, \textit{Jan.}~24\textit{th} [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, perseverance 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.

  \Signature{Your affectionate Father,}{Ch.~Darwin.}
\end{Letter}

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
\index{Darwin, Sir George, genealogy!friendships}%
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 remembers
her father (not long before his death) saving 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
\DPPageSep{016}{xiv}
advantage of knowing Lord Rayleigh at Cambridge, and retained his friendship
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\footnotemarkN, and in October~1869
\footnotetextN{He was called in 1874 but did not practise.}%
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
\index{Darwin, Sir George, genealogy!ill health}%
his growing inclination towards science\footnotemarkN. Thus in the summer of~1869, when
\footnotetextN{As a boy he had energetically collected Lepidoptera during the years 1858--64, 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 \textit{Nature}, Dec.~1, 1870, that
  George was told off to make sketches of the Corona.}%
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~1874, ``February to July very ill.'' In spite of unwellness he began in 1872--3
to write on various subjects. He sent to \textit{Macmillan's Magazine}\footnoteN
  {\textit{Macmillan's Magazine}, 1872, vol.~\Vol{XXVI.} pp.~410--416.}
an entertaining
article, ``Development in Dress,'' where the various survivals in modern
\DPPageSep{017}{xv}
costume were recorded and discussed from the standpoint of evolution. In
1873 he wrote ``On beneficial restriction to liberty of marriage\footnotemarkN,'' a eugenic
\footnotetextN{\textit{Contemporary Review}, 1873, vol.~\Vol{XXII.} pp.~412--426.}%
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\footnoteN{Not published.}---on
which he writes in his diary, ``found it was old.'' Besides another paper in the
\textit{Messenger of Mathematics}, he reviewed ``Whitney on Language\footnotemarkN,'' and wrote
\footnotetextN{\textit{Contemporary Review}, 1874, vol.~\Vol{XXIV.} pp.~894--904.}%
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\footnotemarkN. It
\footnotetextN{\textit{Journal of the Statistical Society}, 1875, vol.~\Vol{XXXVIII.} pt~2, pp.~158--182, also pp.~183--184,
  and pp.~344--348.}%
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 Equipotentials
\index{Darwin, Sir George, genealogy!association with Lord Kelvin}%
\index{Kelvin, associated with Sir George Darwin}%
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 \textit{Philosophical
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 \textit{Phil.\ Trans.} in~1879. It was in
regard to this paper that his father wrote to George on October~29th, 1878\footnotemarkN:
\footnotetextN{Probably he heard informally at the end of October what was not formally determined till
  November.}%
\index{Darwin, Charles, ix; letters of}%

\begin{Letter}{}{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.'\ldots\  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)\ldots\footnotemarkN.
\end{Letter}
\footnotetextN{\textit{Emma Darwin, A Century of Family Letters}, 1915, vol.~\Vol{II.} p.~233.}%

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
\DPPageSep{018}{xvi}
not merely into scientific co-operation\DPnote{[** TN: Hyphenated in original]} but into a close friendship. Sir~Joseph
\index{Darwin, Sir George, genealogy!friendships}%
Larmor has recorded\footnoteN
  {\textit{Nature}, Dec.~12, 1912.}
that George's ``tribute to Lord Kelvin, to whom he
dedicated volume~\Vol{I} of his Collected Papers\footnotemarkN\ldots gave lively pleasure to his
master and colleague.'' His words were:
\footnotetextN{It was in 1907 that the Syndics of the Cambridge University Press asked George to prepare
  \index{Darwin, Sir George, genealogy!at Cambridge}%
  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 Rayleigh and other men of distinction.}%
\begin{Quote}
Early in my scientific career it was my good fortune to be brought
into close personal relationship with Lord Kelvin. Many visits to Glasgow
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.
\end{Quote}

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 Whittingehame, or of luncheons at
Mr~Balfour's house in Carlton Gardens for which George's scientific committee
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 eminently
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\footnotemarkN,
\footnotetextN{Thus in 1872 he was in Homburg, 1873~in Cannes, 1874~in Holland, Belgium, Switzerland
  and Malta, 1876~in Italy and Sicily.}%
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 friendships
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 \textit{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.

\Section{}{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
\DPPageSep{019}{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\footnoteN
  {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 Raverat, 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.'\,"}
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 connected
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~1900--1 to~1903--4
and on the Council of the Senate in 1905--6 and~1908--9. But he never
became a professional syndic---one of those virtuous persons who spend their
lives in University affairs. In his obituary of George (\textit{Nature}, Dec.~12, 1912),
Sir~Joseph Larmor writes:
\index{Darwin, Sir George, genealogy!university work, described by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}%
\index{Larmor, Sir Joseph, Sir George Darwin's work on university committees}%
\index{University committees, Sir George Darwin on, by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}%
\begin{Quote}
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
\DPPageSep{020}{xviii}
of the Cambridge University Association, to promote the further endowment
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.
\end{Quote}
The Master of Christ's writes:
\index{Darwin, Sir George, genealogy!university work, described by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}%
\index{Master of Christ's, Sir George Darwin's work on university committees}%
\index{Newall, Prof., Sir George Darwin's work on university committees}%
\index{University committees, Sir George Darwin on, by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}%
\begin{Letter}{\textit{May}~31\textit{st}, 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\DPnote{[** TN: Hyphenated in original]} 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.
\end{Letter}
Professor Newall has also been good enough to give me his impressions:
\begin{Quote}
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 formulating
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 responsible
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.
\end{Quote}
\DPPageSep{021}{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.
\index{Darwin, Sir George, genealogy!marriage}%
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\footnoteN
  {\textit{Emma Darwin, A Century of Family Letters}, Privately printed, 1904, vol.~\Vol{II.} p.~350.}
from George's mother, dated April~15th, 1884:
\begin{Quote}
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\ldots 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.
\end{Quote}

Later, probably in June, George's mother wrote\footnoteN
  {\textit{Emma Darwin, A Century of Family Letters}, 1912, vol.~\Vol{II.} p.~266.}
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
\index{Darwin, Sir George, genealogy!house at Cambridge}%
Grange\footnotemarkN, which remained his home to the end of his life. It stands at the
\footnotetextN{At that time it was known simply as \textit{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.}%
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.
\DPPageSep{022}{xx}
It was formerly the house of the Beales family\footnotemarkN, one of the old merchant
\footnotetextN{The following account of Newnham Grange is taken from C.~H. Cooper's \textit{Memorials of
  Cambridge}, 1866, vol.~\Vol{III.} p.~262 (note):---``The site of the hermitage was leased by the Corporation
  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.}%
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\footnotemarkN.'' The house is conveniently close to the town, yet has a most
\footnotetextN{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.}%
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 \textit{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,
\index{Darwin, Sir George, genealogy!children}%
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
\DPPageSep{023}{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 accompanied
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 \textit{Romeo and Juliet}, in which the hero and heroine
were scions of the rival factions of Trinity and St~John's.

\Section{}{Games and Pastimes.}
\index{Darwin, Sir George, genealogy!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 \textit{History}. I think George must have had pleasure in the
obvious development of the tennis court from some primaeval court-yard in
which the \textit{pent-house} was the roof of a shed, and the \textit{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 \textit{Tambour} and
the \textit{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
\DPPageSep{024}{xxii}
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 painstaking,
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 \textit{The Archer's Register}\footnotemarkN. George's point was
\footnotetextN{\textit{The Archer's Register} for 1912--1913, by H.~Walrond. London, \textit{The Field} Office, 1913.}%
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 \textit{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.''

\Section{}{Scientific Committees.}
\index{Committees, Sir George Darwin on}%
\index{Darwin, Sir George, genealogy!work on 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:---
\index{Meteorological Council, by Sir Napier Shaw}%
\index{Shaw, Sir Napier, Meteorological Council}%
\begin{Quote}
It was in February~1885 upon the retirement of Warren De~la~Rue
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.
\DPPageSep{025}{xxiii}

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 examination
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 Commission
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
\DPPageSep{026}{xxiv}
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 fundamental
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 inspection
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
\DPPageSep{027}{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 influence
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 contributions
to meteorological literature. He gave me a note on a curious
point in the evaluation of the velocity equivalents of the Beaufort Scale
\DPPageSep{028}{xxvi}
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.
\end{Quote}

On May~6, 1904, he was elected President of the British Association---the
\index{British Association, South African Meeting, 1905}%
\index{South African Meeting of the British Association, 1905}%
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:
\begin{center}
\footnotesize
\begin{tabular}{cl}
August 19 & Embark, \\
\Ditto 22 & Arrive at Durban, \\
\Ditto 23 & Mount Edgecombe, \\
\Ditto 24 & Pietermaritzburg, \\
\Ditto 26 & Colenso, \\
\Ditto 27 & Ladysmith, \\
\Ditto 28 & Johannesburg.
\end{tabular}
\end{center}

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 \textit{The Observatory},
Jan.~1913, p.~58:
\begin{Quote}
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.
\end{Quote}
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.~Poincaré.
\DPPageSep{029}{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\footnoteN
  {See Prof.~Brown's Memoir, \Pageref{xlix}.}
of the Royal Society, his subject
being ``Tidal Prediction.'' This annual prælection 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
\index{Tides, The@\textit{Tides, The}}%
the origin of his book on \textit{Tides}, published in the following year. Of this
Sir~Joseph Larmor says\footnoteN
  {\textit{Nature}, 1912. See also Prof.~Brown's Memoir, \Pageref{l}.}
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.

\Section{}{International Associations.}

During the last ten or fifteen years of his life George was much occupied
\index{Geodetic Association, International}%
\index{Larmor, Sir Joseph, Sir George Darwin's work on university committees!International Geodetic Association}%
with various International bodies, e.g.~the International Geodetic Association,
the International Association of Academics, 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 (\textit{Nature}, December~12, 1912) gives the following
account of the origin of the Association:
\begin{Quote}
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\ldots. 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.
\end{Quote}
\DPPageSep{030}{xxviii}

The further history of George's connection with the Association is told in
\index{Bakhuyzen, Dr Van d.\ Sande, Sir George Darwin's connection with the International Geodetic Association}%
\index{Geodetic Association, International}%
the words of its Secretary, Dr~van~d.\ Sande Bakhuyzen, to whom I am greatly
indebted.
\begin{Quote}
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 Association,
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 theoretical
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 Copenhagen
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.
\DPPageSep{031}{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.
\index{Mathematicians, International Congress of, Cambridge, 1912}%

With the utmost care he prepared everything to render the Conference
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.
\end{Quote}
Sir~Joseph Larmor writes\footnoteN
  {\textit{Nature}, Dec.~12, 1912.}:
\index{Larmor, Sir Joseph, Sir George Darwin's work on university committees!International Congress of Mathematicians at Cambridge 1912}%
\index{Congress, International, of Mathematicians at Cambridge, 1912!note by Sir Joseph Larmor}%
\begin{Quote}
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 \textit{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 \textit{verve} the deliberations of the final assembly of what turned
out to be a most successful meeting of that important body.
\end{Quote}
\DPPageSep{032}{xxx}

\Section{}{Personal Characteristics.}
\index{Darwin, Sir George, genealogy!personal characteristics}%
\index{Darwin, Margaret, on Sir George Darwin's personal characteristics}%
\index{Raverat, Madame, on Sir George Darwin's personal characteristics}%

His daughter, Madame Raverat, writes:
\begin{Quote}
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 \textit{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 Schönbrunn 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\footnotemarkN.
\footnotetextN{Compare Mr~Chesterton's \textit{Twelve Types}, 1903, p.~190. He speaks of Scott's critic in the
  \textit{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.''}%

The books he used to read to us when we were quite small,
and which we adored, were Percy's \textit{Reliques} and the \textit{Prologue to the
Canterbury Tales}. He used often to read Shakespeare to himself,
I think generally the historical plays, Chaucer, \textit{Don Quixote} in Spanish,
and all kind of books like Joinville's \textit{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 understanding
and appreciation of the glory and tragedy of this war would
be like no one else's.
\end{Quote}
His daughter Margaret Darwin writes:
\begin{Quote}
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 \textit{with} them in a way that
\DPPageSep{033}{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 important
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.
\end{Quote}

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\footnoteN
  {\textit{Emma Darwin, A Century of Family Letters}, 1915, vol.~\Vol{II.} p.~146.}
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:
\begin{Quote}
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.
\end{Quote}

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:
\DPPageSep{034}{xxxii}
\begin{Quote}
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 \textit{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.
\end{Quote}

Of his energy we have evidence in the \textit{amount} of work contained in
\index{Darwin, Sir George, genealogy!energy}%
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 defined 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.
\DPPageSep{035}{xxxiii}


\Heading{Honours, Medals, Degrees, Societies, etc.}
\index{Darwin, Sir George, genealogy!honours}%

\Subsection{Order. \upshape K.C.B. 1905.}

\Subsection{Medals\footnotemarkN.}
\footnotetextN{Sir~George's medals are deposited in the Library of Trinity College, Cambridge.}

1883. Telford Medal of the Institution of Civil Engineers.

1884. Royal Medal\footnotemarkN.
\footnotetextN{Given by the Sovereign on the nomination of the Royal Society.}

1892. Royal Astronomical Society's Medal.

1911. Copley Medal of the Royal Society.

1912. Royal Geographical Society's Medal.

\Subsection{Offices.}

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\footnotemarkN, Royal Astronomical
\footnotetextN{Re-elected in 1912.}
Society, British Association.

\Subsection{Doctorates, etc.\ of Universities.}

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

\Subsection{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 (Philosophical
Society), Rome (Lincei), Stockholm (Swedish Academy), Toronto
(Physical Society), Washington (National Academy), Wellington (New
Zealand Inst.).

\Subsection{Correspondent of Academies, etc.\ at}

Acireale (Zelanti), Berlin (Prussian Academy), Buda Pest (Hungarian
Academy), Frankfort (Senckenberg.\ Natur.\ Gesell.), Göttingen (Royal Society),
Paris, St~Petersburg, Turin, Istuto Veneto, Vienna\footnotemarkN.
\footnotetextN{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.}
\DPPageSep{036}{xxxiv}


%[** TN: Changed the running heads; original splits the title]
\Chapter{The Scientific Work of Sir George Darwin}
\BY{Professor E. W. Brown}
\index{Brown, Prof.\ E. W., Sir George Darwin's Scientific Work}%
\index{Darwin, Sir George, genealogy!scientific work, by Prof.\ E. W. Brown}%
\index{Darwin, Sir George, genealogy!characteristics of his work}%

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 Poincaré, 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
\DPPageSep{037}{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 mathematicians
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 speculative
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.
\DPPageSep{038}{xxxvi}

The influence which Darwin exerted has been felt in many directions.
\index{Cosmogony, Sir George Darwin's influence on}%
\index{Darwin, Sir George, genealogy!his first papers}%
\index{Darwin, Sir George, genealogy!his influence on cosmogony}%
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
\index{Darwin, Sir George, genealogy!his relationship with his pupils}%
\index{Pupils, Darwin's relationship with his}%
friend. First meeting them at his courses on some geophysical or astronomical
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.

\tb

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
\DPPageSep{039}{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 \textit{à~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\footnotemarkN, ``may array facts
\footnotetextN{\textit{Science}, Sept.~4, 1914, p.~333.}%
\index{Barrell, Prof., Cosmogony as related to Geology and Biology}%
\index{Cosmogony, Sir George Darwin's influence on!as related to Geology and Biology, by Prof.\ Barrell}%
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\ldots. 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 unrestrained
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
\index{Darwin, Sir George, genealogy!association with Lord Kelvin}%
\index{Kelvin, associated with Sir George Darwin}%
into close connection with Lord Kelvin, then Sir~William Thomson. The
\DPPageSep{040}{xxxviii}
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 statistical
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 therefore
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
\index{Earth-Moon theory of Darwin, described by Mr S. S. Hough}%
from the Notice in the \textit{Proceedings of the Royal Society}\footnoteN
  {Vol.~\Vol{89\;A}, p.~i.}
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.
\DPPageSep{041}{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' (\textit{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-equilibrating
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' (\textit{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\ldots for moderate degrees of viscosity, the position of zero
\DPPageSep{042}{xl}
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\ldots 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 practically
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' (\textit{Phil.\
Trans.}, 1880, vol.~171), Darwin summarises the results derived in his
different memoirs. Various factors ignored in the earlier investigations,
\DPPageSep{043}{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 \textit{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'
(\textit{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 constituted
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
\DPPageSep{044}{xlii}
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 he 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
\index{Poincaré, reference to, by Sir George Darwin!on equilibrium of fluid mass in rotation}%
\index{Equilibrium of a rotating fluid}%
\index{Rotating fluid, equilibrium of}%
memoir by M.~Poincaré, `Sur l'équilibre d'une masse fluide animée d'un
mouvement de rotation' (\textit{Acta Math.}, vol.~7).

``The figures of equilibrium known as Maclaurin's spheroid and Jacobi's
\index{Jacobi's ellipsoid}%
\index{Maclaurin's spheroid}%
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, Poincaré succeeded in enunciating and
applying to this problem the principle of exchange of stabilities. This principle
may be 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.~Poincaré 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 coefficients
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. Poincaré 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 \textit{vice versâ}.
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.
\DPPageSep{045}{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. Poincaré 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 Poincaré
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 Poincaré, 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
\index{Ellipsoidal harmonics}%
\index{Harmonics, ellipsoidal}%
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' (\textit{Phil.\ Trans.},~A, 1901, vol.~197), apart from the application
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 Poincaré, `On the Pear-shaped Figure of
\index{Pear-shaped figure of equilibrium}%
Equilibrium of a Rotating Mass of Liquid' (\textit{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.~Lia\-pou\-noff pointed to an
\index{Liapounoff's work on rotating liquids}%
opposite conclusion. Darwin again reverts to this point in a further paper
`On the Figure and Stability of a Liquid Satellite' (\textit{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
\index{Roche's ellipsoid}%
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
\index{Jeans, J. H., on rotating liquids}%
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 Poincaré, but involves far greater
\DPPageSep{046}{xliv}
simplicity of the conditions. Jeans finds solutions of his problem strictly
analogous to the spheroids of Maclaurin, the ellipsoids of Jacobi, and the
pear of Poincaré, 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
\index{Hecker's observations on retardation of tidal oscillations in the solid earth}%
evidence of any considerable amount of retardation in the tidal oscillations
of the solid earth, for, within the limits of error of observation, the
phase of the 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
\index{Tidal friction as a true cause of change}%
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\ldots.

``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 load to a slight difference
in the conclusion; moreover such a real cause as the secular increase
in the masses of the earth and moon through the accumulation of meteoric
dust, and possibly other causes, are left out of consideration.

``The exact nature of the process by which the moon was detached from
the earth must remain even more speculative. I suggested that the fission
of the primitive planet may have been brought about by the synchronism of
the solar tide with the period of the fundamental free oscillation of the
\DPPageSep{047}{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
\index{Kelvin, associated with Sir George Darwin!cooling of earth}%
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 contended
that the actual distribution of land and sea proves that the planet
solidified at a time when the day had nearly its present length. If this
were true the effects of tidal friction relate to a period antecedent to the
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.~\Vol{XX} of the third edition~(1910) of his book
on the Tides, Darwin gives a popular summary of this evidence which had
\DPPageSep{048}{xlvi}
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 homogeneous
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
\index{Numerical work on cosmogony}%
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 communicated
to his pupils, one of whom, Mr~J.~H. Jeans, as mentioned above, is
continuing with success to solve those gravitational problems on similar
lines.

The nebular hypothesis of Kant and Laplace has long held the field as
\index{Kant, Nebular Hypothesis}%
\index{Laplace, Nebular Hypothesis}%
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 hypothesis
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,''
\index{Mechanical condition 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.
\DPPageSep{049}{xlvii}
In his preface to volume~\Vol{IV} he says with respect to this paper: ``Cosmogonists
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
\index{Chamberlain and Moulton, Planetesimal Hypothesis}%
\index{Moulton, Chamberlain and, Planetesimal Hypothesis}%
\index{Planetesimal Hypothesis of Chamberlain and Moulton}%
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
\index{Tidal problems, practical}%
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
\DPPageSep{050}{xlviii}
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 computation
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 \textit{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 automatically
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
\textit{Proceedings of the Royal Society}.

A little later he published the description of his now well known abacus,
\index{Abacus}%
designed to avoid the frequent rewriting\DPnote{[** TN: Not hyphenated in original]} of the numbers when the harmonic
analysis for many different periods is needed. Much care was taken to obtain
\DPPageSep{051}{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 computation
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
\index{Bakerian lecture}\Pagelabel{xlix}%
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 predictions
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 \textit{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 \textit{Discovery}, concludes Darwin's
published work on practical tidal problems. But he was constantly in correspondence
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
\index{Rigidity of earth, from fortnightly tides}%
\index{Tide, fortnightly}%
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
\DPPageSep{052}{l}
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 produced
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
\index{Hecker's observations on retardation of tidal oscillations in the solid earth}%
whose results confirmed his first estimate. Since his death the remarkable
experiment of Michelson\footnoteN
  {\textit{Astrophysical Journal}, March,~1914.}
\index{Michelson's experiment on rigidity of earth}%
\index{Rigidity of earth, from fortnightly tides!Michelson's experiment}%
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
\index{Tides, The@\textit{Tides, The}}%
\index{Tides, articles on}\Pagelabel{l}%
work of himself and others. He wrote several articles on Tides, notably for
the \textit{Encyclopaedia Britannica} and for the \textit{Encyclopaedie der Mathematischen
Wissenschaften}, but he will be best remembered in this connection for his
volume \textit{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 \textit{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.
\textit{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
\index{Hill, G. W., Lunar Theory}%
moon were published. The first of these, \textit{Researches in the Lunar Theory},
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
\DPPageSep{053}{li}
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 \textit{vice
versâ}) 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,
\DPPageSep{054}{lii}
however, saw that it was of more general interest than its particular application
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. Poincaré 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; \textit{Les méthodes nouvelles de la Mécanique
Céleste}. His treatment throughout is highly theoretical. He shows that
\index{Poincaré, reference to, by Sir George Darwin!\textit{Les Méthodes Nouvelles de la Mécanique Céleste}}%
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 \textit{résumé} 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
\index{Glaisher, Dr J. W. L., address on presenting the gold medal of the R.A.S. to G. W. Hill}%
\index{Hill, G. W., Lunar Theory!awarded gold medal of R.A.S.}%
\index{Hill, G. W., Lunar Theory!lectures by Darwin on Lunar Theory}%
contains an illuminating analysis of the methods employed and the ideas
which are put forward. Probably both Darwin and Adams had a considerable
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 recommendation
in the summer of 1888 that the writer took up the study of Hill's
papers. ``They seem to be very 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
Poincaré had gone to the foundations. Darwin felt, however, that the work of
\DPPageSep{055}{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 Poincaré'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
\index{Numerical work, great labour of}%
\index{Periodic orbits, Darwin begins papers on}%
\index{Periodic orbits, Darwin begins papers on!great numerical difficulties of}%
\index{Periodic orbits, Darwin begins papers on!stability of}%
the differential equations in harmonic series. It was obvious that the slowness
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 $x$-axis with an arbitrary speed in a direction
parallel to the $y$-axis. Tracing the orbit step-by-step, he again reaches the
$x$-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 $\frac{3}{4}$~hr.\ (allowing for
correction of small mistakes---which sometimes is tedious). You will see
that it is far from periodic\ldots. I 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
\DPPageSep{056}{liv}
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
\index{Hough, S. S., Darwin's work on Earth-Moon Theory, xxxviii; Periodic Orbits}%
\index{Periodic orbits, Darwin begins papers on!classification of, by Jacobi's integral}%
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 Poincaré'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 $x$-axis, and some later
corrections of the classification partly by himself and partly by Mr~S.~S. Hough
were necessary. In volume~\Vol{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
\DPPageSep{057}{lv}
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 Poincaré, 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.
\DPPageSep{058}{lvi}
%[Blank Page]
\DPPageSep{059}{1}
\index{Orbits, periodic|see{Periodic orbits}}


\Chapter{Inaugural Lecture}
\index{Inaugural lecture}%
\index{Cambridge School of Mathematics}%
\index{Lecture, inaugural}%
\index{Mathematical School at Cambridge}%

\Heading{(Delivered at Cambridge, in 1883, on election to the
Plumian Professorship)}

\First{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\DPnote{[** TN: Double word OK]} those, who
by their energy and ability have raised up all the activity which we now see?
\DPPageSep{060}{2}

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\footnotemarkN. His pupil Balfour had perhaps a less
\footnotetextN{Sir Michael Foster was elected the first Professor of Physiology a few weeks after the
  delivery of this lecture.}%
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 development
of practical work in various branches within these buildings. It must
suffice to say that each school has had its difficulties, and that those difficulties
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 \textit{nothing}
to say worth hearing by students of mathematics. Granting the possibility
\DPPageSep{061}{3}
that a distinguished man may lack the power of exposition, yet it is inadmissible
that they are \textit{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
\index{Mathematical School at Cambridge!Tripos}%
\index{Tripos, Mathematical}%
Tripos, you must bear in mind that I have hitherto taken no part in mathematical
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 Mathematical
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 \textit{coup d'{\oe}il} 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
\DPPageSep{062}{4}
to remember. A first step in this direction has been taken by the introduction
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
\index{Mathematical School at Cambridge!text-books}%
\index{Text-books, mathematical}%
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 got 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 \textit{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 \textit{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
\DPPageSep{063}{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 generally
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
\DPPageSep{064}{6}
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\footnotemarkN; Adams' manuscript is as much a model of
\footnotetextN{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 \textit{Electricity}
  was crowded with errata, which have now been weeded out one by one.}%
neatness in mathematical writing as Porson'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
\DPPageSep{065}{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. Whatever
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 examination
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 \textit{only}, and
\DPPageSep{066}{8}
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 beginning
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.
\DPPageSep{067}{9}


\Chapter{Introduction to Dynamical Astronomy}
\index{Introduction to Dynamical Astronomy}%
\index{Dynamical Astronomy, introduction to}%

\First{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 communicated
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, however,
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, gravitation
will not be transmitted to infinity, but only to a limited distance.
\DPPageSep{068}{10}
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 surrounding
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 experimentally
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
\index{Gravitation, theory of}%
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
\DPPageSep{069}{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 electrodynamic
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 Göttingen, 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 conservative
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 Poincaré.

Two leading divisions of dynamical astronomy are the planetary theory
\index{Lunar and planetary theories compared}%
\index{Planetary and lunar theories compared}%
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
\DPPageSep{070}{12}
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 differing
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.
\DPPageSep{071}{13}
As an illustration of the simplest kind suppose that the equation of motion
of a system under no forces were
\[
\frac{d^{2}x}{dt^{2}} + n^{2}x = 0.
\]
Then we know that the solution is
\[
x = A \cos nt + B \sin nt,
\]
that is to say the free period is~$\dfrac{2 \pi}{n}$. Suppose then such a system be acted on
by a perturbing force $F\cos(n - \epsilon)t$, where $\epsilon$~is small; the equation of motion is
\[
\frac{d^{2}x}{dt^{2}} + n^{2}x = F\cos(n - \epsilon)t,
\]
and the solution corresponding to such a disturbing force is
\[
x = \frac{F}{-(n - \epsilon)^{2} + n^{2}} \cos(n - \epsilon)t
  = \frac{F}{2n\epsilon - \epsilon^{2}} \cos(n - \epsilon)t.
\]
If $\epsilon$~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
\textit{à~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
\[
\frac{n'}{in ± i'n'} \text{ and } \left(\frac{n'}{in ± i'n'}\right)^{2},
\]
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
\DPPageSep{072}{14}
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 subjected
to the most searching analysis by Chandler and others, has shown
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, apparently
well founded, which prevented the detection of the small inequality
in question. It was Chandler who first searched for an inequality of unknown
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.
\index{Latitude, variation of}%
\index{Variation, the!of latitude}%
The 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
\index{Earth's figure, theory of}%
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.
\DPPageSep{073}{15}

The theory of Saturn's rings is another branch. The older and now
\index{Saturn's rings}%
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 equilibrium
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
\index{Gravitation, theory of!universal}%
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 explanation
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 show how vast it is, and I cannot
hope in these lectures to do more than touch on some of the leading topics.
\DPPageSep{074}{16}


\Chapter{Hill's Lunar Theory}
\index{Hill, G. W., Lunar Theory!lectures by Darwin on Lunar Theory}%
\index{Hill, G. W., Lunar Theory!characteristics of his Lunar Theory}%
\index{Lunar Theory, lecture on}%

\Section{§ 1. }{Introduction\footnotemark.}

\footnotetext{The references in this section are to Hill's ``Researches in the Lunar Theory'' first published
  (1878) in the \textit{American Journal of Mathematics}, vol.~\Vol{I.} pp.~5--26, 129--147 and reprinted in
  \textit{Collected Mathematical Works}, vol.~\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 \textit{Acta Mathematica}, vol.~\Vol{VIII.} pp.~1--36, 1886 and in \textit{Collected
  Mathematical Works}, vol.~\Vol{I.} pp.~243--270.}

\First{An} account of Hill's \textit{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?\ldots\ 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.''
\DPPageSep{075}{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.

\Section{§ 2. }{Differential Equations of Motion and Jacobi's Integral.}
\index{Differential Equations of Motion}%
\index{Equations of motion}%

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 rectangular
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
the axes through~$G$ are clearly~$\dfrac{E}{E + M} x$, $\dfrac{E}{E + M} y$, $\dfrac{E}{E + M} z$; those of~$E$ are
$-\dfrac{E}{E + M} x$, $-\dfrac{E}{E + M} y$, $-\dfrac{E}{E + M} z$. The distances $EM, E\m', M\m'$\DPnote{** TN: Inconsistent overlines in original} are denoted
\DPPageSep{076}{18}
by $r, r_1, \Delta$ respectively. It is assumed that $G$~describes a Keplerian ellipse
round~$\m'$ so that $x', y', z'$ are known functions of the time. The accelerations
of~$M$ relative to~$E$ are shewn in the diagram.
\begin{figure}[hbt!]
\centering
\Input[0.75\textwidth]{p018}
\caption{Fig.~1.}
\end{figure}

We have
\begin{gather*}
r^{2} = x^{2} + y^{2} + z^{2}, \\
\begin{aligned}
r_{1}^{2}
  &= \left(x' + \frac{Mx}{E + M}\right)^{2}
   + \left(y' + \frac{My}{E + M}\right)^{2}
   + \left(z' + \frac{Mz}{E + M}\right)^{2}, \\
\Delta^{2}
  &= \left(x' - \frac{Ex}{E + M}\right)^{2}
   + \left(y' - \frac{Ey}{E + M}\right)^{2}
   + \left(z' - \frac{Ez}{E + M}\right)^{2}.
\end{aligned}
\end{gather*}

Hence
\begin{gather*}
\frac{\dd r}{\dd x} = \frac{x}{r}, \\
\begin{aligned}
\frac{E + M}{M}\, \frac{\dd r_{1}}{\dd x}
  &= \frac{x' + \dfrac{Mx}{E + M}}{r_{1}}, \\
-\frac{E + M}{M}\, \frac{\dd \Delta}{\dd x}
  &= \frac{x' - \dfrac{Ex}{E + M}}{\Delta};
\end{aligned}
\end{gather*}
\begin{alignat*}{3}
\text{$\therefore$ the direction cosines of }& EM &&\text{ are }&&
  \frac{\dd r}{\dd x},\ \frac{\dd r}{\dd y},\ \frac{\dd r}{\dd z},\\
%
\Ditto[the ]\Ditto[direction ]\Ditto[cosines of ]& E\m' &&\text{ are }&&
  \Neg\frac{E+M}{M}\left(\frac{\dd r_{1}}{\dd x},\ \frac{\dd r_{1}}{\dd y},\ \frac{\dd r_{1}}{\dd z}\right),\\
\Ditto[the ]\Ditto[direction ]\Ditto[cosines of ]& M\m' &&\text{ are }&&
  -\frac{E+M}{M}\left(\frac{\dd \Delta}{\dd x},\: \frac{\dd \Delta}{\dd y},\: \frac{\dd \Delta}{\dd z}\right).
\end{alignat*}

If $X, Y, Z$ denote the components of acceleration of~$M$ relative to axes
through~$E$,
\DPPageSep{077}{19}
\[
\left.
\begin{aligned}
 X &= -\frac{E+M}{r^{2}}\, \frac{\partial r}{\partial x}
  - \frac{\m'}{\Delta^{2}}\, \frac{E + M}{E}
    \frac{\partial\Delta}{\partial x} - \frac{\m'}{r_{1}^{2}}\, \frac{E + M}{M}\,
    \frac{\partial r_{1}}{\partial x}\\
  &= \frac{\partial F}{\partial x},\\
&
\lintertext{where}{\phantom{= -\frac{E+M}{r^{2}} \frac{\partial r}{\partial x}
  - \frac{\m'}{\Delta^{2}} \frac{E+M}{E}
   \frac{\partial\Delta}{\partial x} - \frac{\m'}{r_{1}^{2}} \frac{E+M}{M}
   \frac{\partial r_{1}}{\partial x}}} \\
F &= \frac{E+M}{r} + \frac{\m'}{\Delta}\, \frac{E+M}{E}
   + \frac{\m'}{r_{1}}\, \frac{E + M}{M}. \\
&\lintertext{\indent Similarly,}{\phantom{= -\frac{E+M}{r^{2}} \frac{\partial r}{\partial x}
  - \frac{\m'}{\Delta^{2}} \frac{E+M}{E}
   \frac{\partial\Delta}{\partial x} - \frac{\m'}{r_{1}^{2}} \frac{E+M}{M}
   \frac{\partial r_{1}}{\partial x}}} \\
Y &= \frac{\partial F}{\partial y},\
Z =\frac{\partial F}{\partial z}.
\end{aligned}
\right\}
\Tag{(1)}
\]

Let $r'$~be the distance between $G$~and~$\m'$, and let $\theta$~be the angle~$\m'GM$;
then
\begin{align*}
r'^{2} &= x'^{2} + y'^{2} + z'^{2} \text{ and }
  \cos\theta = \frac{xx' + yy' + zz'}{rr'}, \\
r_{1}^{2} &= r'^{2} + \frac{2M}{E + M}\, rr' \cos\theta + \left(\frac{Mr}{E + M}\right)^{2}, \\
\Delta^{2} &= r'^{2} - \frac{2E}{E + M}\, rr' \cos\theta + \left(\frac{Er}{E + M}\right)^{2}.
\end{align*}

Since $r$~is very small compared with~$r'$,
\begin{gather*}
\begin{aligned}
\frac{1}{r_{1}}
  &= \frac{1}{r'} \left\{1 - \frac{M}{E + M}\, \frac{r}{r'} \cos\theta
  + \left(\frac{M}{E + M} · \frac{r}{r'} \right)^{2}
    (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots\right\}, \\
%
\frac{1}{\Delta}
  &= \frac{1}{r'} \left\{1 + \frac{E}{E + M}\, \frac{r}{r'} \cos\theta
  + \left(\frac{E}{E + M} · \frac{r}{r'} \right)^{2}
    (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots\right\}.
\end{aligned} \\
%
\therefore \frac{1}{E\Delta} + \frac{1}{Mr_{1}}
  = \frac{E + M}{EM} · \frac{1}{r'}
  + \frac{1}{E + M} · \frac{r^{2}}{r'^{3}}
    (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots.
\end{gather*}

Hence
\[
F = \frac{E + M}{r} + \frac{\m'(E + M)^{2}}{EMr'}
  + \frac{\m'r^{2}}{r'^{3}} (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots.
\]

But the second term does not involve $x, y, z$, and may be dropped.
\[
\therefore
F = \frac{E + M}{r}
  + \frac{\m'r^{2}}{r'^{3}} (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2}),
\Tag{(2)}
\]
neglecting terms in~$\dfrac{r^{3}}{r'^{4}}$.

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 $G\m'$~rotates with
uniform angular velocity~$n'$. 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 $x, y$
\DPPageSep{078}{20}
rotating with angular velocity~$n'$ in the plane of the sun's orbit round the
earth; the $x$-axis being so chosen that it passes through the sun. When
required, a $z$-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 approximately
$\a', 0, 0$. In this approximation $r\cos\theta = x$ and
\[
F = \frac{E + M}{r}
  + \tfrac{3}{2}\, \frac{\m'}{\a'^{3}} x^{2}
  - \tfrac{1}{2} \m' \frac{r^{2}}{\a'^{3}}.
\]

This suggests the following general form for~$F$, instead of that given in
equation~\Eqref{(2)}:
\begin{align*}
F = \frac{E + M}{r}
  &+ \tfrac{3}{2}\, \frac{\m'}{\a'^{3}} x^{2}
  + \tfrac{3}{2} \m' \left( \frac{r^{2} \cos^{2}\theta}{r'^{3}} - \frac{x^{2}}{\a'^{3}} \right) \\
  &- \tfrac{1}{2}\, \frac{\m'}{\a'^{3}} (x^{2} + y^{2})
  - \tfrac{1}{2}\, \frac{\m'}{\a'^{3}} z^{2}\DPnote{** Why aren't previous terms combined?}
  + \tfrac{1}{2} \m' r^{2} \left(\frac{1}{\a'^{3}} - \frac{1}{r'^{3}}\right).
\end{align*}

For the sake of future developments, we now introduce a new notation.
Let $\nu$~be the moon's synodic mean motion and put $m = \dfrac{n'}{\nu} = \dfrac{n'}{n - n'}$\footnotemark. In the
\footnotetext{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.}%
case of our moon, $m$~is approximately~$\frac{1}{12}$: 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 $\m' = n'^{2} \a'^{3}$, whence $\dfrac{\m'}{\a'^{3}} = n'^{2} = \nu^{2} m^{2}$. Let us also
write $E + M = \kappa \nu^{2}$, and then we get
\begin{align*}%[** TN: Re-broken]
F &+ \tfrac{1}{2} n'^{2} (x^{2} + y^{2}) \\
  &= \nu^{2} \biggl[\frac{\kappa}{r} + \tfrac{1}{2} m^{2} (3x^{2} - z^{2})
    + \tfrac{3}{2} m^{2} \left(\frac{\a'^{3}}{r'^{3}}\, r^{2} \cos^{2}\theta - x^{2}\right)
    + \tfrac{1}{2} m^{2} r^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right)\biggr].
\end{align*}

For convenience we write
\Pagelabel{20}
\[
\Omega
  = \tfrac{3}{2} m^{2} \left(\frac{\a'^{3}}{r'^{3}} r^{2} \cos^{2}\theta - x^{2}\right)
  + \tfrac{1}{2} m^{2} r^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right),
\]
and then
\[
F + \tfrac{1}{2} n'^{2} (x^{2} + y^{2})
  = \nu^{2} \left[\frac{\kappa}{r} + \tfrac{1}{2} m^{2} (3x^{2} - z^{2}) + \Omega\right].
\]

The equations of motion for uniformly rotating axes\footnote
  {See any standard treatise on Dynamics.}
are
\[
\left.
\begin{alignedat}{3}
\frac{d^{2}x}{dt^{2}} &- 2n' \frac{dy}{dt} &&- n'^{2} x
  &&= \frac{\dd F}{\dd x}\Add{,} \\
\frac{d^{2}y}{dt^{2}} &- 2n' \frac{dx}{dt} &&- \DPtypo{n'}{n'^{2}} y
  &&= \frac{\dd F}{\dd y}\Add{,} \\
\frac{d^{2}z}{dt^{2}} & &&
  &&= \frac{\dd F}{\dd z}\Add{,}
\end{alignedat}
\right\}
\]
\DPPageSep{079}{21}
\index{Jacobi's ellipsoid!integral}%
which give
\begin{alignat*}{5}
&\frac{d^{2}x}{dt^{2}}-2n'\,\frac{dy}{dt}
 &&=\frac{\dd}{\dd x}\left[F + \tfrac{1}{2}n'^{2}(x^{2} + y^{2})\right]
 &&=\nu^{2}\biggl[-\frac{\kappa x}{r^{3}} &+{}&& 3m^{2}x &+ \frac{\dd \Omega}{\dd x}\biggr],\\
%
&\frac{d^{2}y}{dt^{2}}+2n'\,\frac{dx}{dt}
 &&=\frac{\dd}{\dd y}\left[F + \tfrac{1}{2}n'^{2}(x^{2} + y^{2})\right]
 &&=\nu^{2}\biggl[-\frac{\kappa y}{r^{3}} &&&&+\frac{\dd \Omega}{\dd y}\biggr],\\
%
&\frac{d^{2}z}{dt^{2}}
 &&=\frac{\dd}{\dd z}\left[F + \tfrac{1}{2}n'^{2}(x^{2} + y^{2})\right]
 &&=\nu^{2}\biggl[-\frac{\kappa z}{r^{3}} &-{}&& m^{2}z &+ \frac{\dd \Omega}{\dd z}\biggr].
\end{alignat*}

We might write $\tau = \nu t$ and on dividing the equations by~$\nu^2$ use $\tau$~henceforth
as equivalent to time; or we might choose a special unit of time such
that $\nu$~is unity. In either case our equations become
\[
\left.
\begin{alignedat}{4}
\frac{d^{2}x}{d\tau^{2}}
  & - 2m\frac{dy}{d\tau}
  &&+ \frac{\kappa x}{r^{3}}
  &&-& 3m^{2}x
  =& \frac{\dd \Omega}{\dd x}\Add{,} \\
%
\frac{d^{2}y}{d\tau^{2}}
  & + 2m\frac{dx}{d\tau}
  &&+ \frac{\kappa y}{r^{3}} &&
  &=& \frac{\dd \Omega}{\dd y}\Add{,} \\
%
\frac{d^{2}z}{d\tau^{2}} &
  &&+ \frac{\kappa z}{r^{3}}
  &&+& m^{2}z
  =& \frac{\dd \Omega}{\dd z}\Add{.}
\end{alignedat}
\right\}
\Tag{(3)}
\]

If we multiply these equations respectively by $2\dfrac{dx}{d\tau}$, $2\dfrac{dy}{d\tau}$, $2\dfrac{dz}{d\tau}$ and add
them, we have
\begin{multline*}%[** TN: Slightly wide]
\frac{d}{d\tau}\Biggl\{
  \left(\frac{dx}{d\tau}\right)^{2} +
  \left(\frac{dy}{d\tau}\right)^{2} +
  \left(\frac{dz}{d\tau}\right)^{2}\Biggr\}
  - 2\kappa \frac{d}{d\tau}\left(\frac{1}{r}\right)
  - 3m^{2} \frac{d}{d\tau}(x^{2})
  +  m^{2} \frac{d}{d\tau}(z^{2})\\
  =2\left(\frac{\dd \Omega}{\dd x}\,\frac{dx}{d\tau}
        + \frac{\dd \Omega}{\dd y}\,\frac{dy}{d\tau}
        + \frac{\dd \Omega}{\dd z}\,\frac{dz}{d\tau}\right).
\end{multline*}

The whole of the left-hand side is a complete differential; the right-hand
side needs the addition of the term $2\dfrac{\dd \Omega}{\dd \tau}$.

Let us put for brevity
\[
V^{2}
  = \left(\frac{dx}{d\tau}\right)^{2}
  + \left(\frac{dy}{d\tau}\right)^{2}
  + \left(\frac{dz}{d\tau}\right)^{2}.
\]

Then
\[
V^{2} = \frac{2\kappa}{r} + 3m^{2}x^{2} - m^{2}z^{2}
  + 2\int_{0}^{\tau} \left[
      \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau}
    + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}
    + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau} \right]d\tau + C.
\Tag{(4)}
\]

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
\[
x' = r' = \a',\quad
y' = z' = 0\Add{,}
\]
and
\[
r\cos\theta = \frac{xx' + yy' + zz'}{r'} = x,
\]
\DPPageSep{080}{22}
giving
\[
\frac{\a'^{3}}{r'^{3}} r^{2} \cos^{2}\theta - x^{2} = 0.
\]

In this case $\Omega$~would vanish. It follows that $\Omega$~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
\[
\left.
\begin{alignedat}{5}
\frac{d^{2}x}{d\tau^{2}}
  &- 2m \frac{dy}{d\tau} &+ \frac{\kappa x}{r^{3}} &-& 3m^{2} x &= 0\Add{,} \\
\frac{d^{2}y}{d\tau^{2}}
  &+ 2m \frac{dx}{d\tau} &+ \frac{\kappa y}{r^{3}} &&         &= 0\Add{,} \\
\frac{d^{2}z}{d\tau^{2}}
  &                  &+ \frac{\kappa z}{r^{3}} &+&  m^{2} z &= 0\Add{.}
\end{alignedat}
\right\}
\Tag{(5)}
\]

Of these equations one integral is known, viz.\ Jacobi's integral,
\[
V^{2} = 2\frac{\kappa}{r} + 3m^{2} x^{2} - m^{2} z^{2} + C.
\]

\Section{§ 3. }{The Variational Curve.}
\index{Variational curve, defined}%

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 $z$~disappears. If the path of a body whose motion satisfies
\[
\left.
\begin{alignedat}{2}
\frac{d^{2}x}{d\tau^{2}} - 2m \frac{dy}{d\tau}
  &+ \left(\frac{\kappa}{r^{3}} - 3m^{2} \right) x &&= 0\\
\frac{d^{2}y}{d\tau^{2}} + 2m \frac{dx}{d\tau}
  &+ \frac{\kappa y}{r^{3}}  &&= 0
\end{alignedat}
\right\}
\Tag{(6)}
\]
intersects the $x$-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, $x = f(\tau^{2})$, $y = \tau f(\tau^{2})$; for if in the differential
equations the signs of $y$~and~$\tau$ 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~$\tau$ have signs changed, but $y$~is unaltered, the equations are unaltered.
\DPPageSep{081}{23}

Now it is evident that the body may start from a given point on the
$x$-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 $y$-axis at right angles.

If the body move from one axis to the other, crossing both at right
\index{Variational curve, defined!determined}%
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~$\tau$ and $y$~an odd one, both of
period~$2\pi$, it must be possible to expand $x$~and~$y$ by Fourier Series---thus
\begin{alignat*}{4}
x &= A_{0} \cos \tau &&+ A_{1} \cos 3\tau &&+ A_{2} \cos 5\tau &&+ \ldots\ldots, \\
y &= B_{0} \sin \tau &&+ B_{1} \sin 3\tau &&+ B_{2} \sin 5\tau &&+ \ldots\ldots.
\end{alignat*}

When $\tau$~is a multiple of~$\pi$, $y = 0$; and when it is an odd multiple
of~$\dfrac{\pi}{2}$, $x = 0$: also in the first case $\dfrac{dx}{d\tau} = 0$ and in the second $\dfrac{dy}{d\tau} = 0$. Thus
these conditions give us the kind of curve we want. It will be noted that
there are no terms with even multiples of~$\tau$; such terms have to be omitted
if $x, \dfrac{dx}{d\tau}$ are to vanish at $\tau = \pi/2$,~etc.\DPnote{** Slant fraction}

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~$3\tau$. It is
also convenient to change the constants into another form. Thus we write
\begin{align*}
A_{0} &= a_{0} + a_{-1},\quad A_{1} = a_{1}, \\
B_{0} &= a_{0} - a_{-1},\quad B_{1} = a_{1}.
\end{align*}
We have one constant less than before, but it will be seen that this is
sufficient, for in fact $A_{1}$~and~$B_{1}$ only differ by terms of an order which we
are going to neglect. We assume $a_{1}, a_{-1}$ to be small quantities.

Hence
\begin{align*}
x &= (a_{0} + a_{-1}) \cos\tau + a_{1} \cos 3\tau, \\
y &= (a_{0} - a_{-1}) \sin\tau + a_{1} \sin 3\tau.
\end{align*}
\DPPageSep{082}{24}

Since
\begin{alignat*}{4}
\cos 3\tau &=  && 4\cos^{3}\tau - 3\cos\tau &&=  &&\cos\tau(1 - 4\sin^{2}\tau), \\
\sin 3\tau &= -&& 4\sin^{3}\tau + 3\sin\tau &&= -&&\sin\tau(1 - 4\cos^{2}\tau),
\end{alignat*}
we have
\[
\left.
\begin{aligned}
x = a_{0} \cos\tau &\left[1 + \frac{a_{1} + a_{-1}}{a_{0}}
    - \frac{4a_{1}}{a_{0}} \sin^{2}\tau\right]\Add{,} \\
y = a_{0} \sin\tau &\left[1 - \frac{a_{1} + a_{-1}}{a_{0}}
    + \frac{4a_{1}}{a_{0}} \cos^{2}\tau\right]\Add{.}
\end{aligned}
\right\}
\]

Neglecting powers of $a_{1}, a_{-1}$ higher than the first, we deduce
\begin{align*}
r^{2} &= a_{0}^{2} \left[1 + 2\frac{a_{1} + a_{-1}}{a_{0}} \cos2\tau\right],
\Allowbreak
\frac{1}{r^{3}}
  &= \frac{1}{a_{0}^{3}} \left[1 - 3\frac{a_{1} + a_{-1}}{a_{0}} \cos2\tau\right] \\
  &= \frac{1}{a_{0}^{3}} \left[1 - 3\frac{a_{1} + a_{-1}}{a_{0}} + 6\frac{a_{1} + a_{-1}}{a_{0}} \sin^{2}\tau\right] \\
  &= \frac{1}{a_{0}^{3}} \left[1 + 3\frac{a_{1} + a_{-1}}{a_{0}} - 6\frac{a_{1} + a_{-1}}{a_{0}} \cos^{2}\tau\right];
\Allowbreak
\frac{\kappa x}{r^{3}}
  &= \frac{\kappa}{a_{0}^{2}} \cos\tau
       \left[1 - \frac{2a_{1} + 2a_{-1}}{a_{0}}
         + \frac{2a_{1} + 6a_{-1}}{a_{0}} \sin^{2}\tau\right], \\
\frac{\kappa y}{r^{3}}
  &= \frac{\kappa}{a_{0}^{2}} \sin\tau
       \left[1 + \frac{2a_{1} + 2a_{-1}}{a_{0}}
         - \frac{2a_{1} + 6a_{-1}}{a_{0}} \cos^{2}\tau\right],
\Allowbreak
%[** TN: Added breaks at second equalities]
\frac{d^{2} x}{d\tau^{2}}
  &= -\left[\left(a_{0} + a_{-1}\right) \cos\tau + 9a_{1} \cos3\tau\right] \\
  &= -\cos\tau \left[a_{0} + 9a_{1} + a_{-1} - 36a_{1} \sin^{2}\tau\right],
\Allowbreak
\frac{d^{2} y}{d\tau^{2}}
  &= -\left[\left(a_{0} - a_{-1}\right) \sin\tau + 9a_{1} \sin3\tau\right] \\
  &= -\sin\tau \left[a_{0} - 9a_{1} + a_{-1} - 36a_{1} \cos^{2}\tau\right].
\end{align*}

With the required accuracy
\[
-2m \frac{dy}{d\tau} = -2m a_{0}\cos\tau,\
 2m \frac{dx}{d\tau} = -2m a_{0} \sin\tau, \text{ and }
 3m^{2} x = 3m^{2} a_{0} \cos\tau.
\]

Substituting these results in the differential equations,~\Eqref{(6)}, we get
\begin{multline*}
a_{0}\cos\tau
  \biggl[-1 - \frac{9a_{1} + a_{-1}}{a_{0}} + \frac{36a_{1}}{a_{0}}\sin^{2}\tau - 2m \\
    + \frac{\kappa}{a_{0}^{3}} \left(1 - \frac{2a_{1} + 2a_{-1}}{a_{0}}
      + \frac{2a_{1} + 6a_{-1}}{a_{0}} \sin^{2}\tau\right) - 3m^{2}\biggr] = 0,
\end{multline*}
\begin{multline*}
a_{0}\sin\tau
  \biggl[-1 + \frac{9a_{1} + a_{-1}}{a_{0}} - \frac{36a_{1}}{a_{0}}\cos^{2}\tau - 2m \\
    + \frac{\kappa}{a_{0}^{3}} \left(1 + \frac{2a_{1} + 2a_{-1}}{a_{0}}
      - \frac{2a_{1} + 6a_{-1}}{a_{0}} \cos^{2}\tau\right)\biggr] = 0.
\end{multline*}
\DPPageSep{083}{25}

Equating to zero the coefficients of $\cos\tau$, $\cos\tau \sin^{2}\tau$, $\sin\tau$, $\sin\tau \cos^{2}\tau$,
we get
\[
\left.
\begin{gathered}
\begin{alignedat}{2}
&-1 - \frac{9a_{1} + a_{-1}}{a_{0}} - 2m
  + \frac{\kappa}{a_{0}^{3}} \left(1 - \frac{2a_{1} + 2a_{-1}}{a_{0}}\right)
  & -3m^{2} &= 0\Add{,} \\
&-1 + \frac{9a_{1} + a_{-1}}{a_{0}} - 2m
  + \frac{\kappa}{a_{0}^{3}} \left(1 + \frac{2a_{1} + 2a_{-1}}{a_{0}}\right)
  &&= 0\Add{,}
\end{alignedat}
\\
%
\frac{36a_{1}}{a_{0}}
  + \frac{\kappa}{a_{0}^{2}} \left(\frac{2a_{1} + 6a_{-1}}{a_{0}}\right) = 0\Add{.}
\end{gathered}
\right\}
\Tag{(7)}
\]

As there are only three equations for the determination of $\dfrac{\kappa}{a_{0}^{3}}$, $\dfrac{a_{1}}{a_{0}}$, $\dfrac{a_{-1}}{a_{0}}$
our assumption that $A_{1} = B_{1} = a_{1}$ is justified to the order of small quantities
considered.

Half the sum and difference of the first two give
\begin{gather*}
-1 - 2m - \tfrac{3}{2} m^{2} + \frac{\kappa}{a_{0}^{3}} = 0, \\
\frac{9a_{1} + a_{-1}}{a_{0}} + \frac{2\kappa}{a_{0}^{3}}\, \frac{a_{1} + a_{-1}}{a_{0}}
  + \tfrac{3}{2} m^{2} = 0.
\end{gather*}

Therefore
\begin{align*}
&\frac{\kappa}{a_{0}^{3}} = 1 + 2m + \tfrac{3}{2} m^{2}, \\
&\frac{11a_{1}}{a_{0}} + \frac{3a_{-1}}{a_{0}} = -\tfrac{3}{2}m^{2},
  \text{ to our order of accuracy, viz.~$m^{2}$}; \\
\intertext{also}
&\frac{19a_{1}}{a_{0}} + \frac{3a_{-1}}{a_{0}} = 0,
  \text{ from the third equation;}
\end{align*}
\begin{gather*}
\therefore \frac{8a_{1}}{a_{0}} = \tfrac{3}{2} m^{2}, \\
\left.
\begin{aligned}
\frac{a_{1}}{a_{0}}
  &= \tfrac{3}{16} m^{2},\quad \frac{a_{-1}}{a_{0}}
   = -\tfrac{19}{16} m^{2}\Add{,} \\
\frac{\kappa}{a_{0}^{3}}
  &= 1 + 2m + \tfrac{3}{2} m^{2}\Add{.}
\end{aligned}
\right\}
\Tag{(8)}
\end{gather*}

Hence
\begin{align*}
x &= a_{0}\left[(1 - \tfrac{19}{16} m^{2}) \cos\tau
       + \tfrac{3}{16} m^{2} \cos 3\tau\right], \\
y &= a_{0}\left[(1 + \tfrac{19}{16} m^{2}) \sin\tau
       + \tfrac{3}{16} m^{2} \sin 3\tau\right],
\end{align*}
or perhaps more conveniently for future work
\[
\left.
\begin{aligned}
x &= a_{0}\cos\tau
  \left[1 - m^{2} - \tfrac{3}{4} m^{2} \sin^{2}\tau \right]\Add{,} \\
y &= a_{0}\sin\tau
  \left[1 + m^{2} + \tfrac{3}{4} m^{2} \cos^{2}\tau \right]\Add{.}
\end{aligned}
\right\}
\Tag{(9)}
\]

It will be seen that those are the equations to an oval curve, the semi-axes
of which are $a_{0}(1 - m^{2})$, $a_{0}(1 + m^{2})$ along and perpendicular to the line
joining the earth and sun. If $r, \theta$~be the polar coordinates of a point on the
curve,
\begin{align*}
r^{2} &= a_{0}^{2}[1 - 2m^{2} \cos 2\tau], \\
\intertext{giving}
r &= a_{0}[1 - m^{2} \cos 2\tau].
\Tag{(10)}
\end{align*}
\DPPageSep{084}{26}
Also
\begin{gather*}
\begin{aligned}
\tan\theta &= \frac{y}{x} = \tan\tau \bigl[1 + 2m^{2} + \tfrac{3}{4} m^{2}\bigr] \\
  &= \bigl(1 + \tfrac{11}{4}\bigr) \tan\tau.
\end{aligned} \\
\therefore \tan(\theta - \tau)
  = \frac{\tan\tau}{1 + \tan^{2}\tau} · \tfrac{11}{4} m^{2}
  = \tfrac{11}{8} \sin 2\tau,
\end{gather*}
giving
\[
\theta = \tau + \tfrac{11}{8} m^{2} \sin 2\tau.
\Tag{(11)}
\]

If $\a$~be the mean distance corresponding to a mean motion~$n$ in an
undisturbed orbit, Kepler's third law gives
\[
n^{2}\a^{3} = E + M = \kappa \nu^{2}.
\Tag{(12)}
\]

But
\[
\frac{n}{\nu} = \frac{n - n' + n'}{n - n'} = 1 + m.
\]
Hence
\begin{gather*}
(1 + m)^{2} \a^{3} = \kappa = a_{0}^{3} (1 + 2m + \tfrac{3}{2} m^{2}), \\
\frac{a_{0}^{3}}{\a^{3}} = \frac{1 + 2m + m^{2}}{1 + 2m + m^{2} + \tfrac{1}{2} m^{2}}, \\
\intertext{and}
a_{0} = \a(1 - \tfrac{1}{6} m^{2}).
\Tag{(13)}
\end{gather*}

This is a relation between $a_{0}$ and the undisturbed mean distance.


\Section{§ 4. }{Differential Equations \texorpdfstring{\protect\\}{}
for Small Displacements from the Variational Curve.}
\index{Small displacements from variational curve}%
\index{Variational curve, defined!small displacements from}%

If the solar perturbations were to vanish, $m$~would be zero and we should
have $x = a_{0}\cos\tau$, $y = a_{0}\sin\tau$ 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 $\Omega = 0$, 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~$\frac{1}{20}$ 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~\Eqref{(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
\DPPageSep{085}{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 $x + \delta x$, $y + \delta y$ 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~$\delta x, \delta y$ are
\[
\left.
\begin{alignedat}{2}
&\frac{d^{2}}{d\tau^{2}}\, \delta x - 2m \frac{d}{d\tau}\, \delta y
  + \kappa \delta \left(\frac{x}{r^{3}}\right) &- 3m^{2}\, \delta x &= 0\Add{,} \\
%
&\frac{d^{2}}{d\tau^{2}}\, \delta y + 2m \frac{d}{d\tau}\, \delta x
  + \kappa \delta \left(\frac{y}{r^{3}}\right) &&= 0\Add{.}
\end{alignedat}
\right\}
\Tag{(14)}
\]

\begin{wrapfigure}[14]{r}{1.75in}
  \centering
  \Input[1.75in]{p027}
  \caption{Fig.~2.}
\end{wrapfigure}
Now it is not convenient to proceed immediately
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~$\delta x, \delta y$ by normal
and tangential displacements~$\delta p, \delta s$.

Suppose then that $\phi$~denotes the inclination
of the outward normal of the variational curve to
the $x$-axis. Then we have
\[
\left.
\begin{aligned}
\delta x &= \delta p \cos\phi - \delta s \sin\phi\Add{,} \\
\delta y &= \delta p \sin\phi + \delta s \cos\phi\Add{.}
\end{aligned}
\right\}
\Tag{(15)}
\]

Multiply the first differential equation~\Eqref{(14)} by~$\cos\phi$ and the second by~$\sin\phi$
and add; and again multiply the first by~$\sin\phi$ and the second by~$\cos\phi$
and subtract. We have
\[
\left.
\begin{aligned}
\cos\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}}
  &+ \sin\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}}
   - 2m \left[\cos\phi\, \frac{d\, \delta y}{d\tau}
             - \sin\phi\, \frac{d\, \delta x}{d\tau}\right] \\
%
  &+ \kappa \cos\phi\, \delta\left(\frac{x}{r^{3}}\right)
   + \kappa \sin\phi\, \delta\left(\frac{y}{r^{3}}\right)
   - 3m^{2}\cos\phi\, \delta x = 0, \\
%
-\sin\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}}
  &+ \cos\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}}
   + 2m \left[\sin\phi\, \frac{d\, \delta y}{d\tau}
   + \cos\phi\, \frac{d\, \delta x}{d\tau}\right] \\
%
  &- \kappa \sin\phi\, \delta\left(\frac{x}{r^{3}}\right)
   + \kappa \cos\phi\, \delta\left(\frac{y}{r^{3}}\right)
   + 3m^{2}\sin\phi\, \delta x = 0.
\end{aligned}
\right\}
\Tag{(16)}
\]

Now we have from~\Eqref{(15)}
\[
\delta p =  \delta x \cos\phi + \delta y \sin\phi,\quad
\delta s = -\delta x \sin\phi + \delta y \cos\phi.
\]

Therefore
\begin{align*}
\frac{d\, \delta p}{d\tau}
  &= \Neg\cos\phi\, \frac{d\, \delta x}{d\tau}
   + \sin\phi\, \frac{d\, \delta y}{d\tau}
   + (-\delta x \sin\phi + \delta y \cos\phi)\, \frac{d\phi}{d\tau}, \\
 %
\frac{d\, \delta s}{d\tau}
  &= -\sin\phi\, \frac{d\, \delta x}{d\tau}
    + \cos\phi\, \frac{d\, \delta y}{d\tau}
    - (\Neg\delta x \cos\phi + \delta y \sin\phi)\, \frac{d\phi}{d\tau}.
\end{align*}
\DPPageSep{086}{28}

Hence the two expressions which occur in the second group of terms of~\Eqref{(16)}
are
\begin{align*}
\cos\phi\, \frac{d\, \delta y}{d\tau} - \sin\phi\, \frac{d\, \delta x}{d\tau}
  &= \frac{d\, \delta s}{d\tau} + \delta p\, \frac{d\phi}{d\tau}, \\
%
\sin\phi\, \frac{d\, \delta y}{d\tau} + \cos\phi\, \frac{d\, \delta x}{d\tau}
  &= \frac{d\, \delta p}{d\tau} - \delta s\, \frac{d\phi}{d\tau}.
\end{align*}

When we differentiate these again, we obtain the first group of terms in~\Eqref{(16)}.
Inverting the order of the equations we have
\begin{align*}
\cos\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}}
  &+ \sin\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} \\
  &= \frac{d^{2}\, \delta p}{d\tau^{2}}
   - \frac{d\, \delta s}{d\tau}\, \frac{d\phi}{d\tau}
   - \delta s\, \frac{d^{2}\phi}{d\tau^{2}}
   - \left(\cos\phi\, \frac{d\, \delta y}{d\tau}
         - \sin\phi\, \frac{d\, \delta x}{d\tau}\right) \frac{d\phi}{d\tau} \\
  &= \frac{d^{2}\, \delta p}{d\tau^{2}}
  - 2\frac{d\, \delta s}{d\tau}\, \frac{d\phi}{d\tau}
   - \delta p\, \left(\frac{d\phi}{d\tau}\right)^{2}
   - \delta s\, \frac{d^{2}\phi}{d\tau^{2}},
\Allowbreak
-\sin\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}}
  &+ \cos\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} \\
  &= \frac{d^{2}\, \delta s}{d\tau^{2}}
   + \frac{d\, \delta p}{d\tau}\, \frac{d\phi}{d\tau}
   + \delta p\, \frac{d^{2}\phi}{d\tau^{2}}
   + \left(\sin\phi\, \frac{d\, \delta y}{d\tau}
         + \cos\phi\, \frac{d\, \delta x}{d\tau}\right) \frac{d\phi}{d\tau} \\
  &= \frac{d^{2}\, \delta s}{d\tau^{2}}
  + 2\frac{d\, \delta p}{d\tau}\, \frac{d\phi}{d\tau}
   - \delta s\, \left(\frac{d\phi}{d\tau}\right)^{2}
   + \delta p\, \frac{d^{2}\phi}{d\tau^{2}}.
\end{align*}

Substituting in~\Eqref{(16)}, we have as our equations
\[
\left.
\begin{aligned}
\frac{d^{2}\, \delta p}{d\tau^{2}}
  &- \delta p \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m \frac{d\phi}{d\tau}\right]
   - 2\frac{d\, \delta s}{d\tau} \left(\frac{d\phi}{d\tau} + m\right)
   - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} \\
&\qquad
  + \kappa\cos\phi\, \delta\left(\frac{x}{r^{3}}\right)
  + \kappa\sin\phi\, \delta\left(\frac{y}{r^{3}}\right)
  - 3m^{2}\cos\phi\, \delta x = 0\Add{,} \\
%
\frac{d^{2}\, \delta s}{d\tau^{2}}
  &- \delta s \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m \frac{d\phi}{d\tau}\right]
   + 2\frac{d\, \delta p}{d\tau} \left(\frac{d\phi}{d\tau} + m\right)
   + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} \\
&\qquad
  - \kappa\sin\phi\, \delta\left(\frac{x}{r^{3}}\right)
  + \kappa\cos\phi\, \delta\left(\frac{y}{r^{3}}\right)
  + 3m^{2}\sin\phi\, \delta x = 0\Add{.}
\end{aligned}
\right\}
\Tag{(17)}
\]

Variation of the Jacobian integral
\[
V^{2}
  = \left(\frac{dx}{d\tau}\right)^{2}
  + \left(\frac{dy}{d\tau}\right)^{2}
  = \frac{2\kappa}{r} + 3m^{2}x^{2} + C
\]
gives
\[
\frac{dx}{d\tau}\, \frac{d\, \delta x}{d\tau} +
\frac{dy}{d\tau}\, \frac{d\, \delta y}{d\tau}
  = -\frac{\kappa}{r^{3}}\, \delta r
    + 3m^{2}x\, \delta x.\footnotemark%[** TN: Moved mark after period]
\]
\footnotetext{We could introduce a term~$\delta C$, but the variation of the orbit which we are introducing
  is one for which $C$~is unaltered.}

Now
\[
\frac{dx}{d\tau} = -V\sin\phi,\quad
\frac{dy}{d\tau} =  V\cos\phi,
\]
\DPPageSep{087}{29}
and
\begin{alignat*}{4}
  \frac{d\, \delta x}{d\tau}
  &= \cos\phi\, \frac{d\, \delta p}{d\tau}
  &&- \delta s \cos\phi\, \frac{d\phi}{d\tau}
  &&- \sin\phi\, \frac{d\, \delta s}{d\tau}
  &&- \sin\phi\, \delta p\, \frac{d\phi}{d\tau}, \\
%
  \frac{d\, \delta y}{d\tau}
  &= \sin\phi\, \frac{d\, \delta p}{d\tau}
  &&- \delta s \sin\phi\, \frac{d\phi}{d\tau}
  &&+ \cos\phi \frac{d\, \delta s}{d\tau}
  &&+ \cos\phi\, \delta p\, \frac{d\phi}{d\tau}.
\end{alignat*}

Hence
\[
\frac{dx}{d\tau}\, \frac{d\, \delta x}{d\tau} +
\frac{dy}{d\tau}\, \frac{d\, \delta y}{d\tau}
  = V \left(\frac{d\, \delta s}{d\tau}
          + \delta p\, \frac{d\phi}{d\tau}\right).
\]

Also
\begin{align*}
-\frac{\kappa\, \delta r}{r^{2}}
  &= -\frac{\kappa}{r^{3}}(x\, \delta x + y\, \delta y) \\ %[** TN: Added break]
  &= -\frac{\kappa x}{r^{3}}(\delta p \cos\phi - \delta s \sin\phi)
     -\frac{\kappa y}{r^{3}}(\delta p \sin\phi + \delta s \cos\phi) \\
  &= -\frac{\kappa}{r^{3}} \bigl[\delta p\, (x \cos\phi + y \sin\phi)
       + \delta s\, (-x \sin\phi + y \cos\phi)\bigr].
\end{align*}

Thus, retaining the term $3m^{2} x\, \delta x$ in its original form, the varied Jacobian
integral becomes
\Pagelabel{29}
\begin{multline*}
V\left(\frac{d\, \delta s}{d\tau} + \delta p\, \frac{d\phi}{d\tau}\right) \\
  = -\frac{\kappa}{r^{3}} \bigl[\delta p\, (x \cos\phi + y \sin\phi)
      + \delta s\, (-x \sin\phi + y \cos\phi)\bigr] + 3m^{2} x\, \delta x.
\Tag{(18)}
\end{multline*}

Before we can solve the differential equations~\Eqref{(17)} for $\delta p, \delta s$ we require to
express all the other variables occurring in them, in terms of~$\tau$ by means of
the equations obtained in~\SecRef{3}.


\Section{§ 5. }{Transformation of the equations in \SecRef{4}.}

We desire to transform the differential equations~\Eqref{(17)} so that the only
variables involved will be $\delta p, \delta s, \tau$. We shall then be in a position to solve
for $\delta p, \delta s$ in terms of~$\tau$.

We have
\[
r\, \delta r = x\, \delta x + y\, \delta y
  = ( x \cos\phi + y \sin\phi)\, \delta p
  + (-x \sin\phi + y \cos\phi)\, \delta s.
\]

Hence
\begin{align*}
\cos\phi\, &\delta\left(\frac{x}{r^{3}}\right) +
\sin\phi\, \delta\left(\frac{y}{r^{3}}\right) \\
  &= \frac{1}{r^{3}} (\delta x \cos\phi + \delta y \sin\phi)
   - \frac{3}{r^{5}} (x \cos\phi + y \sin\phi) r\, \delta r
\Allowbreak
  &= \frac{\delta p}{r^{3}} - \frac{3}{r^{5}}
     \biggl[(x^{2} \cos^{2} \phi + y^{2} \sin^{2} \phi
       + 2xy \sin\phi \cos\phi)\, \delta p \\
  &\qquad \rlap{$\displaystyle
       + (- x^{2} \sin\phi \cos\phi
          + xy \cos^{2}\phi
          - xy \sin^{2}\phi
          + y^{2} \sin\phi \cos\phi)\, \delta s\biggr]$}
\Allowbreak
  &= \frac{\delta p}{r^{3}} - \frac{3}{r^{5}} \biggl[
    \bigl\{\tfrac{1}{2}(x^{2} + y^{2})
         + \tfrac{1}{2}(x^{2} - y^{2}) \cos 2\phi
         + xy \sin 2\phi\bigr\}\, \delta p \\
  &\qquad\qquad\qquad
  + \bigl\{-\tfrac{1}{2}(x^{2} - y^{2}) \sin 2\phi
         + xy \cos 2\phi\bigr\}\, \delta s \biggr]
\Allowbreak
  &= \frac{\delta p}{r^{3}} \left[
    -\tfrac{1}{2} - \tfrac{3}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \cos 2\phi
    - \frac{3xy}{r^{2}} \sin 2\phi
  \right] \\
  &\qquad\qquad\qquad
  - \frac{3\delta s}{r^{3}} \left[
    \frac{xy}{r^{2}} \cos 2\phi
    - \tfrac{1}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \sin 2\phi
  \right],
\Tag{(19)}
\Allowbreak
\DPPageSep{088}{30}
-\sin\phi\, &\delta\left(\frac{x}{r^{3}}\right) +
 \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) \\
  &= \frac{1}{r^{3}} (-\delta x \sin\phi + \delta y \cos\phi)
   - \frac{3}{r^{3}} (-x \sin\phi + y \cos\phi)r\, \delta r
\Allowbreak
  &= \frac{\delta s}{r^{3}} - \frac{3}{r^{5}} \biggl[
     (-x^{2} \sin\phi \cos\phi
     - xy \sin^{2}\phi + xy \cos^{2}\phi
     + y^{2} \sin\phi \cos\phi)\, \rlap{$\delta p$} \\
  &\qquad\qquad\qquad
     + (x^{2} \sin^{2}\phi + y^{2} \cos^{2}\phi
     - 2xy \sin\phi \cos\phi)\, \delta s\biggr]
\Allowbreak
  &= \frac{\delta s}{r^{3}}
   - \frac{3}{r^{5}} \biggl[
     \bigl\{-\tfrac{1}{2}(x^{2} - y^{2}) \sin 2\phi + xy\cos 2\phi\bigr\}\, \delta p \\
  &\qquad\qquad\qquad
    + \bigl\{\tfrac{1}{2}(x^{2} + y^{2}) - \tfrac{1}{2}(x^{2} - y^{2})\cos 2\phi
    - xy\sin 2\phi\bigr\}\, \delta s \biggr]
\Allowbreak
  &= -\frac{3\, \delta p}{r^{3}} \biggl[\frac{xy}{r^{2}}\cos 2\phi
       - \tfrac{1}{2} \frac{x^{2} - y^{2}}{r^{2}} \sin 2\phi\biggr] \\
  &\qquad\qquad\qquad
    + \frac{\delta s}{r^{3}} \biggl[
      -\tfrac{1}{2} + \tfrac{3}{2} \frac{x^{2} - y^{2}}{r^{2}} \cos 2\phi
     + \frac{3xy}{r^{2}} \sin 2\phi \biggr].
\Tag{(20)}
\end{align*}

We shall consider the terms $3m^{2}\, \delta x \begin{array}{@{\,}c@{\,}}\cos\\ \sin\end{array} \phi$ later (\Pageref{33}).

The next step is to substitute throughout the differential equations~\Eqref{(17)}
the values of~$x, y$ and~$\phi$ which correspond to the undisturbed orbit. For
simplicity in writing we drop the linear factor~$a_{0}$. It can be easily
introduced when required.

We have already found, in~\Eqref{(9)},
\begin{alignat*}{2}
x &= \cos\tau (1 - \tfrac{19}{16} m^{2}) + \tfrac{3}{16} m^{2}\cos 3\tau
  &&= \cos\tau (1 - m^{2} - \tfrac{3}{4} m^{2}\sin^{2}\tau), \\
x &= \sin\tau (1 + \tfrac{19}{16} m^{2}) + \tfrac{3}{16} m^{2}\sin 3\tau
  &&= \sin\tau (1 + m^{2} + \tfrac{3}{4} m^{2}\cos^{2}\tau).
\end{alignat*}

Then
\begin{align*}
\frac{dx}{d\tau}
  &= -\sin\tau(1 - \tfrac{7}{4} m^{2} + \tfrac{9}{4} m^{2}\cos^{2} \tau)
   = -\sin\tau(1 + \tfrac{1}{2} m^{2} - \tfrac{9}{4} m^{2}\sin^{2} \tau), \\
%
\frac{dy}{d\tau}
  &= \Neg\cos\tau(1 + \tfrac{7}{4} m^{2} - \tfrac{9}{4} m^{2}\sin^{2} \tau)
   = \Neg\cos\tau(1 - \tfrac{1}{2} m^{2} + \tfrac{9}{4} m^{2}\cos^{2} \tau).
\end{align*}

Whence
\begin{align*}
V^{2}
  &= \left(\frac{dx}{d\tau}\right)^{2} + \left(\frac{dy}{d\tau}\right)^{2} \\
%[** TN: Added break]
  &= \sin^{2}\tau (1 + m^{2} - \tfrac{9}{2} m^{2}\sin^{2}\tau)
   + \cos^{2}\tau (1 - m^{2} + \tfrac{9}{2} m^{2}\cos^{2}\tau) \\
%
  &= 1 - m^{2} \cos 2\tau + \tfrac{9}{2} m^{2}\cos 2\tau
   = 1 + \tfrac{7}{2} m^{2}\cos 2\tau \\
%
  &= 1 + \tfrac{7}{2} m^{2} - 7 m^{2}\sin^{2}\tau
   = 1 - \tfrac{7}{2} m^{2} + 7 m^{2}\cos^{2}\tau.
\end{align*}

Therefore
\[
\frac{1}{V}
  = 1 + \tfrac{7}{4} m^{2} - \tfrac{7}{2} m^{2}\cos^{2}\tau
  = 1 - \tfrac{7}{4} m^{2} + \tfrac{7}{2} m^{2}\sin^{2}\tau
  = 1 - \tfrac{7}{4} m^{2} \cos 2\tau.
\]
\DPPageSep{089}{31}

Now
\[
\sin\phi = -\frac{1}{V}\, \frac{dx}{d\tau},\quad
\cos\phi =  \frac{1}{V}\, \frac{dy}{d\tau}.
\]

Therefore
\begin{align*}
\sin\phi
  &= \sin\tau(1 + \tfrac{1}{2} m^{2} - \tfrac{9}{4} m^{2}\sin^{2}\tau
                - \tfrac{7}{4} m^{2} + \tfrac{7}{2} m^{2}\sin^{2}\tau) \\
  &= \sin\tau(1 - \tfrac{5}{4} m^{2} + \tfrac{5}{4} m^{2}\sin^{2}\tau)
   = \sin\tau(1 - \tfrac{5}{4} m^{2}\cos^{2}\tau),
\Allowbreak
\cos\phi
  &= \cos\tau(1 - \tfrac{1}{2} m^{2} + \tfrac{9}{4} m^{2}\cos^{2}\tau
                + \tfrac{7}{4} m^{2} - \tfrac{7}{2} m^{2}\cos^{2}\tau) \\
  &= \cos\tau(1 + \tfrac{5}{4} m^{2} - \tfrac{5}{4} m^{2}\cos^{2}\tau)
   = \cos\tau(1 + \tfrac{5}{4} m^{2}\sin^{2}\tau);
\Allowbreak
\sin2\phi
  &= \sin2\tau(1 - \tfrac{5}{4} m^{2}\cos2\tau), \\
%
\cos2\phi
  &= \cos2\tau + \tfrac{5}{4} m^{2}\sin^{2}2\tau);
\Allowbreak
\cos\phi\, \frac{d\phi}{d\tau}
  &= \Neg\cos\tau(1 - \tfrac{5}{4} m^{2} + \tfrac{15}{4} m^{2} \sin^{2}\tau), \\
%
\sin\phi\, \frac{d\phi}{d\tau}
  &= -\sin\tau(1 + \tfrac{5}{4} m^{2} - \tfrac{15}{4} m^{2} \cos^{2}\tau).
\end{align*}

Summing the squares of these,
\begin{align*}
\left(\frac{d\phi}{d\tau}\right)^{2}
  &= \cos^{2}\tau(1 - \tfrac{5}{2} m^{2} + \tfrac{15}{2} m^{2} \sin^{2}\tau)
   + \sin^{2}\tau(1 + \tfrac{5}{2} m^{2} - \tfrac{15}{2} m^{2} \cos^{2}\tau) \\
  &= 1 - \tfrac{5}{2} m^{2} \cos2\tau,
\end{align*}
and thence
\[
\frac{d\phi}{d\tau} = 1 - \tfrac{5}{4} m^{2} \cos2\tau.
\Tag{(21)}
\]

Differentiating again
\[
\frac{d^{2}\phi}{d\tau^{2}} = \tfrac{5}{2} m^{2} \sin 2\tau.
\]

We are now in a position to evaluate all the earlier terms in the
differential equations~\Eqref{(17)}.

Thus
\[
\left.
\begin{aligned}%[** TN: Re-broken]
\frac{d^{2}\, \delta p}{d\tau^{2}}
  &- \delta p \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m\frac{d\phi}{d\tau}\right]
  - 2\frac{d\, \delta s}{d\tau}\left(\frac{d\phi}{d\tau} + m\right)
  - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} \\
%
&= \frac{d^{2}\, \delta p}{d^{2}} + \delta p \bigl[-1 + \tfrac{5}{2} m^{2}\cos 2\tau - 2m\bigr] \\
  &\qquad\qquad
   - 2\frac{d\, \delta s}{d\tau}(1 + m - \tfrac{5}{4} m^{2}\cos2\tau)
  - \tfrac{5}{2} m^{2}\sin2\tau\, \delta s\Add{,} \\
%
\frac{d^{2}\, \delta s}{d\tau^{2}}
  &- \delta s \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m\frac{d\phi}{d\tau}\right]
  + 2\frac{d\, \delta p}{d\tau}\left(\frac{d\phi}{d\tau} + m\right)
  + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} \\
%
&= \frac{d^{2}\, \delta s}{d^{2}} + \delta s \bigl[-1 + \tfrac{5}{2} m^{2}\cos 2\tau - 2m\bigr] \\
  &\qquad\qquad
  + 2\frac{d\, \delta p}{d\tau}(1 + m - \tfrac{5}{4} m^{2}\cos2\tau)
  + \tfrac{5}{2} m^{2}\sin2\tau\, \delta p\Add{.}
\end{aligned}
\right\}
\Tag{(22)}
\]
\DPPageSep{090}{32}

We now have to evaluate the several terms involving $x$~and~$y$ in \Eqref{(18)},~\Eqref{(19)},~\Eqref{(20)}.

\begin{align*}
x \cos\phi + y \sin\phi
  &= \cos^{2}\tau (1 - m^{2} - \tfrac{3}{4} m^{2} \sin^{2}\tau + \tfrac{5}{4} m^{2} \sin^{2}\tau) \\
  &\,+ \sin^{2}\tau (1 + m^{2} + \tfrac{3}{4} m^{2} \cos^{2}\tau - \tfrac{5}{4} m^{2} \cos^{2}\tau) \\
  &= 1 - m^{2} \cos 2\tau,
\Allowbreak
%
-x \sin\phi + y \cos\phi
  &= -\sin\tau \cos\tau (1 - m^{2} - \tfrac{3}{4} m^{2} \sin^{2}\tau - \tfrac{5}{4} m^{2} \cos^{2}\tau) \\
  &\quad+  \sin\tau \cos\tau (1 + m^{2} + \tfrac{3}{4} m^{2} \cos^{2}\tau + \tfrac{5}{4} m^{2} \sin^{2}\tau) \\
  &= 2m^{2} \sin 2\tau;
\Allowbreak
%
r^{2} = x^{2} + y^{2} &= 1 - 2m^{2} \cos 2\tau,
\Allowbreak
%
x^{2} - y^{2} &= \cos^{2}\tau(1 - 2m^{2} - \tfrac{3}{2} m^{2}\sin^{2}\tau) \\
  &\,- \sin^{2}\tau (1 + 2m^{2} + \tfrac{3}{2} m^{2}\cos^{2}\tau) \\
  &= \cos 2\tau - 2m^{2} - \tfrac{3}{4} m^{2}\sin^{2} 2\tau,
\Allowbreak
%
xy &= \tfrac{1}{2}\sin 2\tau(1 + \tfrac{3}{4} m^{2}\cos 2\tau);
\Allowbreak
%
(x^{2} - y^{2}) \cos 2\phi
  &= \begin{aligned}[t]
    \cos^{2}2\tau - 2m^{2} \cos 2\tau
      &- \tfrac{3}{4} m^{2} \sin^{2}2\tau \cos 2\tau \\
      &+ \tfrac{5}{4} m^{2} \sin^{2}2\tau \cos 2\tau
    \end{aligned} \\
  &= \cos 2\tau (\cos 2\tau - 2m^{2} + \tfrac{1}{2} m^{2} \sin^{2}2\tau),
\Allowbreak
%
(x^{2} - y^{2}) \sin 2\phi
  &= \sin 2\tau (\cos 2\tau - 2m^{2} - \tfrac{3}{4} m^{2} \sin^{2}2\tau - \tfrac{5}{4} m^{2} \cos^{2}2\tau) \\
  &= \sin 2\tau (\cos 2\tau - \tfrac{11}{4} m^{2} - \tfrac{1}{2} m^{2} \cos^{2}2\tau);
\Allowbreak
%
xy \cos 2\phi
  &= \tfrac{1}{2} \sin 2\tau (\cos 2\tau + \tfrac{5}{4} m^{2} \sin^{2}2\tau + \tfrac{3}{4} m^{2} \cos^{2}2\tau) \\
  &= \tfrac{1}{2} \sin 2\tau (\cos 2\tau + \tfrac{5}{4} m^{2} - \tfrac{1}{2} m^{2} \cos^{2}2\tau), \\
%
xy \sin 2\phi
  &= \tfrac{1}{2}\sin^{2}2\tau (1 - \tfrac{5}{4} m^{2}\cos 2\tau + \tfrac{3}{4} m^{2}\cos 2\tau) \\
  &= \tfrac{1}{2}\sin^{2}2\tau (1 - \tfrac{1}{2} m^{2}\cos 2\tau).
\end{align*}

Therefore
\begin{gather*}
\begin{aligned}
&\tfrac{1}{2}(x^{2} - y^{2})\cos 2\phi + xy \sin 2\phi \\
%
  &= \tfrac{1}{2}\cos^{2}2\tau - m^{2}\cos 2\tau + \tfrac{1}{4} m^{2}\sin^{2}2\tau \cos 2\tau
   + \tfrac{1}{2}\sin^{2}2\tau - \tfrac{1}{4} m^{2}\sin^{2}2\tau \cos 2\tau \\
%
  &= \tfrac{1}{2}(1 - 2m^{2}\cos 2\tau) = \tfrac{1}{2}r^{2},
\end{aligned} \\
%
  \therefore
    -\tfrac{1}{2} \mp \tfrac{3}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \cos 2\phi \mp \frac{3xy}{r^{2}}\sin 2\phi
  = -\tfrac{1}{2} \mp \tfrac{3}{2} = -2 \text{ or } +1.
\end{gather*}

These are the coefficients of~$\dfrac{\delta p}{r^{3}}$ in the expression~\Eqref{(19)} for
\[
\cos\phi\, \delta \left(\frac{x}{r^{3}}\right) +
\sin\phi\, \delta \left(\frac{y}{r^{3}}\right),
\]
and of~$\dfrac{\delta s}{r^{3}}$ in the expression~\Eqref{(20)} for $-\sin\phi\, \delta\left(\dfrac{x}{r^{3}}\right) + \cos\phi\, \delta\left(\dfrac{y}{r^{3}}\right)$.
\DPPageSep{091}{33}

Again
\begin{align*}
-\tfrac{1}{2}(x^{2} - y^{2}) \sin 2\phi &+ xy \cos 2\phi \\
  &=
  \begin{alignedat}[t]{3}
  -\tfrac{1}{2} \sin 2\tau
  &(\cos 2\tau &&- \tfrac{11}{4} m^{2} &&- \tfrac{1}{2} m^{2} \cos^{2} 2\tau) \\
  +\tfrac{1}{2} \sin 2\tau
  &(\cos 2\tau &&+ \tfrac{5}{4}  m^{2} &&- \tfrac{1}{2} m^{2} \cos^{2} 2\tau)
  \end{alignedat} \\
  &= 2m^{2} \sin 2\tau.
\end{align*}

Then since to the order zero, $r^{3} = 1$, we have
\[
3\left(\frac{xy}{r^{2}} \cos 2\phi - \tfrac{1}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \sin 2\phi\right)
  = 6m^{2} \sin 2\tau.
\]

This is the coefficient of~$-\dfrac{\delta s}{r^{3}}$ in $\cos\phi\, \delta\left(\dfrac{x}{r^{3}}\right) + \sin\phi\, \delta\left(\dfrac{y}{r^{3}}\right)$ and of~$-\dfrac{\delta p}{r^{3}}$ in
$-\sin\phi\, \delta\left(\dfrac{x}{r^{3}}\right) + \cos\phi\, \delta\left(\dfrac{y}{r^{3}}\right)$.

Hence we have
\[
\left.
\begin{aligned}
\cos\phi\, \delta\left(\frac{x}{r^{3}}\right) +
\sin\phi\, \delta\left(\frac{y}{r^{3}}\right)
  &= -2\frac{\delta p}{r^{3}} - \frac{6m^{2}}{r^{3}}\, \delta s \sin 2\tau \\
  &= -2\delta p\, (1 + 3m^{2} \cos 2\tau)
     - 6m^{2}\, \delta s \sin 2\tau\Add{,} \\
%
-\sin\phi\, \delta\left(\frac{x}{r^{3}}\right) +
 \cos\phi\, \delta\left(\frac{y}{r^{3}}\right)
  &= -\frac{\delta p}{r^{3}} · 6m^{2} \sin 2\tau + \frac{\delta s}{r^{3}} \\
  &= -6m^{2}\, \delta p \sin 2\tau + \delta s\, (1 + 3m^{2} \cos 2\tau)\Add{.}
\end{aligned}
\right\}
\Tag{(23)}
\]

These two expressions are to be multiplied by~$\kappa$ in the differential
equations~\Eqref{(17)}.

{\stretchyspace
The other terms which occur in the differential equations are $-3m^{2}\cos\phi\, \delta x$
and~$+3m^{2}\sin\phi\, \delta x$.\Pagelabel{33}}

Since $m^{2}$~occurs in the coefficient we need only go to the order zero of
small quantities in $\cos\phi\, \delta x$ and~$\sin\phi\, \delta x$.

Thus
\begin{align*}%[** TN: Added two breaks]
3m^{2}\, \delta x \cos\phi
  &= 3m^{2} (\delta p \cos\tau - \delta s \sin\tau) \cos\tau \\
  &= \tfrac{3}{2} m^{2}\, \delta p\, (1 + \cos 2\tau)
   - \tfrac{3}{2} m^{2}\, \delta s \sin 2\tau, \\
%
3m^{2}\, \delta x \sin\phi
  &= 3m^{2} (\delta p \cos\tau - \delta s \sin\tau) \sin\tau \\
  &= \tfrac{3}{2} m^{2}\, \delta p \sin 2\tau
   - \tfrac{3}{2} m^{2}\, \delta s\, (1 - \cos 2\tau).
\end{align*}

Now $\kappa = 1 + 2m + \frac{3}{2} m^{2}$, and hence
\begin{align*}
     \kappa \cos\phi\, \delta\left(\frac{x}{r^{3}}\right)
  &+ \kappa \sin\phi\, \delta\left(\frac{y}{r^{3}}\right) - 3m^{2}\, \delta x \cos\phi \\
  &= -2\delta p\, (1 + 3m^{2} \cos 2\tau + 2m + \tfrac{3}{2} m^{2})
     - 6m^{2}\, \delta s \sin 2\tau \\
  &\quad -\tfrac{3}{2} m^{2}\, \delta p\, (1 + \cos 2\tau) + \tfrac{3}{2} m^{2}\, \delta s \sin 2\tau \\
  &= -2\delta p\, [1 + 2m + \tfrac{9}{4} m^{2} + \tfrac{15}{4} m^{2} \cos 2\tau]
     - \tfrac{9}{2} m^{2}\, \delta s \sin 2\tau,
\Allowbreak
\DPPageSep{092}{34}
-\kappa \sin\phi\, \delta\left(\frac{x}{r^{3}}\right)
  &+ \kappa \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + 3m^{2}\, \delta x \sin\phi \\
%
  &= -6m^{2}\, \delta p \sin 2\tau
     + \delta s\, (1 + 2m + \tfrac{3}{2} m^{2} + 3m^{2} \cos2\tau) \\
  &\quad + \tfrac{3}{2} m^{2}\, \delta p \sin 2\tau
     - \delta s\, (\tfrac{3}{2} m^{2} - \tfrac{3}{2} m^{2} \cos2\tau) \\
%
  &= -\tfrac{9}{2} m^{2}\, \delta p \sin2\tau
     + \delta s\, (1 + 2m + \tfrac{9}{2} m^{2} \cos2\tau).
\end{align*}

Hence
\begin{multline*}
\frac{d^{2}\, \delta p}{d\tau^{2}}
  - \delta p \left[\left(\frac{d\phi}{d\tau}\right)^{2}
              + 2m\left(\frac{d\phi}{d\tau}\right)\right]
  - 2 \frac{d\, \delta s}{d\tau} \left(\frac{d\phi}{d\tau} + m\right)
  - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} \\
%
  + \kappa \cos\phi\, \delta\left(\frac{x}{r^{3}}\right)
  + \kappa \sin\phi\, \delta\left(\frac{y}{r^{3}}\right)
  - 3m^{2} \cos\phi\, \delta x = 0
\end{multline*}
becomes
\begin{multline*}
\frac{d^{2}\, \delta p}{d\tau^{2}}
  - \delta p\, [1 + 2m - \tfrac{5}{2} m^{2} \cos 2\tau]
  - 2 \frac{d\, \delta s}{d\tau}(1 + m - \tfrac{5}{4} m^{2} \cos 2\tau)
  - \tfrac{5}{2} m^{2}\, \delta s \sin 2\tau \\
%
  - 2\delta p\, [1 + 2m + \tfrac{9}{4} m^{2} + \tfrac{15}{4} m^{2} \cos 2\tau]
  - \tfrac{9}{2} m^{2}\, \delta s \sin 2\tau = 0
\end{multline*}
or
\begin{multline*}
\frac{d^{2}\, \delta p}{d\tau^{2}}
  - \delta p\, [3 + 6m + \tfrac{9}{2} m^{2} + 5m^{2} \cos 2\tau]
  - 2 \frac{d\, \delta s}{d\tau} (1 + m - \tfrac{5}{4} m^{2} \cos 2\tau) \\
%
  - 7m^{2}\, \delta s \sin 2\tau = 0.
\Tag{(24)}
\end{multline*}

This is the first of our equations transformed.

Again the second equation is
\begin{multline*}
\frac{d^{2}\, \delta s}{d\tau^{2}}
  - \delta s \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m \frac{d\phi}{d\tau}\right]
  + 2 \frac{d\, \delta p}{d\tau} \left(\frac{d\phi}{d\tau} + m \right)
  + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} \\
%
  - \kappa \sin\phi\, \delta\left(\frac{x}{r^{3}}\right)
  + \kappa \cos\phi\, \delta\left(\frac{y}{r^{3}}\right)
  + 3m^{2} \sin\phi\, \delta x = 0,
\end{multline*}
and it becomes
\begin{multline*}
\frac{d^{2}\, \delta s}{d\tau^{2}}
  + \delta s\, (-1 - 2m + \tfrac{5}{2} m^{2} \cos 2\tau)
  + 2 \frac{d\, \delta p}{d\tau}(1 + m - \tfrac{5}{4} m^{2} \cos 2\tau)
  + \tfrac{5}{2} m^{2}\, \delta p \sin 2\tau \\
%
  - \tfrac{9}{2} m^{2}\, \delta p \sin 2\tau
  + \delta s\, (1 + 2m + \tfrac{9}{2} m^{2} \cos 2\tau) = 0.
\end{multline*}

Whence
\[
\frac{d^{2}\, \delta s}{d\tau^{2}}
  + 7m^{2}\, \delta s \cos 2\tau
  + 2 \frac{d\, \delta p}{d\tau} (1 + m -\tfrac{5}{4} m^{2} \cos 2\tau)
  - 2m^{2}\, \delta p \sin 2\tau = 0.
\Tag{(25)}
\]

This is the second of our equations transformed.

The Jacobian integral gives
\begin{align*}%[** TN: Rebroken]
\frac{d\, \delta s}{d\tau} &+ \delta p\, \frac{d\phi}{d\tau} \\
  &= \frac{3m^{2} x\, \delta x}{V}
  - \frac{\kappa}{V r^{3}} \bigl[\delta p\, (x \cos\phi + y \sin\phi) + \delta s\, (-x \sin\phi + y \cos\phi)\bigr]
\Allowbreak
  &= 3m^{2} \cos\tau (\delta p \cos\tau - \delta s \sin\tau) \\
  &\qquad\qquad
    - (1 + 2m + \tfrac{3}{2} m^{2} - \tfrac{7}{4} m^{2} \cos2\tau
    + 3m^{2} \cos2\tau) \\
  &\qquad\qquad\qquad\qquad\Add{·}
  \bigl[\delta p\, (1 - m^{2} \cos2\tau) + 2m^{2}\, \delta s \sin2\tau\bigr]
\Allowbreak
\DPPageSep{093}{35}
  &= \frac{3m^{2}}{2}\, \delta p\, (1 + \cos 2\tau)
   - \frac{3m^{2}}{2}\, \delta s \sin 2\tau \\
  &\qquad -\delta p\, (1 + 2m + \tfrac{3}{2} m^{2}
   + \tfrac{5}{4} m^{2} \cos2\tau - m^{2} \cos2\tau) - 2m^{2}\, \delta s \sin2\tau
\Allowbreak
  &= -\delta p\, (1 + 2m - \tfrac{5}{4} m^{2} \cos2\tau)
   - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau.
\end{align*}

Substituting for~$\dfrac{d\phi}{d\tau}$ its value from~\Eqref{(21)}
\begin{align*}
\frac{d\, \delta s}{d\tau}
  &= -\delta p\, (1 + 2m - \tfrac{5}{4} m^{2} \cos2\tau)
    - \delta p\, (1 - \tfrac{5}{4} m^{2} \cos2\tau)
    - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau \\
%
  &= -\delta p\, (2 + 2m - \tfrac{5}{2} m^{2} \cos2\tau)
    - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau \\
%
\frac{2d\, \delta s}{d\tau}
  &= -4\delta p\, (1 + m - \tfrac{5}{4} m^{2} \cos2\tau)
    - 7m^{2}\, \delta s \sin2\tau \\
\frac{2d\, \delta s}{d\tau} &(1 + m - \tfrac{5}{4} m^{2} \cos2\tau)
  = -4\delta p\, (1 + 2m + m^{2} - \tfrac{5}{2} m^{2} \cos2\tau)
     - 7m^{2}\, \delta s \sin2\tau \\
%
\frac{2d\, \delta s}{d\tau} &(1 + m - \tfrac{5}{4} m^{2} \cos2\tau)
    + 7m^{2}\, \delta s \sin2\tau
  = -4\delta p\, (1 + 2m + m^{2} - \tfrac{5}{2} m^{2} \cos2\tau).
\Tag{(26)}
\end{align*}

This expression occurs in~\Eqref{(24)}, and therefore can be used to eliminate
$\dfrac{d\, \delta s}{d\tau}$ from it.

Substituting we get
\begin{gather*}
\frac{d^{2}\, \delta p}{d\tau^{2}}
  + \delta p\, \bigl[-3 - 6m - \tfrac{9}{2} m^{2} - 5m^{2} \cos2\tau
                    + 4 + 8m + 4m^{2} - 10 m^{2} \cos2\tau\bigr] = 0,
\Allowbreak
\left.
\begin{gathered}
\lintertext{i.e.}
{\frac{d^{2}\, \delta p}{d\tau^{2}} + \delta p\, \bigl[1 + 2m  - \tfrac{1}{2} m^{2} - 15m^{2} \cos 2\tau\bigr] = 0.} \\
\lintertext{And}{\frac{d\, \delta s}{d\tau}
  = -2\delta p\, (1 + m - \tfrac{5}{4} m^{2} \cos2\tau)
    - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau.}
\end{gathered}
\right\}
\Tag{(27)}
\end{gather*}

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
\Pagelabel{35}
\begin{align*}%[** TN: Rebroken]
\frac{d^{2}\, \delta s}{d\tau^{2}}
  + 7m^{2}\, \delta s \cos2\tau
  &+ \tfrac{7}{2} m^{2} \sin 2\tau\, \frac{d\, \delta s}{d\tau} \\
  &+ 2 (1 + m - \tfrac{5}{4} m^{2} \cos 2\tau)\, \frac{d\, \delta p}{d\tau}
   + 5 m^{2}\, \delta p \sin 2\tau = 0,
\end{align*}
and eliminating~$\dfrac{d\, \delta s}{d\tau}$
\begin{align*}
\frac{d^{2}\, \delta s}{d\tau^{2}}
  + 7m^{2}\, \delta s \cos2\tau
  &- 7m^{2}\, \delta p \sin 2\tau \\
  &+ 2 (1 + m - \tfrac{5}{4} m^{2} \cos2\tau)\, \frac{d\, \delta p}{d\tau}
  + 5m^{2}\, \delta p \sin 2\tau = 0,
\end{align*}
\DPPageSep{094}{36}
or
\[
\frac{d^{2}\, \delta s}{d\tau^{2}}
  + 7m^{2}\, \delta s \cos 2\tau
  + 2(1 + m - \tfrac{5}{4} m^{2} \cos 2\tau)\, \frac{d\, \delta p}{d\tau}
  - 2m^{2}\, \delta p \sin 2\tau = 0,
\]
and this is as might be expected our second differential equation which was
found above. Hence we only require to consider the equations~\Eqref{(27)}.

\Section{§ 6. }{Integration of an important type of Differential Equation.}
\index{Differential Equation, Hill's}%
\index{Hill, G. W., Lunar Theory!Special Differential Equation}%

The differential equation for~$\delta p$ belongs to a type of great importance
in mathematical physics. We may write the typical equation in the form
\[
\frac{d^{2}x}{dt^{2}}
  + (\Theta_{0} + 2\Theta_{1} \cos 2t + 2\Theta_{2} \cos 4t + \dots) x = 0,
\]
where $\Theta_{0}, \Theta_{1}, \Theta_{2}, \dots$ 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 $\Theta_{1}, \Theta_{2}, \dots$, we get as a first approximation
\[
x = A \cos(t \sqrt{\Theta_{0}} + \epsilon).
\]

Using this value for~$x$ in the term multiplied by~$\Theta_{1}$, and neglecting $\Theta_{2},
\Theta_{3}, \dots$, we get
\[
\frac{d^{2}x}{dt^{2}}
  + \Theta_{0} x + A\Theta_{1} \left\{
      \cos\bigl[t(\sqrt{\Theta_{0}} + 2) + \epsilon\bigr]
    + \cos\bigl[t(\sqrt{\Theta_{0}} - 2) + \epsilon\bigr]\right\} = 0.
\]

Solving this by the usual rules we get the second approximation
\begin{align*}%[** TN: Rebroken]
x = A\biggl\{\cos\left[t\sqrt{\Theta_{0}} + \epsilon\right]
  &+ \frac{\Theta_{1} \cos\left[t(\sqrt{\Theta_{0}} + 2) + \epsilon\right]}
          {4(\sqrt{\Theta_{0}} + 1)} \\
  &- \frac{\Theta_{1} \cos\left[t(\sqrt{\Theta_{0}} - 2) + \epsilon\right]}
          {4(\sqrt{\Theta_{0}} - 1)}
  \biggr\}.
\end{align*}

Again using this we have the differential equation
\[
\begin{split}
\frac{d^{2}x}{dt^{2}}
  &+ \Theta_{0} x + A\Theta_{1}\left\{
       \cos\bigl[t(\sqrt{\Theta_{0}} + 2) + \epsilon\bigr]
     + \cos\bigl[t(\sqrt{\Theta_{0}} - 2) - \epsilon\bigr]
  \right\} \\
%
  &+ \frac{A\Theta_{1}^{2}}{4(\sqrt{\Theta_{0}} + 1)} \left\{
       \cos\bigl[t(\sqrt{\Theta_{0}} + 4) + \epsilon\bigr]
     + \cos(t\sqrt{\Theta_{0}} + \epsilon)
  \right\} \\
%
  &- \frac{A\Theta_{1}^{2}}{4(\sqrt{\Theta_{0}} - 1)} \left\{
       \cos(t\sqrt{\Theta_{0}} + \epsilon)
     + \cos\bigl[t(\sqrt{\Theta_{0}} - 4) + \epsilon\bigr]
  \right\} \\
%
  &+ A\Theta_{2} \left\{
       \cos\bigl[t(\sqrt{\Theta_{0}} + 4) + \epsilon\bigr]
     + \cos\bigl[t(\sqrt{\Theta_{0}} - 4) + \epsilon\bigr]
  \right\} = 0.
\end{split}
\]

Now this equation involves terms of the form~$B \cos(t\sqrt{\Theta_{0}} + \epsilon)$; on
integration terms of the form~$Ct\sin(t\sqrt{\Theta_{0}} + \epsilon)$ 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 $\Theta_{0}$~alone and is independent of~$\Theta_{1}, \Theta_{2}, \dots$. But the
\DPPageSep{095}{37}
appearance of secular terms leads us to revise this assumption and to take as
a first approximation
\[
x = A \cos (ct \sqrt{\Theta_{0}} + \epsilon),
\]
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
\[
x = A \cos (ct + \epsilon),
\]
where $c$~is nearly equal to~$\Surd{\Theta_{0}}$.

Using this value of~$x$ in the term involving~$\Theta_{1}$, our equation becomes
\[
\frac{d^{2}x}{dt^{2}}
  + \Theta_{0} x + A\Theta_{1}\left\{
      \cos\bigl[(c + 2)t + \epsilon\bigr] + \cos\bigl[(c - 2)t + \epsilon\bigr]
  \right\} = 0,
\]
and the second approximation is
\begin{align*}
x = A \cos (ct + \epsilon)
  &+ \frac{A\Theta_{1}}{(c + 2)^{2} - \Theta_{0}} \cos\bigl[(c + 2)t + \epsilon\bigr] \\
  &+ \frac{A\Theta_{1}}{(c - 2)^{2} - \Theta_{0}} \cos\bigl[(c - 2)t + \epsilon\bigr].\footnotemark
\end{align*}
\footnotetext{This is not a solution of the previous equation, unless we actually put $c=\sqrt{\Theta_{0}}$ in the
  first term.}%

Proceeding to another approximation with this value of~$x$, we get
\[
\begin{split}
\frac{d^{2}x}{dt^{2}}
  &+ \Theta_{0}x + A\Theta_{1}\left\{
     \cos\bigl[(c + 2)t + \epsilon\bigr] + \cos\bigl[(c - 2)t + \epsilon\bigr]
  \right\} \\
%
  &+ \frac{A\Theta_{1}^{2}}{(c + 2)^{2} - \Theta_{0}} \left\{
     \cos\bigl[(c + 4)t + \epsilon\bigr] + \cos(ct + \epsilon)\right\} \\
%
  &+ \frac{A\Theta_{1}^{2}}{(c - 2)^{2} - \Theta_{0}} \left\{
     \cos(ct + \epsilon) + \cos\bigl[(c - 4)t + \epsilon\bigr]\right\} \\
%
  &+ A\Theta_{2}\left\{
     \cos\bigl[(c + 4)t + \epsilon\bigr] + \cos\bigl[(c - 4)t + \epsilon\bigr]
  \right\} =0.
\end{split}
\]

We might now proceed to further approximations but just as a term in
$\cos (ct + \epsilon)$ generates in the solution terms in
\[
\cos\bigl[(c ± 2)t + \epsilon\bigr]\quad \text{and}\quad
\cos\bigl[(c ± 4)t + \epsilon\bigr],
\]
terms in
\[
\cos\bigl[(c ± 2)t + \epsilon\bigr]\quad \text{and}\quad
\cos\bigl[(c ± 4)t + \epsilon\bigr]
\]
will generate new terms in~$\cos(ct + \epsilon)$, 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
\[
\frac{d^{2}x}{dt^{2}}
  + x\sum_{-\infty}^{+\infty} \Theta_{i} e^{2it\Surd{-1}} = 0,
\Tag{(28)}
\]
where
\[
\Theta_{-i} = \Theta_{i},
\]
\DPPageSep{096}{38}
and the solution is assumed to be
\[
x = \sum_{-\infty}^{+\infty} A_{j} e^{(c + 2j)t \sqrt{-1}},
\]
where the ratios of all the coefficients~$A_{j}$, and~$c$, are to be determined by
equating coefficients of different powers of~$e^{t\sqrt{-1}}$.

Substituting this expression for~$x$ in the differential equation, we get
\[
-\sum_{-\infty}^{+\infty} (c + 2j)^{2} A_{j} e^{(c + 2j)t\sqrt{-1}} +
 \sum_{-\infty}^{+\infty} A_{j} e^{(c + 2j)t \sqrt{-1}}
 \sum_{-\infty}^{+\infty} \Theta_{i} e^{2i t\sqrt{-1}} = 0,
\]
and equating to zero the coefficient of~$e^{(c + 2j)t \sqrt{-1}}$,
\begin{multline*}
-(c + 2j)^{2}A_{j} + A_{j}\Theta_{0}
  + A_{j-1}\Theta_{1} + A_{j-2}\Theta_{2} + A_{j-3}\Theta_{3} + \dots \\
  + A_{j+1}\Theta_{-1} + A_{j+2}\Theta_{-2} + A_{j+3}\Theta_{-3} + \dots = 0.
\end{multline*}

Hence the succession of equations is
\index{Hill, G. W., Lunar Theory!infinite determinant}%
\index{Infinite determinant, Hill's}%
\iffalse
\begin{align*}
\dots &+ \bigl[\Theta_{0} - (c-4)^2\bigr]A_{-2} + \Theta_{-1}A_{-1} + \Theta_{-2}A_{0} + \Theta_{-3}A_{1} + \Theta_{-4}A_{2} + \dots = 0, \\
\dots &+ \Theta_{1}A_{-2} + \bigl[\Theta_{0} - (c-2)^2\bigr]A_{-1} + \Theta_{-1}A_{0} + \Theta_{-2}A_{1} + \Theta_{-3}A_{2} + \dots = 0, \\
\dots &+ \Theta_{2}A_{-2} + \Theta_{1}A_{-1} + (\Theta_{0} - c^2)A_{0} + \Theta_{-1}A_{1} + \Theta_{-2}A_{2} + \dots = 0, \\
\dots &+ \Theta_3A_{-2} + \Theta_{2}A_{-1} + \Theta_{1}A_{0} + \bigl[\Theta_{0} - (c+2)^2\bigr]A_{1} + \Theta_{-1}A_{2} + \dots = 0, \\
\dots &+ \Theta_4A_{-2} + \Theta_3A_{-1} + \Theta_{2}A_{0} + \Theta_{1}A_{1} + \bigl[\Theta_{0} - (c+4)^2\bigr]A_{2} + \dots = 0.
\end{align*}
\fi
{\small
\[
\begin{array}{@{\,}*{17}{c@{\,}}}
\hdotsfor{17} \\
\dots &+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c-4)^2\bigr]A_{-2}} &+& \Theta_{-1}A_{-1} &+& \Theta_{-2}A_{0} &+& \Theta_{-3}A_{1} &+& \Theta_{-4}A_{2} &+& \dots &=& 0, \\
\dots &+& \Theta_{1}A_{-2}&+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c-2)^2\bigr]A_{-1}} &+& \Theta_{-1}A_{0} &+& \Theta_{-2}A_{1} &+& \Theta_{-3}A_{2} &+& \dots &=& 0, \\
\dots &+& \Theta_{2}A_{-2}&+& \Theta_{1}A_{-1} &+& \multicolumn{3}{c}{(\Theta_{0} - c^2)A_{0}} &+& \Theta_{-1}A_{1} &+& \Theta_{-2}A_{2} &+& \dots &=& 0, \\
\dots &+& \Theta_3A_{-2}  &+& \Theta_{2}A_{-1} &+& \Theta_{1}A_{0} &+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c+2)^2\bigr]A_{1}} &+& \Theta_{-1}A_{2} &+& \dots &=& 0, \\
\dots &+& \Theta_4A_{-2}  &+& \Theta_3A_{-1} &+& \Theta_{2}A_{0} &+& \Theta_{1}A_{1} &+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c+4)^2\bigr]A_{2}} &+& \dots &=& 0. \\
\hdotsfor{17}
\end{array}
\]}

We clearly have an infinite determinantal equation for~$c$.

If we take only three columns and rows, we get
\begin{multline*}
\bigl[\Theta_{0} - (c - 2)^{2}\bigr] \bigl[\Theta_{0} - c^{2}\bigr] \bigl[\Theta_{0} - (c + 2)^{2}\bigr]
  - \Theta_{1}^{2} \bigl[\Theta_{0} - (c - 2)^{2}\bigr] - \Theta_{1}^{2} \bigl[\Theta_{0} - (c + 2)^{2}\bigr] \\
%
  - \Theta_{2}^{2}(\Theta_{0} - c^{2}) + 2\Theta_{1}^{2} \Theta_{2} = 0, \\
%
\bigl[(\Theta_{0} - c^{2} - 4)^{2} - 16c^{2}\bigr] \bigl[\Theta_{0} - c^{2}\bigr]
  - 2\Theta_{1}^{2}(\Theta_{0} - c^{2} - 4)
  -  \Theta_{2}^{2}(\Theta_{0} - c^{2}) + 2\Theta_{1}^{2} \Theta_{2} = 0.
\end{multline*}

If we neglect $(\Theta_{0} - c^{2})^{3}$ which is certainly small
\begin{multline*}
\bigl[-8(\Theta_{0} - c^{2}) + 16 + 16(\Theta_{0} - c^{2}) - 16\Theta_{0}\bigr] \bigl[\Theta_{0} - c^{2}\bigr] \\
%
  \shoveright{ -(\Theta_{0} - c^{2}) \bigl[2\Theta_{1}^{2} + \Theta_{2}^{2}\bigr] + 8\Theta_{1}^{2} + 2\Theta_{1}^{2} \Theta_{2} = 0,} \\
%
  \shoveright{8(\Theta_{0} - c^{2})^{2} + (\Theta_{0} - c^{2})(16 - 16\Theta_{0} - 2\Theta_{1}^{2} - \Theta_{2}^{2}) + 8\Theta_{1}^{2} + 2\Theta_{1}^{2} \Theta_{2} = 0,} \\
%
(\Theta_{0} - c^{2})^2 + 2(\Theta_{0} - c^{2})(1 - \Theta_{0} - \tfrac{1}{8}\Theta_{1}^{2} - \tfrac{1}{16}\Theta_{2}^{2}) + \Theta_{1}^{2} + \tfrac{1}{4}\Theta_{1}^{2} \Theta_{2} = 0.
\end{multline*}

Since $\Theta_{1}^{2}, \Theta_{2}^{2}$ are small compared with~$1 - \Theta_{0}$, and $\Theta_{2}$~compared with~$1$, we
have as a rougher approximation
\[
(c^{2} - \Theta_{0})^{2} + 2(\Theta_{0} - 1) (c^{2} - \Theta_{0}) = -\Theta_{1}^{2},
\]
\DPPageSep{097}{39}
whence
\begin{gather*}
c^{2} - \Theta_{0}
  = -(\Theta_{0} - 1) ± \sqrt{(\Theta_{0} - 1)^{2} - \Theta_{1}^{2}}, \\
%
c^{2} = 1 ± \sqrt{(\Theta_{0} - 1)^{2} - \Theta_{1}^{2}}.
\end{gather*}

Now $c^{2} = \Theta_{0}$ when $\Theta_{1} = 0$. Hence we take the positive sign and get
\[
c = \sqrt{1 + \sqrt{(\Theta_{0} - 1)^{2} - \Theta_{1}^{2}}},
\Tag{(29)}
\]
which is wonderfully nearly correct.

For further discussion of the equation for~$c$, see Notes~1,~2, pp.~\Pgref{note:1},~\Pgref{note:2}. %[** TN: pp 53, 55 in original]

\Section{§ 7. }{Integration of the Equation for $\delta p$.}

We now return to the Lunar Theory and consider the solution of our
differential equation. Assume it to be
\[
\delta p = A_{-1}\cos\bigl[(c - 2)\tau + \epsilon\bigr]
  + A_{0}\cos(c\tau + \epsilon)
  + A_{1}\cos\bigl[(c + 2)\tau + \epsilon\bigr].
\]

On substitution in~\Eqref{(27)} we get
\begin{align*}
 A_{-1} \bigl[(1 + 2m - \tfrac{1}{2} m^{2} - 15 m^{2}\cos2\tau) &- (c - 2)^{2}\bigr]\cos\bigl[(c - 2)\tau + \epsilon\bigr] \\
%
+ A_{0} \bigl[(1 + 2m - \tfrac{1}{2} m^{2} - 15 m^{2}\cos2\tau) &- c^{2}\bigr]\cos(c\tau + \epsilon) \\
%
+ A_{1} \bigl[(1 + 2m - \tfrac{1}{2} m^{2} - 15 m^{2}\cos2\tau) &- (c + 2)^{2}\bigr]\cos\big[(c + 2)\tau + \epsilon\bigr] = 0.
\end{align*}

Then we equate to zero the coefficients of the several cosines.

1st~$\cos(c\tau + \epsilon)$ gives
\[
-\tfrac{15}{2} m^{2}A_{-1}
  + A_{0}(1 + 2m - \tfrac{1}{2} m^{2} - c^{2})
  - \tfrac{15}{2} m^{2}A_{1} = 0.
\]

2nd~$\cos \bigl[(c - 2)\tau + \epsilon\bigr]$ gives
\[
A_{-1} \bigl[1 + 2m - \tfrac{1}{2} m^{2} - (c - 2)^{2}\bigr]
  - \tfrac{15}{2} m^{2}A_{0} = 0.
\]

3rd~$\cos \bigl[(c + 2)\DPtypo{t}{\tau}\bigr] + \epsilon]$ gives
\[
-\tfrac{15}{2} m^{2}A_{0} + A_{1}\bigl[1 + 2m - \tfrac{1}{2} m^{2} - (c + 2)^{2}] = 0.
\]

If we neglect terms in~$m^{2}$ the first equation gives us $c^{2} = 1 + 2m$, and
\Pagelabel{39}
therefore $c = 1 + m$, $c - 2 = -(1 - m)$, $c + 2 = 3 + m$.

The second and third equations then reduce to
\[
4m A_{-1} = 0;\quad A_{1}(-8 - 4m) = 0.
\]

From this it follows that $A_{-1}$~is at least of order~$m$ and $A_{1}$~at least of
order~$m^{2}$.

Then since we are neglecting higher powers than~$m^{2}$, the first equation
reduces to
\[
A_{0}(1 + 2m - \tfrac{1}{2} m^{2} - c^{2}) = 0,
\]
so that
\[
c^{2} = 1 + 2m - \tfrac{1}{2} m^{2}\quad \text{or}\quad
c = 1 + m - \tfrac{3}{4} m^{2}.
\]

Thus
\[
(c - 2)^{2} = (1 - m + \tfrac{3}{4} m^{2})^{2}
  = 1 - 2m + \tfrac{5}{2} m^{2},
\]
and
\[
1 + 2m - \tfrac{1}{2} m^{2} - (c - 2)^2
  = 4m - 3m^{2}.
\]
\DPPageSep{098}{40}

Hence the second equation becomes
\[
A_{-1}(4m - 3m^{2}) = \tfrac{15}{2} m^{2}A_{0};
\]
and since $A_{-1}$~is of order~$m$, the term~$-3m^{2}A_{-1}$ is of order~$m^{3}$ and therefore
negligible. Hence
\[
4m A_{-1} = \tfrac{15}{2} m^{2} A_{0} \quad \text{or}\quad
A_{-1} = \tfrac{15}{8} m A_{0},
\]
and we cannot obtain $A_{-1}$~to an order higher than the first.

The third equation is
\[
-\tfrac{15}{2} m^{2} A_{0} + A_{1}[1 - 9] = 0,
\]
or
\[
A_{1} = -\tfrac{15}{16} m^{2} A_{0}.
\]

We have seen that $A_{-1}$~can only be obtained to the first order; so it is
useless to retain terms of a higher order in~$A_{1}$. Hence our solution is
\[
A_{-1} = \tfrac{15}{8} m A_{0},\quad
A_{1} = 0.
\]

Hence
\[
\delta p = A_0 \left\{\cos(c\tau + \epsilon) + \tfrac{15}{8} m \cos\bigl[(c - 2)\tau + \epsilon\bigr]\right\}.
\Tag{(30)}
\]

In order that the solution may agree with the more ordinary notation we
write $A_{0} = -a_{0}e$, and obtain
\[
\left.
\begin{gathered}
\delta p = -a_{0}e \cos(c\tau + \epsilon) - \tfrac{15}{8} m a_{0} e \cos\bigl[(c - 2)\tau + \epsilon\bigr]\Add{,} \\
\lintertext{where}
{c = 1 + m - \tfrac{3}{4} m^{2}\Add{.}}
\end{gathered}
\right\}
\Tag{(31)}
\]

To the first order of small quantities the equation~\Eqref{(27)} for~$\delta s$ was
\begin{align*}
\frac{d\, \delta s}{d\tau}
  &= -2(1 + m)\, \delta p \\
  &=  2(1 + m)a_{0}e \cos(c\tau + \epsilon)
   + \tfrac{15}{4} m a_{0} e \cos\bigl[(c - 2)\tau + \epsilon\bigr].
\end{align*}

If we integrate and note that $c = 1 + m$ so that $c - 2 = -(1 - m)$, we have
\Pagelabel{40}
\[
\delta s = 2a_{0} e \sin(c\tau + \epsilon)
  - \tfrac{15}{4} m a_{0} e \sin\bigl[(c - 2)\tau + \epsilon\bigr].
\Tag{(32)}
\]

We take the constant of integration zero because $e = 0$ will then correspond
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
\[
\delta p = -a_{0} e \cos (c\tau + \epsilon),\quad
\delta s = 2a_{0} e \sin (c\tau + \epsilon).
\]

In the undisturbed orbit
\[
x = a_{0} \cos\tau,\quad
y = a_{0} \sin\tau \quad \text{so that}\quad
\phi = \tau,
\]
and
\begin{gather*}
\begin{aligned}
\delta x &= \delta p \cos\phi - \delta s \sin\phi, \\
\delta y &= \delta p \sin\phi + \delta s \cos\phi;
\end{aligned} \\
\begin{aligned}
\delta x &= -a_{0} e \cos(c\tau + \epsilon)\cos\tau - 2a_{0} e \sin(c\tau + \epsilon)\sin\tau, \\
\delta y &= -a_{0} e \cos(c\tau + \epsilon)\sin\tau + 2a_{0} e \sin(c\tau + \epsilon)\cos\tau.
\end{aligned}
\end{gather*}
\DPPageSep{099}{41}

Therefore writing $X = x + \delta x$, $Y = y + \delta y$, $X = R \cos\Theta$, $Y = R \sin\Theta$,
\begin{alignat*}{3}
X &= a_{0}\bigl[\cos\tau &&- e \cos(c\tau + \epsilon)\cos\tau
                        &&- 2e \sin(c\tau + \epsilon)\sin\tau\bigr], \\
%
Y &= a_{0}\bigl[\sin\tau &&- e \cos(c\tau + \epsilon)\sin\tau
                        &&+ 2e \sin(c\tau + \epsilon)\cos\tau\bigr].
\end{alignat*}

Therefore
\[
R^{2} = a_{0}^{2} \bigl[1 - 2e \cos(c\tau + \epsilon)\bigr]
\]
or
\[
R = a_{0} \bigl[1 - e \cos(c\tau + \epsilon)\bigr]
  = \frac{a_{0}}{1 + e \cos(c\tau + \epsilon)}.
\Tag{(33)}
\]

Again
\begin{alignat*}{2}
\cos\Theta &= \cos\tau &&- 2e \sin (c\tau + \epsilon)\sin\tau, \\
\sin\Theta &= \sin\tau &&+ 2e \sin (c\tau + \epsilon)\cos\tau.
\end{alignat*}

Hence
\[
\sin(\Theta - \tau) = 2e \sin(c\tau + \epsilon),
\]
giving
\[
\Theta = \tau + 2e \sin(c\tau + \epsilon).
\Tag{(34)}
\]

It will be noted that the equations for $R, \Theta$ 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 $m = 0$
we see that $e$~is to be identified with the eccentricity and $c\tau + \epsilon$~with the
mean anomaly.

\footnotemark~We can interpret~$c$ in terms of the symbols of the ordinary lunar theories.
%[** TN: Minor rewording coded using \DPtypo]
\footnotetext{\DPtypo{From here till the foot of this page}
  {In the next three paragraphs} a slight knowledge of ordinary lunar theory is
  supposed. The results given are not required for the further development of Hill's theory.}%
When no perturbations are considered the moon moves in an ellipse. The
\index{Apse, motion of}%
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
$\dfrac{d\varpi}{dt}$~the mean motion of the line of apses, the argument entering into the
elliptic inequalities is~$\left(n - \dfrac{d\varpi}{dt}\right)t + \epsilon$. This must be the same as~$c\tau + \epsilon$, i.e.~as
$c(n - n')t + \epsilon$.

Hence
\[
n -\frac{d\varpi}{dt} = c(n - n'),
\]
giving
\begin{align*}
\frac{d\varpi}{n\, dt}
  &= 1 - c \frac{n - n'}{n} \\
  &= 1 - \frac{c}{1 + m}\quad \text{since} \quad
m = \frac{n'}{ n - n'}.
\end{align*}

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
\index{Perigee, motion of}%
\[
\frac{d\varpi}{n\, dt} = \tfrac{3}{4} m^{2}.
\]
\DPPageSep{100}{42}

Returning to our solution, and for simplicity again dropping the factor~$a_{0}$,
we have from \Eqref{(31)},~\Eqref{(32)}
\begin{align*}
\delta p &= -\tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr] -  e \cos(c\tau + \epsilon), \\
%
\delta s &= -\tfrac{15}{4} m e \sin\bigl[(c - 2)\tau + \epsilon\bigr] + 2e \sin(c\tau + \epsilon).
\end{align*}

Also $\cos\phi = \cos\tau$, $\sin\phi = \sin\tau$ to the first order of small quantities, and
\[
\delta x = \delta p \cos\phi - \delta s \sin\phi,\quad
\delta y = \delta p \sin\phi + \delta s \cos\phi.
\]
Therefore
\begin{multline*}
\delta x
  = -\tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr]\cos\tau
  -  e \cos(c\tau + \epsilon) \cos\tau \\
%
   + \tfrac{15}{4} m e \sin\bigl[(c - 2)\tau + \epsilon\bigr]\sin\tau
  - 2e \sin(c\tau + \epsilon) \sin\tau,
\end{multline*}
\begin{multline*}
\delta y
  = -\tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr] \sin\tau
  -  e \cos(c\tau + \epsilon) \sin\tau \\
%
   - \tfrac{15}{4} m e \sin\bigl[(c - 2)\tau + \epsilon\bigr] \cos\tau
  + 2e \sin(c\tau + \epsilon) \cos\tau.
\end{multline*}

Now let $X = x + \delta x$, $Y = y + \delta y$ and we have by means of the values of $x,
y$ in the variational curve
\begin{align*}
X &= \cos\tau \bigl[1 - m^{2}
    - \tfrac{3}{4} m^{2} \sin^{2}\tau
    - \tfrac{15}{8} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\}
    - e \cos(c\tau + \epsilon)\bigr] \\
  &\qquad\qquad\qquad\qquad
   + \sin\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\}
   - 2e \sin(c\tau + \epsilon)\bigr], \\
%
Y &= \sin\tau \bigl[1 + m^{2}
   + \tfrac{3}{4} m^{2} \cos^{2}\tau
   - \tfrac{15}{8} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\}
   - e \cos(c\tau + \epsilon)\bigr] \\
  &\qquad\qquad\qquad\qquad
   - \cos\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\}
   - 2e \sin(c\tau + \epsilon)\bigr].
\end{align*}

Writing $R^{2} = X^{2} + Y^{2}$, we obtain to the requisite degree of approximation
\begin{align*}
R^{2} &= \cos^{2}\tau \bigl[1 - 2m^{2}
    - \tfrac{3}{2} m^{2} \sin^{2}\tau
    - \tfrac{15}{4} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\}
    - 2e \cos(c\tau + \epsilon)\bigr] \\
%
  &+ \sin^{2}\tau \bigl[1 + 2m^{2}
    + \tfrac{3}{2} m^{2} \cos^{2}\tau
    - \tfrac{15}{4} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\}
    - 2e \cos(c\tau + \epsilon)\bigr] \\
%
  &+ \sin 2\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\}
    - 2e \sin(c\tau + \epsilon)\bigr] \\
%
  &- \sin 2\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\}
    - 2e \sin(c\tau + \epsilon)\bigr], \\
%
R^{2} &= 1 - 2m^{2} \cos 2\tau
  - \tfrac{15}{4} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\}
  - 2e \cos(c\tau + \epsilon).
\end{align*}

Hence reintroducing the factor~$a_{0}$ which was omitted for the sake of brevity
\[
R = a_{0}\bigl[1 - e \cos(c\tau + \epsilon)
    - \tfrac{15}{8} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\}
    - m^{2} \cos 2\tau\bigr].
\Tag{(35)}
\]

This gives the radius vector; it remains to find the longitude.

We multiply the expressions for $X, Y$ by~$1/R$,\DPnote{** Slant fraction} i.e.~by
\[
1 + e \cos(c\tau + \epsilon)
  + \tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr]
  + m^{2} \cos 2\tau,
\]
and remembering that
\[
m^{2} \cos 2\tau
  = m^{2} - 2m^{2} \sin^{2}\tau
  = 2m^{2} \cos^{2}\tau - m^{2},
\]
we get
\begin{align*}
\cos\Theta
  &= \cos\tau \bigl[1 - \tfrac{11}{4} m^{2} \sin^{2}\tau\bigr]
   - \sin\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\}
   - 2e \sin(c\tau + \epsilon)\bigr], \\
%
\sin\Theta
  &= \sin\tau \bigl[1 + \tfrac{11}{4} m^{2} \cos^{2}\tau\bigr]
   - \cos\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\}
   - 2e \sin(c\tau + \epsilon)\bigr].
\end{align*}

Whence
\[
\sin(\Theta - \tau)
  = \tfrac{11}{8} m^{2} \sin 2\tau
  - \tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\}
  + 2e \sin(c\tau + \epsilon),
\]
\DPPageSep{101}{43}
or to our degree of approximation
\[
\Theta = \tau + \tfrac{11}{8} m^{2} \sin 2\tau
  - \tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\}
  + 2e \sin(c\tau + \epsilon).
\Tag{(36)}
\]

We now transform these results into the ordinary notation.
\index{Equation, annual!of the centre}%
\index{Latitude of the moon}%

\footnotemark~Let $l, v$ be the moon's mean and true longitudes, and $l'$~the sun's mean
\footnotetext{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.}%
longitude. Then $\Theta$~being the moon's true longitude relatively to the moving
axes, we have
\[
v = \Theta + l'.
\]

Also
\begin{gather*}
\tau + l' = (n - n')t + n't =l, \\
\therefore \tau = l - l'.
\end{gather*}

We have seen that $c\tau + \epsilon$ is the moon's mean anomaly, or~$l - \varpi$,
\[
\therefore (c - 2)\tau + \epsilon = l - \varpi - 2(l - l') = -(l + \varpi - 2l').
\]

Then substituting these values in the expressions for $R$~and~$\Theta$ and
adding~$l'$ to the latter we have on noting that $a_{0} = \a(1 - \frac{1}{6} m^{2})$
\index{Evection}%
\[
\left.
\begin{aligned}
R &= \a\bigl[1 - \tfrac{1}{6} m^{2}
    - \UnderNote{e \cos(l - \varpi)}{equation of centre}
    - \UnderNote{\tfrac{15}{8} m e \cos(l - 2l' + \varpi)}{evection}
    - \UnderNote{m^{2} \cos 2(l - l')\bigr]}{variation}\Add{,} \\
%
v &= l + \UnderNote{2e \sin (l - \varpi)}{equation of centre}
    + \UnderNote{\tfrac{15}{4} m e \sin(l - 2l' + \varpi)}{evection}
    + \UnderNote{\tfrac{11}{8} m^{2} \sin 2(l - l')}{variation}\Add{.}
\end{aligned}
\right\}
\Tag{(37)}
\]

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.

\Section{§ 8. }{Introduction of the Third Coordinate.}
\index{Third coordinate introduced}%
\index{Variation, the}%

Still keeping $\Omega=0$, consider the differential equation for~$z$ in~\Eqref{(5)}
\[
\frac{d^{2}z}{d\tau^{2}} + \frac{\kappa z}{r^{3}} + m^{2}z = 0.
\]

From~\Eqref{(8)}
\[
\frac{\kappa}{a_{0}^{3}} = 1 + 2m + \tfrac{3}{2} m^{2},
\]
and from~\Eqref{(10)}
\[
\frac{a_{0}^{3}}{r^{3}} = 1 + 3m^{2} \cos 2\tau.
\]

The equation may therefore be written
\[
\frac{d^{2}z}{d\tau^{2}} + z(1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau) = 0.
\]

This is an equation of the type considered in~\SecRef{6} and therefore we
assume
\[
z = B_{-1} \cos\bigl\{(g - 2)\tau + \zeta\bigr\}
  + B_{0}  \cos(g\tau + \zeta)
  + B_{1}  \cos\bigl\{(g + 2)\tau + \zeta\bigr\}.
\]
\DPPageSep{102}{44}

On substitution we get
\begin{align*}
B_{-1}  &\bigl[-(g - 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau \bigr] \cos\bigl[(g - 2)\tau + \zeta \bigr] \\
%
+ B_{0} &\bigl[-g^{2} + 1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau \bigr] \cos(g\tau + \zeta) \\
%
+ B_{1} &\bigl[-(g + 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau \bigr] \cos \bigl[(g + 2)\tau + \zeta \bigr] = 0.
\end{align*}

The coefficients of $\cos(g\tau + \zeta)$, $\cos \bigl[(g - 2)\tau + \zeta\bigr]$, $\cos \bigl[(g + 2)\tau + \zeta\bigr]$ give
respectively
\[
\left.
\begin{alignedat}{2}
&\tfrac{3}{2} m^{2} B_{-1} + B_{0} \bigl[-g^{2} + 1 + 2m + \tfrac{5}{2} m^{2}\bigr] + \tfrac{3}{2} m^{2} B_{1} &&= 0\Add{,} \\
%
&B_{-1} \bigl[-(g - 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2} ] + \tfrac{3}{2} m^{2} B_{0} &&= 0\Add{,} \\
%
&\tfrac{3}{2} m^{2} B_{0} + B_{1} \bigl[-(g + 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2}\bigr] &&= 0\Add{.}
\end{alignedat}
\right\}
\Tag{(38)}
\]

As a first approximation drop the terms in~$m^{2}$. The first of these equations
then gives $g^{2} = 1 + 2m$. The third equation then shews that $\dfrac{B_{1}}{B_{0}}$~is of
order~$m^{2}$. But a factor~$m$ can be removed from the second equation shewing
that $\dfrac{B_{-1}}{B_{0}}$~is of order~$m$ and can only be determined to this order. Hence
$B_{1}$~can be dropped. [Cf.~pp.~\Pgref{39},~\Pgref{40}.]

Considering terms in~$m^{2}$ we now get from the first equation
\[
g^{2} = 1 + 2m + \tfrac{5}{2} m^{2}.
\]

Therefore
\begin{gather*}
g = 1 + m + \tfrac{5}{4} m^{2} - \tfrac{1}{2} m^{2}
  = 1 + m + \tfrac{3}{4} m^{2}, \\
(g - 2)^{2} = (1 - m)^{2} = 1 - 2m, \text{ neglecting terms in~$m^{2}$}.
\end{gather*}

The second equation then gives
\[
B_{-1} = -\tfrac{3}{8} m B_{0},
\]
and the solution is
\[
z = B_{0} \bigl[\cos(g\tau + \zeta) - \tfrac{3}{8} m \cos\bigl\{(g - 2)\tau + \zeta\bigr\}\bigr].
\Tag{(39)}
\]

We shall now interpret this equation geometrically. To do so we neglect
the solar perturbation and we get
\[
z = B_{0} \cos(g\tau + \zeta).
\Tag{(40)}
\]

\begin{wrapfigure}{r}{1.5in}
  \centering
  \Input[1.5in]{p044}
  \caption{Fig.~3.}
\end{wrapfigure}
Now consider the moon to move in a plane orbit inclined at angle~$i$ to
the ecliptic and let $\Omega$~be the longitude of the lunar
node, $l$~the longitude of the moon, $\beta$~the latitude.

The right-angled spherical triangle gives
\[
\tan\beta = \tan i \sin(l - \Omega)
\]
and therefore
\[
z = r \tan\beta = r \tan i \sin (l - \Omega).
\]
\DPPageSep{103}{45}

As we are only dealing with a first approximation we may put $r = a_{0}$ and
so we interpret
\begin{gather*}
B_{0} = a_{0} \tan i, \\
g\tau + \zeta = l - \Omega -\tfrac{1}{2}\pi.
\end{gather*}

\footnotemark~We can easily find the significance of~$g$, for differentiating this equation
\footnotetext{From here till end of paragraph is a comparison with ordinary lunar theories.}%
with respect to the time we get
\begin{gather*}
g(n - n') = n - \frac{d\Omega}{dt}, \\
\begin{aligned}
\therefore \frac{d\Omega}{n\, dt}
  &= 1 - \frac{g(n - n')}{n} \\
  &= 1 + \frac{g}{1 + m} \\
  &= -\tfrac{3}{4} m^{2} \text{ to our approximation.}
\end{aligned}
\end{gather*}
Thus we find that the node has a retrograde motion.

We have
\begin{align*}
g\tau + \zeta
  &= l - \Omega - \tfrac{1}{2}\pi, \\
%
(g - 2)\tau + \zeta
  &= l - \Omega - \tfrac{1}{2}\pi - 2(l - l') \\
%
  &= -(l - 2l' + \Omega) - \tfrac{1}{2}\pi.
\end{align*}

If we write $s = \tan\beta$, $k = \tan i$, we find
\[
s = k \sin(l - \Omega) + \tfrac{3}{8} m k \sin(l - 2l' + \Omega).
\Tag{(41)}
\]

The last term in this equation is called the evection in latitude.
\index{Evection!in latitude}%

\Section{§ 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~$\Omega$\footnotemark. This gave
\footnotetext{The $\Omega$~of \Pageref{20}, not that of the preceding paragraph.}%
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 projection
on the ecliptic. We will denote by~$r_{1}$ the curtate radius vector, so
that $r_{1}^{2} = x^{2} + y^{2}$, $r^{2} = r_{1}^{2} + z^{2}$; the $x, y$~axes rotating as before with angular
velocity~$n'$ in the plane of the ecliptic. In determining the variational curve,~\SecRef{3},
we put $\Omega = 0$, $r = r_{1}$. It will appear therefore that in finding the actual
motion of the moon we shall require to consider not only~$\Omega$ but new terms in~$z^{2}$.
In the next section we shall discuss the actual motion of the moon, making
use of the approximations we have already obtained.
\DPPageSep{104}{46}

\Section{§ 10. }{General Equations of Motion and their solution.}
\index{Equations of motion}%

We have
\[
r_{1}^{2} = x^{2} + y^{2} \text{ and }
r^{2} = r_{1}^{2} + z^{2}.
\]

Hence
\[
\frac{1}{r^{3}}
  = \frac{1}{r_{1}^{3}} \left(1 - \frac{3}{2}\, \frac{z^{2}}{r_{1}^{2}}\right);
  \text{ and }
\frac{1}{r}
  = \frac{1}{r_{1}} \left(1 - \frac{1}{2}\, \frac{z^{2}}{r_{1}^{2}}\right),
\]
to our order of accuracy.

The original equations~\Eqref{(3)} may now be written
\[
\left.
\begin{alignedat}{4}
\frac{d^{2}x}{d\tau^{2}}
  &- 2m\, \frac{dy}{d\tau} &&+ \frac{\kappa x}{r_{1}^{3}} &&- 3m^{2}x
  &&= \frac{\dd \Omega}{\dd x} + \frac{3}{2}\, \frac{\kappa z^{2}x}{r_{1}^{5}}\Add{,} \\
%
\frac{d^{2}y}{d\tau^{2}}
  &+ 2m\, \frac{dx}{d\tau} &&+ \frac{\kappa y}{r_{1}^{3}} &&
  &&= \frac{\dd \Omega}{\dd y} + \frac{3}{2}\, \frac{\kappa z^{2}y}{r_{1}^{5}}\Add{,} \\
%
\frac{d^{2}z}{d\tau^{2}}
  & &&+ \frac{\kappa z}{r_{1}^{3}} &&+ m^{2}z
  &&= \frac{\dd \Omega}{\dd z} + \frac{3}{2}\, \frac{\kappa z^{2}}{r_{1}^{5}}\Add{.}
\end{alignedat}
\right\}
\Tag{(42)}
\]

If we multiply by $2\dfrac{dx}{d\tau}$, $2\dfrac{dy}{d\tau}$, $2\dfrac{dz}{d\tau}$ and add, we find that the Jacobian
integral becomes
\[
V^{2} = 2\frac{\kappa}{r_{1}} + m^{2}(3x^{2} - z^{2})
  - \frac{\kappa z^{2}}{r_{1}^{3}}
  + 2\int_{0}^{\tau} \left(
      \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau}
    + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}
    + \frac{\dd \Omega}{\dd y}\, \frac{dz}{d\tau}
  \right) d\tau + C,
\Tag{(43)}
\]
where
\[
V^{2} = V_{1}^{2} + \left(\frac{dz}{d\tau}\right)^{2}
  = \left(\frac{dx}{d\tau}\right)^{2}
  + \left(\frac{dy}{d\tau}\right)^{2}
  + \left(\frac{dz}{d\tau}\right)^{2}.
\]

Now
\[
\Omega = \tfrac{3}{2} m^{2} \left(\frac{\a'^{3}}{r'^{3}}\, r^{2}\cos^{2} - x^{2}\right)
  + \tfrac{1}{2} m^{2} r^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right),
\]
and
\[
\cos\theta = \frac{xx' + yy' + zz'}{rr'}
  = \frac{xx' + yy'}{rr'}, \text{ since $z' = 0$}.
\]

Hence
\[
\Omega = \tfrac{3}{2} m^{2} \left\{\frac{\a'^{3}}{r'^{3}}(xx' + yy')^{2} - x^{2}\right\}
  + \tfrac{1}{2} m^{2} (x^{2} + y^{2}) \left(1 - \frac{\a'^{3}}{r'^{3}}\right)
  + \tfrac{1}{2} m^{2} z^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right).
\]

When we neglected $\Omega$~and~$z$, we found the solution
\begin{alignat*}{2}
x &= a_{0}\bigl[(1 - \tfrac{19}{16} m^{2})\cos\tau
  &&+ \tfrac{3}{16} m^{2}\cos 3\tau\bigr], \\
y &= a_{0}\bigl[(1 + \tfrac{19}{16} m^{2})\sin\tau
  &&+ \tfrac{3}{16} m^{2}\sin 3\tau\bigr].
\end{alignat*}

We now require to determine the effect of the terms introduced on the
right, and for brevity we write
\[
X = \frac{\dd \Omega}{\dd x} + \frac{3}{2}\, \frac{\kappa z^{2}x}{r_{1}^{5}},\quad
Y = \frac{\dd \Omega}{\dd y} + \frac{3}{2}\, \frac{\kappa z^{2}y}{r_{1}^{5}}.
\]

When we refer to~\SecRef{4} and consider how the differential equations for~$\delta p, \delta s$
were formed from those for~$\delta x, \delta y$, we see that the new terms~$X, Y$ on
the right-hand sides of the differential equations for~$\delta x, \delta y$ will lead to new
terms $X\cos\phi - Y\sin\phi$, $-X\sin\phi + Y\cos\phi$ on the right-hand sides of those
for~$\delta p, \delta s$.
\DPPageSep{105}{47}

Hence taking the equations \Eqref{(24)}~and~\Eqref{(25)} for $\delta p$~and~$\delta s$ and introducing
these new terms, we find
\begin{multline*}
\frac{d^{2}\, \delta p}{d\tau^{2}}
  + \delta p\, \bigl[-3 - 6m - \tfrac{9}{2}m^{2} - 5m^{2}\cos 2\tau\bigr]
  - 2\frac{d\, \delta s}{d\tau}\, (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) \\
\shoveright{-7m^{2}\, \delta s \sin 2\tau = X\cos\phi + Y\sin\phi,} \\
%
\shoveleft{\frac{d^{2}\, \delta s}{d\tau^{2}}
  + 7m^{2}\, \delta s \cos 2\tau
  + 2\frac{d\, \delta p}{d\tau}\, (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau)
  - 2m^{2}\, \delta p \sin 2\tau} \\
  = -X\sin\phi + Y\cos\phi.
\end{multline*}

In this analysis we shall include all terms to the order~$m k^{2}$, where $k$~is the
small quantity in the expression for~$z$. Terms involving~$m^{2}z^{2}$ will therefore
be neglected. In the variation of the Jacobian integral the term~$\dfrac{dz}{d\tau}\, \dfrac{d\, \delta z}{d\tau}$ can
obviously be neglected. The variation of the Jacobian integral therefore
gives (cf.~pp.~\Pgref{29},~\Pgref{35})
\begin{multline*}
\frac{d\, \delta s}{d\tau}
  = -2\delta p\, (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau)
   - \tfrac{7}{2}m^{2}\, \delta s \sin 2\tau \\
%
  + \frac{1}{V_{1}} \biggl[\int_{0}^{\tau}\!\!
     \left(\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau}
         + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}
         + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau}\right) d\tau
   + \tfrac{1}{2} \biggl\{\delta C - \frac{\kappa z^{2}}{r_{1}^{3}}
       - \left(\frac{dz}{d\tau}\right)^{2}\biggr\}
  \biggr],
\Tag{(44)}
\end{multline*}
where $\delta C$~will be chosen as is found most convenient. [In the previous work
we chose $\delta C = 0$.]

By means of this equation we can eliminate~$\delta s$ from the differential
equation for~$\delta p$. For
\begin{align*}
2\frac{d\, \delta s}{d\tau}\, (1 &+ m - \tfrac{5}{4}m^{2}\cos 2\tau) + 7m^{2}\, \delta s \sin 2\tau \\
%
  &= -4\delta p\, (1 + 2m + m^{2} - \tfrac{5}{2}m^{2} \cos 2\tau) \\
%
  &+ \frac{2}{V_{1}} (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau)
     \biggl[\int_{0}^{\tau}\left(
       \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau}
     + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}
     + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau} \right) d\tau \\
%
  &+ \tfrac{1}{2} \left\{\delta C - \frac{\kappa z^{2}}{r_{1}^{3}} - \left(\frac{dz}{d\tau}\right)^{2}\right\}
  \biggr],
\end{align*}
and therefore
\begin{align*}
\frac{d^{2}\delta p}{d\tau^{2}}
  &+ \delta p\, (1 + 2m  - \tfrac{1}{2}m^{2} - 15m^{2}\cos 2\tau)
   = X\cos\phi + Y\sin\phi \\
%
  &+ \frac{2}{V_{1}} (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau)
     \biggl[\int_{0}^{\tau} \left(
       \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau}
     + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}
     + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau}\right) d\tau \\
%
  &+ \tfrac{1}{2}\left\{\delta C - \frac{\kappa z^{2}}{r_{1}^{3}} - \left(\frac{dz}{d\tau}\right)^{2}\right\}
  \biggr].
\Tag{(45)}
\end{align*}

We first neglect~$\Omega$ and consider $X, Y$~as arising only from terms
in~$z^{2}$, i.e.\
\begin{gather*}
X = \frac{3}{2}\, \frac{\kappa z^{2}x}{r_{1}^{5}},\quad
Y = \frac{3}{2}\, \frac{\kappa z^{2}y}{r_{1}^{5}}. \\
%
\therefore X\cos\phi + Y\sin\phi
  = \frac{3}{2}\, \frac{\kappa z^{2}}{r_{1}^{5}}(x\cos\phi + y\sin\phi).
\end{gather*}
\DPPageSep{106}{48}

To the required order of accuracy.
\begin{gather*}
z = ka_{0} \cos(g\tau + \zeta),\quad \frac{\kappa}{a_{0}^{3}} = 1 + 2m, \\
%
r_{1} = a_{0},\quad \phi = \tau,\quad x = a_{0}\cos\tau,\quad y = a_{0}\sin\tau. \\
%
\therefore X \cos\phi + Y \sin\phi
  = \tfrac{3}{4}(1 + 2m)k^{2}a_{0} \bigl[1 + \cos 2(g\tau + \zeta)\bigr].
\end{gather*}

Also to order~$m$
\begin{align*}
\frac{\kappa z^{2}}{r_{1}^{3}} + \left(\frac{dz}{d\tau}\right)^{2}
  &= (1 + 2m) k^{2}a_{0}^{2} \cos^{2}(g\tau + \zeta)
    + g^{2}k^{2}a_{0}^{2} \sin^{2}(g\tau + \zeta) \\
%
  &= (1 + 2m) k^{2}a_{0}^{2},
\end{align*}
since $g^{2} = 1 + 2m$.

The equation for~$\delta p$ becomes therefore, as far as regards the new terms
now introduced,
\begin{multline*}
\frac{d^{2}\, \delta p}{d\tau^{2}} + \delta p\, (1 + 2m)
  = \tfrac{3}{4}(1 + 2m) k^{2}a_{0} \bigl[1 + \cos 2(g\tau + \zeta)\bigr] \\
  + \frac{(1 + m)}{a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr].
\end{multline*}

Hence
\[
\delta p - \tfrac{3}{4} k^{2}a_{0}
  - \frac{(1 - m)}{a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr]
  = \tfrac{3}{4}\frac{1 + 2m}{1 + 2m + 4g^{2}} k^{2}a_{0} \cos 2(g\tau + \zeta), \footnotemark
\]
\footnotetext{It is of course only the special integral we require. The general integral when the right-hand
  side is zero has already been dealt with,~\SecRef{7}.}%
but
\begin{gather*}
g^{2} = 1 + 2m, \text{ and therefore }
1 + 2m - 4g^{2} = -3(1 + 2m), \\
%
\therefore \delta p = \tfrac{3}{4} k^{2}a_{0}
  + \frac{(1 - m)}{a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr]
  - \tfrac{1}{4} k^{2}a_{0} \cos 2(g\tau + \zeta).
\end{gather*}

Again the varied Jacobian integral is
\begin{align*}
\frac{d\, \delta s}{d\tau}
  &= -2(1 + m)\, \delta p
   + \frac{1}{2a_{0}} \bigl[\delta C - (1 - 2m) k^{2}a_{0}^{2}\bigr] \\
%
  &= -\tfrac{3}{2}(1 + m) k^{2}a_{0}
   - \frac{3}{2a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr]
   + \tfrac{1}{2}(1 + m) k^{2}a_{0} \cos 2(g\tau + \zeta).
\end{align*}

In order that $\delta s$~may not increase with the time we choose~$\delta C$ so that the
constant term is zero,
\begin{align*}
\therefore \delta C &= m k^{2}a_{0},
\intertext{and}
\frac{d\, \delta s}{d\tau}
  &= \tfrac{1}{2}(1 - m) k^{2}a_{0} \cos 2(g\tau + \zeta), \\
%
\intertext{giving}
\delta s &= \tfrac{1}{4} k^{2}a_{0} \sin 2(g\tau + \zeta),
\Tag{(46)}
\end{align*}
as there is no need to introduce a new constant\footnotemark. Using the value of~$\delta C$ just
\footnotetext{Cf.\ same point in connection with equation~\Eqref{(32)}.}%
found we get
\[
\delta p = -\tfrac{1}{4} k^{2}a_{0}
  - \tfrac{1}{4} k^{2}a_{0} \cos 2(g\tau + \zeta).
\Tag{(47)}
\]

Having obtained $\delta p$~and~$\delta s$, we now require~$\delta x, \delta y$. These are
\begin{align*}
\delta x &= \delta p \cos\phi - \delta s \sin\phi, \\
\delta y &= \delta p \sin\phi + \delta s \cos\phi.
\end{align*}
\DPPageSep{107}{49}

In this case with sufficient accuracy $\phi = \tau$,
\begin{alignat*}{3}
\delta x
  &= - \tfrac{1}{4} a_{0}k^{2} \cos\tau
  &&- \tfrac{1}{4} a_{0}k^{2} \cos\tau \cos 2(g\tau + \zeta)
  &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \sin 2(g\tau + \zeta), \\
%
\delta y
  &= - \tfrac{1}{4} a_{0}k^{2} \sin\tau
  &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \cos 2(g\tau + \zeta)
  &&+ \tfrac{1}{4} a_{0}k^{2} \cos\tau \sin 2(g\tau + \zeta).
\end{alignat*}

Dropping the recent use of~$X, Y$ in connection with the forces and using
as before $X = x + \delta x$, $Y = y + \delta y$ we have
\begin{alignat*}{3}
X &= a_{0}\cos\tau(1 - \tfrac{1}{4}k^{2})
  &&- \tfrac{1}{4} a_{0}k^{2} \cos\tau \cos 2(g\tau + \zeta)
  &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \sin 2(g\tau + \zeta), \\
%
Y &= a_{0}\sin\tau(1 - \tfrac{1}{4}k^{2})
  &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \cos 2(g\tau + \zeta)
  &&+ \tfrac{1}{4} a_{0}k^{2} \cos\tau \sin 2(g\tau + \zeta), \\
%
R^{2} &= \rlap{$X^{2} + Y^{2}
  = a_{0}^{2}(1 - \tfrac{1}{2}k^{2})
  - \tfrac{1}{2} a_{0}^{2}k^{2} \cos 2(g\tau + \zeta)$,}&&&& \\
%
R &= \rlap{$a_{0}\bigl[1 - \tfrac{1}{4}k^{2}
            - \tfrac{1}{4}k^{2} \cos 2(g\tau + \zeta)\bigr]$.}&&&&
\Tag{(48)}
\end{alignat*}

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

Again
\begin{alignat*}{2}
\cos\Theta &= \frac{X}{R}
  &&= \cos\tau - \tfrac{1}{4} k^{2} \sin\tau \sin 2(g\tau + \zeta), \\
%
\sin\Theta &= \frac{Y}{R}
  &&= \sin\tau + \tfrac{1}{4} k^{2} \cos\tau \sin 2(g\tau + \zeta), \\
%
\Theta - \tau
  &= \rlap{$\sin(\Theta - \tau) = \tfrac{1}{4} k^{2} \sin 2(g\tau + \zeta)$.}&&
\Tag{(49)}
\end{alignat*}

Hence we have as a term in the moon's longitude $\frac{1}{4}k^{2}\sin 2(g\tau + \zeta)$. Terms
\index{Reduction, the}%
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~$\Omega$. We have seen
that $\Omega$~vanishes when the solar eccentricity,~$e'$, is put equal to zero. We shall
only develop~$\Omega$ 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
\begin{align*}
r' &= \a' \bigl\{1 - e'\cos(n't - \varpi')\bigr\}, \\
v' &= n't + 2e'\sin(n't - \varpi').
\end{align*}

Hence
\begin{alignat*}{2}
x' &= r'\cos(v' - n't) = r' &&= \a' \bigl\{1 - e'\cos(n't - \varpi')\bigr\}, \\
y' &= r'\sin(v' - n't)      &&= 2\a'e' \sin(n't - \varpi').
\end{alignat*}
And
\begin{gather*}
n't = m\tau; \\
\begin{aligned}
\therefore \frac{xx' + yy'}{\a'}
  &= x - e'x \cos(m\tau - \varpi) + 2e'y \sin(m\tau - \varpi), \\
\left(\frac{xx' + yy'}{\a'}\right)^{2}
  &= x^{2} - 2e'x^{2} \cos(m\tau - \varpi) + 4e'xy \sin(m\tau - \varpi), \\
\frac{\a'^{5}}{r'^{5}}
  &= 1 + 5e' \cos(m\tau - \varpi),
\end{aligned}
\Allowbreak
\DPPageSep{108}{50}
\frac{3m^{2}}{2} \left\{\frac{\a'^{3}}{r'^{5}} (xx' + yy')^{2} - x^{2}\right\}
  = \frac{9m^{2}}{2} e' x^{2} \cos(m\tau - \varpi')
   + 6m^{2} e'xy \sin(m\tau - \varpi'), \\
%
\tfrac{1}{2} m^{2} (x^{2} + y^{2} + z^{2}) \left(1 - \frac{\a'^{3}}{r'^{3}}\right)
  = -\tfrac{3}{2} m^{2} (x^{2} + y^{2} + z^{2}) e' \cos(m\tau - \varpi'), \\
\Omega
  = m^{2} e' \bigl[3x^{2} \cos(m\tau - \varpi')
  + 6xy \sin(m\tau - \varpi') - \tfrac{3}{2} y^{2} \cos(m\tau - \varpi') \bigr],
\end{gather*}
for we neglect~$m^{2}z^{2}$ when multiplied by~$e'$,
\begin{align*}
\frac{\dd \Omega}{\dd x}
  &= 6m^{2}e' \bigl[x \cos(m\tau - \varpi') + y \sin(m\tau - \varpi')\bigr], \\
%
\frac{\dd \Omega}{\dd y}
  &= 6m^{2}e' \bigl[x \sin(m\tau - \varpi') - \tfrac{1}{2} y \cos(m\tau - \varpi')\bigr].
\end{align*}

It is sufficiently accurate for us to take
\begin{align*}
x &= a_{0} \cos \tau,\quad
y  = a_{0} \sin \tau, \\
\phi &= \tau;
\end{align*}
\begin{multline*}
\therefore
\frac{\dd \Omega}{\dd x} \cos\phi +
\frac{\dd \Omega}{\dd y} \sin\phi
  = 6m^{2} e' a_{0} \bigl[\cos^{2}\tau \cos(m\tau - \varpi')
    + \cos\tau \sin\tau \sin(m\tau - \varpi') \\
%
\shoveright{+ \cos\tau \sin\tau \sin(m\tau - \varpi')
    - \tfrac{1}{2} \sin^{2}\tau \cos(m\tau - \varpi') \bigr]} \\
%
\shoveleft{= 3m^{2} e'a_{0} \bigl[\cos(m\tau - \varpi')
    + \cos 2\tau \cos(m\tau - \varpi') + 2\sin 2\tau \sin(m\tau - \varpi') \bigr]} \\
%
\shoveright{- \tfrac{1}{2} \cos(m\tau - \varpi') + \tfrac{1}{2} \cos2\tau \cos(m\tau - \varpi')} \\
%
\shoveleft{= 3m^{2} e' a_{0} \bigl[\tfrac{1}{2} \cos(m\tau - \varpi')
  + \tfrac{3}{4} \cos \bigl\{(2 + m)\tau - \varpi' \bigr\}
  + \tfrac{3}{4} \cos \bigl\{(2 - m)\tau + \varpi' \bigr\}} \\
%
\shoveright{+ \cos \bigl\{(2 - m)\tau + \varpi' \bigr\}
            - \cos \bigl\{(2 + m)\tau - \varpi' \bigr\} \bigr]} \\
%
\shoveleft{= \tfrac{3}{2} m^{2} e'a_{0} \bigl[\cos(m\tau - \varpi')
  - \tfrac{1}{2} \cos \bigl\{(2 + m)\tau - \varpi' \bigr\}
  + \tfrac{7}{2} \cos \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr].}
\end{multline*}

Again
\begin{multline*}
\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} +
\frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}
  = 6m^{2} e'a_{0} \bigl[-\sin\tau \cos\tau \cos(m\tau - \varpi')
    - \sin^{2} \tau \sin(m\tau - \varpi') \\
%
\shoveright{+ \cos^{2} \tau \sin(m\tau - \varpi')
  - \tfrac{1}{2} \sin\tau \cos\tau \cos(m\tau - \varpi') \bigr]} \\
%
\shoveleft{= 3m^{2} e'a_{0} \bigl[-\tfrac{3}{2} \sin 2\tau \cos(m\tau - \varpi')
  + 2 \cos 2\tau \sin(m\tau - \varpi') \bigr]} \\
%
\shoveleft{= \tfrac{3}{2} m^{2} e'a_{0} \bigl[-\tfrac{3}{2} \sin \bigl\{(2 + m)\tau - \varpi' \bigr\}
  - \tfrac{3}{2} \sin \bigl\{(2 - m)\tau + \varpi' \bigr\}} \\
%
\shoveright{+ 2\sin \bigl\{(2 + m)\tau - \varpi' \bigr\}
            - 2\sin \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr]} \\
%
\shoveleft{= \tfrac{3}{2} m^{2} e'a_{0} \bigl[\tfrac{1}{2} \sin \bigl\{(2 + m)\tau - \varpi' \bigr\}
  - \tfrac{7}{2} \sin \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr],} \\
%
\shoveleft{2 \int \left(\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau}
                      + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}\right) d\tau
  = -\tfrac{3}{2} m^{2} e'a_{0} \bigl[\tfrac{1}{2} \cos \bigl\{(2 + m)\tau - \varpi' \bigr\}} \\
%
\shoveright{- \tfrac{7}{2} \cos \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr];} \\
%
\shoveleft{\therefore
    \frac{\dd \Omega}{\dd x} \cos\phi
  + \frac{\dd \Omega}{\dd y} \sin\phi
  + 2 \int \left(\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau}
               + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}\right) d\tau
  = \tfrac{3}{2} m^{2} e'a_{0} \bigl[\cos(m\tau - \varpi')} \\
%
  -  \cos \bigl\{(2 + m)\tau - \varpi' \bigr\}
  + 7\cos \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr].
\end{multline*}
\DPPageSep{109}{51}

Hence to the order required
\begin{multline*}
\frac{d^{2}\, \delta p}{d\tau^{2}} + (1 + 2m)\, \delta p = \tfrac{3}{2} m^{2} e'a_{0}
\bigl[
\cos(m\tau - \varpi') - \cos \left\{(2 + m) \tau - \varpi'\right\} \\
  + 7 \cos \left\{(2 - m)\tau + \varpi'\right\}\bigr],
\end{multline*}
\[
\begin{aligned}
\delta p &= \tfrac{3}{2} m^{2} e'a_{0}
  \left[\frac{\cos(m\tau - \varpi')}{-m^{2} + 1 + 2m}
    - \frac{ \cos\left\{(2 + m)\tau - \varpi'\right\}}{-(4 + 4m) + 1 + 2m}
    + \frac{7\cos\left\{(2 - m)\tau + \varpi'\right\}}{-(4 - 4m) + 1 + 2m}\right] \\
%
  &= \tfrac{3}{2} m^{2} e'a_{0}
  \left[\cos(m\tau - \varpi')
    + \tfrac{1}{3} \cos \left\{(2 + m)\tau - \varpi'\right\}
    - \tfrac{7}{3} \cos \left\{(2 - m)\tau + \varpi'\right\}\right]\Add{,}
\end{aligned}
\Tag{(50)}
\]
{\setlength{\abovedisplayskip}{0pt}%
\setlength{\belowdisplayskip}{0pt}%
\begin{multline*}
\frac{d\, \delta s}{d\tau}
  = -2\delta p\, (1 + m)
  + \frac{1}{V}\int \left(\frac{d\Omega}{dx}\, \frac{dx}{d\tau}
                         +\frac{d\Omega}{dy}\, \frac{dy}{d\tau}\right) d\tau \\
%
\shoveleft{= -3m^{2} e'a_{0} \left[\cos(m\tau - \varpi')
    + \tfrac{1}{3}\cos\left\{(2 + m)\tau - \varpi'\right\}
    - \tfrac{7}{3}\cos\left\{(2 - m)\tau + \varpi'\right\}\right]} \\
%
\shoveright{- \tfrac{3}{4} m^{2} e'\left[\tfrac{1}{2} \cos\left\{(2 + m)\tau - \varpi'\right\}
   - \tfrac{7}{2} \cos \left\{(2 - m)\tau + \varpi'\right\}\right]} \\
%
\shoveleft{= -3m^{2}e'a_0 \bigl[\cos(m\tau - \varpi')
   + \tfrac{11}{24} \cos\left\{(2 + m)\tau - \varpi'\right\}} \\
%
\shoveright{-\tfrac{77}{24} \cos\left\{(2 - m)\tau + \varpi'\right\}\bigr];} \\
\end{multline*}
\begin{multline*}
\therefore \delta s = - 3m e'a_{0} \sin(m\tau - \varpi')
  - 3m^{2} e'a_{0} \bigl[\tfrac{11}{48} \sin \left\{(2 + m) \tau - \varpi'\right\} \\
  - \tfrac{77}{48} \sin\left\{(2 - m)\tau + \varpi'\right\}\bigr]\Add{.}
\Tag{(51)}
\end{multline*}}

Hence to order~$m e'$, to which order only our result is correct,
\[
\delta p = 0, \quad
\delta s = -3m e'a_{0} \sin (m\tau - \varpi').
\]

And following our usual method for obtaining new terms in radius vector
and longitude
\begin{align*}
\delta x &= \delta p \cos \phi - \delta s \sin \phi, \quad
\delta y  = \delta p \sin \phi + \delta s \cos \phi, \\
\delta x &=
%[** TN: Hack to align second equation with previous second equation]
  \settowidth{\TmpLen}{$\delta p \cos \phi - \delta s \sin \phi$,\quad}
  \makebox[\TmpLen][l]{$- \delta s \sin \tau$,}\,
\delta y = \delta s \cos \tau, \\
X &= a_{0} \left[\cos \tau + 3m e' \sin \tau \sin (m\tau - \varpi')\right], \\
Y &= a_{0} \left[\sin \tau - 3m e' \cos \tau \sin (m\tau - \varpi')\right], \\
R^{2} &= a_{0}^{2} \left[1 + 3m e' \sin 2\tau \sin (m\tau - \varpi')
  - 3m e' \sin 2\tau \sin (m\tau - \varpi')\right] = a_{0}^{2}, \\
\Tag{(52)}
\end{align*}
and to the order required there is no term in radius vector
\begin{align*}
\cos \Theta &= \cos \tau + 3m e' \sin \tau \sin (m\tau - \varpi'),\\
\sin \Theta &= \sin \tau - 3m e' \cos \tau \sin (m\tau - \varpi'),\\
\sin (\Theta - \tau) &= - 3m e' \sin (m\tau - \varpi'),\\
\Theta &= \tau - 3m e' \sin(m\tau - \varpi').
\Tag{(53)}
\end{align*}

The new term in the longitude is~$-3m e' \sin (l' - \varpi')$. This term is called
the annual equation.
\index{Annual Equation}%
\index{Equation, annual}%
\DPPageSep{110}{52}

\Section{§ 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
\begin{align*}
v &= \settowidth{\TmpLen}{longitude}%
  \UnderNote{\makebox[\TmpLen][c]{$l$}}{%
  \parbox[c]{\TmpLen}{\centering(mean\\ longitude\\ ${}= nt + \epsilon$)}}
  + \UnderNote{2e \sin (l - \varpi)}{%
  \settowidth{\TmpLen}{equation to}%
  \parbox[c]{\TmpLen}{\centering equation to\\ the centre}}
  + \UnderNote{\tfrac{15}{4} m e \sin(l - 2l' + \varpi)}{evection}
  + \UnderNote{\tfrac{11}{8} m^2 \sin2(l - l')}{variation} \\
%
  &\qquad\qquad\qquad
  \UnderNote{-\tfrac{1}{4} k^{2} \sin 2(l - \Omega)}{reduction}
  - \UnderNote{3m e' \sin(l' - \varpi')}{annual equation}, \\
%
s &= k \sin(l - \Omega)
  + \UnderNote{\tfrac{3}{8} m k \sin(l - 2l' + \Omega)}{evection in latitude}.
\end{align*}

For~$R$, the projection of the radius vector on the ecliptic, we get
\begin{multline*}
R = \a\bigl[1 - \tfrac{1}{6} m^{2} - \tfrac{1}{4} k^{2}
  - \UnderNote{e \cos(l - \varpi)}{%
      \settowidth{\TmpLen}{equation to the}%
      \parbox[c]{\TmpLen}{\centering equation to the\\ centre}}
  - \UnderNote{\tfrac{15}{8} m e \cos(l - 2l' - \varpi)}{evection}
  - \UnderNote{m^{2} \cos 2(l - l')}{variation} \\
%
  + \UnderNote{\tfrac{1}{4} k^{2} \cos 2(l - \Omega)}{reduction}\bigr].
\Tag{(54)}
\end{multline*}

To get the actual radius vector we require to multiply by~$\sec\beta$, i.e.~by
\[
1 + \tfrac{1}{2} k^{2} \sin^{2}(l - \Omega) \text{ or }
1 + \tfrac{1}{4} k^{2} - \tfrac{1}{4} k^{2} \cos 2(l - \Omega).
\]

This amounts to removing the terms $-\frac{1}{4}k^{2} + \frac{1}{4}k^{2}\cos2(l - \Omega)$. The radius
vector then is
\[
\a \bigl[1 - \tfrac{1}{6} m^{2} - e \cos(l - \varpi)
  - \tfrac{15}{8} m e \cos(l - 2l' + \varpi) - m^{2} \cos2(l - l')\bigr].
\]

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
$m = 0$ 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
\[
\frac{1}{\a}\bigl[1 + \tfrac{1}{6} m^{2} + e \cos(l - \varpi)
  + \tfrac{15}{8} m e \cos(l - 2l' + \varpi) + m^{2}\cos2(l - l')\bigr].
\]

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.
\DPPageSep{111}{53}

\Note{1.}{On the Infinite Determinant of \SecRef{5}.}
\index{Hill, G. W., Lunar Theory!infinite determinant}%
\index{Infinite determinant, Hill's}%

We assume (as has been justified by Poincaré) 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 $c_{0}$~is a root $-c_{0}$~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 $c_{0}$~is a root, $± c_{0} ± 2j$~are also roots.

But these are the roots of $\cos \pi c = \cos \pi c_{0}$.

Therefore the determinant must be equal to
\[
k(\cos \pi c - \cos \pi c_{0}).
\]

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~$\Theta$'s. Let us put $\Theta_{1} = \Theta_{2} = \Theta_{3} = \dots = 0$.

The determinant then becomes equal to the product of the diagonal terms
and the equation is
\[
\dots \bigl[\Theta_{0} - (c - 2)^{2}\bigr]
      \bigl[\Theta_{0} - c^{2}\bigr]
      \bigl[\Theta_{0} - (c + 2)^{2}\bigr] \dots = 0.
\]

$c_{0} = ±\Surd{\Theta_{0}}$ is one pair of roots, and all the others are given by~$c_{0} ± 2i$.

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 multipliers
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~$-c^{2(2i + 1)}$. We wish to associate
the infinite determinant with~$\cos \pi c$. Now
\[
\cos \pi c
  = \left(1 - \frac{4c^{2}}{1}\right)
    \left(1 - \frac{4c^{2}}{9}\right)
    \left(1 - \frac{4c^{2}}{25}\right) \dots.
\]
\DPPageSep{112}{54}

The first $2i + 1$~terms of this product may be written
\[
\left(1 - \frac{2c}{4i + 1}\right)
\left(1 - \frac{2c}{4i - 1}\right) \dots
\left(1 + \frac{2c}{4i - 1}\right)
\left(1 + \frac{2c}{4i + 1}\right),
\]
and the highest power of~$c$ in this product is
\[
\frac{4c^{2}}{(4i)^{2} - 1} · \frac{4c^{2}}{\bigl\{4(i - 1)\bigr\}^{2} - 1} \dots \frac{4c^{2}}{(4i)^{2} - 1}.
\]

Hence we multiply the $i$th~row below or above the central row by~$\dfrac{-4}{(4i)^{2} - 1}$.
The $i$th~diagonal term below the central term will now be~$\dfrac{4\bigl[(2i + c)^{2} - \Theta_{0}\bigr]}{(4i)^{2} - 1}$
and will be denoted by~$\{i\}$. It clearly tends to unity as $i$~tends to infinity by
positive or negative values. The $i$th~row below the central row will now
read
\[
\dots
\frac{-4\Theta_{2}}{(4i)^{2} - 1},\quad
\frac{-4\Theta_{1}}{(4i)^{2} - 1},\quad \{i\},\quad
\frac{-4\Theta_{1}}{(4i)^{2} - 1},\quad
\frac{-4\Theta_{2}}{(4i)^{2} - 1},\dots.
\]

The new determinant which we will denote by~$\nabla (c)$ has the same roots
as the original one and so we may write
\[
\nabla (c) = k' \{\cos \pi c - \cos \pi c_{0}\},
\]
where $k'$~is a new numerical constant. But it is easy to see that~$k' = 1$.
This was the object of introducing the multipliers and that it is true is easily
proved by taking the case of $\Theta_{1} = \Theta_{2} = \dots = 0$ and $\Theta_{0} = \frac{1}{4}$, in which case the
determinant reduces to~$\cos \pi c$. We thus have the equation
\[
\nabla (c) = \cos \pi c - \cos \pi c_{0},
\]
which can be considered as an identity in~$c$.

Putting $c = 0$ we get
\[
\nabla (0) = 1 - \cos \pi c_{0}.
\]

$\nabla (0)$~depends only on the~$\Theta$'s; written so as to shew the principal elements
it is
\[
\left\lvert
\begin{array}{@{}c *{5}{r} c@{}}
\multicolumn{7}{c}{\dotfill} \\
\dots & \tfrac{4}{63}(16-\Theta_{0}),&    -\tfrac{4}{63}\Theta_{1},& -\tfrac{4}{63}\Theta_{2},&    -\tfrac{4}{63}\Theta_{3},&     -\tfrac{4}{63}\Theta_{4},& \dots \\
\dots &     -\tfrac{4}{15}\Theta_{1},& \tfrac{4}{15}(4-\Theta_{0}),& -\tfrac{4}{15}\Theta_{1},&    -\tfrac{4}{15}\Theta_{2},&     -\tfrac{4}{15}\Theta_{3},& \dots \\
\dots &                  4\Theta_{2},&                 4\Theta_{1},&              4\Theta_{0},&                 4\Theta_{1},&                  4\Theta_{2},& \dots \\
\dots &     -\tfrac{4}{15}\Theta_{3},&    -\tfrac{4}{15}\Theta_{2},& -\tfrac{4}{15}\Theta_{1},& \tfrac{4}{15}(4-\Theta_{0}),&     -\tfrac{4}{15}\Theta_{1},& \dots \\
\dots &     -\tfrac{4}{63}\Theta_{4},&    -\tfrac{4}{63}\Theta_{3},& -\tfrac{4}{63}\Theta_{2},&    -\tfrac{4}{63}\Theta_{1},& \tfrac{4}{63}(16-\Theta_{0}),& \dots \\
\multicolumn{7}{c}{\dotfill}
\end{array}
\right\rvert
\]

{\stretchyspace
If $\Theta_{1}, \Theta_{2}$,~etc.\ vanish, the solution of the differential equation is $\cos(\Surd{\Theta_{0}} + \epsilon)$
or~$c = \Surd{\Theta_{0}}$. But in this case the determinant has only diagonal terms and
the product of the diagonal terms of~$\nabla (0)$ is~$1 - \cos \pi \Surd{\Theta_{0}}$ or~$2 \sin^{2} \frac{1}{2}\pi\Surd{\Theta_{0}}$.}
\DPPageSep{113}{55}

Hence we may divide each row by its diagonal member and put
$2 \sin^{2} \frac{1}{2} \pi \Surd{\Theta_{0}}$ outside.

If therefore
{\small
\begin{align*}
\Delta(0) &= \left\lvert
\begin{array}{@{}c *{5}{>{\ }c@{,\ }} c}
\multicolumn{7}{c}{\dotfill} \\
\dots &  1                                & -\dfrac{\Theta_{1}}{16-\Theta_{0}}& -\dfrac{\Theta_{2}}{16-\Theta_{0}}& -\dfrac{\Theta_{3}}{16-\Theta_{0}}& -\dfrac{\Theta_{4}}{16-\Theta_{0}} & \dots \\
\dots & -\dfrac{\Theta_{1}}{4-\Theta_{0}} &  1                                & -\dfrac{\Theta_{1}}{4-\Theta_{0}} & -\dfrac{\Theta_{2}}{4-\Theta_{0}} & -\dfrac{\Theta_{3}}{4-\Theta_{0}}  & \dots \\
\dots &  \dfrac{\Theta_{2}}{\Theta_{0}}   &  \dfrac{\Theta_{1}}{\Theta_{0}}   &  1                                &  \dfrac{\Theta_{1}}{\Theta_{0}}   &  \dfrac{\Theta_{2}}{\Theta_{0}}    & \dots \\
\dots & -\dfrac{\Theta_{3}}{4-\Theta_{0}} & -\dfrac{\Theta_{2}}{4-\Theta_{0}} & -\dfrac{\Theta_{1}}{4-\Theta_{0}} &  1                                & -\dfrac{\Theta_{1}}{4-\Theta_{0}}  & \dots \\
\multicolumn{7}{c}{\dotfill}
\end{array}
\right\rvert
\\
\nabla(0) &= 2 \sin^{2} \tfrac{1}{2} \pi\Surd{\Theta_{0}} \Delta(0).
\end{align*}}

Now since
\[
\cos \pi c_{0} = 1 - \nabla (0)
  = 1 - 2 \sin^{2} \frac{1}{2} \pi \Surd{\Theta_{0}} \Delta(0),
\]
we have
\Pagelabel{55}
\[
\frac{\sin^{2} \frac{1}{2} \pi c_{0}}{\sin^{2} \frac{1}{2} \pi \Surd{\Theta_{0}}}
  = \Delta(0),
\]
an equation to be solved for~$c_{0}$ (or~$c$).

Clearly for stability $\Delta(0)$~must be positive and $\Delta(0) < \cosec^2 \frac{1}{2} \pi \Surd{\Theta_{0}}$.
Hill gives other transformations.

\Note{2\footnotemark.}{On the periodicity of the integrals of the equation
\footnotetext{This treatment of the subject was pointed out to Sir~George Darwin by Mr~S.~S. Hough.}
\begin{gather*}
\frac{d^{2}\, \delta p}{d\tau^{2}} + \Theta\, \delta p = 0, \\
\lintertext{where}
{\Theta = \Theta_{0}
  + \Theta_{1} \cos 2\tau
  + \Theta_{2} \cos 4\tau + \dots.}
\end{gather*}}
\index{Differential Equation, Hill's!periodicity of integrals of}%
\index{Hill, G. W., Lunar Theory!periodicity of integrals of}%
\index{Periodicity of integrals of Hill's Differential Equation}%

Since the equation remains unchanged when $\tau$ becomes~$\tau + \pi$, it follows
that if $\delta p = F(\tau)$ is a solution $F(\tau + \pi)$ is also a solution.

Let $\phi(\tau)$~be a solution subject to the conditions that when
\[
\tau=0,\quad
\delta p = 1,\quad
\frac{d\, \delta p}{d\tau} = 0; \text{ i.e.\ } \phi(0) = 1,\quad
\phi'(0) = 0.
\]

Let $\psi(\tau)$~be a second solution subject to the conditions that when
\[
\tau=0,\quad
\delta p = 0,\quad
\frac{d\, \delta p}{d\tau} = 1; \text{ i.e.\ } \psi(0) = 0,\quad
\psi'(0) = 1.
\]
\DPPageSep{114}{56}

It is clear that $\phi(\tau)$ is an even function of~$\tau$, and $\psi(\tau)$~an odd one, so
that
\begin{alignat*}{2}
\phi (-\tau) &= \Neg\phi(\tau),&\qquad \psi(-\tau)&= -\psi(\tau),\\
\phi'(-\tau) &= -\phi(\tau),&\qquad \psi'(-\tau)&= \Neg\psi(\tau).
\end{alignat*}
Then the general solution of the equation is
\[
\delta p = F(\tau) = A\phi(\tau) + B\psi(\tau),
\]
where $A$~and~$B$ are two arbitrary constants.

Since $\phi(\tau + \pi)$, $\psi(\tau + \pi)$ are also solutions of the equation, it follows
that
\[
\left.
\begin{aligned}
\phi(\tau + \pi) &= \alpha\phi(\tau) + \beta \psi(\tau)\Add{,} \\
\psi(\tau + \pi) &= \gamma\phi(\tau) + \delta\psi(\tau)\Add{,}
\end{aligned}
\right\}
\Tag{(55)}
\]
where $\alpha, \beta, \gamma, \delta$ are definite constants.

If possible let $A : B$ be so chosen that
\[
F(\tau + \pi) = \nu F(\tau),
\]
where $\nu$~is a numerical constant.

When we substitute for~$F$ its values in terms of $\phi$~and~$\psi$, we obtain
\[
A\phi(\tau + \pi) + B\psi(\tau + \pi) = \nu\bigl[A\phi(\tau) + B\psi(\tau)\bigr].
\]

Further, substituting for $\phi(\tau + \pi)$, $\psi(\tau + \pi)$ their values, we have
\[
A\bigl[\alpha\phi(\tau) + \beta \psi(\tau)\bigr] +
B\bigl[\gamma\phi(\tau) + \delta\psi(\tau)\bigr]
  = \nu\bigl[A\phi(\tau) + B\psi(\tau)\bigr],
\]
whence
\[
\bigl[A(\alpha - \nu) + B\gamma\bigr] \phi(\tau)
  + \bigl[A\beta + B(\delta - \nu)\bigr] \psi(\tau) = 0.
\]

Since this is satisfied for all values of~$\tau$,
\begin{align*}
A(\alpha - \nu) + B\gamma &= 0,\\
A\beta + B(\delta - \nu) &= 0,\\
\therefore(\alpha - \nu)(\delta - \nu) - \beta\gamma &= 0,\\
\text{i.e.}\quad
\nu^{2} - (\alpha + \delta)\nu + \alpha\delta - \beta\gamma &= 0,
\end{align*}
an equation for $\nu$ in terms of the constants $\alpha, \beta, \gamma, \delta$. This equation can be
simplified.

Since
\[
\frac{d^{2}\phi}{d\tau^{2}} + \Theta\phi = 0,\qquad
\frac{d^{2}\psi}{d\tau^{2}} + \Theta\psi = 0,
\]
we have
\[
\phi \frac{d^{2}\psi}{d\tau^{2}} - \psi \frac{d^{2}\phi}{d\tau^{2}} = 0.
\]
On integration of which
\[
\phi\psi' - \psi\phi' = \text{const.}
\]
But
\[
\phi(0) = 1,\quad
\psi'(0) = 1,\quad
\psi(0) = 0,\quad
\phi'(0) = 0.
\]

Therefore the constant is unity; and
\[
\phi(\tau)\psi'(\tau) - \psi(\tau)\phi'(\tau) = 1.
\Tag{(56)}
\]
\DPPageSep{115}{57}
But putting $\tau = 0$ in the equations~\Eqref{(55)}, and in the equations obtained by
differentiating them,
\begin{alignat*}{3}
\phi(\pi) &= \alpha\,\phi\,(0) &&+ \beta\,\psi(0) &&= \alpha,\\
\psi(\pi) &= \gamma\,\phi\,(0) &&+ \delta\,\psi\,(0) &&= \gamma,\\
\phi'(\pi) &= \alpha\phi'(0) &&+ \beta\psi'(0) &&= \beta,\\
\psi'(\pi) &= \gamma\phi'(0) &&+ \delta\,\psi'(0) &&= \delta.
\end{alignat*}

Therefore by~\Eqref{(56)},
\[
\alpha\delta - \beta\gamma = 1.
\]
Accordingly our equation for~$\nu$ is
\[
\nu^{2} - (\alpha + \delta)\nu + 1 = 0
\]
or
\[
\tfrac{1}{2} \left(\nu + \frac{1}{\nu)}\right) = \tfrac{1}{2} (\alpha + \delta).
\]

If now we put $\tau = -\frac{1}{2}\pi$ in~\Eqref{(55)} and the equations obtained by
differentiating them,
\begin{align*}
&\begin{alignedat}{4}
\phi(\tfrac{1}{2}\pi)
   &= \alpha\phi(-\tfrac{1}{2}\pi) &&+ \beta\psi(-\tfrac{1}{2}\pi)
   &&= \Neg\alpha\phi(\tfrac{1}{2}\pi) &&- \beta\psi(\tfrac{1}{2}\pi), \\
%
\psi(\tfrac{1}{2}\pi)
   &= \gamma\phi(-\tfrac{1}{2}\pi) &&+ \delta\psi(-\tfrac{1}{2}\pi)
   &&= \Neg\gamma\phi(\tfrac{1}{2}\pi) &&- \delta\psi(\tfrac{1}{2}\pi), \\
%
\phi'(\tfrac{1}{2}\pi)
   &= \alpha\phi'(-\tfrac{1}{2}\pi) &&+ \beta\psi'(-\tfrac{1}{2}\pi)
   &&= -\alpha\phi'(\tfrac{1}{2}\pi) &&+ \beta\psi'(\tfrac{1}{2}\pi), \\
%
\psi'(\tfrac{1}{2}\pi)
   &= \gamma\phi'(-\tfrac{1}{2}\pi) &&+ \delta\psi'(-\tfrac{1}{2}\pi)
   &&= -\gamma\phi'(\tfrac{1}{2}\pi) &&+ \delta\psi'(\tfrac{1}{2}\pi),\\
\end{alignedat}
\Allowbreak
&\frac{\phi(\tfrac{1}{2}\pi)}{\psi(\tfrac{1}{2}\pi)}
  = \frac{\beta}{\alpha - 1}
  = \frac{\delta + 1}{\gamma},\quad
\frac{\psi'(\tfrac{1}{2}\pi)}{\phi'(\tfrac{1}{2}\pi)}
  = \frac{\alpha + 1}{\beta}
  = \frac{\gamma}{\delta - 1}, \\
%
&\frac{\phi(\tfrac{1}{2}\pi) \psi'(\tfrac{1}{2}\pi)}
      {\psi(\tfrac{1}{2}\pi) \phi'(\tfrac{1}{2}\pi)}
  = \frac{\alpha + 1}{\alpha - 1} = \frac{\delta + 1 }{\delta - 1}.
\end{align*}

But since $\phi(\frac{1}{2}\pi)\psi'(\frac{1}{2}\pi) - \phi'(\frac{1}{2}\pi)\psi(\frac{1}{2}\pi) = 1$ we have
\[
\alpha = \delta = \tfrac{1}{2}(\alpha + \delta)
  = \phi (\tfrac{1}{2}\pi) \psi'(\tfrac{1}{2}\pi)
  + \phi'(\tfrac{1}{2}\pi) \psi (\tfrac{1}{2}\pi).
\]

Hence the equation for~$\nu$ may be written in five different forms, viz.\
\begin{align*}
\tfrac{1}{2}\left(\nu + \frac{1}{\nu}\right)
  &= \phi(\pi) = \psi'(\pi)
   = \phi (\tfrac{1}{2}\pi)\psi'(\tfrac{1}{2}\pi)
   + \phi'(\tfrac{1}{2}\pi)\psi (\tfrac{1}{2}\pi) \\
  &= 1 + 2\phi'(\tfrac{1}{2}\pi)\psi(\tfrac{1}{2}\pi)
   = 2\phi(\tfrac{1}{2}\pi) \psi'(\tfrac{1}{2}\pi) - 1.
\Tag{(57)}
\end{align*}

It remains to determine the meaning of~$\nu$ in terms of the~$c$ introduced in
the solution by means of the infinite determinant.

The former solution was
\[
\delta p = \sum_{-\infty}^{+\infty}
  \bigl\{A_{j} \cos(c + 2j)\tau + B_{j} \sin(c + 2j)\tau\bigr\},
\]
where
\[
A_{j} : B_{j} \text{ as } -\cos\epsilon : \sin\epsilon.
\]
In the solution $\phi(\tau)$ we have $\phi(0) = 1$, $\phi'(0) = 0$, and $\phi(\tau)$~is an even
function of~$\tau$. Hence to get~$\phi(\tau)$ from~$\delta p$ we require to put $\sum A_{j} = 1$, and
$B_{j} = 0$ for all values of~$j$.
\DPPageSep{116}{58}

This gives
\begin{align*}
\phi(\pi) &= \sum \bigl\{A_{j} \cos(c + 2j)\pi\bigr\} \\
  &=\cos\pi c \sum A_{j} = \cos\pi c.
\end{align*}
Similarly we may shew that $\psi'(\pi) = \cos\pi c$.

It follows from equations~\Eqref{(57)} that
\begin{align*}
\cos\pi c &= \phi(\pi) = \psi'(\pi),\\
\cos^{2} \tfrac{1}{2}\pi c
  &=  \phi(\tfrac{1}{2}\pi)\psi'(\tfrac{1}{2}\pi);\quad
\sin^{2} \tfrac{1}{2}\pi c
   = -\phi'(\tfrac{1}{2}\pi)\psi(\tfrac{1}{2}\pi).
\end{align*}

We found on \Pageref{55} that $\sin^{2} \frac{1}{2}\pi c = \sin^{2} \frac{1}{2}\pi  \sqrt{\Theta_{0}} · \Delta(0)$, where $\Delta(0)$~is a
certain determinant. Hence the last solution being of this form, we have
the value of the determinant~$\Delta(0)$ in terms of $\phi$~and~$\psi$, viz.\
\[
\Delta(0) = - \frac{\phi'(\frac{1}{2}\pi)\psi(\frac{1}{2}\pi)}
                   {\sin^{2} \frac{1}{2}\pi \sqrt{\Theta_{0}}}.
\]

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 $\phi$~and~$\psi$.
\DPPageSep{117}{59}


\Chapter{On Librating Planets and on a New Family
of Periodic Orbits}
\SetRunningHeads{On Librating Planets}{and on a New Family of Periodic Orbits}

\Section{§ 1. }{Librating Planets.}
\index{Brown, Prof.\ E. W., Sir George Darwin's Scientific Work!new family of periodic orbits}%
\index{Librating planets}%
\index{Periodic orbits, Darwin begins papers on!new family of}%

\First{In} Professor Ernest Brown's interesting paper on ``A New Family of
Periodic Orbits'' (\textit{M.N.}, \textit{R.A.S.}, vol.~\Vol{LXXI.}, 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'' (\textit{Scientific Papers}, vol.~\Vol{IV.}, or \textit{Acta Math.}, vol.~\Vol{LI.}). ``Jove,''~J, of
mass~$1$, revolves at distance~$1$ about the ``Sun,''~S, of mass~$\nu$, and the orbital
angular velocity is~$n$, where~$n^{2} = \nu + 1$.

{\stretchyspace
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\DPnote{[** TN: Hyphenated in original]} are
respectively $r, \theta$ and $\rho, \psi$. The potential function for relative energy is~$\Omega$.}

The equations of motion and Jacobian integral, from which Brown
proceeds, are
\[
\left.
\begin{gathered}
\begin{aligned}
\frac{d^{2}r}{dt^{2}}
  - r \frac{d\theta}{dt} \left(\frac{d\theta}{dt} + 2n\right)
  &= \frac{\dd \Omega}{\dd r}\Add{,} \\
%
\frac{d}{dt} \left[r^{2} \left(\frac{d\theta}{dt} + n\right)\right]
  &= \frac{\dd \Omega}{\dd \theta}\Add{,} \\
%
\left(\frac{dr}{dt}\right)^{2}
  + \left(r \frac{d\theta}{dt}\right)^{2} &= 2\Omega - C\Add{,}
\end{aligned} \\
\lintertext{where}{2\Omega
  = \nu\left(r^{2} + \frac{2}{r}\right) + \left(\rho^{2} + \frac{2}{\rho}\right)\Add{,}}
\end{gathered}
\right\}
\Tag{(1)}
\]

The following are rigorous transformations derived from those equations,
virtually given by Brown in approximate forms in equation~(13), and at the
foot of p.~443:---
\DPPageSep{118}{60}
\begin{align*}
\left(\frac{d\theta}{dt} + n\right)^{2}
  &= A + \frac{1}{r}\, \frac{d^{2}r}{dt^{2}},
\Tag{(2)}
\Allowbreak
%
\frac{dr}{dt} \left(L + 3 \frac{d^{2}r}{dt^{2}}\right)
  &= B + D \left(\frac{d\theta}{dt} + n\right) - r \frac{d^{3}r}{dt^{3}},
\Tag{(3)}
\Allowbreak
%
\frac{d^{2}r}{dt^{2}} \left(L + 3 \frac{d^{2}r}{dt^{2}}\right)
  &= E \left(\frac{dr}{dt}\right)^{2}
   + F \frac{dr}{dt}\, \frac{d\theta}{dt}
   + G \left(\frac{d\theta}{dt}\right)^{2}
   + H \frac{dr}{dt} + J \frac{d\theta}{dt} + K \\
   &\qquad\qquad\qquad\qquad
   - 4 \frac{dr}{dt}\, \frac{d^{3}r}{dt^{3}} - r \frac{d^{4}r}{dt^{4}},
\Tag{(4)}
\end{align*}
where
\begin{align*}
A &= n^{2} - \frac{\dd \Omega}{r\, \dd r}
   = \frac{\nu}{r^{3}} + 1
     - \frac{1}{r} \left(\rho - \frac{1}{\rho^{2}}\right)\cos(\theta-\psi),
\Allowbreak
%
B &= -nr \frac{\dd^{2}\Omega}{\dd r\, \dd \theta}
   = -n \sin\psi \left[\left(\rho - \frac{1}{\rho^{2}}\right) + \frac{3r}{\rho^{3}} \cos(\theta - \psi)\right],
\Allowbreak
%
D &= r \frac{\dd^{2}\Omega}{\dd r\, \dd \theta} + 2 \frac{\dd \Omega}{\dd \theta}
   = 3  \sin\psi \left[\left(\rho - \frac{1}{\rho^{2}}\right) + \frac{r}{\rho^{3}} \cos(\theta - \psi)\right],
\Allowbreak
%
%[** TN: Added break]
L &= 4n^2r - r \frac{\dd^{2} \Omega}{\dd r^{2}} - 3 \frac{\dd \Omega}{\dd r} \\
  &= \frac{\nu}{r^{2}} + 3r + \frac{r}{\rho^{3}}
     - 3\left(\rho - \frac{1}{\rho^{2}}\right) \cos(\theta - \psi)
     - \frac{3r}{\rho^{3}} \cos^{2}(\theta - \psi),
\Allowbreak
%
E &= r \frac{\dd^{3} \Omega}{\dd r^{3}} + 4 \frac{\dd^{2} \Omega}{\dd r^{2}} - 4n^{2} \\
   &= \frac{2\nu}{r^{3}} + \frac{4}{\rho^{3}}\bigl[3 \cos^{2}(\theta - \psi) - 1\bigr] % \\
%
+ \frac{3r}{\rho^{4}} \cos(\theta - \psi) \bigl[3 - 5\cos^{2}(\theta - \psi) \bigr], \\
%
F &= 2r \frac{\dd^{3} \Omega}{\dd r^{2}\, \dd \theta}
   + 4\frac{\dd^{2} \Omega}{\dd r\, \dd \theta} - 4 \frac{\dd \Omega}{r\, \dd \theta}
   = \frac{6}{\rho^{4}} \sin\psi \bigl[5r \sin^{2}(\theta - \psi) - 4\cos\theta\bigr], \\
%
G &= r \frac{\dd^{3} \Omega}{\dd r\, \dd \theta^{2}} + 2\frac{\dd^{2} \Omega}{\dd \theta^{2}}
   = \frac{3r}{\rho} \left[\left(\rho - \frac{1}{\rho^{2}}\right)\cos\theta
     - \frac{r}{\rho^{3}} \sin\psi(5 \sin^{2}(\theta - \psi) - 1)\right],
\Allowbreak
%
H &= -\frac{4n}{r}\, \frac{\dd \Omega}{\dd \theta}
   = 4n\left(\rho - \frac{1}{\rho^{2}}\right) \sin(\theta - \psi), \\
%
J &= 2n \frac{\dd^{2} \Omega}{\dd \theta^{2}}
   = \frac{2nr}{\rho} \left[\left(\rho - \frac{1}{\rho^{2}}\right) \cos\theta
     - \frac{3}{\rho^{2}} \sin\psi \sin(\theta - \psi)\right], \\
%
K &= \frac{\dd \Omega}{r^{2}\, \dd \theta} \left(r\frac{\dd^{2} \Omega}{\dd r\, \dd \theta} + 2\frac{\dd \Omega}{\dd \theta}\right)
   = \frac{3}{r} \left(\rho - \frac{1}{\rho^{2}}\right)
     \sin\theta \sin\psi \left(1 + \frac{1}{\rho^{4}}\cos\psi\right).
\end{align*}
A great diversity of forms might be given to these functions, but the foregoing
seemed to be as convenient for computations as I could devise.

It is known that when $\nu$~is less than~$24.9599$\footnote
  {``Periodic Orbits,'' \textit{Scientific Papers}, vol.~\Vol{IV.}, p.~73.}
the vertex of the equilateral
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~$\nu$ there are no small closed periodic orbits of the kind
considered by Brown. But certain considerations led him to conjecture that
\DPPageSep{119}{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 $\nu$~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 $\nu$~equal to~$10$.

Brown's system of approximation, which he justifies for large values of~$\nu$,
may be described, as far as it is material for my present object, as follows:---

We begin the operation at any given point~$r, \theta$, such that $\rho$~is greater
than unity.

Then in \Eqref{(2)}~and~\Eqref{(3)} $\dfrac{d^{2}r}{dt^{2}}$ and $\dfrac{d^{3}r}{dt^{3}}$ are neglected, and we thence find
$\dfrac{dr}{dt}$,~$\dfrac{d\theta}{dt}$.

By means of these values of the first differentials, and neglecting $\dfrac{d^{3}r}{dt^{3}}$
and $\dfrac{d^{4}r}{dt^{4}}$ in~\Eqref{(4)}, we find~$\dfrac{d^{2}r}{dt^{2}}$ from~\Eqref{(4)}.

Returning to \Eqref{(2)}~and~\Eqref{(3)} and using this value of~$\dfrac{d^{2}r}{dt^{2}}$, we re-determine the
first differentials, and repeat the process until the final values of $\dfrac{dr}{dt}$ and $\dfrac{d\theta}{dt}$
remain unchanged. We thus obtain the velocity at this point~$r, \theta$ on the
supposition that $\dfrac{d^{3}r}{dt^{3}}$, $\dfrac{d^{4}r}{dt^{4}}$ are negligible, and on substitution in the last of~\Eqref{(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 $\nu$~is large. It does not concern me to
follow him here, since his process could scarcely be applicable for small values
of~$\nu$. 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~$\nu$.

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
\begin{alignat*}{2}
x_{0} &= -.36200,\quad& y_{0} &= .93441, \\
\intertext{which give}
r_{0} &= 1.00205,& \rho_0 &= 1.65173.
\end{alignat*}
\DPPageSep{120}{62}
The successive approximations to~$C$ were found to be
\[
33.6977,\quad 33.7285,\quad 33.7237,\quad 33.7246,\quad 33.7243.
\]
I therefore took the last value as that of~$C$, and found also that the direction
of motion was given by $\phi_{0} = 2°\,21'$. These values of $x_{0}, y_{0}, \phi_{0}$, and~$C$ then
furnish the values from which to begin the quadratures.

\FigRef[Fig.]{1} 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.
\begin{figure}[hbt!]
  \centering
  \Input{p062}
  \caption{Fig.~1. Results derived from Professor Brown's Method.}
  \Figlabel{1}
\end{figure}

From~A back to perijove and from~C on to~J the orbit was computed as
undisturbed by the Sun\footnotemark. Within the limits of accuracy adopted the body
\footnotetext{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 $t = 0$ when solar
  perturbation is to be neglected, be given by $x_{0}, y_{0}$ (or $r_{0}, \theta_{0}$, and $\rho_{0}, \psi_{0}$), $\phi_{0}, V_{0}$; and generally
  let the suffix~$0$ to any symbol denote its value at this epoch. Then the mean distance~$\a$, mean
  motion~$\mu$, and eccentricity~$e$ are found from
  \begin{gather*}
    \frac{1}{\a}
    = \frac{2}{\rho_{0}}
    - \bigl[V_{0}^{2} + 2\pi \rho_{0} V_{0} \cos(\phi_{0}
      - \psi_{0}) + n^{2} \rho_{0}^{2}\bigr],\quad
    \mu^{2} \a^{3} = 1, \\
    %
    \a (1 - e^{2})
    = \bigl[V_{0} \rho_{0} \cos(\phi_{0} - \psi_{0}) + n \rho_0^{2}\bigr]^{2}.
  \end{gather*}
  Let $t = \tau$ be the time of passage of perijove, so that when $\tau$~is positive perijove is later than the
  epoch $t = 0$.

  At any time~$t$ let $\rho, v, E$ be radius vector, true and eccentric anomalies; then
  \begin{align*}
    \rho  &= \a(1 - e \cos E), \\
    \rho^{\frac{1}{2}} \cos \tfrac{1}{2} v
      &= \a^{\frac{1}{2}}(1 - e)^{\frac{1}{2}}\cos \tfrac{1}{2} E, \\
%
    \rho^{\frac{1}{2}} \sin \tfrac{1}{2} v
      &= \a^{\frac{1}{2}}(1 + e)^{\frac{1}{2}}\sin \tfrac{1}{2} E, \\
%
    \mu(t - \tau) &= E - e \sin E, \\
    \psi &= \psi_{0}- v_{0} + v - nt.
  \end{align*}

  On putting $t = 0$, $E_{0}$~and~$\tau$ may be computed from these formulae, and it must be noted that
  when $\tau$~is positive $E_{0}$~and~$v_{0}$ are to be taken as negative.

  The position of the body as it passes perijove is clearly given by
  \[
  x - 1 = \a(1 - e)\cos(\psi_{0} - v_{0} - n\tau),\quad
  y = \a(1 - e)\sin(\psi_{0} - v_{0} - n\tau).
  \]
  Any other position is to be found by assuming a value for~$E$, computing $\rho, v, t, \psi$, and using the
  formulae
  \[
  x - 1 = \rho \cos\psi,\quad y = \rho \sin\psi.
  \]

  In order to find $V$~and~$\phi$ we require the formulae
  \[
  \frac{1}{\rho}\, \frac{d\rho}{dt} = \frac{\a e\sin E}{\rho} · \frac{\mu \a}{\rho};\quad
  \frac{dv}{dt} = \frac{\bigl[\a(1 - e^{2})\bigr]^{\frac{1}{2}}}{\rho} · \frac{\a^{\frac{1}{2}}}{\rho} · \frac{\mu \a}{\rho}, \\
  \]
  and
  \begin{align*}
    V\sin \phi
      &= -\frac{(x - 1)}{\rho}\, \frac{d\rho}{dt}
        + y\left(\frac{dv}{dt} - n\right), \\
    %
      V\cos \phi &= \Neg\frac{y}{\rho}\, \frac{d\rho}{dt}
        + (x - 1) \left(\frac{dv}{dt} - n\right).
  \end{align*}

  The value of~$V$ as computed from these should be compared with that derived from
  \[
  V^{2} = \nu\left(r^{2} + \frac{2}{r}\right)
        + \left(\rho^{2} + \frac{2}{\rho}\right) - C,
  \]
  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 $\psi$~will be measured in the positive direction.

  A similar investigation is applicable, \textit{mutatis mutandis}, when the body passes very close to~S\@.}%
collides with~J\@.
\DPPageSep{121}{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~$dy/dt$\DPnote{** slant fractions} will have as a divisor the
function
\[
Q = \left(4n^{2} - \frac{\dd^{2} \Omega}{\dd x^{2}}\right)
    \left(4n^{2} - \frac{\dd^{2} \Omega}{\dd y^{2}}\right)
  - \left(\frac{\dd^{2} \Omega}{\dd x\, \dd y}\right)^{2}.
\]
The method will then fail if~$Q$ vanishes or is small.
\DPPageSep{122}{64}

I find that if we write $\Gamma = \dfrac{\nu}{r^{3}} + \dfrac{1}{\rho^{3}}$, the divisor may be written in the
form
\[
Q = (3n^{2} + \Gamma)(3n^{2} - 2\Gamma) + \frac{\rho \nu}{r^{5}\rho^{5}} \sin\theta \sin\psi.
\]

Now, Mr~T.~H. Brown, Professor Brown's pupil, has traced one portion of
the curve $Q = 0$, corresponding to $\nu = 10$, 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.

\Section{§ 2. }{Variation of an Orbit.}
\index{Orbit, variation of an}%
\index{Variation, the!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 indeterminate
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 $\delta q\, V^{-\frac{1}{2}}$ along the outward normal and
$\delta s$~along the positive tangent, then we must have
\begin{gather*}
\frac{d^{2}\, \delta q}{ds^{2}} + \Psi\, \delta q = 0, \\
%
\frac{d}{ds}\left(\frac{\delta s}{V}\right)
  = -\frac{2\delta q}{V^{\frac{3}{2}}} \left(\frac{1}{R} + \frac{n}{V}\right),
\end{gather*}
where
\[
\Psi = \frac{5}{2} \left(\frac{1}{R} + \frac{n}{V}\right)^{2}
  - \frac{3}{2V^{2}} \left[\frac{\nu}{r^{3}}\cos^{2}(\phi - \theta)
    + \frac{1}{\rho^{3}}\cos^{2}(\phi - \psi)\right]
  + \frac{3}{4} \left(\frac{dV}{V\, ds}\right)^{2},
\]
and
\[
\frac{dV}{V\, ds}
  = \frac{\nu}{V^{2}} \left(\frac{1}{r^{2}} - r\right)\sin(\phi - \theta)
  + \frac{1}{V^{2}} \left(\frac{1}{\rho^{2}} - \rho\right) \sin(\phi - \psi).
\]
Also
\[
\delta \phi = -\frac{1}{V^{\frac{1}{2}}}
    \left[\frac{d\, \delta q}{ds}
      - \tfrac{1}{2}\, \delta q \left(\frac{dV}{V\, ds}\right)\right]
  + \frac{\delta s}{R}.
\]
\DPPageSep{123}{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 $\Psi$~may be computed for each of those
points.

At the point chosen as the starting-point the variation may be of two
kinds:---
\begin{alignat*}{2}
(1)\quad \delta q_0 &= \a, \qquad
   \frac{d\delta q_{0}}{ds} &&= 0, \text{ where $\a$ is a constant}, \\
%
(2)\quad \delta q_0 &=0, \qquad
   \frac{d\delta q_{0}}{ds} &&= b, \text{ where $b$ is a constant}.
\end{alignat*}
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 $\delta q$~and~$\delta s$ 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~$\Delta s$, and we may take four successive values of~$\Psi$, say $\Psi_{n-1},
\Psi_{n}, \Psi_{n+1}, \Psi_{n+2}$, as affording a sufficient representation of the march
of the function~$\Psi$ throughout the arc~$\Delta s$ between the points indicated by
$n$~to~$n+1$.

If the differential equation for~$\delta q$ be multiplied by~$(\Delta s)^{2}$, and if we
introduce a new independent variable~$z$ such that~$dz = ds/\Delta s$,\DPnote{** slant fractions} and write
$X = \Psi(\Delta s)^{2}$, the equation becomes
\[
\frac{d^{2}\, \delta q}{dz^{2}} = -X\, \delta q,
\]
and $z$~increases by unity as the arc increases by~$\Delta s$.

Suppose that the integration has been carried as far as the point~$n$, and
that $\delta q_{0}, d\, \delta q_{0}/dz$ are the values at that point; then it is required to find $\delta q_{1},
d\, \delta q_{1}/dz$ at the point~$n + 1$.

If the four adjacent values of~$X$ are $X_{-1}, X_{0}, X_{1}, X_{2}$, and if
\[
\delta_{1} = X_{1} - X_{0},\quad
\delta_{2} = \tfrac{1}{2} \bigl[(X_{2} - 2X_{1} + X_{0}) + (X_{1} - 2X_{0} + X_{-1})\bigr],
\]
Bessel's formula for the function~$X$ is
\[
X = X_{0} + (\delta_{1} - \tfrac{1}{2}\delta_{2})z
  + \tfrac{1}{2}\delta_{2}z^{2}\DPtypo{}{.}
\]
\DPPageSep{124}{66}
We now assume that throughout the arc $n$~to~$n + 1$,
\[
\delta q = \delta q_{0} + \frac{d\, \delta q_{0}}{dz} z
  + Q_{2} z^{2} + Q_{3} z^{3} + Q_{4} z^{4},
\]
where $Q_{2}, Q_{3}, Q_{4}$ have to be determined so as to satisfy the differential
equation.

On forming the product~$X\, \delta q$, integrating, and equating coefficients, we
find $Q_{2} = -\frac{1}{2} X_{0}\, \delta q_{0}$, and the values of~$Q_{3}, Q_{4}$ are easily found. In carrying out
this work I neglect all terms of the second order except~$X_{0}^{2}$.

\pagebreak[1]
The result may be arranged as follows:---\pagebreak[0] \\
Let
\begin{align*}
A &= 1 - \tfrac{1}{2} X_{0} - \tfrac{1}{6}\delta_{1}
   + \tfrac{1}{24} (\delta_{2} + X_{0}^{2}), \\
%
B &= 1 - \tfrac{1}{6} X_{0} - \tfrac{1}{12} \delta_{1} + \tfrac{1}{24} \delta_{2}, \\
%
C &= X_{0} + \tfrac{1}{2} \delta_{1} + \tfrac{1}{12} \delta_{2} - \tfrac{1}{6} (\delta_{2} + X_{0}^{2}), \\
%
D &= 1 - \tfrac{1}{2} X_{0} - \tfrac{1}{8} \delta_{1} + \tfrac{1}{6} \delta_{2};
\end{align*}
then, on putting $z =1$, we find
\begin{align*}
\delta q_{1} &= \Neg A\, \delta q_{0} + B \frac{d\, \delta q_{0}}{dz}, \\
\frac{d\, \delta q_{1}}{dz} &= -C\, \delta q_{0} + D \frac{d\, \delta q_{0}}{dz}.
\end{align*}

When the~$\Psi$'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 $\delta q$~and~$d\, \delta q/dz$ from point to
point both backwards and forwards. The necessary change of procedure when
$\Delta s$~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 $\delta q$~and~$d\, \delta q/dz$ have been found at each point, a further
integration has to be made to determine~$\delta s$, and this has, of course, to be done
for each of the solutions. Next, we have to find the normal displacement~$\delta p$
(equal to~$\delta q\, V^{-\frac{1}{2}}$), and, finally, $\delta p, \delta s$~have to be converted into rectangular
displacements~$\delta x, \delta y$.

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,
\DPPageSep{125}{67}
but it is also possible to vary~$C$. If $C$~becomes $C + \delta C$, it will be found that
the equations assume the form
\begin{align*}
\frac{d^{2}\, \delta q}{ds^{2}} + \Psi\, \delta q
  + \frac{\delta C}{V^{\frac{3}{2}}} \left(\frac{1}{R} + \frac{n}{V}\right) &= 0, \\
%
\frac{d}{ds}\left(\frac{\delta s}{V}\right)
  + \frac{2\delta q}{V^{\frac{3}{2}}} \left(\frac{1}{R} + \frac{n}{V}\right)
  + \frac{\delta C}{2V^{2}} &= 0.
\end{align*}

But this kind of variation cannot be used with much advantage, for
although it is possible to evaluate $\delta q$~and~$\delta s$ for specific initial values of~$\delta C,
\delta q, d\, \delta q/ds$ 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.

\Section{§ 3. }{A New Family of Periodic Orbits.}
\index{Periodic orbits, Darwin begins papers on!new family of}%

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~\SecRef{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~\SecRef{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 \FigRef{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 \FigRef{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~SJ\@. In the completed orbit positively described
\DPPageSep{126}{68}
half-circuits are interposed between negative loops described alternately on
the positive and negative sides of~SJ\@.

%[** TN: Moved up two paragraphs to accommodate pagination]
\begin{figure}[hbt!]
  \centering
  \Input{p068}
  \caption{Fig.~2. Orbits computed for the Case of $C = 33.25$.}
  \Figlabel{2}
\end{figure}
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 \FigRef{2}, from which this orbit may be constructed
with substantial accuracy.

In \FigRef{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
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 $x = .2$, $y = 1.3$. It became clear ultimately that the horizontal tangent
ought to have been chosen with a somewhat larger value for~$y$. From these
\DPPageSep{127}{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 \FigRef{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~$33.75$ shewn
in \FigRef{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
\begin{figure}[hbt!]
  \centering
  \Input{p069}
  \caption{Fig.~3. Three Periodic Orbits.}
  \Figlabel{3}
\end{figure}
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 \FigRef{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.
\DPPageSep{128}{70}

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'' (\textit{Scientific Papers}, vol.~\Vol{IV.}, p.~11, or \textit{Acta Math.}, vol.~\Vol{XXI.}
(1897)). Now, for $\nu = 10$ 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 $C$~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 \FigRef{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~$33.75$ 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~SJ\@.

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
\DPPageSep{129}{71}
which Lord Kelvin drew as an illustration of his graphical method of curve-tracing\footnotemark.
\footnotetext{\textit{Popular Lectures}, vol.~\Vol{I.}, 2nd~ed., p.~31; \textit{Phil.\ Mag.}, vol.~\Vol{XXXIV.}, 1892, p.~443.}%

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 \FigRef{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~$0°$
to a symmetrical one on the negative side Jove has turned through~$960°$ round
the Sun, that is to say, through $2\frac{2}{3}$~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
$C = 33.5$, the angle~$nT$ (where $T$~is the period) is~$1115°$ or $3.1$~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 $C = 33.5$ 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 remarkable
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,
\DPPageSep{130}{72}
before it ceases to circulate round the triangular point, is in fact a librating
planet. It may be conjectured that when the Sun's mass~$\nu$ is yet smaller
than~$10$, no such orbit as those traced is possible. When $\nu$~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 $\nu$~has reached~$24.9599$ a loop described an
infinite number of times must have become possible, since this is the smallest
value of~$\nu$ which permits oscillation about the triangular point. If this idea
is correct, and if $\mathrm{N}$~denotes the number expressing the multiplicity of the
loop, then as $\nu$~increases $d\mathrm{N}/d\nu$~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 Poincaré, 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.}

\Heading{$C = 33.5$.}

\noindent\begin{minipage}{\textwidth}
\centering\footnotesize
\settowidth{\TmpLen}{Time from perijove up to $s=-2.1$ is given by $nt=9°\, 25'$.}%
\parbox{\TmpLen}{Perijove $x_0=1.0171$, $y_0=-.0034$, taken as zero. \\
Time from perijove up to $s=-2.1$ is given by $nt=9°\, 25'$.}
\end{minipage}
\begin{longtable}{.{1,3}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} ,{6,2}!{\extracolsep{\fill}} .{2,3}}\\
\ColHead{s} & \ColHead{x} & \ColHead{y} & \ColHead{\Z\Z\Z\Z\phi} & \ColHead{2n/V}\\
      &         &         & \ColHead{\AngleHeading}  & \\
\endhead
-2.1  &+\Z.8282&+\Z.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,\Z3  &  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  &+\Z.2814&+\Z.5306&   {}+57, 56  &  2.613 \\
\DPPageSep{131}{73}
-1.3  &+\Z.1948&+\Z.5805&   {}+62,\Z8  &  2.728 \\
 1.2  &  .1049 &  .6243 &      65, 51  &  2.930 \\
 1.1  &+\Z.0126&  .6628 &      68, 38  &  3.251 \\
 1.0  &-\Z.0810&  .6979 &      69, 46  &  3.760 \\
  .9  &  .1747 &  .7330 &      68,\Z7  &  4.598 \\
  .85 &  .2207 &  .7526 &      65, 13  &  5.240 \\
  .8  &  .2653 &  .7754 &      60,\Z1  &  6.133 \\
  .75 &  .3068 &  .8035 &      50, 51  &  7.377 \\
  .725&  .3252 &  .8203 &      44,\Z2  &  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 &  {}+\Z0, 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,\Z4  &  6.961 \\
  .2  &-\Z.0565& 1.1626 &      76, 47  &  6.730 \\
-\Z.1 &+\Z.0421& 1.1791 &      83, 58  &  6.647 \\
  .0  &  .1419 & 1.1842 &   {}-90,\Z0  &  6.633 \\
+\Z.05&  .1919 & 1.1830 & 180°+87, 21  &  6.630 \\
  .1  &  .2418 & 1.1797 &      84, 54  &  6.626 \\
  .15 &  .2915 & 1.1742 &      82, 38  &  6.609 \\
  .2  &  .3410 & 1.1669 &      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,\Z8  &  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 &     \Z9, 38  &  4.326 \\
 1.35 & 1.1642 &  .4821 &180°+\Z3, 16  &  4.250 \\
 1.4  & 1.1641 &  .4322 &180°-\Z3, 40  &  4.141 \\
 1.45 & 1.1577 &  .3826 &      11,\Z5  &  3.983 \\
 1.5  & 1.1448 &  .3343 &      18, 44  &  3.758 \\
 1.55 & 1.1257 &  .2881 &      26,\Z8  &  3.460 \\
 1.6  & 1.1011 &  .2446 &      32, 39  &  3.100 \\
 1.65 & 1.0723 &  .2038 &      37, 33  &  2.701 \\
 1.7  & 1.0408 &  .1650 &      40,\Z4  &  2.291 \\
+1.75 &+1.0087 &+\Z.1267& 180°-39, 12  &  1.893 \\
\end{longtable}
\noindent\begin{minipage}{\textwidth}
\centering\footnotesize
\settowidth{\TmpLen}{Time from $s = 1.75$ to perijove given by $nt = 5°58'$.}%
\parbox{\TmpLen}{Time from $s = 1.75$ to perijove given by $nt = 5°58'$. \\
Coordinates of perijove $x = .9501$, $y = -.0029$.}
\end{minipage}
\DPPageSep{132}{74}

The following additional positions were calculated backwards from a perijove at
$x = .95$, $y = 0$, $\phi = 180°$.

\[
\begin{array}{.{1,4} c<{\qquad} .{1,4} c<{\qquad} ,{6,2}}
\ColHead{x} && \ColHead{y} && \ColHead{\Z\Z\Z\Z\phi} \\
            &&             && \ColHead{\AngleHeading} \\
+\Z.9500 && +.0000 && 180°+\Z0, \Z0 \\
   .9512 &&  .0531 && 180°- 22, 30 \\
   .9647 &&  .0797 &&       30, 52 \\
   .9756 &&  .0966 &&       34, 48 \\
   .9874 &&  .1127 &&       37, 37 \\
  1.0128 &&  .1436 &&       40, 37 \\
  1.0390 &&  .1738 &&       40, 56 \\
  1.0649 &&  .2043 &&       39, 12 \\
  1.0893 &&  .2360 &&       35, 51 \\
  1.1114 &&  .2693 &&       31, 16 \\
  1.1463 &&  .3412 &&       20, 10 \\
+ 1.1661 && +.4186 && 180°-\Z8, 40 \\
\end{array}
\]

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
$nT = 1115°.4 = 3.1$ revolutions of Jove.

\Heading{$C = 33.75$.}

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 $x_{0} = 1.0106$, $y_{0} = .0006$, which may be
taken as virtually on the line of syzygies. The motion from perijove is direct.

\begin{longtable}{.{1,3}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} ,{6,2}!{\extracolsep{\fill}} .{2,3}}\\
\ColHead{s} & \ColHead{x} & \ColHead{y} & \ColHead{\Z\Z\Z\Z\phi} & \ColHead{2n/V}\\
      &         &         & \ColHead{\AngleHeading}  & \\
\endhead
\ColHead{\text{perijove}}
        &+1.0106 &+\Z.0006&     \Z0, \Z0& \ColHead{\text{very nearly}}  \\
-\Z.3   &  .9652 &  .0403 &      66, 38 &  1.140 \\
-\Z.3   &  .9184 &  .0578 &      71, \Z6&  1.635 \\
-\Z.2   &  .8713 &  .0744 &      69, 27 &  2.075 \\
-\Z.2   &  .8251 &  .0936 &      65, \Z3&  2.447 \\
-\Z.1   &  .7391 &  .1444 &      54, 15 &  2.882 \\
  0.0   &  .6625 &  .2084 &      47, \Z0&  2.946 \\
   .1   &  .5911 &  .2785 &      44, 44 &  2.850 \\
   .2   &  .5202 &  .3490 &      46, \Z0&  2.749 \\
   .3   &  .4465 &  .4165 &      49, 13 &  2.686 \\
   .4   &  .3685 &  .4791 &      53, 29 &  2.675 \\
   .5   &  .2858 &  .5352 &      58, 10 &  2.723 \\
   .6   &  .1987 &  .5844 &      62, 52 &  2.838 \\
   .7   &  .1081 &  .6265 &      67, 13 &  3.036 \\
   .8   &+\Z.0147&  .6622 &      70, 49 &  3.348 \\
   .9   &-\Z.0805&  .6929 &      73, 11 &  3.834 \\
  1.0   &  .1764 &  .7213 &      73, 25 &  4.631 \\
  1.1   &  .2713 &  .7525 &      69, 17 &  6.090 \\
  1.15  &  .3173 &  .7721 &      63, 50 &  7.333 \\
  1.2   &  .3601 &  .7977 &      53, 25 &  9.236 \\
  1.225 &  .3791 &  .8140 &      45, \Z6& 10.360 \\
  1.25  &-\Z.3951&+\Z.8332&      33, 54 & 11.840 \\
\DPPageSep{133}{75}
  1.275 &-\Z.4064&+\Z.8553&   {}+19, 53 & 12.955 \\
  1.3   &  .4118 &  .8796 &   {}+\Z4, 42& 13.412 \\
  1.325 &  .4108 &  .9046 &   {}-\Z9, 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   &-\Z.0377& 1.1696 &      72, \Z2&  7.618 \\
  2.0   &+\Z.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, \Z1&  2.682 \\
  3.8   & 1.4797 &  .3168 & 180°+\Z4, 28&  2.670 \\
  4.0   & 1.4674 &  .1181 & 180°-12, 14 &  2.733 \\
  4.1   & 1.4377 &+\Z.0227&      23, 43 &  2.806 \\
  4.2   & 1.3894 &-\Z.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, \Z8&  2.477 \\
  4.6   & 1.0448 &  .1399 &      49, 32 &  2.101 \\
  4.65  & 1.0108 &  .1034 &      36, 18 &  1.683 \\
  4.7   &  .9867 &-\Z.0598&      21, \Z1&  1.234 \\
\ColHead{\text{perijove}}
        & +\Z.990&+\Z.011 & \llap{\text{about }} 49, & \\
\end{longtable}

The orbit is not vigorously periodic, but an extremely small change at the beginning
would make it so. The period is given by $nT = 1234°.6 = 3.43$ revolutions of Jove.

\normalsize
\DPPageSep{134}{76}


\Chapter{Address}
\index{Address to the International Congress of Mathematicians in Cambridge, 1912}%
\index{Cambridge School of Mathematics}%
\index{Congress, International, of Mathematicians at Cambridge, 1912}%
\index{Mathematical School at Cambridge}%
\index{Mathematicians, International Congress of, Cambridge, 1912}%

\Heading{(Delivered before the International Congress of Mathematicians
at Cambridge in 1912)}

\First{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
\index{Specialisation in Mathematics}%
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.
\DPPageSep{135}{77}

Up to a few weeks ago there was one man who alone of all mathematicians
\index{Poincaré, reference to, by Sir George Darwin}%
might have occupied the place which I hold without misgivings as to his
fitness; I mean Henri Poincaré. 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 ``Poincaré 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 Poincaré 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
\index{Galton, Sir Francis!analysis of difference in texture of different minds}%
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\DPnote{[** TN: Not hyphenated in original]} 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
\DPPageSep{136}{78}
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. Poincaré has well
remarked in his \textit{Science et Méthode} (p.~57):
\index{Poincaré, reference to, by Sir George Darwin!\textit{Science et Méthode}, quoted}%

``On peut s'étonner de voir invoquer la sensibilité apropos de démon\-stra\-tions
mathématiques qui, semble-t-il, ne peuvent intéresser que l'intelligence.
Ce serait oublier le sentiment de la beauté mathématique, de
l'harmonie des nombres et des formes, de l'élégance géometrique. C'est un
vrai sentiment esthétique que tous les vrais mathématiciens connaissent.
Et c'est bien là de la sensibilité.''

And again he writes:

``Les combinaisons utiles, ce sont précisément les plus belles, je veux dire
celles qui peuvent le mieux charmer cette sensibilité spéciale que tous les
mathématiciens connaissent, mais que les profanes ignorent au point qu'ils
sont souvent tentés 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 Poincaré'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 Poincaré 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
\DPPageSep{137}{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
\index{Kelvin, associated with Sir George Darwin!cooling of earth}%
Lord Kelvin's celebrated investigation as to the cooling of the earth. He
was 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
\index{Darwin, Sir George, genealogy!on his own work}%
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 Störmer\footnote
  {\textit{Videnskabs Selskab}, Christiania, 1904.}
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
Poincaré.

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.
\DPPageSep{138}{80}
\backmatter
\phantomsection
\pdfbookmark[-1]{Back Matter}{Back Matter}

\Pagelabel{indexpage}

\printindex

\iffalse
%INDEX TO VOLUME V

%A

Abacus xlviii

Address to the International Congress of Mathematicians in Cambridge, 1912#Address 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#British xxvi

Brown, Prof.\ E. W., Sir George Darwin's Scientific Work xxxiv
  new family of periodic orbits 59

%C

Cambridge School of Mathematics 1, 76

Chamberlain and Moulton, Planetesimal Hypothesis xlvii

Committees, Sir George Darwin on xxii

Congress, International, of Mathematicians at Cambridge, 1912#Congress 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

%D

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
  his first papers xxxvi
  characteristics of his work xxxiv
  his influence on cosmogony 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

%G

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 R.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, l

Hill, G. W., Lunar Theory l
  awarded gold medal of R.A.S. lii
  lectures by Darwin on Lunar Theory lii, 16
  characteristics of his Lunar Theory 16
  Special Differential 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
%\DPPageSep{139}{81}

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

%L

Laplace, Nebular Hypothesis xlvi

Larmor, Sir Joseph, Sir George Darwin's work on university committees xvii
  International Geodetic Association xxvii
  International Congress of Mathematicians at Cambridge 1912#Cambridge 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#Cambridge xxix, 76

Mechanical condition of a swarm of meteorites xlvi

Meteorological Council, by Sir Napier Shaw xxii

Michelson's experiment on rigidity of earth l

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

%O

Orbit, variation of an 64

Orbits, periodic, |see{Periodic}

%P

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

Poincaré, reference to, by Sir George Darwin 77
  on equilibrium of fluid mass in rotation xlii
  \textit{Les Méthodes Nouvelles de la Mécanique Céleste} lii
  \textit{Science et Méthode}, quoted 78

Pupils, Darwin's relationship with his xxxvi

%R

Raverat, Madame, on Sir George Darwin's personal characteristics xxx

Reduction, the 49

Rigidity of earth, from fortnightly tides xlix
  Michelson's experiment l

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 Association, 1905#British 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

\textit{Tides, The} xxvii, l

Tides, articles on l

Tripos, Mathematical 3

%U

University committees, Sir George Darwin on, by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall xvii, xviii

%V

Variation, the 43
  of an orbit 64
  of latitude 14

Variational curve, defined 22
  determined 23
  small displacements from 26
\fi
\DPPageSep{140}{82}
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% Darwin, by George Darwin                                                %
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manyfoot.sty    2005/09/11 v1.10 Many Footnote Levels Package (NCC)
nccfoots.sty    2005/02/03 v1.2 NCC Footnotes Package (NCC)
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multicol.sty    2006/05/18 v1.6g multicolumn formatting (FMi)
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graphicx.sty    1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
  keyval.sty    1999/03/16 v1.13 key=value parser (DPC)
graphics.sty    2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)
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graphics.cfg    2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
  pdftex.def    2007/01/08 v0.04d Graphics/color for pdfTeX
 wrapfig.sty    2003/01/31  v 3.6
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hyperref.sty    2007/02/07 v6.75r Hypertext links for LaTeX
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hyperref.cfg    2002/06/06 v1.2 hyperref configuration of TeXLive
kvoptions.sty    2006/08/22 v2.4 Connects package keyval with LaTeX options (HO
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everysel.sty    1999/06/08 v1.03 EverySelectfont Package (MS)
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    umsa.fd    2002/01/19 v2.2g AMS font definitions
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