LIBRARY

OF THK

UNIVERSITY OF CALIFORNIA.

Class

THE MATHEMATICAL

AND OTHEE WETTINGS

OF

ROBERT LESLIE ELLIS, M.A.

PRINTED BY C. J. CLAY. M.A.
AT THE UNIVEE8ITY PRESS.

iri

THE MATHEMATICAL AND
OTHER WRITINGS

OP

EGBERT LESLIE ELLIS, M.A.

LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE.

EDITED BY

WILLIAM WALTON, M.A.

TRINITY COLLEGE ; MATHEMATICAL LECTURER AT
MAGDALENE COLLEGE, CAMBRIDGE.

a Jfltograpjfcal

BY

THE YERY REYEREND HARYEY GOODWIN, D.D.

DEAN OF ELY.

CAMBRIDGE :
DEIGHTON, BELL, AND CO.

LONDON: BELL AND DALDY.
1863.

DEDICATORY LETTER.

DEAR LADY AFFLECK,

Having undertaken with your sanction the publication
of the writings of your lamented brother in a collected form,
I may be permitted to address to you a few observations on the
principles by which I have been guided in the fulfilment of an
act of piety to the memory of a friend of many years, whom
I shall ever remember with affection and veneration.

The greater part of the writings contained in this volume
had previously appeared in Scientific Journals, and Keports,
some few in Educational Treatises. I have had no scruple
whatever in reprinting these works, since their publication had
already taken place by the author's own act or permission.

The question of the publication of manuscripts presented to
my mind greater difficulty. During his long years of suffering
he was in the habit of dictating to his friends various speculations
in adaptation to their different tastes and pursuits, evidently for
the most part not intended for the press. A great many mathe-
matical investigations, which may be described as interesting
problems, communicated to me on numerous occasions either by
dictation or by letter, are in my possession. Out of these ma-
nuscripts I have ventured to publish only two, the one on the
[Retardation of Sunrise, and the other a new t Solution of a
Problem in the First Book of Newton's Principia. All the
manuscripts entrusted to me by you, classical, philological,
botanical, and mathematical, works of a more elaborate cha-
racter, I have not hesitated to include in this volume, from a
conviction that they will be interesting to many readers, and

217977

vi DEDICATORY LETTER.

from an impression, grounded on internal evidence, that the pro-
bability of their ultimate publication may have been contem-
plated by their author.

The difficulty under which an amanuensis labours, in trans-
ferring accurately to paper the words of one afflicted by severe
illness, is at all times considerable. In the manuscripts placed
in my hands the errors were necessarily, by reason of the pecu-
liar nature of the subjects, veiy numerous. The obstacles,
arising from this source, in the way of preparing for the press
some portions of the work with proper emendations, I should
have regarded as not a little formidable, had it not been for the
zealous assistance of Mr Munro, Fellow of Trinity College, to
whom I am indebted for the corrections of the text in all the
philological and classical writings which had not previously
been published.

I may mention also that to the Dean of Ely, who at my
request and with your entire approbation undertook most
heartily the composition of the Biographical Memoir, I am
under obligation for occasional advice, and for much kindly
interest in the progress of the work through the press.

The engraving has been taken from an admirable portrait by
Samuel Lawrence, in the possession of Professor Grote, by
his kind permission.

Hoping that I have adequately discharged my duties as
Editor of this Collection of your brother's writings, and at any
rate conscious that I have done my best, I beg to dedicate to
you this volume, and to subscribe myself,

Your faithful servant,

WILLIAM WALTON.

CHESTERTON, Oct. 16, 1863.

CONTENTS.

PAGE

BlOGBAPHIOAL MEMOIR ix

On the Foundations of the Theory of Probabilities .... I

On the Method of Least Squares . 12

Some Remarks on the Theory of Matter 38

Remarks on the Fundamental Principle of the Theory of Probabilities 49

Remarks on an Alleged Proof of the Method of Least Sqxtares . . 53
Note to a Former Paper on an Alleged Proof of the Method of Least

Squares 62

On some Properties of the Parabola 63

On the Existence of a Relation among the Co-efficients of the Equation

of the Squares of the Differences of the Roots of an Equation . 68

On the Achromatism of Eye- Pieces of Telescopes and Microscopes . 71
On the Condition of Equilibrium of a System of Mutually Attractive

Fluid Particles -75

Mathematical Notes . . . .81, 92, 130, 142, 149, 157, 191, 223

Variation of Node and Inclination . ...... 82

Investigation of the Aberration in Right Ascension and Declination . 84

On the Lines of Curvature on an Ellipsoid ..... 86

On the Tautochrone in a Resisting Medium 94

On the Integration of Certain Differential Equations. No. I. . 97

„ „ » „ » No. II. . . 108

Analytical Demonstrations of Dr Matthew Stewart's Theorems 118

Note on a Definite Integral 124

Remark on the Distinction between Algebraical and Functional Equa-
tions 126

On the Solution of Functional Differential Equations . . . 132

Evaluation of Certain Definite Integrals 14 3

On the Evaluation of Definite Multiple Integrals .... 150

Note on a Definite Multiple Integral 160

Notes on Magnetism. No. 1 163

No. II . 186

On a Multiple Definite Integral 169

On a Question in the Theory of Probabilities *73

On the Balance of the Chronometer . . . . 180

viii CONTENTS.

PAGE

Memoir of the late D. F. GREGORY, M.A., Fellow of Trinity College,

Cambridge 193

On the Solution of Equations in Finite Differences .... 202

General Theorems on Multiple Integrals 212

On the Area of the Cycloid . 224

Sur lea Integrates aux Differences Finies 726

Report on the Recent Progress of Analysis (Theory of the Comparison

of Transcendentals) 238

Solution of a Dynamical Problem . . . . . . . .324

On the Tautochronism of the Cycloid 326

On Napier's Rules 328

On the Retardation of Sunrise 336

A Solution of Problem IX. of the First Book of Newton's Principia . 337

On Roman Aqueducts 339

On the Form of Bees' Cells 353

On the Theory of Vegetable Spirals 358

Some Thoughts on Comparative Metrology 373

Notes on Boole's Laws of Thought 391

Remarks on Certain Words in Diez's Etymologisches Wbrterbuch der

Romanischen Sprachen 395

Some Thoughts on the Formation of a Chinese Dictionary . . 400

Value of Roman Money 415

The Course of Mathematical Studies 417

ERRATUM,
p. i, 4, 5. For Bernoulli!, read Bernoulli.

BIOGRAPHICAL MEMOIR

OF

ROBERT LESLIE ELLIS, M.A.

LATE FELLOW OF TKINITY COLLEGE, OAMBBIDGE.
BY

HAKVEY GOODWIN, D.D.

DEAN OF ELY.

" H avait une promptitude infinie a tout saisir, une me'moire prodigieuse et une
facult^ me'thodique et rectifiante pour tirer, comme par une chimie naturelle,
quelque chose de preVieux de tout ce qui s'offrait a lui, soit dans la conversation,
soit dans la lecture. Tout sujet d'entretien lui e"tait bon ; il acceptait volontiers
celui qu'on mettait sur le tapis, et il e"tonnait les indiffe"rents par les tre'sors qu'il
tirait a 1'instant de la mine qu'ils lui avaient offerte sans y songer. Son esprit
e*tait comme une bibliotheque encyclop^dique bien ordonne'e, qu'il suflSsait d'ou-
vrir a la lettre qu'on voulait, pour en faire sortir des richesses."

BIOGKAPHICAL MEMOIR

OP

ROBERT LESLIE ELLIS.

THE publication in a collected form of the papers which this
volume contains is to be regarded, to a certain extent, as a
tribute of affection. From this point of view the work would
hardly seem to be complete without some notice of the life
and character of the author. That he was a man of no ordi-
nary attainments, in at least one field of knowledge, will be
sufficiently evident to those who know no more of him than
they can gather from this portion of his writings ; that he had
powers distinct from those of mathematical research, is evident
from what he has done as the editor of the philosophical works
of Bacon ; but even these published records of his intellect will
perhaps fail to convey to readers that impression of remarkable
and various ability which was made, I believe, upon all those
who were brought into personal contact with him. It may
therefore be interesting to the general reader, besides completing
the memorial character of the volume, if an attempt be made
by one of his contemporaries to give some account of what
he was.

The mere facts of the life are few and simple. The ex-
ternal picture of it may be very easily drawn. It was short,

xii BIOGRAPHICAL MEMOIR

quiet, uneventful, but very full of suffering. The plan which
I shall adopt in the following memoir will be this: I shall
first give the story of the life in as compact a form as may be
possible, and then endeavour to lay before the reader some
estimate of the mind and character.

EGBERT LESLIE ELLIS was born at Bath, August 25, 1817,
being the youngest of a family consisting of three sons and three
daughters. His mother's health was not good, and from her
he appears to have inherited that highly nervous constitution,
which became, during a considerable portion of his life, as we
shall see hereafter, the medium of great suffering. His father
was a man of cheerful disposition, of active and well cultivated
intellect, fond of speculative inquiry, and in worldly circum-
stances independent. His character and his mode of dealing
with Robert, as a child, had a great influence upon him through-
out his life : he became his father's companion from a very
early age, and the affection with which he referred in later
life to his father's care and to the happy days of his boyhood,
could not fail to strike those who had the pleasure of know-
ing him intimately,

I do not find that as a child he exhibited any extraordinary
symptoms of precocity1, though it is manifest, from records of
his boyish doings made by himself, that he was very forward
in his studies, and that he took an interest in his work, and

1 With reference to what is said in the text, and possibly the reader may
think in contradiction to it, I insert here a memorandum, which I find, amongst
the papers intrusted to me, and which appears to be in his father's hand.

"The following numerical theorem, if not curious in itself, may perhaps be
•esteemed so, as coming from a boy of eight years old, who was not far advanced
in the ordinary rales of arithmetic.

" If any number be added to its equal, subtracted from its equal, multiplied by
its equal, and divided by its equal, then the sum, the difference, the product, and
the quotient of these equal numbers, added together, will equal the square of the
next higher number."

That is to say, if n be the number,

OF ROBERT LESLIE ELLIS. xiii

exerted his mind upon the subjects to which it was directed
in a manner by no means usual.

He was never at school, but had the advantage of two
tutors at Bath, one in classics, the other in mathematics1. He
worked for them with great earnestness, and I find from his
own memoranda, that in the year 1827, when he was about ten
years old, he was doing equations, and reading Xenophon and
Virgil, besides giving some attention to French and drawing.
These same memoranda shew that at this time, in addition to
his ordinary work with his tutors, he was reading books not
usually read by boys at such an age, Cuvier's Theory of the Earth,
The Edinburgh Journal of Science, The Edinburgh Review, &c.

One remark which is suggested by the boyish records left
behind him is, that it is clear that from an early age Ellis had
an extreme delight in knowledge for its own sake : he had not
the ordinary stimulus of school emulation, indeed he was singu-
larly free from the influence of competition until his college
days : but it is manifest, from his own account, that his pro-
gress in knowledge, and perhaps especially in mathematical
knowledge, was a source of very keen delight.

In 1829, that is, when twelve years old, he began to read
Mechanics. In the early part of 1830 he commenced the Dif-
ferential Calculus; from which he rapidly proceeded to the
Integral Calculus; and towards the middle of the year he
speaks of being engaged with his tutor in finding the lengths
and areas of curves.

Meanwhile his general reading, for which he was dependent
upon his father's library and upon that of the Bath Institution,
was most multifarious; but each particular subject seems to
have been carefully studied, and an opinion formed upon it.

Thus his education proceeded quietly and also rapidly under
his father and private tutors for several years.

1 His mathematical tutor was Mr T. S. Davies, afterwards of Woolwich ; his
classical, Mr H. A. S. Johnstone.

b

xiv BIOGRAPHICAL MEMOIR

This home education had, I think, a perceptible effect upon
his future character. The effect was not bad in the sense in
which tli at epithet is generally believed to be applicable to
home education ; but there might be observed in him a kind of
elderly sobriety of manner, not amounting to stiffness, but con-
veying the impression that he had been accustomed to converse
with those older than himself, and standing out in marked
contrast with that lively boyish freedom and gaiety which is
especially the characteristic of young men educated at the great
public schools.

In October, 1834, he became the pupil of the Eev. James
Challis, then Eector of Papworth St Everard, in Cambridge-
shire, who soon after was appointed and still remains Plumian
Professor of Astronomy in the University of Cambridge. His
residence at Papworth was, however, very short; his health
gave way, and at the end of six weeks he was compelled to
return home. Here he remained for about two years, not
coming up to the University in 1835, as originally intended,
but postponing the event, on the ground of health, to the follow-
ing year.

He came into residence as a Pensioner of Trinity College,
in October 1836, being entered as a pupil of the Eev. G.
Peacock, afterwards Lowndean Professor and Dean of Ely.
During his undergraduate career his health was not strong, but
I think he was never compelled by illness to desist from his
course of study. He was very much in advance of the men
of his year in mathematical acquirement, and had already read
most of the subjects which usually occupy an undergraduate's
time. He was himself much amused at the surprise expressed
by his tutor, Mr Peacock, when at an early stage of his College
life in answer to the question, « What are you chiefly reading
now?" he replied, " Woodhouse's Isoperimetrical Problems."
He read mathematics, chiefly without the aid of a private tutor,
but in his third year and his last term had the advantage of

OF ROBERT LESLIE ELLIS. xv

Mr Hopkins' s direction1. I was myself a pupil of Mr Hopkins'
at the time ; but Ellis never read with the class of which I was
one; in fact, he did not need the kind of lecture which was
adapted to myself and others ; he required only that his reading-
should be arranged, and put in a form suitable for the Cam-
bridge examinations.

The only occasion upon which I was brought into contact
with him as a fellow-student was in attending Professor Peacock's
lectures on Plane Astronomy. I remember well the astonish-
ment with which I witnessed his demeanour during the lectures :
he made no note, he asked no question ; but he quietly remarked
as we left the lecture-room together one day, "It saves one the
trouble of reading these things up."

It was in fact a great advantage to him to be able to sub-
stitute the use of his ears for that of his eyes. His sight was
very tender, and during the latter part of his undergraduate
career he regularly employed a person to read to him high
mathematical subjects. He once mentioned to me incidentally
that the theory of the Earth's Figure, as given in Pratt's
Mechanical Philosophy, was in this manner read to him ; which
instance I here record as an indication of a power of mental
effort possible to very few, and the magnitude of which ma-
thematicians will appreciate.

During his undergraduate career I was not intimately ac-
quainted with him : probably he had no desire to increase his
circle of friends beyond that which was naturally brought round
him in his own college: and his manner was not such as to
encourage rapid intimacy. I do not think that at this period
the number of his intimate friends was large even within Trinity
College, and sometimes a feeling of desolation and want of

1 In a note to me Mr Hopkins says, with reference to his recollections of
Ellis as a pupil, " On one point he always seemed to puzzle me. The extent and
definiteness of his acquirement, and his maturity of thought, were so great, so
entirely pertaining to the man, that I could hardly conceive when he could have
been a boy."

xvi BIOGRAPHICAL MEMOIR

sympathy oppressed him painfully. He once described to me
the forcible manner in which he was affected in College Chapel
by those words of the Psalm, " I had no place to flee unto, and
no man cared for my soul." This melancholy feeling, which
sometimes assumed a very painful intensity, was no doubt con-
nected with the weak state of his bodily health, and the highly
nervous temperament which naturally belonged to him : it was
the source of much suffering, I fear, even in that period which
preceded the most distressing portion of his life.

Ellis was to be seen sometimes at the debates of the
Union Society, but he seldom took any part in them. On one
or two occasions, however, when domestic troubles had arisen,
and squabbles of a somewhat personal kind ran high, he stood
up as a pacificator ; he was heard with marked respect, and his
suggestions were readily adopted.

In January, 1840, he passed his examination for B.A.
degree. In consequence of complaints which had been made
of the coldness of the Schools, in which the Candidates for
Mathematical Honours were then examined, the examination
took place for two or three years in the Lecture Rooms of
Trinity College. The difficulty was afterwards solved by the
proper warming of the Senate-House. When we visited the
rooms on the day before the Examination to inspect our places,
I found that the alphabetical arrangement of names combined
with the conditions imposed by the length of the tables had
brought Ellis and myself almost immediately opposite to each
other, and I was rather pleased with the thought of seeing him
in actual work. He however made a special request that his
seat might be changed, (I do not exactly know why,) and was
allowed to be placed in a different room ; so that I saw nothing
of him during the examination.

Those who knew anything of the relative powers of the
men of the year had no doubt as to which place Ellis must
occupy, if only his health should enable him to do himseir

OF ROBERT LESLIE ELLIS. xvii

justice in the examination. His health of course introduced an
element of uncertainty; but when the examination was con-
cluded, and it was found that he had been able to take every
paper, the result was quite sure. He was Senior Wrangler;
and I can truly say, that for myself I had almost as much
satisfaction in seeing his name at the top of the list, as in
seeing my own next to it, for I felt convinced that it was
his rightful position, and that nothing but the accident of ill
health could have put any of his competitors above or even
near him.

His appearance in the Senate-House when he took his
degree was very striking. He looked very pale and ill, but
this perhaps enhanced the intellectual beauty of his counte-
nance. A person who was present remarked to me very
pithily, "If I had seen him before, I could have told you
you could not beat him."

In October, 1840, he was elected Fellow of Trinity College.
He retained his fellowship until the year 1849, that is, for
seven years after the degree of M.A., when as a layman he
ceased to be a fellow in due course.

His intention after taking his degree was to read for the
bar, and at one time there was a notion of his entering upon
political life by becoming a candidate for his native city Bath.
His name was publicly discussed with reference to the election,
but the design was given up on the ground of the weakness of
his health. Had he been a candidate, it would have been
on Whig principles; he was not a very earnest politician,
but always professed himself a Whig, a profession which
was probably strengthened by his intimacy with Sir William
Napier, to whom he always expressed himself as much at-
tached.

With regard to the bar, he was duly called, but did not
study long with the intention of practising. The fact is that
his worldly position was unexpectedly altered. Both of his

xviii BIOGRAPHICAL MEMOIR

elder brothers died, and he thus became heir in expectation
of considerable property, and soon by the death of his father
heir in possession. He was thus deprived of the chief induce-
ment to labour as a lawyer; and had it been otherwise, it is
clear that his health would never have enabled him to undergo
the necessary drudgery. Nor indeed would the actual practice
of law-courts have been very congenial to his feelings and
tastes: law in the abstract he loved exceedingly, as we shall
see presently, but law as it is concerned with the actual strifes
and quarrels of mankind would have been eminently distasteful
to him.

As a Fellow of Trinity he made his College his home,
except for a short period after his election. Here he continued
his mathematical reading, but not with any very definite pur-
pose. He became very intimate with the late D. F. Gregory,
who was then Fellow of Trinity College, and who did good
service to mathematics by the establishment of the Cambridge
Mathematical Journal; when Gregory resigned the editorship
shortly before his death, Ellis took the office, and edited part
of the third and fourth volumes of the journal, in the latter of
which he inserted a short biographical memoir of his friend1.

In January, 1844, Ellis was Moderator. On this occasion,
being myself one of the Examiners, I was thrown into closer
relations with him than before, and commenced that real inti-
macy which lasted as long as his life. His problem paper
on this occasion was singularly elegant, but perhaps too refined
for its purpose. His fellow moderator, O'Brien of Caius Col-

1 Ellis's name appears as editor on the title-page of the fourth volume of the
Journal. I may take this opportunity of observing that the parallel drawn be-
tween Gregory and Ellis in a very kind and warmhearted obituary notice of the
latter, which was inserted in the Atken&um of Feb. n, 1860, seems to me not
justified. They resembled each other, no doubt, in the fact that both were good
mathematicians and both real philosophers and lovers of truth: but beyond
this very general resemblance the parallel does not hold. They were much
attached, and Ellis felt the loss of his friend keenly : but neither in mind nor
in manner was there much likeness between them.

OF ROBERT LESLIE ELLIS. xix

lege, and he, published their problems with their own solutions,
soon after the Examination. He bore the labour of the Exami-
nation better than could have been expected; he was a very
pleasant workfellow, being always ready to fill up the intervals
of work with that rich and varied conversation which his
friends remember so well. Shall I be pardoned if I mention
that his fees as Moderator were transferred to Addenbrooke's
Hospital ?

He worked for the Senate-House again in 1845. It is cus-
tomary for the Moderator of one year to act as Examiner the
next1, but he was desirous of escaping the labour, and had
declined to serve. There was, however, a difficulty in finding a
substitute ; I was myself one of the Moderators and felt anxious
that he should serve, which at my earnest entreaty he at length
consented to do. It was in this .year that Professor W. Thom-
son took his degree ; great expectations had been excited con-
cerning him, and I remember Ellis remarking to me with a
smile, " You and I are just about fit to mend his pens." He
again got through the Examination much better than could have
been expected ; in fact, the effort seemed to do him good ; when
he had consented to act he said, " I feel all the better for having
done something plucky."

It has already been mentioned that the study of the prin-
ciples of Law was very agreeable to him. At this period of his
life he de voted ~much time to the study of the Civil Law, and he
has left behind him several volumes of notes made in the course
of his reading. The only appointment concerning which I ever
heard him express any strong wish was that of the Professor-

1 It may be mentioned, for the benefit of readers not acquainted with Cam-
bridge customs, that the distinction between Moderator and Examiner is prac-
tically merely this, that the papers of original problems are set wholly by the
Moderators. Constitutionally the difference is, that the office of Moderator
is an old statutable office, whereas the Examiners were added by Grace of the
Senate, in consequence of the increase of the number of candidates for mathema-
tical honours.

xx BIOGRAPHICAL MEMOIR

ship of Civil Law. He acknowledged to me that he would
have felt gratified by the tenure of this office, which is the more
remarkable when taken in connection with the fact that on the
occasion of one of the Mathematical Professorships being vacant
he expressed no desire to be appointed, but, on the contrary,
declared that he would not consent to be nominated as a candi-
date. Indeed it is a mistake to suppose that Ellis was in any
exclusive or even preponderating degree devoted to mathematics :
his mathematical power was no doubt very great, but I think
not greater than several other powers, and certainly his taste by
no means exclusively leaned in this direction, as his intimate
friends very well knew. But of this more hereafter.

He did not give himself in any degree to tuition during his
Cambridge residence. So far as I know he never had a private
pupil ; he gave a few College Lectures upon high mathematical
subjects, but he did this only as locum-tenens for friends upon
whom the task devolved. Probably his health would have
interfered with any regular occupation of this kind ; but besides
this, he had not, I think, any taste or any special fitness for
imparting knowledge to average minds; his remarks were
always suggestive, and he could throw light upon almost any
subject which could be brought forward, but he usually assumed
a considerable amount of knowledge on the part of those with
whom he conversed, and sometimes (as it seemed to me) he was
obscure, in consequence perhaps of the neatness and conciseness
which were so remarkable in his conversation.

At the request of the British Association, which held its
annual meeting at Cambridge in 1845, he undertook a Eeport
upon the progress of certain branches of pure mathematics.
This Report is reprinted in the present volume. It represents a
great amount of labour and research, and I have no doubt that
the preparation of it was a source of pleasure to him, as it
refers to a department of mathematics which was with Ellis
a special favourite.

OF ROBERT LESLIE ELLIS. xxi

It was during his residence as a Fellow in Trinity College
that he undertook, in conjunction with Mr James Spedding and
Mr Douglas Denon Heath, to edit the works of Bacon. The
philosophical section of the works was the share allotted to Ellis.
No literary occupation could have been more congenial to his
taste, and the prefaces to the several treatises which he was able
more or less to complete, especially the " General Preface to the
Philosophical Works," are perhaps the most valuable thing
which he has left behind him. He was engaged upon the preface
to the Novum Organum, when he was stopped by illness ; and so
complete and sudden was the break in his health that he never
completed it. The last sentence that he wrote will be found on
page 100 of the first volume of his edition of Bacon's Works.
It is an affecting monument, and as such I here produce it.
"Again he affirms that he does not inculcate, as some might
suppose, a — ;" to which Mr Spedding has appended a note,
" Mr Ellis had written thus far when the fever seized him."

The mention of Bacon has led me to anticipate the course
of events. In the years 1847 and 1848 he visited Malvern for
the benefit of his health, still making Trinity College his head-
quarters, and he certainly appeared to be strengthened by the
course of treatment to which he was submitted. He had, I
believe, always intended, at the expiration of the tenure of his
fellowship, to go abroad ; partly perhaps for the general advan-
tages of travel, and partly with the belief that his health
would be improved by residence in a warmer climate. He
desired to settle himself in some place possessing a good
library, where he might complete his work for the edition of
Bacon. Accordingly in the autumn of 1849 he went to Nice.
After remaining there some little time he started, not well in
health, for the journey by post along the Eiviera. The first
night he slept at Mentone, and, as he believed, in a damp bed.
The next day he arrived early at S. Remo, but feeling indis-
posed determined to proceed no further that day. In the

xxii BIOGRAPHICAL MEMOIR

evening lie took the last walk which he was ever able to take,
otherwise than as a cripple ; he described afterwards to one of
his friends the profound effect produced upon his mind, possibly
rendered more sensitive by approaching illness, by the loveli-
ness of the scene. That night, which was one of horrors to
him, he was seized with a rheumatic fever, which for several
days put his life in great danger. A physician, who was
called in, seems to have exerted himself with great kindness
and to the utmost of his skill, to do all that could be done.
Ellis always retained an affectionate remembrance of him. He
ordered his patient to be bled extensively, and after a few days
the imminent danger was passed. Rheumatism however re-
mained fixed hopelessly upon him; he was ever after in con-
stant pain, with very little use of any part of his body; and
the rest of his life, ten years, may be described as a long
process of gradual dissolution.

After a residence of nearly three months at S. Remo, he
was brought home by easy stages. He visited several places
after his return, London, Brighton, Bath, Malvern, Tunbridge
Wells, consulting various physicians, with no apparent result.
At length giving up all hope of amendment, he fixed himself
in 1853 at Trumpington, a village two miles from Cambridge,
of which his friend Professor Grote was Vicar, for the sake of
being near the University and his old friends. When he arrived
he was unable to walk, but could drive out in a carriage, and in
the house he could move from one place to another on the same
floor by means of a chair set upon wheels : after some time
however he became entirely confined to the house ; then to his
bed, where he remained in a sadly suffering condition till the
day of his death.

He wrote me a letter shortly before his arrival at Trumping-
ton, telling me that he had taken Anstey Hall (the name of
his house1 at Trumpington), and adding that he was coming

1 When Ellis first came into residence at Cambridge his father had some

OF ROBERT LESLIE ELLIS. xxm

to leave his bones amongst us, and he trusted we should " give
him a little earth for charity." During his lingering illness
I saw him not unfrequently, and am able to record, from
personal recollection, some few things which may be pleasant
to those who knew him, and not unprofitable perhaps to the
general readers of this volume. In the early [part of his
residence in Anstey Hall, he was well enough to enjoy" the
society of his friends to a very considerable extent ; he sat in
his invalid chair, with sometimes two or three persons present,
pouring forth his varied stores of knowledge as in olden days ;
in fact, it should be stated here once for all, that during the
whole of his lingering sickness his mental powers never ap-
peared to be in the smallest degree impaired ; it was a wonder
to note the perfect action of the mind, at a time when the body
was a mere distorted and attenuated heap of skin and bones.
But this brightness of intellect doubtless made the suffering
more acute. His life for several years was a constant looking
of death in the face, with scarcely an interval of ease or ob-
liviousness. By degrees the resource of the society of his
friends began to be diminished ; frequently we called and
found him unable to see us ; and even relatives staying in the
house could not be admitted into his chamber for days
together.

In the earlier part of his illness he was able to give some
attention, but not much, to Bacon; some of the notes which
have been since printed were dictated at this time. The thought
of leaving his work imperfect could not fail to be painful to
him, and the pain would be increased by the very high standard
of excellence which he set up for himself in all matters which
he undertook. Latterly he could not bear the subject of Bacon
to be alluded to: if it happened to be introduced, he would

intention of engaging Anstey Hall, in order that lie might be near his son,
whose health even then was, as we have seen, not strong. This circumstance
had made Ellis always feel an interest in the house.

xxiv BIOGRAPHICAL MEMOIR

say, "We don't talk about it in this room." He amused
himself also with mathematical investigations, which he was
able to carry on in a remarkable manner, without paper or
figure, "in his head," as the common phrase is. It was in
this way that he discovered for himself what he believed to
be a new view of Napier's rules for the solution of right-
angled spherical triangles. His discovery involves so curious
a piece of history that I shall venture for a moment to dwell
upon it. Ellis found out in his illness, that Napier's rules,
instead of being, as they have been stated to be in Cam-
bridge books, from Professor Woodhouse downwards, a mere
memoria techmca, were all capable of being deduced from one
geometrical construction. He sent me a paper which he dic-
tated on the subject, and which I requested him to allow me
to communicate to the Cambridge Philosophical Society. Hav-
ing gained his permission, I thought it well to examine the
literature of the subject, and above all to see what Napier
had himself said. On turning to Napier's famous tract, Miri-
fici Logarithmorum Canonis Descriptio, I found that Ellis had
in fact rediscovered Napier's own original conception of the
problem.

It was during his illness that he dictated his remarks con-
cerning the construction of bees' cells ; 1 believe also that he
thought out at this time his demonstration of the tautochronous
quality of the cycloid. Indeed he had usually some mathe-
matical question running in his head, which served him for
recreation during his easier moments. Nor were other subjects
excluded; it was in this season of extreme bodily weakness
that he corresponded with the late Dr Gilly on the Eomaunce
language, discussed the date of the "Noble Lesson," and criti-
cized Dr Gilly's edition of the Yaudois Gospel of S. John. I
have also before me a considerable number of letters dictated to
friends, dealing with subjects so different, according to the tastes
of the persons to whom they were sent, that it seems difficult

OF ROBERT LESLIE ELLIS. xxv

to believe them to be the production of the same person. He
dictated also papers on Vegetable spirals, on Comparative Me-
trology, and on various points of Etymology.

But the most remarkable effort of his illness was the dic-
tation of a pamphlet (reprinted in this volume) on the subject of
a Chinese Dictionary, and the best mode of constructing such a
work. This pamphlet was in the form of a letter to the Rev.
J. Power, the Librarian of the University, who had kindly as-
sisted him in his literary researches, and had supplied him with
the most recent literature bearing upon the Chinese language.
I can give no opinion of its value, but can hardly be wrong in
regarding it as a marvellous exertion of mental vigour under
very depressing conditions.

In truth his taste for language was as marked as that for
mathematics. I have just now remarked incidentally upon his
study of the Eomaunce language ; Gothic also appears from
his letters as having received considerable attention; he has
left a paper on Sanscrit; and amongst the modern languages,
besides the usual acquirements of French and German, I per-
ceive that he was well versed in Italian, and that he had given
attention to Danish and Spanish. He thoroughly enjoyed the
study of a language, and I remember very well the pleasure which
he expressed at having had the courage to communicate a
French memoir to Liouville's Journal: it seemed to me that
the writing mathematics in a foreign language gave him al-
most as much satisfaction as the mathematical results them-
selves *. I ought, perhaps, to mention that I found him one
day reading the New Testament in Swedish, which he told
me he had " picked up " since he had been ill2.

I have said that through his long illness Ellis retained his

1 He was very fond of translating. Amongst his papers are some translations
of Danish ballads, Spanish ballads, Andersen's Tales, &c.

2 During his illness his notes to his physician were usually written in
Latin.

xxvi BIOGRAPHICAL MEMOIR

vigour of mind. It is wonderful that in such suffering, and in
the consciousness of approaching dissolution the mind should
have been capable of dwelling calmly upon subjects of abstract
interest, such as investigations in pure mathematics ; he himself
felt that there was something strange in the occupation. In a
note accompanying a mathematical paper he writes to me : "I
have been very miserable all this week. God will mend it,
when His will is. It seems strange that my mind still runs at
all upon triangles, and I am not at all sure that it is right it
should. I need not tell you to think charitably of me in this
as in other respects." His mind by no means however dwelt
upon triangles to the exclusion of more solemn subjects, as I
shall have occasion to shew more fully presently ; but it ap-
peared to have a vigour of action and a fulness of matter which
no external circumstances could affect, and so far as my obser-
vation went he conversed with the same facility and command
of his subject during his illness, as in earlier days.

It would be only painful to draw as vivid a picture as might
be easily drawn of the protracted sufferings which he had to
endure. Medicine could do nothing more for him than mitigate
the severity of the disease, which seemed to claim as its own
one muscle after another, as it slowly approached the heart.
" One twitch there" said he to me one day, speaking of his
heart, " and then I shall know the great secret." For a con-
siderable period, reading (as might be supposed) was a relief;
the weary hours of night, sometimes rendered horrible by the
fear of dreams if sleep should come upon him1, were beguiled
with books, a lamp suspended over his head giving him the
necessary light. By and bye however this resource also failed :
the eyes began to be affected by the complaint, and for about

1 He one day represented his condition to me in words curiously resembling
those of Job: "When I say, My bed shall comfort me, my couch shall ease my
complaint ; then Thou scarest me with dreams, and terrifieat me through visions."
Job vii. 13, 14.

OF ROBERT LESLIE ELLIS. xxvii

two years before his death he was almost entirely blind1.
Books were read to him, and he dictated occasionally to an
amanuensis, but the loss of sight. was a very severe addition to
his sufferings.

The following lines, which may be taken as a sample of the
love of epigram 2 which belonged to him, were sent to Dr- Paget,
his physician, when the blindness was gaining upon him.

Contortos artus nunc culcita celat, at olim

Terra teget melius : sit modo et ilia levis.
Et quam vix possunt oculi tolerare dolentes
Lux fugit, ac tenebris mox adopertus ero.
Mar. 9, 1857.

In the earlier part of his illness, I made the remark one day
that he appeared to me to be a little better; he at once said

1 His eyes were first attacked in April, 1856 ; he was unable to read in July,
1857.

2 This love of epigram was very striking. Here is an instance. During his
illness an old friend wrote to him asking him for some new conundrums. It so
happened that on the day of receiving this request he fancied that he had disco-
vered from Dr Paget, that he was labouring under Bright's disease ; he sent the
following answer :

Si petis hinc aenigma novum, si ludicra poscis,

Quod nuper didici scribere cur dubitem ?
Morbus, qui clarum fecit qui nomine clarus

Semper erat, solvet vincula queis teneor.

On the same day he wrote on the same subject in a different style :
DEAR PAGET,

I think it well to thank you for your most kind note. It came a few
minutes before my dinner. That over, I told W — * its purport, and desired him
to take notice how little it disturbed me.

Of course, no such communication can ever be matter of indifference, and least
of all to a person like me, in whom the power of suffering and of being anxious
has been but little impaired by years of suffering and of anxiety.

But, to use John Bradford's words, "He who has helped me till now will not
leave me when I have most need, for His truth and mercy sake ;" and of neces-
sity I am less anxious about many things than I have long been.

Yours,

R L. ELLIS.
Feb. 19, (1857).

* His servant.

xxviii BIOGRAPHICAL MEMOIR

very earnestly, " 0 pray do not say so !" He recognized from
the first the sure character of his disease, and he desired that
his sufferings might not be protracted ; at the same time he was
perfectly patient, even cheerful, and reverently acknowledged in
all his afflictions the governing hand of God.

But it is time that I should leave the story of my friend's
sickness, upon which affectionate remembrance tempts me to
linger, in order that I may endeavour to give such estimate as
I can of his mind and character.

Speaking generally I should say that his intellect was the
most remarkable that I have known. It was made up of a
combination of powers so delicately balanced, and working
together in such perfect harmony, that it would be difficult to
say that any one predominated over the rest. Popularly in
Cambridge he might be regarded as specially a mathematician,
because he was Senior Wrangler, but those who knew anything
of him were fully aware that mathematics was only one of his
acquirements, and that in conversation mathematics by no
means presented itself as the chief or even the favourite subject
of his thoughts. Indeed, I think, it would be difficult to say
that there was any one subject in which his mental powers were
manifested more decidedly than in others ; and the only marked
deficiency which I ever detected was in respect of music, for
which he had no special taste. I do not mean that he had no
sense of melody : this was far from being the case : but his con-
versation never ran upon the great masters of music and their
works, as it did upon almost all other subjects.

Doubtless, however, mathematical power belonged to him
in a very large degree. With the present volume in the
reader's hands it would be superfluous to say much concerning
the special departments of mathematical investigation in which
his taste impelled him; but I may remark that he seemed
most naturally to associate his mathematics with the past ; he

OF ROBERT LESLIE ELLIS. xxix

delighted to discuss the principles of investigations already
known, to trace the history of processes, to examine the phi-
losophy of a subject, to hunt up its literature, or to simplify
its treatment. His memoir on the Foundations of the Theory
of Probabilities and that on the Method of Least Squares,
which stand at the opening of this volume, appear to me to
represent as well as possible his special taste, so far as he
had a special taste, with regard to mathematics. He always
seemed to talk on the subject of Probabilities with great plea-
sure, and as one in which he was thoroughly at home. The
remarkable little essay on the Theory of Matter was also one
which I think gave him much satisfaction. His taste did
not seem to lead him much in the direction of elaborate phy-
sical experiment, nor do I remember that on any occasion he
worked in this path of investigation. His mind in fact was
rather that of the philosopher than the physicist ; his impulse
was rather to contemplate existing knowledge, than to take up
a particular line of physical investigation and press forward
knowledge upon that one line.

This characteristic of mind gave a great charm to his con-
versation : his thoughts were, so to speak, set in a rich historical
framework, and they were produced with an ease and readiness
which I have never seen equalled. In referring to his conver-
sational powers generally, I may record the singular accuracy
of his speech ; this was perhaps partly a natural gift, and partly
the result of early education ; certainly it was very wonderful ;
his sentences were not only full of thought and of references
to literature of all kinds, but they were so remarkably correct
in their construction and elegant in diction. He was one of
the few men who could have borne a Boswell with great
advantage to their reputation1.

1 This remark was made to me by one very intimate with him, and I cordially
adopt it. In fact, with obviously wide differences, there was a good deal of curious

xxx BIOGRAPHICAL MEMOIR

He was a good scholar, and very fond of the Greek and
Roman literature. I believe I am justified in saying, that his
knowledge of that literature was really more extensive and
thorough than that of many whose reputation as scholars has
been much greater. He could enjoy a discussion of a point
of classical philology as keenly as one on scientific subjects, and
when so engaged no one would have supposed that mathematics
was his favourite study. Indeed, as I have already intimated,
mathematics could not in any proper sense be so described :
Civil Law was certainly as much a favourite : and he seemed
to be most happy in conversation, when the subject was
one of a philosophical character. I have often felt disposed
to compare his mental constitution in many respects with that
of Leibnitz. Each was the philosopher quite as emphatically
as the mathematician. Leibnitz, I may observe, was one of
his favourites, and he mentioned to me one day with some
feeling of amusement that a Fellow of Trinity had spoken
to him of Leibnitz, under the title " your Leibnitz," as
though the old feeling of jealousy were still lurking in the
College.

His love of philosophy fitted him especially to be the editor
of Bacon. It was a work, I believe, which he undertook with all
his heart, and relinquished with extreme pain and only under
a sense of imperious necessity. He was assisted too by his
familiarity with the philosophical speculations of the middle
ages: there was something congenial to his own cast of mind
in the discussions of the schoolmen, and I think he appreciated
Bacon all the more in virtue of his appreciation of those, whose
processes of thought and methods of argument it was Bacon's
task to supersede.

similarity between Ellis and Dr Johnson. Ellis was an excellent conversationalist:
he not only expressed himself with singular precision, but he was a patient listener,
and readily caught up and retained in his memory the remarks of those with whom
he conversed.

OF ROBERT LESLIE ELLIS. xxxi

In producing his knowledge he was much assisted by the
strength and clearness of his memory. All his knowledge
seemed to be as it were in hand; it was not merely that he
knew where to look for information, although that is a great
thing and as much as many clever men are satisfied to accom-
plish ; but information upon the most various subjects, the most
trivial and the most important, seemed to be at call upon all
occasions. His long painful illness had no apparent effect upon
this faculty. On one occasion, some years after he had been
confined to the house, he wished to describe a certain picture
in a room which he had only once entered, and that merely for
a morning call : to my astonishment he mentioned the pictures
one after another as they hung on the wall, and so identified
that to which he desired to refer.

Ellis was very fond of expressing his thoughts in verse.
Some of his notebooks are full of poetical scraps, generally
somewhat melancholy in their tone, but expressing (I have no
doubt) the feelings of his mind at the time of writing. I will
produce two or three of these scraps in this place: it will be
seen that they all belong to the period preceding his last long
illness.

i.

E'en in the days when life is dear
And we would fain live on for aye,
Then let it not forgotten be
That death draws near.

2.

And when we fall on sadder hours,
And gladly would lie down and die,
Remember that we live to do

God's will ; not ours.
DINANT, 1846,

Written April g.

The two following were written, I presume, in Trinity
College-.

xxxii BIOGRAPHICAL MEMOIR

i.

The flower of life is gone — 'tis well
We know each flower will have its day.
It were not wise did you repine
Because the spring has passed away.

2.

The flower was cankered in the bud,
It was a sickly faded flower—
What recks it now? however fair
It still had died before this hour.

3-

'Aye, but it died and left no fruit,
And therefore I must needs lament.'
'Lament not for the buried past,
For all in vain your grief is spent.'

4-

"Tis true, and days to come may bring
A sharper woe than all the past,
And every passing year may seem
To me more bitter than the last.'

5-

'Think not of this, but turn to Him
Who bids His wearied follower rest,
About the altars of whose house
The swallow builds itself a nest.

6.

Turn you to Hun whose voice can calm
The working of our troubled sea,
Who brings the way-worn traveller
Unto the port where he would be.

7-

To Him who said these solemn words,
"Blessed are they that weep,"
Who now as in the days of old
Gives His beloved sleep.'

Jan. 12, 1848.

I do lament me for the days gone by,

All spent in loving frail mortality :

Else I had been what now I cannot be,

For I had wings wherewith to mount on high.

OF ROBERT LESLIE ELLIS. xxxiii

Thou king of heaven eternal, hear my cry I
Thou knowest all my grief and misery :
Recal, O God, my wandering soul to Thee,
And its shortcomings with Thy grace supply !
O let me die in harbour and at rest,
Though I have lived in tempest and in strife :
So when I journey hence at Thy behest
My death will be less worthless than my life.
Vouchsafe in life and death to succour me ;
Thou knowest that I trust in nought but Thee.

Amen and Amen.

Miserere mei Domine et nunc et in hor£ mortis.
Feb. 24, 1848.

I have no wish to indulge in any extravagant eulogy of my
friend, but I should leave a sad blank in this brief memorial
of him if I did not say that his moral qualities were not below
his intellectual. His manner might be accounted by some
persons cold, and he was certainly not one with whom familiar
intercourse and the thorough freedom of friendship were attained
rapidly ; but those who knew him well knew that he possessed
one of the most gentle of hearts, with a delicate consideration
for the feelings of others, and a most grateful sense of any
kindness shewn to himself. Above all his sense of honour and
propriety was perfect ; nothing shabby or mean could exist in
the same place with Leslie Ellis. This is a description which
I am convinced that all who knew anything of him will certify
to be correct, and I think I need not add more to it: his in-
tellectual faculties were undoubtedly his most striking charac-
teristics, and the fruits of his intellect, which he was permitted
to gather during his brief period of health and activity, remain
as an indication of what he might have done ; let it suffice to
say that those qualities of which his works can tell but little,
were such as we delight to contemplate in union with great
intellectual gifts.

There is one other point which must not be omitted, but which
must be treated with delicacy. It would be unjust not to add
a word upon the more definitely religious aspect of his charac-

xxxiv BIOGRAPHICAL MEMOIR

ter; I say unjust, because the remarkable keenness of his mind
concerning mathematical and other questions, during an illness
in which his life was hanging constantly by a thread, may give
those who are disposed to do so the occasion of remarking that
his mind might very well have been occupied with more solemn
thoughts. Let such persons then be satisfied by knowing that
more solemn thoughts did occupy his mind. I think that as
his sickness advanced and his bodily powers were diminished,
his mind gradually found more settled peace and rested more
surely upon the love of God and the merits of the Saviour.
Certainly there was much to tempt him to murmur, but I never
noticed any murmuring propensity or any tendency to do
otherwise than bow to God's will and accept with patience a
mysterious and painful dispensation. In the early part of his
illness, he asked me whether I had ever thought much or heard
a sermon upon Habakkuk iii. 17, 18: "Although the fig-tree
shall not blossom, neither shall fruit be in the vines ; the labour
of the olive shall fail, and the fields shall yield no meat ; the
flock shall be cut off from the fold, and there shall be no herd
in the stalls : yet I will rejoice in the Lord, I will joy in the
God of my salvation." He made no application of the verses
to his own circumstances, only remarking how striking the
language was ; but it was evident to me that his own case was
in his mind.

He always begged me to read prayers with him. I usually
introduced the Collect from the Visitation of the Sick ; on one
occasion I omitted it ; he noticed the omission at my next visit,
and begged me to use it. Anyone who remembers the substance
of that Collect will see the value of this simple anecdote.

His own bodily weakness and utter abstraction from all
works of active piety intensified his desire of doing something
for the benefit of his fellow-creatures, and made him grieve over
his forced indolence. I do not mean that his charitable feelings
first germinated in his sick room : this was very far from being

OF ROBERT LESLIE ELLIS. xxxv

the case: but his sense of his own inability to discharge any
active duty made him more keenly sensible of the privilege of
being permitted to exert ourselves for God and for our brethren.
On reading the account of the death of Captain Gardiner in his
noble but not wisely-arranged effort to found a mission in Pata-
gonia, he expressed a wish that his own life might have had a
similar termination. Indeed a sense of the spiritual needs of
mankind appeared to grow upon him as his own bodily weak-
ness brought him nearer to the great realities of existence: I
can never forget the earnestness with which he said to me at a
late period of his illness, " The thing above all others which
strikes me, as I lie here on my bed, is the intense wickedness of
mankind." "I feel," he continued, "as if I should be con-
strained, did God ever raise me up again, to rush in amongst
them, as Barnabas and Paul did amongst the people of Lys-
tra, and rend my clothes and say, Sirs! why do ye these
things1?"

His speculative mind, acting under the peculiar conditions
to which it was subjected by his diseased body, could hardly
fail to look sometimes anxiously into the future, and guess what
might be the nature of the life prepared for him in the world to
which he was brought so near: as time went on however the
keen discipline of affliction seemed to have taught him that he
must "stand and wait," and simply look forward with calm
hope.

1 I ventured to introduce this reminiscence into a sermon preached before the
University, March 2-2, 1863, and since published.

I find a similar anecdote amongst some recollections, which have been
kindly put in my hands by the Rev. J. P. Norris. Speaking of children,
and his strong disapproval of giving them prizes for mere cleverness, he added,
"There is another point connected with children that I feel with an intensity
which I would give much to have felt years ago, the sacred duty of keeping
them pure. Wounded Arthur, speaking to Sir Bedivere, threatens to arise and
slay him with his hands if he fail in his behest ; and I feel sometimes as if I could
arise from this bed and tear to shreds some of the books that are left in children's
way."

xxxvi BIOGRAPHICAL MEMOIR OF R. L. ELLIS.

In fact it was impossible not to observe that throughout his
illness he perceived, that it was to be regarded in the light of
a divine discipline, however mysterious such discipline might
be. " Aches and pains," he said to a friend, " have been my
teachers of late." "Fiat voluntas Tua," he observed to the
same friend, taking up his De Imitatione Christi, " is after all
the only prayer. Domine, modo sum in tribulatione, et non est

cordi meo bene, sed miiltum vexor a praesenti passion e Et

mine inter haec quid dicam ? Domine, fiat voluntas Tua ; ego
bene merui tribulari et gravari."

His last days were days of peace ; and his last words were
so striking that I think it right to put them here upon record.
It will be remembered that for a considerable period before his
death he had been quite blind : just before his departure, which
was in a certain sense sudden though so long expected, he ex-
claimed, " I see a light !" and so expired. Possibly some phy-
sical explanation of this exclamation may be given : for myself
I would rather look upon it as indicative of something spiritual,
and as announcing the arrival of a glorious change for which his
imprisoned soul had long waited and earnestly prayed \

He was released from his sufferings May 12, 1859, and was
buried at Trumpington. His grave is at the South-East corner
of the churchyard, and bears the simple inscription —

ROBERT LESLIE ELLIS

BORN 25 AUGUST 1817
DIED 12 MAY 1859.

BLESSED is THE MAN THAT HATH SET HIS HOPE IN THE LORD.

Psm. 40. v. 5.

1 He once observed that he wondered no one had ever chosen for an epitap
the words of Psalm cxvi. 14, "Thou hast broken my bonds in sunder."

ON THE FOUNDATIONS OF THE THEORY
OF PROBABILITIES*.

THE Theory of Probabilities is at once a metaphysical and a
mathematical science. The mathematical part of it has been
fully developed, while, generally speaking, its metaphysical
tendencies have not received much attention.

This is the more remarkable, as they are in direct opposition
to the views of the nature of knowledge, generally adopted at
present.

2. The theory received its present form during the
ascendancy of the school of Condillac. It rejects all reference
to h priori truths as such, and attempts to establish them as
mathematical deductions from the simple notion of probability.
Are we prepared to admit, that our confidence in the regularity
of nature is merely a corollary from Bernouilli's theorem?
That until this theorem was published, mankind could give no
account of convictions they had always held, and on which
they had always acted? If we are not, what refutation have
we to give? For these views are entitled to refutation, from
the general reception they have met with, from the authority of
the great writers by whom they were propounded, and even
from the imposing form of the mathematical demonstration in
which they are invested.

* Transactions of the Cambridge Philosophical Society, Yol. vm. [Read
Feb. 14, 1842.]

1

ON THE FOUNDATIONS OF

I shall be satisfied if the present essay does no more than
call attention to the inconsistency of the theory of probabilities
with any other than a sensational philosophy.

3. As the first principles of the mathematical theory are
familiar to every one, I shall merely recapitulate them.

If on a given trial, there is no reason to expect one event
rather than another, they are said to be equally possible.

The probability of an event is the number of equally pos-
sible ways in which it may take place, divided by the total num-
ber of such ways which may occur on the given trial.

If tfj, 5j, m1? denote equally possible cases which may

occur on one trial, a2b2 \ those which may occur on a

second trial, «353 .p3 those belonging to a third, &c. : then

afaa^ «!«2^3 &c. &c. are all equally possible complex

results.

Hence it follows that on the repetition of the same trial
k times, the probability that an event whose simple probability
is m will occur p times is

1 .2. ..pi .2...(k-p)

this follows merely by the doctrine of combinations. These are
all the propositions to which I shall have occasion to refer.

4. If the probability of a given event be correctly deter-
mined, the event will, on a long run of trials, tend to recur with
frequency proportional to this probability.

This is generally proved mathematically. It seems to me
to be true a priori.

When on a single trial we expect one event rather than
another, we necessarily believe that on a series of similar trials
the former event will occur more frequently than the latter.
The connection between these two things seems to me to be an
ultimate fact, or rather, for I would not be understood to deny
the possibility of farther analysis— to be a fact, the evidence of
which must rest upon an appeal to consciousness. Let any one
endeavour to frame a case in which he may expect one event on
a single trial, and yet believe that on a series of trials another
will occur more frequently ; or a case in which he may be able

THE THEORY OF PROBABILITIES. 3

to divest himself of the belief that the expected event will occur
more frequently than any other.

For myself, after giving a painful degree of attention to the
point, I have been unable to sever the judgment that one event
is more likely to happen than another, or that it is to be
expected in preference to it, from the belief that on the long
run it will occur more frequently.

5. It follows as a limiting case, that when we expect two
events equally, we believe they will recur equally on the long
run. In this belief we may of course be mistaken : if we are,
we are wrong in expecting the two events equally, and in think-
ing them equally possible. Conversely, if the events are truly
equally possible, they really will tend to recur equally on a
series of trials. But this proves the proposition placed at the
head of the section : for if any event can occur in a out of b

equally possible ways, its probability is j-i and if all these

b cases tend to recur equally on the long run, the event must
tend to occur a times out of b ; or in the ratio of its probability.
Which was to be proved.

6. Let us now examine the mathematical demonstration
of this proposition. In entering upon it, we are supposed to
have no reason whatever to believe that equally possible events
tend to occur with equal frequency.

It is well known that what is called Bernouilli's theorem,
relates to the comparative magnitudes of the several terms of
the binomial expansion.

The general term of

which is the probability that an event whose simple probability
is m will recur p times on Jc trials ; and hence the connexion
between the binomial expansion and the theory of probabilities.

7. A particular example will suffice to illustrate what
seems to me to be the essential defect of the mathematical proof
of the proposition in question.

1—2

4 ON THE FO UNDA TIONS OF

A coin is to be thrown 100 times: there are 2100 definite
sequences of heads and reverses, all equally possible if the coin
is fair. One only of these gives an unbroken series of 100
heads. A very large number give 50 heads and 50 reverses;
and Bernouilli's theorem shows that an absolute majority of the
2100 possible sequences give the difference between the number of
heads and reverses less than 5.

If we took 1000 throws, the absolute majority of the 2100(
possible sequences give the difference less than 7, which is pro-
portionally smaller than 5. And so on.

Now all this is not only true, but important.

But it is not what we want. We want a reason for believing
that on a series of trials, an event tends to occur with frequency
proportional to its probability ; or in other words, that generally
speaking, a group of 100 or 1000 will afford an approximate
estimate of this probability.

But, although a series of 100 heads can occur in one way
only, and one of 50 heads and 50 reverses in a great many,
there is not the shadow of a reason for saying that therefore the
former series is a rare and remarkable event, and the latter,
comparatively at least, an ordinary one.

Non constatj but the single case producing 100 heads may
occur so much oftener than any case which produces 50 only,
that a series of 100 heads may be a very common occurrence,
and one of 50 heads and 50 reverses may be a curious anomaly.

Increase the number of trials to 1000, or to 10,000. Pre-
cisely the same objection applies : namely, that in Bernouilli's
theorem, it is merely proved that one event is more probable
than another, i.e. by the definition can occur in more equally
possible ways, and that there is no ground whatever for saying,
it will therefore occur oftener, or that it is a more natural occur-
rence. On the contrary, the event shown to be improbable may
occur 10,000 times for once that the probable one is met with.

To deny this, is to admit that if an event can take place in
more equally possible ways, it will take place more frequently.
But if this is admitted, Bernouilli's theorem is unnecessary. It
leaves the matter just where it was before, and introduces no
new element into the question.

8. Thus, both by an appeal to consciousness, and by the

THE THEOR Y OF PROBABILITIES. 5

impossibility of dispensing with such an admission, we are led
to recognize the principle, that when an event is expected rather
than another, we believe it will occur more frequently on the
long run. And thus we perceive that we are in the habit of
forming judgments as to the comparative frequency of recur-
rence of different possible results of similar trials. These
judgments are founded, not on the fortuitous and varying cir-
cumstances of each trial, but on those which are permanent — on
what is called the nature of the case. They involve the funda-
mental axiom, that on the long run, the action of fortuitous
causes disappears. Associated with this axiom is the idea of an
average among discordant results, &c.

I conceive this axiom to be an a priori truth, supplied by
the mind itself, which is ever endeavouring to introduce order
and regularity among the objects of its perceptions.

9. With a view to conciseness, I omit several interesting
points which here present themselves — namely, the connection
between the axiom just stated, and the inductive principle ; the
real utility of Bernoulli's theorem ; and what seems to me to be
the true definition of probability, founded on a reference to the
ratios developed on the long run.

I proceed to illustrate what has been said by a few passages
from Laplace's " Essai Philosophique sur les Probabilites"

10. It seems obvious that no mathematical deduction from
premises which do not relate to laws of nature, can establish
such laws. Yet it is beyond doubt that Laplace thought Ber-
nouilli's theorem afforded a demonstration of a general law of
nature, extending even to the moral world.

At p. xlii. of the Essay, prefixed as an Introduction to the
third edition of the Th^orie des Probabilites, after giving some
account of the theorem of James Bernouilli, Laplace proceeds:
" On peut tirer du the"oreme precedent cette consequence qui
doit 6tre regardee comme urie loi generale, savoir que les rap-
ports des effets de la nature, sont & fort peu pres constants,
quand ces effets sont considered en grand nombre....Je n'excepte
pas de la loi pre'ce'dente, les effets dus aux causes morales."

It appears not to have occurred to Laplace, that this theorem
is founded on the mental phenomenon of expectation. But it is

6 ON THE FOUNDATIONS OF

clear that expectation never could exist, if we did not believe in
the general similarity of the past to the future, i.e. in the
regularity of nature, which is here deduced from it.

A little further on,... "II suit encore de ce theoreme que
dans une scrie d'eVenemens inde'finiment prolonged, 1'action des
causes r^gulieres et constantes doit 1'emporter a. la'longue, sur
celle, des causes irrdgulieres....Ainsi des chances favorables et
nombreuses etant constamment attaches a 1'observation des
principes eternels de raison de justice et d'humanite, qui fondent
et qui maintiennent les societes ; il y a un grand avantage a se
conformer & ces principes, et de graves inconveniens a s'en
e'carter. Que Ton consulte les histoires, et sa propre experience
on y verra tous les faits venir a 1'appui de ce re*sultat du calcul."
Without disputing the truth of the conclusion, we may doubt
whether it is to be considered as a "resultat du calcul."

The same expression occurs immediately afterwards in another
passage, in which the writer seems to allude to the history of his
own times, and to the ambition of the great chieftain whom he
at one time served.

Indeed it would seem as if to Laplace all the lessons of his-
tory were merely confirmations of the " r&sultats du calcul."
We are tempted to say with Cicero — "hie ab artificio suo non
recessit."

11. The results of the theory of probabilities express the
number of ways in which a given event can occur, or the pro-
portional number of times it will occur on the long run : they
are not to be taken as the measure of any mental state ; nor are
we entitled to assume that the theory is applicable wherever a
presumption exists in favour of a proposition whose truth is
uncertain.

Nevertheless it has been applied to a great variety of induc-
tive results ; with what success and in what manner, I shall
now attempt to enquire.

12. Our confidence in any inductive result varies with a
variety of circumstances ; one of these is the number of particular
cases from which it is deduced. Now the measure of this confi-
dence which the theory professes to give, depends on this
number exclusively. Yet no one can deny, that the force of the

TEE THEOR Y OF PROBABILITIES. 7

induction may vary, while this number remains unchanged.
This consideration appears almost to amount to a reductio ad
absurdum.

13. If, on m occasions, a certain event has been observed,
there is a presumption that it will recur on the next occa-
sion. This presumption the theory of probabilities estimates at

— . But here two questions arise ; What shall constitute a
in ~r ^

" next occasion" ? What degree of similarity in the new event
to those which have preceded it, entitles it to be considered a
recurrence of the same event ?

Let me take an example given by a late writer : —

Ten vessels sail up a river. All have flags. The presumption

that the next vessel will have a flag is — . Let us suppose the

ten vessels to be Indiamen. Is the passing up of any vessel
whatever, from a wherry to a man of war, to be considered as
constituting a " next occasion" ? or will an Indiaman only satisfy
the conditions of the question ?

It is clear that in the latter case, the presumption that the
next Indiaman would have a flag is much stronger, than that,
as in the former case, the next vessel of any kind would have

one. Yet the theory gives — as the presumption in both cases.
If right in one, it cannot be right in the other. Again, let all the

flags be red. Is it - - that the next vessel will have a red flag,
12

or a flag at all? If the same value be given to the presumption
in both cases, a flag of any other colour must be an impossibility.
It is to be noticed, that I only refer to the visible differences
among different kinds of vessels, and not to any knowledge we
may have about them from previous acquaintance.

14. I turn to a more celebrated application of the theory.
All the movements of the planetary system, known as yet,

are from west to east. This undoubtedly affords a strong pre-
sumption in favour of some common cause producing motion in
that direction. But this presumption depends not merely upon
the number of observed movements, but also on the natural

8 ON THE FO UNDA TIONS OF

affinity which in a greater or less degree appears to exist among

them.

This is so natural a reflection, that Lacroix, in calculating
the mathematical value of the presumption, omits the rotatory
movements, and, I believe, those of the secondary planets, in
order, as he expressly says, to include none but similar move-
ments. But in the admission thus by implication made, that
regard must be had to the similarity of the movements, too
much is conceded for the interests of the theory. For are the
retained movements absolutely similar ? The planets move in
orbits of unequal eccentricity and in different planes : they are
themselves bodies of various sizes ; some have many satellites
and others none. If these points of difference were diminished
or removed, the presumption in favour of a common cause deter-
mining the direction of their movements would be strengthened;
its calculated value would not increase, and vice versa.

Again, up to the close of 1811, it appears (Laplace) that
100 comets had been observed, 53 having a direct and 47 a
retrograde movement. If these comets were gradually to lose
the peculiarities which distinguish them from planets — we should
have 64 planets with direct movement, 47 with retrograde. The
presumption we are considering would, in such a case, be very
much weakened. At present, we unhesitatingly exclude the
comets on account of their striking peculiarities: in the case
supposed we should with equal confidence include them in the
induction. But at what precise point of their transition-state
are we abruptly, from giving them no weight at all in the
induction, to give them as much as the old planets ?

15. It is difficult to acquiesce in a theory which leads to
so many conclusions seemingly in opposition to the common
sense of mankind.

One of the most singular of them may, perhaps, serve as a
key to explain their nature. When any event, whose cause is
unknown, occurs, the probability that its ci priori probability
was greater than % is f . Such at least is the received result.
But in reality, the <i priori probability of a given event has no
absolute determinate value independent of the point of view in
which it is considered. Every judgment of probability involves
an analysis of the event contemplated. We toss a die, and an

THE THEOR Y OF PROBABILITIES. 9

ace is thrown. Here is a complex event. We resolve it into,
(1) the tossing of the die ; (2) the coming up of the ace. The
first constitutes the 'trial,' on which different possible results
might have occurred ; the second is the particular result which
actually did occur. They are in fact related as genus and
differentia. Besides both, there are many circumstances of the
event ; as how the die was tossed, by whom, at what time,
rejected as irrelevant.

This applies in every case of probability. Take the case of
a vessel sailing up a river. The vessel has a flag. What was
the a priori probability of this? Before any answer can by
possibility be given to the enquiry, we must know (1) what cir-
cumstances the person who makes it rejects as irrelevant. Such
as, e.g. the colour of which the vessel is painted, whether it is
sailing on a wind, &c. &c. ; (2) what circumstances constitute in
his mind the ' trial ;' the experiment which is to lead to the
result of flag or no flag; must the vessel have three masts?
must it be square rigged? (3) What idea he forms to himself
of a flag. Is a pendant a flag ? Must the flag have a particular
form and colour ? Is it matter of indifference whether it is at the
peak or the main ? Unless all such points were clearly under-
stood, the most perfect acquaintance with the nature of the case
would not enable us to say what was the a priori probability of
the event : for this depends, not only on the event, but also on
the mind which contemplates it.

The assertion therefore that f is the probability that any
observed event had on an a priori probability greater than -J, or
that three out of four observed events had such an & priori
probability, seems totally to want precision. A priori proba-
bility to what mind? In relation to what way of looking at
them?

16. Let us see if this will throw any light on the ques-
tion. Let h be a large number. And suppose we took h trials
and that the probability of a certain event from each (considered

in a determinate manner) was — ; let us take a second set of h

m

trials for which the same quantity is — : and so on to -

77? m

and 1.

10 ON THE FOUNDATIONS OF

When the trials have taken place, we shall have approxi
mately,

+^+ ...... +—

m in

of the souht events. Of these

*«

had d priori a probability greater than J. Summing these series
and dividing the second by the first, we get ^^ , for the ratio

which the latter class of events bears to the total number. The
limit of this, when m is infinite, or when we take an infinite
number of sets of trials, is f , which is the received result.

17. Thus, it appears this result is based upon some thing
equivalent to the following assumption : — There are an infinity
of events whose simple probability a priori is x, and another
infinite number for which it is x'. These two infinities bear to
one another the definite ratio of equality (x and x' may repre-
sent any quantity from 0 to 1). Now in reality, as we have seen,
these numbers are not only infinite, but in rerum naturd inde-
terminate, and therefore the assumption that they bear to one
another a definite ratio is illusory.

And this assumption runs through all the applications of the
theory to events whose causes are unknown.

This position could be directly proved only by an analysis of
the various ways in which this part of the subject has been con-
sidered, which would require a good deal of detail. Those who
take an interest in the question, may without much difficulty
satisfy themselves, whether the view I have taken (which at
least avoids the manifest contradictions of the received results) is
correct.

18. I will add only one remark. If in (16) instead of
taking one event from each of the trials there specified, we had
taken p in succession, and kept account only of those sequences
of p events each, which contained none but events of the kind
sought ; we should have had of such sequences

THE THEOR Y OF PROBABILITIES. 11

of which

would have belonged to trials where the simple d, priori proba-
bility was > - : the ratio of these two expressions is ultimately

This is the expression applied to determine the probability of a
common cause among similar phenomena, as in the case already
mentioned of the planets.

But this application is founded on a petitio principii : we
assume that all the phenomena are allied : that they are the
results of repetitions of the same trial, that they have the same
simple probability ; all that, setting other objections aside, we
really determine, is the probability, that this simple probability

common to all these allied phenomena is > - .

But how does this determine the force of the presumption
that the phenomena are allied, or, to use Condorcet's illustration,
that they all come out of the same infinite lottery ?

19. The object of this little essay being to call attention
to the subject rather than fully to discuss it, I have omitted
several questions which entered into my original design.

The principle on which the whole depends, is the necessity
of recognizing the tendency of a series of trials towards regu-
larity, as the basis of the theory of probabilities.

I have also attempted to show that the estimates furnished by
what is called the theory a posteriori of the force of inductive
results are illusory.

If these two positions were satisfactorily established, the
theory would cease to be, what I cannot avoid thinking it now
is, in opposition to a philosophy of science which recognizes
ideal elements of knowledge, and which makes the process of
induction depend on them.

ON THE METHOD OF LEAST SQUARES*.

THE importance attached to the method of least squares is
evident from the attention it has received from some of the
most distinguished mathematicians of the present century, and
from the variety of ways in which it has been discussed.

Something, however, remains to be done — namely, to bring
the different modes in which the subject has been presented into
juxta-position, so that the relations which they bear to one
another may be clearly apprehended. For there is an essential
difference between the way in which the rule of least squares
has been demonstrated by Gauss, and that which was pursued
by Laplace. The former of these mathematicians has in fact
given two different demonstrations of the method, founded on
quite distinct principles. The first of these demonstrations is
contained in the Theoria Motus, and is that which is followed
by Encke in a paper of which a translation appeared in the
Scientific Memoirs. At a later period Gauss returned to the
subject, and subsequently to the publication of Laplace's investi-
gation gave his second demonstration in the Theoria Combina-
tionis Observationum.

The subject has been also discussed by Poisson in the
Connaissance des Terns for 1827, and by several other French
writers. Poisson's analysis is founded on the same principle
as Laplace's : it is more general, and perhaps simpler. It is not,
however, my intention to dwell upon mere differences in the
mathematical part of the enquiry.

The consequence of the variety of principles which have been
made use of by different writers has naturally been to pro-
duce some perplexity as to the true foundation of the method.
As the results of all the investigations coincided, it was natural
to suppose that the principles on which they were founded were

* Transactions of the Camlridge Philosophical Society Vol. vra. [Read
March 4, 1844]

ON THE METHOD OF LEAST SQUARES. 13

essentially the same. Thus Mr Ivory conceived that if Laplace
arrived at the same result as Gauss, it was because in the process
of approximation he had introduced an assumption which re-
duced his hypothesis to that on which Gauss proceeded. In this
I think Mr Ivory was certainly mistaken ; it is at any rate not
difficult to show that he had misunderstood some part at least of
Laplace's reasoning: but that so good a mathematician could
have come to the conclusion to which he was led, shows at once
both the difficulty of the analytical part of the inquiry, and also
the obscurity of the principles on which it rests. Again, a recent
writer on the Theory of Probabilities has adopted Poisson's
investigation, which, as I have said, is the development of La-
place's, and which proves in the most general manner the supe-
riority of the rule of least squares, whatever be the law of pro-
bability of error, provided equal positive and negative errors are
equally probable. But in a subsequent chapter we find that he
coincides in Mr Ivory's conclusion, that the method of least
squares is not established by the theory of probabilities, unless
we assume one particular law of probability of error.

These two results are irreconcilable ; either Poisson or
Mr Ivory must be wrong. The latter indeed expressed his
dissent from all that had been done by the French mathematicians
on the subject, and in a series of papers in the Philosophical
Magazine gave several demonstrations of the method of least
squares, which he conceived ought not to be derived from the
theory of probabilities. In this conclusion I cannot coincide ;
nor do I think Mr Ivory's reasoning at all satisfactory.

From this imperfect sketch of the history of the subject, we
perceive that the methods which have been pursued may be
thus classified.

1. Gauss's method in the Theoria Motfis, which is followed
and developed by Encke and other German writers.

2. That of Laplace and Poisson.

3. Gauss's second method.

4. Those of Mr Ivory.

I proceed to consider these separately, and in detail.
For the analysis of Laplace and Poisson, I have substituted
another, founded on what is generally known as Fourier's theo-

14 ON THE METHOD OF LEAST SQUARES.

rem, having been first given by him in the Theorie de la Chaleur.
It will be seen that the mathematical difficulty is greatly dimin-
ished by the change.

GAUSS'S FIEST METHOD.

This method is founded on the assumption that in a series of
direct observations, of the same quantity or magnitude, the
arithmetical mean gives the most probable result. This seems so
natural a postulate that no one would at first refuse to assent to
it. For it has been the universal practice of mankind to take
the arithmetical mean of any series of equally good direct ob-
servations, and to employ the result as the approximately true
value of the magnitude observed.

The principle of the arithmetical mean seems therefore to be
true a priori. Undoubtedly the conviction that the effect of
fortuitous causes will disappear on a long series of trials, is an
immediate consequence of our confidence in the permanence of
nature. And this conviction leads to the rule of the arithmetical
mean, as giving a result which as the number of observations
increases sine limite, tends to coincide with the true value of the
magnitude observed. For let a be this value, x the observed
value, e the error, then we have

xl-a=el

x2-a = e2

&c. = &c.

And as on the long run the action of fortuitous causes disappears,
and there is no permanent cause tending to make the sum of the
positive differ from that of the negative errors, Se = 0, and
therefore

or, a-5

which expresses the rule of the arithmetical mean, and which is
thus seen to be absolutely true ultimately when n increases sine
limite.

In this sense therefore the rule in question is deducible from
& priori considerations. But it is to be remarked, that it is not
the only rule to which these considerations might lead us. For

ON THE METHOD OF LEAST SQUARES. 15

not only is Se = 0 ultimately, but S./e = 0, where /e-is any func-
tion such that /£ = — / ( — e) ; and therefore we should have

as an equation which ultimately would give the true value of x
when the number of observations increases sine limite, and
which therefore for a finite number of observations may be looked
on in precisely the same way as the equation which expresses
the rule of the arithmetical mean. There is no discrepancy
between these two results. At the limit they coincide : short of
the limit both are approximations to the truth. Indeed, we
might form some idea how far the action of fortuitous causes had
disappeared from a given series of observations by assigning
different forms to f, and comparing the different values thus
found for a.

No satisfactory reason can be assigned why, setting aside
mere convenience, the rule of the arithmetical mean should be
singled out from the other rules which are included in the general
equation S/ (x — a) = 0.

Let us enquire, therefore, whether there is any sufficient
reason for saying that the rule of the arithmetical mean gives the
most probable value of the unknown magnitude. In the first
place, it is only one rule out of many among which it has no
prerogative but that of being in practice more convenient than
any other: in the second place, if this were not so, it would not
follow that in the accurate sense of the words it gave the most
probable result. This objection I shall defer for a moment, and
proceed to consider the manner in which Gauss makes use of the
postulate on which his method is founded.

From the first principles of what is called the theory of
probabilities a posteriori, it appears that the most probable value
which can be assigned to the magnitude which our observations
are intended to determine, is that which shall make the a priori
probability of the observed phenomena a maximum. That is to
say, if a be the true value sought, a?x being the value observed
at the first observation, xz the corresponding quantity for the
second, and so on, the errors at the first, second, &c. observation
must be xv — a, x2 — a, &c., respectively; and if <fte . de be the
probability of an error e in any observation of the

16 ON THE METHOD OF LEAST SQUARES.

quantity which is to be made a maximum for a is propor-

tional to

<t> fo - «) <t> (*,-«)— * (*•-«)•

Equating to zero the differential of this with respect to a,
we find

It

as the equation for determining a in a;. Let — = ^, then it be-

comes

2j ^ (a? - a) = 0.

Now we have assumed that the most probable value of a is given

by the equation

2»(a>-a) = 0:

and it is impossible to make these equations generally coincident,
without assuming that

= me, m being any constant ;

6'e

^—
<Pe

6'e
hence —

Now as the error e is necessarily included in the limits
oo + co , we must have

r 0e^e=

2?r
or if we adopt the usual notation, and replace m by 2A2,

Consequently, we are thus led to adopt one particular law of
probability of error as alone congruent with the rule of the
arithmetical mean.

But, in fact, we are perfectly sure that in different classes of

ON THE METHOD OF LEAST SQUARES. 17

observations the law of probability of error must vary, and we
have no direct proof that in any class it coincides with the form
assigned to it. Therefore one of two things must be true,
either the rule of the arithmetical mean rests on a mere illusory
prejudice, or, if it has a valid foundation, the reasoning now
stated must be incorrect. Either alternative is opposed to
Gauss's investigation. For the reasons already given, we are,
I think, led to adopt the latter, and then the question arises,
wherein does the incorrectness of the reasoning reside ? It
resides in the ambiguity of the words most probable. For let
us consider what they imply in the theory of probabilities
cl posteriori.

Suppose there were m different magnitudes «1«2...am, and
that each of these were observed n times in succession. Let
this process be repeated p times, p being a large number which
increases sine limite. Thus we shall have pm sets of observa-
tions each containing n observations.

Of these a certain number K will coincide with the set of
observations supposed to be actually under discussion ; and we
shall have the equation

where k is that portion of K which is derived from observations
of %.

Then, ultimately, the most probable value which the given
series of observations leads us to assign to a, is (supposing a is
susceptible only of the values a^...^) equal to ar, r being
such that the corresponding quantity kr is the maximum
value of k.

To make the case now stated entirely coincident with the
one which we are in the habit of considering, we have only to
suppose (making m infinite) that the series of magnitudes
a^ . . . am includes all possible magnitudes from — GO to + co .

Now from this statement, it is clear there is no reason for
supposing that because the arithmetical mean would give the
true result if the number of observations were increased sine
limite, it must give the most probable result the number of
observations being finite.

2

18 ON THE METHOD OF LEAST SQUARES.

The two notions are heterogeneous : the conditions implied
by the one may be fulfilled without introducing those required
by the other : and we have already seen that by losing sight of
this distinction, we are led to the inadmissible conclusion, that a
principle recognised as true ci priori necessarily implies a result,
viz. the universal existence of a special law of error, not only
not true a priori, but not true at all.

Having stated what seem to me to be the objections in point
of logical accuracy to this mode of considering the subject,
I will briefly point out the manner in which, from the law of
eiTor already obtained, the method of least squares is to be
deduced.

Let

e = ac + ~b + &c. — ~l

e = ax + + &c. — F2

&c. = &c.

en = anx + bny + &c. - Vn J

be the system of equations of condition, which are to be com-
bined together so as to give the values of x, y, &c. The error
committed at the first observation is €v at the second e2, and so
on ; each observation corresponding to an equation of condition.
The probability of the concurrence of all these errors is,
(according to the law of error already arrived at) propor-
tional to

e - h2[(aix + liy + &c. - Fi)2 + (azx + Izy + &c. - F2)2 + &c.]

and it is to be made a maximum by the most probable values of
x, y, &c. These values will therefore make

(a1x + bly + &c. - FJ2 + (azx + b2y + ... - F2)2+...,

a minimum: that is to say, they will make the sum of the
squares of errors a minimum.

Hence the method of least squares. The conditions of the
minimum give the linear equations :

............ 09),

&c. =&c J

ON THE METHOD OF LEAST SQUARES. 19

in which system there are always the same number of equations
as there are unknown quantities to be determined.

The next investigation of the principle of the method of
least squares which I shall attempt to analyze is that of Laplace.

LAPLACE'S DEMONSTRATION.

If, in order to determine x from the equations of condition
stated in the last paragraph, we multiply the first by /^, the
second by //2, &c., and add : (/^2, &c. fulfilling the conditions

= 0, &c. = 0)
we find x —

and if we assume that Syt-te is equal to zero, then the resulting
value of x is Syu-F: the error of this determination being the
quantity 2/-te, which we have assumed to be equal to zero,
without knowing whether it really is so or not.

Now supposing there are n equations of condition, and p
quantities to be determined, and that n is greater than p, then
we see that there are n factors ^, yu-2.../i<n, and p conditions for
them to fulfil. They may therefore be subjected to n — p addi-
tional conditions.

This being premised, let us consider the probability that the
quantity 2^e will not be less than a, or greater than /3, a and /5
being any quantities whatever. The law of probability of error
at each observation being given, the question is evidently analo-
gous to the common problem of finding the chance, that with a
given set of dice the number of points thrown shall not be less
than one given number or greater than another.

We may therefore suppose that the probability in question
has been determined: call it P. Suppose also that we have
taken a = — I and /3 = Z, I being any positive quantity.

Then P is a function of Z, and of ^. ../*n.

Let us now so determine ^r..//,n, (subject to the conditions
already specified,) that P may be a maximum. When this is
done, it follows that there is a greater probability that the error

2—2

20 ON THE METHOD OF LEAST SQUARES.

in our determination of cc, viz. 2/Lte, lies within the limits ± Z,
than if we had made use of any other set of factors whatever.

On this principle Laplace determines what he calls the most
advantageous system of factors.

It does not follow that the value thus obtained for x is the
most probable value that could be assigned for it. But if we
consider a large number of sets of observations, (the quantities
a, b, &c. being the same for all) then the error which we commit
by using Laplace's factors will in a greater proportion of cases
lie between ± I, than if wre had used any other system of factors.

The investigation has reference merely to the different ways
in which by the method of factors a given set of linear equations
may be solved.

We now enter on the analysis requisite to determine P.

Let the probability that 2/^e will be precisely equal to u,
be pdu. Then manifestly

P =

and we have therefore only to determine p.

Let et €2...en be the errors which occur at the first second &c.
observation; (f>ie1d61, $2e2e?<-2...</>nenden be the probabilities of
their occurrence: the form of the function <f> determining the
law of probability of error, which, for greater generality, we
suppose different at each observation. The probability of the
concurrence of these errors is of course

and the first principles of the theory of probabilities show that
the value of pdu will be obtained by integrating (1), er..en
being subjected to the condition £/ze = u.
Thus

with the relation

ON THE METHOD OF LEAST SQUARES. 21

Consequently

pdu =

Now by Fourier's theorem
A "-A*tC»---A

= i f "«k (

^ •'-«

cos a

which, replacing — by a, becomes

/M^r",. _cosa(»--2/*)<fcu.

r r

I dai. .

Therefore

7 /in6?6n r °° 7 r +o° 7 r +

lu = -^-\ dx deL...

" J f\ <* —m •> —rr

Now if w and ew are to vary together

Jw = -tc?6 and therefore

-| !• 00 /« +00 /• +00

^ = —1 c^a J^... I d6n<f>1el...<j)n€ncosa(u-2,}j,6) ...... (5).

I? J 0 J -ao J -oo

Arid finally,

1 r+i r°° r+co r+0°

P=_| du\ da.} de...\ . d€n6l€1...6n

7Tj_l J0 ^-oo /-•

Now let us suppose that equal positive and negative errors
are equally probable. In this case $e = (£(— e), and conse-

quently,

,*«

I $e sin apede = 0.

J -oo

Hence (6) will become

2 rz /•« /•+« r+*

=- c?w cosaw^a ^ cos dfjuledel . . . </>n

TTJo Jo J-oo *-»

22 ON THE METHOD OF LEAST SQUARES.

The next step is to find an approximate value of this
expression.

/• +•*> /-+30

When « = 0 I <£e cos a//,ecZe == I ^e^e = 1,

J -ao ^ -oo

as the error e must have some value lying between + oc .

It is clear this is the greatest value the integral in question
can have, and therefore as n increases sine limite, the continued
product

/• +« /• +00

^>161 cos /^ae, Jej . . . / <£nen cos f*naenden

» —00 » -00

decreases sine limite, (being the product of n factors each less
than unity) except for values of a differing infinitesimally from
zero.

eVe, K* = f
^o

and develope each of the cosines in the above-written continued
product. It is thus seen to be equal to

- a

- &c.

Again, n being very large and ultimately infinite, it is
evident that 2//,V is of the same order of magnitude as n, while
^/^a&i^a is of the order of w2, the former term of the coefficient of
a4 may therefore be neglected in comparison with the latter,
which again may be replaced by \ (2^2&2)2, from which it differs
by a quantity of the order of n. Similar remarks apply with
respect to the higher powers of a.

Thus the continued product may be replaced by

or by e^2^2*7; a function which is coincident with it when a is
infinitesimal. When a is finite both are, as we have seen, in-
finitesimal.

Consequently,

=-fldu

*^" J o J n

(8),

OAT THE METHOD OF LEAST SQUARES. 23

or

where we have supposed

u = 2

It is evident, that whatever I may be, this expression for P
is a maximum when

2/^2&2 is a minimum.

Hence we get the following remarkable conclusion : When the
number of observations increases sine limite the most advantage-
ous system of factors are those which make

a mnmum.

It remains to determine JJL from the condition of the minimum
taken in connexion with those already stated, viz.

2yua = l, 2/^ = 0, &c. = 0.
We have

(A).

= 0

&c. = 0 J
Let X15 X2... \ be indeterminate factors, then we may put

............ (B).

&c. = &c. J

From the n equations (B) we deduce a new system of p
equations. To obtain the first of these, we multiply equations

(B) by % , j5, &c. respectively, and add the results. For the

&1 KZ

second, we employ instead of the factors yf , &c., the factors

#i

j\, T| , &c. and then proceed as before. And similarly for the

K! K%

others.

24 ON THE METHOD OF LEAST SQUARES.

In consequence of the relations

2/*a=l, 2/*5 = 0, &c.=0,
the new system of equations will be

0 = &c.

(C).

These p equations determine \, X2,... X,, and thus in virtue of
(B) the values of ^ /v pn become known. Finally as

x will be completely determined.

Now let us recur to the original equations of condition stated
in the last paragraph.

&c. - 7 ^

.- F

&c. = &c.
e = ac

— Fn J

From this system we deduce a new one, containing p equations.
The first of these is got by multiplying equations (a) by

r| > p > &c-> and adding the results : the second by using the
factors j~ , ~, &c.: and so on as before. The resulting system

1 **^S

will be, neglecting all errors,

&c.

= &c.

ON THE METHOD OF LEAST SQUARES. 25

The system (ft') contains as many equations as there are un-
known quantities x, y, &c. I proceed to show that if x be
determined from this system, its value will be the same as if it
had been obtained from the most advantageous system of fac-
tors, namely, that which is determined by means of (B) and ((7).
In order to prove this, we multiply equations (ft') by X1? X2, &c.,
and add the results. Then, in virtue of (G)

Or,

V V

x = (X, at + \11+ &c.) jj- + (\ at + X2 £2 + &c.) Ff + &c.

**l K2

that is to say, as is seen on referring to (B),

as before; which proves that the system (ft') gives the same
value for x as the most advantageous system of factors. More-
over, as (ft?) is symmetrical in x and a, y and 5, &c. it is clear
that it .will also give the most advantageous values for y and
the other unknown quantities.

When the law of probability of error is the same at every
observation ^ = £2 = &c. and (ft1) reduces itself to (ft) given at
p. 18 as the result of the method of least squares. In the
general case, it expresses the modification which the method of
least squares must undergo, when all the observations are not of
the same kind, namely, that instead of making the function
2 (ax + by + &c. — F)2 a minimum with respect to xy, &c., we

must substitute for it the function 2 ™ (ax+ ly + &c.— F)2, and

K

then proceed as before.

Such, in effect, is Laplace's demonstration, except that he
supposed the law of error the same at each observation. The
form in which I have presented it is wholly unlike his. The
introduction of Fourier's theorem enables us to avoid the theory
of combinations, and also the use of imaginary symbols. It
must be admitted that there are few mathematical investigations

26 ON THE METHOD OF LEAST SQUARES.

less inviting than the fourth chapter of the Theorie des Proba-
bilites, which is that in which the method of least squares is
proved.

It may be worth while to recur to the general formula:

j r+l r00 /•+« /•+»

P = -I du I da. \ del . .. I den^ e1 ...(f>n en cos a (u —

TTjji Jo' •>_„ J-oo

It is certain that 2/^e lies between the limits + co . There-
fore when I = co , P should be equal to unity. I proceed to show
that this is the case.

doi f de,...! Jen^,1e1.

J-oo •'-w

P =-

when m — 0.

Effecting the integration for M,

P L =

1 f00 _ *2 /%+'° r+*

I e~4^2 ja I J6i0>% den<j>^ ... </>nen cos a 2/*e ...... (10)

VTT *• *-• •'-»

wr
when m — 0,

since

/"f°° VTT. •

/•+«

e-m*

•/_«

r+o°

and I e~ m2u2 sin ocw du - 0.

•^ -00

Integrating for a, we see that when m = 0
P. = f" «*£,... f"^^...^^

^ -00 •/ -00

Or,

And as each of these integrals is separately equal to unity,
P = 1, which was to be proved.

ON THE METHOD OF LEAST SQUARES. 27

I proceed to show that in a particular case in which the
value of P can be accurately determined, Laplace's approxima-
tion is correct. It has sometimes been thought that the intro-
duction of the negative exponential involves a petitio principii,
and is equivalent to assuming a particular law of error. It is
therefore desirable, and I am not aware that it has hitherto
been done, to verify his result in an individual case.

Let the law of error be the same in all the observations,
and such that <£e = ^ e+% the upper sign to be taken when e is
positive.

Let ^ = p 2 = &c. = 1 , then

(4c*i) de. ...... e+e« c?ecos a «*-

or

coswa

The value of p is thus given by a known definite integral, which
has been discussed by M. Catalan in the fifth volume of
Liouvilles Journal.

It may be developed in a series of powers of u. Up to
w2(n l) no odd power of u can appear in this development, for

Q?p

5-^- da is finite while p is less than n, and therefore the

integral may be developed by Maclaurin's theorem. For higher
powers the method ceases to be applicable, and we must com-
plete the development by other means. But as we suppose
n to increase s. L the integral tends to become developable in a
series of even powers only of u. Thus

r cosua , r d* 2 p a2 ,

J.(M^^(^

Let

Then

28 O.V THE METHOD OF LEAST SQUARES.

and generally,

Now

,, , _7T 1 . 3. ..271-3

/W~2'2.4...27i-2
TT 1 . 3. ..271-5

_ , _ 7T 1 . 3. ..271-7

and generally,

"~ '(2w-l-2p). ..(271-3) '

Thus

f^cosuoLda -(11
Jo TTWT =>r "2 ' 2S=3 "
The coeflficient of u*p is

1.8...(2p-l)

XJ * "

1 1.3

or,

. _J _ £ _ 1 _
±^H 1.2...^ '2P' (2n-l-2p)...(2n-3) "

Let 7i become infinite, this becomes

/M 1 J_
^ Wl .2...^ (4n)p>

and we have only to determine what f(ri) then becomes.
Now by Wallis's theorem

2t4...2n-2 -

Therefore jfo = - ( - J when n is infinite,

ON THE METHOD OF LEAST SQUARES. 29

Consequently,

rcosuoida 1 __ &c

J n^?f=2 (n "p'

e~4n when % is infinite.

==_[_)
2 V^/

Therefore,

1 _ —

2Vw7r

-j rl «2

and P= —j= e~^ du.

Vn-TrJo

Now the value given for P at p. 23 is
P=-. = \e

In the present case //, = 1 ;

Ar2 = -| /* e~*ezd€ = 1 ; and consequently 2yu.2&2 = n.
Thus

P= — = e ~^ duj as before.

V/iTT /,

Thus Laplace's approximation coincides with the result
obtained by an independent method.

This example serves to show distinctly the nature of the
approximation in question.

The function p having been developed in a series of powers
of u, we take the principal term in the coefficient of each power
of w; that is, the term divided by the lowest power of n. We

neglect for instance every such term as — ^ uZp, because we have

U
a term in w2p divided by np. Thus we retain — and neglect

uz(p~8
— ^— , although, unless u be large, the former term is of the

same or a lower order of magnitude than the latter. That La-
place's method does in a very general manner give an approxima-

30 ON THE METHOD OF LEAST SQUARES.

tion of this kind cannot, 1 think, be questioned, especially after
the verification we have just gone through. But some doubt
may perhaps remain, whether such an approximation to the
form of the function P, if such an expression may be used, is
also an approximation to its numerical value, when we consider
that in obtaining it we have neglected terms demonstrably larger
than those retained.

For two recognized exceptions to the generality of Laplace's

investigation, viz. where <£e=-- - -z, and the case in which

/A4, //.2..., decrease in infinitum sine limite, I shall only refer to
p. 10 of Poisson's paper in the Connaissance des Terns for 1827.
Neither affects the general argument. We now come to Gauss's
second method, which is given in the Theoria Combinalionis
Olservationum.

GAUSS'S SECOND DEMONSTRATION.

The connexion between the method of Laplace, and that
which Gauss followed in the Theoria Combinationis Observa-
tionum, will be readily understood from the following remarks.

After determining ^...^ by the condition that P should be
a minimum, Laplace remarked that the same result would have
been obtained (viz. that 2/&2£* must be a minimum), if the
assumed condition had been that the mean error of the result,
^*. e. the mean arithmetical value of 2/ie should be a minimum.
(I should rather say that he makes a remark equivalent to this,
and differing from it only in consequence of a difference of nota-
tion, &c.) It is in fact easy to see that the mean value in ques-
tion is equal to

and as

. r ,
2 Jo updu

F*

2

,- ,
VTT

which is of course a minimum when 2//,2&2 is so.

0/V THE METHOD OF LEAST SQUARES. 31

Gauss, adopting this way of considering the subject, pointed
out that it involved the postulate that the importance of the
error S^e, i.e. the detriment of which it is the cause, is propor-
tional to its arithmetical magnitude. Now, as he observes, the
importance of the error may be just as well supposed to vary
as the square of its magnitude : in fact, it does not, strictly
speaking, admit of arithmetical evaluation at all. We must
assume that it is represented by some direct function of its
magnitude, such that both vanish together. One assumption is
not more arbitrary than another. Let us suppose, therefore,
that the importance of the, error is represented by (2/^e)2. That
is, that (S/u-e)2 is the function whose mean value is to be made a
minimum. I now proceed to find it.

(^6)2=S/,V+2S^26^2 ......... (13).

The mean value of e2 is /^ e2 $ede = 2&2.

Hence, that of 2/&V is 2 2^k\

The mean value of S/u^e^e, is zero, positive and negative
errors of the same magnitude occurring with equal frequency on
the long run.

Consequently,

mean of (2/*e)2 = 2 S^2 ......... (14) ;

and therefore, as before, 2/u.2 &2 is to be made a minimum. The
rest of the investigation is of course the same as that of La-
place.

Nothing can be simpler or more satisfactory than this
demonstration. It is free from all analytical difficulty, and
applicable whatever be the number of observations, whereas that
of Laplace requires this number to be very large.

Recurring to equation (11), differentiating it for m2, and then
making m — 0, we find

r x *« = r " *«, *i • • • f

^ -00 ^ -03 J -

and as the first member of this equation is evidently the mean
value of w2 or of (S/*e)2, this is a new verification of our analysis.

32 ON THE METHOD OF LEAST SQUARES.

As an illustration of Gauss's principle, let the fourth power
of the error be taken as the measure of its importance :
(2/-te)4 = 2//V -f GS/^'V/e/e/ + terms involving odd powers of e.
Therefore,

mean of (2/*e)4 = 2V *4 + 24 S/O^&i'V (15)

and /Ltr.. jJbn must be so determined that this may be a minimum.

I have already said that the results given by what Laplace
called the most advantageous system of factors are not strictly
speaking the most probable of all possible results.

As the distinction involved in this remark seems to me to be
essential to a right apprehension of the subject, I will endeavour
to illustrate it more fully.

Kecurring to the equations of condition, as given in p. 18,
we see that the values Laplace assigns to the factors HJL^ &c.
are independent of V^VZ &c. They depend merely on the
coefficients ab &c., which are quantities known a priori, i.e.
before observation has assigned certain more or less accurate
values to the magnitudes FtF2 &c. All we then can say is,
that if we employ Laplace's system of factors, and also any
other, in a large number of cases (the coefficients ab &c., being
the same in all) we shall be right within certain limits in a
larger proportion of cases when the former system of factors is
made use of than when we employ the latter. And this conclu-
sion is wholly irrespective of the values of Vl F2 &c., and conse-
quently of those which we are led in each particular case to assign
to xy &c. The comparison is one of methods, and not at all one
of results. But when Vl F2 &c. are known, another way of con-
sidering any particular case presents itself. We can then com-
pare the probability of different results. For, let us consider a
large number of sets of equations of condition (in each of which
not only are ab &c. equal, as in the former case, but also F1 V2
&c.). The true values of the elements xy &c. may be different
in each. But in affirming that £77 &c., are the most provable
values of xy &c., we affirm that the true values of xy &c. are
more frequently equal to fy) &c. than to any other quantities
whatever. Here we have no concern with the method by which
the values £rj &c. were obtained. The comparison is merely one
of results.

ON THE METHOD OF LEAST SQUARES. 33

As for one particular law of error (that considered in p. 15),
the results of the method of least squares are the most probable
possible ; and as the function by which this law of error is ex-
pressed occurs in Laplace's demonstration of that method, it has
been thought that his approximation involved an undue assump-
tion, and that in fact his proof was invalid unless that particular
law of error was supposed to obtain.

It is easily seen that the method of least squares can give the
most probable results only for that law of error (if we except
another which involves a discontinuous function). Mr Ivory
attempted to show that Laplace's conclusions might be applied
to prove that the results of the method were, in effect, the most
probable possible, and thence drew the inference which I have
already mentioned. After some consideration, I have decided on
not entering on an analysis of his reasoning, which it would be
difficult to make intelligible, without adding too much to the
length of this communication. It is set forth with a good deal
of confidence ; Laplace's conclusions are pronounced invalid on
the authority of an indirect argument, and without any examina-
tion of the process by which he was led to them. I may just
mention that in the whole of Mr Ivory's reasoning, the proba-
bility that 2/xe is precisely equal to any assigned magnitude, is,
to all appearance at least, considered a finite quantity, though it
is perfectly certain that it must be infinitesimal.

It would seem as if he had taken Laplace's expression of the
probability in question, viz.

without being aware that in Laplace's notation I and a are in-
finite, and that consequently the expression is infinitesimal.
(Vide TillocJis Magazine, LXV. p. 81.)

MR IVORY'S DEMONSTRATIONS.

They are three in number. Two appeared in the sixty-fifth,
and a third in the sixty-seventh volumes of TillocJis Magazine.

The aim of all three is the same, namely, to demonstrate the
rule of least squares without recourse to the theory of proba-

3

34 ON THE METHOD OF LEAST SQUARES.

bilities, which appeared to him to be foreign to the question.
The grounds of this opinion he has not clearly developed :
perhaps the best refutation of it will be found in the unsatis-
factory character of the demonstrations which he proposed to
substitute for the methods of Laplace and Poisson. In common
with many others, Mr Ivory appears to have looked with some
distrust on the results obtained by means of this theory : a not
unnatural consequence of the extravagant pretensions sometimes
advanced on its behalf.

The first of his demonstrations rests upon what I cannot help
considering a vague analogy. In the equation of condition

e = ax — V,

lie remarks that the influence of the error e on the value of x
increases as a decreases, and versd vice : that consequently the
case is precisely similar to that of a lever which is to produce a
given effect, as of course the length of the arm must vary in-
versely as the weight which it supports.

Consequently, he argues, the condition to be fulfilled, in
order that the equations of condition may be combined in the
most advantageous manner, is the same as what would be the
condition of equilibrium, were a a a" &c. weights on a lever,
acting at arms e e e' &c. This condition is of course

%ae = 0, whence S(a# — F) a = 0,
the result given by the method of least squares.

But, granting that the influence of an error e, ought to be
greater when a is less, and versd vice, how are we entitled to
assume that the case is precisely similar to that of equilibrium
on a lever ? Apart from this assumption, there seems to be no
reason for inferring that because this influence increases as a
decreases, it must therefore vary inversely as a. By what
function of a the influence of e ought to be represented, is the
very essence of the question ; to determine, by introducing the

extraneous idea of equilibrium on a lever, that - is the function

a

required, seems to be little else than a petitio principii, concealed
by a metaphor*.

* I have omitted to notice some remarks which Mr Ivory appends to this
demonstration, as they do not appear to affect the view taken in the text.

ON THE METHOD OF LEAST SQUARES. 35

The second demonstration may be thus briefly stated.

The values of different sets of observations might be com-
pared if we knew the average error in each set, or if we knew
the average value of the squares of the errors in each. In either
case that would be the best set of observations in which the
quantity taken as the measure of precision was the smallest.

Similarly, by assigning different values to the unknown
quantities x, y, &c. involved in a system of equations of condi-
tion, we can make it appear that the mean of the squares of the
errors has a greater or less value. Therefore as of sets of obser-
vations, that is the best in which this quantity is least ; so of
different sets of results deduced from one set of observations, the
same is also true ; and therefore the sum of the squares of the
apparent errors is to be made a minimum.

There seems to be involved in this reasoning a confusion of
two distinct ideas; the precision of a set of observations is
undoubtedly measured by the average of the errors actually
committed, and if we knew this average, we should be able
to compare the values of different sets of observations. But
it is not measured by the average of the calculated errors,
namely, those which are determined from the equations of
condition when particular values have been assigned to x, y, &c.

The problem to be solved may be stated thus. Given that
the single observations of which the set is composed are liable
to a certain average of error, to combine them so that the re-
sulting values of the unknown quantities may be liable to the
smallest average of error.

This problem Laplace and Gauss have both solved. Their
solutions differ, because they estimated the average error in
different manners.

But how are we justified in assuming that to be the best
mode of combining the observations which merely gives the
appearance of precision not to the final results, but only to the
individual observations, and which, with reference to them, gives
no estimation of the probability that this appearance of accuracy
is not altogether illusory ?

The third of Mr Ivory's demonstrations is not, I think, more
satisfactory than the other two.

The kind of observations to which the method of least

3—2

36 ON THE METHOD OF LEAST SQUARES.

squares is applicable, are such, Mr Ivory observes, that there
exists no bias tending regularly to produce error in one direc-
tion, and that the error in one case is supposed to have no
influence whatever on the error in any other case.

From this principle he attempts to show that the method of
least squares is the only one which is consistent with the
independence of the errors.

When, however, we speak of the errors as being independent
of one another, only this can be meant, that the circumstances
under which one observation takes place do not affect the others.
In rerum naturd the errors are independent of one another.
Nevertheless, with reference to our knowledge they are not so,
that is to say, if we know one error we know all, at least
in the case in which the equations of condition involve only
one unknown quantity, which is that considered by Mr Ivory.
For the knowledge of one error would imply the knowledge
of the true value of the unknown quantity, and thence that
of all the other errors.

Mr Ivory states the following equations of condition :
e — ax — m
e1 = ax — m
&c. = &c.

He thence deduces the following value of x :

x =<*— o + <> 2 » an<i those of ee' are

He remarks that these errors are not independent of one
another, as all depend on the single quantity 2ae, which may be
eliminated between any two of the last-written equations : but
that there is one case in which they are independent of one
another, namely, when we assume 2ae = 0, which of course leads
to the method of least squares, and that in this case, as we shall
have

ON THE METHOD OF LEAST SQUARES. 37

each error is determined by " the quantities of its own experi-
ment." But this reasoning is perfectly inconclusive. In the
case supposed, ee &c. are as much connected together as in
any other, as may be shown by eliminating 2am between the
equations

and besides, apart from any mathematical reasoning, it is clear
that as if we know one error we know all, so also if we assign
any value to one, we have in effect assigned values to all, whe-
ther we use the method of least squares or any other.

Moreover, e is not determined by the quantities of its own
experiment alone, since Sara involves the results of all the ex-
periments ; there is no difference between this and the general
case, except that ^ae has ceased to appear in the equations. But
suppose we multiplied the equations of condition by any func-
tion of a, we might deduce the following values of x and e :

. m

~~

e = — m +

5' = -W + ^

a . (f>a

Mr Ivory's reasoning would apply word for word as before,
and would show that the best mode of combining the equations
of condition was to employ the factors </>a, (pa, &c. whatever be
the form of </>. As it thus would serve to establish, at least
apparently, an infinity of contradictory results, the inference is
that in no case has it any validity.

I have now completed, though in an imperfect manner, the
design indicated at the outset of this paper, namely, to give
an account of the different modes in which the subject has been
treated, and to simplify the analytical investigations. If I have
succeeded in doing this, the present communication may tend
to make a very curious subject more accessible than it has
hitherto been.

SOME REMARKS ON THE THEORY OF
MATTER.*

IN the present state of Science, there are few subjects of greater
interest than the enquiry whether all the phenomena of the uni-
verse are to be explained by the agency of mechanical force,
and if not whether the new principles of causation, such as
chemical affinity, and vital action, are to be conceived of as
wholly independent of mechanical force, or in some way not
hitherto explained cognate and connected with it. One reason
among many which makes this enquiry interesting is the cir-
cumstance that the application of mathematics to natural philo-
sophy has, up to the present time, either been confined to
phenomena, which were supposed to be explicable without
assuming any other principle of causation than ordinary " push
and pull" forces, or as in Fourier's theory of heat and Ohm's
theory of the galvanic circuit, has been based on proximate
empirical principles.

2. The intention of the remarks which I have the honour
to offer to the Society is to suggest reasons for believing that
while on the one hand it is impossible not merely from the
short-comings of our analysis but from the nature of the case to
reduce, as it appears that Laplace wished to do, all the pheno-
mena of the universe to one great dynamical problem, we cannot
recognise the existence of any principle of causation wholly
disconnected with ordinary mechanical force, or of which the
nature could be explained without a reference to local motion :
in other words, that the idea of "qualitative action" in the
sense which the phrase naturally suggests must be rejected. It
will be seen from the explanations I am about to attempt that
the objection which Leibnitz has opposed to the atomic, and in
effect to any mechanical philosophy, namely, that on such prin-

* Transactions of the Cambridge Philosophical Society, Vol. vui. p. 600.
[Read May 22, 1848.]

SOME REMARKS, &c. 39

ciples a finite intelligence might be conceived to exist by which
all the phenomena of the universe would be fully comprehended,
does not (whatever may be thought of its validity) appear to
apply to the views which I have been led to entertain. For
these views essentially depend on the conception of what may
be called a hierarchy of causes, to which we have no reason for
assigning any finite limit. Of this series of principles of causa-
tion, ordinary mechanical force is the first term.

3. With respect to the first point, namely, the impossibility
of explaining all phenomena mechanically, it may be remarked,
that we are met, in the attempt to discuss it, by the difficulty
which always attends the establishment of a negative proposition.
It is clear that as in the present state of our knowledge we are far
from being able to enumerate and classify the phenomena which
are or which might be produced by the combined agency of con-
ceivable mechanical forces, we are not in a position to decide
a priori that any given phenomenon might not be thus produced.
Non constatj but that the impossibility we find in the attempt to
explain the causes of its existence may have no higher origin
than the imperfect command which we have as yet obtained of
the principles of mechanical causation. We meet, it may be
said, with a multitude of ordinary dynamical problems which
have as yet received no adequate solution — why then should we
have recourse to new kinds of causes, while we have not as yet
exhausted the resources, if the expression may thus be used, of
those which we already recognise? To this enquiry no con-
clusive answer can be given, but the following considerations
will I think naturally suggest themselves.

4. In the first place, no even moderately successful attempt
has, I think, yet been made to explain any chemical phenomenon
on mechanical principles. It is quite true that we are unable,
to take a particular instance, fully to comprehend the mechanical
constitution of the luminiferous ether ; the determinations which
have as yet been attempted of the law of attraction between its
molecules cannot, I apprehend, be accepted as any thing more
than hypothetical or provisional results, and there are other
points involved in yet greater obscurity. Nevertheless the un-
dulatory theory of light has, as we all know, given consistent
and satisfactory explanations of a great variety of phenomena.

40 SOME REMARKS ON THE

Thus it appears, and tlie same remark might be educed from
other though similar considerations, that we are by no means
absolutely estopped by the imperfection of our mechanical phi-
losophy, from explaining phenomena really due to mechanical
forces, even when these phenomena are connected with subjects
not as yet fully comprehended : why then cannot some progress
be made in the mechanical explanation of chemical phenomena,
or of those, to mention no other class, which we are in the habit
of referring to vital action? In these cases, we see or seem to
see that the action of mechanical laws is modified or suspended ;
and though it is not demonstrably impossible that this is not
really the case, and that no other causes are at work beside the
"push and pull" forces of ordinary mechanics, yet we are at
least much tempted to believe, that the difficulties we meet with
do not arise from what may be called the disguised action of
mechanical forces but from the presence of an agency of a dis-
tinct nature. And to this view we find that most of those in-
cline who have made themselves familiar with the science of
chemistry or with that which has been called biology; and
further that, (with reference to the latter science) the insuffi-
ciency not only of a mechanical but even of a chemical phy-
siology has been generally admitted.

Secondly, it is to be observed that even if it be considered
doubtful whether a mechanical philosophy be not after all
sufficient for the explanation of all phenomena, it is at least
certain that it has not been proved to be so: and that by reject-
ing other conceivable modes of action than those which are
recognised by it, we unnecessarily and arbitrarily limit the
problem which the universe presents to us ; falling thereby into
an error similar to that of the atomists, who starting from the
assumption that the ap^ai, or first principles of all things, are
atoms and a vacuum proceeded to construct an imaginary world,
in accordance with this arbitrary hypothesis. At the same time
it must be granted that a purely mechanical* system such as
that of Boscovich is more self-consistent and contains, so to

The word mechanical is of course not used in antithesis to dynamical, in the
sense in which the latter is commonly employed by the philosophical writers of
Germany. The antithesis in question is foreign to the scope of the present essay,
and I have accordingly elsewhere used the word dynamical in its ordinary ac-
ceptation.

THEORY OF MATTER. 41

speak, less that is discontinuous, than any which should recog-
nise other principles, for instance chemical affinity, distinct from
force without enquiring into the relation which subsists between
them.

5. It may however be asserted that this enquiry is alto-
gether superfluous — that the power of exerting attractive or
repulsive force is one property of matter, that chemical affinity
(and so in other cases) is another — that the two are not merely
distinct, but absolutely independent and heterogeneous. But to
this view the arguments which seem to have led to the adoption
of a purely mechanical system, appear to prevent our assenting.
I shall therefore attempt to state what I conceive these argu-
ments to have been.

6. It is a fundamental principle of the secondary mechanical
sciences, for instance of the theory of light, that the secondary
qualities of bodies are to be explained by means of the primary.
Every substance, to use for a moment the language of Leibnitz,
is essentially active ; in other words it is to be conceived of as
the formal cause of the sensible qualities which are referred to it.
If we ask why gold is yellow and silver white, the answer at
once presents itself that the diiference of colour corresponds and
is due to a difference between the essential constitution of the
two substances. Now the essential constitution here spoken of,
and consequently the differences which individuate it in different
cases, may conceivably be something altogether incognisable to
the human intellect. The notion that it is so was expressed
scholastically by saying that substantial forms are not cog-
noscible. But if, setting aside this opinion, we affirm that the
essential constitution of each substance is a matter of which the
mind can take cognisance, we are led at once to the distinction
between primary and secondary qualities. The first are ascribed
to each substance as its essential attributes, in virtue of which
it is that which it is — the second result from the primary*, (by
which as we have said the essential or formal constitution of the
substance in question is determined,) and have reference to the
mind by which they are perceived, while the primary are
ascribed to it independently of any reference to a percipient

* Or that which in its formation it was to be, rb ri ty cli>at.

42 SOME REMARKS ON THE

mind: and a distinction, analogous or identical with that be-
tween primary and secondary qualities, has accordingly been
expressed by the antithesis between that which is a parte
hominis and that which is a parte universi. That the dis-
tinction between primary and secondary qualities is necessary
on the hypothesis on which we are proceeding, appears at once
from the consideration that if we affirm that all the qualities
of bodies of which we can form any conception are equally
subjective and phenomenal, nothing will remain of which the
mind can take cognisance, and by means of which our con-
ception of the nature of any one substance can be discriminated
from that of any other*. Let it be granted therefore that the
distinction of primary and secondary qualities is a necessary
element of physical science. It follows from this that the
secondary qualities in a manner disappear when we look at
the universe from the scientific point of view. Instead of
colours we have vibrations of the luminiferous ether — instead of
sounds vibrations of the ambient air, and so on. Now from
hence it follows that all the phenomena which we. see produced,
of whatever nature they may be, are all in reality dependent on
the primary qualities of matter. Furthermore, these primary
qualities themselves all involve the idea of motion or of a ten-
dency to motion. A body changes its form in virtue of tlxe
local motion (absolute or relative) of some of its parts; 'and
when I press a stone between my hands, I find that I can
produce no sensible change of form, while contrariwise the
stone reacts against my hands, tending to make them move in
opposite directions. I then say that the stone is hard as a
mode of expressing this, viz. that when an attempt is made to
produce relative local motion of its parts, it resists it in virtue
of its reactive tendency to produce motion in that which acts
upon it. Again, a body whose parts are readily susceptible of
relative local motion is said to be soft or fluid, and when a
sensible change of form is accompanied by a tendency to such
motion as shall restore the original form, it is said to be elastic,
and so on. We thus arrive at a point of view at which all

* The doctrine of the cognoscibility of substantial forms, which is intimately
connected with this distinction, is, as Leibnitz in effect remarks, as it were the
common character of those who with more or less success attempted in the seven-
teenth century, the restoration of science. Vid. Leibnitz, Epist. ad Thomas, i.

THEORY OF MATTER. 43

secondary qualities having disappeared, and all primary ones*
having been resolved into motion and tendency to motion, the
sciences which relate to phenomena appear to be resolved into
the general doctrine of motion. But if this be true the universe
can it is said present to us nothing but one great dynamical
problem. Motion, and force the cause of motion, belong essen-
tially to the domain of mechanics: and if chemical affinity be a
cause of local motion, that is, if in virtue of its action f a
particle of matter finds itself at a given time in a position dif-
ferent from that which it would else have occupied, chemical
affinity is not really distinct from mechanical force (which
looked at from the dynamical point of view includes everything
which is a cause of motion); whereas if it be not a cause of
motion the enquiry at once presents itself of what is it? In illus-
tration of this view we may refer to any chemical experiment. If
an acid is dropped into a glass containing any vegetable blue, the
colour is changed to red. But to say this is to say that the liquid
when the acid is introduced into it begins to act on the luminifer-
ous vibrations which exist near it in a different manner from that
in which it had previously acted. The whole change, whether we
call it a chemical phenomenon or not, consists in the introduction
of new forms of motion in virtue of the action of mechanical force.

7. From considerations of this kind it appears to follow
that a complete explanation of all phenomena would introduce
no principles beyond those with which the science of mechanics
is conversant. And in truth if the conclusion drawn had been
that all phenomena might, if our knowledge of nature were
sufficiently extensive, be reduced to cinematical considerations
(using the word cinematics in the large sense in which it is
equivalent to the doctrine of motion), I do not see how on our
fundamental hypothesis we could refuse to assent to it. But the
conclusion drawn by the maintainers of the all-sufficiency of a
mechanical philosophy is something different from this — and as
I conceive the error they appear to have committed is to be
sought for in this discrepancy. But before entering into the
discussion of this point, I will make a few remarks on certain
points in the history of what may be called the theory of matter.

* That is, all that are commonly enumerated as primary qualities,
t As, for instance, in the phenomenon of crystallization.

44 SOME REMARKS ON THE

8. If we suppose the maxim that secondary qualities are to
be explained by means of the primary to have been accepted
(either in that or in some equivalent form), or if not formally
accepted, at least unconsciously assumed, at a time when the
idea of mechanical force was as yet very imperfectly appre-
hended— the natural result of this state of things is the forma-
tion of an atomic theory. For in order to individuate the
constitution of any given body, we could only have had recourse
to the configuration or motion of its parts. Gold, to return to
our previous example, was said to be yellow in virtue of such
and such a configuration of its parts; since except configuration
there appeared to be no disposable circumstance*, if I may so
speak, whereby gold was in its intimate constitution to be
distinguished from silver or from anything else. But this
configuration must be independent of the body's visible and
external form, since changes of the latter do not affect the
body's sensible qualities. Hence it must be a configuration of
small parts, and we are thus at once led to the primitive form
of the atomic theory. In this the atoms possess the primary
qualities of larger bodies — they are of various forms and act
if the expression may be used by their forms, not by being
centres of attractive forces. Such was the atomistic system of the
school of Democritusf — a system which we know found ro
little favour among the scientific reformers of the seventeenth
century \. As an instance of the influence it exerted, I need
only mention the great work of Cudworth, in which it is pre-
sented apart from the atheistical doctrines with which it had
often been connected. Cudworth goes so far as to affirm that
Democritus and his followers had corrupted and degraded the
atomistic system which was originally altogether free from any
irreligious tendency, and which he sought to restore to its first
estate.

But as the imperfections of the atomic system became mani-

• Specific differences of motion seem for more than one reason not to have
been used in giving an account of the differences of bodies.

f See for a more favourable and, I think, a juster view of the philosophy of
Democritus than that which we commonly meet with in the writings of modern
historians of philosophy, Zeller's Philosophic Der Griecken, I. § 10.

+ The physical theories of Des Cartes, though not properly atomistic, since he
proceeded on the hypothesis of a plenum, yet in many respects are akin to those of
which we are speaking.

THEORY OF MATTER. 45

fest, and on the other hand mechanical conceptions came to be
more developed, a new form of this system arose. The atoms,
retaining their forms and those which are commonly called
their primary qualities, were now supposed to act as centres
of attractive force, in other words, each atom was to the rest
a cause of motion. But as the ordinary "primary qualities"
of bodies may as we have seen be analysed into conceptions
which involve nothing beside motion and force, this new form
of the doctrine may clearly be considered merely as a state of
transition to that which is now known by the title of Boscovich's
theory *. To Boscovich appears to belong the credit of having
perceived that if the atoms were conceived of simply as unex-
tended centres of force the primary qualities of bodies might
sufficiently be accounted for without supposing them to result
from the primary qualities of their constituent atoms — a mode
of explanation of which, though there has been something like
a return to it in some recent speculations, it may be observed
that it explains nothing. Boscovich's theory seems to have
been so completely in accordance with the direction in which
mathematical physics have of late been moving, that it was
adopted as it were unconsciously — almost all modern investiga-
tions on subjects connected with molecular action are in effect
based on his views, though his name is, comparatively speaking,
but seldom mentioned. And this theory, (whether or not the
hypothesis of the existence of discrete centres of action be or be
not essential to it, a question connected with that which in
former times caused so much perplexity, namely, the nature of
continuity, and which it is not necessary to my present purpose
to consider), is in truth the highest developement which the
mathematical theory of matter has as yet received — it is that on
which the pretensions of mathematical physicists to vindicate
for their own methods the right, so to speak, if not the power, to
explain all phenomena mainly depend. Adopting for the sake
of definite conception the received form of this theory, that

* It is, I believe, known that Boscovich's fundamental idea was deduced by a
not unnatural filiation from the monadism of Leibnitz. Yet the scope and limits
which he proposed to himself differ essentially from those of the German phi-
losopher, inasmuch as they are essentially physical. Moreover, the latter would
have objected on the principle of sufficient reason to the want of any thing to indi-
viduate the atoms of Boscovich ; and, at least in the latter years of his life, to the
"Feme Wirkung," on which the whole theory depends.

46 SOME REMARKS ON THE

namely in which the centres of force are discrete and at in-
sensible distances from each other, I now shall attempt to show
what ulterior developements it admits of, and how by means of
these the error noticed at the close of the last Section, namely,
the confounding the admission that all phenomena are to be
explained cinematically with the assertion that they can all be
explained mechanically may be met, and, as it seems to me,
sufficiently refuted.

9. I begin by observing that though we speak and shall
continue to do so of the action of matter on matter, yet that no
part of the views I am about to state depends on the hypothesis
we adopt touching the nature of causation. They would remain
unchanged whether we accept a theory of pre-established har-
mony, or one of physical influence, or whether we abstain from
all theories on the subject. This being understood, we may,
I think, lay down the axiom that whatever property we ascribe
to matter, we may also ascribe to it, the property of producing
in other portions of matter the former property. Of this axiom
the present state of Boscovich's theory affords a familiar illustra-
tion. Every portion of matter is locally moveable, therefore we
may ascribe to any portion of matter the power of producing
motion in any other, hereby giving rise to the whole doctrine of
attractive and repulsive forces. At this point we have hitherto
stopped, but for no satisfactory reason. We may proceed farther,
and we are therefore bound, in constructing the most general
possible hypothesis, to do so : we may ascribe to each portion of
matter the power of engendering in any other that which we
call force, in other words the power of producing the power of
actuating the potential mobility of matter. It is not a priori at
all more easy to conceive that A should have the power of
setting B in motion, or of changing the velocity it already has,
than that G should have the power of enabling A to act on B,
or of changing the mode of action which A already possesses.
And let it be observed, that the new power thus ascribed to G
is as distinct from force, as force is from velocity. The two are
related as cause and effect, but formally are wholly independent.
Now unless this hypothetically possible mode of action can be
shown to have 110 existence in rerum naturd, it is clear that
the inference from the conclusion that no phenomenon can be

THEORY OF MATTER. 47

imagined not resoluble en derntire analyse, into local motion
to the assertion that mechanical force is the only agency to be
recognised in the material universe is altogether illusory. For
matter may act on matter in a manner wholly distinct from
force, and yet this kind of action shall, ultimately and indi-
rectly, manifest itself in modifications of local motion. Further-
more, if for an instant we call this kind of action (force)2, we
shall at once be led to recognise a hypothetically possible mode
of action of matter on matter which in accordance with analogy
we shall call (force)3, which consists in the power of modifying
(force)2. And so on, sine limite.

10. If we compare the language in which the relation
between mechanical force and chemical affinity is commonly
spoken of, we shall I think perceive its analogy with that which
I have used in describing the mode of action which we have
called (force)2. Its chemical affinity is spoken of as something
which suspends or modifies the action of force, as something
distinct from it, but which yet interferes with its effects. Or
again, if in physiological writings we observe the manner in
which vital action* is described we recognise, or seem at least
to do so, the possibility of referring its effects to that mode of
action which we have called (force)3. I do not however wish to
lay much stress on these similarities, because I think the kind
of reasoning we have pursued shows more satisfactorily than they
can do, that if chemical affinity and vital action are not resoluble
into force, they must be referred to some of the modes of action
we have pointed out.

It would be useless to remark on the many points of specu-
lation which here present themselves. The expansion of bodies
by heat may however be particularly mentioned, because not-
withstanding what has been learnt with relation to the theory of
heat, nothing like a mechanical explanation of this phenomenon
has as yet been discovered. It seems to depend not on the
introduction of new mechanical forces, but on a modification of
those which already exist ; such modification, in cases of ordi-
nary conduction, being propagated from one part of the body to
that which is next it. — It is easy to conceive that by an altera-

* I am, of course, not to be understood as suggesting a materialistic explana-
tion of phenomena of thought or volition.

48 SOME REMARKS, &c.

tion in the function which expresses the mutual action of the
molecules, the body may pass into a new state of equilibrium in
which the average distance between adjacent molecules may be
increased or diminished. If such an explanation could be esta-
blished, we should have a case of the action of (force)2.

11. In conclusion, it may be well to remark that mathe-
matical analysis is conceivably as applicable to these new
modes of action of matter on matter as to ordinary questions in
dynamics. It is, however, easily seen that as in these we deal
chiefly with differential equations of the second order, and in
merely cinematical questions with equations of the first only, so
contrariwise when we introduce higher powers of force (so to
call them) we shall correspondingly have to do with equations
of higher orders. I venture to predict with a degree of confi-
dence, which doubtless I shall not communicate to many, that if
we ever succeed in establishing a mathematical theory of che-
mistry, it will be as much conversant with equations of the third
or of a higher order, as physical astronomy is with equations
of the second.

REMARKS

ON THE FUNDAMENTAL PRINCIPLE OF
THE THEORY OF PROBABILITIES*.

I WISH to make an addition to the remarks on the foundation
of the theory of probabilities which were offered some years since
to the notice of the Societyf. My intention in doing so is to
consider, in what way the proposition, which I conceive to be
the fundamental principle of the theory, may be the most clearly
and conveniently expressed. This principle may for the moment
be thus stated : " On a long run of similar trials, every possible
event tends ultimately to recur in a definite ratio of frequency."
Our conviction of the truth of this proposition is, I think, intui-
tive,—the word being used, as in all similar cases, with reference
to the intuitions of a mind, which has fully and clearly appre-
hended the subject before it, and to which therefore to have
arrived at the truth and to perceive that it has done so are
inseparable elements of the same act of thought. If we endeavour
to translate the proposition just stated into ordinary philosophical
language, we may in the first place remark that the phrase
"similar trials," expresses the notion of a group or genus of
phenomena to which the different results are subordinated as
distinct species. If the trial is the throwing of a die, this may be
regarded as the generic character; the occurrence of ace, deuce,
&c. constituting different species. Thus much is clear ; but it
is less obvious how the idea expressed by a " long run of trials"
in " definite series of experiments," and the like, is to be ex-
pressed, so as to make the analogy between the fundamental
principle of the theory of probabilities and those of other sciences
more obvious than it has hitherto been. The idea in question

* Transactions of the Cambridge Philosophical Society, Vol. ix. p. 605. [Read
Nov. 13, 1854.]

t Transactions of the Cambridge Philosophical Society, Vol. vm. p. r. [p. i of
this volume.]

4

50 REMARKS ON THE FUNDAMENTAL PRINCIPLE

is not readily expressed in any way, because in its own nature it
is negative and indefinite. The phrases I have just quoted imply
merely the absence of the limitations inseparable from individual
cases, or from any finite number of such 'cases, whether contem-
plated as actually existent or as about to be developed within
definite limits of space and time.

When individual cases are considered, we have no conviction
that the ratios of frequency of occurrence depend on the circum-
stances common to all the trials. On the contrary, we recognise
in the determining circumstances of their occurrence an extra-
neous element, an element, that is, extraneous to the idea of the
genus and its species. Contingency and limitation come in (so
to speak) together ; and both alike disappear when we consider
the genus in its entirety, or (which is the same thing), in what
may be called an ideal and practically impossible realization of
all which it potentially contains. If this be granted, it seems to
follow that the fundamental principle of the theory of probabilities
may be regarded as included in the following statement ; — "The
conception of a genus implies that of numerical relations among
the species subordinated to it."

2. But in what relation, it may be asked, do these con-
ceptions stand to outward realities ? How can they be made the
foundation of a real science, that is, of a science relating to things
as they really exist? We are by such questions led back to
what was long the great controversy of philosophy ; — I mean
the contest between the realists and the nominalists. The former
in asserting the reality of universals did not maintain that what
we think of when we use a general term is an actually existing
thing. Like every one else they admitted, that in one sense
nothing can exist but the individual, nevertheless they held that
universals are not mere figments of the mind, but that they have
a reality of their own which is the foundation of the truth of
general propositions. To assert therefore that the theory of pro-
babilities has for its foundation a statement touching genera
and their species, and is at the same time a real science, is to
take a realistic view of its nature. And this I believe is what,
on consideration, we cannot avoid doing.

If it be said that the grouping phenomena together is merely
a mental act wholly disconnected from outward reality and alto-

OF THE THEORY OF PROBABILITIES. 51

gether arbitrary, it may be replied that no mental act can be so.
Why and how facts and ideas correspond is no doubt one of the
great questions of philosophy ; but the answer to it is surely to
be developed from the consideration, that man in relation to the
universe is not spectator ah extra, but in some sort a part of that
which he contemplates, and that the rebus avolsa ratio, which is
in truth the fundamental postulate of nominalism, is therefore
inconcessible. The thoughts we think are, it is true, ours, but
so far as they are not mere error and confusion, so far as they
have anything of truth and soundness, they are something and
much more. The veritas essendi (to recur to the language of the
schoolmen) is the fountain from whence the veritas cognoscendi
is derived. The meaning which these phrases were intended
to convey is expressed in more modern language by Leibnitz
in the passage which I have cited in the note*. In every science
the fact and the idea correspond because the former is the reali-
zation of the latter, but as this realization is of necessity partial
and incomplete — or rather because in the same fact are simulta-
neously realized a variety of separate ideas, separate, that is, as
we conceive them — this correspondence is but imperfect and
approximate. It is only when in thought we remove the action
of disturbing causes to an indefinite distance, that we can con-
ceive the absolute verification of any a priori law. Only on the
horizon of our mental prospect earth and sky, the fact and the
idea, are seen to meet, though in reality the atmosphere is
everywhere present. Everywhere it surrounds and interpene-
trates the 777 fj,e\cuva on which we stand ; — making it put forth
and sustain all the numberless forms of organization and of life.
The indefinitely prolonged series of trials, which enters into
the ordinary statement of the fundamental principle of the theory
of probabilities, is analogous to the infinite and infinitely smooth
horizontal plane, which would enable us to verify the first law of
motion.

3. The simple negative notion of the absence of disturbing
forces is perpetually confounded with that of a tendency inherent

* C'est Dieu qui est la derniere raison des choses, et la connaissance de Dieu
n'est pas moins le principe des sciences, que son essence et sa volonte" sont les
principes des dtres. [Erdniann, p. 106.] A little further on he adds : C'est sanc-
tifier la philosophic, que de faire couler ses ruisseaux de la fontaine des attributs de
Dieu.

4—2

52 REMARKS, Ac.

in a series of successively developed results to restore the balance
of frequency of occurrence, when this has been by accidental cir-
cumstances temporarily deranged. It is commonly thought that
this notion, which, as we know, is the foundation of many
unsuccessful attempts to circumvent fortune, is sufficiently refuted
by saying, that what is past can exert no influence on what is
yet to come. But in reality the past influences the future in a
thousand different ways ; and it is only in idea that we can
secure the possibility of an indefinite series of trials, of which
those which we regard as the permanent circumstances are not
progressively, however slowly, undergoing alteration. The dice
box for example wears smooth, and the edges of the die are
rounded ; and though, in this example, we cannot say what
result is facilitated by the change, yet this is not always the case.
Such progressive alterations may tend so to alter the ratio of
frequency of occurrence, as to restore the balance which the result
of past trials has disturbed. There is thus nothing absurd in the
notion of a restorative and balancing tendency, though the
grounds on which it is commonly assumed indicate much con-
fusion of thought. It would for instance be perfectly reasonable
to inquire, whether in the succession of seasons hot years are not
oftener followed by cold and cold by hot, than vice versd. Such
questions indicate a branch of the theory of methods of observa-
tion to which hitherto but little attention has been paid.

REMARKS ON AN ALLEGED PROOF OF THE METHOD
OF LEAST SQUARES, CONTAINED IN A LATE
NUMBER OF THE EDINBURGH REVIEW. IN
A LETTER ADDRESSED TO PROFESSOR J. D.
FORBES*

MY DEAR SIR,

THE review of Quetelet's Letires h S. A, R. le Due regnant de
JSaxe Cobourg et Gotha, which appeared in the July Number
of the Edinburgh Review, contains a new demonstration of the
method of least squares which ought not, I think, to pass
unnoticed. If it is correct, it is so much simpler than those
which have hitherto been received, that it ought to supersede
them ; and if not, the sooner its incorrectness is pointed out the
better.

Some years since, in a paper published in the Cambridge
Transactions for 1844, I made an analysis of all the demon-
strations, or professed demonstrations of the method of least
squares, with which I was then acquainted, and I therefore
read this new one with more attention than you perhaps have
given to it.

The reviewer gives some account of the history of the sub-
ject, and remarks that the demonstration of the least squares
was first attempted by Gauss, but that his proof is no proof
at all, because it assumes that in the case of a single element
the arithmetical mean of the observed values is in all cases the
most probable value, " a thing to be demonstrated, not as-
sumed." Gauss afterwards gave another demonstration, which
is perfectly rigorous ; but of this the reviewer takes no notice,
though it i.3 mentioned in at least one of the works on the
theory of probabilities which he has recommended to the
attention of students. However, in the proof which the re-
viewer refers to, which is contained in the tract entitled Theoria

* Philosophical Magazine, November, 1850.

54 REMARKS ON AN ALLEGED PROOF OF

Motus Elliptic^ Gauss undoubtedly does assume that the arith-
metical mean is the most probable value in the case of direct
observations of a single element. From this assumption, he
shows that the probability, that the magnitude of an error lies
between x and x + dx, must be

•

N

TT

h being an indeterminate constant. It follows from this, that
the results of the method of least squares are always the most
probable values that can be assigned to the unknown elements.
Without referring to the Theoria Motus, you can see the de-
tails of Gauss's reasoning in a paper by Bessel, of which a
translation appeared in Taylor's Scientific Memoirs. The re-
viewer is right in saying that Gauss was not entitled to assume
that the arithmetical mean is the most probable value. But
when he speaks of this as a thing to be proved, and not as-
sumed, we are led to suppose that he believes that subsequent
writers have actually proved it. In truth this appears, not
only from his statements, but also from the illustrations of
which he has made use. Thus he states that if shots are fired
at a wafer which is afterwards removed, and we are asked to
determine from the position of the shot-marks the most pro-
bable position of the wafer, " the theory of probabilities affords
a ready and precise rule, applicable not only to this but to far
more intricate cases;" and he goes on to say that it may be
shown that the most probable position of the wafer is the
centre of gravity of the marks. Now this result is only then
true when the law of probability of error, which is implied in
Gauss's assumption, really obtains ; so that, according to the
reviewer, the demonstration of the principle of least squares
must amount to showing that this law obtains universally ; or,
which is the same thing, that the arithmetical mean is always
the most probable value in the case of direct observations of
a single element. If this can be proved, it is doubtless a very
curious conclusion; but it is at any rate certain that Laplace
has not proved it, of whom however the reviewer asserts that
he has given a rigorous demonstration of the principle of
least squares. From one end of Laplace's great work to
the other, there is nothing to justify the assertion that the

THE METHOD OF LEAST SQUARES. 55

centre of gravity of the shot-marks is the most probable
position that can be assigned for the wafer; that is, that the
concurrent existence of the deviations or errors which must
have taken place if the wafer really occupied this position
is more probable than that of those which are similarly implied
in any other hypothesis as to its place. If you find anybody
sceptical as to this, pray ask them to point out the passage,
either in the introductory essay, or in the work itself, or in the
supplements.

What, then, did Laplace demonstrate? something so unlike
this, that one is disposed to wonder how he can have been
thus misunderstood. The method of least squares is simply a
method for the combination of linear equations, of which the
unknown quantities are the elements to be determined ; the
constant term of each being a direct result of observation, and
therefore affected by an unknown error, while the coefficients
are supposed absolutely known.

If there are more equations than requisite, that is, more than
elements to be determined, what is the best way of combining
them? In the first place, they must clearly be combined by
some system of constant multipliers, else the resulting equa-
tions, not being linear, would generally be insoluble. This
condition, however, though absolutely necessary in practice, is
in no way derived from the theory of probabilities. It is a
merely practical limitation. The question thus narrowed is
simply to determine the system of factors to be employed for
obtaining the value of any particular element. The factors
must of course be such that, in the final equation, the coeffi-
cient of this element may be unity, and those of the others
severally equal to zero.

These conditions being fulfilled, we get a value for the ele-
ment in question which is affected by an unknown error, namely
the sum of the errors of observation multiplied respectively
by the corresponding factors. The mean arithmetical value
of this sum may in theory at least be determined, if we
know the law of probability of error for each observation ; and
Laplace calls that system of factors the most advantageous
which makes this mean value a minimum. If, however, the
law of probability of error is unknown, the mean value of the
error cannot be determined. Nevertheless, if the number of

56 REMARKS ON AN ALLEGED PROOF OF

observations is very large, this mean value approximates to a
certain limit, the form of which is independent of the law of
probability. The essence of Laplace's demonstration consists
in its enabling us to determine this limi't. When this is done,
it may easily be shown that the most advantageous system of
factors, those, namely, which make this limiting mean value
of the error a minimum, will give the same value to the
element to be determined as the system of final equations
obtained by employing the method of least squares, provided
equal positive and negative errors are equally probable. And
the same is of course true with respect to the remaining
elements. Thus this system of final equations gives to each
element a value affected by a smaller average error than any
other linear system, if the number of observations is sufficiently
large. It nowise follows that these values are the most pro-
bable ; that is, that the errors which must have been com-
mitted if these are the true values, form a combination a priori
more probable than the errors which in like manner have been
committed if any other set of values are the true ones. The
most advantageous set of factors for determining any element
depends only on the coefficients of the equations to be dis-
cussed, and not on their constant terms, which are the direct
result of observation. Thus these factors are determinable, ^
priori, before the observations are made. But it is only after
the observations have been made that the most probable values
of the elements can be found, and then only if we know the
law of probability of error. Laplace has pointed out the differ-
ence between the two investigations.

This difference, however, the reviewer does not seem to
have apprehended. He plainly supposes that Laplace proves
the results of the method of least squares to be the most pro-
bable results, which can only be the case, as Gauss had in
effect shown, if a special law of error obtains. He therefore
undertakes to prove, that for all kinds of observations this is
actually the only possible law.

But for the supposed authority of Laplace, he would pro-
bably have perceived that nothing can be more unlikely than
that the errors committed in all classes of observations should
follow the same law; and that at any rate this proposition, if
true, could only be proved inductively, and not by an d priori

THE METHOD OF LEAST SQUARES. 57

demonstration. For it is beyond question distinctly conceiv-
able, that different laws may exist in different classes of ob-
servation ; and that which is distinctly conceivable is a priori
possible. So that we cannot prove it to be impossible, though
we may be able to show empirically that it is not true.

You will probably agree with me in thinking that a wrong
notion of Laplace's reasoning lies at the root of the reviewer's
new demonstration. But we now come to the demonstration
itself. The assumption that the law of error is in all cases
the same, is, we are told, justified by our ignorance of the
causes on which errors of observation depend. The law
" must necessarily be general, and apply alike to all cases,
since the causes of error are supposed alike unknown in all.'7
Two remarks are suggested by this statement: in the first
place, that our ignorance of the causes of error is not so great
but that we have exceedingly good reason to believe that they
operate differently in different classes of observations ; and in
the second, that mere ignorance is no ground for any infer-
ence whatever. Ex nihilo nihil. It cannot be that because
we are ignorant of the matter we know something about it.
Or are we to believe that the assumption is legitimate, inas-
much as it in a manner corresponds to and represents our
ignorance? But then what reason have we for believing that
it can lead us to conclusions which correspond to and repre-
sent outward realities? And yet the reviewer at the conclu-
sion of his proof asserts, that, on the long run, and exceptis
excipiendis, the results of observation " will be found to group

themselves according to one invariable law." Thus the

assumption, though "it is nothing more than the expression
of our state of complete ignorance of the causes of error and
their mode of action," leads us by a few steps of reasoning to
the knowledge of a positive fact, and makes us acquainted
with a general law, which is as independent of our knowledge
or our ignorance as the law of gravitation.

Let us, however, suppose it to be true that the .law of error
is always the same, and that equal positive and negative errors
are equally probable. To determine the special form of the
law, the reviewer employs a particular case — he supposes a
stone to be dropt with the intention that it shall fall on a
given mark. Deviation from this mark is error ; and the pro-

58 REMARKS ON AN ALLEGED PROOF OF

bability of an error r may be expressed by the function ,/(r2)
or /(#* + ;*/*), the origin of co-ordinates being placed at the mark.
It is of course supposed that equal errors in all directions are
equally probable. We have now only to determine the form
of/. This the reviewer accomplishes in virtue of a new as-
sumption, namely, that the observed deviation is equivalent to
two deviations parallel respectively to the co-ordinate axes,
" and is therefore a compound event of which they are the sim-
ple constituents, therefore its probability will be the product
of their separate probabilities. Thus the form of our unknown
function comes to be determined from this condition, viz. that
the product of such functions of two independent elements is
equal to the same function of their sum." Or in other words,
we have to solve the functional equation

But it is not true that the probability of a compound event
is the product of those of its constituents, unless the simple
events into which we resolve it are independent of each other ;
and there is no shadow of reason for supposing that the oc-
currence of a deviation in one direction is independent of that
of a deviation in another, whether the two directions are at
right angles or not. Some notion of an analogy with the
composition of forces probably prevented the reviewer from
perceiving that, unless it can be shown that a deviation y
occurs with the same comparative frequency when x has one
value as when it has another, we are not entitled to say that
the probability of the concurrence of two deviations x and y
is the product of the probabilities of each. "Without this sub-
sidiary proof, the rest of the demonstration comes to nothing.
The conclusion to which it leads is in itself a reductio ad ab-
surdum. Of the above written functional equation the solution
is f(x*} = emx\ m being a constant, so that the probability of
an error of the precise magnitude a? is a finite quantity; and
I need not point out to you that it follows from hence, that
the probability of an error whose magnitude lies between any
assigned limits is equal to infinity, — a result of which the
interpretation must be left to the reviewer. He may have
thought that the exponential factor is the essential part of the
expression

THE METHOD OF LEAST SQUARES. 59

and that the others might, for the sake of simplicity, be dropt
out. But whatever his views may have been, his conclusion
is unintelligible.

The demonstration may, however, be amended so as to
avoid this difficulty, and we will suppose that the reviewer
meant something different from what he has expressed. Let
/ (x2) dx be the probability of a deviation parallel to the axis
of abscissae, of which the magnitude lies between x and x + dx.
Then/(?/2) dy is similarly the probability of a deviation parallel
to the axis of ordinates, and lying between y and y 4- dy. Thus
the probability that the stone drops on the elementary area
dxdy, of which the corner next the origin has for its co-ordi-
nates x and ?/, seems to be f(&2)f(y2) dxdy\ and as all devia-
tions of equal magnitude are equally probable, this probabi-
lity must remain unchanged as long as the sum of the squares
of x and y remains the same ; so that we have for determining
the unknown function the equation

=/(o) />'+/), .

of which the solution is

and as the deviation must of necessity have some magnitude
included between positive and negative infinity, we must have

f

» —a

Hence m must be negative; if we call it - #*, it is easy to

show that A is equal to —p? ; so that finally

VTT

;

which is what may be called Gauss's function.

But to this demonstration, though it leads to an intelligible
conclusion, the original objection still applies: the probability
that the stone drops on the elementary area dxdy is not, gene-

60 REMARKS ON AN ALLEGED PROOF OF

rally speaking, equal to/(a;2)/(/) dxdy ; so that the equation
for determining the form of the function, namely,

is not legitimately established.

To illustrate this, let TZ (xy) dxdy be the probability that the
stone falls on the elementary area in question ; then the con-
dition that the probability of a deviation of given magnitude
is constant will be expressed by

cr (xy) = <sr (\^T72 . 0) ..................... (A).

Moreover, we shall plainly have

and

and in order that the demonstration may be valid, we must have

or

/.+« .+«

•w(xy)dy\ v (xy) dx = «r (xy) ............... (B).

J -oo J -oo

If this be true, then, and then only, equation (A) may be
replaced by

But in order that (B) may be true, CT (xy) must evidently be
the product of two factors ; one of them a function of y only,
and the other of #, and the integral of each factor taken be-
tween infinite limits must be equal to unity. Combining this
conclusion with (A), we find that

and consequently

ay) «-

/(**)= A ,-

Consequently the equation for determining the form of / results
from a tacit predetermination of that function.

THE METHOD OF LEAST SQUARES. 61

The assumption expressed by

is therefore either a simple mistake or a petitio principii: the
former, if it is deduced from the general principle that the pro-
bability of a compound event is equal to the product of those
of its elements ; the latter, if it is made to depend on the parti-
cular form assigned to /(a?2).

After all, too, if the demonstration were right instead of
wrong, it would not prove what is wanted. For if the law of
probability of a deviation parallel to a fixed axis is expressed
by the function

—«-**&,

VTT

which is what the amended demonstration tends to show, the
probability that the stone falls on the area dxdy is plainly

Transforming this to polar co-ordinates, and integrating from
0 to 2-7T for the angle vector, we get 2h2e~h*r* rdr for the pro-
bability that the deviation from the mark lies between r and
r + dr ; a result which may be verified by integrating for r
from zero to infinity, the integral between these limits being
equal to unity. Thus if the deviations measured parallel to
fixed axes follow the law which the reviewer supposes to be
universally true, the deviations from the centre or origin fol-
low quite another; and hence it appears that his illustration
is altogether wrong. For if 2Ji2e~h2r2 rdr is the probability of
an error lying between r and r + dr, the centre of gravity of
the shot-marks is not the most probable position of the wafer.
So that his hypothesis is self-contradictory.

The original source of his error was probably the analogy
between Gauss's law, and the limiting function in Laplace's
investigation.

I am, my dear Sir,

Most truly yours,

K. L. ELLIS.

BRIGHTON, Sept* 19.

NOTE TO A FORMER PAPER ON AN ALLEGED
PROOF OF THE METHOD OF LEAST
SQUARES*.

To the Editors of the Philosophical Magazine and Journal.

GENTLEMEN,

ALLOW me to correct an error in my letter to Professor Fortes,
published in your last Number. The Edinburgh reviewer, on
whose proof of the method of least squares I was commenting,
says that the most probable position of the wafer is the centre
of gravity of the shot-marks ; of course on the supposition that
in this, as in all other cases, the probability of a deviation or
error r is equal or proportional to a certain constant base raised
to the power — rz.

Now, admitting this supposition to be true, the centre of
gravity is not the most probable position of the wafer. But,
on the contrary, if the function mentioned at the close of my
former communication, viz. Zffe**1*1* rdr, expresses the proba-
bility of an error r, then the centre of gravity is the most pro-
bable position. I thus not only omitted to notice that the
reviewer's conclusion would not follow from his own hypo-
thesis, but by this omission was led to introduce an error of
my own.

It is unnecessary to trouble you with a proof of what I have
now said, as the matter does not affect the general question.

I am, Gentlemen,

Your obedient Servant,

E. L. ELLIS.

BRIGHTON, Nov. 7.

* Philosophical Magazine, December, 1850.

ON SOME PROPERTIES OF THE PARA-
BOLA*

THERE are many very interesting properties of the Conic
Sections which are not to be found in the usual works on the
subject, but are scattered through various memoirs in scientific
Journals. Those relating to the properties of polygons inscribed
in and circumscribed round conic sections, have been investi-
gated by a great many writers both in France and England.
Pascal was the first who engaged in these researches, and 'was
led by the curious properties which he discovered to call one of
these polygons the " hexagramme mystique." After him Mac-
laurin gave a proof of a theorem which is not only beautiful in
itself, but also very fertile in its consequences. In more recent
times Brianchon has demonstrated the remarkable theorems,
that in all hexagons either inscribed in or circumscribed round
a conic section, the three diagonals joining opposite angles will
intersect in one point. Subsequently, Davies in this country,
and Dandeliri in Belgium, proved in different ways the same
propositions along with others. The latter adopted a very
peculiar method, deducing these and many other properties of
sections of the cone by considering the cone as a particular case
of the " hyperboloide gauche." Generally speaking, the Geo-
metrical method is more easily applied than the Analytical to
these cases, and accordingly all the proofs given have depended
on geometry, with the exception of one published by Mr Lub-
bock in the Number of the Philosophical Magazine for August
1838. He has there demonstrated, by analysis, Brianchon's
Theorem for a circumscribing hexagon in the particular case
where the conic section is a parabola; but his method is tedious,
and not remarkable for symmetry and elegance, so that another
proof is still desirable. The following one is founded on the

* Cambridge Mathematical Journal, No. V. Vol. I. p 204, February, 1839.

64 ON SOME PROPERTIES OF THE PARABOLA.

form of the equation to the tangent of the parabola which is
given in Art. 2 of our first Number*.

Let the parabola be referred to its vertex, then the equation
to its tangent by that article is

x

y = - + ™*>

where a is the tangent of the angle which the tangent makes
with the axis of y. If a! be the corresponding quantity for
another tangent, its equation will be

x ,

y — -7 + ma. .

Combining these equations, we shall find for the co-ordinates
of the point of intersection of the two tangents

x — ma.*, y = m (a + a').

We shall distinguish the tangents which form the different
sides of the hexagon by suffixing numbers to the a which deter-
mines their position, and we shall likewise distinguish the co-
ordinates of the summits of the hexagon by suffix letters.

The equations to the three diagonals are these :

= m {«2 + a,) ae«6 - (ag

(3) yfcfr-wJ-xfa + ^-ti-aJ

= m {(a3 + aj ai«6 - (ax + a6) aaotj.

Expressions which, as they ought to be, are symmetrical with
respect to the a's.

Multiply (1) by oe, (2) by - or4, (3) by cr2, and add. Then y
will disappear, and we shall find

Again, multiply (1) by a,, (2) by -of1? (3) by «6, and add:
as before, y will disappear, and we shall find the same value for
x. Consequently two straight lines whose equations are

* Cambridge Mathematical Journal, Art. i, No. I. Vol. I. p. 9.

ON SOME PROPERTIES OF THE PARABOLA. 65

(1) o8-(2) «4 = 0,
and (1) «3 - (2) cq = 0,

and which have a point in common, cut (3) in points whose ab-
scissas are equal, and which therefore coincide. Hence either
two straight lines enclose a space, or (3) passes through the in-
tersection of (1) and (2). Thus the existence of the point
common to the three diagonals has been proved, and its abscissa
found. To determine its ordinate, add (1), (2), (3), when x dis-
appears, and we have

y = m foa, (a, + a5) - <?2«3 (a, + a6) + «3a4 (ore + aj - a4as (a, + Cf8)

+ a5*6(a2 + a3)+a6a1(a3 + a4)}
divided by

ai?2 - «2a3 + V4 - a4«5 + VG ~ ««*1'

If we call the co-ordinates of the point where the third and
sixth sides of the hexagon meet o\u ysu, and so of the other two
points, these expressions for x and y become

These expressions, as of course we should expect, are sym-
metrical.

In the last Number of this Journal a demonstration was given
of a property of a parabola : That the circle which passes through
the intersections of three tangents also passes through the focus.
Although six demonstrations of this theorem have already ap-
peared, yet the following is so simple that its insertion here may
not be inappropriate.

Referring the parabola to the focus as origin, we can put the
equation to the tangent under the form

_^= fm -\
** m \ in) '

where a is one-fourth of the parameter, and m the trigonometri-
cal tangent of the angle which the tangent makes with the axis
of y . Hence if xt , yl be the co-ordinates of the point of inter-
section of

x f l\

= a (m + — ) ,
m \ m)

y -- = a
y

66 ON SOME PROPERTIES OF THE PARABOLA.

x
* m

we have xv - a (mm - 1),
y^ — a(m + m),

_ sin a , _ sin a'

OOo Ot CyUQ CA

cos (a + a')
x. — a — — , ,

cos a cos a

sin (a + a')
"l cos a cos a'

To simplify these expressions turn the axes through an
angle =— (a + a! + a"), and if a?", y" be the new values of the
co-ordinates, we find, after some simple reductions,

a cos a" „ a sin a"

y =-

cos a cos a cos a cos a

Squaring these and adding,

"2, "«= *

* ~~

cos2 a cos2 a! cos a cos a' cos a" cos a cos a
aa"

cos a cos a cos a

Now this being symmetrical between a, a', a", will hold
equally true of the three points of intersection, and it is the
equation to a circle passing through the origin which is the
focus, whose diameter coincides with the axis of a?, and whose
radius is

2 cos a cos a' cos a"

The chief advantage of this method besides its simplicity is, that
it gives us very readily the radius of the circle, and the position
of the diameter which passes through the focus.

It is easily seen that the distances from the focus of the three
points of intersection of the tangents are respectively
„ a r a a

y — n* — A. _____________

cos a cos a' ' cos a cos a" ' cos a' cos a" "
The area of the triangle formed by the intersection of the tan-
gents, can be expressed by an elegant symmetrical function of

ON SOME PROPERTIES OF THE PARABOLA. 67

tan«, tan a', tan a", that is, of w, m', m". Since the lines joining
the origin with the vertices of the triangle make angles a, a', a"
with the diameter of the circle or the axis of x, the angles they
make with each other are a! — a, a" — a', a" — a, and the area
of the triangle will be

£ rr sin (a' -a) + \rr" sin (a" — a') - \ rr" sin (a" — a).
Substituting for r, r, and r" their values, this becomes

a2 f sin (a' — a) sin (a" — a') sin (a" — a) )

¥ [cos2 a" cos a cos a' cos2 a cos a' cos a ' cos2 a' cos a cos a" j "

Expanding the sines and making obvious reductions, we get

a2 tan a — tan ot tan a" — tan a' tan a — tan a"
2 " 2 * '

or grouping differently, and putting sec2 a for — ^- , and so on,

COS Oi

— (tan a (sec2 a' - sec2 a") + tan a' (sec2 a" - sec2 a)

+ tan a" (sec2 a — sec2 a')).

Lastly, putting 1 + tan2 a = 1 + m* for sec2 a, and so on, we
find the area of the triangle to be

^ [m (m'z - m"2) + m' (m"2 - m2) + m" (m2 - m'2)},
2,

which is quite symmetrical with respect to m, m', m".

It will be easily seen, that the sides of the triangle are re-

spectively

m" — m m — m" m' — m

cos a ' ' cos a' ' cos a"

If these be called p, p, p", and if p be the radius of the circle,
by reduction, we obtain

p = 2p sin (a" — a') , p' = 2p sin (a — a"), p" = 2p sin (a' — a).

If the values of the sines derived from these equations be substi-
tuted in the first expression for the area, it becomes

___ __
2 cos a cos a' cos a"/

ON THE EXISTENCE OF A RELATION AMONG
THE COEFFICIENTS OF THE EQUATION OF
THE SQUARES OF THE DIFFERENCES OF
THE ROOTS OF AN EQUATION*.

THE equation of the squares of the differences of the roots
gives the means of ascertaining whether any assigned equation
has all its roots real ; for if they be so, all the roots of the equa-
tion of differences must be real and positive, and consequently,
by Descartes' rule of signs, all its coefficients must be alternately
positive and negative. Accordingly, Waring applied it to this
purpose, and in the Philosophical Transactions for 1763 gave the
conditions of the reality of all the roots in equations of the
fourth and fifth degree.

There will be as many conditions as there are coefficients —
that is, as there are units in the degree of the equation of the
squares of the differences ; and therefore, for an equation of the
wth order the equation of the squares of the differences will of

course be of the - th order. Thus, in the third order

there would be three conditions, in the fourth, six, and so on.

Lagrange remarked, however, that the number of conditions
in these two cases reduced itself to two and three respectively ;
and he suggested, that a similar simplification might be possible
in the ten conditions of the fifth order.

Sturm's theorem, which, however different in form, is still in
substance intimately connected with the theory of the equation
of the squares of the differences, enables us to ascertain the true
number of independent conditions.

By this theorem, we deduce from a given equation f(x) = 0
a series of n functions. These, with the original f(x), make

* Cambridge Mathematical Journal, No. VI. Vol. I. p. 256, May, 1839.

EXISTENCE OF A RELATION, &c. 69

n + 1 functions of x. We substitute in them the limits a
and b, and the number of changes of sign lost between these
limits is the number of real roots of w, which are to be found
in this interval. Consequently we have only to write plus and
minus infinity in Sturm's functions, to get the whole number of
real roots belonging to the equation.

The signs of each function, when ± - is put for x, will of

course be that of the first term, supposing each function to be
arranged in a series of decreasing powers of x. And if the first
term of each be positive, the series of signs at the superior limit
will be all permanences, and at the inferior all alternations ; that
is, all the roots of the equation will be real.

Hence the reality of all the roots depends on the signs of
n+ I terms. But of these, the sign of/(#) is determined at the

limits + - ; so is that of ~?~ , which is Sturm's first function.
Consequently there remain but n — 1 terms on the sign of which

A> A> -r^ 1

the reality of roots depends. Instead, therefore, of '

conditions, there are in reality but n — 1 .

Thus, in the equation of the third degree we find two condi-
tions, in that of the fourth, three, and so on, agreeing with what
Lagrange found in these cases, and suspected in that of the fifth
degree.

It is not very difficult to see why some of the coefficients of
the equations of differences must be so connected with the rest,
as not to give any independent condition.

In order to get an idea of this connection, let us imagine
n — 1 independent conditions, that is, n — 1 functions of the
coefficients off(x) — 0, which have a definite sign when all the
roots are real. These functions are coefficients of the equation
of the squares of the differences. Let them all become equal to
zero, then we have n — 1 relations among the n roots, which
will give every one of these roots, except one, in terms of that
root. Now the relations b = a, c = a, . . . k = a, will fulfil our
n — I equations, because these evidently make all the coefficients
of the equation of the squares of the differences vanish. Hence
these are the relations implied in the n — 1 equations we have
assumed; and these would, it has just been said, make all

70 EXISTENCE OF A RELATION, &c.

the coefficients = 0. Consequently, n — 1 independent relations
are all we can have, and it follows, that if n — 1 independent
coefficients of the equation of differences become = 0, all will
be so.

We may take the matter somewhat differently, still using
the case of equal roots in f(x) = 0 to show the relations
among the coefficients of the equation of the squares of the
differences.

= 0 has m equal roots, there will be — ~ — - roots

a

of the equation of the squares of the differences, or A = 0, equal
to zero. Let/j(a?) = 0 get another root equal to these m roots by

any change in its coefficients, then there will be - - - — roots

in A = 0 equal to zero ; the difference is m. Thus, a single
fresh relation among the coefficients of f(x] = 0 makes m coeffi-
cients of A = 0 vanish ; for obviously the last coefficients of this
equation disappear whenever it gets roots equal to zero.

We may easily see, too, that the constant function (the last
in Sturm's process) is the same as the term independent of u in
A = 0.

The equation A = 0 may, theoretically, be got by eliminating
x between the two equations

(Lagrange, p. 7) ; A will be the term independent of x : put, then,
u — 0, A reduces itself to its last term, and the process becomes
simply that of finding the common measure of f(x) and/' (#),
which, abstracting the changes of sign, is exactly Sturm's pro-
cess ; hence his term independent of a?, will be " aux signes
pres" the constant term in A = 0.

The development of this idea would undoubtedly lead to the
general theory of Sturm's method, and would make it more than
a happy artifice, by showing its intimate connection with the
equation of the squares of the differences. As is generally the
case, the different ways in which the subject may be viewed,
ultimately coalesce.

ON THE ACHROMATISM OF EYE-PIECES OF
TELESCOPES AND MICROSCOPES*.

MR AIRY, in a paper published in the Second Volume of the
Cambridge Transactions, has investigated the conditions under
which a system of lenses is achromatic, per se— that is, when
only one kind of glass is made use of. The enquiry, on account
of its immediate application to the eye-pieces of telescopes and
microscopes, is one of considerable importance ; and as the way
in which it is conducted in the paper just mentioned appears to
be more complicated than necessary, and does not lead to the
most general solution of the problem, perhaps the following
attempt may be not wholly without interest.

The difficulty of the question consists in finding the angle
which a ray of light makes with the axis of the system of lenses
after having passed through it. When this is done, we have
only to take the chromatic variation, and equate it to zero, to
get the general equation of achromatism for the system.

In what follows, lines are considered positive when measured
in a direction opposite to that of the incident ray.

Let yn be the tangent of the angle which the ray makes with
the axis before its incidence on the nth lens of the system ; let
zn be the distance from the axis of the point where it impinges
on that lens; and take an_t to signify the distance of the nih

from the n — 1th lens. Then — and — — are the distances of

the conjugate foci from the nih lens, and by the ordinary
formula

p being the reciprocal of the focal length ; therefore

#«+l-#«=^n-
* Cambridge Mathematical Journal, No. VI. Vol. I. p. 269, May, 1839.

72 ACHROMATISM OF EYE-PIECES

Again, we have the simply geometrical relation

or

The advantage gained by introducing yn is, that we thus have
precisely similar equations for the optical and geometrical con-
ditions of the problem. Nothing is now easier than by suc-
cessive substitutions to determine the value of yn in terms of y
and z, and yn is the tangent of the " visual angle," which we
are seeking. As an instance, let it be proposed to determine
the conditions of achromatism in a system of three lenses. Mr
Airy has done this only for rays originally parallel to the axis :
the method here proposed applies with equal facility to the
general case.

The equations required are these,

Hence

«i/> J yi

Taking the chromatic variations of the two terms in the
usual way, and equating each to zero, we find

S = 0

= 0.

The first of these equations becomes unnecessary in the par-
ticular case considered by Mr Airy, viz. that in which yt = 0;
the second is identical with that given by him at p. 245, when
attention is paid to the signs. Taken together they determine

OF TELESCOPES AND MICROSCOPES. 73

the relative positions of three given lenses, which shall form a
combination achromatic for rays of any degree of obliquity.

In the particular case in which the focal distances of all the
lenses are equal, and the intervals a^ «2, &c. are also equal, the
general equations degenerate into a system of simultaneous equa-
tions in finite differences. They are then

Eliminating zn, we get

y*»-(pa
The general solution of this will be

A being a root of the recurring quadratic equation

xz - (pa + 2) x + 1 = 0,
c and Cj are to be found by the conditions

y1 = cA + ^ ^l"1, yl + pz1 = cA* + c, A"*.

The general solution of the system of quasi-equations em-
ployed in the enquiry must involve some functional operation
which degenerates into the radical contained in A.

It would be perhaps worth considering how far we might be
able to present this operation in a distinct form, defined and
distinguished by a particular symbol; but the subject is not
one which can be discussed at present. At any rate, we see
that the research of the general expression for yn is one of
considerable difficulty.

The greater part of the investigation given by Mr Airy in
the conclusion of his paper, with respect to the achromatism
of microscopes, becomes unnecessary by employing the general
expression given above for^4. His object is to determine the
distance of an object-glass of given focal length from a diaphragm
whose distance from the field-glass of a given eye-piece of three
lenses is given.

Let aQ be the distance of the diaphragm from the field-glass ;
therefore we have ^ = aQy^ and putting this value for zl9
we get an expression for y^ of the form y4 = yv~R. The chro-
matic variation of this is to be zero, and consequently that of
its logarithm ;

74 ACHROMATISM OF EYE-PIECES, &c.

Now Mr Airy has shown that, (adopting the notation of this
paper)

a; being the distance of the object-glass from the diaphragm, and
pQ its vergency, or the reciprocal of its focal length. Putting
for R its value, we get at once xpQ =

divided by

[1 + a [ft + pt + p^ + at [>2 + p J

which is identical with his result.

ON THE CONDITION OF EQUILIBRIUM OF
A SYSTEM OF MUTUALLY ATTRACTIVE
FLUID PARTICLES*.

THE generally received theory of the Equilibrium of Fluids,
(due in its present form to Euler,) assigns one condition as
necessary and sufficient in every case. Mr Ivory conceives,
that when a fluid is acted on by forces arising from the mutual
attraction of its particles, a second condition is requisite for
equilibrium, and has developed the considerations which have
led him to this result, in several papers published in the Phil.
Trans., and also in the Phil. Mag. The authority of Mr Ivory
on any point of mathematical physics is very great : his decision
on one to which he has long directed his attention, would be
almost final, were it not opposed to the views of Euler, Laplace,
and Poisson. The object of this paper is, to examine how far
Mr Ivory, in a paper published in the Phil. Mag. Vol. xni.
p. 321, has demonstrated the necessity of the subsidiary con-
dition in question. The writer feels it unnecessary to express
the diffidence with which he attempts to consider so difficult
a subject; he regrets also his inability to discuss Mr Ivory's
views more at large than the present limits would permit.

In the paper just mentioned, Mr Ivory states the principal
steps of the investigation by which Clairaut was led to the con-
dition of equilibrium of a fluid acted on by forces directed to
fixed centres ; and proceeds to consider the modifications re-

* Cambridge Mathematical Journal, No. VII. Vol. n. p. 18, November, 1839.

76 EQUILIBRIUM OF MUTUALLY ATTRACTIVE

quired to adapt the method to the case of a fluid whose particles
are mutually attractive. Clairaut first supposes a mass of fluid
in equilibrium, and conceives an infinitesimal stratum added
to it, which shall produce equable pressure over the whole sur-
face ; — the equilibrium of the original mass A will not be dis-
turbed, and the increased mass A + SA will be in equilibrio,
when the forces acting on its surface are normal to it. This
principle, that forces acting on a free surface must be normal
to it, was laid down by Huygens, and is confessedly true. By
a repetition of this process, the original mass can be enlarged
to any extent ; and the condition that the nucleus must be in
equilibrio becomes, Mr Ivory observes, unnecessary, by con-
ceiving it diminished sine limite. The mathematical condition
of equilibrium is, therefore, the expression of the possibility of
adding a stratum which shall produce equable pressure, and at
the free surface of which the forces shall be normal to it.

Let us endeavour to put this symbolically. Let the force
at the original free surface be F\ at the point #, y, z produce

the normal, and take a length on it = ^, &> being infinitesimal :

thus we get a stratum producing an equal pressure o>.

Let f (x, y , z) = c be the equation of the free surface ; then
F being a function of (a:, y, z), all that is requisite for the force
at any point of the new free surface to be normal is, that

shall be its equation.

Let F= J(fx)*+(f'yf+(f*)* ; then

5, &> fx ~ G> fy . a> fz

" =

and f(x, y,z) = c =/(*', y', z') - [fxSx +fySy +/
(where x' = x+ &») by Taylor's theorem ;

therefore c =/(*', 3,', *') -1

therefore c = c + Sc — 0)-=,:

FLUID PARTICLES. 77

y

or -^= a constant, which we may take for unity; therefore

F= V. Resolving this force along the axes,

X=V'f^-; whence X=fx, and so Y=fy, Z=fz.

to is the increment of pressure = &p; multiplying the three
equations (1) by X, Y, Z} and adding, we get

or putting d for S,

dp = -Xdfe + Ydy + Zdz ............ . ..... (2),

the equation of equilibrium of an homogeneous and incom-
pressible fluid, whose density is unity.

An objector to Clairaut's reasoning might urge, that this
result, though certainly sufficient, was not shown to be neces-
sary : he might argue, that a way has been shown of building
up a fluid mass; but that it has not been proved that every
fluid mass is capable of resolution into the smaller masses, by
means of which alone Clairaut investigates the conditions of
equilibrium. Unless it be made a direct postulate, that every
fluid mass in equilibrio will continue in equilibrio, when the
part of it contained between the free surface and any level sur-
face is removed, it is difficult to see how this objection can be
met, except by showing that the property assigned by Huygens
to a free surface, viz. that the force is normal to it, belongs to
every surface of equal pressure, and that consequently Clairaut's
reasoning is in reality independent of any construction or reso-
lution of a fluid mass into successive strata. When we assert,
with Clairaut, that a fluid mass in equilibrium is not disturbed
by the addition of a stratum producing equal pressure, we imply
that the reaction produced at any point of the surface of A9 by
the pressures exerted over the rest of the surface, i. e. the effect
of the transmitted pressures, is normal to it. For we know
that the forces at the surface are so ; and unless the inference
first stated is correct, there could be no equilibrium. It hence
appears, that Clairaut's axiom is equivalent to this — Equable
pressure produces a reaction normal to the surface on which
it is applied. But if the force at a surface of equal pressure
were not normal to it, there could be no equilibrium, because

78 EQUILIBRIUM OF MUTUALLY ATTRACTIVE

it is only by the transmitted pressures that it can be esta-
blished.

Clairaut, as his views are represented by Mr Ivory, says
nothing of the transmission of pressure ; but it is impossible
to investigate fluid equilibrium without tacit or expressed re-
ference to some distinctive character of fluidity; and in the
principle he makes use of, the idea of the transmission of pres-
sure is essential. It appears, then, that the force at a surface
of equal pressure is normal to it; and this conclusion is little
else than a different way of putting the principles employed by
Clairaut. We are now enabled to dispense with any process of
constructing a fluid mass.

On referring to the mathematical reasoning employed above,
we shall easily see that, substituting two infmitesimally near
surfaces of equal pressure for the consecutive free surfaces of
Clairaut, the result we arrive at is simply the symbolical ex-
pression of the principle just laid down, viz. that the force at
a surface of equal pressure is normal to it. A very little at-
tention will show, that (2) is true in every case of fluid equi-
librium, and that it is completely equivalent to the principle
which it represents. In translating, so to speak, his funda-
mental idea from the infinitesimal to the fluxionary conception,
that namely of successive generation, Clairaut has tacitly intro-
duced a new condition, namely, that a surface of equal pressure
will necessarily be a free surface of equilibrium, the superin-
cumbent part being removed.

Mr Ivory remarks — " The investigation of Clairaut is clear
and definite. It evidently assumes that there is no cause tend-
ing to disturb the equilibrium of A, except the action of the
forces at the surface of A upon the matter of $A. On this
account his method fails when there is a mutual attraction be-
tween the mass A and the stratum SA. If the mass A attract
the matter of the stratum 8 A, and cause it to press, it follows
necessarily that the matter of &A will react, and by its attrac-
tion will urge the particles of A to move from their places. In
this case, therefore, the equilibrium of A is disturbed by a force
which Clairaut has not attended to; and unless the effect of
this new force is counteracted, the body of fluid A + SA will not
be in equilibrium. The principle of the method suggests a
remedy for this omission, for it is easy to prove that the equi-

FLUID PARTICLES. 79

librium of A will not be disturbed by the attraction of the
stratum $A, if the resultant of that attraction on every particle
in the surface of A be directed perpendicularly to it."

This reasoning satisfactorily shows, that if a fluid mass of
attractive matter be increased by a stratum producing equal
pressure over the free surface, the equilibrium will be destroyed
unless a certain condition is fulfilled, of which the symbolical
expression is

P, Q, R being the attractions, parallel to the axes of co-ordi-
nates, of an element of that part of a fluid mass which is external
to a given level surface. But the necessity of this condition
cannot be proved, unless it is shown to be impossible in any
way to increase the mass A, without destroying the equilibrium,
supposing it not fulfilled. All that has been shown is, that
the mass cannot be increased by a stratum producing equable
pressure over the free surface. Now, generally speaking, the
mass so increased will not fulfil the condition of having the
forces at the new free surface normal to it, those acting at the
original free surface being of course so. We cannot, therefore,
affirm that we have fallen on a case in which the ordinary con-
dition is fulfilled, without producing equilibrium. If, however,
we dispense with the limitation, that the stratum added shall
produce equable pressure, we lose the simplicity of Clairaufs
method, nor can we make any use of his principle, except by
setting aside the construction he employs, which confines him
to the particular case in which a surface of equal pressure is
potentially a free surface.

This has already been done, and the result is the general
equation of equilibrium. It remains to show, that it is in all
cases sufficient. It is admitted to be sufficient in the case of a
fluid acted on by forces tending to fixed centres. We shall
endeavour to reduce the general case to this. Conceive a body
acted on by a force directed to a fixed point. It may be so
placed, as to remain at rest under the action of the force, that is,
the resultant of the force upon it is equal to zero. In this
position of the body, the centre of force is some point within it.
Let the body, remaining in the same position, diminish sine
limite, being always similar to itself, the resultant of the force
upon it is always equal to zero ; and ultimately, when the body

80 EQUILIBRIUM OF MUTUALLY ATTRACTIVE, &c.

becomes a physical point, it coincides in position with the centre
of force, and is in the same state with respect to the action of
other forces upon it, as if this force did not exist.

This being granted, conceive a homogeneous mass of fluid
composed of mutually attractive particles, the free surface of
which fulfils the required equation

Let the attractive power of each particle be conceived trans-
ferred to a fixed centre of force coinciding with it. Then the
action of all the other particles on one particle is precisely re-
placed by that of the fixed centres ; and it has been shown, that
the resultant of the action of the centre coinciding with a par-
ticle on that particle, equals zero. Hence, the supposition we
have made does not change, in any way, the forces acting on
any particle of the mass. Were the system in its present and
former state respectively to move, the motions would be widely
different; but in the arbitrary position we have placed it in,
the action on it is precisely the same in the two cases. Now,
the single equation given above assures its equilibrium, when
we regard it as a system acted on by forces directed to fixed
centres; and as the hypothesis by which we are enabled to
look upon it in this way nowise affects the forces acting on it,
it follows, that the system considered as acted on by mutual
attraction must be in equilibrium. Consequently, a mass of
homogeneous fluid, the particles of which are mutually attrac-
tive, will always be in equilibrium when the free surface fulfils
the single condition implied in the general equation obtained
above. The same reasoning applies to the case of any fluid,
elastic or incompressible.

If this demonstration be thought satisfactory, the question
raised by Mr Ivory, as to the sufficiency of the general equation,
must be looked upon as settled. The suggestions here made
with respect to the new condition tacitly introduced in Clairaut's
reasoning, will, it is thought, enable us to trace the source of
the difference of the view taken by Mr Ivory, and that gene-
rally entertained. In one form or other, it seems to recur in
eveiy way in which that distinguished mathematician has treated
the subject.

MATHEMATICAL NOTE*.

IN Vol. I. p. 205 f, there were found for the co-ordinates of
the point of intersection of two tangents to a parabola, the ex-
pressions

y as 0} (a -f a*}) a? = maa',

a, a' being the tangents of the angles which the tangents to the
curve make with the axis of y. From these expressions it fol-
lows, that if y^y^ &c. xl9 #2, &c. be the co-ordinates of the
angles of any re-entering polygon of 2n sides circumscribing a
parabola,

and

Also, the continued product of the abscissae of the points of
intersection of any number of tangents, is equal to the continued
product of the abscissae of the points of contact, provided no
three points of intersection lie in the same straight line.

Let a;', x" ', a;"', &c. be the abscissae of the points of contact,
then it is easily seen, from the equation to the parabola, that

x' = ma!\ x" = ma."*, x'" = mv!"\ &c.
the continued product of which is

x'x'x'" . . . x(n] = m W V'2 . . . a(n)2.

And if xlt #2, a;s, &c. be the co-ordinates of the points of inter-
section of the tangents, we have

Xj^ — moLO.", o?2 = wa"a"', x3 = ma."a!"t &c.
the continued product of which is

x,x,x3 . . . xn = m* . a W"2 . . . a(w)2,

which is equal to the preceding expression. It is necessary to
limit the intersections in such a way that no three shall lie in
the same line, because otherwise some one of the a's in the
second series would appear more than twice.

* Cambridge Mathematical Journal, No. VII. Vol. n. p. 48, November, 1839.
t Page 63 of this Volume.

6

VARIATION OF NODE AND INCLINATION*.

THE following method of finding the variations of the in-
clination and longitude of the node, is more convenient than
that given in Pratt's Mechanical Philosophy, p. 336.

Adopting the notation usual in the lunar theory, we have
s = &sin(0-7) ........................ (1),

, d*z IJLZ dR
also -j- +rjr + -r=Q ....................... (2).

0F r dz

In the disturbed orbit, (1) and its first derived equation will
be true, as if the elements were invariable ; which gives the
equation

(3),

and differentiating (1) a second time, there is
d's .d& d .J0

'

The second and third terms are those due to perturbation.
Also, the inclination being very small, the effect of perturbation

°n ~d? ' °r w^^c^ ^s ^e same tn^nS? on — TT- > w^ ^e sensible

only in the term p -^ . Hence, equating the perturbation and
its effect, we have

* Cambridge MathemalicalJournal, No. IX. Vol. n. p. 113, May, 1840.

VARIATION OF NODE AND INCLINATION. 83

and eliminating in turn ^- , -^ , by (3), we get
d07dy dR .

^ar^C'-i)-^

de dk ^

Again, the inclination being small,
dR dR dz dR ds dR .

, dR dR dz dR ds , dR

and -j- = -=- . -j~ = p -j— . -=- = — OK -j- cos (6 — 7) ;
ay dz dy r dz dy dz

, Z7
and p -7rA;-j7 = -j-,

p

_

dk _ 1 no, dR /fiv

M^k^JT^'dy'' "^''

which agree with the known results, & being = tan 7, or sin 7,
quam proximd, and 7 being what Mr Pratt denotes by O. (The
squares, &c. of 7 are neglected throughout.)

6—2

INVESTIGATION OF THE ABEREATION IN
RIGHT ASCENSION AND DECLINATION*.

THE following investigation of the formulas for Aberration
in Right Ascension and Declination will be found to be more
simple than that given in Maddy's Astronomy, p. 214.

Let T L be the ecliptic, <r E the equator, P its pole, T a
point 90° behind the place of the Sun, S the place of the star ;
then 8 T will be the plane of aberration.

Let or N= a, PN= 8, Sun's longitude = 0 , and L T E= a>.

1st. For the Aberration in Declination : produce SN to a
point Q, such that SQ = 90°, and join TQ, TN. If A be the
coefficient of aberration, and AS the aberration in declination,

AS = - A sin ST. cos TSQ = - A cos TQ,
as TSQ is a quadrantal triangle.

* Cambridge Mathematical Journal, No. IX. Vol. II. p. 120, May, 1840.

INVESTIGATION OF THE ABERRATION, &c. 85
But cos QT= cos TN. cos QN+ sin TN. sin QN. sin TN«r ,

and cos TN= cos Tnr . cos nr JV+ sin TV . sin or JV. cos T<r N,

= sin O cos a — cos O sin a cos o>.

Also, sin TN. sin 2Wv = sin TV . sin TT JV= cos 0 sin «.
Substituting these values, and putting 90° — S for QN, we find
"' A8 = — ^t{sinS(sinO cos a — cos O sin a cos w) + cos S cos Q sinw}.

2nd. For the Aberration in Right Ascension : produce
to a point B, such that ^B = 90°, and join R8, RT. Then,
if Aa be the aberration in right ascension,

A A

Aa = - -— K sin ST. cos TSR = - — ^ cosET.

COS 0 COS §

But cos ^T= cos JV . cos R <r + sin TV . sin .5 nn cos R v T,
= sin 0 sin a + cos a sin Q cos w.

Consequently,

£

Aa = -- ^ {sin 0 sin a + cos a sin O cos &>].
coso c

ON THE LINES OF CURVATURE ON AN
ELLIPSOID*.

THE following investigation of the Lines of Curvature on an
Ellipsoid has the advantages of symmetry and of giving a
distinct geometrical conception. The artifice on which it de-
pends may, it is thought, be found useful on other occasions.

The symmetrical equation to the lines of curvature is
d?-c*}xdydz+(c*-a*}ydzdx+(a*-tf)zdxdy = Q ...... (1),

(see Mathematical Journal, Vol. I. p. 142), where xyz are con-
nected by the equation to the surface,

j+f?*?-1 ........................ «••

= «, !' = „, J = w .................. (A).

Then

vv vw ±vuvw

Hence, after the substitution and multiplying by 4 vw , (1)
becomes

(J'-c2) udvdw+ (c2-a2) vdwdu+(a*-t>*) wdudv = 0...(3),
with the relation

•\-v + w — \ (4).

Difierentiate (3) ; then, since

* Cambridge Mathematical Journal, No. IX. Vol. n. p. 133, May, 1840.

ON THE LINES OF CUR VA TURE ON AN ELLIPSOID. 87

we get

(&2 - c2) ud (dvdw) + (c2 - a*) vd (dwdu)

+ (a*-tf)wd(dudv)=Q ...... (5).

Now this is satisfied by the assumptions

^, -, j- ........... (B),

f, g, h, being constants.
But from (4) we deduce

du + dv + dw = Q ........................ (6),

and (B) gives

du —fdu dv div, dv = g du dv dw, dw = hdudv dw.
Hence f+g + h = Q ........................ (7),

which establishes a relation among the otherwise arbitrary con-
stants fj g, h.

Now (B) implies the existence of two linear equations in
u, v, w. Hence, a particular solution of (1) is two linear equa-
tions connecting the three variables. But the given equation
(4) is linear ; hence the solution in question is the one congruent
to the problem.

To find the other relation in u, v, w, eliminate the differen-
tials from (3) by means of (B), and there is

Equations (4) and (5), with the relation (7), contain the com-
plete solution of the problem. It is obvious that the apparent
want of homogeneity of (B) is wholly immaterial.

Keeping in mind the values of u, v, w> given by (A), we see
that the geometrical interpretation of (8) is, every line of cur-
vature on an ellipsoid lies on a conical surface of the second
order, of which the vertex is the centre of the ellipsoid.

To determine the constants, let the line of curvature pass
through a point, for which the values of u, v, w, are wt, v^ w^
we have

(&« _ <.«) u\ + (<? - «») ^ + (a2 - £2) ^ = 0,

£ = 0.

ON THE LINES OF CURVATURE
Hence, after a slight reduction,

(&« - c*) Wi 2 + (c' - a2) ^ ^ - (a2 - Z>2) w,
j y

+ (£2-c2K+(c2 -«>! = <) ......... (9),

a quadratic in , of which the roots are real and of unlike signs.

This is obvious, for ut, vlt are essentially positive, and a, Z>, c,
being in order of magnitude, the signs of b2 — c2 and c2 — a2 are

opposite. Similarly, ^ is determined by a quadratic, whose roots

are always real and of opposite signs. Thus two lines of cur-
vature pass through every point on the surface of the ellipsoid.
Let us now consider the envelope of the surfaces represented

by (8).

Differentiating (7) and (8) for/, g, h, we get

-o (ii).

I being an indeterminate factor, we may put

7 _ (7,2 2\ U 1 _ f 2 2\ V 7 _ / 2 tf\W .

/* ' g*' K"

whence, taking the values of /#, ^, to substitute them in (8),
we deduce

As the signs of the radicals are independent, this represents
four planes ; but c2 — a2 is negative. Hence the possible part of
these planes is their traces on the plane of xz, for which v = 0.
Thus we get the two straight lines

V6a — c2 "Ju + 'Jc? — fr2 *Jw — 0 (13),

and for the points where they meet the ellipsoid,

« + »=•! (14),

whence

ON AN ELLIPSOID. 89

These values belong to the umbilici of the ellipsoid ; a result
easily anticipated. When they are introduced in (9), it becomes

2 2

§2 = 0, and similarly j* = 0.

Hence (8) reduces to

i> = 0 (16),

and represents the principal section of the ellipsoid, which passes
through the greatest and least axes. In this case then, as our
analysis would lead us to anticipate, the lines of curvature
coincide; a result which, although well known, seems not very
accurately demonstrated by Leroy. After having shown (p. 309
of the second edition) that the two directions of curvature
coincide at the umbilical points, he proceeds to integrate, and

passes from ^ = 0 to y = ^, and thence, determining the con-
stant, to y — 0 ; which last represents the line of curvature
sought. But -j- has been shown to have the value 0, only for

u/ZC

the umbilical points, and we are therefore not at liberty to pass
by integration from these to any other points at which this
may not hold. Were the process legitimate, it would lead to
the strange conclusion, that the lines of curvature through an
umbilicus are necessarily plane curves.

As there appears to be still some difficulty with regard to
the theory of these singular points, we may enquire whether,
in order to determine the lines of curvature through any point
whatever, more is requisite than to substitute its co-ordinates in
the general equation of the lines of curvature, and thus to get
two values for the arbitrary constant; whether the result can
ever be indeterminate, except when the lines, as at the extremity
of an axis of revolution, are so in reality. In this view we see
at once, that the process given by Leroy after Poisson for de-
termining the directions of curvature at an umbilicus, is simply
the ordinary method for ascertaining the position of the branches
of any curve at a multiple point ; and that the result arrived at,
is not that more than two lines of curvature pass through an
umbilicus, but that every point which, with reference to the
surface, is umbilical, is, with reference to the lines of curvature,
a multiple, or more generally a singular point. These sug-

90 ON THE LINES OF CURVATURE

gestions may, perhaps, show how we must determine the lines
of curvature which pass through an umbilicus, a problem dis-
tinct from that solved by Leroy of finding the directions of
curvature.

Many curious properties may be deduced from the equations
we have arrived at. Thus, if we take on two concentric and
confocal ellipsoids, a series of pairs of corresponding points,
(such as are spoken of in the enunciation of Ivory's theorem,)
and if the locus of the points on one of the ellipsoids is a line of
curvature, then that of those on the other is so too. Again, the
traces on the tangent planes at the extremities of the three axes,
made by one of the cones represented by (8), are an ellipse and
two hyperbolas respectively. The areas of this ellipse, and of
the ellipses conjugate to the two hyperbolas, are so related that
their continual product is constant for the same ellipsoid, and
for all ellipsoids of the same volume. The method of demon-
strating these two theorems is so obvious, that it seems unneces-
sary to. enter more fully on either.

It still remains to be shown how we pass from (8) to the
projections of the lines of curvature on the co-ordinate planes.
The symmetry of the problem is destroyed by the transition ;
but as it is in this shape that the results are commonly ex-
hibited, we shall dwell rather more upon it than would other-
wise have been necessary.

Putting C=az—b*, B = <? — cf, A = b* — c2, and eliminating
u, v, w, successively between (4) and (8), there result

0 A\ (B & =

(17),

(A B\ (C A\ A

[-2 }V — -y- — -?] W = ~?

\f gi U // /

(B C\ (A B\ B

r w — ( ~? } u= —

\ff h) \f g) g

Put

C A . B C A B

Then k-l =

ON AN ELLIPSOID.

Thus we get the relations

91

lh -m/=0

mg - Jch = 0 )

A B l[A B\

Hence, — > = 1 = 7 T — f ) >

w/ mg h\l k)

and consequently

.(19).

A B^

similarly,

C A
9~m I

(20).

* k m*

By means of (18) and (20) the equations (17) become

ku

Iv —

VCk I

WH

kw

Ak-Bl
Amk

7™

Bm-Ck
Blm

jnu —

Gl-Am.

(21);

which, restoring their values to w, v, w} may be written

V ktf

Put y -2 = m; .'.

It &

, &C.

,(22).

, and then we get

f=mr-v —

with similar equations for the projections on the other co-ordi-
nate planes. This result is identical with the known one in
Leroy, p. 304, or Hymers, p. 201.

It is hoped that the novelty of treating symmetrically a non-
integrable equation in three variables, will be admitted as an
excuse for the length to which this paper has extended itself.

MATHEMATICAL NOTES*.

1. THE area of a polygon of a given number of sides, circum-
scribing a given oval figure, will be the least possible when each
side is bisected in the point of contact.

This elegant proposition, given in the Senate-House Problems
for 1836, may be easily demonstrated as follows : —

Let AB, BC, CD, be consecutive sides of the polygon.
Produce AB, DC, to meet in E; then BC must, by the con-
dition of the minimum, be in such a position that EBC is a
maximum.

Eefer the oval to EA, ED, for axes, then the equation to
the tangent BC is

ydx — x'dy — ydx — xdy,

y and x being the co-ordinates of the point of contact P.
Put aj' = 0;

and so

Also area of EBC = %xQyti sin E.

(ydx — xdy)* . . .. . . . .

Hence, - , , — * - is a maximum (the minus sign is im

material).

Differentiate, considering x as independent ; then
fydx-xdy

xy fyx-xy \ =
dxdy y \ dy )

The last factor only gives a solution ;

.. ydx — xdy
••X = -±—dy- *• = **•>

* Cambridge Mathematical Journal., No. IX. Vol. II. p. 142, May, 1840.

MATHEMATICAL NOTES. 93

that is, PM being parallel to EG, EM= \EB, and /. PB=PC,
or BG is bisected in the point of contact P. The same is true
of any other side, and therefore every side is bisected in the
point of contact.

2. Let p, p, be two forces into which a given system on a
rigid body may be resolved, a, 6, their least distance, and in-
clination of their directions ; pp a sin 6 is invariable. (Senate-
House, 1833.)

Let the line a meet the directions of p and p in P and P'
respectively. At P apply two forces equal and parallel to p',
and opposite each other. Thus the system of forces is replaced
by the couple pa, and by the force at P, which is the resultant
of p and p. Kesolve this, the resultant, along the axis of the
couple and in its plane. Then the former component can arise
only from the resolved part of p, as^>' is wholly in the plane of
the couple. Also, as the shortest distance is perpendicular to
both lines, it follows that the arm of the couple is perpendicular
at P to the plane which contains the two forces p and p.
Hence 6, their mutual inclination, is that of p on the plane of
the couple, and therefore p sin 6 is the part of the general re-
sultant resolved along the axis of the couple. Then, if the
general resultant makes an angle (f> with the axis, we have in
the usual notation

pp a sin 6 = GR cos <f> = B.G cos </>.

Now G cos <f>, as is known, or as may be easily shown, = Gv,
the minimum maximorum moment of the system ;

therefore ppa sin 0 — GJEl,
which is constant.

ON THE TAUTOCHRONE IN A RESISTING
MEDIUM*.

OUR object is to reduce the problem of the tautochronous
curve, when the resistance is equal to Jcv* or to hv + kvz, to the
cases in which it is a cycloid, viz. when the resistance is equal
to zero or to hv.

In vacuo

_ f*1 ds

JQ VflJj — X '

and the necessary and sufficient condition of tautochronism is
s2 = Ax. Hence generally

f«i dz

J0 V.FZ — FZ

is independent of zl9 provided z* = A . Fz. Now, when R = lev*,
there is

vdv , o dx

therefore

= - 2^ f e^ ~ &.
^J ds

therefore er**t?* = 2^ (^ — fs),
and

Put ^ = e"*8^, and let 2? and 5 be equal to zero together,

therefore z = i (1 - «r*) , fs = -P«,
* Cambridge Mathematical Journal, No. X. Vol. n. p. 153, November, 1840.

ON THE TA UTOCHRONE IN A RESISTING MEDIUM. 95

. — pi dz
andV2<^=Jo^—

Therefore for tautochronism

But Fz = ( ~ (1 - fa) dz,
J0ds *

therefore -r z = az — -j- (1 — fa),

^ r, ^ dx I — e~ki
therefore k -j- = a — ^-— ,
ds e k8

the equation of the tautochronous curve.
We shall have, if t = 0 when s = sv

— z^ cos
ds* ek°-l

must become, when s is expressed in z,

d*z
W~~*9*>

a result easily verified, for

(1),

The coefficient of dz in (a) is of course that of d2z in
which shews that if the equation of motion were

dzs ds 7^_ e*°-l
dt^ hdrkd?- '"ff~k~>

or E = hv + A;v8

96 ON THE TAVTOCHRONE IN A RESISTING MEDIUM.
the equation in z would be

dzz , dz

This is precisely the form of the equation of motion on a
cycloid when R = hv. And as in that case ^ deduced from it
is independent of sv so in this it will be independent of zl9 and
consequently of 5r Hence the curve whose equation is (1), is
tautochronous, not only when R = lev*, but also when

For Laplace's abstruse solution, see the first book of the
Mecanique Celeste, or Mr Whewell's Dynamics.

ON THE INTEGRATION
OF CERTAIN DIFFERENTIAL EQUATIONS*.

No. I.

IT is shown in the theory of the earth's figure, that if the
pressure and density at any point be connected by the equation

dp = kp dp,

where Jc is a constant, then the ellipticity of the surface may be
deduced from the solution of the equation

This equation is not easily integrated. La Place, in the
eleventh book of the Mecanique Celeste (v. 51), gives a solution
of it, but without demonstration ; and the lacuna thus left is not
supplied in the works on the subject generally made use of in
Cambridge.

Mr Gaskin has however effected the integration of

is integral, in finite terms (vide Hymers' Diff. JEq. p. 53),
and the proposed equation is a case of this one. But perhaps a
more direct analysis is preferable, as it enables us to extend our
method to two or three classes of equations of all orders. One
of these will be considered in the present paper — another, the
solution of which admits of a remarkable symbolical form, will
be given in the next number of the Journal.

We shall begin with the particular equation which occurs in
the theory of the earth's figure, both because from its physical

* Cambridge Mathematical Journal, No. X. Vol. n. p. 169. November, 1840.

7

98 INTEGRATION OF DIFFERENTIAL EQUATIONS.

application it has an interest for some who care but little for
pure analysis, and because it will exemplify the general method.

Let y = ^anxn ........................... (2),

... {%(w_i)_6K + 2V2 = 0 ............... (3),

n (n - 1) - 6 = n (n - 1) - 3 (3 - 1) = (n - 3) (n + 2) ,

... (w-3)(w+2K4V«.M, = 0 ............. (4).

To get rid of the factor (n — 3), assume

(n-l)l>n ..................... (5),

— 3

= 0 .................. (6).

Hence bn is made to depend on bl or bQ as n is odd or even,
and we see at once that

^bnxn = b0 cos qx + Z>j sin qx,
or changing the constants,

2l>nxn= Csm(qx+a) ...................... (7).

Also by (5),

a» = ^"

for by (6),

/. 2aBccn = 2bnxn + 2 (w - 1)

(9),

the complete solution, which may be written thus,

l.cos(^ + a)| ...... (10).

INTEGRATION OF DIFFERENTIAL EQUATIONS. 99
We now proceed to the more general equation,

As before, we shall get

{n(n-l)-p(p-l)}an + fa^ = 0 .......... (12).

Now n(n-l) -p (p-l) = (n-p) (n+p-l) ...... (13),

which is the fundamental principle of our analysis,

.-. (n-p)(n+p-i)an+fa^ = 0 ......... (14).

Assume (n+p-l) an = (n-p + 2) bn .... ........ (15),

n— ,

and (n -p + 2) (n+p -3)bn + fb^z = 0 ......... (16).

Again, assume

(n +p - 3) bn = (n -^ + 4) cn ............... (17),

and so on successively. , Thus we shall get a series of equations,
of which

(n-p + ri^+p-fjL-m + fl^O ......... (18),

is the general type, where p is even.

If p is even, let p = ^, .'. p — p — 1 = — 1.

If it is odd, let^> = p + 1, ,\ — p + p = — 1 :
and in "both cases (18) becomes

n(n-l)ln + 2*1^ = 0,
and therefore

2lnxn=Cain(2X+a) ..................... (19).

Let {n-p+(n-<*)}(n+p-p + l}in + fin_z = ^
(n-p + fjL)(n+p-fj,-l')kn + flc^ = 0,
be any two consecutive equations ; then

(w+p-A* + l)«. = (»-dp + /A)^... ......... (20),

n —p + /JL = n+p — fj, + l-2(p — p) — 1,

_I ....... (21),

7—2

100 INTEGRATION OF DIFFERENTIAL EQUATIONS.

i 2 (n -p

Now (n - p + A*) V2 = a^"1 (" -^ +

By the application of this formula, y or 2&n#n may be de-
duced by a series of regular operations from C sin (yx + a).

\ip is even, 2 (^> — //,) + 1 gives the series 1, 5, 9, &c.
If it is odd, the series is. 3, 7, 11, &c.

Particular cases may be solved by (22) with considerable facility.
By inspection we have

( . 1 )

y = C -jsin (qx + a) H -- cos (qx + a) >

for the solution of

The solution of -j4 + 22y = — » ? where p = 5, is easily .seen

X*

to be

Lastly, the solution of

s

50

^sin feaj + a) + ^ cos

INTEGRATION OF DIFFERENTIAL EQUATIONS. 101

The second line is equivalent to

jpg j^cos (qx + a.) - — ; jcos fe# + a) --sin faaj + a)

2 2 )

- - sin (qx + a) -- 8 cos (qx + a) K

CC <7iC j

and thus

/ /•» 1C

?/ = 0-jsin (go; + a) H -- cos (^a? + a) — 5-5 sin (^a? + a)

[ O'iC O' X

15

These examples will sufficiently illustrate the general formula.
The same method is applicable to the equation

Here we have

n[(n-i)(n - 2) -p (p - 1)} an + fan_, = 0,

Let

and generally

n (n -p - 1 + v) (n +^ - 2 - z/) ?„ + g8^ = 0 ...... (24),

where v is divisible by 3.
If p is so too, let p = v,

.'. w — ^> — l + v = w — 1 and n + p — 2 — v = w— 2.
If p — 1 is divisible by 3, let j? — 1 = z/,

/. -^ - 1 + v = - 2 and ^-2-z/ = -l;
and in both cases (24) becomes

w(w_l)(n_2)ZM + <?X-3 = 0,
and therefore ^,lnxn fulfils the equation

= 0 ........................ (25).

102 INTEGRATION OF DIFFERENTIAL EQUATIONS.

Again, if

n(n-p-l + *-3) (n+_p-2-v + 3) i + g%_« = 0,
rc (n -^ - 1 + ?) (n + p - 2 - *>) Jcn + tfk^ = 0,
be any consecutive equations, we have

« (n -p - 1 + v)

as may easily be seen h priori, or verified by differentiation.
The formula (26) is used in the same way as (22), to which it
is analogous.

We will give one instance of its application,
d3y 6 dy

*?<**«•

The solution of

daz

. + ^-O, is

rx /V3 \

and by (26),

2 . (3 + 1 - 3)

01 =

lor the complete solution of the proposed eauation.

INTEGRATION OF DIFFERENTIAL EQUATIONS. 103

It is obvious that analogous equations exist in all orders, and
that when p is of certain forms,

may be integrated in finite terms.

It will be sufficient, after what has been said for the cases of
m = 2 and = 3, to state the results of the general investigation ;
they may be very readily deduced by the same method as that
we have already used.

The process succeeds when either of the factors p or p — 1 is
divisible by w, and the general formula of which (22) and (26)
are cases, is

...... (27).

Particular cases may however be easily solved without re-
ference to this formula ; thus, if we had
d*y 4 _ 12 d*y

dtf~zy-—* dx*>

we should proceed as follows :

n(n-l){(n-2)(n-8)-4.8K-2V. = 0,

n(n-l}(n- 6) (n + !)«»- tfa^ = °>

(n + l)an = (n-2)l>n,

/. n (n - 1) (n - 2) (n - 3) bn - gfb^ = 0,

.'. ^bnxn = G/x + C2e~qx + C3 sin (qx + a),

O •» •• O / * \ j

and an=Jtt- _.._ _

- -4 ^n (n - 1) (n - 2) £naT4,

+ <73 -jsin (^# + a) H -- cos (^a? -f a) [• .
( g'a? ;

104 INTEGRATION OF DIFFERENTIAL EQUATIONS.

The principle of our analysis, it has already been remarked,
is contained in the equation

n (n - 1) -p (p-l) = (n-p) (n+p-l);
and this consideration suggests an extension of it.

For it is obvious that the coefficients of an in

\ dm~'-*

differ only in this, that where the first has the factors

(n - m + s + 2) (n - m + s + 1 ) ,
the second has p (p — 1).

Thus the same transformation applies; and if either p, or
p — 1 is divisible by m, the solution of

may be made to depend on that of

and thus effected in finite terms.

The formula of reduction in this case is a little more com-
plicated than those already given, and we will not dwell longer
upon it, our object being rather to point out the integrability of
certain classes of equations than actually to integrate them.

The equation

n (n - 1) -p (p - 1) = (n -p) (n +p - I)
is a particular case of

and the latter will give us various formulae of reduction accord
ing to the value of JJL. Thus

INTEGRATION OF DIFFERENTIAL EQUATIONS. 105

may be reduced to

d*y 1 dy >
S-SaS + rt'88'0'

for the coefficient of an in the former is

n (n - 2) - jp (p - 2) = (w -^) (n +^ - 2),

which, provided p is even, may be reduced to n (n — 2). But in
this, and in analogous cases, the auxiliary equation is, appa-
rently, insoluble.

The applicability of our transformation would, it is evident,
not be affected, if the equation were, instead of (25),

and, provided p or p — 1 were divisible by m — t, (29) might be
reduced to

But this case requires more care than those already con-
sidered, as if certain factors which apparently disappear are
neglected, our solution is incomplete, or erroneous.

An instance will make this clear,

dy y

Here (n - 2) (n 4- 1) an + (n - l)q a^ = 0 .. ........ (A).

Let (n + 1) an = (n - 1) bn .................. (a),

... (w-2)(»-l)n5n+(»-2)(»-l)«^1 = 0 ...... (B),

The factor n — 2 may be safely neglected. But n — 1 is
essential, because the solution of the auxiliary equation

dz

gives (n - 1) (nln + qb^ = 0,

and would be incomplete if we omitted the first factor.
From (a) we get

106 INTEGRATION OF DIFFERENTIAL EQUATIONS.
and, as except when n = 1, there is

nbn + gb^ = 0, ^-~ = - - b^l9 unless n = 0,

a

.-. «. = &. + -&»« ........................ (a').

Now the solution of the auxiliary equation is

z = c1 + cze~*x;
and from (a') we deduce

v — z -\ -- z :
qx

therefore y = (l + — } (ct + W**)

\ 2*E/

is apparently the solution of the proposed equation. But it will
be found not to satisfy it, unless ct = 0, and then

is only a particular solution. The reason is, that in laying down
(a') as generally true, we imply that

nbn + gb^ = 0

is true for n = 1 ; whereas the equation which contains tlxe

, ,. „ d*z dz
solution of -j-5 + q -y- = 0, viz.
dx* s dx

shows that bt is not necessarily connected with b0 ; and that if
we assume such connection, we get only a particular solution.
Hence our formula of reduction implies the connection of b^ and
b0 ; while their independence is implied in the general solution
of the auxiliary equation, to which this formula is applied j and
these contradictory suppositions lead to an erroneous result.
To put ^ = 0, is to connect bQ and blt or, which is the same
thing, to neglect the factor n - 1 ; and the value of y thus got
is therefore a solution, but not the complete solution of the pro-
posed equation.

To complete it, we must, bearing in mind the independence
of 50, recur to (a), which is always true,

INTEGRATION OF DIFFERENTIAL EQUATIONS. 107

and from (a'), which is true for n — — 1, we get

Now £_! is obviously = 0,

_2

and these two quantities are independent of alt «2, &c.,
is a particular solution, and

is the complete solution of the proposed equation.

The method of proceeding suggested by this example is to
obtain a solution, neglecting all factors analogous to (n — 1), and
then to complete it by reference to the assumptions of trans-
formation, such as (a), which have been made use of.

The equations which we have solved are not a very numerous
nor perhaps an important class. But one of them, at least, is
susceptible of a physical application of great interest ; and so
few equations of the higher orders are integrable in finite terms,
that the discussion of those which are, has always some degree
of value.

&>

^ °v-

if9T

ON THE INTEGRATION
OF CERTAIN DIFFERENTIAL EQUATIONS*.

No. II.

IN the last number of the Journal f, a method was given for
the investigation of a class of differential equations, by means of
successive reductions.

The present communication contains solutions of some ana-
logous equations effected by a similar process. The results will
however be exhibited in a very different form.

We begin by taking a particular case of the equations in
question,

(I),

x
where p is an integer.

Let y = S<vett,

.*. n(n— l)...(w— m + 2) (n — m + l —pm) an + kan_m = 0...(2).

Assume
an= {n — m + 1 — (p— 1) m] {n — m+ 1 — (p — 2) m]

...(n-m + l)/(fc)J. ...... (3),

f(k) being some function of Jc, to be determined hereafter.
Then

{n — m + 1 — (p — 1) m}

...... (4).

* Cambridge Mathematical Journal, No. XI. Vol. n. p. 193, February, 1841.
t Page 96 of this volume.

INTEGRATION OF DIFFERENTIAL EQUATIONS. 109

If we substitute these values in (2), every factor of (4) and
every factor, except the last, of (3), will disappear, and the re-
sulting equation will be

n(n-l)...(n-m + 2)(n-m + l)ln + kbn_m = Q ...... (5).

This is what (2) would be, were p = 0. Hence y = 2 . bnxn

dmy
fulfils the equation ~^ + Jcy = 0, to which (1) would, in that

case, be reduced. Let y — X be the ordinary form of the solu-
tion of the last-written equation. Then we must obviously
have

where (f> (k) may be any function of Jc. A very little attention
to the mode of integrating linear equations with constant co-

efficients will show, that in X, x always occurs in conjunction

j
with 7cm.

If we put

we must consequently have

An =

where N is a function of n, except for values of n < m, when it
is an arbitrary constant ; therefore

bn = N<t> (K) 7c™.

Recurring to (3), inverting the factors and multiplying and
dividing by mp, we shall easily deduce the following equation,

m-l

m

The form of these p decreasing factors naturally suggests the

idea of making f(Jc) — k~p ; and if we then put <f> (Jc) = Jc
we get

— m+l

l\ fn
j\

mm

110 INTEGRATION OF DIFFERENTIAL EQUATIONS.

/. a» = mP -jj-jbn, and /. 2anxm = mp -^ ^bnxn.

The factor mp may obviously be neglected, and we shall

j^
therefore have, on replacing %bnxn by <f> (k) X, i. e. by -^-j , the

Ji™^
following equation,

dp X

for the solution of (1), y = X being that of — + ty = 0.

Vfl •""• 1

If 7?i = 2, X= (7 sin {VW « + «}, and - = J. Hence

dp Cs

is the solution of

Jl+%=^.

rfa;2 x dx

This result is given in Hymers' Diff. Equations, and is, I believe,
due to Mr Gaskin.

If the proposed equation were

^

we should immediately conclude, from analogy, that its solution
must be

_ dT* X f

y — T.-P ™-i ?

but it may be as well to establish this conclusion by an inde-
pendent investigation.

Equation (2) will, in this case, be

n (n — 1) ... (n — m + 2) (n — m + 1 +pm) an + ka^ = 0.
Assume

' a = f(1r\ — _ n~
-'V )

INTEGRATION OF DIFFERENTIAL EQUATIONS. Ill
Therefore as before,

and an = m~p N

tn-vi + l \
\ m J

Here we make f(k) — Jcp, and <^> (Jc)

m

m-i

no complementary function being added.
Hence, precisely as before, we find that
_ d~p X

V — JT.-P m-l

tf* A;-

is the solution required.

Let us now consider the more general equation,

If 3^ = S . ana?n, there will be
n... (n — s + l)(n—s—j}m) (n — s — 1)...

an+^an_m = 0 ...... (7).

This equation is analogous to (2); but m-l is replaced
by s. Assume, therefore,

5 - (^ - 1) m}...(n - s + m) Jc~pbn_mJ
and n...(n-s + l)(n-s)(n-s-l)...(n-m+ l)bn + kbn_m = 0.
Hence, as before,

2bnxn = 0 (k) X,

where X denotes the same function of x that it did in the former

case.

2»
Consequently bn = ^V<j£> (A;) km, and

112 INTEGRATION OF DIFFERENTIAL EQUATIONS.

i
We must, it is evident, make <£ (k) = k "l, and then

Tp -rr

Hence # = -77^ ^7 is the solution of the equation

;

for as before the factor ra* may be neglected.

A particular case of this result is that in which 5 = 0. If
with this value of 5 we have m = 2, the equation to be integrated
takes the form

and the solution is

Equation (7) is the most general one in which the coefficient
of an differs in one factor only from what it is in the case of

But our method is applicable in other cases.

Let us resume the equation discussed in the last number of
the Journal,

^-*k-l), ............... (8).

By the usual method of making y — '£anxn, we get
n (n - 1) ... (» - m + 3) {(» - m + 2) (n - m + 1) -p . (p - l)j a

or n (n - l)...(n - m + 3) {(n - m + 2 -jp) (n-m + 1 +^)) an

+ kan_m = 0 ...... (9).

It may be remembered that we found it necessary that p or
p — 1 should be divisible by. m. Suppose then that p is so
divisible, and that the quotient is q.

INTEGRATION OF DIFFERENTIAL EQUATIONS. 113

Let

~fk '/ * T7 i 1 — ' — vT 7 * \" ^n

(10),

_ _ ...... _

(tliere are ^ factors in both numerator and denominator) ; equa-
tion (9) becomes

n (n-l)...(n — m + 3) (n — m + 2)(n — m + 1) &» + &5M_,n = 0;

and, as in the two preceding cases, we shall have
2bnxn = <£ (k) X,

and lm**N.&4>(ty.

(10) may be written thus, as^> = ^,

(n — m + 2\ /w — m + 2

- ...... -- q + l

V m J \ m _ ^_

XT
Now

mm

ft~* f n 1 f7q

therefore ^.

m J \ m -1- ' '

But if we make </> (&) = A; w , then

Let us also put f(k) = kq m ; therefore

'n - m + 2'

) — <KC. w-«m

-*•!

\

) - &c-

y JVr^"^~

114 INTEGRATION OF DIFFERENTIAL EQUATIONS.

Hence

is the solution required.

It admits also of another form, which it may be worth while
to remark.

m

'

/n -m+l \
\ m V

fjq

- ( k~ m k

-g + l)

/ | "I \ /.y- .yyj |_ "I

f^— +1) (-^r— +

~w+l -g+i

Here we must make <f>(k) = k m and f(k) —k m, and then

n—w-fl

~ and a :. = #(..,

therefore a=

The value of ^, deduced from the development of (12), can
of course differ only in a factor of some function of k from that
which is given by (11), and it will easily appear, on comparing

the values of an in the two cases, that this factor is k~**+m.

Let us now consider the case in which p is not divisible by
m, while p — 1 is so. And let p — 1 = qm. The two factors of
(9) on which our reduction operates, viz.

(n — m + 2 -p) (n- m + l +p),
may be written thus,

{n - m + 2 + (p - 1)} {n - m + 1 - (p - 1)},
or (n — m + 2 + qm) (n — m + 1 — qm).

INTEGRATION OF DIFFERENTIAL EQUATIONS. 115

The change which this will introduce in the process, is not
difficult to perceive. We must assume

n— m+1 n—m+1

an =/(&) - I

J ^ ' n-m + 2 n — m + 2

1-1 \-q

m m

as before, the transformed equation will be
n (n — m + 2) (n — m + l}\

Now

m

..

therefore

l\ n — m + 1

\
+)

m m

I -mfl

If then we make /(&) = tf m and <j>(Jc) = km ,WQ get

fn-m + l\ &

I / "*" tt-wi+2

\ m / _ Z.~»«~

/n_m + 2 v "*

I -- 1-1 ... &c.
V m J

Hence, finally,

As an illustration, let us take the case of the equation which
occurs in the theory of the figure of the earth,

Here m = 2, p = 3, p — 1 = 2 : hence q = 1, and as p is not
a multiple of m, the formula (13) is to be used.

8—2

116 INTEGRATION OF DIFFERENTIAL EQUATIONS.

It is in this case, as X— C sin {V(&) x + a}>

)a; + gn
J '

a sm {V (K) x + a} _ . x , ... , ^ sm { v (K) x -

dk $ k $

therefore

y = i^^i [&k* cos (V(&) x + a) ~ sm (V(&) # + «}]>
or integrating the first term by parts,

d'1
= JcC sin

But

sn > x + a = — cos

dk~l *' ' ' ' x

2 . . .... ,

-f -« sm |v (A?) a; + a],

/y»* I * \ / J '

therefore, if kC^G^

y = Cl [sin (V(&) ^ + a}

3 3

+ — r cos y(k) x + a} - y-j sin {VW fl? + all,
^2 A:a?

which is the required solution.

Equation (13) corresponds to (11) : but there is another form
of the solution in the case of p — 1 = gin, which we shall just
mention, and which is the counterpart of (12). It is

It would be a needless repetition to go through the steps
which lead to this result.

All the operations indicated in these symbolical solutions are
practicable. This will appear by considering the nature of the
function JT, which, in its most general form, consists of the sum
of terms, of which the type is

INTEGRATION OF DIFFERENTIAL EQUATIONS. 117

If we make k — Km, this will become

which may be integrated any number of times for K, and con-
sequently, if it is multiplied by any rational and integral func-

tion of K, it may still be integrated by parts as often as we

• flfc
please. Now -7 will become mK^s~ldK] and as m and s are

&

integral, the method of parts applies, provided s is not greater
than m — 1, which it is in none of our formulas.

Fourier's expression, by means of definite integrals for the
tth differential coefficient of any function, would enable us to
extend our solutions to the cases in which p is fractional. But
merely analytical transformations of the results at which we
have arrived are not of much interest, and the methods of effect-
ing them are direct and obvious.

Equation (1) admits of another symbolical solution besides
the one already given.

It is easily seen, that if

S«X = *"* JL _L . Jl _L. . . . Itjf, (p factors),

an = (n —pm -f 1) . . . (n — m + 1) 5n,

which is what (3) is, when f(k) = 1. (-r applies to all that
follows it.J

Hence we shall clearly have

^ d 1 d I * d 1 Y

y~ dx^dxtf^- "Txx™-^
for the solution of (1).

Similarly, the solution of (6) is

,, _ «.<*-i)«-H«-i) _ 1 T

y~ 'dxx™-1" "dxx1

Many applications and modifications of the method we have
employed will readily present themselves, but the subject is not
of sufficient importance to deserve a fuller discussion. It is not
difficult to multiply artifices, by means of which particular
equations may be solved, but the results will, generally speak-
ing, be of little value.

ANALYTICAL DEMONSTRATIONS
OF DR MATTHEW STEWART'S THEOREMS*.

IN 1746 Dr Matthew Stewart, the father of Dugald Stewart,
published his "General Theorems." He was at that time a
candidate for the chair of mathematics at Edinburgh, then vacant
by the death of Maclaurin ; and his success is attributed to the
celebrity which these remarkable propositions immediately ac-
quired. They were enunciated by Dr Stewart without demon-
strations, and remained undemonstrated till 1805. Mr Glenie,
in the Edinburgh Transactions for that year, has given a geo-
metrical method by which the General Theorems and other
similar results may be established.

But as yet they have not, I believe, been proved, except by
Geometry; and in an article in the 17th volume of the Edinburgh
Review, ascribed to Playfair, they are strongly recommended to
the attention of analysts. It is hoped, therefore, that the follow-
ing attempt will have some degree of interest.

We shall begin by establishing a general proposition, from
which all the theorems in question, and many others, may be
deduced.

LEMMA. If f<f> is a rational and integral function of sin^>
and cos <£, then a value may always be assigned to n, such that

shall be independent of <£.

The preceding expression is equivalent to

(1 . + D + . . . Dn~1} /</>, (where D$ = (/> + ^

* Cambridge Mathematical Journal, No. XII. Vol. n. p. 371, May, 1841.

STEWART'S THEOREMS. 119

and therefore to

Now A"1 0 = 2am sin mn$ + S6m cos mnfa (m integral) .
Hence /(<£) + . . . +/( <£ + - - 2?r J = S«m sin rarc<£ + &c.

Let the index of the highest power of sin <j> or cos <j> in f<j>
be p ; then it is easily seen that when /</> is developed, as it may
always be, in a series of sines and cosines of the multiple arcs,
p$ will be the largest arc that can enter into the development.
But if n is greater than p, mn<j) will be greater than p<f>, except
when m is zero. Hence the development

2am sin mn<j> + &c.

cannot coincide with that obtained by summing the separate
developments of /<£, f( <j> + — ) , &c., unless am and bm are = 0

\ 71 /

in every case, except when m = 0. Hence as sin mn<f) = 0 when
m = 0, the expression will be reduced to 50, and we shall have,
when n>p,

+ .../(<£ + ^— - 2?r) = bQ ...

a constant.

Q. E. D.

The constant 1>0 will of course be the sum of the constant
parts of the developments of /<£, &c. ; and as these are all equal
and are n in number, it will be n times the constant term in f<f>.

1 f2ir
Now by Fourier's theorem this is equal to — I f<j> . d(j), as in-

27T J Q

deed is obvious. Hence

which is our fundamental formula.

The first of Stewart's propositions is the following :

From any point in the circumference of a circle draw per-
pendiculars p, pv &c. to the sides of a regular w-sided polygon
circumscribed about it ; then, if r is the radius,

3 = 5wr8 ...... (1) ...... w>3.

120 STEWARTS THEOREMS.

DEM. Let the assumed point subtend at the centre an
angle <£> from the adjacent point of contact. Then

p = r (1 — cos <£), pl = r -1 1 - cos ( <f> + — -J }• , &c. &c.

-cos <;
and by the general formula,

w f2*
2 (1 - cos <£)3 = — I (I — cos $)3 d(j> ...... n being > 3.

Now f ^(1 - cos </>)3 <fy = 24 Tain6 0 dO . . . (0 = J<£) ,

Jo *•

^ir ^\ Q 1 K

and I sin6^^ = g-Lj12.7T = -j27r;

.•.S(1-C08*>'=£,

and therefore 2S3 = 5wrs. Q. E. D.

This is a particular case of the second proposition in which
the assumed point is not confined to the circumference of the
circle, but may have any position whatever. Let I be its dis-
tance from the centre ; then

22/ = 2ws + 3/iZV ... n > 3 ............... (2).

DEM. In this case p = r — I cos <£, &c. = &c.

.-. 2/ = nr3 - 3r2? 2 cos </> + 3rZ2 2 cos2 <£ - Z8 2 cos8 </>.

T2ir r2ir /*2ir

But I cos (f>d<f> = 0, I cos2 <pd(f> = 7r, I cos3 <j>d<j> = 0;

J o J o «/

and 228 = 2wr3 + 3w?V. Q. E. D.

In the third proposition, a regular w-sided polygon is in-
scribed in the circle, and lines c, ct, &c. are drawn from its
corners to a point assumed in the circumference ; then

(3).

STEWARTS THEOREMS. 121

DEM. The assumed point and adjacent corner subtending an
angle </> at the centre, we have

.'. 2c4 = 4/2(1- cos <£)*,

^ /., » \2 w-

/. 2(l-cos<£)2 = — ;

/. 2c4 = 6ft/. Q. E. D.

The fourth proposition includes the third. The assumed
point may now have any position we please. Let I be its dis-
tance from the centre. Here we have

c2 = r2 + Z2 — 2rZ cos <£,
and 2c4 = n (/ + Z4 + 2r2Z2) - 4r? (r2 + Z2) 2 cos $ + 4r2Z2 2 cos2 <£.

By the values above given for 5 cos 0 and 2 cos2 <£, this
becomes

2c4 = nr* + 4nr2/2 + nZ* ................... (4),

which is the proposition in question.

In the fifth proposition we return to the circumscribed poly-
gon, and our object is to determine the sum of the fourth powers
of the perpendiculars. As before,

and

therefore 82/ = 35w.r4 ........................... (5).

Q. E. D.

In the general case, when I is the distance of the assumed
point from the centre,

cos <j> + 6rT 2 cos2 <£ - ±rl3 2 cos3 0 + Z4 2 cos4 <£,

,
and .

122 STEWARTS THEOREMS.

or 8Sp4 = 8nr*+24w*Z* + 3nZ4 .................. (6).

This is the sixth proposition.

The seventh is, for the mih power of the perpendiculars, what
the first arid fifth are for the third and fourth powers respec-
tively. It is this :

^ _ 2m — 1 . 2m — 3 ... 1

m.m — 1...1

,_»
rm ...... (7) ......

DEM. V* = rm- (! - cos $)m d<f>.

*

Let 0 =

= 2wl+1 ["si

^ o

sm

0

2 w - 1 . 2m - 3 . . . 1 2m - 1 ... 1

7T = 2?T .

2m. 2m -2... 2 'm.m — 1...1'

1
m.m — 1 ... 1

Q. E. D.

If the assumed point is at I distance from the centre,

This, it is easily seen, will reduce into the following form,
.m — I I ,„,„ m...m — 33.1

m_474

r 1+&C'
...... (8),

which is the eighth proposition ...... (n> m).

Lastly, let us consider the 2mth powers of the chords in the
case of the inscribed polygon: we have already in the third
proposition found the value of their sum when m = 2.

As before, c2 = 2r2 (1 — cos </>) ;

/. 2c2w = 2V" S (1 - cos </>)"*.
That is, as we have already seen,

m . m — 1 . . . 1

STEWART'S THEOREMS. 123

We have thus gone through Dr Stewart's properties of the
circle, and have arrived at his results by a simple and uniform
method.

It is evident that there is no limit to the number of geome-
trical theorems which may be deduced from the general formula :
almost every curve will afford interpretations, if the word may
be so used, of our analytical conclusions.

Thus in the ellipse : If any n radii vectores be drawn from
the centre at equal angles to one another, the sum of the squares
of their reciprocals is equal to n times the square of the reciprocal
of that radius vector which is equally inclined to the major and
minor axes. For we have

-2 = p(l-e2cos2<£);

and J = cos2 — ;

therefore &c. Q. E. D.

It is to be regretted that we have hardly any idea by what
considerations Dr Stewart was led to the curious theorems which
bear his name. It is said, indeed, that he was engaged on geo-
metrical porisms when he discovered them, and we are told that
he would have published them under the title of porisms, but
for his unwillingness to interfere with a subject which the re-
searches of his friend, Dr Simson, seemed to have appropriated.
Whether they are in reality porismatic, is a question on which
it would not be worth while to enter.

The fundamental formula of our analysis is perhaps not new ;
the geometrical applications which we have made of it appear to
be original.

NOTE ON A DEFINITE INTEGRAL.

THE value of / log sin 6 d0, obviously the same as that of

IT ./O

rlog cos 6d0, was first assigned by Euler, ,and may be obtained
„
in the following manner.

By Cotes's theorem,

/ i \ / m— i \

Z^-l^fz2- 1) Z2- 2Z COS - 7T+ 1 ... (Z*- 2Z COS 7T + 1

' \ m J \ m J

w-

Let « = 1, then

^

m 2 m 2

Take the logarithms of both sides, and divide by m, then

log m + 2 (m — 1) log J
2m

A . 1 TT , . m — 1 7r\ 1

= log sin — -+...+ log sm — .

\ * m 2 m 2J m

Let m become infinite, and = ^- : the first side becomes equal

to log - , for ( - — j = 0 when m = - ; and the second is trans-
A \ ffi / 0

formed into the definite integral I log sin x ^ . dx ; therefore

•'o ^

I log sin x — dx = log J.
^o *

Let ^ = »|;

* Cambridge Mathematical Journal, No. XII. Vol. n. p. 282, May, i84f.

NOTE ON A DEFINITE INTEGRAL. 125

(1).

COR. 1. Integrating by parts, we get
[log sin 0d0 = 0 log sin 0 -
The integrated part vanishes at both limits,

COR. 2. Let sin 6 = e~& ; therefore the limits of x are 0
and oo,

COR. 3. In this last integral, if we expand the denominator

and as I e""1* x dx = -j » , we find

111 1.311 1.3.5

I. - 32+ — ^ 5,+ j^-g 23

REMARKS ON THE DISTINCTION BETWEEN

ALGEBRAICAL AND FUNCTIONAL

EQUATIONS*.

THE distinction which it is usual to make between alge-
braical and functional equations will not, I think, bear a strict
examination. It is generally said that an algebraical equation
determines the value of an unknown quantity, while a func-
tional equation determines the form of an unknown function.
But, in reality, the unknown quantity in the former case is a
function of the coefficients of the equation, and our object in
solving it is simply to ascertain the form of this function.
Thus it appears, that in both cases the forms of functions are
what we seek.

Let us therefore consider the subject in a more general
manner, and endeavour to find a more decided point of dis-
tinction. The science of symbols is conversant with opera-
tions, and not with quantities; and an equation, of whatever
species, may be defined to be a congeries of operations, known
and unknown, equated to the symbol zero. Every operation
implies the existence of a base, or something on which the
operation is performed — in the language of Mr Murphy, a
subject. But the base of an operation is often the result of
a preceding one. Thus, in log £c2, the base of the operation
log is cc2, itself the result of the operation expressed by the
index on the base x. This in its turn may be considered as
the result of an operation performed on the symbol unity. But
in every kind of equation there is a point at which the
farther analysis of symbols into operations on certain bases
becomes irrelevant ; and thus we are led in every case to recog-
nize the existence of ultimate bases.

* Cambridge Mathematical Journal, No. XIV. Vol. in. p. 92, February,
1842.

NATURE OF FUNCTIONAL EQUATIONS. 127

To solve an equation of any kind, is to determine the un-
known operations by means of the known. If one symbol is
said to be a function of another, it is, in reality, the result of
an operation performed upon it. Thus the idea of functional
dependence pervades the whole science of symbols, and on this
idea the following remarks are based.

In order to classify equations, we can make use of two
considerations : 1st. The nature of the operations which are
combined together; 2nd. The order in which they succeed
one another in the congeries of operations which is made equal
to zero.

Let us illustrate these remarks by some examples.

If we have an equation of the form

x*+ax+t> = 0 ........................ (1),

the bases are a and b ; the operations are, first, the unknown
one denoted by x, and then certain known ones denoted by the
index, the coefficient, &c. All these are what are called alge-
braical operations.

If again we have an equation of the form

«.

the base is x\ the operations are, first, the unknown one
denoted by #, which is a function of x, then the operation

-T-, and lastly, certain algebraical operations. From the pre-

UX

sence of the operation -y- , this is called a differential equa-

tion. Equations (1) and (2) are discriminated by the nature
of the operations combined, on our first principle of classifi-
cation.

But in one important point these equations agree. In
both, the unknown operation is performed immediately on the
bases ; the known are subsequent to the unknown : but in what
are called functional equations this is not so. Thus, in the
equation

$(mx) + x = Q ........................ (3),

the base is x, the unknown operation is <f>, which is performed,
not on x, but on the result of a previous operation. In the

128 NATURE OF FUNCTIONAL EQUATIONS.

preceding example the previous operation is known ; but this is
not essential. Thus in

+#=0 ........................... (4)

the previous operation denoted by the right-hand <j> is unknown.
The operation -*- may enter into equations where the un-

known operation is not performed on the base. Thus we may
have an equation of the form

Equations (3), (4), (5) are functional equations; (3), (4) are
ordinary functional equations ; (5) is a differential functional
equation ; (3) is said to be of the first order, (4) of the second.

The introduction of the functional notation appears to be
sometimes taken as the essence of functional equations ; but if
we wrote (1) and (2) thus,

}<£(a£)}2 + a<£(a£) + & = 0 .................. (1)',

3£*(a)-a = 0 .................. (2)',

they would still be perfectly distinct from (3) or (4) or (5).
The name functional equation is not happy; it refers to the
notation, and not to the essence of the thing.

A question now arises : To what class shall we refer equa-
tions in finite differences ? These are generally of the form

*>,y,,y.« •••)=<> .................... (6),

where yx is an unknown function, say <£ (x) of x ; so that (6)
may be written thus,

Here the unknown operation is <£, which in the case of <f> (x + 1)
is performed, not upon the base x, but on x+1. Thus it
appears, that equations in finite differences are only a case of
ordinary functional equations of the first order : and this is the
reason why, in researches on functional equations, we per-
petually meet with cases in which they may be reduced to
equations in finite differences.

NATURE OF FUNCTIONAL EQUATIONS. 129

The preceding remarks contain, I think, the outline of a
natural arrangement of the science of symbols. It is not difficult
to overrate the importance of a mere classification ; but I hope
to be able to show, that the considerations now suggested are
not without some degree of utility.

As the distinction between functional and common equations
depends on the order of operations, it follows that, when part
of the solution of an equation does not vary with the nature
of the operation subjected to the resolving process, this part
is applicable as much to functional equations as to any other.
The special application of this principle to the discussion of a
class of differential functional equations will be the object of a
subsequent paper.

In the preceding remarks, operations of derivation, such as
D, A, &c. are supposed to be replaced by functional operations
in every case in which this can be effected.

9

MATHEMATICAL NOTES*.

1. IN the Examination Papers for 1834, the following
problem is given: "If the chord of a conic section, whose
eccentricity is e, subtend at its focus a constant angle 2a, prove
that it always touches a conic section having the same focus
whose eccentricity is e cos a." A solution of this problem by
a peculiar analysis will be found in a preceding article ; but the
following method may be found not uninteresting.

Let rl5 rz be radii vectores to the ends of the chord, <£ — a,
<£ 4- a, the corresponding angles vectores, p the perpendicular
from the focus on the chord ;

.'. p x chord = r^ sin 2a,
. lV^ + r,2- 2r,r>cos2a)

' p rfa sin 2a

// 1 1 2 \

= cosec 2a » / f -jH — 2 cos 2aj ,

V * 1 2 12 '

1 _ 1 H- e cos (^ — a) JL — <fr

For cos 2a put 1—2 sin2 a ; then, by a few obvious steps,

- = , fJ(l + 2e cos a cos <b + e2 cos2 a).

^? t cos a

Put a = 0 ; then the chord becomes the tangent, and
11

Po I

But the general form coincides with this, if we put

I cos a = X and e cos a = e ;
for then

* Cambridge Mathematical Journal, No. IX. Vol. in. p. 94, February, 1844.

MATHEMATICAL NOTES. 131

Hence p is generally the perpendicular on a tangent of an ellipse
of eccentricity e cos a. Hence the chord touches such an ellipse.
The latus rectum is diminished in the same ratio as the ec-
centricity.

2. The relation between the long inequalities of two mu-
tually disturbing planets, may be easily found without having
recourse to the development of the disturbing function.

Let w, m\ be the masses of the planets, a, a, the major axes
of their orbits, n, ri, their mean motions, A, A', twice the areas
described in 1" ; then we have

a*

fju being the mass of the Sun, in comparison with which the
masses of the planets are neglected, so that it is the same for
both. Taking the logarithmic differentials of these equations,
and replacing the differentials by differences, we find

A^__3Aa Ari _ 3 Aa'
n ~ 2~cT' ~nT~ "~2~oT*

But by the principle of the conservation of areas,

mh + m'ti = const.

so that wAA + m'AA' = 0.

Now the orbits being supposed circular, we have

AA 1 Aa AA' 1 Aa
hence T" = o ~ • -rr = «— r'-

h 2 a h 2 a

Therefore we have

Aw t An' _ Aa a _ AA h' _ mf a'* t
n ' n' Aa' " a AA' * A ~ m #4 '

and - - and — r are the inequalities due to the disturbances, so
n n

that their ratio is thus given.

9—2

ON THE SOLUTION OF FUNCTIONAL
DIFFERENTIAL EQUATIONS*.

IT is well known that the solution of a considerable class of
differential equations may be effected by means of differentiation.
Clairaut's equation is a particular case of this class. We will
begin by considering it.

y=px+fp...(l)

where /denotes any given function.
Differentiating (1), we get

(*+/»2 = 0 ........................ (2),

hence ^ = 0, or x+fp = 0 ........................ (3).

The first of these equations gives the complete integral.
Being twice integrated it becomes y — ax + b ; and on substi-
tution in (1), we get b=fa, therefore y — ax -\-fa . . . (4) is the
complete integral of (1).

It has always been supposed, in this and similar cases, that
f must necessarily be a given function. But this condition is
not essential: a differential equation, e.g. such as (1), will,
when solved, give y as a function of x. Now the function f,
which enters into (1) may, instead of being given, as is usually
the case, be in some way dependent on the function which y
is of x. Thus the form of f is unknown, until that of the
latter function has been determined. It is evident that accord-
ing to the classification proposed in the last number of the
Journal, (1) is in all such cases a functional equation. For the
unknown operation / is performed on p, which is itself the result
of the unknown operation tf performed on x (we suppose

* Cambridge Mathematical Journal, No. XV. Vol. in. p. 131, May, 1847.

ON THE SOLUTION OF FUNCTIONAL EQUATIONS. 133

To differential functional equations, ordinary methods of
solution do not, generally speaking, apply, because they re-
quire a knowledge of the forms of the functions on which they
operate. But in the case before us, the differentiation and
subsequent substitution, by which (4) was derived from (1), are
independent of any knowledge of the nature of/. Consequently
(4) is always true.

Let us suppose, for instance, that / = — m<*jr, m being a
constant; then

^x — xty'x + m^Jr^'x = 0 .................. (5).

We are of course obliged to introduce a functional notation :
(4) in this case becomes

^rx = ax — m^ra ........................ (6).

In order to determine -*Jra, put x = a ;

then a^,

and ilrx = ax — - - a2 ............... (7),

1 +m

which is a solution of (5).

In the ordinary cases of Clairaut's equation, the factor
x+f'p = 0 leads to the singular solution; and so it does when
/is an unknown function.

Thus, in the example just considered, as /' = — mty', we
shall have

m^-^'x = x ........................... (8) .

Of this a solution is

Hence we get, by integration,

^. = _£-.+ <7.
2VW

On substitution it is found that (7 = 0, therefore

is a new solution of (5), and perfectly distinct from (7).

134 ON THE SOLUTION OF

If m— 1, (5) and (7) become respectively

tyx - x^rx + ^r^'x = 0 .................. (5').

•frx = ax-^a2 ............................ (7');

in this case, (8) admits of a variety of simple solutions. Thus
we shall have

&c. = &c.
as singular solutions of (5').

The preceding remarks are sufficient to indicate the exist-
ence of a class of functional equations, to which a considerable
portion of the theory of singular solutions may be applied.
They appear therefore to possess some interest with reference
to this theory, independently of the method they suggest for
the solution of such equations.

In fact the theory can hardly be considered complete, unless
some notice is taken of the equations of which we have been
speaking. They have been excluded from it, because the func-
tion /, which they involve, is not, as in the ordinary case, a
known function. But this, it has been already remarked, is not
an essential distinction.

On the other hand, the method by which the singular
solution is in the common theory deduced from the complete
integral, does not apply to the cases now considered. It appears
unnecessary to point out the reason of this difference.

With regard to the class of differential equations, which,
like Clairaut's, separate into factors on differentiation, we may
refer to Lagrange's Legons sur le Calcul des Fonctions, 1. 16me.
He there shows that if a differential equation of the first order
can be put into the form M=fN, where M and JVare the values
of a and b deduced from

then, when differentiated, it will resolve itself into two factors,
one of which leads to the singular solution, and the other to

FUNCTIONAL DIFFERENTIAL EQUATIONS. 135

the complete integral. (The latter is, as may readily be seen,

The demonstration of this proposition is probably familiar
to the majority of my readers, and I shall therefore not dwell
upon it. Similar considerations apply to equations of higher
orders.

Generalizing the remarks already made, we see that in the
equation

M=fN,

the function / need not be a given one ; it may be, in any
way we please, dependent on the function which, in virtue of this
equation, y is of x. In all such cases the equation in question
is functional. Nevertheless, Lagrange's reasoning applies as
much in these as in other cases. Let us take one or two ex-
amples of what has been said.

The following problem may be proposed.

Any point P of a certain curve is referred to the axis of x
in M, and to that of y in N. MP is produced to Q ; PQ is
taken equal to a, and NQ touches the curve. Find its equa-
tion.

Let x, tyx be the co-ordinates of the point where NQ
touches the curve.

a

and as P is a point in'the curve,

, or

(10).

This is the equation of the problem. Differentiating it, we
get

The former equation gives the complete integral, but, for a
reason I shall hereafter notice, leads to no tangible solution of the
problem ; the latter corresponds to the singular solution.

136 ON THE SOLUTION OF

In order to solve it, assume

^=**;

t, a a , x yx
then ilr —7- = -5- and — = ^- .
Tjra; ^x ^a; jfx

Let a? = we, x# = wm , and therefore

Then — — = — — = -~ , where C is arbitrary ;
u*n V3 0

therefore %a? = (7a?.

0 is Ibfunction of 0, which does not change when z + 1 is
substituted for z.

We confine ourselves to the only simple case, that in which
it is an absolute constant ; then

r — -~- — - ...
Cx x

and tyx — b log - ... c being an arbitrary constant.
c

On substitution, we find b — ae; therefore

# = aelog^ ........................... (11)

o

is a solution of the problem.

This is the equation of a logarithmic curve, which has there-
fore the required property. The method employed to resolve
the equation in yjc, namely,

is applicable to every equation of the form

0 .................. (12).

Every such equation may be at once reduced to the follow-
ing equation in finite differences,

F(uKuK+l...u^n}.= Q ...... . ............ (13).

This reduction is in reality a particular case of an important
transformation due to Mr Babbage, which often enables us to
solve functional equations of the higher orders.

FUNCTIONAL DIFFERENTIAL EQUATIONS. 137

In (12) we may write for %c

Hence %2a;= ffi^x &c. =&c., and (12) becomes

= 0 .................. (14),

by putting <j>x for x • f being a known function, (14) is a
functional equation of the first order.

Such is Mr Babbage's method. Let fx = 1 4- x ; (14) be-
comes

and if we denote $x by ux, and replace x by 2, we shall ob-
tain (13).

It must be admitted, that it is difficult to prove that the
generality of (12) is not restricted by these transformations.
They are however often useful, and serve to illustrate what
was remarked in the last number, with respect to the affinity of
functional equations, and equations in finite differences.

If, instead of (10), we had taken the more general equation

where A is an arbitrary constant, precisely the same method
would have applied. In this case the factor ty"x — 0 would
have led to the result

tyx = a.x 4- ft,

and by substitution fB = a + /3 + A,

therefore a'+A = Q, or /8= oo .

Now in the case we have been considering, the former con-
dition is not fulfilled ; hence we must have ft = co , and the
geometrical interpretation of the complete integral is a right
line at an infinite distance from the axis of abscissa?.

We not unfrequently meet with similar cases, in which the
complete integral becomes nugatory or impossible in the pro-
cess of introducing the necessary relation between its constants.
Under particular conditions, however, this difficulty does not
occur, and then we obtain, what in the ordinary methods of
discussing functional differential equations, appears to be a
conjugate solution, unconnected with any other ; (15) would
be an instance of this, were a + A = 0.

138 ON THE SOLUTION OF

I shall next consider a celebrated problem, first proposed
by Euler, in the Petersburg!! memoirs.

In a certain class of curves, the square of any normal ex-
ceeds the square of the ordinate drawn from its foot by a
certain quantity a.

Let y2 = tyx be the equation of the curve. The subnormal
is therefore fafrx, and the equation of the problem conse-
quently is

+ (x + W*)=i™ + lWx)*-* ............ (16).

Differentiating this, we get

or

The first of these two equations leads to the singular solu-
tions. In order to solve it, let

x -f fflx — %x,
then %*x — %xx + # = 0.
Hence by the transformation already noticed,

whence UK

where Pz and P^z are functions of z, which remain unchanged
when z increases by unity ;

therefore ug+l = Pz + (z + 1 ) P^z,
Hence we have

* t for the required solution.
x = Pz + zPf)

therefore ydy = Pf (Pz + Ptz + zP^z) dz ;
and integrating by parts, we get

f = Pf (2Pz + zPj) +J(PsY &1 , .

"

for a general solution of the proposed problem. (The parame-
ter a is involved in P,z.)

FUNCTIONAL DIFFERENTIAL EQUATIONS. 139

Let us suppose Pz and P^z constant ;

x = I + a.z ;

therefore y*=2ax + C.
On substitution we find

Thus, in order to a real result, we must suppose a negative,
e.g. let a = — &2 ; then

(7 ....................... (18),

the equation to a parabola, which accordingly is a solution of
the problem, and the only simple one it admits of.

When a = 0, it becomes two straight lines parallel to the
axis.

The other factor 1 + ^"x = 0 gives, on integration,

..................... (19),

the equation to a circle ; but on substitution, we find

Ja2 = Ja2 - a,
which leads to no result, unless a = 0.

A solution of this problem, by Poisson, is given at p. 591
of the last volume of Lacroix's great work. It is apparently
equivalent in point of generality to (17) ; and the author
points out its incompleteness in the case of a = 0. The pre-
ceding views show distinctly the nature and origin of the new '
solution which then presents itself. Mr Babbage also has
considered this problem at the end of his second essay on the
Calculus of Functions (vide Phil. Trans. 1816, p. 253). But
I believe it will be found that his solution is erroneous.

Notwithstanding the length this paper has already reached,
I must endeavour to point out, as briefly as possible, my reasons
for thinking so.

Mr Babbage confines himself to the case of a = 0. He
begins by demonstrating the existence of a relation, equivalent,
excepting a difference of notation, to ty [x + ^ifr'x] = ty'x, but in
doing this, loses sight of the other factor 1 -f ffl'x — 0.

140 ON THE SOLUTION OF

This relation shows that ^r'x is constant, for a series of points
in the curve, and therefore, Mr Babbage reasons, we may con-
sider it as a constant in (16), which thus becomes an equation
in finite differences. He integrates it on this supposition, and
adds an arbitrary function of ty'x, which has been treated as an
absolute constant. The result is therefore

which is an ordinary differential equation.

This process appears to have been suggested by an incorrect
analogy with the way in which arbitrary functions are intro-
duced into partial differential equations.

A little consideration would have convinced Mr Babbage,
that by integrating (16) as an equation in finite differences, he
only passed discontinuously from one ordinate of the curve to
another, and therefore could not obtain a continuous relation
between x and y. The legitimate result of his process is
merely,

where n is any positive or negative integer. This is quite
different from

In exemplifying equation (20), Mr Babbage first supposes

dy\ dy

-f-\ =00 x y -r- +

dxj * ax

and thus obtains the equation of a straight line parallel to Ox,
as a solution of the problem, which undoubtedly it is.

In his next example fly ~j = a2. By making the con-
stant of integration imaginary, he gets #2 = a* — x2, the equation
to a circle. But although this is also a real solution, it has
no connection with the relation -^r [x + ^ yfr'x] = ^'x, from which
it appears to be derived. It is, as we have seen, a particular
case of the complete integral. Consequently if the method

FUNCTIONAL DIFFERENTIAL EQUATIONS. 141

pursued had been correct, it could not have given this solu-
tion.

The preceding pages appear to contain the germ of a general
theory of differential functional equations ; a subject of great
extent, and ultimately, perhaps, of considerable importance.
But it cannot be denied, that hitherto the Calculus of Functions
has not led to many results of much interest. Its value arises
chiefly from the wide views it gives of the science of the com-
bination of symbols.

MATHEMATICAL NOTE*.

Problem from the Papers of 1842. — If F(x, y, z) = $ (w, v, to),
where F is homogeneous of the nih degree in x, y, 0, and

dF dF dF

u= ^- t v= T~> w~ ~T~
dx ' _ dy dz

Since 7^ is homogeneous of n dimensions in a?, ?/, z9 we have

dF

nF= x -j- + y -j- + z -j- = xu + yv + zw.
dx 9 dy dz

Hence ndF= xdu + ydv -f- zdw + wc?a; + vdy + wdz,
or (n — 1) dF— xdu + #cfo

But (n-

Therefore equating the coefficients of the differentials,

.-(-i)g, ,-(.-i)g, .-(-1)2-

* Cambridge Mathematical Journal, No. XV. Vol. m. p. 152, May, 1842.

EVALUATION OF CERTAIN DEFINITE
INTEGRALS*.

WHEN the value of a definite integral is known, we may, if
it involve an arbitrary parameter, integrate it (under certain
conditions) with respect to this quantity. The result thus ob-
tained involves an arbitrary constant of integration; in order
to eliminate it, we may ascribe two different values to the
quantity for which the integration has been effected, and then,
of the two corresponding equations thus got, subtract one from
the other. In other words, we integrate between limits for the
arbitrary parameter, and thus get a new definite integral, in-
volving two arbitrary quantities, namely, the two limiting
values ascribed to the single one involved in the original in-
tegral. We may integrate again, with respect to either of these,
and so on. But this method of proceeding, though it will lead
to a variety of particular results, is not well fitted to show the
nature of the class of definite integrals to which they all belong,
and which may be obtained by repeated integrations for an
arbitrary parameter.

If we integrate n times successively, we shall introduce n
constants. These may be eliminated at once, in the manner
I am about to point out. The result thus got, includes for
every original definite integral, all that can be deduced from
it by n integrations for an arbitrary parameter.

The following theorem will serve to illustrate the general
method.

If Fx is a rational and integral function of circular func-
tions of x (sines and cosines), then we may express in finite

r+MFx
terms the value of I — - dx, n being a positive integer, and

J-oo X

such that — -[ is not infinite.
anJo

* Cambridge Mathematical Journal, No. XVI. Vol. in. p. 185, Nov. i8j

144 EVALUATION OF CERTAIN DEFINITE INTEGRALS.

This theorem applies to several remarkable definite integrals,
some of which occur in the theory of probabilities; there are
others which do not seem to have been noticed.

DEM. Fx, as every function of x, may be considered the
sum of two functions, one of which remains unchanged when
x changes its sign, and the other changes its sign with that of
x; its value aux signes pres remaining unaltered. Hence,
whether n is odd or even, we may write

fx_fa fa
xn~xn + xn'

where _

It is obvious that

I
J

and that if n is odd, fa, which is of course a rational and in-
tegral function of circular functions (sines and cosines) of x,
must be developable in a series of sines exclusively, and if n
is even in a series of cosines exclusively.
Thus, we may assume

Now, as J— is not infinite when x = 0, the lowest power of

x which can enter into fx must be not < n, call it m, and
develope in powers of x every sine or cosine which appears on
the second side of the last written equation. We must have
?<Aam~* = 0, S-4am~4 = 0, &c. ... J m — 1 equations if m is even,
and £ (m — 1) if it is odd.

Let us now consider the definite integral

Integrating it repeatedly for r, we get

^LX sm rx , _j r

e - cto = tan l - ,

x a'

EVALUATION OF CERTAIN DEFINITE INTEGRALS. 145

A*

= — r tan"1 --fa log \/(a2 + ?*2) + C,

and generally

sin

(2),

where Fl (ra) does not become infinite for a = 0.

Replace r by every quantity represented in (1) by the
general symbol a. Multiply each result by the corresponding
coefficient A, and add.

Then, in virtue of the conditions,

2L4am~* = 0, ^AcT'* = 0, &c.
we shall have

'£*•-** frS] tanJ I + a*AF> (°a)'

Put a = 0, then tan"1 - = ± - , according to whether a is
> or < than zero.
Thus we have

From hence, the truth of our theorem is obvious.

Of the ambiguous signs outside the symbol of summation,
the upper is to be taken when n is of the forms 4p, or 4p + l.

When a is positive, we must take the upper of the am-
biguous signs under the 2.

It will be remarked, that in obtaining (3) we have elimi-
nated all the constants at once, instead of getting rid of them
one by one by particular conditions at each successive inte-
gration, and that the generality of this method enables us to
recognize a class of definite integrals, which are all deduced

-00

from the known value of I e~a* cos rx dx.

J o

10

146 EVALUATION OF CERTAIN DEFINITE INTEGRALS.

Equation (3) admits of several remarkable applications.
Thus let us suppose n = 3 and fx = sin ax sin bx sin ex : then

fx = — J {sin (a + b + c) a; — sin (— a + b + c) x — sin (a - & + c) a?

— sin (a + 5 — c)a:}.
Consequently, we have by (3),

sin ao? sin foe sin <%c , 7r_, _ N-

where, as in trigonometrical formulas,
25 = a + b + c :

the upper sign is to be taken when the quantity to which it is
affixed is > 0.

T ., . /""sin ax sin bx sin ex , TT ,.,_.,_.,_., N
Likewise I - ax = - (1 + 1 + 1 + 1),

J 0 &

where the signs follow the same rule as in the former case;
the different unities involved being the zero powers of s,
s — a, &c.

Let us now suppose that fx = smmx cos zx ; the correspond-

. , , f00 smmx coszx j -At. it r

ing integral, viz. j — ^ — - ao; occurs in the theory oi

JQ x

probabilities. Its value is given at p. 170 of Laplace's Theorie
des Prolabitites, where it is obtained by a method founded on
a transition from real to imaginary quantities. The nature of
what are called imaginary quantities is certainly better under-
stood than it was some time since ; but it seems to have been
the opinion of Poisson, as well as of Laplace himself, that
results thus obtained require confirmation. In this view I
confess I do not acquiesce; but if only in deference to their
authority, it may be desirable to show how readily imaginary
quantities may be avoided in estimating the value of the integral
in question.

sinm# = ± —^ -jcos mx - — cos (m — 2) x + &c.i if m is odd,

and = + — j jsin mx — — sin (m — 2) x + &c. !• if it is even.

TT T. /o\ f °° sinma; cos zx ,

Hence, by (3), | - - - dx =

J o ^

)"-'± («-*-2)"' J+&CJ

EVALUATION OF CERTAIN DEFINITE INTEGRALS. 147

Let us suppose z > m ; then the lower of each pair of am-
biguous signs must be taken, and the expression within the
brackets may be written thus

(m+z)"-1- ~ (m+z-2)«-*+&c. +^ (m^z-2)n-l^(m-z)n'\..(q).

As, from the nature of the case, m and n are either both
odd or both even, if m is even, n - I is odd, and therefore
(m - z}n~l = - (z — m}n~l'j and thus, in every case, (#) equals

(1 - D~T (* + m)n~1... {D$z = <t>(z + 2) say,}
and this is ^mD~^n (z + m}n~l = Aw (z - m)n~l = 0,
since mis>n—l. Consequently

sinwl x ,

— cos zx ax — 0, when z is > m,

i.

a remark not made by Laplace; when m = n = l9 its truth is
known.

As (q) = 0, add it, multiplied by f - -^ — ^ , to the value

\n — IJ 2

of the integral already found, therefore

where the series stops whenever the next term would intro-
duce a negative quantity raised to the power n — 1. This is
easily seen to be true, for every such term will have a different
sign in (q)9 and in the definite integral, and thus on addition,
all such terms will disappear. Equation (4) is Laplace's form ;
the discontinuity of the function is now expressed, not by
ambiguous signs but by the stopping short of the series at
different points.

By a similar method, we find that

... (5),

which might have been deduced from (4) by integrating both
sides without introducing any complementary quantities. This
remark is general; having once established the general form
of (4) for any given value of w, we may deduce from it that
which corresponds to any other value of w, simply by differ-
entiation or by integration, without bringing in any constants.

10—2

148 EVALUATION OF CERTAIN DEFINITE INTEGRALS.

I conceive that this remark is general, and if so we may dif-
ferentiate on both sides with fractional indices. Let the index
be — p, then as

d* cos zx = xp cos \zx + p ^ J dzp,
the first side of (4) will become

and the second will be

Thus, we get

....(6).

If we take w = m this is equivalent to Laplace's general
formula (p), at p. 168 of the Theorie.

The method of this paper leads to some elegant results when

/CO
e^dx, but it is enough
j
to point out this application, which involves no difficulty what-

ever.

MATHEMATICAL NOTE*.

Stability of Eccentricities and Inclinations. The equation
for proving the stability of the eccentricities and inclinations
of the planetary orbits may, as has been shown by Laplace,
be deduced from the principle of the conservation of areas, joined
to the fact of the invariability of the major axes.

Let m be the mass of a planet, h be twice the area de-
scribed by the radius vector about the sun in an unit of time
projected on the ecliptic, and i the inclination of its orbit to
the ecliptic ; then, by the principle of the conservation of
areas,

2 (mh) = const.

Every term under the sign of Summation is positive, be-
cause all the planets move round the sun in the same direction.
But

h = {a (I - e2)}* cos t = a* (1 - e8)4 (1 + tan2*)"*.

Now 6 and i are small at the present time; hence, if we
neglect their fourth power and the products of their squares,
we have

£ = <^{l-Je2-itan2;}.

Hence 2 {wa* (I - Je2 - \ tan2 *)} = const.

But since the major axes have no secular inequalities

2 (mat) = const.;
hence the preceding equation is equivalent to

2 (ma£ e* + ma? tan2*) = const.

Now the left-hand side of the equation being small at the
present time, the second side is also small, and therefore the
first side is always small, and therefore

2 (wa^e2) and 2, (ma* too? f)
are both always small.

* Cambridge Mathematical Journal, No. XVIII. Vol. m. p. 290, May, 1843.

ON THE EVALUATION OF DEFINITE
MULTIPLE INTEGKALS*.

THE following pages contain some general results obtained
by means of Fourier's theorem. A few words will be sufficient
to explain the manner in which it has been applied.

A definite multiple integral, where the limits are given by
the inequality

may be treated as if the limits of the different variables were
independent of one another, provided the function under the
signs of integration be considered discontinuous, and equal to
zero whenever f(xy ...... ) transgresses the assigned limits \

and h. This idea has been made use of by M. Lejeune Dirichlet.
It had, however, occurred to me before I was acquainted with
his paper on multiple integrals, and the way in which I have
applied it is, I believe, new.

Suppose the function to be integrated were of the form
<f) (xy ...... }^{f(xy ...... )): the limits being given by the in-

equality already mentioned. Then, by Fourier's theorem, writing

1 /"* [k

i|r/. = - I doil -^ru . cos a (/. - u)

7T J o J HI

du,

for all values of f. which lie between \ and h ; moreover for
the purposes of integration the formula may be applied, except
in particular cases, so as to include these limiting values
(see Poisson, Theorie de la Chaleur) ; while for all values of/.
which lie without these limits, the second side of the equation is
equal to zero. Consequently the integral

* Cambridge Mathematical Journal, No. XIX. Vol. iv. p. i, November, 1843.

EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 151
Sdxjdy... * to...)* {/to»0}

= — / ^rudu I da I dx I dy . .. $ (xy . . .) cos a (/. — u) ... (1) ;

"If J h j J 0 •/ »/

the limits on the first side being given in the manner already
mentioned. Those of the integrations with respect to x, y, &c.
are arbitrary, provided they include all values of the variables
which satisfy the given inequality.

Again, if in the given multiple integral the limits were

determined by the single relation f> ^ h, joined to the con-

ditions that x} y, &c. were to have no values less than certain
assigned limits; e.g. if we were to consider only positive
values of the variables, the formula (I) would still apply with
a slight modification. The inferior limit of integration with
respect to u would be arbitrary, provided it included all the
values which could be given to /. by admissible values of
the variables, while the inferior limits of integration for
x, y, &c. would be determined by the particular conditions of
the case.

Let us take, as an example of the method, the integral

fdxfdy...xa-1yb-1zc-\..f(mx + ny + ...) ......... (A),

m, n, &c. being all positive, and the limits being given by

mx + ny + . . . — h}

no negative values of the variables being admitted. In this
case (1) becomes

fdxjdy ... ory1 ...f(mx + ny + ...)

= -( fuduf dal dxl

7T J hi JQ JQ JQ

and the integrations with respect to #, y, &c. may be con-
veniently extended to infinity ; (\ it should be observed is an
arbitrary quantity < 0).

I first seek the value of

/•co /•»

dxl dy ...x^y*'1.,. COBOL (mx + ny+ ...-u) = (B).

^0 ^0

Integrating first for x, we get

152 EVALUATION OF DEFINITE MULTIPLE INTEGRALS.

This follows from tlie formula

a r(«) *O

x cos a xdx = — V^ cos a -

a

m

/ xa~l sin a xdx = — —^ sin a ^

Integrating in a similar manner for y, &c. successively, we get
ultimately

Now by (7), we easily see that

f°° F (a+b + ...) (f TT \

i va+frf---~1cosa (v— u) dv = — • — £+£JT ^-cos -1(^+0+...)— — aw j- ,

' 0 V '

and therefore

Hence Ja (B) = -A- .^ ' V ;'" w1

for all positive values of w, and = 0 for all negative values, by
Fourier's theorem. Consequently the second side of (2) be-
comes

1 r(a)r(fc) ...
tt.T(a+b+.. i

Thus finally

fdxjdy ... xa~l^~l ...f(mx + ny+ ...

which is equivalent to Liouville's extension of Dirichlet's
theorem.

I proceed to evaluate the definite integral
t dx I dy...e^M^'^f(mx + ny+ ...) ......... (E),

J o Jo

ab ... and mn . . . being all positive, and the limits being given by

EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 153

By the general formula
E= - jhfudu ("da 1° °dx \ dy...e-(WMX+nby+---}cosa(mx+ny+...-u)

77V hi J o •/ o ^ o

&«)).

Let F= [ *dx {^ dy ... e-(mtta***+-> cos a (mx 4 ny + ... - u) ;

•'o J o

/•oo /•»

then .F= cos aw cfce eft/ ... e-<1Baa*nw1--) Cos a (ra# -f w# + ...)

J Jo

--} sin a (mx

7x| Jy ..

J o

which we may put equal to G cos au + H sin aw.
First to find the value of Gr

= [ dxi dy ...e-(max+nby+-} cos a (mx + ny +...).

J o ^o

Develope the cosine ; the result is composed of terms, each
containing sines or cosines of all the variables.

Also by the formulae

a

/;

e~max sin amxdx =

we see that every factor whether sine or cosine introduces on

integration a factor of the form — ^ ^ . Moreover a sine

ffi \o> ~r a )

factor introduces a in the numerator, a cosine factor a or b or, &c.
Let P represent the continued product of a, b, c, &c. and

D that of w (a2 + a2) n (52 + a2), &c. Then G = ^ S(7^> ,

where //. is a positive integer less than the whole number of
variables in E, and equal to the number of factors in the

denominator of -j — , and C is some coefficient.
ab ...

A little consideration shows, that if we develope

154 EVALUATION OF DEFINITE MULTIPLE INTEGRALS.

in a series of powers and products of

1 1

a b "
and neglect all terms involving powers above the first of these

quantities, the result will be = SO-y — .
Consequently

ab ... a o a

2 - T denoting the combinations two and two of the quan-

tities - , T . . . ; and so of the rest. For in the development

CL O

of (A + B + . . . )m where m is not greater than the number of
quantities A, B, &c. there is always a term involving no power
above the first of any of these quantities, and its coefficient
is 1 . 2 ... m. This term is obviously the sum of the com-
binations m and m together of A, B, &c., and that its coefficient
is equal to 1.2.3 . . . m, appears from the polynomial theorem,
viz.

Thus C -*-2 -*-. , ,

-mra...(a' + a2)(62 + a2)... "

where pt, pt_^ &c. are the alternate coefficients of the equation

whose roots are a, Z>, c, '&c. (I suppose t to be the number of
variables in E.} In precisely the same way we should find

77=

Next to find the values of

/•CO -00

I G cos a uda. and I H cos a wcfa.

./O J o

Let -ZT= I 84.^^ g 2>> —

3cosawc?a

EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 155

the upper sign is to be taken when u is > 0.

By differentiating this for a, we have the value of
a. sin a. uda. TT ^ 1

=att

0 (a2+a2) (62+a%. L 2 ^ (62-a2) (c2-a2) ...
and so by repeated differentiations we find the values of all the
integrals which enter into I G cos a uda and J H sin a. uda.
Thus

when w is > 0, and

when w is < 0.

But ^ — a^_x + &c. + a* = 0.

Hence ^=0, when w is < 0, and therefore in (E) we may
make \ — 0, so that the limits of integration for u are 0 and h.

Again a* +^>1a*~1 + &c. = (a + a) (a + b) . . .

and 2a(J2-a2)(c2-a2)... = {2a.

Therefore

2a (&2-«2) (c2-a2) ... (& - a) (c -a) ...

— cvu

and jF= STT r-, r — (10),

mn ... (b — a) (c — a) . . .

and therefore finally

{QdxfQdy...e-(mMby+-)f(m

l I fue~audu

V

(^> — a) (c — a) ...

If a = Z> = c = &c. = ^4, the first side of this equation

156 EVALUATION OF DEFINITE MULTIPLE INTEGRALS.

As a verification of our analysis we may remark, that in this
case

2__^__=e-,^);

for we have, what is probably a known result, and which at any
rate may be easily proved,

-. a

where F(p) (A) denotes the pih derived function of F(A) : and
this formula applied to the case where F (a) = e~au gives the
above written result.

By differentiating (11) for a. b, c, &c. X, //,, i>, &c. times
respectively, X, //,, v, &c. being integral or fractional, and dividing
by rax, n11, p", &c. we should obtain the value of

mx + ny + . . . x . . .

which would include every case of (3). But the investigation
would be complex, and I shall therefore only indicate it.

In a future number of the Journal I may perhaps apply the
method to some other cases, and particularly with regard to
such multiple integrals as

the limits being given by the series of inequalities,

&c. > &c. < &c.

In theory, such an integral is reducible to a multiple integral of
as many variables as there are limiting inequalities. But it is
not easy to find cases in which this reduction can be actually
effected.

MATHEMATICAL NOTES*.

1. IF a plane passes through any point of a surface, and
makes any function of the intercepts it cuts off from the axes,
a maximum or a minimum when it touches the surface, this
maximum or minimum value is constant for all points of the
surface ; and, conversely, if for every point of a surface, a given
function of the intercepts of the tangent plane is constant, this
function is, with reference to any single point of the surface,
a maximum or minimum for the tangent plane.

This appears at once from the following considerations:
Xj y, z being a point in the surface, 050, y0, ZQ, the three inter-
cepts, <£ (j»0, y0, z0) the given function, if we seek to determine
the surface so that <f> shall be a maximum or minimum, we have
the equations

<«• IK? ......... <»»

fjb being a factor. From these equations we get

and therefore the differential equation of the surface is
dx

_0 r,x.

"

* Cambridge Mathematical Journal, No, XIX. Vol. rv. p. 47, November, 1843.

1 58 MA THEM A TICAL NOTES.

for by the ordinary equation of the tangent plane we have

I u I m 1 mm dF m dF m dF

xo ' #0 ' ^o " dx ' fy ' dz '
F=Q being the equation to the surface.

Again, if we seek to determine the surface, so that $ shall be
constant, i. e. to find the envelope of all the planes represented
by (1), we have (1), (2), (3), (4), as before, and in addition

Thus, as before, «?««/„ y0 = /„ *0 =/3, and the equation of
the surface may be got by integrating (5), and determining
the constant so that the result may coincide with (6). And the
identity of the equations connecting #0, #0, ZQ, and x, y, z, in the
two cases proves our proposition and its converse. Take as an
example the ellipsoid

a* b* c2

0=-,

Therefore the tangent plane to any point of the ellipsoid

2 T2 2

makes — 5 + — 2 + — 2 , a minimum with reference to any plane

^o y" *!
passing through that point.

2. To find the value of

A A

+&c ,.A

when at = a2 = &c. = a.

Let a = a + zy a = a + z &c.

and so of the rest ;

•'• A =fa k-,,)..1...^-^ + K-^)..1...^-^ + &cj

+ &C.

MATHEMATICAL NOTES. 159

'

l&c

]
>

+ &c. by Taylor's theorem.

Now, when 2J1=2J2=&c.=0, the coefficient off(p}a will vanish
if p> n. And whatever the values of z, the coefficient off(p}a
vanishes if p <n: for we know that

0
" '

being < n, and = 1 when & = w ;

which was to be found.

NOTE ON A DEFINITE MULTIPLE
INTEGRAL*.

IN the XVIIIth number of this Journal Mr Boole pointed
out the incorrectness of a theorem given by M. Catalan. The
following pages contain a brief demonstration of the result to
which he was led. Both he and M. Catalan made use of what
is generally known as Liouville's theorem, and thus perhaps
rendered their analysis less simple than it would otherwise
have been.

Let us transform the integral

/<*BI ...... !&*>*/(<*& + ...... +anxn)

by the assumption

a^xv + ...... anxn — OM^ ...... (a2 = Sa2),

and by (n — 1) other linear relations connecting^ ...... xn and

u^ ...... un, and such that

Then, as is well known, dxt ...... dxn is to be replaced by

duv ...... dun, and thus

fdxl ...fdxnf(alxl + ... +anxn) = fdujaujduz ...fdun ... (1).

Let the integrations on the first side of this equation include
all values of the variables which do not transgress the limits

were B is supposed to be greater than A. Then, as So;2 = S
the corresponding limits on the second side of the equation are

* Cambridge Mathematical Journal, No. XX. Vol. iv. p. 64, February, 1844.

NOTE ON A DEFINITE MULTIPLE INTEGRAL. 161

Transform the integral in x by assuming

xv = r cos 0j , #2 = r sin 0t cos 02 . . . #n = r sin #x . . . sin On_l ;
and that in u, by similar assumptions,

u^—r cos 0'j, w2 = r sin #'t cos 0'2 ... un — r sin 0't ... sin 0'^.

I may be allowed to mention that this transformation, which
appears to have been given for the first time by Mr Boole, in
the last number of the Journal, had occurred to me before I had
seen his paper. His analysis leads at once to the conclusion
that dxl . . . dxn is to be replaced by

rn~l sin"'2 0t . . . sin 0M_2 drd6, . . . d0u_l .

This may be also proved by successive substitutions in the man-
ner pointed out in the case of three variables by Mr A. Smith,
in the first volume of the Journal.

Thus (1) becomes, since 2a?2 = 2w2 = rz,

n-:dr /sin""2 0l d0l... Jd0n_lf{r (^ cos ^ + ... an sin 0l... sin 0^) }

r
J

=

cos

The limits for 6 and & are the same.

That this equation may subsist for all values of A and 5, it
is necessary and sufficient that

/sin"^^ ... JW<9n_1(/{r (ax cos 0, + ... an sin 0l ... sin 0n_J}
= /sinw-26>'1/(ar cos^) dffjam*r*0'td0\ ...fdO'^ ....... (2).

With respect to the limits of 6 and 0', it is not difficult to
perceive that if Ol . . . On_^ are taken between the limits 0 and TT,
Xj^ ... #„_! will receive all the values of which they are capable,
namely, all that included between — r and + r ; and that the
same set of values cannot occur more than once. But in
order that xn may vary from 0 to — r, it is necessary to extend
the superior limit of dn_^ from TT to 2?r. Thus the limits of
0n-1 are 0 and 2?r, while those of the other variables 0t . . . 6n-^
are 0 and TT. And similarly for & '.

On the second side of (2) we have the factor

J n

(A).
11

162 NOTE ON A DEFINITE MULTIPLE INTEGRAL.

Let n be odd, then we shall have

(K -4). ..3.1 (ft -3) ...4.2 2?r

' ~

Similarly

'in"-^ dP. = , &c. = &c.

Lastly j*

J n

n-1

27T 2

Again, if n be even, we get

n-:
7T 2

-f-OPr -)-•»'

And this, multiplied by Wr , or unity, gives, as before,

n-1

and thus (2) becomes, making r = 1,

n_J(av cos (9, + ... &c.)

= -rr/mn-2^^1/(acos^1) (3).

V 2 J °

This is the general result given at the end of Mr Boole's paper,
and which includes the others.

NOTES ON MAGNETISM*.

No. I.

A GEOMETRICAL construction, by means of which the action
of a small magnet on a distant particle of free magnetism may
be readily determined, is mentioned, in a memoir by Weber,
on the Bifilar Magnetometer (Scientific Memoirs, II. p. 270).
It is due to Gauss, but I do not know where he has demon-
strated it. A proof of it may be acceptable to some readers
of the Journal.

I begin by enunciating the construction in question, which
will be easily understood without a figure. Let AB be a small
bar magnet, c its centre, P the particle of magnetism on which
it acts. Join cP, draw PD perpendicular to it, meeting cP
produced in D. Let cQ = JcD. Join PQ. Then PQ or QP
(according to the sign of the magnetism of P and the direction
of the poles of AB) is the direction of the action of A B on P;

and — p — y is its magnitude, M being the measure of the mag-
netism of AB, m that of the magnetism of P.

The dimensions of the magnet being small, and its length
in the direction of its axis being much greater than its breadth
or thickness, we may proceed as follows.

Conceive the magnet to be composed of a series of intense
particles ranged along its axis. Let s be the distance of any
one of them from c, y^ds the measure of its magnetism. Also
let cP= r, and call the angle cP makes with cD, 6.

* Cambridge Mathematical Journal, No. XX. Vol. iv. p. 90, February, 1844.

11—2

164 NOTES ON MAGNETISM.

Then the distance of the particle in question from P is
— 2rs cos#)^, and consequently its action on Pis

r2 + s2 — 2rs cos 0 '

magnetic attraction being supposed to follow the ordinary law.
The component of this action along cD, is

muds , n x

(r cos 6 — s).

— 2rs cos 0)*
This is approximately equal to

— j- (1 + 3 - cos 0) (r cos 0 — s) ds,

or to — ^ {r cos 0 + (3 cos2 0 — 1) s} ds;

and therefore the total action of AB on P, parallel to cZ>, is
wcos0 f 3cos20-l

f , , 3 cos2 (9-1 r
l/tas + w ^ — l/isas,

the integrals being taken along the whole length of the magnet.
Consequently

since the aggregate magnetism of a magnet, each element being
taken with its proper sign, is zero. Again, ffisds being the
moment of the magnetism of AB> is the measure of its magnetic
power, or what we have called M.

Consequently the action parallel to cD is

20-l) (1).

-r

The action of the element at 5, perpendicular to cD, is

= r sin 0,

or, approximately,

Hp (1 + 3 - cos 0) sin 6ds.
The total action perpendicular to cD is, therefore,

sin0cos0 (2).

NOTES ON MAGNETISM. 165

The equation to the resultant of these two forces is, since the
line passes through P,

y—T sin 6 _ x — r cos 6
3sin0cos0~~3cos20-l '

For y = 0, or at the point §,

XQ — r cos 6 — Jr sec d (3 cos2 6 — 1),

or XQ = ^r sec 0.
Now cD = r sec 0, therefore

which proves the first part of the construction.

Next, to find the magnitude of the whole action, square and
add (1) and (2), then

^ (1 - 6 cos20 + 9 cos*0 + 9 sin20 cos20)*,

or

T"

is the magnitude sought. Now

PQ = r {sin2 0 + (cos 0 - J sec 0)"}*

and c

Therefore = (1 + 3 cos2 0) *.

c ty

Consequently, if E be the magnitude of the resultant sought,

p _Mm PQ
"cP3 cP'
which was to be proved.

The preceding formulas enable us to determine all the cir-
cumstances of tl^e mutual action of two magnets, which are such
as to fulfil the conditions of our hypothesis.

For instance, in the memoir Intensitas vis Magneticce Terres-
triSj Gauss has shown that if a magnet and a needle be placed
at right angles to one another, then the moment of rotation of
the needle due to the action of the magnet is approximately
twice as great when the line of the axis of the magnet passes

166 NOTES ON MAGNETISM.

through the centre of the needle, as it is when the line of the
axis of the needle passes through the centre of the magnet.

In neither case has the transverse force (2) any tendency to
produce rotation. Its effect is destroyed by the fixed centre
of the needle.

Let y be the distance of any element of the needle from the
centre, mdy its magnetism. Then, in the first case, we shall

have 6 = ^ approximately, (R being the distance between the

centre of the magnet and that of the needle). Consequently the
force (1) may be expressed by the formula

f 2 - 3 } since cos2 0 = I - 6* nearly ;

\ W

which, neglecting ?/2, becomes

2 Mmd

the moment of this round the centre of the needle is - f ^ ,
and the total moment is, therefore,

2MM' ,

— ™ — where M = jmydy.

In the second case, 6 — 0 for every element of the needle, and
the moment sought is, therefore,

MM' . Mm
P3 , since (1) then becomes 5- ,

JLl T

and the sign is immaterial. The moment in this case is, there-
fore, one half of what it was before, which was to be proved.

This result is included in the following general investiga-
tion, in which we shall ascertain the moment of rotation due
to the action of one magnet or needle upon another, whatever be
their relative positions.

The data by which we shall suppose the position of the
magnets to be determined, are the distance between their
centres, the angles which the axis of each respectively makes
with the line joining their centres, and the angle between the
axes themselves. The last element may be readily replaced
by the dihedral angle between two planes which intersect in the
line joining the centres of the magnets, and one of which passes

NOTES ON MAGNETISM. 167

through the axis of one of the magnets while the other passes
through that of the other.

Let the axis of the magnet, whose action on the other is to
be calculated, be taken as axis of x, the centre of the magnet
being the origin of co-ordinates. Let x, y, z be the co-ordinates
of any point in the other magnet. Then, if r*=

and sin 0 cos fl

rz

Again, let a, b, c be the co-ordinates of the centre of the second
magnet, a, /?, 7 the cos angles its axis makes with the co-ordi-
nate axes, p the distance of any element dp from the centre,
X, Yj Z the forces on dp parallel to the .axes of co-ordinates,
m the intensity of dp. Then

„ 7

Z = — 5— xzaa.

r

Let @, H, K be the moments of these forces about lines drawn
through the centre of the second magnet parallel to the axes of
co-ordinates,

~ . . . ., ,

G = —- x {y(z-c]-z(y-V}} dp,

(x-a)- (Zx* - rz) (z-c}\ dp,

K^~{(^~ O (y - 1} - Say (x - a)} dp.

Now x = a + ap, y = b + @p, z

Hence, neglecting the square, &c. of p,

„ 3Mm (, 0.

G = —^- a (by -c/3) pdp,

H= ~ }3aca - (3a2 - R*) 7) pdp,

pdp,
where £2 = a2 +

168 NOTES ON MAGNETISM.

Integrating for /o, we find, for the total moments,

.
« (c« -07) +

3MM'

being for the second magnet what 3/ is for the first.
Let L be the resultant of these three moments, then

Consequently, since a? + /3? + 7*= 1, we shall have

+ 6 -- {aa (aa + ^ + 07) - «

Now, let ^, ^' be the angles which the axes of the two magnets
make respectively with the line joining their centres, and let </>
be the angle between the axes themselves. Then

a = R cos 6 and aa + b/3 + cy = R cos 6', also a = cos $.
Consequently

L* = -{9 cos20 sin20' + 6 cos 0 (cos & coscjb - cos0) + sin2$,
or L = —3- {1 + 3 cos2 9 - (3 cos 0 cos & - cos <£)2}*.

Let ^ be the dihedral angle already mentioned, then

cos <f) = cosO cos & + sin 0 sin 0' cos ^.
Consequently the last equation becomes

L = -^3- 11 + 3 cos2 0 - (2 cos (9 cos 0' - sin 6 sin ^ cos x)2}*,
the required expression.

ON A MULTIPLE DEFINITE INTEGEAL*.

IN the eighteenth number of the Journal, I pointed out the
mode in which Fourier's theorem may be employed in the
evaluation of certain definite multiple integrals. The theorem
generally known as Liouville's, and another of the same de-
gree of generality, were readily deduced from the considera-
tions then suggested. I proceed to another application of the
same method.

fadu

ab...

the integral being supposed to involve v variables x, y, &c., and
the limits being given by the inequalities

mx + ny + . . . > h and ^ h'. f

(Negative as well as positive values of the variables are
admissible.)

DEM. Recurring to the general theorem stated at the com-
mencement of the paper already mentioned, we see that the
integral whose value is sought is equal to

r& r * r* ... "»$

J0 J-~ J-oo - (a2 +

Develope the cosine in a series of products of sines and cosines
of simple arcs. Every term involving a sine disappears on

* Cambridge MathematicalJournal, No. XXI. Vol. iv. p. 116, May, 1844.

170 ON A MULTIPLE DEFINITE INTEGRAL.

integration, as the limits extend from — GO to + co . Consequently
the last written expression becomes

1 [* s J r j [+c°cosamx 7 f+00cosawy 7
— / fudul cos au da — = y-dx I ™ f dv...

•n-Jh JQ J-« a'-t-or J-M I +y

AT [+MCOSOimX , 7T _wma p

^low — g- — 5- du = —e &c. = &c.,

J_w a +x a

and thus the integral becomes

_."-! rh' r <»

-? — A <&j cos a^~(Wia+n6+- • -)a da,
06. ..J*" Jn

or, Tr"-1

which was to be proved.

It may be well to verify this result in a particular case.
Let v = 2 ; then we have to prove that

/* f(mx + ny) ma + nb [h> fudu

j y (a* + x*)(V + y*)~ ~ab~~h

for simplicity, we will suppose that

Let mx + ny =u, and therefore x = mu -f wv,
nx — w^ = v, y — nu — mv,

u and v being two new variables. As u* + v2 = a?2 + ^2, efo c?y is
to be replaced by du dv, and thus

jdxjdy

!+y)
^

^comes

w J {a2 + (mw + H2} {^2 + (nu - <mv)*} '

The limits are easily seen to be + co — co for v ; h' and h for u.
For the integral expresses the volume of that portion of a solid
bounded by the surface, whose equation is

-

which is included between the plane of (#?/), the bounding surface,

0.2V A MULTIPLE DEFINITE INTEGRAL. 171

and two planes parallel to one another and perpendicular to (xy).
The equations of these planes are respectively

mx +ny = h, and mx + ny — h',

and the introduction of u and v is equivalent to changing the
axes of co-ordinates, so that one of the new axes, that of u9
is perpendicular to these planes, while the other is parallel to
them.

In order to find the value of
dv

/+OO
_oo [a* + (mu + nv)'2} {b2 + (nu — mv)
I

27 , assume

{a2 + (mu + nvY\ {& + (nu - ™VY]

_ A (mu -f nv) + Bna C (nu — mv) + Dmb
a2 + (mu + nv)* ~b2 + (nu-mvf '

It is evident that the terms in A and C will disappear on
integration between infinite limits : those in B and D become
respectively nrB and TrD, and the integral in question is
therefore

Now it may be shown that

D n
Jo = —

a {w2 + (ma + nb)*} {u* + (ma - nb}

m

"

b {u*+(ma + nbY} [u*+(ma-nb)*} '

Consequently B+D = = r, — -. 7^2 ;

ao u + (ma + no)

and thus the integral sought is seen to be equal to

ma + nb fh' fudu,

av ]h w2+ (ma + nb)*'

which was to be proved.

Similar considerations apply in the case of more variables,
and doubtless by induction our general result might be established.
But the method we have followed, besides being more analytical,
is also very much simpler.

172 ON A MULTIPLE DEFINITE INTEGRAL.

Another result of this same kind of analysis I shall indicate
without a demonstration, which there will be no difficulty in
supplying.

fdx Jdy... e'^-W ' ' '/(max + riby+...)

the limits being given by

max + nby + ... > h and < h'.

In conclusion, it may be well to remark, that the analysis
of which we have made use is not unfrequently applicable to
questions which tl*3ugh not difficult in principle are neverthe-
less somewhat perplexing in practice.

ON A QUESTION IN THE THEORY OF
PROBABILITIES*.

THE following question affords a good illustration of the
methods employed in the more difficult parts of the theory of
probabilities. In a paper presented to the Philosophical Society,
I applied the kind of analysis we are about to make use of, to
the celebrated Rule of Least Squares. There is, in fact, a close
analogy between the two investigations. Laplace's solution of
the present question is obtained by a process similar to that
which he had employed when treating of the best method of
combining discordant observations.

What is the probability that the sum of the times which
each of n persons has respectively yet to live will amount to
a given time T?

Let $pxpdxp be the probability that thepth person will live
precisely a time xp longer, (f>p denoting some function of xp,
which is necessarily such that

<l>pxpdxp = l,
as it is certain that he will die at some time or other.

Let xlt a?2, ... xn be so related that

The probability of this particular combination is
<frixi <t>A — faXndx^ dx2... dxn,
or <Mi<M2 — *. (T-XI - ••• aV-i) dxi "• dx»>

and the aggregate probability sought is the integral of this
expression obtained by giving all possible positive values to

* Cambridge Mathematical Journal, No. XXI. Vol. iv. p. 177. May, 1844.

174 THEORY OF PROBABILITIES.

x^ ... xn_l , which do not make T— xl — ... xn_l negative. Thus
we have

P= dxnf0 . . . f^x, fax9 ...<f>n(T-xl-... x^) dx^.. dx^ .
Now, by Fourier's theorem,

i f°° r°°

4>n(T-xl-...x^_1) = --l da <t>nxncosa(T-xl-...xn_l-xn)dxn,

" J o J 0

T— £Cj . . . #„_! being supposed to lie between 0 and GO ; for all
negative values of this quantity, the second member of the
equation is equal to zero. Consequently, all the integrations
may now be taken from zero to infinity; and thus, since as
T and xn vary together, dT=dxn

^rp /-oo /• ,.00

P— -- I doi I dxl ... I dx^x^ . . . <f>nxn cos a ( T- Jsc).

7T J Q J Q J Q

Now it may be shown that the greatest value of

r°° r00

dxl ... dx^x^... (f>nxn cosa (T-2x) ... (a)

^o J o

corresponds to a = 0, and is, therefore, unity ; and that when
n is large (a) diminishes rapidly as a increases. Consequently

/.oo

the value of I (a) da. depends, when n is very large, on the

•'o

elements for which a is very small. This consideration enables
us to employ an approximate value of (a).

Let T=t + m, m being a disposable quantity ; then

/•<» /"»

(a) = cos at I dxl ... I dx^x^ . . . (f>nxn cos a (m —
Jo J o

-CO ,0,

-f sin cat I dxl ... I dxn fax^ . . . <f)nxn sin a (m — S#),
^o Jo

which may be thus written,

(a) = cos OitGr + sin CLtH.

In order to obtain approximate values of 6r and H, expand
cos a (m — 2x) and sin a(m- 2#) ; they become respectively,
a being very small,

1 - Jot2 (m - 2x)* and a (m -

THEORY OF PROBABILITIES. 175

Now let f $xxdx = K\ <j>x x*dx =. tf ';

J o J o

/.QO

then, since / $xdx = 1, we shall have

^

G = 1 - Jot2 (m2-
H=a,(m — S-ST) approximately.
Let w = 2j5T, then

and thus 0 = 1- ^a22 (Aj2 - /iT2),

J?=0;
and therefore, while a is very small,

We have next to show that 2 (&2 — Kz] is a positive
quantity.

Consider the definite integral

-00 /-

I I

^0 ''O

g — xfdx dz ;

it is necessarily positive, since every element is so, <f>x dx being
the expression of a probability, and therefore essentially positive.

Expanding (s — a?)2, we find for the value of this integral
kz-'2Kz + 7c2 or 2(tf-Kz).

Hence &2 - K*, and therefore 2 (k* - ^2), is positive.

(This demonstration is due to Poisson, Con. des Terns, 1827.)
Returning to the value we have found for (a), we see that.

we may in all cases represent (a) by cos aJe"^**2 (**""**), since

when a is very small, the two expressions tend to coincide;

and when a is not so, both are sensibly zero, 2 (k* — E?} being

a large quantity of the order n. Consequently

176 THEORY OF PROBABILITIES.

P being the probability that the required sum T shall be
precisely equal to 27T+ t. The greatest value of P corresponds
to t = 0 ; consequently the most probable value of T is ^K.

It is to be remarked that the approximate formula (p) is
independent of the law of probability expressed by the func-
tion <f>x : it depends merely on the two definite integrals

/•* r°

I x<f>xdx and I

J o J o

We have stopped the approximation to the value of G at
the second power of a. Had we gone farther, and retained
only the principal term in the coefficient of each power of a,
a similar result, viz. one which may be assumed as coincident
with the exponential function, would, there is little doubt,
have been obtained; while the coefficient of each power of a
in H would be negligible in comparison of the corresponding
power in @. Some remarks on this, or at least on a cognate
question, will be found in the paper already mentioned.

As a verification of the approximation we have employed,
which is in effect the same as that of Laplace, let us suppose
that the functions <£1? <£2 ... are all of the same form <£, and that
<frx = e"*. Then, as we have seen, the required probability is
obtained by integrating

for all positive values of xlt x2 ... a?B_l which do not transgress
the limits

Now

and thus we have P— e TdTf0dxl . . . f0dxn_lt

the limits being given by

tf1 + *2...o;n_1 = 77.
Hence, it is easily seen that

e-T

P— •— - Tn~l JT

~ T (n) *

In order to compare this with the approximate expression
(p), I remark that T=n — l renders P a maximum; assume,
therefore,

T=n-l+t,

THEORY OF PROBABILITIES. 177

-,

thenP=-

r(n)

Now, by Stirling's theorem, or by that which Binet proposes to
substitute for it (vide Journal de I* Ecole Poly technique , xvi.
p. 226), we have, when n is very large,

Consequently

(w) V[27T (n - 1)}

being a certain function of 2 and n. Therefore

Now the coefficient of tf in ft is

1

tnat ot t is \ —• ~ -f" y 1 ^r "ij ^ A >

\. VI ~" 1 ( 91 ~— 1)J 1 » « • O

and that of 2* is

L_l_ _iL_ 6 I J_

( w~l (n-l)2 (w-l)8j 1.2.3.4"
Hence the coefficient of t* in e~'/£ is

1

that of ^ is

111 1

2(n-l) ' 3(n-l)3 ^S^-l)27

12

178 THEORY OF PROBABILITIES.

and, lastly, that of t" is

1 2 2 1

" 4 (n - 1) + 4 (n - 1) 1 . 2 . 3 (n - I)2 " 4 (n - 1)

n i

or

1.2.3.4 (n-l)2 4(n-l)!
1

1.2 12(»-1)J 4 (n-l)3'

Now if, in forming the approximate expression, we reject all

{t }p 1
-7-r^rr —q , where q is different from zero,
vWJ w

«*. e. if we look on .. . as a quantity all whose powers are to

be retained, except when divided by any power of n, the

/ t \n~l

value of e~M 1 H --- -J may be taken as equal to

t _L ** _*

2(w-l) + 1.2*4(n-l)2

which, as similar results would have been obtained had we
pursued the investigation farther, might be shown to be equal to

ff

c-2(n-l) ^
1 -J2—

and thus P = -77- — N e~2 ("-1) dt,

P is the probability that T is equal to n — I -f t : writing

^
t + 1 for t in — — — and reducing, we find that, within the

limits of the approximation, it may also be assumed as the

f f
probability that t is equal to n+t: also— — [=- ••• <L-P">

and thus

which, as in our case, k* = 2 and K= 1 is precisely equivalent to
the result deduced from the general. formula (p).

The legitimacy of some parts of the preceding approximation
may be questioned; as quantities which are neglected may,

THEORY OF PROBABILITIES. 179

under certain conditions, be larger than those which are retained :
and, as the result coincides with that of the general method, the
doubt thus suggested appears to extend to the latter. The sub-
ject of approximation by means of definite integrals is certainly
not free from obscurity.

The method of this paper extends in . m . to the case in which
we seek to determine the degree of improbability that the average
length of the reigns of a series of kings shall exceed by a given
quantity the average deduced from authentic history. The ap-
plication of considerations of this nature to historical criticism
appears to have been first made in Sir Isaac Newton's Chrono-
logy. They are doubtless entitled to much attention ; but any
attempt to evaluate their legitimate influence, would, for more
than one reason, be unsatisfactory.

12—2

ON THE BALANCE OF THE CHRONO-
METER*.

IT is well known that a common watch goes more slowly
when its temperature is raised, and versa vice. The reason of
this is that the elasticity of the balance-spring decreases with
the increment of temperature and increases with its decrement.
Neglecting the mass of the spring and the connection of the
balance *fith the other parts of the watch, we may take as the
equation for determining the oscillations of the balance,

eO_

dt* + 1~

where e depends on the form and elasticity of the spring, and
/ is the moment of inertia of the balance. The time of oscilla-

tion depends, of course, on the ratio y, e being, as we have

said, a function of t the temperature. In order, therefore,
to the equable rate of the watch,- it would be necessary that

/ should be such a function of t, that -j may be constant. In

the balance of a common watch I is sensibly constant. Hence
the inequality of which we have spoken.

In the chronometer the balance is so constructed that its
figure alters when the temperature varies. The figure repre-

» Cambridge MathematicalJournal, No. XXI. Vol. iv. p. 133, May, 1844.

BALANCE OF THE CHRONOMETER. 181

sents a common form of the chronometer balance. The arc AB,
which carries a weight at 0, is formed of two concentric laminae
of different kinds of metal, the outer lamina being the most ex-
pansible. These two laminae are securely united in their whole
length, so that an increase of temperature necessarily distorts
the arc AB into some form like AB'. Similarly for ab. Con-
trary effects are produced by a decrease of temperature. Thus,
the moment of inertia I decreases as t increases ; as e also does.

/>

And thus we are enabled, by suitable adjustments, to make j ,

at least approximately, constant.

It would, I believe, be impossible, without some hypothesis,
to determine the form which AB assumes under the influence
of a change of temperature. The following suppositions are
probably sufficiently near the truth to be applicable when the
variations of t are not excessive.

Let us suppose the laminse to be cylindrical and concentric,
and bounded by four plane surfaces, two of which are perpen-
dicular to the axis of the cylinder, while the other two, which
form the boundaries at A and B7 pass through the axis. These
conditions being fulfilled, whatever the value of t may be, it is
clear that the variation of form can depend on two elements
only, namely, the radius of the cylinder, and the angle which
AB subtends at its centre. To determine these, we assume
that the middle filament of each lamina expands as it would do
if free.

In the normal state, let 2e, 2e' be the thicknesses of the outer
and inner laminae respectively, r the radius of the boundary of
the two laminee, //., fju the coefficients of expansibility of the
outer and inner laminae (yit > //), 6 the angle subtended at the
centre.

The radii of the middle filaments are, therefore, r + e, r — e' ;
let their lengths be I and Z', then

For an increase of temperature t, let r and 6 become TI and 0% :
then we shall have

= r -

and e' being so small that their variations may be neglected.

182 BALANCE OF THE CHRONOMETER.

rt + e I \+p*

Hence vl?"ri*7ft'

and therefore

which, as //, and // are very small, is approximately

When« = 0, r-6'=(6 + e')r

Consequently, for a first approximation,

(1),

where Ar = ^ — r, A/A = ft — //, and T = e +
Again T^ = I - 11 + (pi - /*T) £.

When * = 0, rd = 1-1'-,

and therefore rA# = (/*? - /x7') ^.

But fd - pi' = r0&fjL + (fie - tie] 0,

and the last term is negligible. Therefore

(2)

We distinctly perceive from (1) and (2) why the effects of
distortion are so considerable in comparison with those of simple
expansion ; it is because the expressions of Ar and A0 have the
small quantity T in the denominator. AB becomes a larger arc
of a smaller circle.

To apply these results : we suppose that when t = 0 the
centre of AB coincides with the central point 0; and AB,
being securely fastened at A, continues perpendicular at that
point to the line OA, consequently its centre remains in that
line. Let 0' be its new position, then 00'= — Ar. Ifwjbe
the mass of AB, its moment of inertia about 0 was ??2r8;

BALANCE OF THE CHRONOMETER. 183

3

about 0' it is m^ — Zm^p - 1 nearly. Let G be the centre
of gravity of AB' : then, in the triangle 00' Gr, we have

OG*= oof2+ (0' ay +200'. a a cos \e^ since * GO A = $0.

00'* or (Ar)2 may be neglected, then, approximately,
(OGy=(0'G?- 2Ar.<7£ cos $0,

-, ~, ~ sin

and, as 0 G = rl *

we have (OG)Z-

Now the moment of inertia round 0 is equal to that round 0'
increased by m {(OG)2- (O'G)*} ; hence, finally,

/x being the moment of inertia of the arc AB.

(In accordance with the rest of the approximation the ex-
pansion of A 0 is neglected.)

Again, we will suppose the weight at C to be a material
particle, and that the angle AOG is equal to <f>. Then, 72 being
the moment of inertia of this weight, whose mass we will de-
note by w2, we shall have

-£(l- cos<£) ......... ..... (4).

Consequently, as the inertia of the bar OA does not undergo
any sensible alteration, and as every thing which has been proved
of OAB is true of Oab, we have, finally,

1 -cos <£) ...... (5).

It appears that the variation of e is exactly proportional to t :
so that e becomes e (1 - vt)9 v being some constant. Consequently

. i • T ,-, e (1 — vt] e

we must have, in order that -j — -^ = -j. ,

m, (1 - cos </>) ...... (6).

In calculating the value of / we may take into account the
moment of inertia of a OA • moreover, instead of the approxi-

184 BALANCE OF THE CHRONOMETER.

mate expression m:rz for the moment of inertia of AB, we may
employ a more accurate one involving the quantities e and e';
the approximate expression is sufficiently accurate for the deter-
mination of A/.

The adjustment for compensation is effected by shifting the
weight m2 along AB ; that is, by altering the value of <j> until
(6) is fulfilled. On the hypothesis we have made, the value
of / for t = 0 is not affected by the change of <f>.

In determining the approximate expressions (5) and (6), we
have neglected all terms in which A/z, occurs not divided by T ;
all terms involving A/t multiplied by e or e'; all terms into
which any power of /t or // enters. In consequence of the
last restriction t can only rise to the first power in the result.
If this were absolutely correct it would follow that, if the
compensation were effected for a particular value of 2, it would
subsist accurately for all values of t. For instance, if we
give t equal values, positive and negative, the decrease of /
in the one case ought to be equal to its increase in the other.
But when t is considerable, it is found that there is a sensible
deviation from this result; and, assuming that the expression
for Ae does not in any perceptible manner involve powers of t,
it follows that that of A/ must do so. Any term involving f
(and, a fortiori, any higher powers of that quantity), must be
very small, since t always occurs multiplied by /A or ///; but it
may, nevertheless, sensibly affect the chronometer's daily rate.
On the usual construction, the balance oscillates 216,000 times
in twenty-four hours. Consequently a very slight change in the
moment of inertia of the balance will become perceptible in
that period.

In order to obviate the consequent error, it has been proposed
by Mr Dent, a distinguished chronometer-maker of the present

BALANCE OF THE CHRONOMETER. 185

day, to alter the form of the balance. The figure represents
one of those which he proposes to substitute for that in common
use. It would be easy to determine the corresponding expression
for A J, to the degree of approximation of our previous results.
As, however, the comparison of the merits of the two forms
must depend on the terms involving £2, it may be well to reserve
it for another opportunity. If there appears reason to believe
that our hypotheses represent the facts with sufficient accuracy
to encourage us to proceed farther, I hope to resume the subject
in the next number of the Journal.

NOTES ON MAGNETISM*.

No. II.

IN order to a distinct understanding of the results obtained
in the last number of the Journal, it will be desirable to con-
sider the established conventions with respect to the signs of the
symbols which we had occasion to employ.

North magnetism is assumed to be positive ; hence, of course,
south magnetism must be considered as negative.

The measure M of the magnetic power of a bar magnet is,
as we have seen, equal to fpsds, p being the magnetism of the
element ds, which is situated at a distance from the origin
equal to s. The limits of the integral are such as to include
the whole length of the magnet.

The position of the origin is arbitrary : we may conveniently
place it at the centre of the magnet, but the value of f/jisds is
the same whether this be done or any other point be taken.
For let the origin be shifted through a distance a, so
that s = s' — a, then

fjjisds = ffi (s — a) ds' = f/j,s'ds' — a/fids' :

and as all the integrals extend throughout the length of the
magnet f/^ds = 0, and therefore

ffjLs'ds' or M—M\

which was to be proved.

But the value of ffjusds changes its sign if the direction in
which s is measured changes. Let I be the length of the magnet ;
then, s being measured in one direction, say from left to right,
we have

= I /Jisds.
J o

* Cambridge Mathematical Journal, No. XXI. Vol. iv. p. 139, May, 1844.

NOTES ON MAGNETISM. 187

Now suppose that s = I — s, then the limits are interchanged
and ds =-ds-} consequently

= f V (l ~ s'} ds> = - f W<&' = - M',

M

s being measured in the direction opposite to that of 5, or from
right to left.

The magnetism of a magnet may thus be always represented
by a positive quantity.

Any two points in the axis of a magnet may be taken as its
poles. But although the position of the poles is matter of con-
vention, yet relatively to one another, one is the north and the
other the south pole.

The physical character by which they are distinguished is
this : if a particle of north magnetism be placed in the pro-
longation of the axis from south to north, it is repelled from
the magnet. Contrariwise, if it be placed in the prolongation
of the axis towards the south. Further, we must integrate
jjbsds from south to north, i. e. s must be taken as positive
when fjids lies to the north of the origin, in order that M may
be positive. This may be shown by supposing a particle of
north magnetism m placed in the prolongation of the axis
towards the north, and at a distance r from the centre of the
magnet. If we assume that from south to north is positive,

the action of the magnet on m is — — ; and as this action is

repulsive its expression will be positive, and therefore M is so.
If we had assumed from north to south to be positive, the

action of the magnet would have been represented by -- 8~ ,

and as this is positive, M will necessarily be negative. So that,
in order to make the measure of the magnet's power positive, we
must take the direction S . . . N as positive.

Consequently the angle 6 must be measured from it. We
suppose it measured in the usual manner, viz. in the unscrew
direction.

The general expression for the moment of rotation due to the
action of one magnet on another is much simplified when the two
magnets are supposed to lie in one plane.

188 NOTES ON MAGNETISM.

The dihedral angle % is then zero, and consequently the
equation

L = -=- (1 + 3 cos20 - (2 cos 0 cos & - sin 0 sin & cos %)2}*
becomes
Z= ~ (1 + 3 cos2 (9 -4 cos2 0 (1 -sin2 0')

-1- 4 sin 0' cos 0 sin 6 cos 0' — sin2 0 sin2 0']* .
The quantity between the brackets is equal to

4 sin2 & cos2 6 + 4 sin & cos 0. sin 0 cos 0' + sin2 0 cos2 0'.
Consequently

Z = -=- (sin 0 cos 0' + 2 sin 0' cos 0).

This result may be readily established by an independent pro-
cess, which the reader will find no difficulty in supplying. The
last result may be put in the following form :

Professor Lloyd, in the 19th volume of the Memoirs of the Eoyal
Irish Academy, has investigated this case of the mutual action
of two magnets. His result is (mutatis mutandis)

L = ~ sin e + ~ 3 sin d ~ e'

This differs from the last written result, merely because, in the
Professor's analysis, 0 and 0' are measured in opposite directions.
If we replace 0 in Prof. Lloyd's result by 2?r — 0, it becomes

L = " f3 sin (e + ff] ~ sin (e ~ ff]}'

as before.

The general formula affords a simple solution of the follow-
ing problem. The position of a magnet, and that of the centre
of a needle being given, to place the needle in the position in
which the moment of rotation due to the action of the magnet is
a maximum.

By the formula established in the last number of the Journal,
we have

L = ~ (1 + 3 cos2 0 - (3 cos 0 cos 0' - cos

NOTES ON MAGNETISM. 189

We suppose the magnet and needle to be in the same plane.
In the figure let 0 be the centre, SON the line of the axis of

the magnet, C the centre of the needle. Project 0 on ON
in D, take OE — 2 OD, draw EN' perpendicular to SN meet-
ing ON', which is at right angles to OC in N', CN' is the
line in which the axis of the needle must be placed, its north
pole being turned towards N'.

In order to prove this, we have only to remark that the
angle 6 or CON is constant, the position of C being given ;
consequently the condition to be fulfilled, in order that the
moment of rotation L may be a maximum, is

3 cos 6 cos & — cos (j> = 0.
Now as (j> is the angle between N' C and 8N9 we have

ED = CN' cos <£,

and therefore OD = J- CN' cos <j>.

Again, 00 = CN' cos & and OD=OC cos 0.
Hence 0Z> = CN' cos 0 cos 0'.

Consequently 3 cos 0 cos & — cos <£ = 0,
or the required condition is fulfilled.

There are three particular cases worth noticing :

(1) 0 = 0. In this case C lies in the axis ON, D coincides
with it, and ON' is perpendicular to ON, and therefore parallel
to EN'. Consequently the point N' is removed to an infinite
distance, and CN' is therefore perpendicular to ON. The

, T . 2MM' . . ..
corresponding value of L is - _j— , and is the maximum

maximorum.

(2) 0 = 0'. In this case the magnet and needle are parallel

190 NOTES ON MAGNETISM.

to one another. The quadrilateral CD EN' is a parallelogram,
EN' is equal to Z>(7, and consequently

tan COD : tsmN'OE :: EO : OD :: 2 : 1.

But COD = 6 and N'OE=^- 0,

since CON' is a right angle. Consequently

tan 0 = 2 cot 0,
or tan0 = V2.

The corresponding value of 3 cos2 6 is therefore unity ; and
consequently we have, in this case,

(3) 0 = — . Here D (and therefore E) coincides with 0,

while ON' lies in the axis OS'. Consequently Nr is at 0, and
the needle is therefore again perpendicular to the magnet.
In this case

T MM'
:

The first and third cases were noticed in the last number of the
Journal. In the second, the value of L is a mean proportional
between what is in the other two cases, in the last of which it
is a minimum maximorum.

If we were required, for a given position of (7, to find the
position in which the needle would be in equilibrium, or the
moment L equal to zero, we might have recourse to Gauss's
construction already mentioned; for if the needle be placed
along the line in which the magnet tends to attract or repel (7,
as the dimensions of the needle are small, every element
would approximately be attracted or repelled along this line,
and therefore the total action would be destroyed by the
resistance of C.

Thus there are always two directions for every position of C';
one of maximum moment and the other of equilibrium : these
two directions are at right angles to one another.

MATHEMATICAL NOTE*.

Mogg+a;)

, l+af

THIS definite integral is evaluated in a curious manner by
M. Bertrand in Liouvilles Journal. The demonstration I am
about to give of his result, is somewhat different in form from
that which he made use of.

The method employed in an ingenious paper which appeared
in the last volume of the Journal (in. p. 168), will apply to the
integral we are about to consider.

xdx

~J~ ' Ju ~~

X U
°W 1 -L 2 4- 2

Consequently

X

(I + ux] (1 + #2) *
x + u

(1 1 ,j»/yA ...

— \J-i~ UJUJ , *\ (-{ \ 2\ *
X + U u

(l+ux) (1 + a;2) (1 + x*) (l + u?) (1 + ux) (I+u*) '
and therefore

xdx u f1 dx u f1 dx

ux

d „ 1 f1 xdx u f1 dx u f1 ds,

du^U ~ 1+1? J01 + x2 + 14- w2 J0 1 + x* 1 + ti* J01 +

1 7T U

Integrate for u, from 0 to 1,

/(I) -/(O) = £ log 2 + f log 2 -/(I),

Cambridge Mathematical Journal, No. XXI. Vol. iv. p. 143, May, 1844.

log2.

M^ & = | log 3.

192 MATHEMATICAL NOTE.

But/(0)=0, since log 1=0. Therefore

|

Therefore

The singularity of this method, and its applicability in other
cases give it interest : but, as the writer of the paper already
noticed pointed out to me, the integral may be got by assum-
ing x = tan y ; it then becomes

IT

I log(l+tany)<Zy,

^0

and, by his fundamental equation,

IT 7T

4 log (1 + tan y} dy = J* log jl + tan (~ - ^J dy :

— tan y 2

anI-2, = + rn^ = rn_,

and therefore

•n

2[*log(l+tan#)%= j log 2,

^0

whence the truth of M. Bertrand's result is obvious.

MEMOIR OF THE LATE D. F. GREGORY, M.A.
FELLOW OF TRINITY COLLEGE, CAMBRIDGE*.

THE subject of the following memoir died in his thirty-first
year. He had, nevertheless, accomplished enough not only to
justify high expectations of his future progress in the science to
which he had principally devoted himself, but also to entitle
his name to a place in some permanent record.

Duncan Farquharson Gregory was born at Edinburgh in
April 1813. He was the youngest son of Dr James Gregory,
the distinguished professor of Medicine, and was thus of the
same family as the two celebrated mathematicians James and
David Gregory. The former of these, his direct ancestor, is
familiarly remembered as the inventor of the telescope which
bears his name ; he lived in an age of great mathematicians, and
was not unworthy to be their contemporary.

Of the early years of Mr Gregory's life but little need be
said. The peculiar bent of his mind towards mathematical
speculations does not appear to have been perceived during his
childhood; but, in the usual course of education, he shewed
much facility in the acquisition of knowledge, a remarkably
active and inquiring mind, and a very retentive memory. It
may, perhaps, be mentioned here, that his father, whom he lost
before he was seven years old, used to predict distinction for
him ; and was so struck with his accurate information and clear
memory, that he had pleasure in conversing with him, as with
an equal, on subjects of history and geography. In his case, as
in many others, ingenuity in little mechanical contrivances seems
to have preceded, and indicated the developement of a taste for
abstract science.

* Cambridge MatliematicalJournal, No. I. Vol. iv. p. 145, November, 1844.

13

194 MEMOIR OF MR GREGORY.

Two years of his life were passed at the Edinburgh Academy ;
when he left it, being considered too young for the University,
he went abroad and spent a winter at a private academy in
Geneva. Here his talent for mathematics attracted attention;
in geometry, as well as in classical learning, he had already
made distinguished progress at Edinburgh.

The following winter he attended classes at the University
of Edinburgh, and soon became a favourite pupil of Professor
Wallace's, under whose tuition he made great advances in the
higher parts of mathematics. The Professor formed the highest
hopes of Mr Gregory's future eminence : those who long after-
wards saw them together in Cambridge, speak with much in-
terest of the delighted pride he shewed in his pupil's success and
increasing reputation.

In 1833, Mr Gregory's name was entered at Trinity College
in the University of Cambridge, and shortly afterwards he went
to reside there. He brought with him a very unusual amount
of knowledge on almost all scientific subjects: with Chemistry
he was particularly well acquainted, so much so that he had
been at Cambridge but a few months when it was proposed to
him by one of the most distinguished men in the University to
act as assistant to the professor of Chemistry ; which for some
time he did. Indeed, it is impossible to doubt that, had not
other pursuits engaged his attention, he might have achieved
a great reputation as a chemist. He was one of the founders
of the Chemical Society in Cambridge, and occasionally gave
lectures in their rooms.

He had also a very considerable knowledge of botany, and
indeed of many subjects which he seemed never to have studied
systematically : he possessed in a remarkable degree the power
of giving a regular form, and, so to speak, a unity to knowledge
acquired in fragments.

All these tastes and habits of thought Mr Gregory cultivated,
to a certain extent, during the first years of his residence in
Cambridge, of course in subordination to that which was the
end principally in view in his becoming a member of the
University, namely, the study of mathematics and natural
philosophy.

He became a Bachelor of Arts in 1837, having taken high
mathematical honours : more, however, might, we may believe,

MEMOIR OF MR GREGORY. 195

have been effected in this respect, had his activity of mind per-
mitted him to devote himself more exclusively to the prescribed
course of study.

From henceforth he felt himself more at liberty to follow
original speculations, and, not many months after taking his
degree, turned his attention to the general theory of the com-
bination of symbols.

It may be well to say a few words of the history of this part
of mathematics.

One of the first results of the differential notation of Leibnitz,
was the recognition of the analogy of differentials and powers.
For instance, it was readily perceived that

dm+n cF d

or, supposing the y to be understood, that

dx ~ \dx \dx
just as in ordinary algebra we have, a being any quantity,

This, and one or two other remarks of the same kind, were
sufficient to establish an analogy between -j- the symbol of

differentiation and the ordinary symbols of algebra. And it
was not long afterwards remarked that a corresponding analogy
existed between the latter class of symbols and that which is
peculiar to the calculus of finite differences. It was inferred
from hence that theorems proved to be true of combinations of
ordinary symbols of quantity, might be applied by analogy to
the differential calculus and to that of finite differences. The
meaning and interpretation of such theorems would of course
be wholly changed by this kind of transfer from one part of
mathematics to another, but their form would remain unchanged.
By these considerations many theorems were suggested, of which
it was thought almost impossible to obtain direct demonstrations.
In this point of view the subject was developed by Lagrange,
who left undemonstrated the results to which he was led, in-
timating, however, that demonstrations were required. Gradually,

13—2

196 MEMOIR OF MR GREGORY.

however, mathematicians came to perceive that the analogy
with which they were dealing, involved an essential identity;
and thus results, with respect to which, if the expression may
be used, it had only been felt that they must be true, were
now actually seen to be so. For, if the algebraical theorems
by which these results were suggested, were true, because the
symbols they involve represented quantities, and such opera-
tions as may be performed on quantities, then indeed the
analogy would be altogether precarious. But if, as is really
the case, these theorems are true, in virtue of certain funda-
mental laws of combination, which hold both for algebraical
symbols, and for those peculiar to the higher branches of
mathematics, then each algebraical theorem and its analogue
constitute, in fact, only one and the same theorem, except quoad
their distinctive interpretations, and therefore a demonstration of
either is in reality a demonstration of both*.

The abstract character of these considerations is doubtless
the reason why so long a time elapsed before their truth was
distinctly perceived. They would almost seem to require, in
order that they maybe readily apprehended, a peculiar faculty —
a kind of mental disinvoltura which is by no means common.

Mr Gregory, however, possessed it in a very remarkable
degree. He at once perceived the truth and the importance of
the principles of which we have been speaking, and proceeded
to apply them with singular facility and fearlessness.

It had occurred to two or three distinguished writers that the
analogy, as it was called, of powers, differentials, &c., might
be made available in the solution of differential equations, and
of equations in finite differences.

This idea, however, probably from some degree of doubt as
to the legitimacy of the methods which it suggested, had not
been fully or clearly developed : it seems to have been chiefly
employed as affording a convenient way of expressing solutions
already obtained by more familiar considerations.

To this branch of the subject Mr Gregory directed his

^ * The values of certain definite integrals are to be looked upon as merely
arithmetical results ; in such cases we are not at liberty to replace the constants
involved in the definite integrals by symbols of operation. In other cases we are
at liberty to do so, and this remarkable application of the principles stated in the
text, has already led Mr Boole of Lincoln, with whom it seems to have originated,
to several curious conclusions.

MEMOIR OF MR GREGORY. 197

attention, and from the general views of the laws of combina-
tion of symbols already noticed, deduced in a regular and
systematic form, methods of solution of a large and important
class of differential equations (linear equations with constant
coefficients, whether ordinary or partial) of systems of such
equations existing simultaneously, of the corresponding classes
of equations in finite and mixed differences ; and lastly, of many
functional equations. The steady and unwavering apprehension
of the fundamental principle which pervades all these applica-
tions of it, gives them a value quite independent of that which
arises from the facility of the methods of solution which they
suggest.

The investigations of which I have endeavoured to illustrate
the character and tendency, appeared from time to time in the
Cambridge Mathematical Journal.

In this periodical publication Mr Gregory took much interest.
He had been active in establishing it, and continued to be its
editor, except for a short interval, from the time of its first ap-
pearance in the autumn of 1837, until a few months before his
death. For this occupation he was for many reasons well
qualified; his acquaintance with mathematical literature was
very extensive, while his interest in all subjects connected with
it was not only very strong, but also singularly free from the
least tinge of jealous or .personal feeling. That which another
had done or was about to do, seemed to give him as much
pleasure as if he himself had been the author of it, and this
even when it related to some subject which his own researches
might seem to have appropriated.

This trait, as the recollections of those who knew him best
will bear me witness, was intimately connected with his whole
character, which was in truth an illustration of the remark of
a French writer, that to be free from envy is the surest indi-
cation of a fine nature.

To the Cambridge Mathematical Journal, Mr Gregory con-
tributed many papers beside those which relate to the researches
already noticed. In some of these he developed certain parti-
cular applications of the principles he had laid down in an Essay
on the Foundations of Algebra, presented to the Eoyal Society
of Edinburgh in 1838, and printed in the fourteenth volume of
their Transactions. I may particularly mention a paper on the

198 MEMOIR OF MR GREGORY.

curious question of the logarithms of negative quantities, a
question which, it is well known, has often been discussed
among mathematicians, and which even now does not appear
to be entirely settled.

In 1840, Mr Gregory was elected Fellow of Trinity College ;
in the following year he became Master of Arts, and was ap-
pointed to the office of moderator, that is, of principal mathe-
matical examiner. His discharge of the duties of this office
(which is looked upon as one of the most honourable of those
which are accessible to the younger members of the University)
was distinguished by great good sense and discretion.

In the close of the year 1841, Mr Gregory produced his
1 'Collection of Examples of the Processes of the Differential
and Integral Calculus;" a work which required, and which
manifests much research, and an extensive acquaintance with
mathematical writings. He had at first only wished to super-
intend the publication of a second edition of the work with
a similar title, which appeared more than twenty-five years
since, and of which Messrs. Herschel, Peacock, and Babbage
were the authors. Difficulties, however, arose, which prevented
the fulfilment of this wish, and it is not perhaps to be regretted
that Mr Gregory was thus led to undertake a more original
design. It is well known that the earlier work exercised a
great and beneficial influence on the studies of the University,
nor was it in any way unworthy of the reputation of its authors.
The original matter contributed by Sir John Herschel is especially
valuable. Nevertheless, the progress which mathematical science
has since made, rendered it desirable that another work of the
same kind should be produced, in which the more recent im-
provements of the calculus might be embodied.

Since the beginning of the century, the general aspect of
mathematics has greatly changed. A different class of problems
from that which chiefly engaged the attention of the great writers
of the last age has arisen, and the new requirements of natural
philosophy have greatly influenced the progress of pure analysis.
The mathematical theories of heat, light, electricity, and magnet-
ism, may be fairly regarded as the achievement of the last fifty
years. And in this class of researches an idea is prominent,
which comparatively occurs but seldom in purely dynamical
enquiries. This is the idea of discontinuity. Thus, for instance,

MEMOIR OF MR GREGORY. 199

in the theory of heat, the conditions relating to the surface of
the body whose variations of temperature we are considering,
form an essential and peculiar element of the problem; their
peculiarity arises from the discontinuity of the transition from
the temperature of the body to that of the space in which it is
placed. Similarly, in the undulatory theory of light, there is
much difficulty in determining the conditions which belong to
the bounding surfaces of any portion of ether; and although
this difficulty has, in the ordinary applications of the theory,
been avoided by the introduction of proximate principles, it
cannot be said to have been got rid of.

The power, therefore, of symbolizing discontinuity, if such
an expression may be permitted, is essential to the progress of
the more recent applications of mathematics to natural philosophy,
and it is well known that this power is intimately connected
with the theory of definite integrals. Hence the principal im-
portance of this theory, which was altogether passed over in the
earlier collection of examples.

Mr Gregory devoted to it a chapter of his work, and noticed
particularly some of the more remarkable applications of definite
integrals to the expression of the solutions of partial differential
equations. It is not improbable that in another edition he
would have developed this subject at somewhat greater length.
He had long been an admirer of Fourier's great work on heat,
to which this part of mathematics owes so much; and once,
while turning over its pages, remarked to the writer, — " All
these things seem to me to be a kind of mathematical paradise."

In 1841, the mathematical Professorship at Toronto was
offered to Mr Gregory: this, however, circumstances induced
him to decline. Some years previously he had been a candidate
for the Mathematical Chair at Edinburgh.

His year of office as moderator ended in October 1842. In
the University Examination for Mathematical Honours in the
following January, he, however, in accordance with the usual
routine, took a share, with the title of examiner, — a position
little less important, and very nearly as laborious, as that of
moderator. Besides these engagements in the University, he
had been for two or three years actively employed in lecturing
and examining in the College of which he was a Fellow. In
the fulfilment of these duties, he shewed an earnest and constant

200 MEMOIR OF MR GREGORY.

desire for the improvement of his pupils, and his own love of
science tended to diffuse a taste for it among the better order
of students. He had for some time meditated a work on Finite
Differences, and had commenced a treatise on Solid Geometry,
which, unhappily, he did not live to complete. In the midst
of these various occupations, he felt the earliest approaches of
the malady which terminated his life.

The first attack of illness occurred towards the close of 1842.
It was succeeded by others, and in the spring of 1843, he left
Cambridge never to return again. He had just before taken
part in a college examination, and notwithstanding severe suffer-
ing, had gone through the irksome labour of examining with
patient energy and undiminished interest.

Many months followed of almost constant pain. Whenever
an interval of tolerable ease occurred, he continued to interest
himself in the pursuits to which he had been so long devoted ;
he went on with the work on Geometry, and, but a little while
before his death, commenced a paper on the analogy of differ-
ential equations and those in finite differences. This analogy
it is known that he had developed to a great length; un-
fortunately, only a portion of his views on the subject can now
be ascertained.

At length, on the 23rd February 1844, after" sufferings, on
which, notwithstanding the admirable patience with which they
were borne, it would be painful to dwell, his illness terminated
in death. He had been for a short time aware that the end
was at hand, and, with an unclouded mind, he prepared himself
calmly and humbly for the great change ; receiving and giving
comfort and support from the thankful hope that the close of
his suffering life here, was to be the beginning of an endless
existence of rest and happiness in another world. He retained
to the last, when he knew that his own connection with earthly
things was soon to cease, the unselfish interest which he had ever
felt in the pursuits and happiness of those he loved.

A few words may be allowed about a character where rare
and sterling qualities were combined. His upright, sincere,
and honourable nature secured to him general respect. By his
intimate friends, he was admired for the extent and variety
of his information, always communicated readily, but without
a thought of display, — for his refinement and delicacy of taste

MEMOIR OF MR GREGORY. 201

and feeling, — for his conversational powers and playful wit;
and he was beloved by them for his generous, amiable dis-
position, his active and disinterested kindness, and steady
affection. And in this manner his high-toned character acquired
a moral influence over his contemporaries and juniors, in a degree
remarkable in one so early removed.

To this brief history, little more is to be added ; for though
it is impossible not to indulge in speculations as to all that
M? Gregory might have done in the cause of science and for his
own reputation, had his life been prolonged, yet such speculations
are necessarily too vague to find a place here ; and even were it
not so, it would perhaps be unwise to enter on a subject so full
of sources of unavailing regret.

ON THE SOLUTION OF EQUATIONS IN
FINITE DIFFERENCES*.

THE partial differential equations which occur in various
branches of mathematical physics are, for the most part, of such
forms that solutions of them may be obtained without much diffi-
culty. As is well known, the great difficulty in almost all such
cases consists in the necessity of determining which of all possi-
ble solutions satisfies the particular conditions of the problem on
which we are engaged. It seems that before the time of Fourier's
researches on heat, the course which mathematicians had uni-
formly followed was, first to obtain the general solution of the
equation of the problem, and then to determine by particular
considerations the arbitrary functions which it involved. This
course undoubtedly would be the most direct and analytical,
were there any general method for determining the form of the
functions in question : as, however, there is none, the analytical
generality of the first part of the process is in many cases sterile
and useless.

Fourier's methods, which depend essentially on the linearity
of the partial differential equations which occur in the theory
of heat, consist in assuming some simple solution of the equation
of the problem, in deducing from hence a more general solution
of it, and in determining successively and by means of particular
considerations the arbitrary quantities thus introduced in such
a manner as to satisfy all the conditions of the question. The
general solution with arbitrary functions does not make its
appearance in his process ; and the reason why it is so much
more manageable than the other appears to be, that it is far
easier to determine arbitrary constants in accordance with certain

* Cambridge Mathematical Journal, No. XXII. Vol. iv. p. 182, November,
1844-

ON THE SOLUTION OF EQUATIONS, ETC. 203

conditions than arbitrary functions. There will, generally speak-
ing, be an infinite number of arbitrary constants, and it is
therefore necessary to treat them in classes. The ingenious
synthesis by which this is effected by Fourier, in the different
problems discussed by him in the Theorie de la Chaleur, forms
one of the most interesting parts of that admirable work. The
same kind of reasoning is made use of by Poisson, in his re-
searches on similar subjects : and there can be little doubt that
the methods of Fourier, developed and extended as they have
been by subsequent writers, will long continue to be an essential
element in the application of mathematics to physical researches.
Similar methods may be made available in the solution of
equations in partial finite differences. Such equations do not,
it is true, present themselves very often, as the continuity of the
causes to which natural phenomena are due, leads rather to
differential equations than to those in finite differences. In
fact, I am not aware of any subject, except the theory of
probabilities, in which we meet with problems whose solution
depends on that of an equation in partial finite differences.

In this theory, however, such problems are not uncommon.
One of the most interesting of them, both in its own nature
and historically, may serve as an illustration of the application
of the methods of Fourier to finite differences. This problem,
which has engaged the attention of several writers on the subject
of probabilities, and of which a solution was among the earliest
efforts of Ampere, is that of the duration of play. Professor
De Morgan has spoken of this solution arid of that of Laplace,
as being of the highest order of difficulty: that which I am
about to enter on has, I think, a decided advantage in this
respect.

The problem itself may be thus stated: — Two persons,
M and N, have between them a number a of counters : they
play at a game at which M' s chance is p, and N'a q. The
losing player gives one counter to the other, and they are to
play on until one or other have lost all his counters. What
is the probability that the party will terminate in M's favour
after any assigned number of games, N being supposed to have
originally x of the a counters ?

Let yxz be the probability that M will win the party at the
(z + l)th game. If he win the next game (of which the pro-

204 ON THE SOLUTION OF EQUATIONS

bability is^?), this becomes y^:,^} if lie lose it (of which the
probability is q), it becomes y^i.,^, and therefore

This is the equation of the problem. It is clear that

y* = 0, 2/a, = 0 ........................ (2),

as the party ceases as soon as M or N has a counters. Again,
y^ = o unless x = 1, and #1-0 = p ............ (3) ;

for if JV have more than one counter he cannot lose them all
at the next game; and if he have only one, his chance of his
being left without any is p.

Let us assume yM = <£vx ........................... (4),

a being arbitrary. Then

Of this a solution is

X

Vx~ \q) i n^a W' ' ^ "

where C and co are arbitrary, and /JL such that

a ..................... (7):

this form of solution is therefore real if a2 is less than
In order that (4) may satisfy the conditions (2), we must have

£L

x~\2

i/0 = C sin a) = 0, va=Ci—j sin (pa + <a) = 0 ...... (8).

It is impossible to satisfy these two conditions without making
C — 0, which would give a nugatory result, unless sin ^a = 0

or fju — — , r being an integer. Let us therefore assume this
value for p ; and then, by (7),

a. = 2*/(pq) cos — ..................... (9).

Gt

In order to satisfy (8), we have now only to make o> = 0, and
then, substituting the values of vx and a in (4), we get

in x cos .......... (10).

OF FINITE DIFFERENCES. 205

This value, in wliich G and r are arbitrary, satisfies (1) and
(2), and in consequence of the linearity of these equations they
will be satisfied by a sum of similar values, and we shall thus
have a more general solution, viz.

sn

If in this we put z = 0, we have

Now, by (3), this is to be equal to p fora?=l, and to 0 for

the a — 2 values of x, 2.3 a — 1. There are thus a — 1

conditions for (12) to fulfil, and therefore we have, extending
the summation S from r = 1 to r = a — 1, the following system
of equations :

TT . a — I

i sn — + -an —^-ir

Q=Ct sin — + + CL, sin 2 — TT

^ ^ ££

0 =

(13).

From these a — I equations we have to determine the a — I
quantities Ct... Ca_v. In order to do this, multiply the first

T T

equation by sin - TT, the second by sin 2 - TT, and so on,

(r being an integer less than a), and add. Then, as may be
easily shown, the coefficient of every one of the quantities (7,
except (7r, will in the resulting sum be equal to zero, while

that of Cr will be-. Consequently (13) is equivalent to the
2

system of equations included in the general formula

and consequently (11) becomes

1 ^ + 1 / ^i\ 2 A. y«££ / A. \ *

ty = - (±<nq\ * [*-} 2**1 sin- TT sin — TT cos -TT) (15),

a \qJ a a \ a J

which is the required probability.

206 ON THE SOLUTION OF EQUATIONS

We may deduce from this formula, by indirect considera-
tions, one or two analytical theorems. For it is obviously im-
possible that the party should terminate in M 's favour in less
than x games, as x is the number of counters he must win
from N. Consequently

T TX f T \Z

S/1"1 sin -TT sin — TTJCOS -TT) =0 (16).

a a \ a J

for all integer values of z less than x — 1.

Again, M may win the party at the 03th game, if he win x
games in succession, the probability of which is px. Hence,
putting z = x — 1, we have

p*= - UpqY f^V 2""1 sin - TT sin — ir( cos - wY '
a v \27 « a \ a j

or Sj0'1 sin - TT sin — TT^COS - TTJ =^ (17).

These formulae may undoubtedly be established by other methods,
but I have thought it worth while to point out this way of
deducing them, from the analogy it bears to that in which many
remarkable theorems are obtained by Poisson, in his Theorie de
la Chaleur, namely by considering the nature of the quantities
which his formulas represent. This mode of establishing analy-
tical theorems by considerations founded on the interpretation of
our results, is one of the most curious features of the more recent
methods of treating physical questions.

To (16) and (17) another theorem may be added, by the
following consideration. Jf, if he win, must win the party
either in x games or in x + and even number of games. For
if he lose k games he must win back Jc games and x more or
there must have been x + 2k games in the party. Hence his
chance is zero whenever z + l=x + 2k + l, and therefore

s,

a-1

r rx f r \x+2k

sin - TT sin — TH cos - TT) =0 (18),

a a \ a J

k being any positive integer whatever.

When a is infinite, the sums contained in the last three
equations become definite integrals. Let

r j x-u *r * *. z a~ 1

- TT = 9, then - = a<f>, and - - TT = TT.

a a a

OF FINITE DIFFERENCES. 207

Consequently (16), (17), (18), become respectively

I sin</> smx(j> (cos </>)* d<j> = 0 (19),

J o

(3 being integral and less than x — 1),

/sin </> sincc<£ (cos (j>)x~l d(j> = ~ (20),
j 2

rsin<£ sin#</> (cos (j>)x+2kd<f> = 0 (21).
-

If, instead of seeking the probability that M will win the
party at the (z + l)th game, we wished to find that of his
winning it after z or more games shall have been played, we
should only have to sum (15) for z from z to infinity. Calling
this new probability u^ we should thus get

PO1

Wy

. r . rx

sin -TT sin — TT

If in (22) we put z equal to zero, we have then the proba-
bility of Jf's winning the party at the first, second, &c.
games, i. e. of his winning it at all. Writing simply ux for u^
we shall thus get

. r . nc

sin - TT sin — TT

Now of this probability we can obtain, as is well known,
a much simpler expression. For it is easily seen that we shall
have

u. =pu^ + qux+l (24)

for every value of a?, provided that, instead of considering ux as
the probability that M will win the party, we make it denote
the probability that he either has won or will win it. As it is
impossible that he can have already won it while x differs
from zero, this alteration does not affect the value represented
by ux except for the case of x =0. In this case the value of ux,
as expressed by (23), will be zero, as the party is at an end,
M having already won it. But according to the proposed

208 ON THE SOLUTION OF EQUATIONS

modification, the new value of UQ will be unity, and therefore
we have for the initial and final values of ux,

M0 = l, wa=0 ...................... (25).

The necessity of this modification arises from this, that otherwise
the relation expressed by (24) would not be in all cases true.
For when x = l, we should have u^ — qu^ whereas the true value
is of course

From (24) we have (introducing the relation p + q = 1),

(26),
a and /3 being arbitrary constants : and thence, by (25), we get

and consequently ux—— - *• ........................ (27).

This expression is therefore, except for x = 0, equivalent to
(23), into which however the relation already mentioned,
viz. that p + q — 1 has not as yet been introduced.
When p and q? are equal, (27) becomes

while (23) similarly becomes

. r . rx

sin - TT sm — TT

1 5* a-l a a

U*~aZl ~ '

1 — COS - 7T

a

or ux = - Sj0'1 cot - ^ sin — TT (29).

a a 2i a

Comparing (28) and (29), we have the following theorem:
writing x for a — x*

x = 2«-l±cot- £sin -TT (30),

a i a

the upper sign to be taken when r is odd.

OF FINITE DIFFERENCES.

209

This theorem, like the preceding ones (16), (17), &c., re-
quires x not to transgress the limits x=l, x — a — 1. In the
case supposed (viz. when p and q are each equal to -J), (22)
becomes

1 <c* «^i , r TT . rx f r \*
UXK — - Z.0*1 cot - - sin — TT COS-TT ......... (31).

a a 2 a \ a J

But as the party cannot be won in less than x games,
UXQ = ua while z is less than x, and therefore

x — ^'1 + cot- ^ sin — TT/COS- TT) ......... (32),

a 2 a V a J

of which (30) is a particular case.

If, instead of seeking the probability that at the (z + l)th
game N would lose the party, by losing the last of his x
counters, we had sought that of his having at the termination
of this game any assigned number of counters &, the following
method might have been made use of.

Let yxz be the probability in question. It is clear that it will
satisfy, as before, equations (1) and (2). But instead of (3),
we shall in this case have

, unless x = k ± 1, and

+ 2 ± (p -?)} ••• (3).

Equation (11) therefore, which depends merely on (1) and (2),
will still obtain; but instead of the system of equations (13),
we shall have the following:

0 = C. sin -
a

&c. = &c.

. Ar-1

sm TT +

= &c.
= (7t sin
&c.=&c.
0 =

k+l
a

. a-l

sm TT

a

sm

sm

sin - — TT + ... +
a

. sn

7T

7T

....(13').

14

210 ON THE SOLUTION OF EQUATIONS

From whence, by the same system of factors as before, we
deduce the general formula

4/2fc+1\* . kr r ,-•*

r = - ^j-i I sm — TT cos - TT ............ (14 ) ;

a \/r / a a

for the factors corresponding to the two equations whose first

(k — 1) r

members are different from zero, are sin - - — TT and

a

. (k + 1) r , , „ , . . kr r

sm f - '— TT, and the sum of these is 2 sin — TT cos - TT.
a a a

Consequently the expression of the probability sought will be
(accenting the y for distinctness),

x-k

2 . ^.fp\ 2 ~ n-i . kr . rx / r ^ ,.._..
y' =- Upn) 2 £] 5« 'sm— TT sm — TT cos- TT ... (15).
a^ \qj a a \ a J

It is an obvious consequence of the discontinuity of the limiting
conditions of the problem, that this expression does not reduce
itself to (15) when k is taken equal to zero. For the same reason
it is not applicable when k is equal to unity : and on the other
hand, it is not to be greater than a — 2.

It is unnecessary to trace the different corollaries deducible
from the last written equation, as it has been introduced merely
to illustrate the facility with which our method discusses any
proposed modification of the question of the duration of play.

One point, which is perhaps worth notice, is the symmetri-
cal manner in which x and ^?, k and q, enter into (15') : the
result, however, which is the interpretation of this symmetry
may probably be obtained by general considerations.

A more general question would arise from supposing it
possible for M to win or lose at each game any number of
counters not greater than a. The method we have been
illustrating would apply to this question, but the solution of it
involves that of an algebraical equation of a degree superior to
the second.

Another part of the subject, namely, the numerical calculation
of the expressions already obtained, would not be consistent with
the design of this paper. When z is sufficiently large, all the
summations with respect to r may be reduced to their first and

OF FINITE DIFFERENCES. 211

last terms, unless a is extremely large, in which case other
methods of approximating (those, namely, of Laplace), may be
made use of.

Enough has probably been said to show the facility which
the method I have proposed is capable of giving to questions
of acknowledged difficulty. I am not aware that it has been
before pointed out ; but as I am not at present able to refer to
any work on the subject, I cannot speak confidently on this
point, [x, a are integral throughout.]

14—2

GENERAL THEOREMS ON MULTIPLE
INTEGRALS-.

IN Liouville's theorem for the reduction of a certain class
of definite multiple integrals, the integrations comprise all
positive values of the variables which do not transgress a
limiting inequality, which either is of, or may easily be re-
duced to, a linear form. Take for illustration the case of two
variables, and let mx + ny < h be the limiting inequality in
question, m, n and h being positive. Then, geometrically,
mx + ny — h is the equation of a straight line which forms the
base of a triangle of which the intercepts of the positive half
axes of co-ordinates are the sides, and our integration extends
over the whole surface of this triangle. A similar interpreta-
tion may of course be given in the case of three variables.
But to return to that of two. Let mx -\-ny-Ti cut the axis
of x in the point M and that of y in the point N: conceive
another straight line mx + ny — h' ; m', n, h' being also all
positive ; and let it cut the axes in M' , N' respectively. Let

us suppose for distinctness that , is greater than — . Then,

if the value of T, be intermediate between those of the two
h,

fractions — 7 and — , it will be easily seen that the two lines

must intersect in some point A, lying in the positive quad-
rant of co-ordinates, and that we shall have a quadrilateral
OMAN', (0 being the origin of co-ordinates,) formed by the
axes and by the two bounding lines. If now we integrate
any function of x and y for all positive values of the variables

* Cambridge and Dublin Mathematical Journal, Vol. I. p. r, 1846.

GENERAL THEOREMS, &c. 213

not transgressing the two inequalities mx + ny < h, m'x + riy ^ h\
we shall in effect integrate over the surface of the quadrilateral
OMAN'. But if the two lines did not intersect within the
positive quadrant, then one or other bounding inequality would
be inoperative, and we should in effect integrate over the sur-
face, not of a quadrilateral, but of a triangle, as in the case
contemplated by Liouville's theorem. It is manifest that we
may have, instead of two limiting inequalities, any larger num-
ber we please, and that our integrations may thus be made to
extend over an irregular polygon of a greater or less number
of sides. I do not believe that any writer on multiple inte-
grals has considered the case in which the limits arc given by
more than one inequality, but the restriction to that of one is
clearly unnecessary.

Let us suppose there are r variables x, y, ... z, and that we
have to evaluate the integral

f0dx...f0dze-ax--cz<j)(mx+...pz) <£, (mtx + ...ptz) ...... (1),

subject to the two inequalities
mx+ ...z^h,

m ...p, h ; ra, ... pt , h, being all positive ; and <f> and (f>t any func-
tions whose values may be represented within the limits of inte-
gration by Fourier's theorem.

Let the value of the integral in question be /; then, by con-
siderations analogous to those of which I made use in a paper
which appeared at the commencement of the last volume of the
Journal* ', we shall have

where

1 rft rh, r00 f°°

I fa du I $tul dut I da. I da, . G,

6r = I dx . . . I dze-ax'"~ czcos a (mx+. . .pz—u) cosa, (mtx+ . . .

J Jo

and the lower limits of integration with respect to u and u may
be any negative quantities.

I remark in the first place, that

/•<» r<» r00 r°°

da I d^ff-fcij da da,H, where

./O «J 0 J -oo ^ -oo

* Page 150 of this volume.

%
214 GENERAL THEOREMS

=l dx...\

Jo J0

and therefore

I rh rh' r°° r00

1= — x- / <bu du I d>u du I dot. I da H.

iTrJ J J-oo J-oo '

Let H= K cos (a.u -f a,M;) + L sin
Then it will easily "be seen that

N N'

where, if we take the case of three variables,

/ am + OLtm^ an + alnl am + atml cup + a,/?,

wL\ - ftOC I 1 -- T

\ a o a c

a b

ap + ajt?y am + a,m/ aw + a/w/ cup + &tp\

D={a*+ (am + a m,)2} {J2 + (aw + a,^)2} }c2 + (op + aj?,)2}.

(Precisely the same law of formation of these quantities
would obtain if we were to take any number of variables. I
have taken the case of three merely for distinctness of repre-
sentation.)

Putting for cos (aw + a,w,) and sin (OLU + atut) their expo-
nential values, we find that

= a... cjl - V(-

and as

/ \2 2 f1 // -vaw + c^wil f

(am + a<m/)2 = a2 |l - V(-l) -- ~JT \ j

ON MULTIPLE INTEGRALS. 215

[a- V(- 1) (am+ a,m,)} ... {c- V(- 1) (a

Now, assume that*

1

(a + am + a,w,) ... (c + a^> + a^J

...... (2);

where -Fa& , Fw , &c. are independent of a and a, . This assump-

r r _ ^
tion is justifiable because it introduces -- - disposable quan-

tities Ft viz. as many as there are combinations two and two of
the r quantities a, b . . . c, and it will be easily seen that there are
the same number of conditions to be satisfied.

Consequently as

e(«u+aV)V(-i) = cos (aw+ a'tt') + V(~ 1) sin (a.u + aV),
we shall have

H= F& [db — (am + arn^} (an + «,w,)} cos (au + a,ut)

+ {a (an + can) + b (am + a,wy)} sin (an + aft) \ „

divided by

{a2 + (am + am')*} {tf + (an + an')2}

Let us next assume u = mx + w?/, wy = w/c + w^, a; and y being
here two new variables ; also a' = am + a^, , and /3' = aw + ayn, :
then the coefficient of F& in the expression of H will become

(aE - a'ff) cos (qya? + ^y) + (aff + bof) sin (g'x

Moreover dudu1dadal will be replaced by dxdyda'dfi ; and
therefore, as we have

1 >A r^ r+CD /-+00

/ = -—2 1 <f>udu I $lutdul I 6?a I da/ Hj we shall have

I = j^2 2^a& M0 (wia; 4- w?/) ^ (mtx + ny] dxdy M,

216 GENERAL THEOREMS

where the sign of summation extends to all the quantities F, and
where

(aft-a'ff) cos (a'a!+ffy)+(aff+5a') sin (a'

From the known integrals

f+00cos aaj. ^a TT ^ f+00asin c.#. dy

— - _ = _ e*aa; - ^ - 2 — =
./.« a +a « •/_„ a + a2

the upper signs to be taken when x is positive, it follows that
M = wV"***!' (1 + 1 ± 1 ± 1).

If x and y are both positive, the bracket becomes 1 + 1 + 1 + 1
or 4 ; if a; only be negative, it becomes 1— 1 — 1 + 1 or 0; if. y
only be negative, it becomes 1 — l + l-l orO; and similarly if
both x and y are negative. Thus generally

or =

There are, indeed, exceptional cases ; as if y be zero, x being
positive, when M = 2-7T2e~aa;, and similarly if x be zero, y being
positive ; and again, if x and y are both zero, when M = TT* :
but of these, as we are about to multiply M by the element
dx dy, it is unnecessary to take account. Therefore, in inte-
grating for x and y, we include only positive values of the
variables ; and as u and u are not to be greater than h and ht
respectively, x and y must be such as not to transgress the
inequalities

mx + ny^ h, mx + nty S hr

Thus we find that

(mx + ny] <£y (mx + nty] e-«*-^,

the limits being given by the two above-written inequalities.
It appears, therefore, that the integral (1), when there are two
limiting inequalities, is reducible to the sum of a series of double
integrals.

This result is analogous to that which is obtained in the case
of the function <f> (mx + ...pz) e~ax--cz, in the paper already re-
ferred to.

It remains to determine the form of the quantity F^. This
is done at once by multiplying equation (2) by

(a + am + o^raj (b + an + a^),

MULTIPLE INTEGRALS. 217

^nd replacing a, a, by values which make both these factors
vanish. It hence appears that

(mnl —

~~

[c (mnt — mtn) + a (npt — ntp) + b (pmt —p?n}} . . . '

the denominator being the continued product of r — 2 factors,
each of the same form as the one written down. Of course the
other quantities F are obtained in the same manner.

Let us now take the more general case in which there are
s limiting inequalities, s being less than r, and in which the
function to be integrated is

...^(m^+.^p^e-^"-
the inequalities in question being

„ <. 7,
• ... pgZ ^ fig .

"We shall arrive at a perfectly analogous result in this more
general case. In the first place the integral sought may be
thus written,

— I 1<f>luldul ...I <j)au8du8l da^... I dagGr, where

«CO * 00

= I dx... I dz e~aaj--c2cosa1(w1a?+. . -p^-u^ . . .cosa8(m,cc+. . .p8z-us

J ^

o o

Now a little consideration will convince us that

.00 .00 ^ - + 00 .+ 00

d^ ... da8G- = - I da: ... da8H, where

^0 Jo "• J -« J-oo

-oo /.oo

H = I dx ... I dz e-™--™ cos [x2<a>m + . . . + z^cup — Saw] :

^0 - 0

for if we take the expression

cos [x Sam -f . .. + z Sap — Saw],

make ax negative, add the resulting expression to the original
one : then in the two terms thus got make «2 negative, and as
before add the results, we shall, continuing this process, get in
all 2* terms, which will be found to be equal to 2' times the
continued product of the cosines involved in Gf.

218 GENERAL THEOREMS

Effecting the integrations indicated in H, we see that

j^cos Saw + Nf sin 2)aw
= D ~ '

where D = (a2 + (Sam)2} . . . {c2 + (Sap)2},

and JV and ^V' follow the same law of formation as in the

particular case already considered, except that for - — , &c.

Ct

we substitute — — , &c. With this remark we perceive that
H=

[a + V(-l)Saw}...{c + V(-l)

(3).

The assumption now to be made is that

i =SJ\. ,2,

(a + Saw) . . . (c + Sap) A /'

where A is the product of every set of s factors taken out of the
whole number of r factors

a + Saw, . . . c + Sap,
and F is independent of at . . . a8 .

r tr — 1. ... r 5+1

There will thus be - — - — '-^— - disposable quantities

F, which will be found to be the number required to make (2')
identically true. Consequently we shall have

,v cos Saw + v sin Saw

where S is the product of s factors of the form a2 + (Sam)2 ; and v
and v are formed just as in the case of 5 = 2: that is to say, we
shall have

* = a...c(l-<7a+(74-&c.), vf = a...c((71-
where Ct is the sum of the products of every combination
that can be made of the s quantities - — , &c., taken t and
t together.

ON MULTIPLE INTEGRALS. 219

In order to simplify the expression

v cos Saw + v sin Saw

let us denote the s quantities Saw, &c., which are involved in it,
by the single symbols /9t ... fa, and assume

the sign of summation S extending only to that set of s out
of the r quantities x ... z, which corresponds to the factors in-
volved in the denominator 8. Of course a?, y, &c. are here, as
before, new variables. (In the case of s = 3, for instance, these
assumptions will be of the form

It follows from this that dut... dugd^... da.8 will be re-
placed by dx ... dy . dfi^ ... d/3a, and that the factors in S will
take the simpler form a2 + /:?/, £2 + yS22, &c. ; while Saw will be-
come ^a? + @2y + ...

The integrations with respect to /5 extend, like those for a,
from — o= to + oc . Let

Then, from the obvious analogy between the forms v and v', and
those of the developments of cos 2/&c and sin S/ifo respectively,
it follows that if x, y, &c. are all positive,

M'^ite-**-*"' (1 + 1 + ...)»

there being twice as many units within the brackets as there are
terms in the development of sin (ft + ,../,), or of cos (J[ + ...ft),
that is to say, twice 28"1 or 28.

Moreover, if any one, as ic, of the quantities x, y, &c., is
negative, M' = 0 ; and this, whether it alone is negative or any
others, are so too. For if x — — xf, let its coefficient fa be as-
sumed equal to — fa, when the expression of M' becomes of the
same form as if x were positive, except that v and v are changed

220 GENERAL THEOREMS

by having — /3/ wherever /^ occurred previously. Now none of
the quantities @ can occur raised to any power, and therefore
every term involving /3t will change sign when /3X is replaced by
— 8. Hence we shall have

there being as many negative units as positive within the brackets,
since in the development of sin (fl + . . . fs) or cos (f^ + . . .fs) there
are 2a~2 terms independent of the sine of f^ and 2S~2 terms which
involve that quantity, and which therefore change sign when ft
does so. Hence the quantity within the bracket, and conse-
quently Mt , is equal to zero if x be negative ; and so, of course,
for the other variables y ... z.

M will, in particular cases analogous to those already noticed,
assume exceptional or limiting values, but of these we need not
take account. And thus we arrive at the following remarkable
theorem :

The definite integral ofi variables x ... z

/Odx ... /Odz<k (m.x + ... ptz) ... <£8 (msx+ ...
whose limits are given by s inequalities

m1x+...p1z<h1, ...msx+... psz5
can generally be expressed as a linear function of

-«— 'cz

1.2...S

integrals of s variables each. The form of each of these integrals
may be deduced from the original integral by omitting from it
any set of r — s of the variables, and similarly the form of the
limiting inequalities may be got by omitting the same set of vari-
ables from the original inequalities (r > s).

In certain cases, however, when the constants a, m, &c. have
particular values, the theorem fails because the assumption (2')
becomes illegitimate. This failure is indicated by certain of the
quantities F becoming infinite. To determine the form of F, we
have merely to multiply (2') by A, and then to equate to zero all
the s factors of which A is composed. All the quantities F,
except the particular one under consideration, will then dis-
appear, and we have s equations determining the s quantities a.

ON MULTIPLE INTEGRALS. 221

Hence it will appear that F is equal to a fraction whose nume-
rator is unity, and denominator equal to the value assumed by
the product of the remaining r-s factors, when the values
already assigned for the quantities a, &c. have been substituted
for them ; a result which it is obvious can be immediately ex-
pressed in the notation of determinants. F will therefore become
infinite if our equating the s factors by which it is divided
in (2') to zero will make one or more of the remaining r — s
factors vanish. Let it make t of these factors vanish; then
equating these t factors also to zero, we get in all s + 1 equations,
which are equivalent to s independent ones. Therefore any
set of s out of these s + t equations will satisfy the remaining t
equations. Hence

1.2 ...I

of the quantities F will become infinite, and therefore the second
side of (2') will consist of finite terms and of a finite quantity
expressed in the form of the sum of that number of infinite
terms. This indetermination of course indicates a change in the
form of the function, the general character of which the reader
will have little difficulty in perceiving. But the consideration of
these particular cases, some of which are interesting, must be de-
ferred to another occasion.

I am inclined to believe that the process developed in this
paper will admit both of simplification and extension. For
the exponential function we may substitute with certain modifi-
cations any function of CKB + ...CZ, in accordance with a result
given by Mr Boole in his very interesting Memoir on a new
Method in Analysis, which is published in the Transactions of
the Royal Society. (This result would include the one which I
obtained in the last volume of the Journal, from which however
it might be deduced.)

Thus, if in the theorem established in this paper we replace
o, 5, ... c by lea, Tcb ... kc, Jc being a wholly arbitrary quantity,
we may, comparing the coefficients of its powers, deduce new
theorems from the given one. Developing the first side of the
equation, the coefficient of Jcn will be

222 GENERAL THEOREMS, &c.

and in the second it will be the sum of a series of terms of the
form

Jf

n8y + ...) (ax

Tfl

as it is manifest that F will become j— . Hence, if

A/T (ax + ... cz)

be such a function that its development may be substituted for it
in the integrations, we shall have

f0dx...f0dz<t>l(m1x+...p1z) . . . <£g (m& + . . . p$ ty (ax + . . . cz)
= 2Ff0dxf0dy...<l>l(m1x + nly+...) ... <f>8 (m8x + nsy + ...)

^(ax + fy + ...),

Jr-i

where -j-^ -^r^ = tyt and all the differential coefficients of ^ of

an order lower than the (r — s)th vanish for t = 0. This is, I be-
lieve, in the case of 5 equal to unity, precisely equivalent to one
of Mr Boole's results. It might also, I imagine, be obtained
without having recourse to developments.

MATHEMATICAL NOTE*.

Solution of a Functional Equation.
To solve the functional equation

,+oo

<t>mx<f>n (a. -x)dx = (^a,

J -00

provided <f>x = <f> (— x).

Let I <f>QLCOSOLzda = '\Irz. Then

J -00

^Wt.3 = I da I dx<t>mx(f>n (a — n) cos ctz.

J -CO J -00

Write for a, a + x, then cos a.z becomes

cos aiz cos xz — sin az sin #2,
and as the sines change sign with their arcs, we get

/.+00 /.+ 00

^mfn^ = I dy. I dx(j>mx<f>na. cos a^ cos a?z.

J -00 J -00

Or <*lrm+nz = ^m^nzt

whence ^=X^

% being independent of m, and therefore, by Fourier's theorem,

1

z cos ^^ «

"^^o

The required solution.

* Cambridge and Dublin Mathematical Journal, Vol. vn. p. 103, 1852.

ON THE AREA OF THE CYCLOID*.

To the Editor of the Cambridge and Dublin Mathematical Journal.

SIR, — The determination of the area of the Cycloid, so
easily effected by modern analysis, was regarded by the geo-
metricians of the seventeenth century as a problem of no smalt
difficulty. Mersenne was the first who attempted a solution :
he was however unsuccessful. It was proposed by him in
despair to Koberval in 1628, who also failed in his attempt
at that time. About seven years afterwards, however, Koberval
overcame the difficulty, and communicated his good fortune
to Mersenne.

In a letter to Descartes, Mersenne made mention of Ro-
berval's discovery of the area of the Cycloid as a great feat
in geometry, simply stating the result obtained by Roberval,
without giving any clue to the method. Descartes, solving
the problem himself with little difficulty, communicated his
method in reply to Mersenne, with some supercilious remarks
about the supposed difficulty of the problem. Fermat and other
mathematicians of that day exercised their ingenuity in the
same question. A solution of the problem by pure geometry,
which was some time ago communicated to me by Mr R. L.
Ellis of Trinity College, possesses so great a superiority over
any of the geometrical methods of these early mathematicians
which I have seen, that I think it may be acceptable to those
readers of your Journal who take an interest in the history
of mathematics.

" The motion of the generating circle may be resolved into
two uniform motions, a motion of translation parallel to the

* Cambridge and Dublin Mathematical Journal, Vol. ix. p. 263, 1854.

ON THE AREA OF THE CYCLOID. 225

directrix and of rotation round its own centre. The area gene-
rated by the describing point may be considered as generated
by these two motions : that of translation nowise affects the
motion of rotation, and the area due to the latter is the same
as if the former did not exist, that is, it is equal to the area
of the generating circle/ Contrariwise the motion of rotation
does affect the area due to that of translation, inasmuch as in
virtue of it the distance of the describing point from the di-
rectrix is varied : the mean distance, viewed as depending on
the motion of rotation, is equal to the radius of the generating
circle, and the corresponding area is therefore a rectangle, the
base of which is the space slided over and altitude that radius ;
and, as this space is the circumference of the generating circle,
the area in question is equal to twice the area of that circle :
on the whole, therefore, the area of the cycloid is equal to three
times that of the generating circle.

" The reason is just the same as that by which what are
called Guldinus's properties are established. We here resolve
the motion of a describing point into motions parallel and
perpendicular to the abscissa; the latter generates no area, the
former generates a rectangular area having for its base the
abscissa and for its altitude the mean value of the ordinates ;
that is, the ordinate of the centre of gravity of the arc, which
is a known result. The only difference to be attended to in
the two cases relates to the mode in which the average is to
be taken."

Mr Ellis has remarked, that the same method may be ex-
tended to the determination of the areas of the hypocycloid and
epicycloid.

I am, Sir,

Your obedient Servant,

WILLIAM WALTON.

Cambridge, July 31, 1854.

15

SUE LES INTEGEALES AUX DIFFERENCES

FINIES*.

ON peut evaluer I'integrale

(1) fdxfdy . . . fdzf (x, y, . . . , z),

dans laquelle les variables x, y, . . . , z doivent prendre toutes les
valeurs positives qui satisfont a 1'inegalite

(2) ^r (a?, y,.. .,«)<*,

en rempla9ant dans la formule (1) la fonction <f> par une fonction
discontinue, qui devient e*gale a zero pour toutes les valeurs
des variables non comprises dans la formule (2). On peut alors
etendre les integrations depuis ze*ro jusqu'a 1'infini, ce qui sim-
plifie beaucoup les calculs.

Je crois que c'est a M. Lejeune-Dirichlet qu'est due 1'idee
de cette maniere d' ^valuer les inte'grales multiples ; c'est ainsi
qu'il a obtenu, il y a quelques annees, une generalisation tres-
remarquable d'un the*oreme du a Euler.

La thdorie des int^grales d^finies nous fournit plusieurs
moyens d'exprimer les fonctions discontinues ; je me suis servi,
pour cet objet, du theoreme de Fourier. Au moyen de ce the*o-
reme, j'ai de'termine', dans un petit M^moire inse're* dans le
Journal de Mathematiques de Cambridge f, les valeurs de deux
integrates multiples. La premiere de ces integrates revient a
la generalisation qu'a donnee M. Liouville du resultat de
M. Dirichlet ; mais je crois que la seconde est nouvelle.

* Extrait du Journal de Mathematiques pures et appliqutes, Tome IX. 1844.
t Page 150 of this volume.

SUE LES INTEGRALES, ETC. 227

La facilit^ avec laquelle j'avais obtenu ces resultats me fit
penser qu'on pourrait peut-etre appliquer une mdthode sembla-
ble aux differences firiies ; les resultats auxquels je suis parvenu
par cette consideration font le sujet de ce qui va suivre.

En suivant F analogic qui existe entre les differences finies
et les differences infmiment petites, on voit qu'a 1'integration
multiple, il faut substituer des sommations par rapport a toutes
les variables qui entrent dans la fonction donnee.

Soit $ (x, y] cette fonction. Je designe par 262?<£ (x, y) la
quantite suivante (b — a et d — c e*tant des nombres entiers et
positifs),

<£ (a, c) +<£(« + 1, c) + •
+ 4>(a, c +1) +

, d).

II est visible que cette notation pourrait s'etendre a un
nombre quelconque de variables.

Le theoreme de Fourier se remplacera par la formule
suivante, dans laquelle I —x et x — a sont des nombres entiers
et positifs,

1 f""

(3) fx = — I da %bafu cos a (x — u).

77V o

On peut done poser

x = a, — a + 1, ... = b;

mais si Ton donne a x (qui doit toujours £tre un nombre entier)
une valeur quelconque non comprise dans ces limites, on aura

fda £

cos oix — u =

La demonstration de ce theoreme est si facile, qu'il n'est pas
ne*cessaire de s'y arr^ter ; je ferai seulement observer en passant
qu'elle suppose que la fonction fu ne devienne infinie pour
aucune des valeurs

De*signons par {a?}* la fonction — 1, ^' : toutes les fois

1 (x)

que p est un nombre entier et positif, nous aurons
{x}p = x.x+l ...x+p-l.

15—2

228 SUR LES INTEGRALES

t

Cela pose*, entrons en matiere.

Je vais chercher la valeur de la somme multiple

dans laquelle l'e*tendue de la sommation est donnee par 1'ine'-
galite*

(5) x + y + ... +z<h.

D'apres 1'id^e fondamentale de notre analyse, je remplace dans
la formule (4) la fonction f(x + y + . . . + z) par

i r77"

— I da. %lfu cos a(x+y + ... + z — u).
irj o

Done nous aurons, en changeant 1'ordre des sommations,
(6) -22

T

(^ est un nombre n^gatif quelconque).

Les sommations par rapport a a?, y^ etc., peuvent a present
s'e*tendre jusqu'a I'mfini.

Nous allons determiner les valeurs de

2J° {xY~l cos ax, et de Sj° l^}1^1 sin ax.
Soit z = ea^-^-} nous aurons, par un the*oreme connu,

_. ^<oo _r J-

2 2,0 a cos ax = —

1-'- 1-1

a az

puisqu'on a

1 + - + etc. = — — , et 1 + — + etc. = — i—

a i • - * 2a i *

1 — — JL •

a za

Pareillement on a

et de la

AUX DIFFERENCES FINIES. 229

En e*crivant dans cette equation z"1 au lieu de z, elle deviendra

Ajoutant cette Equation a la derniere, nous aurons, a cause de

(8) 22" {,ra

(On doit remarquer que {0}p = 0, puisque T (0) a une valeur
infinie.)

Ensuite, a cause de

(a — z] (a — z"1) = 1 — 2a cos a + a2,
nous aurons

Mais, puisque z — cos a + V^~I sin a,

a — z — sin a (a cosec a — cotang a — V— 1).
Posons done

cotang <j> = a cosec a — cotang a,

la valeur de (a — zf deviendra

tandis que celle de (a — z~1}* sera

par consequent

. («_»-«)• + .- (a -.)»- 9 cos (^ + CC).

Mais la valeur de sin <£ est e*gale a -^ ; — ^ . Done,

nous aurons finalement

Dscuc = r

„ .

(1 - 2a coa a + a")'

230 SUR LES INTEGRALES

On trouvera de la me"me maniere que
|2r{*r«

(l-2a cos a + a2)'

A present faisons a = 1 . Alors nous aurons

, 1 — cos a a

cotang <f> = — : — - = tang - ,
sin a & 2 '

c'est-a-dire

et les Equations (9) et (10) deviendront

cos j|(7r-a)+a[
/1 1 \ v?°° ( ~,li>-i **~ ~~* ~P / ^\ ( L

\ii) *4Q [X\

(12)

(2sin

Au moyen de ces Equations, on prouvera facilement que
(13)

cos

=r(p)r(2)...r(r)

n dtant le nombre des variables a?, ^, . . . , z.

Mais en e*crivant, dans l'e*quation (11), p + q + ...r au lieu
de ^, on aura

. ...r

(asm I)

et pareillement

• . »\"^™r

2s»-j

AUX DIFFERENCES FINIES. 231

En combinant ces deux Equations, on trouvera que
20" {a;}^--^1 cos a (a; -i/)

(14)

(p+q+...r . N

COS \r ^ (Tr-^+a-i

(2sinfp""

Mettons done

v — u — n + 1,

et comparons les Equations (13) et (14), nous aurons

En multipliant les deux membres de cette Equation par da. et en
integrant depuis ze*ro jusqu'a TT, nous aurons, par la formule (3),

. . .z-U

(16)

pour toutes les valeurs positives de w — n + 1.

Pour toutes les valeurs negatives de cette quantite*, le second
membre de 1'^quation (16) est e*gal a zero.

Done, en effectuant la sommation par rapport a w, il est inutile
de donner a u des valeurs moindres que n — \. Cela pose*, nous
aurons fmalement, en considerant la formule (4),

(1" --t-*' r--,

1'etendue des sommations ^tant determined par 1'i

Ce theoreme est 1'analogue pour les differences finies du theo-
reme de M. Liouville, dont j'ai deja parle.

En effet, en supposant que 1'inegalite qui determine les
limites des variables soit, comme ci-dessus,

x + y + . . . + z < ^,

232 SUR LES INTEGRALES

voici le theorenie de M. Liouville :

\dx \ dy ... {

J0 JQ J0

« .

II est vrai que cette equation n'est qu'un cas particulier du
resultat qu'a donne* M. Liouville, mais malheureusement nous
ne pouvons pas generalise! la formule (17) en supposant que
l'e*tendue des sommations soit donne*e par rindgalite*

ax + ~by + . . . -f cz ^ A,
sans au moins lui donner une forme beaucoup plus complique*e.

A present, ddsignons, suivant la notation usite*e, par [x]p la
fonction -, _ -- ^ (nous aurons, quand p sera un nombre

entier,

[x]p = x.x-l ... x-p + 1),

et tachons d'dvaluer la somme suivante,

dans laquellea? peut prendre toutes les valeurs^— l,p,p+l, etc.,
tandis que y peut prendre toutes les valeurs q — 1, ^, g' + l, etc.,
et ainsi de suite pour les autres variables. L'^tendue des som-
mations est de'termine'e par 1'indgalite

... ,

dans laquelle h est egale ap + ^+...r + un nombre entier.
Nous allons premierement trouver les valeurs de

^£•1 H1*"1 cos aa?> et ^-e ^i IXF"1 sm aa?*

Puisque nous avons

il s'ensuit que

AUX DIFFERENCES FINIES. 233

Rempla9ons z par z~\ nous aurons, en ajoutant les deux rdsultats
et en posant a = 1 ,

[p ~ 1?"1 cos a — 1 + ~*~* cos a + etc-
1

=2r^

c'est-a-dire nous aurons

cos ^ = r (p +

et pareillement

II est facile de voir, en suivant a peu pres la m6me route
qu'auparavant, que ees deux Equations reviennent a celles-ci :

cos
(18) S;_1[£C

sin (TT — a) — a
(19) S;., [a,]^ sin ax = T Q,)

(2 sin-)

Cela pose, on peut facilement s'assurer que la somme dont nous

cherchons la valeur est e*gale a

Par consequent elle est egale, en vertu des formules (18) et
(19), I

et de la nous aurons finalement

(20) \ T(p)TM...T(r)^ ;> . ,,J^...,..1

234 SUR LES INTEGRALES

Les deux resultats (17) et (20) suffisent pour montrer 1'esprit
de notre analyse, mais je vais encore 1'appliquer a un autre

exemple.
i
Evaluons 1'expression

(21) 20$0...2()a*b»...cKf(mx + ny + ...pz),

mx + ny+...pz^h e*tant rine*galite* qui determine 1'etendue des
sommations.

Je suppose que w, n, ...,p soient des nombres entiers. En

faisant

_L !

mx — x't ny—y\ etc.; a2n = a', bn=b', etc.,

la formule (21) deviendra

Nous pouvons done adraettre que ra, w, ...,p soient ^gales a
Tunit^ ; le r^sultat ge'ne'ral se ddduira facilement de ce cas par-
ticulier.

Nous aurons d'abord

< 1 fir

I =i ^hfu da.^^ ...2"axbv...c*cosai(x+y+...!5-u).

I 7T J0

A present, puisque

^oo x 1 — a cos a

S0 a cos ax

.. .
a sm aa? =

- « ,

1 — 2a cos a + a

a sin a

-- 2 »
1 — 2a cos a + a2

nous pouvons effectuer les sommations indique*es dans le second
membre de 1' Equation (22).

En effet, on verra, avec un peu d'attention, que nous aurons

(23) ^;...2:a^...
D &ant ^gal a

(1- 2acosa + a2) ... (1 - 2c
tandis que N est e*gal a

cos aw — Sa.cosa(w + l) + SaZ>.cosa (w + 2) - ...

+ ab...c cos a (u -f v),

AUX DIFFERENCES FINIES. 235

le signe de sommation S ayant rapport aux v quantite's a, b, . . . , c.

Afin de donner a ^ une forme plus commode, posons 1' equa-

tion

A *.;.*

1) 1 - 2a cos a H- a2 1 - 2c cos a + c8 '

done nous aurons

A — flV~1 *

~ (o^T) 77(a^ ' (l-o5)...(l-oc) '

et ainsi de suite pour B, ..., C.
Or, nous savons que

— o = o (1 + 2a cos a + etc.),

1 — 2a cos a + a2 1 — a2 v

NA

et de la il est visible que le terme de - — — 5 > Qui ne

1 — 2a cos a -1- a

renferme pas a, est e*gal a

77 ^ «wi /i N

(a- 6)... (a — c) (1— a)(l— a6)...(l— ac)

ou a

puisque

1 — a Sa + a2 Sa& — etc. _
(l-o«) (1-ai) ... (l-ac)"

Nous voyons done que, pour toutes les valeurs positives de u
et pour u — 0,

da 2* 20* . . . 2* a«^ . . . c" cos a (a? + y +. . .« - «)

Si w est ne'gatif, faisons u — — u\ nous aurons

N = cos aw' — Sa . cos a (M' — 1) + etc. ±ab ... c cos a (w' — v) ,

236 SUR LES INTEGRALES

NA

et le terme de - - 2 » °llu ne renfermera pas a, sera
1 - 2a cos a + a2

dgal a

En supposant que w' ne soit pas moindre que v, il est visible

que

1 - a~l$a + a~* Sab - etc. = 0.

Si u' est moindre que v, alors la formule precedente n'est pas
6gale a z&o, mais 1' ensemble des termes semblables tels que

l-^Sa + etc.

(j _ a) . . . (b - c) (1 - 62) (1 - ba) ... (1 - be)
disparaitra ; c'est ce que nous allons demontrer.

Ddsignons par SM' 1' ensemble de toutes les combinaisons qu'on
peut faire avec u des v quantites, a, b, ..., c. Si u est moin-
dre que v. le terme de - — — • — -2 , qui ne renfermera pas

1 — 2a cos a + a2

a, sera 6gal a

(25) t^ K'-a«-'S1 + ...±Su-+aS.-+1±...±a^S );
or

a*- a"''1 Sj + ... ± Stt' + a'1 SM-+1 ±...± a"1*1" S, = 0.
II suit de la que la formule (25) est e*gale a

et, en ajoutant toutes les formules semblables, nous aurons
+ S..+1{r^(a-a-1)+r!

i SM'+S
T etc.

± S, 1^ (a-' - a—) +T|p (6"- - }-«) + ...}.

A UX DIFFERENCES FINIES. 237

Or, je dis que chacune de ces quantity's est se*pare*ment e*gale
a z£ro. En rempla^ant 2 cos a par z + z~l, nous aurons

Az

Si nous developpons les deux membres de cette Equation
selon les puissances positives de z, la puissance la moins e'leve'e
qui se trouvera dans le premier membre sera zv. Par consequent,
nous voyons que

^ (<? - 0->) + _IL (if - 1-*) + etc. = 0,

pour toutes les valeurs entieres et positives de p qui sont moindres
que v, ce qu'il fallait demontrer.

Done, finalement, pour toutes les valeurs negatives de w,

(26)
A present, il est facile de voir que

(27) 1-v..... . ....I

ce qui est le resultat que nous clierchions.

J'ai de'montre', dans le Journal de Mathematiques de Cam-
bridge, une Equation que je vais reproduire ici, aim qu'on puisse
la comparer avec liquation derniere. Les limites dtant deter-
minees par 1'ine'galite'

nous aurons

\dx\dy...\dz e-a

J9 J 0 J 0

Les rdsultats que nous avons obtenus sont, ce me semble,
d'un genre nouveau: c'est pourquoi je pense qu'ils pourront
peut-6tre interesser les ge'om&tres.

REPORT ON THE RECENT PROGRESS OF
ANALYSIS (THEORY OF THE COMPARI-
SON OF TRANSCENDENTALS)*.

1. THE province of analysis, to which the theory of elliptic
functions belongs, has within the last twenty years assumed a
new aspect. A great deal has doubtless been effected in other
subjects, but in no other I think has our knowledge advanced
so far beyond the limits to which it was not long since con-
fined.

This circumstance would give a particular interest to a his-
tory of the recent progress of the subject, even did it now
appear to have reached its full development. But on the con-
trary, there is now more hope of further progress than at the
commencement of the period of which I have been speaking.
When, in 1827, Legendre produced the first two volumes of his
1 The*orie des Fonctions Elliptiques,' he had been engaged on
the subject for about forty years ; he had reduced it to a sys-
tematic form ; and had with great labour constructed tables to
facilitate numerical applications of his results. But little more,
as it seemed, was yet to be done ; nor does the remark of
Bacon, that knowledge, after it has been systematized, is less
likely to increase than before, seem less applicable to mathe-
matical than to natural science. Nevertheless, almost immedi-
ately after the publication of Legendre's work, the earlier re-
searches of Abel and Jacobi became known, and it was at once
seen that what had been already accomplished formed but a part,
and not a large one, of the whole subject.

To say this is not to derogate from the merit of Legendre.
He created the theory of elliptic functions ; and it is impossible

* Report of the Sixteenth Meeting of the British Association; held at Southamp-
ton, in September, 1846.

ON THE RECENT PROGRESS OF ANALYSIS. 239

not to admire the perseverance with which he devoted himself
to it The attention of mathematicians was given to other
things, and though the practical importance of his labours was
probably acknowledged, yet scarcely any one seems to have
entered on similar researches.* This kind of indifference was
doubtless discouraging, but not long before his death he had
the satisfaction of knowing that there were some by whom that
which he had done would not willingly be let die.

The considerations here suggested have led me to select the
theory of the integrals o£ algebraical functions as the subject
of the report which I have the honour to lay before the Associa-
tion.

2. The theory of the comparison of transcendental func-
tions appears to have originated with Fagnani. In 1714, he
proposed, in the ' Griornale de Litterati d'ltalia,' the follow-
ing problem: To assign an arc of the parabola whose equa-
tion is

y=x\

such that its difference from a given arc shall be rectifiable.

Of this problem he gave a solution in the twentieth volume
of the same journal.

The principle of the solution consists in the transformation
of a certain differential expression by means of an algebraical
and rational assumption which introduces a new variable. The
transformed expression is of the same form as the original one,
but is affected with a negative sign. By integrating both we
are enabled to compare two integrals, neither of which can be
assigned in a finite form. It is difficult, however, to perceive
how Fagnani was led to make the assumption in question: a
remark which applies more or less to his subsequent researches
on similar subjects.

The theorem which has made his name familiar to all
mathematicians, appeared in the twenty-sixth volume of the
1 Giornale.' In its application to the comparison of hyperbolic
arcs we find some indications of a more general method. We
have here a symmetrical relation between two variables, x and

* Those of M. Gauss, which would doubtless have been exceedingly valuable,
have not, I believe, been published. They are mentioned in a letter from M.
Crelle to Abel. Vide the introduction to the collected works of the latter, p. vii.

240 ON THE RECENT PROGRESS OF ANALYSIS.

z, such that the differential expression f(x) dx may be written
in the form z dx. It follows at once that f(z) dz — xdz^ and
consequently that

ff(x) dx + jf(z) dz = j{x dz + z dx} = xz+C.

The remarkable manner in which the idea of symmetry here
presents itself, suggested to Mr Fox Talbot his ' Kesearches in
the Integral Calculus.'

In applying his methods to the division of the arc of the
lemniscate, Fagnani obtained some very curious results, and has
accordingly taken for the vignette of his collected works a
figure of this curve with the singular motto, ' Deo veritatis
gloria.'

3. In MacLaurin's Fluxions, and in the writings of
D'Alembert, instances are to be found where the solution of a
problem is made to depend on the rectification of elliptic arcs,
or, as we should now express it, is reduced to elliptic integrals.
But of these instances Legendre has remarked that they are
isolated results, and form no connected theory. MacLaurin is
charged, in a letter appended to the works of Fagnani, with
taking from the latter without acknowledgement, a portion of
his discoveries with respect to the lemniscate and the elastic
curve.

4. In 1761, Euler, in the 'Novi Commentarii Petropolitani '
for 1758 and 1759, published his memorable discovery of the
algebraical integral of the equation

mdx _ ndy

(A + Bx + Cxz + Dx3 + Ex4)k ~ (A + By + Gf + Dy* + %4)* '

m and n being any rational numbers.

He says he had been led to this result by no regular method,
' sed id potius tentando, vel divinando elicui,' and recommends
the discovery of a direct method to the attention of analysts.
In effect his investigations resemble those of Fagnani : he begins
by assuming a symmetrical algebraical relation between the
variables, and hence finds a differential equation which it satis-
fies. In this differential equation the variables are separated,
so that each term may be considered as the differential of some

ON THE RECENT PROGRESS OF ANALYSIS. 241

function. With one form of assumed relation we are led to the
differentials of circular, and with another to those of elliptic
integrals, and so on. It is in this manner that Dr Gudermann,
in the elaborate researches which he has published in Crelle's
Journal, has commenced the discussion of the theory of elliptic
functions.

5. In the fourth volume of the Turin Memoirs, Lagrange
accomplished the solution of the problem suggested by Euler.
He integrated the general differential equation already mentioned
by a most ingenious methoo, which, with certain modifications,
has remained ever since an essential element of the theory of
elliptic functions. He proceeded to consider the more general
equation

dx _ dy

where X and Y are any similar functions of x and y respectively,
and came to the conclusion, that if they are rational and integral
functions, the equation cannot, except in particular cases, be
integrated, if they contain higher powers than the fourth. He
also integrated this equation in a case in which X and Y involve
circular functions of the variables. It had been already pointed
out in the summary of Euler's researches, given in the Nov.
Com. Pet. Tom. vi., that if X and Fare polynomials of the sixth
degree, the last-written equation does not in general admit of an
algebraical integral, since, if so, it would follow that the solu-

fJ/Y* // ?/

tion of the equation — '• — § = - — ^-5 , which (as the square of

1 + x3 is a polynomial of the sixth degree) is a particular case
of that which we are considering, could be reduced to an alge-
braical form. Now this solution involves both circular functions
and logarithms, and therefore the required reduction is impossible.
This acute remark* showed that Euler's result did not admit
of generalisation in the manner in which it was natural to
attempt to generalise it. It was reserved for Abel to discover
the direction in which generalisation is possible.

6. The discovery of Euler, of which we have been speak-
ing, is in effect the foundation of the theory of elliptic func-

* M. Richelot, in one of his memoirs on Abelian or [hyper- elliptic integrals,
quotea it, in a slightly modified form, from Eider's Opuscula.

16

242 ON THE RECENT PROGRESS OF ANALYSIS.

tions, as the generalisation of it by Abel, or more properly
speaking, the theory of which Eider's result is an isolated frag-
ment, is the foundation of our knowledge of the higher trans-
cendents. We may therefore conveniently divide the subject
of this report into two portions, viz. the general theory of the
comparison of algebraical integrals, and the investigations which
are founded on it. Mathematicians have been led, by comparing
different transcendents, to introduce new functions into analysis,
and the theory of these functions has become an important
subject of research.

The second portion may again be divided into two, viz.
the theory of elliptic functions, and that of the higher trans-
cendents.

This classification, though not perhaps unexceptional, will,
I think, be found convenient.

7. About sixteen years after the publication of Lagrange's
earlier researches on the 'comparison of algebraical integrals, he
gave, in the New Turin Memoirs for 1784 and 1785, a method
of approximating to the value of any integral of the form

\~~fT ? where P is a rational function of. x and E the square

root of a polynomial of the fourth degree. I shall consider this
important contribution to the theory of elliptic functions in con-
nexion with the writings of Legendre. At present, in order to
give a connected view of the first division of my subject, it will
be necessary to go on at once to the works of Abel, and to those
of subsequent writers. In the history of any branch of science
the chronological order must be subordinate to that which is
founded on the natural connexion of different parts of the
subject.

I shall merely mention in passing, that in 1775, Landen
published in the Philosophical Transactions a very remarkable
theorem with respect to the arcs of a hyperbola. He showed
that any arc of a hyperbola is equal to the difference of two
elliptic arcs together with an algebraical quantity. In 1780 he
published his researches on this subject, in the first volume of
his Mathematical Memoirs, p. 23. This theorem, as Legendre
has remarked, might have led him to more important results.
It contains the germ of the general theory of transformation, the

ON THE RECENT PROGRESS OF ANAL YSIS. 243

eccentricities of the two ellipses being connected by the modular
equation of transformations of the second order*. It is on this
account that in a report on M. Jacobi's Fundamenta Nova,
contained in the tenth volume of the Memoirs of the Institute,
Poisson speaks of Landen's theorem as the first step made in
the comparison of dissimilar elliptic integrals. Several writers
have accordingly given Landen's name to the transformation
commonly known as Lagrange's.

8. We have seen that even Lagrange failed in obtaining
a result more general than that which had been made known by
Euler, and yet, as we now know, Euler's theorem is but a
particular case of a far more general proposition. But in order
to further progress, it was necessary to introduce a wholly new
idea. The resources of the integral calculus were apparently
exhausted; Abel, however, was enabled to pass on into new
fields of research, by bringing it into intimate connexion with
another branch of analysis, namely, the theory of equations.
The manner in which this was done shews that he was not
unworthy to follow in the path of Euler and of Lagrange.

I shall attempt to state in a few words the fundamental idea
of Abel's method.

Let us suppose that the variable # is a root of the algebraical
equation fa = 0, and that the coefficients of this equation are
rational functions of certain quantities a, b, ... c, which we shall
henceforth consider independent variables. Let us suppose also
that in virtue of this equation we can express certain irrational
functions f of x as rational functions of a?, a, b, ... c. For in-
stance, if the equation were x2 + ax + - (a2 — 1) = 0, it follows

that Vl — x' = a + x. So that any irrational function of the
form F (x Vl — x*) can be expressed rationally (F being rational)
in x and a.

* Vide infra, pp. 263 and 289.

"h It must be remembered that an algebraical function is either explicit or
implicit : explicit, when it can be expressed by a combination of ordinary algebrai-
cal symbols ; implicit, when we can only define it by saying that it is a root of an
algebraical equation whose co-efficients are integral functions of x. Thus y is an
implicit function of a; if y5 + xy+ i =o. The remarks in the text apply to all alge-
braical functions, explicit or implicit.

16—2

244 ON THE RECENT PROGRESS OF ANALYSIS.

From the given equation we deduce by differentiation the

following,

dx = ada + fidb + ...

where cc, /3, ... 7 are rational in x, a, £,..., c.

Let y be one of the functions which can be expressed ration-
ally in x, &c.j it follows that

ydx = Ada +,Bdb + . . . + Cdc,
where A, B, . . . C are also rational in x, &c.

The equation jfc = 0 will have a number of roots, which we
shall call x^ xz, ... x^. It follows that

da

where the indices affixed to y, A, &c. correspond to those
affixed to x, so that yl9 for instance, is the same function of xl
that yz is of x2.

Now At + . . . + Ap is rational and symmetrical with respect
toajj ... a?^, therefore it can be expressed rationally in the co-
efficients of f(x) = 0, and therefore in a, b...c. We will call
this sum Ra, and thus with a similar notation for b, &c.
we get

The second side of this equation is from the nature of the case
a complete differential, and it is rational in a, b, c, &c.; it can
therefore be integrated by known methods ; and if we denote

y i|r (ipj, we get

M being a logarithmic and algebraic function of a, Z>, &c., which
we may suppose to include the constant of integration.

^r (x) is in general a transcendental function, while a, b, &c.
are necessarily algebraical functions of x^ , . . . , x^ , and the result
at which we have arrived is therefore an exceedingly general
formula for the comparison of transcendental functions.

The simplicity and generality of these considerations entitle
them to especial attention : it cannot be doubted that the ap-
plication thus made of the properties of algebraical equations to

ON THE RECENT PROGRESS OF ANALYSIS. 245

the comparison of transcendents will always be a remarkable
point in the history of pure analysis.

A very simple example may perhaps illustrate what has been
said. Let us recur to the equation

- (a2-l)=0, ............... (1),

and suppose that

Differentiating the first of these equations, we find that
(2x + a] dx + (x + a) da = 0.

Comparing this with the general expression of dx, we perceive
that

and as

, da

^—

so that

Let #t and x2 be the two roots of our equation, we have thus
to find the value of

y = = ^ _^^ (vide ante, p. 243) ,*

(2x, + a) (2#2 + a)
since x1 + xz = — a.

Hence & cfex + yzdxz = 0,

and \frx1 + ^frx2 = c.

Since xt + x2 = — a,

and ay£2=-(a2-l),

we see that x* + x* = l, or a?2 = Vl — a?^.

* The ambiguous sign of the radical is to our purpose immaterial.

246 ON THE RECENT PROGRESS OF ANALYSTS.

Hence, as ^x = sm~1x, our result is merely this, that the
sum of two arcs is constant if the sine of one is equal to the
cosine of the other.

An infinity of analogous results may be obtained either by
varying the form of y (e.g. by making y = \ll-x*), or by
changing the equation (1). A formula applicable to all forms
of y, and which, for each, includes all the results which can be
established with respect to it, is, it will readily be acknow-
ledged, one of the most general in the whole range of analysis.
Abel's principal result is a formula of this nature ; he developed
at considerable length the various consequences which may be
deduced from it.

Generally speaking, the number of independent variables
a, Z>, . . . c will be less than that of the different roots, xl . . . x^ ;
hence a certain number, say m, of the roots may be looked on as
independent (viz. as many as there are quantities a, b, ... c), and
the rest will be functions of these. It may be shown that it will
always be possible to make the difference p — m constant, so
that the sum of any number of the transcendents *fy is ex-
pressible by a fixed number of them, together with an algebraical
and logarithmic function of the arguments, i. e. of xl , . . . xm .
In the case of elliptic integrals, it had long been known that the
sum of two may be thus expressed by a third ; and Legendre
pointed out that the sum of any number may similarly be ex-
pressed by means of one. Accordingly it appears from the general
theory, that in this case fju — m may be made equal to unity.

9. The history of this important theory is curious. It was
developed by Abel in an essay which he presented to the In-
stitute in the autumn of 1826, when he had scarcely completed
his twenty-fourth year.

In a letter to M. Holmboe, appended to the edition of his
collected works, Abel writes, ' Je viens de finir un grand traite
sur une certaine classe de fonctions transcendantes pour le pre-
senter a 1'Institut, ce .qui aura lieu lundi prochain. J'ose dire
sans ostentation que c'est un traite' dont on sera satisfait. Je
suis curieux d'entendre Topinion de 1'Institut la dessus. Je ne
manquerai pas de t'en faire part.' Long before this memoir was
published, Abel had become * chill to praise or blame.' He
died at Christiania in the spring of 1829.

ON THE RECENT PROGRESS OF ANALYSIS. 247

M. Jacob! mentions in a note in Crelle's Journal, that while
at Paris he represented, and as he "believed not ineffectually, to
Fourier, who was then one of the secretaries of the Institute, that
the publication of this memoir would be very acceptable to
mathematicians. A long period however was still to elapse
before the publication took place. It was possibly retarded by
the death of Fourier. In 1841 the memoir appeared in the
seventh volume of the Memoires des Savans Etrangers. It was
prepared for publication by M. Libri.

Thus for about fifteen years Abel's general theory remained
unpublished ; but in the meanwhile Crelle's Journal was estab-
lished, and to the third volume of this he contributed a paper
which contains a theorem much less general than the researches
he had communicated to the Institute, but far more so than any-
thing previously effected in the theory of the comparison of
transcendents. This is commonly known as Abel's Theorem.
Legendre, in a letter to Abel, speaks thus of the memoir in
which it appeared: — "Mais le memoire...ayant pour titre Re-
marques sur quelques proprietes generates, &c., me parait sur-
passer tout ce que vous avez publi6 jusqu'a present par la
profondeur de 1'analyse qui y regne ainsi que par la beaute
et la ge'ne'ralite' des resultats." In a previous letter, with refer-
ence I believe to the same subject, he had remarked, ' Quelle
tete que celle d'un jeune Norvegien !'

Abel's theorem gives a formula for the comparison of all
transcendental functions whatever whose differentials are irra-
tional from involving the square root of a rational function of x.

In a very short paper in the fourth volume of Crelle's
Journal, which must have been the last written of Abel's pro-
ductions, the chief idea of his general theory is stated; and
in the second volume of his collected works we find a somewhat
fuller development of it, in a paper written before his visit to
Paris, but not published during his life-time.

While Abel's great memoir remained unpublished at Paris,
several mathematicians, developing the ideas which he had
made known in his contributions to Crelle's Journal, succeeded
in establishing results of a greater or less degree of generality.
Eesearches of this kind may be presented in a variety of forms,
because the algebraical function to be integrated, which we have
called y, may be defined or expressed in different ways. For

248 ON THE RECENT PROGRESS OF ANALYSIS

instance, if J/and N are general symbols denoting any integral

Vlf -, M

functions of x, the two suppositions y = — ^- ancl y = "r^ ar(

precisely equivalent, since by an obvious reduction, and by
changing the signification of M and Nt the one may be trans-
formed into the other ; and so in more general cases. Thus the
same function may assume a variety of aspects, and there will
be a corresponding variety in the form of our final results.

In Crelle's Journal we find a good many essays on this part
of the subject : of these I shall now mention several.

M. Broch is the author of a paper in the twentieth volume
of Crelle's Journal, p. 178. It relates to the integration of certain
functions irrational in consequence of involving a polynomial of
any degree raised to a fractional power. For these functions he
establishes formulae of summation, which of course include Abel's
theorem, since the latter relates to cases in which the fractional
power in question is the (J)th. Subsequently to the publication
of this paper he presented to Jthe Institute a memoir on the
same subject, but gave to the functions to be integrated a
different but not essentially more general form. This memoir,
which was ordered to be printed among the Savans Etrangers,
but which will be found in Crelle's Journal (xxin. 145), may
be divided into two portions : the first contains results analogous
to Abel's theorem: the second relates to the discussion and
reduction of the transcendents which they involve. In this part
of his researches M. Broch has followed the method, and oc-
casionally almost adopted the phraseology of a memoir of Abel,
on the reduction and classification of Elliptic Integrals (Abel's
Works, II. p. 93). MM. Liouville and Cauchy, in reporting on
the memoir, conclude by remarking that the author *n'a pas
trop presume* de ses forces en se proposant de marcher sur les
traces d'Abel.'

M. Jiirgenson has contributed two papers to Crelle's Journal
on the subject of which we are speaking. The first, which
is very short, contains a general theorem for the summation
of algebraical integrals* when the function to be integrated is
expressed in a particular form. This paper appears in the

* I have used the expression " algebraical integrals," though perhaps not cor-
rectly, to denote the integrals of algebraical functions.

ON THE RECENT PROGRESS OF ANALYSIS. 249

nineteenth volume, p. 113. In the second (Vol. xxiii. p. 126)
the author reproduces the results he had already obtained,
pointing out the equivalence of one of them to the theorem
established in M. Broch's first essay. Besides this, he dis-
cusses a question connected with the reduction of algebraical
integrals.

M. Ramus, in the twenty-fourth volume of Crelle's Journal,
p. 69, has established two general formulas of summation ;
from the second he deduces with great facility Abel's theorem,
and also another result. ..which Abel mentions in a letter to
Legendre, published in the sixth volume of Crelle's Journal, but
which he left undemonstrated.

M. Rosenhain's researches (Crelle's Journal, XXVIII. p. 249,
and xxix. p. 1) embrace both the summation and reduction of
algebraical integrals. His analysis depends on giving the
function to be integrated a peculiar form, which he conceives
leads to a simpler mode of investigation than any other.

A paper by Poisson will be found in the twelfth volume
of Crelle's Journal, p. 89. It relates to the comparison of alge-
braical integrals, but is not I think so valuable as that great
mathematician's writings generally are.

Beside the memoirs thus briefly noticed, I may mention two
or three by M. Minding : that which appears in the twenty-third
volume of Crelle's Journal, p. 255, is the one which is most
completely developed.

There is also a very brief note by M. Jacobi in the eighth
volume of Crelle's Journal.

10. To the Philosophical Transactions for 1836 and 1837
Mr Fox Talbot contributed two essays, entitled Researches in
the Integral Calculus. These researches may be said to con-
tain a development and generalisation of the methods of Fag-
nani. They are however far more systematic than the writings
of the Italian mathematician, and if they had appeared in the
last century would have placed Mr Talbot among those by
whom the boundaries of mathematical science have been en-
larged. But it cannot be denied that they fall far short of what
had been effected at the time they were published, nor does it
appear that they contain anything of importance not known
before. I have assuredly no wish to speak disparagingly of

250 ON THE RECENT PROGRESS OF ANALYSIS.

Mr Talbot ; his mathematical writings bear manifest traces of
the ability he has shown in so many branches of science*.
But as in this country they seem to have been thought, and
by men not apparently unqualified to judge, to contain great
additions to our knowledge, I cannot avoid inquiring whether
this be true.

Mr Talbot points out in the early part of his first paper,
that if there are n — 1 symmetrical relations among the n vari-
ables x, y . . . z, then the identical equation

{y . . . z] dx + (x . . . z) dy + . . . + {xy ...}dz— d {xy ... z}
will assume the form

$ (x) dx + $ (y) dy + ...+ <f).(z) dz = d{xy ... z},
and thus give us

fo (x) dx +j<t> (y} dy+... + J<£ (z) dz = xy ...z+C.

Precisely the same remark, though expressed in a different
notation, is the foundation of M. Hill's memoir, published in
1834, on what he calls ' functiones iteratge.' It will be found in
Crelle's Journal, XI. p. 193. A much more general theorem
might be established by similar considerations: they are of
course applicable whether the function <f> be algebraical or trans-
cendent.

In the course of his researches, Mr Talbot recognised the
important principle, that the existence of n — I symmetrical
algebraical relations among n variables may be expressed by
treating them as the roots of an equation, one of whose coeffi-
cients at least is variable, the others being either constant or
functions of the variable one. Unfortunately he did not pass
from hence to the more general view, that the existence of n —p
symmetrical relations may be expressed in a similar manner if
we consider p of the coefficients of the equation as arbitrary
quantities. Had he done so, it is possible, though not likely,
that he would have rediscovered Abel's theorem ; but as it is,
he has never introduced, except once, and then as it were by
accident, more than one arbitrary quantity. Thus only one of

* It must be remembered also that Mr Talbot admits himself to have been anti-
cipated to a considerable extent by the publication of Abel's theorem.

ON THE REGENT PROGRESS OF ANALYSIS. 251

liis variables is independent, and consequently, in more than one
instance, his results are unnecessarily restricted cases of more
general theorems.

The character of his analysis will be perceived from what

has been said. If \Xdx be the transcendent to be considered,

X being an algebraical function of x, he makes the following
assumption —

X=f(xv),

v being a new variable, ajnd /a rational function. From this
assumption he deduces an algebraical equation in x, the co-
efficients of which are rational functions of v. This equation
then is one of those of which we have spoken, by means of which
the function to be integrated can be expressed in a rational
form. Taking the sum with respect to the roots of this equation,
we get

It must be remarked that many forms might be assigned to the
function f, which would give rise to a difficulty, of the means of
surmounting which Mr' Talbot has given no idea. If x and v
are mixed up in f(xv), it is manifest that we cannot integrate
f(xv) dx, since v is a function of x, which if we eliminate we
merely return to our function X. We must therefore express
*£f(xv) dx in the form Vdv, V being a function and, as Abel
has shown, an integrable function of v. Abel has given for-
mulas by means of which this reduction may be effected in all
possible cases. But there is nothing analogous to this in the
writings of Mr Talbot, and consequently he could not, setting
aside the defect already noticed, obtain results as general as many
previously known. In Mr Talbot's investigations, f(xv) dx is
such that ^f(xv) dx may be put in the form —

FX2 {faxdx} + F2S Wzxdx] + &c.,

<j>LXj $2x, &c. (of which Q'jX, (j>'2x, &c. are the derived functions)
being rational functions of x. Then ^(frx = a rational function
of v by a well-known theorem. Let the form of this function be
ascertained, and let us denote it by %v. Then differentiating,

/
and hence

+ F2%> + ...] dv,

252 ON THE RECENT PROGRESS OF ANALYSIS.

and the second side of this equation is of course rational and
integrable. But the form of the f unction f(xv) is unnecessarily
restricted in order that this kind of reduction may be possible.
Nevertheless, Mr Talbot's papers, from their fulness of illus-
tration and the clear manner in which particular cases of the
general theory are worked out by independent methods, will be
found very useful in facilitating our conceptions of the branch
of analysis which forms as it were the link between the theory
of equations and the integral calculus.

In Mr Talbot's second memoir (Phil. Trans. 1837, part 2.
p. 1) he has applied his method to certain geometrical theorems.
Three of them relate to the ellipse, and are proved by the three
following assumptions :

vx> or = — 7=- > or

1-0*

These assumptions are all cases of the following :

where a, a1, c, c1 are arbitrary quantities. The results of this
assumption are completely worked out by Legendre (ThSorie
des Fonctions Elliptiques, in. p. 192) in showing how the
known formulae of elliptic functions may be derived from Abel's
theorem. Mr Talbot's first theorem is a case of the fundamental
formula for the comparison of elliptic arcs. This remark has
reference to an inquiry which Mr Talbot suggests as to the
relation in which his theorems stand to the results obtained by
Legendre and others.

In conclusion, it may be well to observe that Mr Talbot
has remarked that, apparently, a solution discovered by Fagnani
of a certain differential equation cannot be deduced from Abel's
theorem ; but as this solution may be easily derived from
the ordinary formula for the addition of elliptic integrals of
the first kind it is manifestly included in the theorem in
question.

II.

11. I now come to the history of researches into the pro-
perties of particular classes of algebraical transcendents. The
earliest, and still perhaps the most important class of these

ON THE RECENT PROGRESS OF ANALYSIS. 253

researches relates to the transcendents which are commonly
called elliptic functions or elliptic integrals. For a reason
which will be mentioned hereafter the latter name seems pre-
ferable, and it is sanctioned by the authority of M. Jacobi,
though the former was used by Legendre. Elliptic integrals
then may be defined as those whose differentials are irrational
in consequence of involving a radical of the form

ex*

But it may perhaps be more correct to say that all such integrals
may be reduced to three standard integrals, to which the name
of elliptic integrals has been given.

In the Turin Memoirs for 1784 and 1785, p. 218, Lagrange
considered, as has been already mentioned, the theory of these
transcendents. He showed that the integration of every func-

tion irrational in consequence of containing a square root may

p
be made to depend on that of a function of the form -^ , P being

rational, and R the radical in question ; and that if under the
sign of the square root x does not rise above the fourth degree,
it may ultimately be made to depend on that of

Ndx

where Nis rational in x\ He thus laid the foundation of that
part of the theory of elliptic transcendents in which a proposed
integral is reduced to certain canonical or standard forms, or
to the simplest combination of such forms of which the case
admits. In Legendre's earliest writings on elliptic functions
there is nothing relating to this part of the subject. Having
thus, in the simple manner which distinguishes his analysis,
reduced the general case to that which admits of the application
of his method, Lagrange proceeded to prove that if we intro-
duce a new variable whose ratio to x is the subduplicate of the
ratio of 1 ± p*x* to 1 ± jV, the last written integral is made to
depend on another of similar form, but in which p and q are
replaced by new quantities p1 and ql. If p is greater than q,
p1 will be greater than p, and ql less than q, and thus by
successive similar transformations we ultimately come to an
integral in which q is so small that the factor 1 + qlx* may be

254 ON THE RECENT PROGRESS OF ANALYSIS.

replaced by unity, and the elliptic integral is therefore reduced
to a circular or logarithmic form. Or by successive transforma-
tions in the opposite direction we come to an integral in which
pl and <f are sensibly equal, in which case also the elliptic
integral is reduced to a lower transcendent. This most in-
genious method is the foundation of all that has since been
effected in the transformation of elliptic integrals, or at least
whatever has been done has been suggested by it. Thus it
is to Lagrange that we owe -the origin of two great divisions of
the theory of these functions.

In the Memoirs of the French Academy for 1786, p. 616,
we find Legendre's first essay on the subject to which he after-
wards gave so much attention. We recognise in it what may
I think be considered the principal aim of his researches in
elliptic functions, namely to facilitate, by the tabulation of
these functions, the numerical solution of mathematical and
physical problems.

He begins, not with a general form as Lagrange had done,

but with the integral IVl — tf sin2 (f)d(f>, which as we know re-
presents an elliptic arc, and shows how other functions, for
instance the value of the hyperbolic arc, may be expressed by
means of it, and of its differential coefficient with respect to the
eccentricity c. The memoir does not contain much that is now
of interest. After writing it he became aware of the existence
of Landen's researches ; and in a second memoir appended to
the first gave a demonstration of Landen's principal theorem.
This demonstration is founded on Legendre's own methods,
and he deduces from it the remarkable conclusion, that if of
a series of ellipses, whose eccentricities are connected by a
certain law, we could rectify any two, we could deduce from
hence the rectification of all the rest. The law connecting the
eccentricities of the ellipses is that which would be obtained
by making use of Lagrange's method of transformation, with
which accordingly this result is closely allied.

Legendre's next work was an essay on transcendents*, pre-
sented to the Academy in 1792 and published separately the
year after. It contains the same general view as that which is

* A translation of it appeared in Leybourne's Mathematical Repository, Vols. II.
and in. The original I have not seen — it has long been scarce.

ON THE RECENT PROGRESS OF ANALYSIS. 255

developed in the first volume of the Exercises de Calcul In-
tegral, which appeared in 1811.

12. The theory of elliptic f auctions, as it is presented to
us by Legendre, may conveniently be considered under the
following heads :

OL. The reduction of the general integral,

Pdx

Va + fix -f- 7Jj* 4- 8x3 4- ex*

in which P is rational to three standard forms, since known as
elliptic integrals of the first, second and third kinds*.

This classification, though the reduction of the general in-
tegral had, as we have seen, been already considered by La-
grange, is I believe entirely due to Legendre. If we consider
how much it has facilitated all subsequent researches, we can
hardly over-rate the importance of the step thus made. It may
almost be said that Legendre, in thus showing us the primary
forms with which the theory of elliptic integrals is conversant,
created a new province of analysis : he certainly gave unity and
a definite form to the whole subject.

For the three species of functions thus recognised Legendre
suggested the names of nome, epinome and paranome, the name
of the first being derived from the idea that it involves, so
to speak, the law on which the comparison of elliptic integrals
depends. But these names do not seem felicitous, nor have
they, I believe, been adopted. To this part of the subject an
important theorem relating to the reduction of elliptic integrals
of the third kind, whose parameters are imaginary, seems
naturally to belong.

« These three forms are

f> /i-#x* ^ f"
-cV'); Jo V ~T^2 :; J0(

0(i+no;8)y(i-

Legendre always replaces x by sin <£, so that the integrals become

'* dd>

/> d<f> /> . f

I fn=i I Vi-c'sin2^;

J o Vi-c2sm20 Jo v J0

(i + n sin2 0) sji-c2 sin3 0

The radical ^/i -c2 sin2 <f> is often denoted by A.

The constant c is called the modulus; the second constant n (in the third kind)
is called the parameter. The modulus may always be supposed less than unity,
and if c = sin e, then c is the angle of the modulus.

256 ON THE RECENT PROGRESS OF ANALYSIS.

@. The comparison of elliptic integrals of the same form
differing only in the value of the variable, or as it is often
called, the amplitude of each. This part of the subject divides
1tself into three heads, corresponding to the three classes of
integrals. The fundamental results are to be found in the
memoirs of Euler, of which we have already spoken. By Le-
gendre however they were more fully developed.

It is interesting to observe that Legendre suggested that the
discovery of Euler (namely that the differential equation

=Q

admits an algebraical integral, f(x) being the polynomial

a + fa + 7#2 + &C3 + ex*)

might be generalised, if we consider the differential equation
dx +_<fy_+ { dz ==Q

^/(*) #(y) V/w

He remarks that this is perhaps the only way in which it can be
generalised.

7. Theorems relating to the comparison of different kinds
of elliptic functions. One of the most remarkable of these is
the relation between the complete integrals (those, namely, in
which the variable x is unity) of the first and second kind, the
moduli of which are complementary; that is, the sum of the
squares of whose moduli is equal to unity. Legendre's demonstra-
tion of it is rather indirect, but many others have been since
given. Another theorem may be mentioned, — that the complete
integral of the third kind can always be expressed by means
of the complete integrals of the first and second. A third and
most important result shows that in elliptic integrals of the
third kind we may distinguish two separate species, and that
to one or other of these any such integral may be reduced. A
memorable discovery of M. Jacobi has greatly increased the
importance of this subdivision, of which we shall hereafter
speak more fully. This part of the subject is, I imagine,
entirely due to Legendre.

8. The evaluation of elliptic integrals by means of ex-
pansions.

e. The method of successive transformations. The idea of

ON THE RECENT PROGRESS OF ANALYSIS. 257

this method originated, as we have seen, with Lagrange. It is
developed at great length by Legendre, with a special reference
to the modifications required in applying it to the different
species of integrals. As Lagrange had shown, the series of
transformed integrals extending indefinitely both ways conducts
us, in whichever direction we follow it, towards a transcendent
of a lower kind than an elliptic integral, or in other words,
towards a logarithmic or circular integral. There are thus twro
modes of approximation, one of which depends on a series of
integrals with increasing moduli, and the other on a series
whose moduli decrease. Thus for the three species of integrals
there will be in all six approximative processes to be considered.
In the case of the elliptic integral of the third kind, we have
to determine the law of formation of the successive parameters
«, nl, &c.

£. Keductions of transcendents not contained in the general

formula (e.g. \ } to elliptic integrals.

77. Lastly, applications to various mechanical and geometri-
cal problems.

This analysis, however slight, will give an idea of the
contents of that part of the Exercices de Calcul Integral which
relates to elliptic functions. In the third volume there are
tables for facilitating the calculation of integrals of the first and
second kind : they are accompanied with an explanation of the
manner in which they were constructed. The ninth table is
one with double entry, the two arguments being the angle of
the modulus and the amplitude.

13. In 1825 Legendre presented to the Academic des
Sciences the first volume of his Traite des Fonctions Ellip-
tiques. A great part of this work is precisely the same as
the Exercices de Calcul Integral. By far the most important
addition to the theory of elliptic functions consists in the dis-
covery of a new system of successive transformations quite dis-
tinct from that of Lagrange.

In the earlier work Legendre had shown that a certain
transcendent might be expressed in two ways by means of
elliptic integrals of the first kind. Comparing the two results,
he obtained a very simple relation between the two elliptic

17

258 0^ THE RECENT PROGRESS OF ANALYSIS.

integrals. Their moduli are complementary ; while the ratio of
the A's in the two integrals can be expressed rationally in terms
of the sine of the amplitude of one. This circumstance seems
to have suggested to Legendre the possibility of generalising
the result. He accordingly assumed a relation between the
amplitudes of two integrals, of which the equation subsisting in
the theorem of which we have been speaking is a particular
case ; and showed from hence that a simple relation perfectly
similar to that which he had obtained in the particular instance
existed between the two integrals, viz. that they bore to each
other a ratio independent of their amplitudes. Their moduli
are connected by an algebraical equation, but are not comple-
mentary. This circumstance therefore now appeared to be un-
essential, though in the Exercices the investigation is intro-
duced for the sake of exhibiting a case in which an integral
may be transformed into another with a complementary mo-
dulus.

Legendre thus obtained a new kind of transformation, which
might be repeated any number of times or combined in an in-
finite variety of ways with that of Lagrange. To illustrate
this he constructed a kind of table — a 'damier analytique.' In
the central cell is placed the original modulus c. All the moduli
contained in the same horizontal row are derivable from one
another by Lagrange's scale of moduli; those in each vertical
row by the newly discovered scale. He seems to have been
very much struck by the infinite variety of transformations of
which elliptic integrals admit. The integral of the first kind
is especially remarkable, because of the simplicity of the relation
which connects it with any of its transformations, viz. that their
ratio is independent of the amplitudes.

Legendre' s second work was, as we have remarked, pre-
sented to the Academy in 1825, but it was not published till
1827. In the summer of 1827 M. Jacobi announced in Schu-
macher's Astronomischen Nachrichten, No. 123, that he was
in possession of a general method of transformation for elliptic
integrals of the first kind. He was not acquainted with Le-
gendre's discovery of a new scale, and as an illustration of the
general theorem gave two cases of it, the first being equiva-
lent to Legendre's method of transformation. Thus much was
announced in a letter to M. Schumacher, dated June 13th ;

ON THE RECENT PROGRESS OF ANALYSIS. 250

but in one of a later date (August 2nd) lie gave a formal
enunciation of his theorem, but without demonstration. The
two communications appear consecutively (Ast. Nacli. vr. p. 33).

In No. 127 of the Nachrichten, VI. p. 133, M. Jacobi gave a
demonstration of his theorem.

If we can so determine y in the terms of x as to satisfy the
differential equation

di 1 dx

-y) (i - xy v (i - &) (i -

(M being constant), it is manifest that we shall have (^denoting
the elliptic integral of the first kind) F (kx) = MF (ty) , pro-
vided that y and x vanish together. The question therefore is,
how may the differential equation be satisfied, for it is clear
that by means of a solution of it we transform the elliptic in-
tegral F(kx) into another, viz. into F(\y).

M. Jacobi shows that if y be equal to y, Z/and F being

integral functions of x, the differential equation will be satisfied,
provided U and F fulfil two general conditions, the second of
which is found to be deducible from the first. He then makes
an assumption which is equivalent to assigning particular forms
to U and F, and thence shows, by a most ingenious method,
that these forms of £7 and F are such as to fulfil the first of the
required conditions, which, as has been said, implies the other.
He thus verifies, d, posteriori, the assumed value of the func-
tion y.

In proving that the forms assigned for U and F have the
required property, it is necessary to pass from an expression of
the value of 1 — y in terms of x to one of 1 — \y in terms of
the same quantity. This is done by means of a remarkable
property of the functions U and F, namely, that if in both

x be replaced by -j— , -p- or y will (the constants being properly

F 1

adjusted) become —j-T or — . Therefore, in any form in which

A< u ^~y

the relation connecting y and x can be put, we may replace
x by j— , provided we at the same time replace y by — .

This has been called the principle of double substitution, and

17—2

260 ON THE RECENT PROGRESS OF ANALYSIS.

by means of it we pass from the expression of 1 — y to that of

1 , and thence obtain that of 1 — A?/. It is to be ob-

\y'

served that this principle is used merely to prove a certain pro-
perty of the functions U and V. Of course, as the change of x

into =— implies that of v into — in the finite relation between
Jcx \y

these quantities, the same thing will be true in the differential
equation by which they are connected, a remark which may
very easily be verified. But, on the other hand, it by no means
follows that because it is true in the differential equation there-
fore any assumed finite relation between y and x having this
property is the integral required. The property in question
therefore does not enable us to verify any assumed value

of y.

This remark has reference to a communication from Le-
gendre which appears in No. 130 of Schumacher's NacJirichten^
VI. p. 201. In it he gives an account of M. Jacobi's researches,
and an outline of the demonstration of which we have been
speaking. I find it impossible to avoid the conclusion that this
great mathematician mistook the character of the demonstration
in question, and that to him it appeared to be in effect a mere
verification of the assumed value of y by means of the principle
of double substitution. He remarks that the direct substitution
of the value of y in the differential equation is impracticable,
but that M. Jacobi had avoided this substitution by means of
1 une proprie'te particuliere de cette equation qui doit 6tre com-
mune aux integrates qui la repre'sentent.' This property is the
principle of double substitution; and after showing that it is
true of the differential equation, the writer proceeds thus :
* Ce principe une fois pose*, rien n'est plus facile que de verifier

1' equation trouvee y = -^, car par la double substitution on

obtient la m6me valeur de y a un coefficient pr&s qui doit etre
£gal a 1' unite ;' and, after a remark to our present purpose im-
material, concludes, ' Ainsi se trouve demontre'e ge'ne'ralement

liquation y = -painsi que, etc.'

As we have seen, such a verification would be wholly in-
conclusive, nor is the essential point of M. Jacobi's reasoning,

ON THE RECENT PROGRESS OF ANALYSIS. 261

namely, that the assumed forms of U and V satisfy the general
condition, laid down at the outset of his demonstration, here
adverted to.

In 1828, Legendre published the first supplement to the
Traite des Fonctions ElUptiques, &c. It contains an account
of the researches of M. Jacobi, and of a memoir by Abel
inserted in the third volume of Crelle's Journal. The account
here given of M. Jacobi' s demonstration is fuller and more ex-
plicit than that already noticed. It leaves, I think, no doubt
of the error into which Leggndre had fallen. No notice what-
ever is taken of the first part of M. Jacobi's reasoning: arid
after remarking that the differential equation is satisfied when
the double substitution is made, he goes on, ' Tout se reduit

done a faire cette double substitution dans I'inte'grale y — --y

et a examiner si elle est satisfaite.' After showing that it is so,
lie adds, ' Par ce proce'de tres simple il est constate que I'e'qua-

x U

tion y — - -y. satis fait ... a 1'dquation differentielle dont 1' in-
tegrate est F (&</>) = pF (h-p), etc.' (Trait, des Fonct. Ell. ill.
p. 10.)

Legendre remarks, that although M. Jacobi's demonstration
rests on ' un principe incontestable et tres inge'nieux,' it is still
desirable to have another verification of so important a theorem.
He accordingly gives an original demonstration of it, which is
however more nearly allied to M. Jacobi's than to him it seemed
to be. This demonstration had already been hinted at in his
communication to the Nachrichten. The principal difference is,
that while M. Jacobi proved generally that if the first of the
two required conditions were satisfied, the second would also be
so, and then showed that the forms assigned to U and V satis-
fied the first condition ; Legendre shows the assigned forms are
such as to satisfy both conditions, on the connection between
which it is therefore unnecessary for him to dwell. In the
third supplement to the Traite des Fonctions Elliptiques, Le-
gendre has given another demonstration of M. Jacobi's theorem,
remarking that it is both more rigorous and more like M.
Jacobi's than that which he had first given. I have thought
it necessary to make these remarks, because it has been said
that it was in the supplements to Legendre's work that the

262 O^V THE RECENT PROGRESS OF ANALYSIS.

demonstration of this theorem received * le dernier degre de
rigueur*.'

14. In 1829 M. Jacobi's great work on elliptic functions,
the Fundamenta Nova Theories Functionum Ellipticarum, was
published at Koenigsberg. It contains his researches not merely
on the theory of transformation, but also with respect to other
parts of the subject. But the great problem of transformation
is the fundamental idea of the whole work ; the other parts are
subordinate to it, or at least derived from it. The subject is
treated with great fulness of illustration and in a manner not
unlike that of Euler.

M. Jacobi begins by considering the possibility of trans-
forming the general transcendent whose differential coefficient
is unity divided by the square root of a polynomial of the fourth
degree. Subsequently, having shown that this transcendent
may be transformed by introducing a new variable y equal to
the quotient of two integral functions of a?, and also that the
general transcendent may be reduced to one of the form

dy

he proceeds to consider the latter in detail.

The first step of this reasoning, viz. the possibility of the
transformation, depends on a comparison of the number of the
disposable quantities in the assumed value of y with that of the
conditions required, in order that the quantity under the radical
in the transformed expression may be equal to the square of an
integral function of x multiplied by four unequal linear factors.
It is shown that the number of disposable quantities exceeds
by three that of the required conditions. But, as Poisson has
remarked in the report already mentioned (Mem. de Vlnstitut.
x. p. 87), and as M. Jacobi himself intimates, this does not
amount to an absolute a priori proof of the possibility of the
transformation ; non constat but that some of these conditions
may be incompatible.

Granting however the possibility of putting the quantity
under the radical in the required form, it is shown, as in
Schumacher's Journal, that this condition is not only necessary

* Verhulst, Traite Elemmtaire des Fonctions EUiptiques.

ON THE RECENT PROGRESS OF ANALYSIS. 263

but also sufficient, or, in other words, that it involves the
second condition already mentioned.

The transcendent I -7= . 8 may be transformed

by assuming y— y, U being composed wholly of odd powers

of x, and F of even powers of it. If the degree of U be greater
than that of F, the transformation is said to be of an odd order,
and of an even order in the contrary case.

This being premised, M. Jacobi discusses the particular
cases of the transformations of the third and of the fifth order.
The first is the same as that of Legeridre. It is shown that if
we put

(v + 2u3] vx -f u?x3
y ~ v* + V*K? (v + 2us) x* '

where u and v are constants connected by the following equa-
tion— •

we shall get

__ _ dy _ v + 2u? dx

in which k = u* and \ = v4. The equation connecting u and v
is called the modular equation.

The ' principle of double substitution' may be illustrated by

writing -- for a? in the expression for y, which, then becomes,
u x

according to the principle in question, - — .

If we seek to show that the assigned value of y actually
satisfies the differential equation just stated, we begin by find-
ing the value of 1 — y. Reducing this value by means of the

7?2
equation between u and v, we can put it in the form (1 — x) -y ,

R being an integral function of x and F, as heretofore the
denominator of the expression for y. The value of 1 + y is hence
got by changing the sign of x and y, while that of 1 — v*y is ob-

tained by simultaneously replacing x and y respectively by -4-
.- u x

and — and reducing. Similarly for 1 + v*y. Hence it will

c/

appear that

264 ON THE RECENT PROGRESS OF ANALYSIS.

where S, like E, is integral. By differentiating and reducing,
we then show that

dy — - - j^2 dx,

v I

and combining these two results obtain the required verification.
The essence of M. Jacobi's demonstration consists in showing
that if the value of y in terms of x is such that an equation of the
form (a) subsists, then necessarily

^ = — (8}

where /z- is a constant ; the existence of the two equations (a) and
(/3) being equivalent to the two conditions of which we have
already spoken (p. 259). In the particular case we are now con-
sidering,

fJL= - — .

v

15. After considering the transformation of the fifth order
(in which the modular equation is

u" - v6 + 5wV {u* - v2} + ±uv [I - wVJ = 0),

M. Jacobi prepares the way for a more general investigation
by introducing a new notation. This step is one of the highest
importance. We have been in the habit of calling <£ the am-
plitude of the integral

d<f>

let this integral be called u. The new notation is contained
in the equation (j> = amw ; or if we call sin <£, x, so that

dx

-i:

then x = sin am u.

A new notation is in itself merely a matter of convenience :
what gives it importance is its symbolizing a new mode of con-
sidering any subject. We had hitherto been accustomed to look
on the value of the elliptic integral as a function of its amplitude,
to make the amplitude (if the expression may so be used) the in-
dependent variable. But in reality a contrary course is on many
accounts to be preferred. We have in the more advanced part

ON THE RECENT PROGRESS OF ANALYSIS. 2G5

of the theory more frequently occasion to consider the value of
the amplitude as determined by the corresponding value of
the integral than vice versa ; and it therefore becomes expedient
to frame a notation by which the amplitude may be expressed as
a function of the integral. In a paper in the ninth volume of
Crelle's Journal by M. Jacobi, which, like many of his writings,
contains in a short compass a philosophical view of a wide
subject, he has made use of the analogy between circular and
elliptic functions to illustrate the importance of the new notation
for the latter. When the rnodulus of an elliptic integral of the
first kind is equal to zero, the integral becomes

dx

which, as we know, is equal to the arc whose sine is x, or to
sin"1 a;. Now this is a function which we have much less often
occasion to express than its inverse sin x, and we accordingly
always look on the latter as a direct, and on the former as an
inverse function. Yet in the case of elliptic functions, the func-
tional dependence for which we had an explicit and recognised
notation, viz. that of the integral on the amplitude, corresponds
to that which in circular functions lias always and almost neces-
sarily been treated merely as an inverse function. The origin of
this discrepancy is obvious; our knowledge of the nature of
circular functions is not derived from the algebraical integrals
connected with them, and therefore these integrals are not
brought so much into view as in the theory of elliptic functions
the corresponding integrals necessarily are ; but it is certain that
while the discrepancy continued to exist the subject could never
be fully or satisfactorily developed. The maxim " verba vestigia
mentis " is as true of mathematical symbols as of the elements of
ordinary language.

We shall see hereafter that Abel took the same step in his
first essay on elliptic functions. At present I shall only re-
mark, that one of the earliest consequences of the new notation
was the recognition of a most important principle, viz. that the
' inverse function' sin am u, that is, the function corresponding
to sin u in circular functions, is doubly periodic, or that it
retains the same value when u increases by any multiple either
of a certain real or of a certain imaginary quantity. Now

266 ON THE RECENT PROGRESS OF ANALYSIS.

M. Jacob! lias shown that no function* can be triply periodic, and
therefore these inverse functions possess the most general kind
possible of periodicity, a property which gives them great
analytical importance.

Following M. Jacobi, we shall henceforth give the name of
elliptic functions to those which are analogous to circular func-
tions. It is on this account better to call Legenclre's functions
elliptic integrals than, as he has done, elliptic functions (vide
ante, p. 253).

By the new notation we are led to consider a great variety
of formulae analogous to those of ordinary trigonometry. The
sine or cosine of the amplitude of the sum of two quantities may
be expressed in terms of the sines and cosines of the amplitudes
of each, &c.f ; and we have only to make the modulus equal to
zero to pass from what has sometimes, though not with much
propriety, been called elliptic trigonometry to the common
properties of circular functions.

" M. Jacobi gives a table of formulas relating to the new
elliptic functions, and proceeds to apply their properties to the
problem of transformation. It was in this manner that he had
treated the problem in the Nachrichten. As in his earlier essay,
he assumes y equal to a rational function of #, whose coefficients

* i. e. no function of one variable.
*f* The fundamental formulae are

sin am u cos am #A am v + sin am v cos am uA am u

— 7 *--•>" — .— « —
i - k2 sin2 am u sin"5 am v

cos am u cos am v - sin am u sin am vA am wA am v
cos am (u + v)= — - 19 . .. — — -r-n — - ;

i - k2 smj am u Bin2 am v

A am -uA am v - k2 sin am u sin am v cos am u cos am v
A am (u + v) — — — ,.. . „ — — r- ^- -

i - k2 sm2 am u am2 am v

being the modulus, and A am u— A/I -fc2 sin2 am u. If

where &2-f &'2 = i, then it may be shown that

sin am (u + ^K) = sin am u,
and

sin am (u + iK' \/— i) = sin am u,

so that 4# is the real and iK' V -^ tne imaginary period of sin am u. Hence it
is obvious that we shall have generally

sin am (M + 4m£+ inK' /J - i.) = sin am u,
m and n being any integers.

ON THE RECENT PROGRESS OF ANALYSIS. 267

are elliptic functions, and shows that this assumption satisfies
the differential equation already mentioned. It may be asked
what is gained by the introduction of elliptic functions into a
problem of which, as we have seen, particular cases (e.g. the
transformations of the third and fifth order) can be solved -by
algebraical considerations. The answer is, that the properties
of these functions enable us to transform the assumed relation
between y and x in a manner which would otherwise be im-
practicable. It is conceivable that any particular case might be
solved by mere algebra, l^ut it does not seem possible to dis-
cover in this way a general theorem for transformations of all
orders, and practically the labour of obtaining the formula
for the transformation of any high order would be intolerable.

Having proved the theorem for transformation in nearly the
same manner as he had already done, M. Jacobi developes the
demonstration which, as we have said, Legendre hinted at in
No. 130 of Schumacher's Journal.

He then proceeds to consider the various transformations of
any given order. We have seen that the modular equation for
those of the third order rises to the fourth degree, that is to say,
for a given value of the modulus of the original integral four
new moduli exist, corresponding to four new integrals, into
which the given one may be transformed. These four trans-
formations are all included in the general formula for the third
order; but it is to be remarked that in general only two of
the roots of the modular equation are real. Thus there are
two real transformations and no more. The same thing is true,
mutatis mutandis, of the transformations of any prime order (to
which M. Jacobi's attention is chiefly directed), that is to say,
there will be n + 1 transformations of the nth. order, n — I of
which are imaginary. The two real transformations are called
the first and the second ; the second is sometimes called the
impossible transformation, because it presents itself in an ima-
ginary form*. Of the formulae connected with these two trans-
formations M. Jacobi gives copious tables.

* Mr Bronwin, in the Cambridge Mathematical Journal and in the Pliil. Mag.,
has made some objections to this transformation ; but from a correspondence which
I have recently had with him, I believe I am justified in stating that he does not
object either to M. Jacobi's result or to the logical correctness of his reasoning,
but only to the form in which the result is exhibited.

268 ON THE RECENT PROGRESS OF ANALYSIS.

He next shows, in a very remarkable manner, that, cor-
responding to a transformation in which we pass from a modulus
Jc to a modulus X, there exists another, whose formulae are
derivable from those of the former, in which we pass from a
modulus Vl — kz to a modulus Vl — A/2, or which connects
moduli complementary to \ and Jc. The latter is accordingly
called, with reference to the former, the complementary trans-
formation. The first real transformation of k corresponds to
the second real transformation Vl — k\ and vice versa.

The next theorem which M. Jacobi demonstrates is not less
remarkable. It is that the combination of the first and second
real transformations gives a formula for the multiplication of the
original integral, or, in other words, that the modulus of the
integral which results from this double transformation is the
same as that of the original integral, so that the two integrals
differ only in their amplitudes. Of this theorem he had in the
earlier part of the work proved some particular cases*.

After fully developing this part of the subject, he next
treats of the nature of the modular equation, and shows that
it possesses several remarkable properties. One is, that all
modular equations, of whatever order, are particular integrals
of a differential equation of the third order, of which the
general integral .can be expressed by means of elliptic tran-
scendents.

16. We now enter on the second great division of M.
Jacobi's researches, the evolution of elliptic functions.

* It may be shown that if we pass from k to X by the first transformation, we
can pass from A/I - X2 to A/I -&2 also by the first transformation. Also, as has
been said, we derive from the transformation {k to X} a transformation {^/i -k*
to v/i-X2}, and similarly from {*/! -X2 to A/I -&2} a transformation {X to k}.
The first and last of these transformations correspond respectively to the diffe-

rential equations

dy _ i dx

Hence, combining these equations and integrating,

and it may also be shown that TT-TT, is an integer.

ON THE RECENT PROGRESS OF ANALYSIS. 269

The evolution of elliptic functions into continued products
with an infinite number of factors presents itself as the limit
towards which M. Jacobi's theorem for the transformation of the
nth order tends as n increases sine limite. It is for this reason
that we may look on the problem of transformation as the lead-
ing idea in M. Jacobi's researches.

We may in some degree illustrate these evolutions by a
reference to circular functions. A sine- is, as we know, an
elliptic function whose modulus is zero. Now if k is zero, X is
also zero. Thus if we apply a formula of transformation to a
sine, we shall be led to another sine either of the same or of
a multiple arc. Accordingly the first real transformation de-
generates in the case in question into the known formula for
the sine of a multiple arc ; while the second, leading us merely
to the sine of the same arc, becomes illusory. Thus in the case
of a sine, transformation is merely multiplication; but from
the formula for multiplication, viz.

we at once deduce, by making (2m + 1) 6 = cf> and 2m + 1 in-
finite, the common formula

This then is a formula of evolution deduced from the first
real transformation. It is however only when k is zero that the
first transformation will give such a formula. In all other cases
it is, for a reason which we cannot here enter on, impossible
to derive from it a formula of this kind. M. Jacobi's formulas
are accordingly derived from the second real transformation, and
therefore are illusory when k is zero, or for the case of the
sine. There is nothing therefore strictly analogous to them in
the theory of angular sections. By means of them we express
the function sin am x in terms of sin mx, m being a certain
constant.

From the fundamental expressions in continued products, of
which there are three, many important theorems may be derived.
This part of the subject seems to admit of almost infinite
increase, and it is difficult to give any general view of it.

270 ON THE RECENT PROGRESS OF ANALYSIS.

I may, however, mention a remarkable transcendental function
of the modulus k which is usually denoted by ^, and which
occurs perpetually in this part of the theory of elliptic func-
tions. If for the moment we denote this function by Fk, so
that q = Fk, then if for k we write 1en , which we suppose to re-
present the modulus of the first real transformation of the nth
order, we find that f = Fkn9 so that if qn is the same function
of kn that q is of k,

<ln = 2n-

This singular property, and others of an analogous character,
are of great use in establishing various formulas*.

Before discussing the evolution of integrals of the third kind,
M. Jacobi has premised some important theorems. He proves
that the elliptic integral of the third kind, though it involves
three elements, viz. the amplitude, the modulus and the para-
meter, can yet be expressed in terms of other quantities severally
involving but two. In order to this we introduce either a new
transcendent t or a definite elliptic integral of the third kind,
whose amplitude is a certain function of its modulus and para-
meter. It is almost impossible to tabulate the values of a
function of three elements, on account of the enormous bulk
of a table with triple entry ; we therefore see the importance of
the step thus made. M. Jacobi announced this discovery as
generally true of elliptic integrals of the third kind, but his
demonstration applies to that subdivision already mentioned,
which was designated by Legendre * Fonctions du troisieme
ordre a parametre logarithmique,' and not to functions ' a para-
metre circulairej.' It is probable that this limitation was in

* A method of calculating elliptic integrals by means of q was suggested by
Legendre. Vide Verhulst, p. 252, and M. Jacobi in Crelle.

1* This transcendent is denoted by T, and is defined by the equation

-/

where E (c0) is the elliptic integral of the second kind. If we introduce Ihe
inverse notation, and make 0 = am u, we can readily establish the following result,

sin2 am udu2.

T= * w3 - c2 If si
is the logarithm

+ In the former species (r +n) f i +- ) is negative, and in the latter positive

The function T, which is the logarithm of ft (vide infra, p. 288), has many re-
markable properties.

ON THE RECENT PROGRESS OF ANALYSIS. 271

M. Jacobi's mind, but he does not seem to have expressed
it. Further on, in the Fundamenta Nova, we find another
mode of expressing integrals of the third kind in terms of
functions of two elements, but this method also applies only to
' fonctions du troisieme ordre a parametre logarithmique,' the
two methods being in fact closely allied.

Legendre appreciated the importance of this discovery of
M. Jacobi. He speaks of it in a letter to Abel, as a ' decou-
verte majeure,' but adds that his attempts to extend M. Jacobi's
demonstration to the other ^class of integrals of the third kind
had been unsuccessful. The same remarks occur in his second
supplement (Traite des Fonct. Ell. ill. p. 141). The distinction
thus made between the two classes of integrals of the third kind
appeared to Legendre sufficient to make it desirable to recognise
in all four classes of elliptic integrals, so as to make the division
between the two species of the third class coordinate with that
between either and the first or second. Legendre says explicitly
that M. Jacobi had announced, in making known his discovery,
that it applied to functions ' a parametre circulaire.' This how-
ever possibly arose from some misconception of M. Jacobi's
meaning. Dr Gudermann, in the fourteenth volume of Crelle's
Journal, has given it as his opinion that the circular species of
integrals of the third kind does not admit of the reduction in
question; and remarks, that it occurs much more frequently
than the other species in the applications of mathematics to
natural philosophy.

After having discussed at some length, and by new methods,
the properties of elliptic integrals of the third kind, M. Jacobi
concludes his work by investigating the nature of two new
transcendents which present themselves in immediate connexion
with the numerator and denominator of the continued product
by which sin am u is expressed. One of them however M.
Jacobi had already recognised by a distinctive symbol, in con-
sequence of its intimate connexion with the theory of integrals
of the third kind.

Such is the outline of this remarkable work : before it ap-
peared M. Jacobi gave in the third and fourth volumes of

•

(vide ante, p. 255). The specific names are derived from the circumstance that for
the former the fundamental formula of addition involves a logarithm, for the
latter a circular arc.

272 ON THE RECENT PROGRESS OF ANALYSIS.

Crelle's Journal (in. pp. 192, 303, 403, IV. p. 185) notices, mostly
without demonstrations, of the progress of his researches. Al-
most everything in the first and second of these notices is
found in the Fundamenta. In the third we find a remark-
able algebraical formula for the multiplication of the elliptic
integral of the first kind. The fourth and last relates to ul-
terior investigations, which it was the intention of the author
to develope in a second part of his work. It contains an in-
dication of a method of transformation depending on a partial
differential equation*; values of the elliptic functions of multiple
arguments ; a method of transforming integrals of the second
and third kinds ; a most important simplification of the method
of Abel for the division of any integral of the first kind, &c.
Of this simplification he had already given some idea in a note
in the preceding volume of the same Journal, p. 86.

17. It may not be improper in this place to observe, that
in 1818, and thus in the interval between Legendre's first and
second systematic works on the theory of elliptic functions,
M. Gauss published the tract entitled Determinatio Attrac-
tionis, &c. The illustrious author begins by remarking that
the secular inequalities due to the action of one planet on
another are the same as if the mass of the disturbing planet
were diffused according to a certain law along its orbit, so
that the latter becomes an elliptic ring of variable but infini-
tesimal thickness. The problem then presents itself of deter-
mining the attraction exerted by such a ring on any external
point. In the solution of this problem M. Gauss arrives at two
definite integrals; they can readily "be reduced to elliptic in-
tegrals of the first and second kinds. For the evaluation of
the integrals to which he reduces those of his problem, M. Gauss
gives a method of successive transformation, analogous in some
measure to that of Lagrange. But the transformation of which
he makes use is a rational one, and is in fact the rational
transformation of the second order. The discovery of this
transformation appears therefore to be due to M. Gauss. He
has remarked, though merely in passing, that his method is
applicable to the indefinite as well as to the definite integral.

* Mr Cayley, to whose kindness I have been, while engaged on the present
report, greatly indebted, has communicated to me a demonstration of the truth
of this equation.

ON THE REGENT PROGRESS OF ANALYSIS. 273

The rational transformation in question leads to a continually
increasing series of moduli, or is, to use an expression of M.
Jacobi, a transformation ' minoris in majorem.' The law con-
necting two consecutive moduli is the same as in Lagrange's,
which is, as we have seen, an irrational transformation ; so
that M. Gauss's method does not afford us a new scale of moduli.
Nevertheless, as no rational transformation had I believe been
noticed when his tract appeared*, his method is, in a historical
point of view, of considerable interest.

18. In the second volume of Crelle's Journal, p. 101, we
find Abel's first memoir on elliptic functions. It was published
in the spring of 1827, and therefore before M. Jacobi' s announce-
ment in No. 123 of Schumacher's Journal. But it contains
nothing which interferes with M. Jacobi's discovery of the
general theory of transformation. Abel's researches on this part
of the subject appeared in the third volume of Crelle's Journal,
p. 160. This second communication is dated, as we are in-
formed by an editorial note, the 12th of February, 1828, and
though it is announced as a continuation of the former memoir,
it is yet in effect distinct from it, as its contents are not
mentioned in the general summary prefixed to the first com-
munication.

These details may not be without interest, though it is not
often that questions of priority deserve the importance sometimes
given to them. There is no doubt that Abel's researches were
wholly independent of those of M. Jacobi ; and though the co-
incidence of some of their results is therefore interesting, yet
the general view which they respectively took of the theory of
elliptic functions is essentially different, as different as the style
and manner of their writings.

With M. Jacobi the problem of transformation occupied the
first place ; with Abel that of the division of elliptic integrals.
Both introduced a notation inverse to that which had previously
been used, and as an immediate consequence recognised the double
periodicity of elliptic functions. Expressions of these functions
in continued products and series were given by both, but those
of Abel were deduced by considering the limiting case of the

* The fundamental formula of his transformation is incidentally mentioned in
Legendre's second work (TraiM des Fonct. i. 61).

18

274 ON THE RECENT PROGRESS OF ANALYSIS.

multiplication of elliptic integrals, those of M. Jacobi, as we
have seen, from the limiting case of their transformation. Hence
Abel's fundamental expressions depend on doubly infinite con-
tinued products, corresponding to the double periodicity of elliptic
functions. On the other hand, M. Jacobi's continued products
are all singly infinite.

Other differences might of course be pointed out, but the
most remarkable is that which we find in the character and
style of their writings. Nothing can be more distinct. In
M. Jacobi's we meet perpetually with the traces of patient and
philosophical induction ; we observe a frequent reference to par-
ticular cases and a most just and accurate perception of analogy.
Abel's are distinguished by great facility of manner, which
seems to result from his power of bringing different classes of
mathematical ideas into relation with each other, and by the
scientific character of his method. We meet in his works with
nothing tentative, with but little even that seems like artifice.
He delights in setting out with the most general conception of
a problem, and in introducing successively the various conditions
and limitations which it may require. The principle which he
has laid down in a remarkable passage of an unfinished essay
on equations seems always to have guided him — that a question
should be so stated that it may be possible to answer it. When
so stated it contains, he remarks, the germ of its solution*.

I do not presume to compare the merits of these two mathe-
maticians. The writings of both are admirable, and may serve
to show that if ever the modern method of analysis seems to be
an €fjL7T6ipLa rather than a re^vrj, it does so, either because it
has not been rightly used, or because it is not duly understood.

To obtain a general view of Abel's writings it may be re-
marked, that his earliest researches related to the theory of equa-
tions. Of the ideas with which he was then conversant he has
made two principal applications. The one is to the comparison

• For instance, Is it possible to trisect an angle by the rule and compass ?
The question thus stated leads us to consider the general character of all problems
soluble by the methods of elementary geometry ; and following the suggestion thus
given, we find that it is to be answered in the negative. But if the last clause
be omitted or neglected, we can only proceed, as many persons have done,
tentatively, i.e. by attempting actually to solve the problem. If we fail, the
question remains unanswered; if we succeed, we do answer it, but as it were
only by accident.

ON THE RECENT PROGRESS OF ANALYSIS. 275

of transcendents in the manner already described ; the other to
the solution of the equations presented by the problem of the
division of elliptic integrals. The second of these applications
is contained in the memoir published in the second volume of
Crelle's Journal.

He begins by introducing an inverse notation <£> (u) corre-
sponding to the function denoted in the Fundamenta Nova
by sin am M, while f(u) and F(u) correspond respectively to
cos am u and A am u. This notation has the defect of appro-
priating three symbols which we cannot well spare. On the
other hand it is certainly more concise than M. Jacobi's.

He then verifies the fundamental formulae for the addition of
the new functions, and goes on to show that they are doubly
periodic*. He next considers the expressions of <£wa, &c. in
<£>a, &c., and proceeds to prove the important proposition that
the equation of the problem of the division of elliptic integrals
of the first kind is always algebraically soluble.

In order to illustrate this, which is one of the most remark-
able theorems in the whole subject, it may be observed, that as
any circular function of a multiple arc can be algebraically ex-
pressed in terms of circular functions of the simple arc, so may
(pna^fna, Fno. be algebraically expressed by means of ^a,^, Fa..

Conversely, as the determination (to take a particular
function) of sin a in terms of sin no. requires the solution of
an algebraical equation, so does that of <j£>a in terms of <£wa. The
equation which presents itself in the former case is, as we know,
of the nth or of the (2w)th degree as n is odd or even. But the
equation for determining <£a rises to the (n2)ih degree in the
former case, and in the latter to the (2w*)th. We may however
confine ourselves to the case in which n is a prime number;

* The formulae in question differ from those already given, only because Abel's
form of the elliptic integral is I . , which becomes the same as

;V('-<**) ('+**)

Legendre's on making e2= - i. The double periodicity of the functions is ex-
pressed by the formula

00=0 {(-i

with similar formulae for /and F. The quantities m and n are integral, and
f« dx p

W=2 I — ? •57=2 I — •=

18—2

276 ON THE RECENT PROGRESS OF ANALYSIS.

since if it be composite the argument of the circular or elliptic
function may first be divided by one of the factors of n, and the
result thus got by another, and so on. Thus setting aside the
particular case of n = 2, we shall have to consider, in order to
determine sin a or </>a, an algebraical equation of the nth or (n*)th
degree respectively.

In consequence of the periodicity of sin a, the roots of the
equation in sin a admit of being expressed in a transcendental

form; they are all included in the formula sin (a + -f—\ , in

which p is integral, and which therefore admits only n different
values.

But elliptic functions are doubly periodic, and therefore the
roots of the equation in <£a are expressible by a formula
analogous to the one just written, but which involves two in-
determinate integers corresponding to the two periodicities of the
function, just as p does to the single periodicity 2?r. Giving
all possible values to these integers, we get n2 different values
for the formula.

The question now is, how are we to pass from the transcen-
dental representation of these roots to their algebraical ex-
pression ? Or, in other words, how are the relations among the
roots deducible from the circumstance of their being all included
in the same formula, to be made available in effecting the solu-
tion of the algebraical equation ?

The answer to this question is to be found in the following
principle : that if ^u be such a rational function of u that

x, y, . . . z being the roots of an algebraical equation, then any
of these quantities may be expressed in terms of the coefficients
of the equation. This follows at once from the consideration
that we shall have

IJL being the number of the roots x,y,...z. For the sum within
the bracket being a rational and symmetrical function of the
roots, is necessarily expressible in the coefficients of the equation,
and the same is therefore of course true of %#, or of any of the
other quantities to which it is equal.

ON THE RECENT PROGRESS OF ANALYSIS. 277

If, therefore, by means of the relations which we know to
exist among the roots of the equation to be solved we can esta-
blish the existence of a system of such functions, ^, ^', %", &c.,
each of which retains the same value of whichever root we
suppose it to be a function ; and if by combining these functions
we can ultimately express x in terms of them, the equation is
solved, since each of these functions may be considered a known
quantity.

Such is the general idea of Abel's method of solution. The
•principle on which it depends, namely, the expressibility of any
unchangeable function %, is* one which is frequently met with
in investigations similar to that of which we are speaking.
M. Gauss's solution of the binomial equation is founded upon it.

I have already remarked that an important simplification
of Abel's process was given by M. Jacobi. The result which
M. Jacobi has stated without demonstration may be proved by
means of a theorem established by Abel in the fourth volume
of Crelle's Journal, p. 194.

M. Jacobi shows the existence of a system of n* functions
^, %', &c., by combining which we can immediately express the
values of the roots. In the last of his Notices on elliptic
functions we find, as has been said, the explicit determination of
all the roots. The formula given for this purpose is, like the
former, undemonstrated, and I do not know whether any demon-
stration of it has as yet been published; but from a note of
M. Liouville, in a recent volume of the Comptes Hendus, we
find that both he and M. Hermite have succeeded in proving it.

But in whatever manner the solution is effected it will
always involve certain transcendental quantities, which are intro-
duced in the expressions of the relation subsisting between the
different roots. The solution can therefore be looked on as
complete, only if we consider these to be known quantities.
They are the roots of a particular case of the equation to be
solved. They relate to the division of what are called the com-
plete integrals. We may therefore say that the general case is
reduced to this particular one. But the latter is not, except
under certain circumstances, soluble, though the solution of the
equation on which it depends can be reduced to the solution of
certain other equations of lower degrees.

But for an infinity of particular values of the modulus, the

278 ON THE RECENT PROGRESS OF ANALYSIS.

case in question is soluble by a method closely analogous to that
used by M. Gauss for the solution of binomial equations. Thus
for all such values the problem of the division of elliptic inte-
grals is completely solved.

The most remarkable of these cases corresponds to the
geometrical problem of the division of the perimeter of the
lemniscate. Abel discovered that this division can always be
effected by means of radicals, and further, that it can be con-
structed by the rule and compass in the same cases (that is for
the same values of the divisor) as the division of the circum-
ference of a circle. Of this discovery we find Abel writing to
M. Holmboe, " Ah qu'il est magnifique! tu verras*."

In order to form an idea of the nature of the difficulty which
disappears in the case of which we are speaking, let us suppose
that we have to solve the algebraical equation which is repre-
sented by the transcendental one <£ (30) = 0, in the same manner
as the equation 4#3 — 3x = 0 is represented by sin (36) = 0.

The roots of 4o?s — 3x = 0, are, setting aside zero,

. 2?r . 4?r
smT, sin — .

Those of the former algebraical equation, which, as we know, is
of the ninth degree, are, beside zero,

2&) 4o)

2m*

~T' T»

2 (w 4- m") 4 (to + m')

+ 2m') , 4 (a) +
-- ' * --

where * = V— 1.

* It is right to mention that M. Libri has disputed Abel's title to the theory
of the division of the lemniscate. I shall, however, not enter on the merits of the
controversy which arose on this point between him and M. Liouville. The reader
will find it in the seventeenth volume of the Comptes Rendus. It appears that
M. Gauss had himself recognised the applicability of his method to the equation
arising out of the problem of the division of the perimeter of the lemniscate (vide
Eecherches Arithmetiques, § vii. p. 419. I quote from the translation published
at Paris, in 1809).

ON THE RECENT PROGRESS OF ANALYSIS. 279

To satisfy ourselves that these are the roots required, we
observe that </> (mw 4- nisi) = 0 for all integral values of m and n.

Hence the general form of the roots of our equation is ^ — - ;

o

but it will be found that if we give any values not included in
the above table to m and ft, the resulting expression can be re-
duced to one or other of the forms we have specified in virtue of
the formula £ (0) =<£{(- l)m+" 0 + may + nvri}. E. g. The non-

,-,-1 ,-, , , 5co + 2m . - . , 4 (co + iifi)

tabulated root <£ -- - -- is equal to our sixth root (j> — • — - -- -,

since the sum of their arguments is 3<w + 2m, and the sum of
3 and 2 is an odd number.

On considering our table, we observe that it consists of 3 + 1
horizontal rows, each containing 3 — 1 terms, and that the argu-
ments of the terms in each row are connected by a simple
relation ; that of the second being double that of the first. If
we were to replace 3 by any odd number p, we should get an
equation of the p2 degree, whose roots, setting aside zero, might
similarly be arranged in p + 1 rows, each of p — 1 terms, the
arguments of the terms in each row being as 1, 2, 3, &c.

Moreover, sin — is rationally expressible in sin — , and

generally sin is so in sin — —- , n and p being any in-

tegers we please. So too are all the terms in each horizontal
row of our table, whether for the particular case we have written
down, or for that of any odd number, rationally expressible in
the first term.

Hence it may be shown that when the divisor 2n + l is a
prime number, an equation whose roots were the terms in any
horizontal row could be solved algebraically, by a method
essentially the same as that of Gauss, just as we can solve the

o .

equation the type of whose roots is sin — — — - . But to construct

J ^ 271 + 1

this equation, i. e. to determine its coefficients, requires the so-
lution of an equation of the same degree as the number of
horizontal rows, i. e. of the degree 2n + 2. And this equation
is in general insoluble. The difficulty we here encounter may
be expressed in general language, by saying that although we

280 ON THE RECENT PROGRESS OF ANALYSIS.

can pass from one root to another along each horizontal row,
yet we cannot pass from row to row.

Our table, however, has the remarkable property, that sup-
posing, as we may always do, 2n + 1 to be a prime number, all
the roots are rationally expressible in terms of any two not
lying in the same row. This depends on a property of the
function </>, which it is very easy to demonstrate, and it is inti-
mately connected with the relations which exist among the
terms of the same row.

If, then, which is the case for an infinite variety of values
of the modulus, we can express any root rationally in terms of

another of a different row, say in 0 - - - , all the roots become

Afl ~}~ i.

rational in terms of <f> - -- - . Moreover, it appears that not

only are the roots all expressible in one, but they are so in such
a manner that the functional dependencies among them fulfil
a certain simple condition, which, as Abel shows in a separate
memoir (Crelle, IV. p. 131 ; or Abel's Works, I. p. 114), renders
every equation, all whose roots are rationally expressible in
terms of one, algebraically soluble.

To take the simplest case, the arc of the lemniscate may be

f dx
represented by the integral I If <f> be the function in-

verse to this integral, we have the simple relation between roots

f TfV JL ., , . . . .

of different rows, 9 - — = tq> — - - , &> being in this case
r 2n + 1 r 2n + 1

equal to OT.

To apply what has been said to the solution of the general
equation for determining <£a in terms of <£ (2n + 1) a, it is suf-
ficient to remark that the transcendents introduced in consider-
ing the relations among the roots of this equation, are simply

(f> — — - and 0 — — , or at least may be algebraically ex-

2,11 -j- 1 271 -f- 1

pressed in terms of these two quantities.

The remainder of the first memoir contains developments of
the functions <£,/, and Fin doubly and. singly infinite continued
products and series. They are derived from the expressions

of 0a, &c. in terms of </> - , &c., by supposing n to increase sine

ON THE RECENT PROGRESS OF ANALYSIS. 281

limite, and are therefore analogous to the expression of sin $ in
terms of <£ which we have already mentioned.

The second contains the development of what had already
been pointed out with respect to the lemniscate, so far as relates
to the division of its perimeter by any prime number of the form
4m + 1. In an interesting note which M. Liouville communicated
to the Institute in 1844, and which is published in the eighth
volume of his Journal, p. 507, he has proved generally that
the division of the perimeter of this curve can always be
effected whether the divisor be a composite or prime number,
real or complex (that is, 01 the form p + V— q, p and q being
integers). In order to do this, it was only requisite to follow
m . m . , the reasoning by which Abel has shown that the equa-
tion which presents itself in the problem of the division of
the circumference of the circle is always resoluble. Thus, as
M. Liouville has remarked, his analysis is implicitly contained
in Abel's.

This memoir also contains Abel's theorem for the transfor-
mation of elliptic integrals of the first kind. It is equivalent
to that of M. Jacobi ; nor is the demonstration, though presented
in quite a different form, altogether unlike M. Jacobi's.

Abel begins by considering the sum of a certain series of <f>
functions whose arguments are in arithmetical progression. He
shows that the sum of this series is a rational function of its first
term. If we call this sum (multiplied by a certain constant) y,
and the first term #, then y is such a function of x as to satisfy
the differential equation already mentioned, viz.

d dx A

or rather an equation of equivalent form. In fact y is m . m . the
same function of x that it is in M. Jacobi's theorem. Thus the
sum of the series of elliptic functions is itself, when multiplied
by a constant, a new elliptic function, having a new modulus,
and whose argument bears a constant ratio to that of the first
term of the series. It appears also that for the sum of the
elliptic functions we may, duly altering the constant factor, sub-
stitute their continued product. Thus, beside the algebraical
expression of y, there are two transcendental expressions of it,
both of which are given by M. Jacobi in the Fundamenta

282 ON THE RECENT PROGRESS OF ANALYSIS.

Nova. At the close of the memoir Abel compares his result
with the one in Schumacher's Journal, No. 123, and mentions
that he had not met with the latter until his own paper was
terminated.

19. In the 138th number of this journal, Abel resumed the
problem of transformation, and treated it in a more general and
direct manner than had yet been done. This memoir appeared
in June 1828. M. Jacobi, in a letter to Legendre, has spoken in
the highest terms of Abel's demonstration of the formulae of
transformation : he says, " Elle est au-dessus de mes eloges,
comme elle est au-dessus de mes travaux." An addition to this
memoir, establishing the real transformations by an independent
method, appeared in Number 148 of the same journal. These
two papers are printed consecutively in the first volume of
Abel's Works, pp. 253, 275.

In the first of these two remarkable essays, Abel makes use
of the periodicity of the function $0, or, as he here denotes it,
X#, to determine h priori what rational function of x, y must be
in order that the differential equation

=.g

may be satisfied. [I have altered his notation for the sake of
uniformity.] Let ^x be the function sought, then considering
y = tyx as an equation determining x in terms of y, he shows
that certain relations necessarily exist among its roots. Let \Q
be one of them and \& another, it will readily be seen that we
may put

dO' = d0,
since each is equal to

Hence & = 6 + a,

a being the constant of integration, or, which is the same thing,
being independent of y. Hence \0 being one root, every other
root is necessarily of the form \(0 + a). Again, we see from
hence that

which is to be true for all values of 6, and which therefore

ON THE RECENT PROGRESS OF ANALYSIS. 283

implies the existence of a series of equations, of which the
type is

where k is an integer. Hence X (6 + ka) is a root, whatever in-
tegral value we may give to k. But the equation y — tyx has
but a finite number of roots, and therefore the values of the
general expression X (6 + Tea.) must recur again and again. This
consideration throws light on the nature of the quantity a ; it
must in all cases be an aliquot part of a period (simple or com-
pound) of the function \0. ^

All the values of X (9 + lea.} got by giving different values
to Jc are roots ; but the converse is not necessarily true ; all the
roots are not necessarily included in this expression. But it is
not difficult to perceive that all the roots are included in a more
general expression, viz. X (6 +Jc^ al + &2«2 . . . &notn), and conversely,
tli at all the values of this expression are roots. The number
n is indeterminate : we may have formulae of the form y = tyx,
in which n is unity, others in which it is two, &c. ; but in all
cases a is an aliquot part of some period of X0, and It is integral.

It is easy when the roots of y — tyx are known, to express y
in terms of 0. For let

fx
tyx =~,f and F being integral functions. Then

is (yp — q being the coefficient of the highest power of x in
yFx—fx) an identically true equation; whence, to determine
y in #, we have only to assign a particular value to a?, or to com-
pare the coefficients of similar powers of it*.

This then determines the form which the function y must
necessarily be of: the question which Abel goes on to discuss is
this : Under what circumstances will a function of the form thus
determined h priori be such a function as we require? The
character of the reasoning by which this question is treated is
similar to that of the method by which Abel had, in his second
memoir on elliptic functions, verified the form which, without
assigning any reason, he had there assumed for the function y.

The second essay is singularly elegant. If <f>k denote the

* I have not noticed an ambiguity of sign at the outset of thia reasoning",
as given by Abel, as for the purposes of illustration it is immaterial.

284 ON THE RECENT PROGRESS OF ANALYSIS.

function inverse to the integral I . — , and <bc the

JV(l-O (1-&V)

corresponding function for the modulus c, then, on introducing
the inverse notation, the differential equation

dy _ dx

= a

becomes of course d& — add, with x = <$>C6 and y = <$>k&. Hence
for a given increment a of 0, that of & is aa.

Let us take the simplest case, and suppose y to be a rational
function of a?; then, as x or $C0 remains unchanged when 6 in-
creases by a period of the function (j>c , y does so too ; that is (j)kO'
remains unchanged when & increases by a times a period of <£c,
or in other words, a times a period of <£c is necessarily one of <£fc.

Suppose now Jc and c to be both real and less than unity ;
then <j)k and (f>c have each a real period, here denoted by 2cok and
2coc respectively, and each an imaginary period vrki and «rc* re-
spectively, Txk and TXC being both real. Let 6 receive first the
increment 2wc, and secondly the increment «r««, then, by what
has been said,

m, ft, p, q being certain integers. But can these two equations
subsist simultaneously? Not generally, since if we eliminate
a and equate possible and impossible parts, we get two relations
among c0c'&ea)k'GTk, which are continuous functions of the two
quantities k and c. Hence both are determinate ; and if we
wish c to remain indeterminate, we must either make m and q
equal to zero, in which case a is impossible, or, making n and^?
equal to zero, assign a real value to it. When a is real we have

(Oj, vrk
a = m — — q — ,
<oc * ™c

and hence the remarkable conclusion, that

0)k COC

- : —:: q: m,

^k ™c

m and q being integers.

» -nr here is in M. Jacobi's notation 2JT, so that <f>6 = <f> (d+ imw F n-Gn), m and
n being any integers.

ON THE RECENT PROGRESS OF ANALYSIS. 285

The commensurability of the transcendental functions -1, -

is therefore a necessary condition, in order that an integral with
modulus c can be transformed into one with modulus 7c, the
regulator a being real and c indeterminate. And it may be
shown that this condition is not only necessary but sufficient.
Similar considerations apply to the case in which a is impossible.
Simple as this view is, it leads to many consequences of great
interest. The function ^, of which we have already spoken

•cs

(p. 270), is merely- 6 w« , an& as we know for the first real trans-

nty

formation of the nth order, it becomes e~"~^~ . Hence in this
case we have ( — j =n(— j according to the general law. It

may be well to remark, that if k = c we have a — m~=m (an

integer). Hence in multiplying an integral, the multiplier must
be an integer, if y is rational in #, except for particular values
of c.

In the paper of which we are speaking Abel has applied
precisely similar considerations to the case in which x and y
are connected by any algebraical equation.

Passing over one or two shorter papers, one of which has
been already referred to at p. 276, we come to a Pr&cis of the
theory of elliptic functions, published in the fourth volume of
Crelle's Journal, p. 236. The work of which it was designed to
be an extract was never written, and the Precis itself is left
unfinished. A general summary was prefixed to it, from which
we learn that the work was to be divided into two parts. In the
first elliptic integrals are considered irrespectively of the limits
of integration, and their moduli may have any values, real or
imaginary. Abel proposes the general problem of determining
all the cases in which a linear relation may exist among elliptic
integrals and logarithmic and algebraical functions in virtue of
algebraical relations existing among the variables*.

His first step is to apply his general method for the com-
parison of transcendents to elliptic integrals, which may be

* In the assumed relation, the amplitude, or rather the sine of the amplitude
of each elliptic integal, is to be one of the variables, and not a function of one
or more of them.

286 ON THE RECENT PROGRESS OF ANALYSIS.

done by what is called Abel's theorem, in at least two different
ways: the one, that of which he now makes use; the other,
that which we have seen is applied to the case of four functions
by Legendre in his third Supplement.

He next determines the most general form of which the
integral of an algebraical differential expression of any number
of variables is capable, provided it can be expressed linearly
by elliptic integrals and logarithmic and algebraical functions.
The result at which he arrives admits of many important ap-
plications. It is, that the integral in question may be expressed
in a form in which the sine of the amplitude of each elliptic
integral and the corresponding A, and also the algebraical and
each logarithmic function are all rational functions of the varia-
bles and of the differential coefficients of the integral with re-
spect to each.

He proceeds by an interesting train of reasoning to establish
the remarkable conclusion, that the general problem which we
are considering may ultimately be reduced to that of the trans-
formation of elliptic integrals of the first kind. The problem
of this transformation is then discussed, and by a method
essentially the same as that of which he had made use in his
paper in Schumacher's Journal. The appearance however of
the two investigations is dissimilar, because no reference is made
to elliptic functions (as distinguished from elliptic integrals) in
the first part of the Pr&cis. The relations therefore which
exist among the roots of y = tyx are established by considera-
tions independent of the periodicity of elliptic functions ; though
it is not difficult to perceive that they were suggested by the
results previously obtained by means of that fundamental pro-
perty. It is shown, that if the equation y — tyx, where tfrx is a
rational function, satisfy the differential equation (A), then this
equation, considered as determining x in terms of y, is always
algebraically soluble. As the multiplication of elliptic integrals
may be considered a case of transformation (that, namely, in
which the modulus of the transformed integral remains un-
changed), this theorem may be looked on as an extension of
that which we have spoken of (p. 275) in giving an account of
Abel's first memoir on elliptic functions. The two theorems
are proved by the same kind of reasoning.

The second part of the memoir was to have related to cases

ON THE RECENT PROGRESS OF ANALYSIS. 287

in which the moduli are real and less than unity ; of this how-
ever only the summary exists. Abel proposed to introduce three
new function's, the first corresponding to that which he had
previously designated by <f)9*. He now denotes it by \0. The
second and third functions are apparently what the second and
third kind of elliptic integrals respectively become, when, in-
stead of x, we introduce the new variable 0 • x and 0 being
of course connected by the equation x = \0. The double period-
icity of the function X and its other fundamental properties
having been established, it was his intention to proceed to more
profound researches. Some of his principal results are briefly
stated. I may mention one, that all the roots of the modular
equation may be expressed rationally in terms of two of them f.

One of the last paragraphs of the summary relates to func-
tions very nearly identical with those which M. Jacobi discusses at
the close of the Fundamenta Nova, and which he has desig-
nated by the symbols H and ®.

The second volume of Abel's collected works consists of
papers not published during his life. Two or three of these
relate to elliptic functions. The longest contains a new and

very general investigation for the reduction of the general trans-

p

cendent, whose differential is of the form -j= , P being, as usual,

V R

rational and E a polynomial of the fourth degree ; together with
transformations with respect to the parameter of integrals of the
third kind.

20. Having now given some account of the revolution
which the discoveries of Abel and Jacobi produced in the
theory of elliptic functions, I shall mention some of the prin-
cipal contributions which have been made towards the further
development of the subject since the publication of the Funda-

* In the Precis Abel has adopted the canonical form of the integral of the
first kind made use of by Legendre and M. Jacobi ; so that the quantity under
the radical is (i-se2) (r-c2*2). It is worth remarking, that in his first paper
in Schumacher's Nctchricliten this quantity is (i — e2£e2) (i — c2ce2), while in the
second it is the same as in the Precis. To this form he appears latterly to
have adhered.

t It is not clear whether by roots of the modular equation we are to under-
stand the transformed moduli themselves, or their fourth roots, i. e. in M. Jacobi's
notation X or v. Vide supra, p. 263.

288 ON THE RECENT PROGRESS OF ANALYSIS.

menta Nova. In Crelle's Journal, IV. p. 371, we find a paper
by M. Jacobi, entitled ' De Functionibus Ellipticis Commentatio.'
It contains, in the first place, a development of the method of
transforming elliptic integrals of the second and third kind, and
introduces a new transcendent H, which takes the place of ®,
with which it is closely connected. M. Jacobi proves that the
numerator and denominator of the value of y, mentioned above,
and which have been denoted by U and V, satisfy a single dif-
ferential equation of the third order. The remainder of the paper
relates to the properties of H (vide ante, note, p. 270). When
this function is multiplied by a certain exponential factor it be-
comes a singly periodic function, and, which is very remarkable,
its period is equal to one of the single or composite periods of
the elliptic function inverse to the integral of the first kind.
By composite period I mean the sum of multiples of the funda-
mental periods. The exponential factor being properly deter-
mined, its product by flt is equal to © multiplied by a constant.
In considering this subject, M. Jacobi is led to introduce the
idea of conjugate periods. These are periods by the combi-
nation of which all the composite periods may be produced. It
is obvious that the fundamental periods are conjugate periods ;
and there are, as may easily be shown, an infinity of others.

In the sixth volume of the same Journal we find a second
part of the ' Commentatio.' It contains a remarkable demon-
stration of the fundamental formulas of transformation of the odd
orders founded on elementary properties of elliptic functions.

In a historical point of view a notice by M. Jacobi in the
eighth volume of Crelle (p. 413) of the third volume of Legen-
dre's Traite des Fonctions Elliptigues is interesting. It was
here, I believe, that M. Jacobi first proposed the name of
Abelian integrals for the higher transcendents, which we shall
shortly have occasion to consider. After some account of the
contents of Legendre's supplements, the first two of which con-
tain the greater part of M. Jacobi's earlier researches, he goes
on to generalise a remarkable reduction given by Legendre at
the close of his work.

21. I turn to one of the very few contributions which
English mathematicians have made to the subject of this report,
namely, to a paper by Mr Ivory, which appeared in the Phil.

ON THE RECENT PROGRESS OF ANALYSIS. 289

Trans, for 1831. His design is to give in a simple form
M. Jacobi's theorem for transformation. The demonstration
is essentially the same as that in the Fundamenta Nova.

But Mr Ivory does not set out with assuming y = -=^ , U and

F being integral functions of x, but with assuming it equal to
the continued product of a number of elliptic functions (whose
arguments are in arithmetical progression), multiplied by a con-
stant factor. This is one of M. Jacobi's transcendental expres-
sions for y, and the two assumptions are therefore perfectly
equivalent in the transformations of odd orders ; but in those of
even orders, or where the continued product consists of an even
number of factors, Mr Ivory's amounts to making y equal to an
irrational function of x. Transformations by irrational substitu-
tions, though long the only kind known (since Lagrange's be-
longs to this class), had not of late been considered in detail.
Abel indeed remarked in the beginning of the general investi-
gation contained in Schumacher's Journal (No. 138), that the
existence of an irrational transformation implied that of a rational
one leading to an integral with the same modulus as the other.
He was, therefore, in seeking for the most general modular
transformation, exempted from considering irrational substitu-
tions ; but in a historical point of view it is interesting to see
the connection between Lagrange's transformation and those
which have been more recently discovered*.

« If y = (i+&)aA/_LL^_, where 6a+c*=i,
V i - c2^3

then
dx

This is Lagrange's direct transformation. The corresponding rational transforma-
tion is

which satisfies the same differential equation as before.
. dy i+c dx

where

is satisfied by if = i + ex9 - V(i - x2) (i - c2x2),

19

290 ON THE REGENT PROGRESS OF ANALYSIS.

The question presents itself, what is the connection between
the irrational transformation (that of which Lagrange's is a
particular case) and the rational transformation of even orders ?
Perhaps the simplest answer to it (though every question of the
kind is included in the general investigations contained in Abel's
Precis) is found in a paper by M. Sanio in the fourteenth
volume of Crelle's Journal, p. 1. The aim of this paper is to
develope more fully than Mr Ivory has done the theory of trans-
formations of even orders, and particularly of the irrational
transformations, which M. Sanio considers more truly analogous
to the rational transformations of odd orders than the rational
transformations of even orders ; and also to discuss the multipli-
cation of elliptic integrals by even numbers, a subject intimately
connected with the other. We have already mentioned the ex-
istence of what are called complementary transformations, each
of which may be derived from the other by an irrational sub-
stitution, by which two new variables are introduced. In the
case of transformations of odd orders, the original transformation
and the complementary one are both rational, and are both in-
cluded in the general formula given by M. Jacobi's theorem ;
but to the rational transformation of any even order corresponds
as its complement the irrational transformation of the same order.
This remark, which, as far as I am aware, had not before been
made, sets the subject in a clear light*.

which may be called Lagrange's inverse transformation, Tc being now the same func-
tion of c, that c was before of Jc. The corresponding rational transformation is

„_(!+£)*

y i+cx*>

which is M. Gauss's, and is termed in M. Jacobi's nomenclature the rational trans-
formation of the second order. It satisfies the equation

dx

* Lagrange's transformation being

where, as before, 1c=

J + c

then we find that tf

while the differential equation becomes

dyr

" 6)

where

ON THE RECENT PROGRESS OF ANALYSIS. 291

22. In the twelfth volume of Crelle's Journal (p. 173), Dr
GuetzlafF has investigated the modular equation of transforma-
tions of the seventh order : it is, as we know from the general
theory, of the eighth degree, and presents itself in a very re-
markable form, which closely resembles that in which M. Jacobi,
at p. 68 of the Fundamenta Nova, has put the modular equa-
tion for the third order. Dr Sohncke has given, at p. 178 of
the same volume, modular equations of the eleventh, thirteenth
and seventeenth orders, none of which apparently can be reduced
to so elegant a form as those^pf the third and seventh. Possibly
the transformation of the thirty-first order might admit of a
corresponding reduction. The whole subject of modular equa-
tions is full of interest. Dr Sohncke has demonstrated his re-
sults in a subsequent volume of the Journal (xvi. 97).

In the fourteenth volume of Crelle's Journal there is a paper
by Dr Gudermann on methods of calculating and reducing in-
tegrals of the third kind. I have already quoted from this
paper the expression of the opinion of its learned author, that it
is impossible to express the value of integrals of the circular
species in terms of functions of two arguments. If this be so,
it is impossible to tabulate such integrals, and therefore our
course is to devise series more or less convenient for determining
their values when any problem, e.g. that of the motion of a
rigid body, to which Dr Gudermann especially refers, requires us
to do so. The formation of such series is accordingly the aim
of this memoir, which contains some remarkably elegant formulae;
one of which connects three integrals of the third kind with
three of the second.

In the sixteenth and seventeenth volumes of the same Jour-
nal, Dr Gudermann has given some series for the development
of elliptic integrals; and he has since published in the same
Journal a systematic treatise on the theory of modular functions
and modular integrals, these designations being used to denote
the transcendents more generally called elliptic. The point of
view from which he considers the subject has been already in-
dicated (vide supra, p. 271). In a systematic treatise there is of
course a great deal that does not profess to be original, and it is
not always easy to discover the portions which are so. Dr
Gudermann's earlier researches are embodied and developed in his
larger work ; and in some of the latter chapters (xxiu. 329, &c.)

19—2

292 ON THE RECENT PROGRESS OF ANALYSIS.

we find some interesting remarks on the forms assumed by the
general transcendent when the biquadratic polynomial in the
denominator has four real roots. Dr Gudermann points out the
existence of a species of correlation between pairs of values of
the variable.

23. The development of the elliptic function <£ in the form
of a continued product may be applied to establish formulae of
transformation. This mode of investigating such formulae was
made use of by Abel in his second paper in Schumacher's
Journal, No. 148, which we have already noticed; and a cor-
responding method is mentioned by M. Jacobi in one of the
cursory notices of his researches which he inserted in the early
volumes of Crelle's Journal. Mr Cay ley, in the Philosophical
Magazine for 1843, has pursued a similar course. Another
and very remarkable application of the same kind of develop-
ment consists in taking it as the definition of the function <£,
and deducing from hence its other properties. It has been re-
marked that the continued products of Abel and M. Jacobi
are derived from considerations which, although cognate, are
yet distinct; those of the latter being singly infinite, while
Abel's fundamental developments consist of the product of an
infinite number of factors, each of which in its turn consists of
an infinite number of simple factors. Thus we can have two
very dissimilar definitions of the function <£ by means of con-
tinued products. M. Cauchy, who has investigated the theory
of what he has termed reciprocal factorials, that is, of continued
products of the form

{(1 +x) (1+te) } {(14- faf) (1 + fx*) }

which is immediately connected with M. Jacobi's developments,
has accordingly set out from the singly infinite system of pro-
ducts, and has deduced from hence the fundamental properties of
elliptic functions (Comptes Rendus, XVII. p. 825).

Mr Cayley, on the other hand, has made use of Abel's doubly
infinite products, and has shown that the functions defined by
means of them satisfy the fundamental formulae mentioned in
the note at page 266, which, as these equations furnish a sufficient
definition of the elliptic functions, is equivalent to showing that
the continued products are in reality elliptic functions. He has

ON THE RECENT PROGRESS OF ANALYSIS. 293

therefore effected for Abel's developments that which M. Cauchy
had done for M. Jacobi's. Mr Cayley's paper appeared in the
fourth volume of the Cambridge Mathematical Journal, but he
has since published a translation of it with modifications in
the tenth volume of Liouville's. On the same subject we
may mention a paper by M. Eisenstein (Crelle's Journal,
XXVII. 285).

24. M. Liouville has in several memoirs investigated the
conditions under which the integral of an algebraical function
can be expressed in an algebraical, or, more generally, in a finite
form. This investigation is of the same character as that which
occurs in the beginning of Abel's last published memoir on
elliptic functions (vide supra, p. 286). But while Abel's re-
searches are more general than M. Liouville's, the latter has
arrived at a result more fundamental, if such an expression may
be used, than any of which Abel has left a demonstration.

He has shown that if y be an algebraical function of x, such

that \ydx may be expressed as an explicit finite function of x,
we must have

\ydx = t + A log u + B log v + . . . + Clog w,

A, B, ... C being constant, and t, u, v, ...w algebraical functions
of x. The theorem established by Abel in the memoir refer-
red to includes as a particular case the following proposition,
that if

+ ... + Clogw,

then t, u, v, . . . w may all be reduced to rational functions of
x and y.

Combining these two results, it appears that if \ydx be ex-
pressible as an explicit finite function of x, its expression must
be of the form

t + A logu + JBlog v+ ... 4- Clogw,

where t, u, v, ... w are rational functions of x and y, or rather
that its expression must be reducible to this form*.

* An equivalent theorem is stated by Abel in his letter to Legendre for im-
plicit as well as explicit functions (Crelle's Journal, vi.).

294 ON THE RECENT PROGRESS OF ANALYSIS.

After establishing these results in the memoir (that on ellip-
tic transcendents of the first and second kinds), which will be
found in the twenty-third cahier of the Journal de TEcole

Polytechnique, p. 37, M. Liouville supposes y to be of the

p
form -jjj , where P and R are integral polynomials, and hence

f P

deduces the general form in which the integral \-pp dx may

J Y ±i

necessarily be put, provided it admit of expression as an ex-
plicit finite function of x.

f P
He shows from hence that if I —p^ dx cannot be expressed

by an algebraical function of x, it cannot be expressed by an
explicit finite function of it, and finally demonstrates that an
elliptic integral, either of the first or second kind, is not ex-
pressible as an explicit finite function of its variable.

In a previous memoir inserted in the preceding cahier, M.
Liouville proved the simpler proposition, that elliptic integrals
of the first and second kinds are not expressible as explicit
algebraical functions of their variable (Journal de VEcole Poly-
technique, t. xiv. p. 137). His attention appears to have been
directed to this class of researches by a passage of Laplace's
* Theory of Probabilities,' in which the illustrious author, after
indicating the fundamental, and, so to speak, ineffaceable dis-
tinctions between different classes of functions, states that he

had succeeded in showing that the integral I—

JV

- s

not expressible as a finite function, explicit or implicit, of x.
Laplace however did not publish his demonstration.

In his own Journal (v. 34 and 441), M. Liouville has since
shown that elliptic integrals of the first and second kinds, con-
sidered as functions of the modulus, cannot be expressed in finite
terms.

25. In the eighteenth volume of the Comptes Rendus
(Liouville's Journal, ix. 353), we find in a communication from
M. Hermite, of which we shall shortly have occasion to speak
more fully, a remarkable demonstration of Jacobi's theorem. It
is stated for the case of the first real transformation, but might
of course be rendered general. This demonstration depends

ON THE RECENT PROGRESS OF ANALYSIS. 295

essentially on the principle already mentioned (p. 276), that any
rational function of a root of an algebraical equation which has
the same value for every root of the equation is rationally ex-
pressible in the coefficients. The equation to which this princi-
ple is applied is that to which we have so often referred, viz.

y = -y , considered as an equation to determine x in terms of ?/,

and by means of it, M. Hermite shows at once that a certain
rational function of x is also a rational function of ?/, the form of
which is subsequently determined.

M. Hermite goes on to prove other theorems relating to ellip-
tic functions.

As elliptic functions are doubly periodic, we may determine
certain of their properties by considering to what conditions
doubly periodic functions must be subject. This view is
mentioned by M. Liouville in a verbal communication to the
Institute (Comptes Rendus, t. xix.). He states that he had
found that a doubly periodic function which is not an absolute
constant and has but one value for each value of its variable
must be, for certain values of it, infinite ; that from hence the
known properties of elliptic functions are easily deduced; and
that by means of this principle he had succeeded in proving the
expressions of the roots of the equation for the division of an
elliptic integral of the first kind, which M. Jacobi had given
without demonstration in Crelle's Journal*. I am not aware
that any development of M. Liouville's view has as yet ap-
peared.

In the recent numbers of Crelle's Journal there are many
papers by M. Eisenstein on different points in the theory of
elliptic functions. Among these I may mention one which con-
tains a very ingenious proof of the fundamental formula for
the addition of two functions, derived from the differential equa-
tion of the second order, which each function must satisfy.

Other contributions to the theory of elliptic functions might
be mentioned ; some of these, not here noticed, are referred to in
the index which will be found at the end of this report. But in
general it may be remarked that the form which the subject has

* M. Liouville has mentioned that M. Hermite had demonstrated the formulae
in question in a different manner.

296 ON THE RECENT PROGRESS OF ANALYSIS.

assumed, in consequence of the discoveries of Abel and M.
Jacobi, is that which it will probably always retain, however
our knowledge of particular parts of it may increase* What has
since been effected relates for the most part to matters of detail,
of which, however important they may be, it is difficult or im-
possible to give an intelligible account.

26. It does not fall within the design of this report to con-
sider the various applications which have been made of the
theory of elliptic functions ; but I shall briefly mention some of
the geometrical interpretations, if the expression may so be used,
which mathematicians have given to the analytical results of the
theory.

The lemniscate has, as is well known, the property that its
arcs may be represented by an elliptic integral of the first kind,

the modulus of which is —= . The problem of the division of its

V2

perimeter is accordingly a geometrical interpretation of that of
the division of the complete integral, and was considered by
mathematicians at a time when the theory of elliptic functions
was almost wholly undeveloped. Besides Fagnani, whose re-
searches with respect to the lemniscate have been already noticed,
we may mention those of Euler, who however did not succeed in
obtaining a solution of the problem. Legendre, who seems to
have attached considerable importance to geometrical illustra-
tions of his analytical results, assigned the equation of a curve
of the sixth order, whose arcs measured from a fixed point
represent the sum of any elliptic integral of the first kind and an
algebraical expression. He showed also that an arc of the curve
might be assigned equal to the elliptic integral, but in order to
this both extremities of the arc must be considered variable, so
that in effect the integral is represented by the difference of
two arcs measured from a fixed point (Traite des Fonctions
Elliptiques, I. p. 36).

M. Serret, in a note presented to the Institute in 1843
(Liouville's Journal, viil. 145), has proved a beautiful theorem,
viz. that the sum and difference of the two unequal arcs, inter-
cepted by lines drawn from the centre of Cassini's ellipse to cut
the curve, are each equal to an elliptic integral of the first
kind, and that the moduli of the two integrals are comple-

ON THE RECENT PROGRESS OF ANALYSIS. 297

mentary. In the lemniscate, which is a case of Cassini's ellipse,
one of these arcs disappears, and the moduli of the two integrals
are equal, each being the sine of half a right angle. So that
M. Serret's theorem is an extension of the known property of
the lemniscate.

M. Serret has since considered the subject of the representa-
tion of elliptic and hyper-elliptic arcs in a very general manner.
His memoir, which was presented to the Institute and ordered to
be published in the Savans Etr angers, appears in Liouville's
Journal, X. 257. He had remarked that the rectangular co-
ordinates of the lemniscate are rationally expressible in terms of
the argument of the elliptic integral which represents the arc,
for if we assume

/- z + zs , /- z —

,

-r z i -f- z

we shall have

— =,

and if between the first two of these equations we eliminate z, we
arrive at the known equation of the lemniscate *. So that if we
state the indeterminate equation

(x, y and Z being real and rational functions of 0), the lemniscate
will afford us one solution of it ; and every other solution will
correspond to some curve whose arc is expressible by an elliptic
or hyper-elliptic integral. Of this indeterminate equation M.
Serret discusses a particular case. He succeeds in solving it
by a most ingenious method, which is applicable to the general
equation, and shows from hence that there are an infinity of
curves, the arcs of which represent elliptic integrals of the first
kind. M. Serret's researches however have not led him to a
geometrical representation by means of an algebraical curve of
any integral of the first kind, though his results are generalized
in a note appended to his memoir by M. Liouville. In order
that the curve may be algebraical, it is necessary and sufficient,

* On reducing the integral I . to the standard form of elliptic integrals,

J vi+24

we find that it is an elliptic integral of the first kind, of which the modulus is the
sine of 45°.

298 0^ THE RECENT PROGRESS OF ANALYSIS.

as M. Liouville lias remarked, that the square of the modulus of
the integral should be rational, and less than unity.

In a subsequent memoir (Liouville's Journal, x. 351) he
has very much simplified the analytical part of his researches,
and in the same Journal (x. 421) has proved some remarkable
properties of one class of what may be called elliptic curves.
In the fourth number of the Cambridge and Dublin Mathe-
matical Journal (p. 187), M. Serret has developed this part of
the subject, and has also given a general sketch of his pre-
vious papers. M. Liouville ( Comptes Eendus, xxi. 1 255, or his
Journal, x. 456) has given a very elegant investigation of an
analytical theorem established by M. Serret.

In the fourteenth volume of Crelle's Journal (p. 217), M.
Gudermann has considered the rectification of the curve called
the spherical ellipse, which is one of a class of curves formed
by the intersection of a cone of the second order with a sphere.
He has shown that its arcs represent an elliptic integral of the
third kind.

In the ninth volume of Liouville's Journal (p. 155), Mr W.
Roberts proves that a cone of the second order, whose vertex
lies on the surface of a sphere, and one of whose external axes
passes through the centre, intersects the sphere in a curve whose
arcs will, according to circumstances, represent any elliptic inte-
gral of the third kind and of the circular species ; or any elliptic
integral of the same kind and of the logarithmic species, pro-
vided the angle of the modulus is less than half a right angle ;
or (subject to the same condition) any elliptic integral of the first
kind ; or lastly, by a suitable modification, any elliptic integral
of the second kind. The cases here excepted may be avoided by
introducing known transformations. The cases in which the
arcs represent elliptic integrals of the first kind, Mr Eoberts has
previously mentioned in the eighth volume of Liouville's Journal
(p. 263). He has since given in the same Journal (x. 297), a
general investigation of the subject, in which it is supposed that
the vertex of the cone may have any position we please. M.
Verhulst has represented the three kinds of elliptic integrals by
means of sectorial areas of certain curves, and the function T by
the volume of a certain solid. It is manifest, however, that it is
incomparably easier to do this than to represent these transcend-
ents by means of the arcs of curves.

THE RECENT PROGRESS OF ANALYSIS. 299

Besides one or two other papers I may mention a tract by
the Abbe Tortolini, on the geometrical representation of elliptic
integrals of the second and third kinds. This tract, however, I
have not seen.

Lag-range long since proved (vide Theorie des Fonctions Ana-
lytiques, p. 85), that by means of a spherical triangle a geome-
trical representation of the addition of elliptic integrals of the
first kind may easily be obtained, and that hence by a series of
such triangles we are enabled to represent the multiplication as
well as the addition of thesejuitegrals.

M. Jacobi ha-s given (Crelle's Journal, ill. p. 376, or vide
Liouville's Journal, X. p. 435) a geometrical construction for the
addition and multiplication of elliptic integrals of the first kind.
It is founded on the properties of an irregular polygon inscribed
in a circle, and the sides of which touch one or more other
circles. It is to be remarked that Legendre, in giving an ac-
count 4of this construction in one of the supplements to his last
work, has only considered its application to multiplication and
not to addition, and has been followed in this respect by M. Ver-
hulst, whose treatise on elliptic functions has been already men-
tioned. In consequence of this, M. Chasles was led to believe
that until the publication of his own researches, no construction
for addition excepting that of Lagrange was known. But he has
recently (Comptes Rendus, January 1846) pointed out the error
into which he had fallen.

27. In the Transactions of the Eoyal Irish Academy (ix.
p. 151), Dr Brinkley gave a geometrical demonstration of
Fagnani's theorem with respect to elliptic arcs, and in the
sixteenth volume of the same Transactions (p. 76), we find Lan-
den's theorem proved geometrically by Professor MacCullagh.

M. Chasles has considered the subject of the comparison of
elliptic arcs by geometrical methods, and with great success.
His fundamental proposition may be said to be, that if from any
two points of an ellipse we draw two pairs of tangents to any
confocal ellipse, the difference of the two arcs of the latter respec-
tively intercepted by each pair of tangents is rectifiable. Or,
what in effect is the same thing, if we fasten a string at two
points in the circumference of an ellipse, arid suppose a ring
to move along the string, keeping it stretched, and winding it

300 ON THE RECENT PROGRESS OF ANALYSIS.

on and off the arc which lies between its two extremities, the
ring will trace out a portion of an ellipse confocal to the former.
If for the first ellipse we substitute an hyperbola confocal with
the second, the sum of the arcs will be constant. From hence
a series of theorems is deduced, remarkable not only for their
elegance, but also for the facility with which they are obtained.
They furnish constructions for the addition and multiplication
of elliptic integrals. The whole of this investigation, of which
an account is given in the Comptes Rendus (Vol. xvn. p. 838,
and Vol. xix. p. 1239), shows, like others of M. Chasles, how
much is lost in treating geometrical questions by an exclusive
adherence to what may be called the method of co-ordination.
Invaluable as this method is, it yet often introduces considera-
tions foreign to the problem to which it is applied*.

III.

28. The first outline of a detailed theory of the higher
transcendents was given by Legendre in the third supplement to
his Trait& des Fonctions Elliptiques. He proposes to classify
the transcendents comprised in the general formula

f /(«)*»_

J (x — a) V</>#

according to the degree of the polynomial (f>x, the first class being
that in which the index of this degree is three or four; the
second that in which it is five or six, and so on. The first class
therefore consists of elliptic integrals; all the others may be
designated as ultra-elliptic. This epithet, however, which was
proposed by Legendre, has not been so generally used as Jiyper-
elliptic, which was, I believe, first used by M. Jacobi. M. Jacobi,
however, has proposed to call the higher transcendents Abelian
integrals.

The principle of Legendre's classification is to be found in
the mininum number of integrals to which the sum of any

• M. Chasles has also considered the subject of spherical conies, as well as that
of the lines of curvature and shortest lines on an ellipsoid. The latter has re.
cently engaged the attention of several distinguished mathematicians— MM. Jacobi,
Joachimsthal, Liouville, Mac Cullagh and Roberts may be particularlv mentioned.

ON THE RECENT PROGRESS OF ANALYSIS. 301

number of them can be reduced. As we know, this number is
unity in the case of elliptic integrals, and by Abel's theorem we
find that it is two in the first class of the higher transcendents,
three in the next, and so on.

Following the analogy of elliptic integrals, Legendre pro-
posed to recognise three canonical forms in each class of hyper-
elliptic integrals, and thus to divide it into three orders. The
sum of any number of functions of the first kind will, when the
required conditions are satisfied, be equal to a constant ; that of
any number of the secondhand third kinds respectively will,
under similar conditions, be equal to an algebraical or logarith-
mic function.

Much the greater part of the remainder of the supplement
consists of a discussion of the particular transcendents

It contains a multitude of numerical calculations, and if the
writer's age be considered (he was then almost eighty), is a very
remarkable production. By means of the numerical calculations
he recognised, as it were empirically, the values to be assigned
in different cases to the above-mentioned constant : what these
values ought to be, he did not attempt to determine a priori.

At the close of the supplement we find a remarkable reduc-
tion of an integral, apparently of a higher order to elliptic in-
tegrals. The method employed has been generalised by M.
Jacobi, in a notice of Legendre's Supplements, inserted in the
eighth volume of Crelle's Journal (p. 413).

29. In the ninth volume of Crelle's Journal (p. 394), we
find a most important paper by M. Jacobi ( Considerationes Ge-
nerates, &c.} which may be said to have determined the direction
in which the researches of analysts in the theory of algebraical
integrals were to proceed.

The writer proposes two questions, both suggested by the
cases of trigonometrical and elliptic functions. First, as in these
cases we consider certain functions to which circular and elliptic
integrals are respectively inverse, and which are such that func-
tions of the sum of two arguments are algebraically expressible
in terms of functions of the simple arguments, what are the coi>

302 ON THE REGENT PROGRESS OF ANALYSIS.

responding functions to which the hyper-elliptic or Abelian
integrals are inverse, and how by means of them can Abel's
theorem be stated?

Secondly, as in the same cases we obtain algebraical inte-
grals of differential equations, whose variables are separated, but
which nevertheless can only be directly integrated by means of
transcendents*, what are the differential equations of which
Abel's theorem gives us algebraical integrals ? These two ques-
tions are, it is obvious, intimately connected.

M. Jacobi first takes the particular case in which the poly-
nomial under the radical is of the fifth or sixth degree. If we
call this polynomial Jf, it follows from Abel's theorem, that if

and fax =

we shall have the equations

(pa -+• (f>b = <px -
faa + fab = fax + fay + fax1 + fay\

where a and b are given as algebraical functions of the indepen-
dent quantities x, y, x1, y1.

fax + fay = v, fax1 + fay1 = v1.

Then x and y are both given as functions of u and v. We
may therefore put

and similarly,

and as (pa + <pb = u + u1

we shall have a = X (u + u1, v + v')

c\ v + v1).

o, of which the algebraical integral is

x xi-y3 +y *i-x* = C.

Each term of this differential equation is a differential of a transcendent function
sin-1^ or Bi*rly.

ON THE REGENT PROGRESS OF ANALYSIS. 303

Hence the functions X (u + u\ v + vl) and \ (u + u1,v+ vl) are
expressible as algebraical functions of

\(uv), \(uv), X(wV), \(wV).

These then are functions to which the integrals are in a cer-
tain sense inverse, and which have the same fundamental pro-
perty as circular and elliptic functions.

In the general case of Abel's theorem, we introduce (when
the degree of the polynomial is 2m or 2m — 1), m — 1 functions
analogous to X, each being a^ function of m — 1 variables. These
functions will, it may easily be shown, have the fundamental
property just pointed out for the case in which m is equal
to three.

Again, the differential equations of which Abel's theorem
gives us algebraical integrals, are, if the degree of the polynomial
X be five or six, the following :

dx dy dz _

++ '

xdx ydy zdz

+ + = 0;

and generally, if the degree of the polynomial be 2m or 2m— 1,
there are m - 1 such equations, the numerators of the last con-
taining the (m — 2)th power of the variables.

M. Jacobi concludes by suggesting as a problem the direct
integration of these differential equations, so as to obtain a proof
of Abel's theorem corresponding to that which Lagrange gave of
Euler's (vide ante, p. 241).

30. Another important paper by M. Jacobi is that which
is entitled De Functionibus duarum Varidbilium quadrupliciter
periodicis, etc. (Crelle, xiil. p. 55). It is here shown that a
periodic function of one variable cannot have two distinct real
periods. In the case of a circular function, though we have for
all values of x

sin x = sin (x + 2m?r)
= sin

m and n being any integers, yet 2m7r and 2mr do not constitute
two distinct periods, since each is merely a multiple of 2?r, which

304 ON THE RECENT PROGRESS OF ANALYSIS.

is the fundamental period of the function. But if we had for
all the values of x

f(x) =/{* + a] =
we should also have

where m and n may be any integers, positive or negative.
Hence my. + nft may, provided a and ft are incommensurable,
which is implied in their being distinct periods, be made less
any assignable quantity, so that we may put
fx=f(x+e),

where e is indefinitely small, and this manifestly is an inad-
missible result. Accordingly we see that one at least of the
periods of elliptic functions is necessarily imaginary.

Again, similar reasoning shows that in a triply periodic func-
tion, that is in one in which we have

f(x)=f{x+m(a.+ft^— l)+mf(oL+ft'>J— l)+m"(a"+/8'V— 1)}
for every value of x, m, m', m" being any integers, and in which
the three periods a + /3V— 1, &c. are distinct, we can make

/(*) =/(»+«)

by assigning suitable values to m, m, m"; e being as before less
than any assignable quantity. Hence as this result is inadmissi-
ble, it follows that there is no such thing as a triply periodic
function. Whenever therefore a function appears to have three
periods they are in reality not distinct, and so a fortiori when it
appears to have more than three. But now we come to a diffi-
culty. For M. Jacobi proceeds to show that if we consider a
function of one variable inverse to the Abelian integral

(a + ftx) dx

X being of the sixth degree in x, this function has four dis-
tinct and irreducible periods. His conclusion is that we cannot
consider the amplitude of this integral as an analytical function of
the integral itself. In the present state of our knowledge, this
conclusion, though seemingly forced on us by the impossibility
of recognising the existence of a quadruply periodic function of
one variable, is not, I think, at all satisfactory. The functional

ON THE RECENT PROGRESS OF ANALYSIS. 305

dependence, the existence of which we are obliged to deny, may-
be expressed by a differential equation of the second order ; and
therefore it would seem that the commonly received opinion that
every differential equation of two variables has a primitive, or
expresses a functional relation between its variables, must be
abandoned, unless some other mode of escaping from the difficulty
is discovered. It is probable that some simple consideration,
rather of a metaphysical than an analytical character, may here-
after enable us to form a consistent and satisfactory view of the
question, and this I believe^ I may say is the opinion of M.
Jacobi himself. The same difficulty meets us in all the Abelian
integrals : as in the case of those of Legendre's first class, namely
where X is of the fifth or sixth degree, so also generally, the in-
verse function has more than its due number of periodicities.

Abel, in a short paper in the second volume of his works,
p. 51, has in effect proved the multiple periodicity of the func-
tions which are inverse to the integrals to which his theorem
relates. The difficulty to which this gives rise did not strike
him, or was perhaps reserved for another occasion.

M. Jacobi next proves that his inverse functions of two
variables are quadruply periodic, but that quadruple periodicity
for functions of two variables is nowise inadmissible.

A difficulty however seems to present itself, which is sug-
gested by M. Eisenstein in Crelle's Journal, viz. that if for each

/* dx
value of the amplitude the integral <f>x or I -= (vide supra, p.

-' Vj£"

302) , has an infinity of magnitudes real and imaginary, and the
same is the case for (f>y, it is by no means easy to attach a defi-
nite sense to the equation u = <frx + <py, or to see how the value
of u is determined by it*.

31. Two divisions of the theory of the higher transcendents
here suggest themselves, which are apparently less intimately
connected than the corresponding divisions in the theory of

* The difficulty here mentioned may perhaps be met by saying that the value of

(j>x determined by the integral I - — is necessarily determinate, and so likewise

Jo i^jX

is that of u. That considerations connected with the conception of a function
inverse to <j>x make the latter quantity appear indeterminate is undoubtedly a
difficulty ; but it is, so to speak, a difficulty collateral to M. Jacobi's theory,_and
therefore need not prevent our accepting it.

20

306 ON THE RECENT PROGRESS OF ANALYSIS.

elliptic functions, viz. the reduction and transformation of the
integrals themselves, and the theory of the inverse functions.

But before considering these I shall give some account of what
has been done in fulfilment of the suggestion made by M. Jacobi
at the close of the Considerations Generates. Mathematicians
have succeeded in effecting the integration of the system of
differential equations to the consideration of which we are led by
Abel's theorem, and which is commonly designated by German
mathematicians as the " Jacobische system;" its existence and
its integrability having been first pointed out by M. Jacobi.

In Crelle's Journal (xxm. 354), M. Eichelot, after modifying
the form in which Lagrange's celebrated integration of the dif-
ferential equation of elliptic integrals is generally presented, ex-
tended a similar method to the system of two differential
equations which occurs when we consider the Abelian transcen-
dents of the first class. He thus obtains one algebraical integral
of the system. In the case of Lagrange's equation one integral
is all we want; but in that which M. Eichelot here discusses
we require two. Now if in the former case we replace each of
the variables by its reciprocal, we obtain a new differential equa-
tion of the same form as the original one, and integrable there-
fore in the same manner; and if in its integral we again replace
each new variable by its reciprocal, that is by the original
variable, we thus, as it is not difficult to see, get the integral of
the original equation in a different form. That the two forms
are in effect coincident may be verified a posteriori. But the
same substitutions being made in M. Eichelot' s equations, which
are of course those we have already mentioned at p. 302, the
first of them becomes similar in form to the second, and vice
versa the second to the first. Thus the system remains similar
to itself; and if in the algebraical integral we obtain of it we
again replace the new variables by their reciprocals, we fall on
a new algebraical integral of the original system ; this integral
being, which is remarkable, independent of that previously got.
Thus the system of two equations is completely integrated.
Extending his method to the general system of any number of
equations, M. Eichelot obtains for each two integrals, but of
course these are not all that we want. At the conclusion of his
memoir M. Eichelot derives from Abel's theorem the algebraical
integrals of the "Jacobische system."

ON THE RECENT PROGRESS OF ANALYSIS. 307

Though in this memoir M. Richelot only obtained by direct
integration two of the m — l algebraical integrals of the " Jaco-
bische system," yet he put the problem of its complete inte-
gration into a convenient and symmetrical form. As there are
m variables and m — 1 relations among them, we may suppose
each to be a function of an independent variable t. Lagrange,
as we know, in integrating the equation

cfo dL_Q

VZ VT '

introduced such an independent variable by the assumption

dx

which of course implied that -jf- = — V Y. This assumption is
unsymmetrical, and it is therefore difficult to see how to gene-

ralise it. But if we assume -r- = - , we shall of course have

dt x — y

J/ — - and therefore t is symmetrically related to x and
dt y-x'

y. Let Fu = 0 be an equation whose roots are x and y, then,
as we know, when u — x^ F'u~x — y, and, when u—y,
F'u = y — x, so that using an abbreviated notation

dx VT dy V?

dt~F\x) a 1 dt~ F'(yY

Nothing is easier than to generalise this result. For instance,
the " Jacobische system" of two equations is

^L , dz A
+" 0>

xdx ydy zdz

Now if Fu — 0 have x, y, z for its roots, the two preceding
equations may, in virtue of a very well-known theorem, be re-
placed by the three following,

_ _ ^

dt ~ F'x ' dt ~ Fy ' dt ~ F'z '

20—2

308 ON THE RECENT PROGRESS OF ANALYSIS.

which introduce an independent variable t, symmetrically re-
lated to x, y and z ; and so in all cases.

M. Richelot* then takes a symmetrical function of x, y, ...z,
viz. their sum, and by means of the last written equations arrives
at an integrable differential equation of the second order, the
principal variable being the said sum and the independent
variable being t. From the first integral of this equation it
is easy to eliminate the differentials, and we have thus an alge-
braical relation in x, y, ... z, from which, in the manner already
mentioned, M. Richelot deduces another. We now see that if
we could find any other symmetrical function which would lead
to an integrable equation we should get a finite relation among
the variables.

In the next volume of Crelle's Journal M. Jacobi took the
following function as his principal variable,

fju being a root of X = 0 or fx — 0 if we suppose X =fx. Call-
ing this function v, we get a simple differential equation in v and
t, and a corresponding integral of the system. Now fx = 0
has 2m or 2m — 1 roots, and we only want m — 1 integrals.
The integrals therefore which we get by making p the first,
second, &c. root of fx = 0 are not all independent.

* As M. Richelot's method of demonstrating Euler's theorem is more sym-
metrical and far more easily remembered than Lagrange's, it ought, I think, to be
introduced into all elementary works on elliptic functions. The equation to be
integrated being

,

V a + fiy + yy* + Sy* + ey*
dx

assume ,

dt y-x

Then

dt x-y

Let p = x + y. Then after a few obvious reductions

Hence

- * a +

the algebraical integral sought. It may easily be expressed in other forms.

ON THE RECENT PROGRESS OF ANALYSIS. 309

In the twenty-fifth volume of Crelle's Journal M. Eichelot
resumed the subject of his former paper, and discussed it in a
very interesting memoir. This fundamental or principal result
may be said to be -a generalisation of M. Jacobi's. It is that in
the function

fjb may have any value whatever. The resulting differential
equation, though rather more complicated than when, with M.
Jacobi, we suppose //, a root of fx — 0, is still very readily in-
tegrable. We have thus an indefinite number of algebraical in-
tegrals, since the quantity /JL is arbitrary, but of course not more
than m — 1 of them are independent.

In the same volume of Crelle's Journal, p. 178, there is a
curious paper by Dr Hsedenkamp, in which the algebraical in-
tegrals of Jacobi's system are for the case of a polynomial of the
fifth degree under the radical deduced from geometrical con-
siderations. It is shown that in a system of curvilinear co-
ordinates (those of which MM. Lame* and Liouville have made
so much use), the equations of. the system are the differential
equations given by the Calculus of Variations for the shortest
line between two points. Consequently the finite equations of
a straight line are the integrals sought. This very ingenious
consideration is afterwards generalised.

32. In the twelfth volume of Crelle's Journal, p. 181, M.
Eichelot has considered the Abelian integral of the first class.
The principal result at which he arrives is, that the only rational
transformation by means of which such an integral may be
changed into one of similar form is linear in both the variables
which it involves. By means of this substitution, he trans-
forms, under certain conditions, the integral in question into
a form analogous to the standard forms of elliptic integrals.
The subject of the division of hyper-elliptic integrals of each
class into three genera is also considered, and the same prin-
ciple of classification as Legendre's is made use of. The paper
concludes by pointing out an error which Legendre committed
in the application of his principle. Legendre had thought that
the formula of summation given by Abel's theorem for integrals

f Te(?3'

of the form I -7= could not involve a logarithmic function.
J Nx

310 ON THE RECENT PROGRESS OF ANALYSIS.

Thus these integrals would belong to the first or second kind,
according to the value of the index e, and of X the degree of (f>x.
But in reality, though the integrals in question are of the first
kind (that is, they admit of summation without introducing
either an algebraical or logarithmic function) if e be less than
a certain limit, yet if it be not so their formula of summation
will in general involve both algebraical and logarithmic func-
tions. Either may, under certain conditions as to the form of
(j)x, disappear, but while <j>x is merely known as the polynomial
of the A,th degree, we cannot decide whether the integral is to be
referred to the second or third kind.

I may mention here a very elegant result due to M. Jacobi.
It appears in the thirtieth volume of Crelle's Journal, p. 121, and
is a generalisation of the fundamental formula for the addition of
elliptic arcs. With a slight modification it may be thus stated.
If <f)x involve only even powers of x, the highest being a?4wl, then

Cx x*mdx
the sum of the integrals I -j= '= is equal to the product of their

arguments, that is of the different quantities denoted by the sym-
bol x. In this case then the logarithmic function disappears,
and the^'integral belongs to the second kind.

In the twenty-ninth volume of Crelle's Journal there is a
paper by M. Eichelot on a question connected with hyper-elliptic
integrals. The reader will find in it a good many fully-de-
veloped results, which may be considered as particular cases of
Abel's theorem. They illustrate the learned author's criticism
of Legendre's classification of hyper-elliptic integrals, though
they are not adduced for that purpose.

The function M (vide ante, p. 244) is a function of the arbi-
trary quantities a, Z>, ... c, which, as has been remarked, may
themselves be considered functions of the arguments a?l5 x2, ... x^.
To determine M as a function of the last-written quantities is a
necessary ulterior step in almost any special application of Abel's
theorem, and this M. Eichelot has done in several interesting
cases, establishing at the same time the relations which exist
among the quantities in question. His investigations, however,
have an ulterior purpose, and are not to be considered merely as
corollaries from Abel's theorem.

Another paper of M. Eichelot, on the subject of the Abelian
integrals, is found in the sixteenth volume of Crelle's Journal,

ON THE RECENT PROGRESS OF ANALYSIS. 311

p. 221. The aim of it is to furnish the means of actually calcu-
lating the value of the Abelian integral of the first class by a
method of successive transformation, that is, by a method analo-
gous to that used for elliptic integrals. M. Eichelot's process
depends essentially on an irrational substitution, by means of
which we can replace the proposed integrals by two others which
differ only with respect to their limits. In the development of this
idea the author confines himself to the first kind of the Abelian
integrals of the first class, though the same method may m . m .
be more generally applied*. ^From the formula which expresses
the proposed integral as the aggregate of two others is deduced
another, in which it is expressed by means of four integrals,
the inferior limits of all being zero. The first and second of
these integrals differ only in their amplitude, and the same is
true of the third and fourth. There are two principal trans-
formations, either of which may be repeated as often as we
please ; and though it might seem that the number of inte-
grals would in the successive transformations increase in a geo-
metrical progression, yet by the application of Abel's theorem
we can always reduce them to the same number. But the de-
velopment of this part of the subject M. Eichelot has reserved
for another occasion f.

At the close of his memoir, M. Eichelot has given some
numerical examples of his method for the case of a complete
hyper-elliptic integral. The third example he had previously
given in a brief notice of his researches, published in No. 311 of
Schumacher's Journal.

33. For many years after the death of Legendre the subject
of the comparison of transcendents was studied principally by
German and Scandinavian writers \: a young French mathema-
tician, M. Hermite, has recently made important discoveries
in this theory ; but as the principal part of what he has done

* The integral to be transformed is —77 — ; - r — -

- r — - — ^ . 9 \
(1-2) (i -/c%) (i -X2z) (i -

M and N, &c. being constant.

"t* His transformations ultimately reduce the hyper- elliptic integral to elliptic
integrals j the latter may be considered known quantities, "vel per paucas ad-
jectas transformationes directe computentur."

+ The papers of M. Liouville, already noticed, may be said to be an exception
to this remark.

312 ON THE RECENT PROGRESS OF ANALYSIS.

is as yet not published, a very imperfect outline is all that
can be given.

In the seventeenth volume of the Comptes Eendus, we find
the report of a commission, consisting of MM. Lame and Liou-
ville, on a memoir presented to the Institute by M. Hermite.
This report is reprinted in the eighth volume of Liouville's
Journal, p. 502. A remark which incidentally occurs in it,
namely, that Abel was the first to give the general theory of the
division of elliptic integrals, led to a very warm discussion be-
tween MM. Liouville and Libri, on the subject of the claims
which, as I have already remarked, the latter had made with
reference to this theory.

It appears from the report, that M. Hermite has succeeded
in solving the problem of the division of hyper-elliptic integrals.
The division of elliptic integrals depends on the solution of an
algebraical equation ; that of the hyper-elliptic integrals (as the
functions inverse to them involve, as we have seen, more than
one variable), on the solution of a system of simultaneous
algebraical equations. This solution can, M. Hermite has
shown, be effected by means of radicals assuming, as in the ana-
logous case of elliptic functions, the division of the complete
integrals. M. Hermite's method depends for the most part on
the periodicity of the functions considered. A transcendental
expression of the roots of the equation of the problem having
been obtained, their algebraical values are deduced from it.

These researches, in themselves of great interest, are yet
more interesting, when we consider how completely they jus-
tify the views of M. Jacobi as to the manner in which Abel's
theorem ought to be interpreted, by showing that his theory
of the higher transcendents is no barren or artificial generali-
sation.

At page 505 of the volume of Liouville's Journal already
mentioned, we find an extract from a letter of M. Jacobi to
M. Hermite, in which, after congratulating him on the im-
portant discovery he had made, he points out that the transcen-
dental functions X (uv), \ (uv) (vide ante, p. 302) are algebraical
functions of transcendental functions which involve but one
variable.

M. Hermite's subsequent researches have embraced a much
more general theory than that of the Abelian integrals, namely,

ON THE RECENT PROGRESS OF ANALYSIS. 313

that of the integrals of any algebraical function whatever. Thus
his views bear the same relation to Abel's general theory, deve-
loped in the Savans Etrangers, that those of M. Jacobi in the
Considerationes Generates do to Abel's theorem.

All that has yet been published with respect to them is con-
tained in the Comptes Rendus, xviil. p. 1133, in the form of an
extract of a letter from M. Hermite to M. Liouville. This ex-
tract is reprinted in Liouville's Journal, IX. p. 353. It was com-
municated to the Institute in June 1844.

Following the course of^M. Jacobi's inquiries, M. Hermite
proposed to determine what are the differential equations of
which Abel's investigations give the complete algebraical inte-
grals. When this is done it suggests the nature of the inverse
functions which are to be introduced. The number of these
functions will of course vary in different cases, just as in M. Ja-
cobi's less general theory. Let us suppose this number to be
denoted by 7, then each function will involve 7 variables. And
if each of these variables be replaced by the sum of two new
variables, then all the functions are given as the roots of an
equation of the 7th degree, whose coefficients are rational in
terms of the corresponding functions of each of the new variables
and of certain known algebraical functions. From hence is de-
rived the theory of the periodicity of these functions.

After some other remarks on the theory of the higher trans-
cendents, M. Hermite states that the method of division of which
he made use in the problem of the division of Abelian integrals
extends also to the new transcendents now considered, but that
in the theory of transformation he had not as yet been success-
ful. The greater part of the remainder of this remarkable com-
munication relates to elliptic functions, and has been already
noticed. The remark just mentioned as having been made by
M. Jacobi for the functions which are inverse to the Abelian
integrals, extends, M. Hermite observes, to the functions which
he considers.

In conclusion, M. Hermite remarks that the method of dif-
ferentiation with respect to the modulus of which Legendre made
so much use in the theory of elliptic functions, may be applied
to all functions of the form

jf(xy] dx,

314 ON THE RECENT PROGRESS OF ANALYSIS.

where y is given by the equation

y-z=o.

In concluding this report, it may be remarked that the sub-
ject of it is still incomplete, and that there is yet much to be
done which we may hope it will not be found impossible to do.
It is however difficult to predict the direction in which progress
will hereafter be made. Yet I think we may reasonably sup-
pose that the question of multiple periodicity, from the para-
doxical aspect in which it has presented itself, and from its
connexion with the general principles of the science of sym-
bols, will sooner or later attract the attention of all philosophical
analysts. M. Liouville's idea of considering the conditions to
which a doubly periodic function must as such be subject, can
scarcely be developed or extended to the higher transcendents
without leading to results of great generality and interest.

The detailed discussion of different classes of algebraical in-
tegrals, their transformations and reductions, form an endless
subject of inquiry. But in this, as in other cases, the increasing
extent of our knowledge will of itself tend to diminish the in-
terest attached to the full development of particular portions of
it; and with reference to analytical problems arising out of
questions of physical science, the theory of the higher trans-
cendents will it is probable never become of so much importance
as the theory of elliptic functions. We have occasion to make
use of circular much more frequently than of elliptic functions,
and similarly we shall, it may be presumed, have less frequently
to introduce the higher transcendents than elliptic functions.
Numerical calculations of the values of the higher transcendents
are therefore less important than similar calculations in the case
of elliptic functions*.

The following index is intended to contain references to all
the papers in the first thirty-one volumes of Crelle's Journal, and
in the first ten volumes of Liouville's Journal, more or less con-
nected with the subject of this report, together with a consider-
able number of others.

* The Academy of Sciences has proposed as the subject of the great mathe-
matical prize for 1846 the following question: — " Perfection ner dans quelque point
essentiel la the'orie des fonctions abeliennes ou plus gdneralement des transcen-
dantes qui re'sultent de la consideration des integrates de quantite*s algdbriques."
The memoirs are to be sent in before the ist of October.

THE REGENT PROGRESS OF ANALYSIS. 315

In the following Index Crelles Journal is denoted by C., Liou-
villes by L., and the present Report by R.

ABEL. Ueber die Integration der differential Formel ~=. wenn R

und p ganze Functionen sind. — C. i. 185. This paper contains
formulae of reduction. It is mentioned by M. Liouville, ' Jour-
nal de FEcole Poly technique,' 23d cah. p. 38. It appears in
French in Abel's collected works, Tom. I. 33.

Recherches sur les Fonctions Elliptiques. — C. n. 101, and in.
160 [1827]. Y. Abel's works, i. 141 ; also R, p. 273.

. -- Remarques sur quelques Proprietes Generales d'une certaine
sorte de Fonctions Transcendantes. — G. in. 313. This paper
contains the theorem commonly known as 'Abel's Theorem.'
V. Abel's works, i. 288 ; also R. p. 247.

-- Sur le Nombre de Transformations Differentes qu'on pent
faire subir a une Fonction Elliptique par la Substitution d'une
Fonction donnee du premier desse. — C. in. 394. V. Abel's
works, i. 309.

-- Theorem e General sur la Transformation des Fonctions Ellip-
tiques de la seconde et de la troisieme espece. — C. in. 402. The
theorem is stated without demonstration. Y. Abel's works,
i. 317.

-- Note sur quelques Formules Elliptiques. — C. iv. 85. This
paper contains developments of elliptic functions, &c. Y. Abel's
works, I. 299.

-- Theore~rnes sur les Fonctions Elliptiques. — C. rv. 194. They
relate to the demonstration of the theorem stated by M. Jacobi
in the third volume of Crelle's Journal, p. 86, by means of which
Abel's method for the division of elliptic integrals is greatly sim-
plified. Y. Abel's works, i. 318 ; also R p. 277.

-- Demonstration d'une Propriety Generale d'une certaine Classe
de Fonctions Transcendantes. — C. iv. 200. This short paper
contains the fundamental idea of the memoir presented to the In-
stitute in 1826. Y. Abel's works, i. 324 ; also R. p. 246.

-- Precis d'une Theorie des Fonctions Elliptiques. — C. iv. 236
and 309. This precis was left unfinished. Y. Abel's works, I.
326 ; also R. p. 286.

- Extracts from Letters to M. Crelle, one of which relates to
the comparison of Transcendents. — C. v. 336. Y. Abel's works,
ii. 253.

- A Letter to M. Legendre.— C. vi. 73. Works, n. 256. It

316 ON THE RECENT PROGRESS OF ANALYSIS.

contains a theorem proved by M. Ram us in the twenty-fourth
volume of Crelle's Journal, p. 78 ; and another, proved by M.
Liouville in the twenty-third cahier of the ' Journal de 1'Ecole
Polytechnique.' Y. R. p. 249 and p. 294.

Memoire sur une certaine classe de Fonctions Transcendantes.

Presented to the Institute, Oct. 30, 1826, published in 1841 in
the 'Memoires des Savans Etrangers,' vn. 176. It is the only
memoir of Abel's not contained in the collected edition of his
works published in 1839; the editor, M. Holmboe, not having
been able to procure a copy of it. V. R. p. 246.

Solution d'un Probleme General concernant la Transformation

des Fonctions Elliptiques. — Schumacher's Astronomische Nach-
richten, No. 138, vi. 365. Y. Abel's works, I. 253; also R.
p. 282.

Addition au Me'moire Precedent. — Schumacher's Astrono-
mische Nach., No. 147, vn. 33. Y. Abel's works, i. 275; also
R. ubi supra.

The following papers were published for the first time in Abel's
collected works. The references are to the second volume : —

Proprie"tes remarquables de la Fonction £/ = <£#, etc., p. 51.

The multiple periodicity of a function inverse to a hyper-elliptic
integral is here mentioned. Y. R. p. 305.

Sur une Propriete" remarquable d'une Classe tres etendue de

Fonctions Transcendantes, p. 54. This paper contains a gene-
ralisation of a theorem relating to elliptic functions.

Extension de la The*orie Precedente, p. 58.

Sur la Comparaison des Fonctions Transcendantes, p. 66.

This paper contains a somewhat fuller development of his general
theory than that which is inserted in the fourth volume of
Oelle's Journal, p. 200. Y. R. p. 247.

Theorie des Transcendantes Elliptiques, p. 93. Y. R. p. 287.

Demonstration de quelques Formules Elliptiques, p. 210.

BROCH. Sur quelques Proprie*tes d'une certaine classe des Fonctions

Transcendantes. — C. xx. 178. An extension of Abel's theorem.
Y. R. p. 248.

Me"moire sur les Fonctions de la forme

(X>-VP-IF(X*) (R(x*})*kdx, etc.— C. xxm. 145 and 201.

This memoir, of which the first part may be considered a gene-
ralisation of the preceding, is accompanied by a report of MM.
Liouville and Cauchy. Y. R. p. 248.

THE RECENT PROGRESS OF ANALYSTS. 317

BRONWIN. On Elliptic Functions. — Camb. Mathematical Journal,
in. 123. Mr Bronwin puts the transcendental formula of trans-
formation in a very neat form.

On M. Jacobi's Theory of Elliptic Functions. — Lond., Ed. and

Dub. Phil. Mag. xxn. 258. V. R. p. 267.

Reply to Mr Cay ley's Remarks. — L., E. & D. Phil. Mag. xxin.

89. Y. R. uU supra.

CATALAN. Sur la Reduction d'une Classe d'Integrales Multiples. — L.
iv. 323.

Sur les Transformations des Variables dans les Integrates

Multiples. Memoires Qouronnes par 1' Academic Royal e de
Bruxelles, xiv. 2de partie, p. 1. The third part contains a trans-
formation of a multiple integral leading to properties of hyper-
elliptic integrals analogous to known properties of elliptic in-
tegrals.

CAUCHY. Comptes Rendus, xvn. 825.— Y. R. p. 292.

CAYLEY. Memoire sur les Fonctions doublement Periodiques. — L.
x. 385. An enlargement of his paper on the inverse elliptic
functions, published in the fourth volume of the Cambridge
Mathematical Journal. Y. R. p. 292.

Remarks on the Rev. B. Bronwin's paper. — L., E. and D. Phil.

Mag. xxii. 358.

Investigation of the Transformation of certain Elliptic Func-
tions.— L., E. and D. Phil. Mag. xxv. 352. Y. R. p. 292.

On the iDverse Elliptic Functions. — Camb. Math. Journal, iv.

257. Y. R. p. 292.
CHASLES. Comptes Rendus de 1'Institut, xvn. 838, and xix. 1239.

M. Chasles in these two communications presents to the Institute

notices of his geometrical researches illustrative of the theory of

elliptic functions. Y. R. p. 300.
CLAUSEN. Schumacher's Nachrichten, xix. 178. On a particular

Integral mentioned by Legendre.

Schumacher's Nachrichten, xix. 181. It is shown that the

arcs of one of the curves, known as the Spirica of Perseus, may
be rectified by means of an elliptic integral.

EISENSTEIN. Theoremes sur les Formes Cubiques. — C. xxvu. 75.
At the end of this paper we find some developments of elliptic
functions in continued fractions. This subject is continued in
the following paper of M. Eisenstein's.

Transformations remarquables de quelque Series. — C. xxvu.

193, and xxvni. 36. See also Theorema, C. xxix. 96.

Bemerkungen zu den elliptischen und Abelschen Transcenden-

318 ON THE RECENT PROGRESS OF ANALYSIS.

ten. — C. xxvii. 185. M. Jacob! has criticised this paper (of
which a translation appears in Liouville's Journal, x. 44 5) in the
thirtieth volume of Ci\ lie's Journal.

Elementare Ableitung einer merkwiirdiger Relation zwischen

zwei unendlichen Producten. — C. xxvii. 285. V. R. p. 293.

Beitrage zur Theorie der Elliptischen Functionen. — C. xxx.

185, 211. This paper contains a demonstration of the funda-
mental formula of elliptic functions. Y. R. p. 295.

GUDERMANN. Integralia Elliptica Tertise Speciei Reducendi Metho-
dus Simplicior, &c. — C. xiv. 159, 185. V. R. p. 291.

Einige Bemerkungen liber Elliptische Functionen. — C. xvi. 78.

Series novae quarum ope Integralia Elliptica Primse et Se-

cundse Speciei computantur, &c. — C. xvi. 366, and xvn. 382.

Theorie der Modular Functionen und der Modular Integrale. —

C. xvm. 1, 142, 220, 303; xix. 46, 119, 244; xx. 62, 103;
XXL 240 ; xxni. 301 ; xxv. 281. A systematic treatise on
elliptic functions. V. R. p. 291.

GUTZLAFF. ^Equatio Modularis pro Transformation Functionum

Ellipticarum Septimi Ordinis. — C. xn. 173. Y. R. p. 291.
HAEDENKAMP. De Transformation Integralis

I!

nV - sin2</> cos2\js) '
— C. xx. 97. It is shown to be the product of two elliptic
integrals.

Tiber Transformation vielfacher Integrale. — C. xxii. 184.

Analogous to the researches of M. Catalan in the * Memoires de
Bruxelles,' which appears to have been previously published.

tJber Abelsche Integrale.— C. xxv. 178. Y. R. p. 309.

HERMITE. Sur la Theorie des Transcendantes a Difierentielles Al-
gebriques. — L. ix. 353. Extracted from the « Comptes Rendus '
[June 1844]. This note, which contains scarcely more than an
indication of M. Hermite's results, may be said to mark the
furthest advance yet made in the theory of the comparison of
transcendents. Y. R. p. 312.

HILL. Exemplum usus Functionum Iteratarum, <fec. — C. xi. 193.
This paper contains some interesting applications of the calculus
of functions to the comparison of transcendents. Y. R. p. 250.

JACOBI. Addition au Memoire de M. Abel sur les Fonctions Ellip-
tiques. — C. in. 86. A short note, containing an important
simplification of Abel's method of solving the equation of the
problem of division. Y. R. p. 277.

Note sur la Decomposition d'un Nombre donne* en quatre

ON THE RECENT PROGRESS OF ANALYSIS. 319

quarr6s. — 0. in. 191. The demonstration referred to is founded
on elliptic functions.

- Note sur les Fonctions Elliptiques. — G. in. 192.

- Suite des Notices sur les Fonctions Elliptiques. — C. in. 303.

- Suite des Notices, etc. — C. in. 403.

- Suite des Notices, etc. — C. iv. 185. These notes contain theo-
rems stated for the most part without demonstration. Y. R
p. 271.

- TJeber die Anwendung der elliptischen Transcendenten auf
ein bekanutes Problem der Elementar-geometrie, u. s. w. — C. ill.
376. This paper contains a geometrical construction for the
addition and multiplication of elliptic integrals of the first kind.
A translation of the most important part appears in Liouville's
Journal, x. 435. V. R p. 272.

- De Functionibus Ellipticis Commentatio. — C. iv. 371. Trans-
formations of integrals of the second and third kinds, &c. Y. R
p. 288.

- De Functionibus Ellipticis Commentatio altera. — C. iv. 397.
We find here an elementary demonstration ofM. Jacobi's theo-
rem. Y. R. p. 288.

- Note sur une nouvelle application de 1'Analyse des Fonctions
Elliptiques a 1'Algebre. — C. vii. 41. It relates to the develop-
ment in continued fractions of a function of the fourth degree.

- Notiz zu Theorie des Fonctions Elliptiques de Legendre, Troi-
sieme Supplement.— C. vm. 413. Y. R. p. 288.

- De Theoremate Abeliano— C. ix. 99. Y. R p. 249.

- Considerationes Generales de Transcendentibus Abelianis. —
C. ix. 394 [1832]. This memoir lays the foundation of the
theory of the higher transcendents. Y. R p. 301.

- De Functionibus Duarum Yariabilium quadrupliciter Periodi-
cis, &c. — C. xin. 55. M. Jacobi here proves the impossibility of
a function of one variable being triply periodic. Y. R. p. 303.

- De usu Theorise Integralium Ellipticorum et Integralium
Abelianorum in Analysi Diophantea. — C. xin. 353. It is here
pointed out that a problem of indeterminate analysis, discussed
by Euler in the posthumous memoirs recently published by the
Academy of St Petersburg, is in effect that of the multipli-
cation and addition of elliptic integrals. Suggestions are made
as to the corresponding application that might be made of the
Abelian integrals.

- Formulae novae in Theoria Transcendentium Fundamentales. —
C. xv. 199. Elegant elementary formulae.

320 ON THE RECENT PROGRESS OF ANALYSIS.

JACOBI. Note von der Geodatischen Lime auf einem Ellipsoid, u. s. w.
— 0. xix. 309. M. Jacobi has here announced the important dis-
covery that the equation to the shortest line on an ellipsoid is
expressible by means of Abelian integrals of the first class. As
this is perhaps the first application made of Abelian integrals
since their recognition as elements of analysis, I have thought it
well to mention it in this place. A translation of the note is
found in Liouville's Journal, vi. 267.

Demonstratio nova Theorematis Abeliani. — C. xxiv. 28. Y. R.

p. 308.

. Zur Theorie der elliptischen Functionen. — C. xxvi. 93. This

paper contains series for the calculation of elliptic functions, and
a table of the function q.

Ueber die Additions-theoreme der Abelschen Integral e zweiter

unddritter Gattung. — C. xxx. 121. We find here some remark-
able formulae. V. R. p. 310.

Note sur les Fonctions Abeliennes. — C. xxx. 183. This note

relates principally to the fact announced in M. Jacobi's letter to
M. Hermite. Y. L. vm. 505.

Ueber einige die elliptischen Functionen betrefienden For-

meln. — C. xxx. 269.

Extrait d'une Lettre 3, M. Hermite.— L. vm. 505. Y. R.
p. 312.

Extraits de deux Lettres de M. Jacobi, &c. — Schumacher's

Nachrichten, vi. 33 [Sept. 1827]. They contain the first an-
nouncement of his theorem.

Demonstratio Theorematis ad Theoriam Functionum Ellipti-

carum Spectantis. — Schumacher, vi. 133. The first published
demonstration of his theorem. See also Legendre at p. 201 of
the same volume. Y. R. pp. 259 and 260.

JURGENSEN. Sur la Sommation des Transcendantes a DifFerentielles
Algebriques. — C. xix. 113.

Remarques Generates sur les Transcendantes a Differentielles

Algebriques.— C. XXIIL 126. Y. R. p. 248.

IVORY. On the Theory of the Elliptic Transcendents. — Phil. Trans.,

1831, p. 349. Y. R. p. 288.
LIBRI. Sur la Theorie des Nombres. — Memoires des Savans Etran-

gers, v. 1.

Sur la Resolution des Equations Algebriques, &c. — C. x. 1 67.

These memoirs are referred to in the controversy between MM.
Liouville and Libri.

LIOUVILLE. Sur lea Integrates de Yaleur Algebrique. — Journal

ON THE RECENT PROGRESS OF ANALYSIS. 321

de 1'Ecole Poly technique, cah. xxn. 124 and 149. These two
memoirs arex printed also in the fifth volume of the ' Memoires
des Savans Etrangers,' pp. 76, 105. Poisson's report on them is
inserted in the tenth volume of Crelle's Journal, v. infra.

LIOUVILLE. Sur les TranscendantesElliptiques de Premiere et de
Seconde Espece. — Journ. de 1'Ecole Polytech., cah. xxtn. 37.
V. R. p. 294.

Note sur la Determination des Integrales dont la Valeur

est Algebrique. — C. x. 347. This note is appended to Poisson's
report.

Sur T Integration d'une Olasse de Fonctions Transcendantes.—

C. xin. 93. On the same general subject as the preceding
memoirs.

Sur la Classification des Transcendantes. — L. n. 56, and in.

523. These papers contain an exposition of the principles on
which this classification is to be effected.

Sur les Transcendantes Elliptiques de Premiere et de Seconde

Espece considerees comme Fonctions de leurs Modules. — L. v.
34 and 441. It is proved that these transcendents so considered
cannot be reduced to algebraical functions.

Rapport fait a 1'Academie des Sciences, &c. — L. vm. 502.

Report on M. Hermite's memoir. V. R. p. 312.

Sur la Division du Perimetre de la Lemniscate. — L. vm. 507.

V. R. p. 281.

Rapport sur le Memoir e de M. Serret sur la Representation

Geometrique des Fonctions Elliptiques et Ultra-elliptiques. —
L. x. 290. A note is appended to this report generalising M.
Serret's theory. V. R. p. 297.

— — Sur un Memoire de M. Serret, &c.— L. x. 456. V. R. p. 73.
LOBATTO. Sur 1'Integration de la Differentielle
dx

Jx* + ax3 + fix? + yx + 8 '
— C. x. 280.

LUCHTERHANDT. De Traiisformatione Expressionis

dy

v/[*(y-")(2/-my-S)]'
—C. xvn. 248.

MAcCuLLAGH. Transactions of the Royal Irish Academy, xvi. 76.

An elegant geometrical proof of Landen's theorem.
MINDING. Theor£me relatif a une certaine Fonction Transcendaiite. —

C. ix. 295. The function in question was shown by M. Richelot

to be reducible to elliptic integrals.

21

322 ON THE RECENT PROGRESS OF ANALYSIS.

MINDING. Sur les Integrates de la forme

f dx P Jp t
y-*-9 &c.

J C — X

— C. x. 195. An addition to this memoir is found at p. 292 of
the same volume.

Recherches sur la Sommation d'un certain nombre de Fonc-
tions Transcendantes, &c. — C. xi. 373. These researches relate
to an extension of Abel's theorem.

Propositiones qusedam de Integralibus Functionum Algebrai-

carum unius variabilis e principiis Abelianis derivatse. — C. xxm.
255. This memoir is mentioned by M. Hermite.

POISSON. Rapport sur deux Memoires de M. J. Liouville, &c. — C. x.
342. Y. supra, Liouville.

Theoremes relatifs aux Integrates des Fonctions Algebriques. —

C. xn. 89. Y. R. p. 249.

RAABE. Bemerkungen zum Principe der doppelten Substitution,
u. s. w. — C. xv. 191.

RAMUS. De Integralibus Differentialium Algebraicarum. — C. xxiv.
69. Y. R. p. 249.

RICHELOT. Note sur le Theoreme, &c. — C. ix. 407. Y. supra, Mind-
ing.

De Integralibus Abelianis Primi Ordinis Commentatio Pri-

ma.— C. xii. 181. Y. R. p. 309.

De Transformatione Integralium Abelianorum Primi Ordinis

Commentatio.— C. xvi. 221 and 285. Y. R. p. 310.

Ueber die Integration eines merkwiirdigen Systems Differ-

ential-gleichungen. — C. xxm. 354. These equations are those
known as the " Jacobische System." Y. R. p. 306.

— — Einige neue Integral-gleichungen des Jacobischen Systems
Differential-gleichungen. — C. xxv. 97. The results contained in
this paper are much more general than those of the preceding
one. Y. R. p. 309.

Nova Theoremata de Functionum Abelianorum cuj usque ordi-

nis Yaloribus, &c.— C. xxix. 281. Y. R. p. 310.

Ueber die auf wiederholten Transformationen beruhende

Berechnung der ultra-elliptischen Transcendenten. — Schumacher
Astr. Nach. xm. 361. [July, 1836]. Y. R. p. 311.

ROBERTS. Sur une Representation Geometrique des Fonctions Ellip-
tiques de Premiere Espece. — L. vm. 263.

Sur une Representation Geometrique des Trois Fonctions

Elliptiques. — L. ix. 155. Mr Roberts's papers relate to curves
formed by the intersection of a cone of the second order with a

ON THE RECENT PROGRESS OF ANALYSIS. 323

sphere. The following paper contains a more general exposition

of his views.
ROBERTS. Memoire sur quelques Proprietes Geometriques relatives

aux Fonctions Elliptiques. — L. x. 297. V. R. p. 298.
ROSENHAIN. Exercitationes Analyticse in Theorema Abeliamim de

Integralibus Functionum Algebraicarum. — C. xxvm. 249, and

xxix. 1. V. R. p. 249.
SANIO. De Functionum Ellipticarum Multiplication et Transforma-

tione quse ad numerum parem pertinet Commentatio. — C. xiv.

1. Y. R. p. 290.
SERRET. Note sur les Fonctions Elliptiques de Premiere Espece. —

L. vm. 145. Y. R. p. 296.
Proprietes Geometriques relatives a la Theorie des Fonctions

Elliptiques. — L. viu. 495.
Note a 1'occasion du Memoire de M. William Roberts, &c. —

L. ix. 1GO.
Memoire sur la Representation Geometrique des Fonctions

Elliptiques et Ultra-Elliptiques. Addition au memoire pre"-

cedent. — L. x. 257 and 286. It was on this memoir that M.

Liouville made so favourable a report to the Institute. Y. R.

p. 297.

Developpemens sur une Classe d'Equations relatives a la Re-
presentation des Fonctions Elliptiques. — L. x. 351.
Note sur les Courbes Elliptiques de la Premiere Classe. — L.

x. 421.
Sur la Representation des Fonctions Elliptiques de Premiere

Espece. — Camb. and Dublin Math. Journ. i. p. 187.
SOHNCKE. .^quationes Modulares pro Transformation Functionum

Ellipticarum et undecimi et decimi tertii et deciini septimi

ordinis. — C. xn. 178. M. Sohncke here gives the results which

he investigates by a general method in the following paper.
^Equationes Modulares, &c. — C. xvi. 97. Y. R. p. 291.
TALBOT. Researches in the Integral Calculus.— Phil. Trans. 1836, p,

177; 1837, p. 1. Y. R, p. 249.

21—2

SOLUTION OF A DYNAMICAL PROBLEM.

A PERFECTLY rough sphere is placed upon a perfectly rough
horizontal plane which is made to rotate with a uniform angular
velocity about a vertical axis : to determine the path described
by the sphere in space*.

A sphere, resting on a perfectly rough horizontal plane,
receives a tangential impulse when the plane is made to move
in its own plane. This impulse gives a velocity to the centre
of the sphere and produces an angular velocity about a hori-
zontal axis. The centre of the sphere moves parallel to the
impulse, the axis of rotation is perpendicular to it; therefore
the point of contact moves parallel to the impulse and therefore
to the direction of motion of the centre. Therefore, as there is
no sliding, the centre moves in the same direction as that of the
motion of the plane supposed rectilineal. Moreover it is easily
seen that the velocity of the centre is to that of the point of
contact, or, which is the same thing, to that of the plane, as

1 : 1 4- p > « being the radius of the sphere, Jc its least radius

of gyration. While the direction and velocity of the plane's
motion remain unaltered, no farther action occurs; when a
change takes place, a new tangential impulse is given to the
sphere, producing a new velocity of the centre parallel to its
own direction, and a new velocity of rotation about an axis at
right angles to it. The new velocity of rotation bearing to the
old the same ratio as the new velocity of the centre to the old,
the result is a compound velocity of the centre bearing the same
ratio as before to the velocity of the point of contact, and as
before parallel to it, and therefore still parallel to the direction

* Walton's Problems in Theoretical Mechanics, p. 540.

SOLUTION OF A DYNAMICAL PROBLEM. 325

of motion of the plane ; and so on ; whether the motion of the
plane varies continuously or discontinuously, in direction, in
velocity, or in both. In the case proposed the motion of the
plane (by which throughout I mean the element thereof in con-
tact with the sphere) is always normal to a line drawn to a
fixed point. Therefore the motion of the centre is so too,
therefore the centre describes a circle whose centre is perpen-
dicularly over the said fixed point. Q.E.D.

ON THE TAUTOCHRONISM OF THE
CYCLOID*.

CONCEIVE two points, not acted on by gravity, to move on
the circumference of a stationary circle towards its lowest point :
the plane of the circle we will suppose to be vertical. Let their
motion be such that the ratio between their distances from the
lowest point may be invariable. Then their velocities towards
that point must be in that invariable ratio. Their heights above
it are in the duplicate ratio: their initial heights above it were
also in the duplicate ratio: so likewise therefore are their ver-
tical descents towards it. In other words, the squares of their
velocities towards the lowest point are as their vertical descents.

Conceive one of the points to be, when the other starts, at
the highest point of the circle, and to move with a c6nstant
velocity (#a)J, a being the radius of the circle. Then, when it
has descended through a vertical space z from its initial position,
the square of its velocity towards the lowest point of the circle
is equal to \gz. On this supposition with respect to one point,
it appears, from what has been said before, that the square of
the velocity of the other point towards the lowest point of the
circle is similarly equal to \gz ', z being the quantity correspond-
ing to z, viz. its vertical descent below its initial position.

Now suppose the circle to move horizontally in its own plane
with a velocity equal at every instant to the velocity, along the
arc, of one of the points, the direction of the motion of the circle
being towards the right or the left accordingly as the point is to
the right or the left of the vertical diameter. If the former point
be chosen, then the velocity of the circle will be constant, if the

* Walton's Probkms in Ekmentary Mechanics, p. 245.

.V THE TAUTOCHRONISM OF THE CYCLOID. 327

latter point, it will be variable. In either case, the path of the
point selected obviously becomes a cycloid, and it is easily seen
that the velocity of the point towards the lowest point of the
circle is destroyed by the motion of the circle itself, while the
velocity at right angles to this direction is doubled : consequently
the whole velocity of the point will have, for its square, 2gz or
2gz, and we have thus a perfect representation of cycloidal mo-
tion under the action of gravity. But it is obvious from the
fundamental hypothesis that the two points will reach the lowest
point of the circle at the same time: that is to say, the descent
to the lowest point of the cycloid is tautochronous.

The analogy to the descent to the lowest point of a circle
along its chords is in the essential point complete, but here the
motion is not along the chord but along the arc, and is compli-
cated with the motion of the circle itself. In both cases it is
easily seen that a medium, the resistance of which varies as the
velocity, does not affect the tautochronism.

ON NAPIER'S RULES*.

To the Editor of the " Quarterly Journal of Mathematics."

Some time ago, my friend Mr K. L. Ellis sent me the fol-
lowing remarks upon Napier's rules.

NAPIER'S rules for the solution of right-angled spherical tri-
angles are generally presented merely as a memoria technica ;
and when so presented do not exhibit the principle upon which
they depend. To investigate and exhibit that principle is the
purpose of this paper.

LEMMA.

If the three sides of a tetrahedron are right-angled triangles,
no right angle being at the apex, then the base is also a right-
angled triangle.

Let OAB be a right-angled triangle in A, and similarly
OAC. Let the third side OCB be right-angled in G.

Then will the base A CB be also right-angled in 0.

• Since Napier's method of investigating his rules was independently discovered
by Mr Ellis, the Editor is of opinion that this Essay, written by the Dean of Ely,
will not be without interest to the readers of this volume.

ON NAPIERS RULES.

329

or the angle at C is a right angle. Q.E.D.

c

Fig. i.

COR. It follows from this that the dihedral angle AC is a
right angle.

Considering the figure OABG, we observe that it is in some
sort symmetrical with regard to the line OB. GB being at
right angles to the plane OA C, and OA at right angles to the
plane CAB ; and as the dihedral angle 0 C is a right angle, so
also is the dihedral angle BA.

Now the three lines OA, OB, OG evidently represent any
right-angled spherical triangle; and similarly the three lines
BO, BA, BG represent another, between which and the former
a certain relation exists. One angle, namely, the dihedral angle
BO, is common ; the side ABC is the complement of the dihe-
dral angle OA ; the hypothenuse OBG is the complement of
the side BOG', the side OB A is the complement of the hypo-
thenuse AOB\ and, lastly, the angle BG is the complement of
the side AOG.

Hence this conclusion. If av a2, as, a4, a5 represent the parts
of a right-angled spherical triangle taken in order, and begin-
ning at the hypothenuse, then | - a3, | - «4, | - aB, | - al9

and aa are the parts also taken in order and beginning at the
hypothenuse of another right-angled spherical triangle. If

330 ON NAPIER'S RULES.

therefore to characterize the former triangle, we introduce a new
set of quantities p, such that a1+p1 = a2+p^ = a5+p5 = — ,

the original triangle being characterized by pl9 p^ p3, pv p5 the
secondary triangle is similarly characterized by p^ p±, J95, plt p^ ;
and as the secondary triangle gives rise to a third, and so on,
we thus see that every right-angled spherical triangle is one of
a system of five such triangles.

It is obvious, that the transformation just employed de-
pends upon the circumstance that the complement of the com-
plement of an angle is the angle itself, and would succeed
equally if the parts of the secondary triangle, which are the com-
plements of those of the first, had been any function (f) of them,
provided that/2= 1.

Keturning to the figure we observe, that if, instead of taking
a point .Z? in OB we had taken one, as^t, in OA, we should by a
similar construction have got another of the four spherical tri-
angles, which with the original one make up the system of
which we have been speaking. Further, if in BG we assume
any point as a first centre related to B as B to 0, and make a
similar assumption of a fifth centre in A C, the system will be
complete. But as a figure so drawn would be complicated, it is
better to adopt a different plan. Since BA is at right angles to
OA, and lies in the plane OAB, it is easy to represent the
corelate spherical triangles, to which the system of lines of which
we have been speaking gives rise.

Let BA C (fig. 2) be the original triangle right-angled in A.

Fig. 2.

NAPIER'S RULES. 331

7T

Produce BG to F, and AC to A', making AA = £B' = -', and

join A'B'.

Then .4'jB' G' is the secondary triangle ; and that it is a
right-angled triangle may be proved, independently of what
has been already said, by joining BA1. For since AA' is a
quadrant, and A a right angle, BA is as well as BB' a
quadrant, and therefore B is the pole of A'B', and therefore
B' a right angle. The hypothenuse of the secondary triangle
A G is the complement of A C, ft side of the first : the angle G is
common to the two triangles: GB' is the complement of the
hypothenuse BG : B'A = angle B'BA and is therefore the
complement of the angle ABG: lastly, the angle BAG has its
complement measured by the arc AB.

Again producing B'A to (9, B' 0 being a quadrant, and
similarly GA to (7, (7(7 being also a quadrant, we obtain a
third triangle OG'A\ and finally completing the figure (the
obvious details of demonstration are omitted) we get a reentering
pentagon, all the angles of which are right angles, each
being the right angle of one of the system of five right-angled
triangles.

From this point of view it is plain that Napier's rules may
just as naturally be considered as the statement of similar pro-
perties of five associated triangles, as in the usual mode, namely,
as the statement of dissimilar properties in one triangle ; and
thus we obtain the rationale of their uniformity. Every relation
between a middle and opposites, is the relation between two
sides and the hypothenuse: every relation between a middle
and adjacents is the relation between two angles and the
hypothenuse. I only give these relations, of course, as instances,
for any other would do as well.

A convenient mode of establishing these relations is afforded
by the Lemma.

If we project OB in fig. 1, into OA, or if we project OB
into OG and then OG into OA, the coincidence of the two
results gives rise to the equation

cos c = cos a cos b,
Again, OA tan c = AB,

and ABmsA = AC= OAt&nb;

332 ON NAPIERS RULES.

.'. tan b — tan c cos A,
or, cos A = tan b cot c,

which gives the relation between any middle part and the two
adjacent ones.

But in whatever way this relation and the preceding one
are established, the point to which I wish to call attention is
this, that by considering the system of five associated triangles
we immediately generalize any particular result without having
to demonstrate it in the separate cases.

It is clear that an equiangular spherical pentagon would in
every case give rise to the system of five equiangular spherical
triangles mutually corelated in a manner analogous to the right-
angled triangles of which we have been speaking. But in
the general case the relation would be too complicated to be
useful.

Having perused the preceding I was anxious to know in
what manner the subject had presented itself to Napier's own
mind, and having referred to a copy of the Mirifici Logarith-
morum Canonis Descriptio in the Cambridge University Li-
brary, I found to my astonishment that the mode of treatment
invented by Mr Ellis was in reality a re-invention of Napier's
own conception of the subject, which owing to some cause or an-
other has dropped out from all Cambridge books upon Spherical
Trigonometry*.

The following is translated from Napier. After giving a
general explanation of the use of circular parts, he proceeds
thus:

" This uniformity of the circular parts becomes very evident
in the case of right-angled triangles formed on the surface of a
sphere by five great circles, of which the first cuts the second,
the second the third, the third the fourth, the fourth the fifth,
and the fifth the first at right angles; the other intersections
taking place at oblique angles.

" Thus, for example, let the meridian DB (fig. 3) cut the hori-
zon BE in the point B. Let the horizon BE cut the circle EC of
which the sun S is the pole in E. Let EC cut the sun's declina-

• This defect has been recently corrected in Mr Todhunter's Treatise on
Spherical Trigonometry.

ON NAPIER S RULES.

333

tion circle CF in C. Let CF cut the equator FD in F. And,
lastly, let FD cut the meridian DB in D. Then all these inter-
sections take place orthogonally in the points B, E, C, F, D;

Fig. 3-

the other intersections taking place obliquely in the points Z, P,
S, 0, Q. And by this means five right-angled triangles will be
formed, namely, PBS, SFO, OEQ, QDZ and ZCP, of which
though the parts are different the circular parts are the same.

"The same uniformity of circular parts for quadrantal triangles
may be made to appear by joining in the preceding figure PQ,
QS, SZ, ZO, OP] by which means five quadrantal triangles will
be formed having different parts with the same circular parts.

" The rules for the solution of the circular parts may be proved
in each case separately, but besides this proof the general truth
of the rules may be seen from what precedes. For the homo-
logous constitution of the circular parts argues the similarity of
the relations connecting them; so that any proposition which
can be enunciated concerning the relation of any middle part to
the adjacents or opposites may be at once concluded to be true of
every other part regarded as the middle part."

Thus it appears that Napier himself did not regard his rules
as a mere memoria technica, but saw them in their mutual
relation, and in fact conceived them in the very best manner
possible. Hence, it is very strange that we should find such
statements as the following : —

334 ON NAPIERS RULES.

Woodhouse. " There is no separate and independent proof
of these rules ; but the rules will be manifestly just, if it can be
shewn that they comprehend every one of the ten results, (1),
(2), (3), &c."

Airy, Encyclopaedia Metropolitana. " These rules are proved
to be true only by shewing that they comprehend all the equa-
tions which we have just formed."

The same view has been adopted (I believe) in all our Cam-
bridge books.

Soon after receiving Mr Ellis' paper, given above, I received
from him the following : —

Addition to the foregoing.

In any spherical rectangular polygon all the sides but three
may have any assigned length, the length of the other three
being functions of them. In the case of the pentagon there are
thus two elements arbitrary, the same number as in a right-
angled spherical triangle, between which and the pentagon
exists a close analogy, which may be developed from the follow-
ing lemmas.

LEMMA 1.

Of three consecutive sides of a right-angled spherical poly-
gon, the extremes cut off quadrants of one another.

LEMMA 2.

The angle between the said extremes at their point of inter-
section (measured by the deflection of the direction of motion of
a point which travels along the boundary of the polygon) is
equal to the supplement of the next ; in other words, the com-
plement of the angle added to the complement of the side is
equal to zero.

Consider any side of a pentagon and the two sides which
are opposite to it and which meet one another at a right angle :
they form with that side a right-angled spherical triangle. We
will for distinction call that side (1), so that the other sides will
similarly be called (3) and (4). Then as (4), (5) and (1) are
consecutive, the angle between (4) and (1) is equal to the sup-
plement of (5), and similarly the angle between (3) and (1) is
the supplement of (2). Also the lengths of the sides of the tri-

ON NAPIER S RULES. 335

angle are by Lemma 1 the negative complements of (3) and (4),
that is, they are those sides diminished by a quadrant. As the
points of intersection of (3) and (4) with (1) are respectively
distant by quadrants from the two extremities of (4), it follows
that the segment they cut off between them is the supplement
of (4). Hence the sides of the triangle and the complements of
its angles and hypothenuse are severally equal to the negative
complements of the sides of the pentagon, taken in the same
order as the five parts of the triangle. The same demonstration
will of course apply to the triarlgle formed by any other side of
the pentagon and its two opposites, the only difference being in
the starting-point of the cycle. Hence whatever relation con-
nects any one part of a right-angled spherical triangle as such
with its two opposites connects every other part with its two
opposites. Q.E.D.

SCHOLIUM.

The general principle that the sines of the sides are as those
of the opposite angles of course gives a relation between a part
and its opposites. But I should prefer to begin from that be-
tween hypothenuse and the sides, proving that, as in p. 331, from
a right-angled tetrahedron, which might be made the foundation
of spherical, as the right-angled triangle is of plane trigono-
metry.

The relation connecting adjacent parts will be most simply
got by considering three consecutive relations of those which
connect opposites, multiplying the extremes and dividing by the
means.

The general theory of spherical polygons must certainly
lead to some curious general results, but of course they are out
of my reach.

The preceding paper was not sent to the Quarterly Journal,
because it was pointed out to me by a friend that a similar
resuscitation of Napier's own conception of his rules and a
similar remark upon the phenomenon of the disappearance of this
conception from modern English books had been already made
in the Course of Mathematics used in the Royal Military College
— Dr Button's Course of Mathematics, edited by the late T. S.
Davies*.

* Tutor to Mr Ellis ; see Biographical Memoir.

ON THE RETARDATION OF SUNRISE*.

SUPPOSE the sun on the eastern horizon : his actual apparent
motion until next sunrise may obviously be replaced by a ficti-
tious one composed of an unscrew rotation round II, the pole of

z

the ecliptic, combined with a screw rotation round P, the pole of
the earth, the magnitude of the latter rotation measuring the
retardation. Combine these rotations supposed small into one :
the resultant axis must lie in some point Q, where PIT produced
cuts ZS, (Z the zenith, S the sun). The angle between the
component rotations is constant, being PEE or w. Therefore
the ratio of inequality between the given rotation round II,
measured by the sun's diurnal motion in his orbit, and rotation,
round P, which measures the retardation, is then greatest when
the resultant axis Q lies as near as possible to II, or, in other
words, when PQ is a minimum. Now, S being the pole of the
arc ZTl, it will easily be seen that the angle $^11 is a right
angle and the angle QZP is an obtuse angle, while near the
equinox the angle ZQP must always be acute. Therefore PQ
is least when Q lies in PZ, and then II does so too. In other
words, the retardation is least when II culminates at sunrise,
that is, at the equinoxes.

The common expression for retardation may easily be de-
duced from the expression furnished by what has been said,
sin II Q

viz.

sin PQ

» Now first published.

A SOLUTION OF PROBLEM IX. OF THE
FIRST BOOK OF NEWTON'S PRINCIPIA*.

TIIE fundamental principle of the proof I gave Dr Goodwin f
is, that as the time varies as the increment of area, or as the
square of the radius into the increment of angle, the force varies
as the increment of angle, so that the acceleration in an instant

dt is equal to dd. The following application of this principle

is perhaps more simple. Let 0 "be the centre of force, P a
position of the moving body. Take PR at right angles to PO

and equal to ~dt. Join HP', P' being the position of the body

at the next instant. Similarly draw P'R at right angles to
PO and equal to PR. Join R'P", P" being a third position
of the body. Draw RQ parallel to RP'. Therefore the angle
at R being equal to the increment of the angle vector, QP is

* Dictated to the Editor of this volume and now first published,
t Goodwin's Course of Mathematics.

22

338 SOLUTION OF PROB. IX. B. I. OF THE PRINCIPIA.

equal to the acceleration into dt produced by the central force
during the instant dt. Likewise PP' is the space described
with the initial velocity in the time dt. Compounding this with
the acceleration we have P' Q equal in magnitude and coincident
in direction with P'P". Therefore in the triangles RQP',
RP'P", we have, neglecting quantities of the third order, EQ
equal and parallel to R'P', QP' equal and in same direction
with P'P", and therefore the third side RP' equal and parallel
to R'P". That is to say, the point so moves as that its velocity
may always be resolved into two elements, both of them con-
stant; one, i.e. PR or P'R', normal to the radius vector, the
other, RP' or R'P", parallel to a fixed straight line.

2. The velocity along the radius vector arises wholly from
the component RP', and varies as P'S. The velocity normal
to RP' arises wholly from the component RP and varies as
PT. Now these lines P'/S, PT, lie in similar triangles and are
as the hypotenuses, and therefore in a constant ratio. But as
the velocity towards the centre of force is proportional to the
velocity in a fixed direction, it is clear the point moves in a
conic section, because the fundamental property is, the distance
of any point from the focus is proportional to that from the
directrix.

SCHOLIUM. It is obvious that if a boat rows at a given
rate through still water so that a line at right angles to the keel
always passes through a given point, the boat moves in a circle,
and the above demonstration shews that if the water is flowing
in a given direction with a constant velocity, the boat will move
as if attracted to the fixed point according to the natural law of
force, that is, it will describe a conic section with major axis
transverse to the stream.

ON ROMAN AQUEDUCTS*.

THE aqueducts by means of which Rome was supplied
with water are in many points of view interesting, not only
from the magnificence of their construction but also from the
problems which they suggest respecting the scientific knowledge
possessed by those who originally formed or subsequently ma-
naged them. The accounts which have come down to us with
respect to the water supply of Rome are unsatisfactory* The
treatise of Frontinus is the most detailed, and, on that account
perhaps, the most disappointing. We find in it a great deal of
information with respect to the sizes of the pipes by means of
which the distribution of water took place, but that is nearly
all ; and, as .Prony has remarked, the fulness of details on this
matter makes the total silence on another not less important
appear more strange than it otherwise would do; I mean the
depth at which the pipes were placed below the level of the
surface of the water in the reservoirs. To this remark we may
add, that there is no estimate of the amount of water which in
a given time was actually received through a pipe of given size.
Prony is of opinion that a general rule must have prevailed,
and that pipes of all sizes must have been inserted at the same
depth, because we find it assumed that the discharge was pro-
portional to the area of the section. The inference being natural,
the same writer is of opinion that the notions of Frontinus,
who of course knew all that was known in his time, were ex-
ceedingly unscientific, and it is difficult to see how this can be
denied considering the account he gives of the attempts he made
.to gauge the different aqueducts, I cannot attempt to enter

* Now first published.

22—2

340 ON ROMAN AQUEDUCTS.

upon the general subject, "but there are one or two points con-
nected with it which admit of a kind of elucidation we have
hitherto perhaps not received.

It was doubtful even in the time of Frontinus why a pipe
called quinaria, which, as is well known, is the fundamental
modulus of the whole system, had received that designation.
Setting aside the explanation, in itself improbable and resting
on no authority, that Agrippa had introduced a modulus equi-
valent to five of the original units, and therefore called quinaria,
two others remain ; the one that of Vitruvius, which Frontinus
mentions in connection with him, the other that which Frontinus
himself preferred. According to Vitruvius leaden pipes receive
their names from the number of fingers (digiti) which the sheet
they were formed of was in width. Thus a centenaria was
formed of a sheet of lead 100 inches in width, and so in all
other cases. Frontinus observes that one opinion touching the
origin of the word quinaria is, that it was introduced by the
plumbarii, and Yitruvius because it was a pipe made of lead
5 digiti in width; and we may remark that it is a way in which
a plumber would be likely to designate a particular pipe, inas-
much as it depends on the way of making it, and not on a
measurement to be taken after it is made. Frontinus goes on
to say that this is uncertain, meaning apparently the size of the
pipe so formed, for that in rolling up the lead the outside surface
was stretched and the inner one contracted. A modern writer
who has devoted much attention to Roman antiquities, without
being, so far as I have seen, happy in his conjectures in respect
to them, namely, Dureau de la Malle, mistranslates the text,
and makes Frontinus say that one part of the lead overlaps the
other ; the inference appears to be not Only that he mistook the
meaning of his author, but that he did not know how pipes were
made. At present of course pipes are made like wires, by ex-
tension, and I do not know whether the old plan is ever used ;
but comparatively speaking this is a modern improvement, and
in not very old books one may find described the process of
cutting the lead into strips, folding it up so as to bring the
edges together and then laying solder over the line of junction.
Frontinus's own account is that the name probably indicated

ON ROMAN AQUEDUCTS. 341

the circumstance that the diameter, being as it appears a clear
internal diameter, was equal to J of a digit : and undoubtedly
in his time the names of a number of other pipes were formed
in analogy with that of quinaria in a way which indicates that
the magnitude of the bore was regarded as the foundation of
the nomenclature. Thus senaria was a pipe of six quarters bore,
and so on ; but it by no means follows that the received opinion
as to the origin of the word is true, and it seems impossible to
set aside the authority of Vitruvius, who speaks without doubt,
and must have been speaking of a matter with which he was
familiar : nor is there any thing improbable in the supposition
that in the interval between his time and that of Frontinus a
new nomenclature may have grown up linked with the old one
at one point only, namely, the word quinaria. Dureau de la
Malle brings an objection against the supposition that the width
of the lead is the cause of the name, that the circumference of
5 digits does not correspond to a diameter of one and a quarter,
adding sagely, what is by no means to any practical purpose,
that the problem of the rectification of the circle has hitherto
been found insoluble. But this objection, rightly considered, leads
to a curious confirmation of the statement of Vitruvius, which
I shall now attempt to explain.

The diameter spoken of is the bore or lumen, whereas the
diameter corresponding to a circumference of 5 inches must be
that of a circle lying somewhere in the thickness of the lead
between its two surfaces, one of which, as Frontinus remarks,
is stretched and the other diminished, either by condensation or
puckering in the process of formation. If we assume, as we
may do without sensible error, that the lead is stretched to one
half of its thickness and no more, we have to calculate the
diameter corresponding to a circumference of five, and to di-
minish this by the thickness of the lead, that is, by twice the
half thickness, in order to get the bore. The question then is,
what data have we for determining the thickness of the lead
from which these pipes are made, and the answer to it is given
by a passage in Vitruvius, and which has been copied by Pliny
arid Palladius.

Vitruvius, in speaking of aqueducts, says that leaden pipes
ought to be made in pieces of 10 feet long, their weight to vary
with the width of the lead of which they are made, from 1200 Ibs.

342 ON ROMAN AQUEDUCTS.

in the case of a centenaria to 60 in that of a quinaria. It will
be observed that these weights and the intermediate ones men-
tioned in the same passage are proportioned to the corresponding
widths, which indicates that the lead employed was always of
the same thickness. We see from these statements that a strip
of lead 10 feet long and 16 digits wide would weigh 192 Ibs. ;
192 being to 16 as 12 to 1. Now it is not necessary to convert
this number of Roman pounds into English pounds, and simi-
larly the digits into inches, in order to obtain the result we
seek. We may make use of a method which, if not perfectly
accurate, has at least the advantage of not depending on any
experimental comparison of ancient and modern standards. For
we know that the Eomans reckoned 80 Ibs, as the weight of a
cubic foot of water. The specific gravity of lead is about 11*4;
but as lead is seldom pure, and when alloyed is alloyed with
substances lighter than itself, we will take that of the pipes
at 11. A cubic foot of lead will therefore weigh 880 Ibs., and
Iby what we have just seen the weight of a prism whose base
is a square foot and height the thickness sought will be (16
digits being of course equal to a foot) 19*2 Ibs. Dividing the
latter number by the former, and multiplying the quotient by 16,
we obtain the .thickness of the lead in digits : it is approximately
0'35. Subtracting this from the diameter corresponding to a
circumference of 5 (that is, from 1*59, the ratio of the circum-
ference of the diameter being taken at 3*14), there remains for
the diameter of the bore l*24r or within a 100th of a digit of
the 5 quarter digits assigned by Frontinus to the diameter
of the quinariar which is a nearer coincidence than we were
entitled to expect. So far therefore from the statement of
Yitruvius being contradicted by that of Frontinus, they appear
to be in perfect harmony, the quinaria being at once the pipe
made of a strip of lead 5 digits in width, and that whose bore
was f of a digit. Not so, however, the other moduli of the
system. The vicenaria of Frontinus is not that made of a strip
of lead 20 digits in width, but that whose bore is equal to 20
quarter digits. The discrepancy between the two things is
not inconsiderable, as may be seen by a little calculation. The
truth is, that the nomenclature used with reference to the
water supply of Rome, appears to have been very unsettled.
We find in the treatise of Frontinus traces more or less deve-

ON ROMAN AQUEDUCTS. 343

loped of four different systems ; that of Vitruvius is a fifth ; and
there are certain exceptional cases mentioned by Frontinus in
speaking of the frauds of the aquarii, which it is not very easy
to connect with any of the five. In the midst of all this com-
plexity it is well to have one point fixed, namely, the historical
significance of the name quinaria applied to what appears in
Frontinus as the fundamental modulus to which in practice
all others were referred. This I think has been effected by
the considerations just suggested, and it is curious to remark
that neither Frontinus nor**any one else was likely to have
fallen on them. He was probably aware of the real state of
the case, but if the difficulty suggested by Dureau de la Malle
had occurred to him, he could only have got rid of it by actual
measurement, and not by a calculation founded on the specific
gravity of the material employed. In conclusion, it may be
well to observe that there are many cases in which the un-
certainty arising from the experimental comparison of ancient
and modern standards may be avoided: for instance, if it is
stated that a modius of a particular kind of grain weighed 26lbs.
(I refer of course to what Pliny says of the far of Clusium),
we may determine at once what an imperial bushel of it would
weigh by very simple considerations. The congius of water
like the gallon weighed lOlbs., and therefore i£ a congius
and a gallon are filled with the same substance, the former
will weigh, whatever the substance may be, as many Roman
pounds as the latter weighs pounds avoirdupois. Now the
modius is 3 congii, and the bushel 8 gallons, and consequently
the grain of which Pliny speaks would weigh 69 Ibs. and a
third the bushel. This is more than our best English wheat,
but some specimens of Australian wheat have been known to
weigh as much as 70 Ibs., and it is said that in Spain this
weight is not unusual. Another point of interest suggested
by the statement of Vitruvius is the comparative thickness
of Roman and English sheet lead. By the process of milling,
that is, of rolling a plate of lead between heavy rollers whose
distance is gradually diminished as the sheet becomes thinner,
the weight corresponding to a given surface may be reduced
much below the minimum possible in the old process of casting.
Thus milled lead sometimes weighs only 4 Ibs. in the square
foot. But it is more natural to compare the Roman lead which

344 ON ROMAN AQUEDUCTS.

was doubtless produced by casting with that which is now
made in a similar manner. Of the thickest kind of the latter
the weight is said to be lllbs. to the square foot, and we
have seen that Vitruvius' estimate gives a weight of 19 '2 Ibs.
to the square foot. Reducing this to English weights and
measures it would appear, speaking in round numbers, that an
English square foot of Roman lead would weigh about 15 Ibs.
avoirdupois. It is therefore considerably thicker than that
which is usually made in England. But it is to be under-
stood that the Romans may have had a thinner kind than
that which they were in the habit of using in the formation
of pipes. It has already been remarked that pipes of moderate
bore are now made by extension. The thickest kind of pipes
of an inch bore are made in lengths of 15 feet,, and weigh 56 Ibs.
Fifteen English feet of fistula quinaria would weigh about 68 Ibs.
avoirdupois. I have thought these details worthy of notice,
because they illustrate the economy which is the result of
improved modes of manufacture, and of our power of calcu-
lating the strength of materials.

II.

The most interesting consideration which the treatise of
Frontinus suggests relates to the state of knowledge current at
the time it was composed of what used to be called Hydraulics.
The details into which he enters are so precise (at least on some
points), and the provisions of the Senatus Consulta so strict,
that one finds it difficult to believe that he and his contempo-
raries were ignorant of any of the circumstances which it is
necessary to attend to in distributing a supply of water (neces-
sary, that is, unless considerations of justice are altogether to be
neglected) ; but then the difficulty arises, why, when so much is
said, so many details, not at all more likely to be generally
known than those mentioned, should have been omitted.

Prony has remarked that it has been the universal practice
to estimate the supply of water by the quantity which in a given
time and under a given pressure is discharged through a pipe of
given bore from a reservoir whose surface is kept at a fixed
level. But although he conceives that this remark applies to
the Roman system, yet neither the text of Frontinus, nor any
inference which can be deduced from it, appears to justify us in

ON ROMAN AQUEDUCTS. 345

supposing that they had any idea of the necessity of making the
water in the reservoir from which the distribution took place
remain at a constant height, or, to use Prony's own expression,
of making it stagnate. It is quite true, that Fabretti asserts
that the measurements were made in the piscinse, where the
water was comparatively speaking at rest, but although mea-
surements in the piscina were undoubtedly made, yet it is quite
clear that Frontinus had no scruple as to the propriety of mea-
suring the section of a running stream, calculating its area in
quinarise by a process of simple reduction, and then assuming
the result as an expression of the number of quinariae which the
stream was capable of supplying. This is so evident, that Prony
remarks that Frontinus could have had but vague ideas as to
the efflux of fluids, and that a method which took no account of
the velocity of the stream must have led to strange results. He
adds, and rightly, that Frontinus appears to have avoided mak-
ing his measurements in places where the velocity was small.
Nor is this inconsistent with Fabretti' s remark as to measure-
ments in piscinas, for we do not in reality know that the water
was in any sense reduced to rest in the places in which these
measurements were made.

The same conclusion, namely, that one of the conditions
which Prony speaks of as essential, was not attended to at all,
appears from the passage in which Frontinus describes the posi-
tion which the calix ought to occupy with respect to the stream of
water in which it is placed. The passage is obscure, but a stream
is plainly spoken of. A third consideration would lead to the
same result. I refer to the description which Yitruvius gives of
the three basins immediately supplied from the castellum. From
two of these there appears to have been a waste pipe into the
third; — a circumstance which it is hard to reconcile with the
supposition that the surface was kept at a constant level by
means of pipes employed in its ulterior distribution.

In this respect then at least the system of the Komans ap-
pears to omit one of the conditions which in modern times have
been found necessary, and there is at least another point in
which the discrepancy between their practice and ours deserves
to be noticed. It is well known, that if water escapes from an
orifice in the side of a vessel, the quantity discharged will, other
things remaining alike, vary with the length of the adjutage, as

346 ON ROMAN AQUEDUCTS.

it is called, through which it passes. If the side of a reservoir
be very thin the discharge is less than if it be thick, or if, which
comes to the same thing, a short horizontal tube is adjusted to
it. Now, at first sight, the Romans seem to have understood
this, for Frontinus states that a calix of not less than 12 digits
in length ought to be applied to the side of the reservoir. This
calix, he says, should be made of brass in order that its aperture
may not be tampered with ; but between the calix and the ad-
jutages of which we have been speaking there is the essential
distinction that the water escapes freely from the extremity of
the one, while from the other, I mean the calix, it passes at
once into the fistula. The Roman system of distribution is
essentially a system of distribution through pipes inserted into
the reservoir from which the erogation was made. To prevent
fraud the commencement of the pipe was made of brass, and
moreover, it was enacted, that for 50 feet the diameter of the
pipe should be the same as that of the calix to which it was
joined. It is one of Frontinus's complaints that the aquarii
would sometimes insert pipes without using a calix, and thereby
gave an opportunity, to use a homely phrase, of playing tricks
with the aperture of the orifice. There is nothing in this which
resembles an adjutage or short pipe through which the water is
allowed to flow freely; and it is, moreover, especially remarkable,
that although the Senatus Consultum would not allow the
grantee of the pipe to vary its size within 50 feet of the calix, it
is silent with respect to the direction in which it was carried,
though nothing can be clearer, that if the pipes slope downwards
the supply of water would be greater than if it were turned in
an upward direction, and of two persons who lived on different
levels one of them must have had an advantage over the other.
Of this, Frontinus appears to have been aware, as he states that
according to the relative position of the acceptorium and the
castellum, the erogation was to be burthened or relieved. But
according to what rule this correction was to be made, he has
not told us.

The question of the nature of the calix is particularly worth
considering, because Prony's hypothesis for determining the
quantity of water brought by the aqueducts to Rome rests upon
an assumption that the calix was in the usual sense of the word
an adjutage.

ON ROMAN AQUEDUCTS. 347

It so happens that in modern Home the unit of distribution
-is a pipe of an uncia in diameter, the pressure on. the depth of
its centre and its length being each r of a palm.

Now, remarks Prony, although Frontinus has given us no
information as to the pressure on the orifice, that is, as to the
depth of its centre below the surface of the water in the reser-
voir, yet the Romans must have had some rule, and as in mo-
dern Rome the rule is to make the depth in question equal to
the length of the adjutage, let us assume that this relation ex-
isted also in his time. The next step is to convert Frontinus's
rule, that the calix must not be less than 12 digits in length, into
a statement that an adjutage must be employed of that precise
length, and then the inference follows at once that the pipes
were placed 12 digits below the surface. But, however inge-
nious this is, there are several objections to it besides the one
already noticed.

In the first place, the argument from tradition is worth little
or nothing in the case of a matter which nowise depends on
popular usage, but is settled at the will and pleasure of persons
in authority, who are free to adopt whatever may seem to them
to be an improvement. In the second place, Italy has for many
reasons long been remarkable for the degree of attention paid to
the subject of hydraulics. The reasons for this are connected
with the physical geography of that country. Not to dwell
upon them, it is sufficient to remark, that in a country in which
so much attention has been paid to the subject there is little
reason to suppose that any point of modern practice is founded
on old tradition; and, as we have seen, the most eminent Italian
writers confess that they cannot discover that the Romans had
any rule as to the pressure on the orifice. Moreover, no one
knew better than Prony how vague were Frontinus's views OD
hydraulics.

Quid te exempta juvat spinis de pluribus una ]

What is gained by getting over one difficulty by an ingenious
assumption, while others, and especially the question as to the
different directions in which water is distributed, remain un-
touched ?

On this hypothesis Prony estimates the quantity of water

348 ON ROMAN AQUEDUCTS.

supplied by a quinaria at about 56 cubic metres in 24 hours,
that is to say, at more than 1200 gallons, an estimate which
appears excessive, notwithstanding all we know of the magni-
ficence of Rome. It is about three times as much as that given
by Dureau de la Malle, who simply assumes, without assigning
any reason for doing so, that the quinaria was subject to the
same pressure as the Pouce de Fontainier in the French system
of distribution, that is, to a pressure of 7 lines on the centre.
The old French system of distribution (and I am not aware
what improvements have of late been introduced) was compara-
tively speaking rude and inartificial, and it is on that account
more likely to represent the Roman practice than the scientific
method to which Prony would compare the latter. No atten-
tion was paid to the presence or absence of an adjutage, and
allowance being made for the thickness of the lead, the upper
part of the outside surface of the pipe would be just at the sur-
face, a mode of placing it which is perhaps indicated by Fron-
tinus's phrases ad libram and ad lineam. The objection at once
occurs that pipes of different bores would have their centres at
different depths below the surface. Prony has remarked that
the Romans must always have subjected the centre of the orifice
to the same pressure (and therefore must have had a different
rule as to depth), because they estimated the discharge of pipes
of different sizes by simply comparing the area of their sections
with that of the quinaria. The inference would be valid if we
had any reason to believe that these estimates were either
founded on or verified by observation. In the present state of
our knowledge it involves a petitio principii, and it is remark-
able that Prony should have attached any weight to it, as he
admits that Frontinus estimated the product of a stream of water
as if it bore a constant ratio to its section.

Several considerations may be suggested which make it
easier for us to believe that neither Frontinus nor any of his
contemporaries were sufficiently acquainted either theoretically
or practically with the principles of hydraulics to be able to
avoid enormous errors. Why, it may be said, was so much
care bestowed on a proper determination of the size of the pipes,
if other elements of the question of supply were neglected?
Surely they must have learned to give up making calculations
on which a little observation would have shown that no reliance

ON ROMAN AQUEDUCTS. 349

was to be placed. In reply to these remarks, which of course
are not without weight, it is to be observed that the questions
with which Frontinus undertook to deal had not pressed them-
selves on the attention of any one at Rome until a comparatively
short time before. Several hundred years had no doubt elapsed
since the first aqueduct was constructed, but until concessions
were made to individuals, and further, until such concessions had
acquired sufficient importance to attract the attention of the
government, there was no occasion to attempt to determine how
much water any aqueduct brought, or how it was distributed —
all went to public uses : private persons were only allowed to
appropriate what was afterwards called aqua caduca, that is,
water which would otherwise have run to waste ; and of this the
amount was probably insignificant.

Not until the time of Agrippa, we are told by Frontinus, did
it become usual to make concessions of water, and he adds, that
it was not until after Agrippa's death that any attempt was
made to reduce these concessions to a system. Moreover, he has
particularly spoken of the substitution of a larger pipe in cases
in which a concession had been made of several quinarise, as of a
comparatively recent practice, and it is obvious that the diffi-
culties of the question could not be felt to their full extent, as
long as no other pipes than quinarias were employed as moduli.
We thus see that not much more than a hundred years, if so
much, can have elapsed during which any attention need have
been paid to the subject. At the end of this period, when Nerva
appointed Frontinus to the office of Curator Aquarum, every-
thing was, according to his own account, in the wildest con-
fusion. He blames the aquarii unsparingly, and no doubt they
had sins enough of their own to answer for ; but that there were
other vices in the system than the frauds with which he charges
them, appears sufficiently from his own statements. All his cal-
culations are completely at variance with the results which his
predecessors had obtained, and which were recorded in the Com-
mentarii Principum. He has no explanation to give of these
discrepancies.

He seems to have been struck by finding that the amount of
water as estimated in quinariae and known to be distributed was
greater than the estimate of the amount supplied. With a strong
impression, therefore, that his predecessors had under-estimated

350 ON ROMAN AQUEDUCTS.

the supply, lie attempted to re-measure it, and formed a new
estimate, greatly exceeding theirs. The supply being now in
excess, he explained the discrepancy by imputing frauds to the
aquarii. Of many they were probably guilty, but if any
scientific knowledge of the subject had existed in his time or
that of his predecessors, there could scarcely have been the
great difference between his estimate of the supply and theirs,
unless the supply varied greatly at different seasons of the year,
and that he and they omitted taking this variation into account.

If I were to guess how the distribution of water was really
made, my conjecture would be something of this kind. The
distribution to private persons having at first been quite a secon-
dary object, their pipes were inserted so as not to receive any
water when the water in the reservoir fell below a certain level,
which was known as the libra or linea. If, therefore, the water
rose above this height, the grantees had the benefit of the in-
creased supply, whatever it may have been. When the pipe
first became full, the level of the water was higher than the
libra by the diameter of the tube, and at this state the phrase
implere mensuram may perhaps have been especially applied.
The determination of the libra was probably made on the prin-
ciple that when the water fell below it, the supply was observed
to be no more than adequate to the public purposes of the
aqueduct. Vitruvius speaks of a reservoir especially devoted
to private purposes, but it does not appear that this arrange-
ment was actually followed ; on the contrary, we know by the
practice being prohibited, that at one time private pipes were
placed, but also by the running streams.

Two or three things would be explained by this hypothesis.
1st, The absence of any statement of the depth below the surface
of the water at which the pipe was inserted. This question
would of course not arise, if the rule were to place the pipe above
what was considered low-water level. 2ndly, The absence of
any statement in Vitruvius as to how the water in the Castella
was to be kept at a constant level. He certainly speaks of the
reservoirs overflowing — that is, of two of them overflowing — but
the third, which in his arrangement was that into which private
pipes were to be inserted, was fed by the other two, and from
the nature of things must have been fed unequally in different
seasons : nor, as I have said, do we know that his arrangement

ON ROMAN AQUEDUCTS. 351

was really adopted ; but if the waters in the reservoirs rose and
fell, the depths of the pipes below the surface must have varied,
for the position of the pipe was plainly fixed. We know so
little about matters of detail, that much stress cannot be laid
upon what I have mentioned, but the difficulty of shifting the
pipes, and the absence of any information as to how the water
in the reservoir was kept at a constant level, are worth consider-
ing. 3rdly, This would account for the absence of any state-
ment as to the quantity of water delivered by a quinaria. The
question, how much does sucji a pipe deliver in twenty-four
hours, must have occurred to the most unthinking person, if any
steps had been taken to make the quantity constant. 4thly,
Frontinus speaks of a case in which the erogation had ceased,
on the ground that the modulus was exhausted — a passage
which would be easily understood, if we suppose that from a
deficient supply the water had sunk below what was accounted
low- water level.

5thly. The fraud imputed to the aquarii of making the
large pipes too large would thus appear to have at least had its
origin in a reasonable principle of compensation. For if the
lowest point of six vicenarias, and that of one pipe of 120 square
digits were all on the same level, the six small pipes, though
nominally equivalent to the one large one, would, by having
their centres lower, deliver a great deal more water, and it
would be easy to calculate the circumstances under which they
would be equivalent to the larger pipe which the aquarii sub-
stituted for one of 120 digits. — Lastly. The old French water
system would then be an improvement on the Roman, and not a
retrocession from it. The former seems to run so far back in
the middle ages that we may fairly suppose it traditionally con-
nected with the Roman. The transition is easy, and may not
improbably have been made at Rome itself. You have only to
take the pipe out of the reservoir so as to allow the efflux of water
to be free, and to let the water just rise so as fairly to cover the
whole aperture. This with making the standard modulus one
inch would in effect be the old French system, which may easily
have suggested itself in answer to a question — what would be
the discharge through a given opening when its mensura was
just filled and no more, and the disturbing influences (due to
the direction of the pipe, &c.) removed ?

352 ON ROMAN AQUEDUCTS.

If what has been said is the correct view, it would follow
that any attempt to estimate the water supply of Rome from the
data given by Frontinus is mere guess-work. Paris appears to
have been ill-supplied with water in 1814, and the supply not to
have exceeded that which a writer quoted by Prony speaks of
as sufficient, namely, an inch of water to each thousand of the
population. Prony himself thinks this to be as much as is
absolutely required, though more may be desirable. It would
appear, if I remember Frontinus's numbers correctly, that in his
time (that is, before three more aqueducts had been added by
Trajan) the water supply of Rome would, on Prony 's theory,
have sufficed at the Paris rate for a population of 42 millions.
Dureau de la Malle, as I have said, gives a far more moderate
estimate ; but there is really no authority for either. My im-
pression is that the supply of London is at least double what
was thought sufficient at Paris. The highest estimate I have
seen of the French water inch is Prony's, who makes it nearly
19500 litres per diem, which would be under 4j gallons to each
head of the population. The Abbe Bossut's estimate would
reduce the amount under 4 gallons. That Eome was far better
supplied than London is clear, but at Eome there were no brew-
eries, no steam-engines, and probably far less manufactures than
in London ; and though there were many more baths there was
much less washing of clothes. A good many fountains, &c.
could be filled with the present London supply if these sources
of expenditure were taken away.

ON THE FORM OF BEES' CELLS*.

MARALDI seems to have been the jifsl person who determined
the form of the rhombs of the pyramidal ending of the bee's
cell : his observations on bees were published in the Memoirs of
the French Academy for 1712. They are very good and clear":
one ambiguous expression, however, has given rise to a miscon-
ception which is still current and is repeated by every writer, or
nearly so, on the subject. Maraldi states that the angles of the
rhomb are 110° and 70°, and that the angles of the trapeziums
which form the sides of the body of the cell, are also 110° and 70°»
I mean, of course, the angles next the apex of the cell, those at
the open extremity of the tube are right angles ; the point to be
observed is this, Maraldi when he gives the results of his mea-
surements makes no attempt to do more than state the value of
the angles to the nearest degree, an amount of accuracy beyond
which it is scarcely possible, under the circumstances, to go, and
which is amply sufficient, considering the irregularities which un«-
doubtedly exist in the structure of individual cells ; but a little
further on in his paper, he remarks, that as the angles of the
rhombs are equal to those of the trapeziums, they must be respec*
tively 109° 28V and 70° 32\ Thus, he remarks, the angles chosen
by the bees have this advantage which conduces to the elegance and
symmetry of the structure, namely, that only two angles are used
throughout. He does not state how he is led to this conclusion,
but an attentive reader will see that after having asserted that
the result of measurement was 110 degrees, Maraldi did not pro-
ceed in the next page to contradict himself by asserting that the
result of his measurements was 109° 28\ There can be no ques-
tion but that, being struck by the equality (so far as his measure-
ments could ascertain it) of the angles in the rhombs and in the
trapeziums, he assumed that in the normal or typical cell, this

* Now first published.

23

354 ON THE FORM OF BEES' CELLS.

equality was absolute, and hence the determination of the angles
became, as I need hardly point out, a matter of spherical trigo-
nometry, namely to determine the sides of the equilateral spheri-
cal triangle whose angles are each equal to 120°, that is, to the
angle between the two adjacent sides of a hexagon. This
angle has for its cosine J, and is, I believe, to the nearest second
109° 28V 16V\

Unfortunately Eeaumur chose to look upon this second deter-
mination of Maraldi's as being, as well as the first, a direct
result of measurement, whereas it is in reality theoretical.
He speaks of it as Maraldi's more precise measurement, and this
error has been repeated in spite of its absurdity, to the present
day: nobody appears to have thought of the impossibility of
measuring such a thing as the end of a bee's cell to the nearest
minute. One can only suppose that Maraldi made so many ob-
servations, varying between 109° and 110°, that the arithmetical
mean came out with this excessive amount of accuracy, a suppo-
sition in itself highly improbable, and at variance with Maraldi's
distinct statement. The subsequent history of the matter is
this, — Eeaumur employed Koenig to determine the form the
rhombs ought to have in order to give the greatest volume to
the cell with the least expenditure of wax. Koenig, for a reason
which will be mentioned by and by, gave as his solution the
following values to the angles of the rhomb, 109° 26' and 70° 34\
He was agreeably surprised, says Eeaumur, to find that his
result agreed within two minutes with Maraldi's measurements,
whereas in reality they only agreed with Maraldi's theoretical
determination. With this determination they ought to have
been absolutely coincident : for it is easy to show that when the
surface of the cell is made a minimum, the angles of the rhombs
are equal to those of the trapeziums. It was soon afterwards
shown that Kcenig's results were wrong, and thus we have been
pleasantly told, that the bees proved to be right and the mathe-
matician wrong, that there was no mistake on the part of the
bees, and so on. Maclaurin was, I believe, the first to correct
Kcenig : he has not always had credit for this priority. Thus Dr
Carpenter in his Physiology, informs us that Lord Brougham,
not satisfied with Kcenig's determination, took into account cer-
tain small quantities previously neglected, and showed that the
coincidence between theory and observation was absolute.

ON THE FORM OF BEES' CELLS. 355

Lord Brougham's own remark is that the deviation between
Koenig and Maraldi had always been ascribed either to an error
of measurement or to an inaccuracy in the construction of the
cells. It is certainly not so ascribed by Maclaurin, who remarks
it could only have arisen from Koenig's not having carried his
computation far enough. Throughout Lord Brougham's elabo-
rate discourse on the matter, one circumstance is omitted, namely,
Maraldi's statement that the angles, according to his measure-
ments, were 110° and 70°.

Either the habit of an advocate's mind, — for Lord Brougham
may be regarded as counsel for the bees, — or his not having
read Maraldi's paper, must have been the cause of this omission.
Nor has he mentioned the advantage which Maraldi conceived
to result from the angles being what they are: namely, that
only two plane angles occur throughout the structure. He
quotes at second-hand the opinion of Boscovich that tfee angles
could not be measured with the supposed degree of precision.

I have not seen what Boscovich says, but have no doubt
that Lord Brougham has misunderstood him : for he makes
Boscovich' s opinion to be, that Maraldi merely deduced his
result by assuming that the dihedral angles are all equal to
120°, that is, he makes Boscovich accuse Maraldi of dishonesty,
whereas, in reality, he probably only repeated what Maraldi
in effect says : that he deduced the precise angles from assuming
that the angles in the trapezium were precisely equal to those
of the rhomboid, his measurements giving the same value to
both, namely, as I have already said, 110° and 70°.

The matter has been so long confused that it is worth while
to quote Maraldi's own words.

* Chaque base d' Alveole est forinee par troisrhombes presqte
toujours e*gaux et semblables, qui suivant les mesures que
nous avons prises, ont les deux angles obtus chacun de 110

degres et par consequent les deux aigus chacun de 70 degres.'
#%##%#

' Ces six memes cotes des trois rhombes sont autant de bases
sur lesquelles les Abeilles elevent des plans qui forment les six
cotes de chaque Alveole. Chacun de ces c6tes est un trapeze
qui a un angle aigu de 70 degres, 1'autre obtus de 110 degre*s,
et les deux angles du trapeze qui sont du cote* de 1'ouverture,
sont droits. II faut remarquer ici que Tangle aigu du trapeze

23—2

356 ON THE FORM OF BEES CELLS.

est egal a Tangle aigu du rhombe de la base, et Tangle obtus

du m6me rhombe e*gal a Tangle obtus du trapeze.7

******

* Outre ces avantages qui viennent du c6te de la figure de
la base, il y en a encore qui dependent de la quantite des angles
des rhombes ; c'est de leur grandeur que depend celle des angles
des trapezes, qui forment les six cdtes de T Alveole ; or on trouve
que les angles aigus des rhombes, e*tant de 70 degrds 32 mi-
nutes et les obtus de 109 degre*s 28 minutes, ceux des trapezes
qui leur sont contigus doivent £tre aussi de la me me grandeur/

The l etant' in the last sentence is plainly the prothesis
of a hypothetical proposition, if they are &c. then so and so.

The best part of Lord Brougham's essay is his argument
against the theory contained in the Article on Bees in the
Penny Cyclopaedia, though rightly or wrongly, he has deprived
himself of the fatal objection to this theory furnished by Bar-
clay's observation of the doubleness of the walls.

A system of associated cells has many curious properties.
The following may perhaps not have been noticed. Round each
corner which is not an apex of any cell, the apices are arranged
in groups of 6 at 6 of the corners of a cube of which the first-
named point is the centre, and each apex belongs to three such
groups.

The peculiar difficulty as to the instinct shown by bees is
this, that one does not see how they perceive when the true form
of their cell is attained. In common cases of instinct, though the
impulse is mysterious, one sees how the animal knows that its
end has been obtained : not so in this case. The following is
my guess. Beside the complex eyes of bees, they have three
single eyes placed lower down, and probably serving for the
vision of near objects. Assume that the axes of these eyes di-
verge so as to be respectively normal to three ideal planes form-
ing a solid angle, each dihedral angle of which is of 120
degrees. Geometry shows that every solid angle of the bee's
cell is precisely similar to this type, so that a bee looking at it
with his three single eyes, might have direct vision with each
eye, of one of the three planes of the solid angle. This direct
vision may correspond to a particular sensation, so that a bee is
not satisfied till it is attained. If we had three eyes, the axes
mutually at right angles, we should, I think, be well able to

ON THE FORM OF BEES' CELLS. 357

judge whether the walls and ceiling of a room were truly at
right angles to each other. And so in the case proposed. To
this guess two objections have been made, the first, that most
if not all hymenoptera have similar eyes, but do not make
similar cells ; the second, that there is very little light inside
a hive : but neither appears to me conclusive.

I remember a little inaccuracy in the way I explained my
notion as to the manner in which bees are guided in making
their cells. I said that all the angles of the cell were of a cer-
tain type ; I should have said* all the trihedral angles. There
are of course three others, each bounded by four sides, but
these also have the fundamental property that all the dihedral
angles are of 120 degrees, so that the bee could always obtain
direct vision of the faces of each dihedral angle.

Take two equal cubes, divide one into six pyramids, the base
of each being the face of the cube, and the apex at the centre of
the cube, fit each of the six pyramids by its base on a face of the
other cube, divide the solid thus got by a plane through the
centre of the central cube, normal to a diagonal. Each half is
the typical form of the bee's cell, except that the prismatic por-
tion of the latter is a little longer than according to this con-
struction it would be. There is of course no theory as to the
length of this portion of the cell, and I believe no accurate obser-
vations.

The above is, I think, the simplest way of conceiving the
form of the cell, and as far as I know it is not given in any book:
it would certainly be the easiest way of modelling a cell.

ON THE THEORY OF VEGETABLE
SPIRALS*.t

THE circumstances under which the following remarks have
been composed would be a sufficient excuse for any defects, if
they were not rather a reason why no such task should have
been attempted. But the error, if it be one, will not be re-
peated, and, like Socrates, ^^KaXv^ra^evo^ €p& \.

The following remarks on the Theory of Vegetable Spirals,
appear to possess some interest, though it is very possible that
it is only my imperfect acquaintance with the history of the
subject which makes me think they contain anything not already
noticed. On the chance however that they are new I shall en-
deavour to put them down, though my state of health obliges
me to do so in a hurried and imperfect manner.

The fundamental principle of what follows is the resolution
of the symmetrical spiral into portions actually unsymmetrical
yet capable of becoming regular without essential alteration.
Symmetry is therefore regarded as something superadded to the
essential principle of the spiral, and we are thus led to trace
the existence of this principle in cases in which it exists apart
from any appearance of symmetry.

A spiral may be divided in two ways : by two azimuths or
by a horizontal circle. We may either obliterate all the leaves

* Now first published.

t Though the nature of the subject and my own ignorance make the whole of
these remarks unsatisfactory, no part of them seems to me more so than that which
relates to the way in which simpler forms are included in the more complex. I
have not expressed my own idea, imperfect as that is. The following statement
would be nearer it : " Every form includes all simpler forms, modifying them alter-
nately in opposite directions, i. e. by expansion and contraction."

t Plat. Phcedr.

ON THE THEORY OF VEGETABLE SPIRALS. 359

which grow, for instance, on the north side of a tree, or all those
belonging to a given spiral which rise above a circle drawn
round the axis of growth. The two modes of resolution give,
as we shall see, similar results ; but we shall set out from the
first. If we were to take off all the leaves on the north side of
a tree, those left could not be arranged in spiral order at all,
except in particular cases : but there is always one mode in
which spirals may be divided by means of two azimuths, so that
the essential principle of the spiral remains untouched. This
will be clear from the figure .which, like all the other figures,

Fig. i.

represents a spiral in a state of development, that is, as if un-
wound from the cylindrical axis on which it grew, so that the
right-hand and left-hand sides of the xectangle are in reality one

360 ON THE THEORY OF VEGETABLE SPIRALS.

and the same line. The spiral in the figure has for its angle
of divergence the fraction -j^-, that is, it consists of 11 leaves
and goes 7 times round the axis ; counting always from left to
right the successive leaves are found at intervals of 7 cells.
Now if we blot out all that lies to the right of the line DD and
consider the first 7 cells by themselves as having been unwound
from a smaller cylinder, the leaves will be found to stand at
equal intervals from each other, namely, at intervals of 3 cells.
There is indeed a dislocation as to height, but height is through-
out the Theory of Vegetable Spirals an unessential, or at least
very variable element. Similarly what lies to the right of the
line DD forms a system of 4 leaves, similarly arranged at in-
tervals of 3 cells. Thus the spiral ^ is resolved into two whose
angles of divergence are respectively f and f . The former is
of course the same as a spiral whose angle of divergence is %
only running in the opposite direction ; and as it is usual to
give the name of Fundamental Spiral to the flattest or most
sloping spiral which can be drawn through the leaves of a given
system, the resolution we have effected may be described by
saying that the original system has been divided into two, whose
fundamental lines run in opposite directions, thus suggesting
the notion of two opposing or antagonistic growths, making up,
by the mutual influence of their antagonism, one symmetrical
whole. Now I would propose, as a sort of postulate, the assump-
. tion that every system which forms a part of a system actually
existing in nature is capable of independent existence, that it
only wants, if the expression may be used, to be allowed to get
possession of the whole axis of growth in order to shape itself
into a symmetrical spiral. The consequences of this assumption
are sufficiently interesting to make it worth while to trace them
in detail. They afford a remarkable instance of the confirmation
by facts of an a priori principle : the principle in question being,
so to speak, an interpretation of Schelling's celebrated maxim*,
that the whole is in every part and every part in the whole ;
a maxim which, though it may have led Oken wrong (I allude
of course to his speculations as to the significance of the bones
of the skull), yet enabled him to lead others right.

2. Let us in the second place make an assumption naturally

* Thi& maxim is in effect equivalent to the fundamental principle of Leibnitz's
philosophy, that each monad represents the universe.

ON THE THEORY OF VEGETABLE SPIRALS. 361

connected with what has been already said, namely, that the
portions into which any real system can be resolved are, as in
the example just given, opposed to one another in the direction
of growth. This assumption, combined with the postulate pre-
viously stated, enables us to determine a priori what systems
are real or possible, and what not.

N
If the angle of divergence is -~ in the original spiral, those

in the resolved portions have the same numerator, namely,
2N—D, the denominators Heing respectively N and D — N,
and the two fundamental spirals run in opposite directions, pro-
vided the original fraction is not greater than f or less than f .
The proof of these propositions I will not stop to give ; a little
consideration will enable any one to supply the omission. Our
assumption therefore excludes all fractions lying beyond the
limits just stated, and in fact we need not consider the second
limit at all, as the first is sufficient to guide us to the conclusions
we require.

In order to determine the successive spirals into which the
original one may be divided and subdivided, we proceed as if
we were seeking the greatest common measure of the numerator
and denominator of the angle of divergence, which of course
have no common measure but unity. As long as the successive
fractions are less than f , the successive quotients are of course
unity; but whenever the fraction transcends that limit, the next
step of the process will introduce 2 or some higher number as
a quotient; and conversely no such quotient can appear until
this limit has been transgressed. Hence this conclusion, every
real system has both its numerator and denominator consecutive
terms in a certain recurring series, namely, 1, 2, 3, 5, 8, 13, 21,
34, 55, 89, &c. of which the law is that each term is the sum
of the two preceding ones. For these are the only fractions
which, according to the method of continued fractions, can pre-
sent themselves when all the quotients are unity. Thus f , |,
and so on, are the angles of real systems, and there can be no
system not similarly included in the preceding series. Every
other system will, when our dichotomizing process is carried far
enough, present us with two sub-systems, which have lost the
character of antagonism, and have their fundamental spirals in
the same direction. We have thus arrived a priori at the con-

362 ON THE THEORY OF VEGETABLE SPIRALS.

elusion established inductively by Schimper and Braun, namely,
that these are the only numbers which present themselves in
vegetable spirals. I believe at least I am justified in saying so,
for though Professor Henslow*, in giving an account of Braun's
researches, does not say that no other numbers occur than those
of the series, yet all his examples belong to it, and the principle
is broadly stated as a general law by linger in his Botanical
Letters|. To this numerical law we can now append an equi-
valent statement, which not being numerical brings us a good
deal nearer to a theory of the actual genesis of vegetable spirals,
for we may substitute the following as an equivalent law, namely,
that every vegetable spiral is resoluble into antagonistic portions,
actually unsymmetrical but potentially symmetrical. In this
form the law serves to give morphological significance to many
obvious phenomena ; as for instance, the three larger petals of
the pansy stand in opposition to the two smaller, the balance of
symmetry not having been fully established. Again, the five
petals of papilionaceous flowers present us with two petals op-
posed to three, the three latter containing a sub-system of two
petals opposed to one. Thus perhaps too we are led to recognise
a certain duplicity in the structure of cruciferous flowers ; but
I will not venture to enter on any but obvious examples.

It is interesting to observe how consciously or unconsciously
we are always influenced by the forms of nature. That man is
her minister and interpreter is true, not only of his knowledge
and his power, but in that which combines both, his works of
art. That volutes and tracery reproduce the forms of the vege-
table world is obvious, but we may be reminded of the numerical
relations of which we have been speaking when we recollect
how often three lights in a western window are opposed to five
in an eastern: the higher symmetry in the holier place and
related to the lower as the flower to the leaf.

Sed fugit interea, fugit irreparabile tempus;
Singula dum capti circumvectamur amore.

3. The result at which we have arrived may perhaps be
more clearly expressed by saying that every natural spiral can
be resolved into two parts, of which the smaller stands in the

* In Lardner's Cabinet Cyclopaedia.

t Naumann appears to have recognised a spiral of 377 elements, thus reaching
the thirteenth term of the series.

ON THE THEORY OF VEGETABLE SPIRALS. 363

same relation to the greater, that the greater does to the whole ;
thus when a system of 8 leaves is divided into 5 and 3, as the
first contains the second once, so does the second the third;
moreover the second is opposed in direction to the first, and so
similarly is the third to the second. Or we might say, though
it is dangerous to use mathematical terms except with mathe-
matical precision, that every natural spiral can be divided in
extreme and mean ratio.

Thus much respecting division Iby azimuths. We now come
to consider horizontal division, that is, division in which all
above and below a given horizontal circle is successively sup-
posed to be obliterated, so as to leave in the first instance only
the lower part of the spiral and in the second only the higher.
It may be easily shown — but as before I shall not stop to give
the mathematical demonstration — that if JVand 1) are respectively
the numerator and denominator of the angle of divergence of any
spiral, and N' and D' respectively are the numerator and deno-

N
minator of the last converging fraction to •_ ; then D' and D — D'

are the number of leaves in the two portions, N' and N—N'
respectively being the corresponding number of times that the
spiral goes round the axis. Thus recurring to the former ex-
ample, the spiral whose angle of divergence is ^ is resoluble
into two whose fractions respectively are | and f . In figure 1
the line AA indicates this division, there being 3 leaves above
and 8 below. Of course the division might have been made so
as to leave 3 rows below and 8 at top. But this is scarcely
worth remarking ; it may be more needful to point out that in
examining the figure we are not, as before, to be guided by
the number of cells, but must look at each set of leaves as we
would at a spiral existing in nature. It will be observed that
there is a deviation from the perpendicular in both parts of the
figure ; a retardation, so to speak, in the one and an acceleration
in the other, and that these, on the whole, balance. As in the
former mode of division there was a certain amount of vertical
displacement, so here there is of horizontal.

Let us now observe the prcerogativa which, with reference to
this mode of division, belongs to the natural systems. It is this,
that whether divided vertically or horizontally the result is the
same. Not so with the case of the figure; 11 was before divided

364 ON THE THEORY OF VEGETABLE SPIRALS.

into 7 and 4, now into 8 and 3. But when the quotients in the
process of finding the greatest common measure are all unity,
then the successive fractions formed by the remainders are them-
selves the series of converging fractions. Thus j is the last
converging fraction to T^, and so is f to f . In consequence of
this the character remarked at the beginning of this section with
respect to vertical division presents itself again with respect to
horizontal, namely, that natural systems can be divided into
parts, of which the first is to the second as the second to the
whole ; whereas, to recur for a moment to the figure, f is not
the last converging fraction to f , although f is so to ^

There is something very interesting in this recurrence of
similar relations among the parts of natural systems. One of
the Bernoullis engraved upon his tomb the equiangular spiral
as an image of Immortality, giving it the motto ' Eadem mutata
resurgo :' a vegetable spiral might similarly be chosen by Braun
or Schimper.

But what gives this mode of division a peculiar interest is
that it brings us nearer to the actual genesis of the spiral. For
the lower leaves are formed before the upper, and as the process
of dichotomy may in this as in the former case be carried on
indefinitely, we thus come to the remarkable conclusion that
every vegetable spiral has been* every simpler spiral before it
becomes what it is. It is impossible not to be reminded of the
similar doctrine which has been held with respect to animal or
rather with respect to organic life in general, namely, that the
higher forms are not merely typically connected with the lower,
but actually developed from them. It would not be wise to
carry our inferences too far, but with respect to vegetable forms,
so far as these consist of arrangements of leaves, the matter
admits of demonstration. It is no more than anybody may
see who will take the trouble to count the scales on a fir
cone.

Though I have said that we must not carry our inferences
too far, it would, I think, be impossible to stop at the limits
which in the present state of our knowledge lie within the range
of precise demonstration. The production of leaves mark, to
use the German phrase, a series of similar moments of develop-

* In saying this, we of course neglect the small acceleration or retardation
already mentioned.

ON THE THEORY OF VEGETABLE SPIRALS. 365

merit in the life of the plant; but what is true of such series
must, we can hardly doubt, be true of all. We can hardly look
at figure (2), in which the spiral T8^ is represented, without seeing

Fig. 2.

at its base the primitive phyton with its two leaves and radicle :
nor having got thus far not to remember that the laws of vege-
table spirals are simply the result of their capability of division,
and that the ideal division of the spiral is only a type of the
actual division of the cell. The Monad, said the Neo-platonists,
produced the Duad, and the Duad all things.

4. Maxims .of this kind may well be called ' oracula mentis;'
they envelope the truth rather than express it, and like oracles
do not teach us how we are to interpret them. They conse-
quently resemble oracles in this also, that for the most part those
who are guided by them are led astray. For it is difficult to
resist the sort of charm which philosophical dicta possess when
having reached the boundary land between the finite and the
infinite, they pass into the region of poetry*. In thinking, for

* See W. von Humboldt on the "Bhagavad Gita."

366 ON THE THEORY OF VEGETABLE SPIRALS.

instance, of the relation of species to the individual, of the whole
to its constituent parts, of the unity of type and the endless
diversity of forms, we feel that our doubts and difficulties can-
not be better expressed than in the words of the old Orphic poet :

7TW5 e pOl f.V Ti TO, TTO.VT COTCtS, KCU ^COptS €Ktt(7TOV J

Is a cell a unit, or is a leaf, or is a whole plant, or is the idea
of unity a relative one, realised in different degrees in lower and
higher organisms ? However unsatisfactory this view certainly
is, so much must be admitted, that the constituent parts of lower
organisms have a greater tendency to commence an independent
existence than those of higher : the bond of unity is weaker.

If, following the train of ideas suggested in the last section,
we attempt to express the difference between monocotyledonous
and dicotyledonous plants, we may perhaps say that the former
assume the condition of independent phyta before they have
raised themselves above the system expressed by the number
two, or if we look to the leaf alone above the first system,
whereas the latter stand one step above them in complexity.
And it is interesting to observe that they keep at the same dis-
tance throughout, at least, in the greater number of cases three
being the prevailing number in the flowers of the former class
and five in those of the latter. There is no real spiral structure
in what for the sake of regularity we call the spiral |-, because
there is nothing to determine which way we are to turn ; growth
is in this case an oscillation in a single plane, and the formation
of a structure possessing three dimensions must be the result of
the superposition of growths in parallel planes. There is no
single spiral growth upwards and outwards, binding together
indissolubly the growth in both horizontal dimensions ; the
primary direction of growth is therefore directly upwards, and
any growth upwards and outwards corresponds to another with
which it may be said to coalesce, directed outwards and down-
wards. Similarly there is no spiral growth downwards and
inwards, no tap-root, and therefore in effect no true root at all ;
what supplies its place is the downward and outward growth
tending to recur at every part of the structure. I do not say
that any of this amounts to a demonstration, but it certainly
coincides in a remarkable manner with what we know, as a
matter of fact, of the growth of monocotyledons, not only with

ON THE THEORY OF VEGETABLE SPIRALS. 367

respect to the absence of a tap-root and to the tendency to put
out adnate roots, but also as relates to Mohl's discovery of the
crossing of the vascular bundles or fibres; a discovery which
might, I think, have been anticipated by an attentive considera-
tion of morphological principles; or which, at any rate, now
that it has been made, seems to confirm them. Figure* (3) is
intended to illustrate what has now been said, and we shall see
hereafter that the same view is confirmed by a consideration of
the leaves of this class of plants. That they stand lower in
point of organization than dicotyledons is admitted by every
one, and seems to be indicated on the principle we have already
noticed by their tendency to put out roots above the ground as
well as by many other circumstances, and we seem to be justified
in inferring from this that the spirals which are numerically
more complex are really types of higher organizations.

It may be well, in order to obviate a misconception of my
meaning, to explain what has been said as to upward and out-
ward growth in dicotyledons. No doubt the stem or trunk of
any tree belonging to this class is a portion of a conical surface
whose apex is above the ground and not below it, and the same
is true of the surface of contact of cambium and wood, that is,
of the surface where fresh wood is, at any given moment, in
process of being formed, so that in one sense the growth is up-
wards and inwards. But the corresponding conical surface a
year ago lay within and below the other, so that the transition
from the one to the other is outwards as well as upwards, and
that it takes place in a spiral as if the surface turned on its axis,
and at the same time enlarged, may reasonably be presumed in
accordance with the views which I have been endeavouring,
though with much diffidence, to support. Of course it is not an
objection to these views to say that the woody fibre does not
consist of spirals, actual growth being one thing and the de-
velopement of forms, or what may be called ideal growth, an-
other.

5. In endeavouring to apprehend the process of transition
from one kind of spiral to another, it is difficult in the present

* The figures (2) and (3) referred to in the text, were not to be found among
the papers placed in the Editor's hands : fig. (2) on page 365 has been drawn by
the Editor, who is however unable to infer with any confidence the form of the lost
figure (3).

368 ON THE THEORY OF VEGETABLE SPIRALS.

state of our knowledge, or perhaps impossible, to arrive at an
explanation not based on morphological grounds or expressed in
morphological language. Yet it is certain that we ought not
to acquiesce in a merely morphological explanation of any phe-
nomenon, though it is going too far to say, that such explana-
tions are merely transitional or provisional, to be accepted only
until something else can be got. For teleological considerations,
whereof morphological are one kind*, neither exclude nor are
excluded by those which relate to actual genesis. The formal
and the final cause alike exist and coincide, if I may presume
to touch upon the subject, never to be mentioned but most
humbly and most reverently, in the Unity of the Divine Wis-
dom, from which both proceed, and the explanations derived
from either are, or rather would be, if our knowledge were com-
plete, equally and alike true, though in different ways. But to
return to the matter in hand. It seems reasonable to suppose
that when vegetable growth is proceeding according to any
given system, say for instance, that whose characteristic frac-
tion is 3%, the tendency to rhythmical recurrence on which the
whole depends still exists among the parts of which the whole
is made up, especially as these two parts are as we have seen
nearer in point of position to constituting a symmetrical spiral
than any other two portions into which the whole could be
divided. The first or last five, or the first or last eight are
within -jL of the circumference of going round the axis an integral
number of times. Now as we have seen, it is the larger portion
of a spiral which possesses a character of antagonism to the whole.
If the interval between adjacent points exceeds the limit towards
which this interval continually tends f, then that between adjacent

* It is surely only from want of attention to the History of Philosophy that the
two are put in opposition to one another, whereas in the sense in which Aristotle
would have used the term, the doctrine of those who speak of the Unity and Modi-
fication of Types, and that of those who refer every thing to the advantage of the
individual or species, are alike based on the Final Cause. Moreover there seems
no sufficient reason for setting the two doctrines in opposition, still less for reject-
ing either. Truth comes to us in fragments, and so we must be content to accept
it, not letting ourselves be deterred by any antinomy from embracing all of it that
we can apprehend.

t Namely four right angles multiplied by the positive root of the equation
xz + x=i. I use the word tend here in the mathematical sense of tendency to a
limit. It is, I believe, Bravais' doctrine that there is in reality but one angle of

ON THE THEORY OF VEGETABLE SPIRALS. 369

points of the spiral next below it falls short of this limit, and
vice versa. In consequence of this, under the influence of circum-
stances, which we cannot appreciate, the last eight points may
repeat themselves in a symmetrical position, and maintain this
form of symmetry permanently. On the contrary, if after having
thus repeated themselves they are again subjected to the in-
fluence of the thirteen preceding points, a sort of compromise
is effected, of which the result is a spiral of twenty-one points.
The two cases resemble a contest between the constitution of
a whole kingdom and that of a particular province ; the pro-
vince may revolt and succeed in establishing its own law
universally, or a mixture of the two may become the common
law of the land. Thirdly, things may remain as they are and
the provincial law be altogether abrogated. Thus the opposi-
tion of the greater part to the whole may under varying cir-
cumstances produce an increase or a decrease in the complexity
of the system, and the same cause, namely the rhythmical ten-
dency which seems to belong to all growth, may be at one time
a cause of change, and at another of permanence. If we try to go
beyond this, " ad ulteriora tendentes ad proximiora recidemus,"
our explanations must become vaguer and yet more unsatis-
factory. Something might be said of the contest between the
principles of recurrence and opposition, but it would too much
resemble the Us and amicitia of Empedocles1.

6. The attempt I am about to make to trace the general
characters of vegetable arrangement in the structure of leaves
is based upon so simple a remark that I can hardly suppose it
has escaped observation. As however I have not met with it, it
seemed worth while to set it down here though I cannot trace
its consequences in detail. The primary division of the nerves
of leaves is into anguli-form and curvi-form nerves, 'that is,
those which have a straight portion at their base and those
which exhibit curvature throughout. Now just as the straight

divergence, namely, this limiting one; and that if botanists have thought they
recognised distinct systems, it has teen because they have stopped sometimes at
one degree of approximation to it and sometimes at another; but this doctrine ap-
pears to be opposed to ascertained facts, the real unity of nature does not in this
way exclude variety.

1 When the spiral passes into a whorl, the tendency to rhythmical recurrence
seems generally to cease and to be replaced by a tendency to symmetrical anti-
thesis or alternation, as if, speaking mathematically, positions of equilibrium pre-
viously stable had become unstable, and vice vend.

24

370 ON THE THEORY OF VEGETABLE SPIRALS.

growth of the stem is the result of a quaquaversal tendency to
curvature, so more generally I think it may be asserted, that
straight growth in any part of a plant is constrained growth,
the result of a tendency to curve in opposite directions. What
characterises, therefore, the venation of such a leaf as that of
the lily is in this sense the absence of constraint. To this
remark an hypothesis is to be added which seems to be justified
by the phenomena presented by compound leaves, namely, that
the nerves are originally boundary lines, showing where the leaf
in its growth paused in its development, just as a range of
pebbles marks the old boundary of sea and land after the sea has
receded. Thus the lily leaf shows us a single growth in various
stages of progress. If we now look at the second division of
monocotyledonous leaves, those which have a mid-rib and angular
formed nerves, we see the successive stages of a double growth
lying on the right hand and left hand respectively of the mid-rib,
which is to be regarded as the straight resultant of the opposite
curvatures of the boundaries of the two portions; the remaining
part of the boundary remaining free and therefore curved. When
therefore we say, that with few exceptions, monocotyledonous
leaves follow one or other of the types just described, we in
effect say that they are either single or double growths, that
they belong to the first or second forms of vegetable develop-
ment, and rise no higher. Let us now look at the commonest
form, namely the penni-nerved of dicotyledonous leaves ; in these
not only the mid-rib but the diverging nerves are, at their base
at least, straight, exhibiting evidence therefore of constraint,
or opposing growth on their two sides. The space between
the apex and the first pair of lateral ribs seems to me to be
a leaf in itself, that between the first pair and the second to be
similarly two independent portions osculating with one another
and with the first. By their osculation with one another, they
form an additional portion of mid-rib, by their osculations with
the central part they form the first pair of lateral ribs, and if
we follow the course of either of these ribs we find that it gives
out secondary branches, which freed from constraint, or com-
paratively so, become curved. In the space they leave, how-
ever, we find a secondary central portion, so that the meeting
of three elements is exhibited at the sides of the leaf as well
as in its central growth: thus the reticulation is throughout

ON THE THEORY OF VEGETABLE SPIRALS. 371

polygonal, and not properly speaking, curved, though from the
shortness of the sides of the polygons it may assume a curvi-
linear appearance. At the sides, however, there often runs a
sort of fringing nerve between which and the edge there is
nothing but parenchyma, and this nerve being free from con-
straint appears truly curvilinear. In a simple leaf the lateral
growths remain subordinate to the central ; the contest between
them being shewn by the jagged edges, the cicatrices veteris
vestigia pugnse. But if the lateral growths prevail, the central
one ceases, the osculations no longer taking place in the manner
described, and thus the leaf becomes compound, the fissures
now reaching to the axis.

From what has been said, it appears that a penni-nerved
leaf belongs to that system of vegetable growth of which the
characteristic number is 3, divided as such a system must be,
into 1 against^ 2, that the 1, so to speak, assimilates 2 and 3,
so as to form a second unit or 1' ; and this entering similar
relations with similar lateral portions 2' 3' forms a third unit,
a wave of form being thus propagated from the apex towards
the base, and increasing in size as it goes along. To enter into
an examination of the cases in which a simple leaf belongs to
a higher system than the third or of the structure of compound
leaves would detain us too long. It is enough for my purpose
to have pointed out the two principles by which I believe all
the details are governed, and by the help of which they may
be traced out. The distinction between a leaf and a floral
whorl which makes the numerical relations so much less easy to
recognise in the former case than in the latter, is, that in a
floral whorl the axis of such symmetry as exists coincides with
that of growth, whereas in the case of an ordinary leaf, the two
are at right angles. It is interesting to observe that the dis-
tance between monocotyledonoiis and dicotyledonous plants
which shows itself in the commonest cases in their respective
phyta and flowers, is also seen in the intermediate growth of
the leaf, the commonest number which presents itself in the
dicotyledonous leaf being that which occurs most frequently in
the monocotyledonous flower. We are, I think, entitled to con-
sider the leaf as intermediate between the phyton and the
flower, since reckoning the sheath as one, it consists of three
elements, as the phyton of two and the flower of four: the

24—2

372 ON THE THEORY OF VEGETABLE SPIRALS.

respective element of all three fulfilling Goethe's law of alternate
contraction and expansion.

The ordinary form of the edge of a dicotyledonous leaf, sup-
posing the jags filled up, is sufficiently well represented by the
projection on a vertical plane of an equiangular spiral, chased
on an inverted cone (I mean of course the outline of the half
leaf, commencing at the base and going thence to the apex).
The curve in question cuts the axis at a greater angle at the
base than at the apex, and has its maximum ordiriate nearer
to the former than to the latter. Something may be said on
theoretical grounds in favour of taking this as the typical leaf
curve in cases in which the petiole and lamina lie in the same
plane, since when they are at right angles, or nearly so, there
is an approximation towards a circular form, as if the plane
of projection had now become parallel to the base of the cone,
and the slope of the spiral were diminished. In truth this
principle of projection might be applied to explain many varia-
tions of form in the flower as well as in the leaf.

7. The preceding remarks can scarcely be free from mis-
takes, and yet they may contain a portion of important truth,
namely, the principle of tracing numerical relations in the un-
symmetrical as well as in the symmetrical aspects of vegetable
growth. Just as spirals are less symmetrical than whorls, and
yet enable us to understand the latter better than we otherwise
could have done, so likewise unsymmetrical portions of spirals
and of whorls may throw new light upon both *.

1 For an account of recent speculations on the theory of vegetable spirals, see
Braun, Betrachungen uber die Ersckdnung der Verjiingung in der Natur, p. 125.

THOUGHTS ON COMPARATIVE
METROLOGY*.

IN many respects it resembles comparative philology; one is,
the necessity of avoiding general hypotheses. Corresponding
to the attempts to deduce all languages from some one assumed
to be primitive, or from certain elementary sounds, such as
Alexander Murray's nine words, are such hypotheses as that of
Gosselin, who, following Bailly, thought that at some early age
in the history of the world, the length of the meridian had been
accurately determined, and that all the guesses which have come
down to us as to the magnitude of the earth were all equally
accurate statements of the result of this primitive geodesy, the
difference being solved by assuming the existence of a number
of different stadia. Touching Bailly's astronomical speculation,
it is hard to deny the truth of De Lambre's judgment, who
describes it as "le plus cru des songes," or by some equiva-
lent phrase. Another general hypothesis seems little better, I
mean that of Boeckh, namely, that the weights and measures of
antiquity all came from Babylon, and were connected with the
use in astronomy of the clepsydra. Thus weight is the primi-
tive element, then capacity is derived from weight, and lastly
linear measure from capacity ; the unit of the last being the edge
of a cube whose volume was the unit of the second ; the unit of
the first being the weight of that volume of water. Starting
from hypotheses so unproved as these, and modifying them as
occasion may require, we may derive anything from anything
and explain everything.

2. In another point of view metrology resembles philology,
namely, that similarity is a very bad test of real connection, and
that much better evidence is furnished by formal relations than

* Now first published.

374 SOME THOUGHTS ON

by material resemblance. The tradition of magnitudes is always
inaccurate even when there is not, as in the case of coinage,
any advantage to be gained by inaccuracy. The French livre,
for instance, though shrunk to a shadow of its former self, pre-
served to the last the due number of sous and deniers. The
tradition of magnitudes is, so to speak, physical, that of relations,
mental.

3. William Von Humbolt wrote an ingenious essay (first
published in the Bevlin Transactions") on- the influence which
is exerted on a language by its having been reduced to writ-
ing. It is easy in a general way to see that writing not only
gives greater fixity to language but also greater influence to the
educated class by whom alone it is familiarly employed. Cor-
responding to this is the reduction of weights and measures
to tables, &c., whieh are made a part of ordinary education,
and here we find an analogy which both the sciences I have
been speaking of bear to jurisprudence. In the famous essay
on the vocation of the age to codification, Savigny has pointed
out that in the earlier period of the history of a nation law is a
living and popular thing, " volitat per ora virum," and does not
fall into the hands of a particular class, at least not to any great
extent, until a later time. It ascribes this mainly to its increas-
ing complexity, but it arises also from the division of labour
inseparable from the progress of civilization, and likewise from
other causes. In English literature it is curious to see how
much more frequent legal allusions are in old than in more
recent writers. Some have fancied that Shakespeare must have
Lad a legal education, but the same thing might have been
thought with regard to Chaucer. Hugo has pointed out that
many phrases of quotation or reference in the Digest relate not
to the maxims of earlier writers but to what may be called legal
proverbs. To these correspond in the history of Metrology such
maxims as " a pound is a pint,'r or the Arabs' saying that a
mile is as far as one can tell a man from a woman. Of much
the same nature are the names given to measures of land from
the quantity of seed which they were supposed to require.
Half legal, half metrological are such things as Mr Justice
Buller's well-known rule, and the curious determinations in the
Sachsen Spiegel, whereby a man was hindered from making his

COMPARATIVE METROLOGY. 375

house his fortress. He was to dig no deeper trench than he
could throw the earth out of with his spade, to raise the bottom of
the door no higher than his knee, to enclose himself by no wall
higher than a man on horseback could look over. Cases of
the limitation of jurisdiction by throwing a stone or shooting
an arrow from a given spot and noting where it falls, are men-
tioned by Grimm, and there is even now a ceremony of the same
kind at Cork, the mayor annually throwing a javelin into the
sea. The meaning of his doing so was doubtless in its origin
some determination of a boundary.

Of popular pieces of metrology I may mention that I have
heard that a certain popular hymn-book passes current in Han-
over as a pound weight. For all these things picturesque and
inaccurate as they are, advancing civilization substitutes ab-
stract numerical determinations, but the natural tendency to
connect standards of different kinds with one another, and all
with natural objects, remains. For the foot or the fathom we
employ the length of the meridian or of the pendulum. It
may be quite true that as things now are it is better to trust
to standards accurately made and carefully preserved, but
the desire to recognize something ideal in practical life and
to connect ourselves with something less perishable than our
own handy-work, is not easily to be dismissed. How interest-
ing, says the report of the French Metrical Commission, for
the father of a family to know what proportion of the sur-
face of the earth is the field which supports his family ! The
remark is a little in the tone of the golden age of Rousseau, but
nevertheless it has its foundation in the realities of human
nature.

4. What the historian of Metrology must do is, dismissing
general hypotheses, to observe the nature of the relations
which the popular mind seeks and establishes between different
standards, making large allowance both for popular inaccuracy
and for the love of simplicity. He must also observe the nature
of the traditions by which different standards are handed down,
and the greater or less respect in which they are held : the curse,
for instance, on him who removes a land-mark, and the unscru-
pulous way in which arbitrary governments have always dealt
with coinage. Then again with regard to what may be called

376 SOME THOUGHTS ON

positive metrology, we must take into account the degree of
scientific culture and of practical accuracy which may be sup-
posed to have existed in different ages and countries. What
follows is merely some remarks on detached points.

5. I do not now recollect Boeckh's explanation, founded on
the general view already noticed of the relation between the
Greek and the Roman foot, but a different one to which I
think he makes no reference, seems to me far more probable.
The Komans starting from the assumption that the pace is five
feet (which is as near the truth as any relation expressed so
simply could be) made a unit of itinerary measure of a thou-
sand paces length. Observe by the way how like this is to
the derivation of the sestertium from the sestertius. Now, it is
a general principle that units tend to divide themselves into
halves, quarters, &c., although they are for the most part mul-
tiplied by tens or twelves, the decimal system seeming much
more natural in the upward than in the downward scale. Thus
the Komans would speak of distances as being a quarter or half-
a-quarter of a mile ; and this last fraction was sufficiently near
the stadium to be identified with it. On the other hand, the
Greeks, starting from another natural assumption, made the
orgyia six feet, and a hundred of the former became their unit of
distance. The stadium would seem less an itinerary than, if
the word may so be used, an athletic measure, for it is cer-
tainly more natural to measure distances by paces than by
fathoms. So it was, however, that 600 Greek feet were identi-
fied with the eighth of a Roman mile — that is — with 625 Roman
feet, which gives the common relation of 25 to 24. I do not
mean that the average Greek foot was not longer than the
Roman. We know that it was, and the difference probably
corresponded to a difference in the two races of men. But what
we have to explain is the origin of the precise relation in ques-
tion. Again, it is not to be supposed that after it was recog-
nized, the measure-makers of Greece and Rome thought them-
selves bound to conform to it. Writers on metrology perpetually
fall into a kind of realism, and speak as if they believed that
there was for instance an ideal Roman foot, an archetype to
which the Romans perpetually endeavoured to conform them-
selves, and which it is our business to discover. Whereas there

COMPARATIVE METROLOGY. 377

was of course in reality no general archetype. One measure-
maker copied another with more or less accuracy, and all we
can do with respect to absolute magnitudes, is to determine as
well as we can their average value. The intention only is ideal,
and this cannot have reference to the tradition of standards of
the same denomination, but to the relations among different
standards. I add a few words of etymology. There can be
no doubt that orgyia is properly oregyia, and is derived from
orego. Thus the word means the ^stretch, which is not, properly
speaking, the meaning of either, of fathom or of the French
brasse, though all three words in effect mean the same thing,
namely, the greatest distance at which the finger-tips of the two
hands can be placed from each other. Brasse comes from em-
brasser, and suggests the idea of a man putting his arms round
a pillar or a tree. Whether any appreciable difference exists
between the girth of a tree which a man could just embrace
and what we call a fathom I do not know, nor am I likely to
try the experiment. Fathom has the same origin as brasse,
as we may see without referring to Anglo-Saxon in the modern
Danish, in which favn, a fathom, is obviously connected with
the verb favne, to embrace. The latter, by the way, we have
in English, in to fawn, though we only use it figuratively,
or with reference to dogs who cannot properly be said to em-
brace those whom they caress. The word orgyia, besides con-
firming the common etymology of agyia from ago suggests
another which I do not remember to have seen, namely, aithyia
from aitho. Nothing can be more natural than to call a sea-
bird swift on the wing, and whose feathers are often sprinkled
with sea-water when it dashes down after its prey, the gleamer
or flasher. I have noticed, probably after a shower, absolute
flashes of light from pigeons wheeling in the sunshine, and the
same thing must happen more frequently in the case of sea-
birds.

6. Nothing of the kini is more permanent than the fix-
ation of land-marks. Many causes which affect the tradition of
other measures produce no effect on that of agrarian. It is
probable that the majority of our fields had their present limits,
or nearly so, at the time Domesday Book was compiled, and yet
the estimation of their magnitude may have varied. The tradi-

378 SOME THOUGHTS ON

tion of agrarian measure may be of two kinds. First, direct, as
when the magnitude of certain pieces of ground being known
and admitted, others are measured in accordance with them ;
or, second, indirect, as when what tradition preserves, is the
relation of square to the units of linear measure, the latter being,
for obvious reasons, the more common case. As an instance of
it I will take the relation of the acre to the jugerum. Of these
seven jugera, which long formed the Roman unit of landed
property, we may reasonably suppose that one was devoted to
house, offices, and garden. Here it is worth noting that Pliny
remarks that hortus originally meant not merely a garden, but
what in his time was called a villa, that is, I presume, a country-
house with its homestead. Something of the same kind still
perhaps exists in the popular Italian use of the word casa.
We know, at any rate, that in Dauphiny casau means hortus,
and that Casaubon when a young man translated his name
into Hortibonus, and was in consequence at a later period
charged with plagiary by those who did not know who Horti-
bonus was. Admitting this hypothesis, six jugera remain, for
what comparatively speaking may be designated by the French
phrase, " la grande culture." Now nothing is more marked than
the tendency of agrarian measures to form themselves as far as
possible (I mean of course ideally) into squares, and to be sub-
divided into similar smaller figures. Thus, and perhaps the
change was connected with some forgotten system of four-course
husbandry, the six jugera may have fallen into four agri as our
acre itself has into four roods. If this be the origin of our
acre, its area would be to that of the jugerum (no regard being
had to the difference of the Roman and English foot) as 3 to
2, and therefore the acre should contain 4800 square yards. It
does contain 4840, but the reason why it has thus been modi-
fied is obvious, in order that the tenth of a rood may be a
square with a whole number of yards for its side, or which is
the same thing, in order that the side of the square perch,
which is the 160th of an acre, may be accurately expressed
without any complex fraction. Had the acre remained what
on my theory it ought to be, the perch would have been 30
square yards; by adding a quarter of a square yard to it, it
becomes a square whose side is 5j yards. Thus, setting aside
this modification and the difference of units, the jugerum is

COMPARATIVE METROLOGY. 379

two-thirds of our acre. The former cause reduces it by about
five-sixths per cent., and the latter by about six, and we thus
get an approximate rule for converting jugera into acres, namely,
to take two-thirds of their number and strike off 7 per cent,
from the result. How the tradition of the jugerum ever got to
Ireland seems very hard to say. Perhaps it was not extinct
in England in the time of Earl Strongbow, at least I am much
disposed to recognize in the Irish acre, not three actus as in
the English, but five, which would of course make it 8000 yards.
But in order that its 160th part should be a square, it under-
went rather a greater modification than the English, losing about
2 per cent, and becoming a square of 7840 yards, the linear
perch being thus not 5j yards but 7. Here again my specu-
lation suggests an approximate rule to reduce Irish acres to
English ; multiply them by ten, divide by six, and diminish
the result by 3 per cent. Between these two acres stands in
point of magnitude the Scotch, but here we seem to come upon
a tradition of the direct kind, and I mean therefore to connect
it with that of the pezza from the jugerum.

7. At page 163 of the 3rd volume of Miss Winkworth's
translation of Niebuhr's letters is one to Savigny, which offers
several points of interest as to the connection of ancient and
modern measures. He tells Savigny that at Eome there is no
manual of weights and measures, nor any information to be got
either from scholars or men of business, but that it having
occurred to him that during the French rule tables must have
been formed connecting their measures with the Roman, he had
been enabled to ascertain by accurate calculation the magnitude
of the pezza, and thus to show its near approach to the ancient
jugerum. Consequently the rubbio being 7 pezze represents the
old land allotment. The remarks on the depourvu state of Rome
as to information are in the querulous tone into which unhappily
Niebuhr often falls. The tables he speaks of having found are
doubtless those published but a very few years before, and of
which Prony has given an account at the end of his work on
the drainage of the Pontine Marshes. If this commission had
ceased to exist in 1818, when Niebuhr wrote, its previous exist-
ence must at any rate have been matter of notoriety, and it is
hard to believe either that all the commissioners had left Rome

330 SOME THOUGHTS ON

or that any of them would have been unable to give the required
information. They were certainly not all Frenchmen. Scar-
pellini, the only person whom Prony mentions, and whom he
speaks of with respect, being, beyond all question, one of them.
Was he ultimus Romanorum? the last person who knew the
length in metres or French feet of a measuring chain? and
might not Niebuhr have borrowed one from any village land-
surveyor and measured it himself? Niebuhr's result is that the
pezza is equal to 24,716 square feet and a fraction. It is not
said what foot he means ; most probably the French foot, which
is the longest of any he was at all likely to make use of, and it
is therefore worthy notice that even so his estimate is too small.
I am obliged to do these things in my head, but a simple piece
of mental arithmetic enables one to see that the pezza is much
more than 25,000 French feet. Prony's statement, which does
not differ from that of the tables, if at all, except by the cor-
rection of an almost inappreciable error (I know not if this error
affected the particular result in question) is, that the pezza is
equal to 2640*6257 square metres. Neglecting the last decimal,
it follows that the pezza is 10 times the square of 1 6 J metres.
Hence if we assume the pezza to be 25,000 feet, 100 linear feet
will be 32 '5 metres ; a result which exceeds the truth by about
the 2000th part. Consequently the pezza is more by about 25
feet than 25,000 feet, and consequently Niebuhr's calculation,
if he meant the French foot, is in defect by about 300. When
we come to his calculations of the jugerum, another difficulty
arises, namely, his not having mentioned what value he assigns
to the ancient Eoman foot. According to Boeckh, Niebuhr has in
his history expressed a strong opinion in favour of Cagnazzi's
determination. Whether he had formed it at the date of the
letter of which I am speaking, I do not know. Certain it is that,
if we adopt this value, which is higher than the average, Niebuhr's
determination of the size of the jugerum errs in excess as that of
the pezza in defect ; and, the latter being the larger measure, both
errors tend the same way, namely, to make the pezza and the
jugerum appear more really equal than they are. The value in
question is 131*325 French lines, and the result is that the
jugerum is something less than 24,000 feet. Niebuhr makes it
24,310 feet and a fraction. Thus unless I am greatly mistaken,
the difference between the two measures instead of being scarcely

COMPARATIVE METROLOGY. 381

more than 400 feet is in reality over 1000. Taking the relations
of the rubbio into account, and also the subdivisions of the pezza,
there can be no reasonable doubt that the latter does really re-
present the jugerum ; but in such a matter an error of Niebuhr's
is worth pointing out. A little farther on he says that no one
can blame him for not having sooner known that the pezza was
not any piece of land, but one of a particular size. No one will
blame him probably, and yet it is only what must have been
known to a great number of persons. In 1760 Cristiani pub-
lished a work which I only know by a table quoted in Diderot's
Encyclopedic (Art. Arpent), in which in the space of a few lines
the values of the pezza and the jugerum are both given (T am
at least sure as to that of the former), much more accurately
than by Niebuhr. The same table, or the materials from which
it was compiled, have probably found their way into many other
works of reference. Niebuhr tells Savigny that rubbio is also
a measure of wheat of about 640 Roman pounds weight, so called
because it was the usual amount of seed for the corresponding
measure of land. But he has not noticed the curious circum-
stance that the rubbio of oats and that of wheat differ, and that
the measure (for it is a measure and not a weight) is less in the
case of oats, although probably in Italy as here a much larger
quantity of seed is used for it than for wheat. Niebuhr has also
remarked that in his opinion the stajolo, which is the smallest
square measure, is derived from sextarius, so as to signify such
measure of land as would require a sextarius of seed, admitting
however that the stajolo must have grown smaller since it ob-
tained its name. Greatly smaller it must have grown, for a
sextarius to the stajolo would be at the rate of about 50 bushels
to the English acre. It is almost incomprehensible that Niebuhr
should not have seen that stajolo can be nothing else than sta-
diolum, whether or no the latter word occurs in dictionaries.
It is moreover properly a linear measure.

Although square measures have strictly no definite form, yet
one form which may be called the normal form is generally
indicated by their subdivisions if not by historical evidence.
In the case of the jugerum both concur in showing that it was
contemplated as a double square, and probably in early times
it never existed in any other form. That of the fundus of
7 jugera is not I apprehend so distinctly marked, though it can

382 SOME THOUGHTS ON

scarcely be doubted but that it formed a rectangle, as the
methods of the gromatici would else have been inapplicable.
The simplest hypothesis appears to be that the 6 jugera of which
I have before spoken were arranged parallelly in a quadrangle,
whose longer side was 480 feet and shorter 360, and that the
7th jugerum (that which I have supposed to be reserved for
house and garden) was treated exceptionally, divided, that is,
into 2 half jugera so as to form a strip 60 feet in width, and in
length equal to and in contact with one of the longer sides of
the before-mentioned rectangle. Or this strip may have been
interposed among the pairs of jugera, but this is immaterial,
what I am going to say depending merely on the assumption
(and it is no more) of the divided jugerum. Of its two halves
one I suppose occupied by house and kitchen-garden, the other
by a vineyard with probably some pot-herbs growing among
the vines, at least the enmity of the vine and the cabbage
(depending probably on the amount of potash they both con-
tain) could hardly have been observed where such a mode of
culture had been uncommon. Half a jugerum would (speaking
quite roughly) produce about 20 dozen yearly on the lowest
computation admitted by Columella, twice as much according
to his own estimate, and yet more according to others. Even
this in a frugal family, the women and slaves drinking no wine,
would have been quite enough, though of course wine was not,
as with us, a luxury. My inference is that the normal form of
a vineyard was in "la petite culture" a rectangle whose longer
side was 4 times the shorter, and that this idea was retained
when the unit of a vineyard was doubled and became in point
of area a jugerum. It is thus I would account for what is
plainly indicated by its divisions. That the pezza or petia
(and it is to be remembered that the word is only used with
reference to garden and vineyards) has to this day such a rect-
angle for its normal form. Perhaps the word replaced its older
synonyme in consequence of this ideal change of figure. The
proof that the pezza is conceived of in the form I have men-
tioned appears to result from the following consideration : it is
divided into 4 quarters and each quarter into 40 ordini or vine-
ranks. Now the length of the surveyor's chain is 10 stajoli,
and therefore no length would be so naturally assigned to each
ordine as this. And with this length the distance between two

COMPARATIVE METROLOGY. 383

adjacent vine-ranks will be 1 stajolo, which is just about what
one might expect to find it, being a little less than 4 feet. It is
obvious that in this way the pezza will contain 1600 vines, and
it is impossible to believe but that the fundamental unit in
laying out a vineyard, namely, the distance between the vines,
should not coincide with the unit of mensuration, namely, the
stajolo. This granted, it appears probable, as I have already
said, or in fact certain, that the length of the ordine is 10 stajoli
or 1 fourth of the 40 stajoli, which would form the longer side
of the pezza ; and hence, on far the most probable supposition,
namely, that the pezza was similar to its fourth part, it results
that the shorter side of the former is 20, and its longer 80 stajoli.
What makes all this very curious is that the length of the
stajolo is of course incommensurable with the side of the old
jugerum, which, for anything we see, was a perfectly convenient
shape for a vineyard. I cannot help thinking therefore that
there was an historical reason for the change, namely, that the
form of the half jugerum was transferred to the whole one, or
pezza. A change which probably occasioned a slight one in
the inter-spacing of the vines. There may have 'been, for in-
stance, in the primitive vineyard 4 quarters, each 30 feet in
width and 120 in length, containing 40 ordini and 400 vines,
the inter-space men t being only -3 feet. All this was preserved,
except that the vines were put at a distance equal to 3 into the
square root of 2 ; a change which has nothing to surprise -us
seeing the extreme discrepancy of opinion as to the best inter-
spacement. Whether the question is settled now I do not know,
but the doctrine was at one time popular that 4 or even 5 feet
was always the best distance, while on the other hand it was
said that even within the limits of France the distance ought to
range from 3 or 4 deci-metres in the north to 2 metres in the
south. Climate however is clearly only one element, and the
question on the whole is probably as hard to settle as the one
which relates to the proper quantity of seed corn. In the ab-
sence of definite grounds for forming an opinion men might very
well have been guided by habit, and the love of the proportions
which habit had made the most agreeable both to the eye and
to the mind. Virgil has alluded to the vintner's love of sym-
metry and of the extreme delight which a man of imagination
may be led to feel in the harmonious arrangement of trees.

334 SOME THOUGHTS ON

Brown's Garden of Cyrus is a good, though perhaps an exag-
gerated, example. Niebuhr, on what grounds does not appear,
speaks of the pezza as a square. If it was, so no doubt was its
fourth part ; and in that case, the actual vine-row being no doubt
equal in length to the side of the square, would have been
20 stajoli; that is, just double the length of the ordine; for
that the width of the latter was one stajolo does not appear to
admit of question. It seems therefore to be a strong objection
to his view that it would not represent the subdivisions of the
pezza. What practical modifications there may have been can-
not be ascertained. Every vine-dresser was of course at liberty
to alter his standard or stajolo. Probably the modern practice
varies less from the ancient in Italy than in France, and a good
deal of information about the former may be gathered from the
agricultural writers. I have forgotten what Columella says on
the subject, and though the book is at hand cannot now refer
to it. As Cervantes says in his last preface, " the thread is
broken here but some other day I may be able to take it up ;"
but the other day did not come.

8. It is well known that within the boundary of a fun-
dus there could be no usucapio for a space of 4 feet, or in
other words, that up to that space there was " asterna auctori-
tas," a perpetual right of reclamation. It is commonly supposed
that it was intended not only to prevent disputes, but to secure
the existence of the numerous rights of way which form parts
of the necessary evils of small property. Hugo has remarked
that it does not follow that the space in question was uncul-
tivated, and he must therefore have regarded the law as being
merely a matter which concerned the owners of contiguous fundi,
for if the space were cultivated at all it must have been by them.
That a law leads to absurdly inconvenient consequences is not
conclusive against its having existed, but on his view, if my
neighbour had encroached in one place 10 feet within the
boundary, he might establish a claim by usucapio to a width
of six feet, while I could at any time re-establish myself in
possession of the outward portion, so that what he required
would be enclave within land with which I could do absolutely
what I pleased. It would seem therefore that the auctoritas lay
with those whose rights of way were interfered with by the

COMPARATIVE METROLOGY. 385

occupation of the space in question, and not in any special
manner with the two marches; and thus we come back to
the common opinion that the space was really unoccupied. Now,
what became of these spaces when the whole system of which
they formed part was forgotten ? I apprehend that the 7 jugera
included only the clear cultivable space, and that consequently
the strip in question came to be an addition to the area of the
fundus. On the hypothesis already mentioned, that the regular
form of the latter was a rectangle of which the longer side
was 480 feet and the shorter 420, the whole circuit would be
1800 feet. The space gained would therefore amount to 7264
feet, or on the average jugerum, to nearly 1038 feet. Of the
residual phenomenon, to use what was once a popular phrase, of
the existence of 1000 Paris feet of difference between the pezza
and the jugerum, my hypothesis would thus explain five-sixths
and leave the difference so small as to require no explanation. It
is difficult to believe that the limiting strip was counted in the
allotment, because it would make the jugera for practical pur-
poses of unequal size, a thing in itself inconvenient and liable
in many cases to produce injustice, and there is therefore a
strong presumption in favour of a view which at once explains
what has become of it and why the pezza exceeds the jugerum.

9. The existence of a rubbio of oats of about four-fifths
the size of that of wheat, while the ordinary proportion of seed
corn used for oats exceeds that for wheat in about the propor-
tion of 8 to 5, suggests the idea that oats, which were certainly
but little cultivated in Italy in old times, (and probably are not
now), were not usually sown over the full surface of the rubbio.
There may have been a simple course of husbandry, namely, the
whole crop of wheat one year, half fallow and half oats the two
next years, and then wheat again, so that every part of the land
was fallow at intervals of 3 years. Twenty other guesses might
be made, but the fact is curious that the measure of oats would
serve for about half as much land as that of wheat. As for the
third rubbio, that of salt, it seems to have been formed on the
principle of weighing as much as that of wheat.

At the end of Niebuhr's letter to Savigny he observes that
7 jugera must have been sufficient for one family, because he
knew a vigneron who, holding 11. pezze as a metayer, supported

25

386 SOME THOUGHTS ON

himself and his family comfortably. The remark, if I may say-
so, appears to be singularly inconclusive. As the man was a
vigneron, we may presume that his chief occupation was growing
grapes ; and how would it have been if his 11 pezze had been all
under glass ? and he had devoted himself to the culture of pine
apples? The question is not within what area a gainful in-
dustry may be carried on, but what acreage per head of the
population is necessary to secure a given degree of comfort, and
whether the best possible use is made of it by dividing it into
allotments. If Sancho's piece of cloth had belonged to five bro-
thers, it might still have been better to make one wearable cap
of it than one for each. The cases are not quite parallel, and
there is doubtless much to be said in favour of small properties,
but the question cannot be settled in the present state of things
by individual instances, even if the question were not compli-
cated by the existence of common rights of pasturage. Colu-
mella, Book III. chap. 5, speaks of rows, ordines, 240 feet in
length and 3 feet apart, as the allowance for a jugerum. In each
row there are 60 vines, and in the whole 2400. Thus if we
divide the jugerum into 4 quarters by lines parallel to its shorter
ends, we get, as in the quarter pezza, 40 rows, and each row of
about the length of an ordine. The transition preserved the num-
ber of rows, but removed an objection to Columella's arrange-
ment, namely, that the interval between trees in the same row is
not equal to that between the rows, whereby Virgil's precept
" to set vines square so that they may have an equal supply of
nourishment all round" is transgressed. The change also in-
creased the amount of ground for each vine from 12 to 18 feet,
diminishing- the number of vines from 600 to 400. I do not
know whether the error in the text of the edition I have has
been corrected. By it he is made to speak of 600 vines in a row
of 240 feet, and of a total of 24000, numbers which are abso-
lutely incredible. In the preceding chapter, when speaking of
the number of cuttings which before the vines have grown up
may be put between the rows and taken up when they have be-
come viviradices, he allows 20000 to be planted. In this there
is nothing which is not quite natural. They may then be put, I
suppose, nearly as thick as asparagus, and perhaps the tran-
scriber was misled by not seeing that Columella was speaking
in the 5th chapter of the permanent arrangement of the vineyard.

COMPARATIVE METROLOGY. 387

That his calculation of its profitableness refers to an extent of 7
jugera as that in which it would be proper to employ one vine-
dresser, is a curious instance of the existence of the idea that
this extent of land is the proper allowance for one person.
Here we have the old allowance passing into the modern rub-
bio. It is useless to look for much accuracy in such matters,
but if we take Pliny's statement that the regular allowance of
wheat seed was 5 modii, and assume that the amphora was 26
litres, which agrees with Cagnazzi's^ value of the Roman pound
in French grammes, and if we also adopt the common statement
that the modius is one-third of the amphora, then 35 modii (the
allowance for a rubbio) will be 30 decalitres and a third, which
agrees very closely with Prony's value of the rubbio of wheat.
That of oats is yet more nearly equal to 4 modii. The perpetu-
ation of the traditionary allowance of seed is in the former case
more close than we could have expected. It amounts to some-
thing less than 2 bushels the acre, and is therefore under the old
practice in England and in Cisalpine countries generally.

10. The divisions of the acre appear to have grown out of
those of the pezza. It is divided first into 4 roods as the other
into 4 quarta, and the 40 perches of the rood correspond to the
40 ordines of the quarta. But with us so small a division as the
staggiolo was useless, and the perch becoming practically the
smallest measure seems to have been regarded as a square.
Properly it was 30 square yards, but for convenience it was
increased by a quarter of a yard, so that its side became 5 yards
and a half. This has been already noticed. I repeat it now
because it leads to the history of the English mile, which got its
name from being not much unlike in magnitude the Roman
mile ; though it is said not to be formally recognized in any-
thing earlier than an act of Elizabeth, What we call a furlong
is properly a fiiiing, or 4th part. Why then is it the 8th part of
a mile ? The answer is, that the original unit was the long acre,
that is, an acre consisting of perches arranged in single file.
One hundred and sixty times 5 yards and a half make eight
hundred and eighty, which is said to be the length of Long Acre
in London. The furlong was therefore the long rood, and we
have thus an instance in which a measure of length has sprung
out of one of surface. Perhaps the word rood has relation to the

25—2

388 SOME THOUGHTS ON

cross-like division of the acre into quarters, the sacred symbol
replacing the cardo and decumanus of the Romans, and the
name being transferred from the dividing figure to the divisions
it produced. K. O. Miiller's idea -that mensuration was a sort of
degeneration from the augurial laying out a space, seems to
invert the natural order of ideas. I should be much more in-
clined to think that the religious ideas which in early times
associated themselves with the transition from a pastoral life to
tillage and individuated property in land, were earlier than any
formal system of divination. The curious story how Attus
Marius found the largest bunch of grapes in a vineyard by
repeated subdivisions of the space and repeated taking auguries
so as to ascertain the row in which it was to be found, illustrates
the connexion between the two things. All this is by the way.
Another incidental remark is, that the reduction»known as cul-
tellatio does not appear to be rightly understood. At least if
I remember what is said on the subject, it is made to depend
on the fact that if corn grows on a hill-side, the horizontal
distance between the straws ought not to be greater than if
they grew on level ground, and that therefore to distribute
land equitably you must conceive it projected on a level sur-
face, but a variety of other considerations ought to be taken
in, and this reason has much the appearance of being invented
to give a favourable colour to a practice, the real motive to
which was that without it it would have been impossible to
keep the boundary lines straight. A hill in the middle of a
level plane would set every line wrong on one or the other
side of it ; on the side, namely, towards which the survey ad-
vanced.

11. It is well known that in Gaul the old native measure,
the league, is never supplanted by the mile, and that a league
equivalent to a mile and a half occurs in the Itineraries. Com-
paring this with the modern French leagues it would seem to
have been a half league introduced by the Romans, in order that
the intervals marked by the league stones might differ less from
those to which they were accustomed. I am inclined to this
view, because the short league would obviously be 5000 cubits.
Now although t here is a reason why 5000 feet should be the
linear unit, as 5 feet make a pace, there is no reason of the same

COMPARATIVE METROLOGY. 389

kind in favour of 5 cubits ; and, considered apart from any such
reason, 10,000 cubits would be a more natural unit than 5000.
And this would give the league which occurs in modern times.
Let it be granted, however, that the Gallo-Roman league contains
as many cubits as the mile feet, it would immediately suggest
itself by analogy that the unit of square measure should be a
square whose side was 120 cubits, so as to correspond to the
actus. Thus we are led to the Arpentum, which continued to
be the unit of French surveying untij. the revolution, varied how-
ever in size by the difference between the French and the Ko-
man foot. Like the actus it is described as a square, the side
being of course 180 feet or 10 perches of 18 feet each. This was
known as the " Arpent de Paris," but there existed another also
much used called the " Arpent des Eaux et F6rets," connected
with the former by an approximately simple relation, but perhaps
resulting from a tradition of the old jugerum which must have
long survived in Provence. At least with the double jugerum it
coincides more closely than the pezza with the jugerum. This
latter arpent is a square of 220 feet, and therefore differs Less than
a half per cent, from being to the Arpent de Paris as 3 to 2, a
circumstance which may have modified its definition. Half of it
would be 24200 square feet, which would be precisely the juge-
rum if the Eoman foot were 132 Paris lines. The calculation in
Niebiihr's letter to Savigny implies a greater value than this of
the Roman foot, but there is no occasion to have recourse to any-
thing doubtful in order to show how nearly the two things coin-
cide. For if we take Gagnazzi's value already mentioned, the
jugerum will not fall short of the half Arpent des Eaux et Fdrets,
much more than one per cent. Two things are worthy of notice
respecting this second arpent. One, that it appears to have led
to a definition of the league as 60 times the side of the arpent, or
2200 toises ; the other, that there can be little doubt of its being
the origin of the Scotch acre, with which its coincidence in size
may be called absolute. Converting this arpent into yards,
according to the logarithms given in Babbage's Constants, I
make it a fraction more than 6108 square yards, the normal
Scotch acre being 6104 yards, though it is said to vary in dif-
ferent counties from 6084 to 6150 yards. Thus, if my fancies
(for I can hardly call them more) are right, we have the pezza
equal to two actus, the English acre to three, the Scotch to four,

390 SOME THOUGHTS, &c.

the Irish to five, and the rubbio to fourteen : all with more or
less modification, but yet so as that these measures may all fairly
be considered to be derived from a single unit. From the inti-
mate connexion between France and Scotland the hypothesis of
the origin of the Scotch acre does not seem improbable, at least
compared with other links in my chain. The passage in the 5th
book of Columella appears to show that the arepennis was
regarded as a half jugerum, which of course it would be if it
were considered an actus. But he speaks of two kinds of cande-
tum, the former used in towns and being a hundred feet square,
the latter agricultural and 150 feet square, that is, I suppose, 100
cubits. The transition from the latter to the arepennis or arpen-
tum of 120 cubits square would thus be the point at which the
influence of Koman ideas showed itself.

NOTES ON BOOLE'S LAWS OF THOUGHT*.

IT appears to be assumed in Chapter m. Section 8, that in
deriving one conception from another the mind always moves,
so to speak, along the line of predicamentation, always passes
from the genus to the species. No doubt everything stands in
relation to something else, as the species to its genus, and con-
sequently the symbolical language proposed is in extent per-
fectly general, that is, it may be applied to all the objects in the
universe. But I venture to doubt whether it can express ex-
plicitly all the relations between ideas which really exist, all the
threads of connexion which lead the mind from one to the other.
It seems to me that the mind passes from idea to idea in accord-
ance with various principles of suggestion, and that in corre-
spondence with the different classes of such principles of sug-
gestion we ought to recognize different branches of the general
theory of inference. This leads me to a further doubt whether
logic and the science of quantity can in any way be put in
antithesis to one another. From the notion of an apple we may
proceed to that of two apples, and so on in a process of aggre-
gation, which is the foundation of the science of discrete quantity.
Or again, from the notion of an apple we may proceed to that
of a red apple, and this movement of the mind in lined predica-
mentali is the foundation of ordinary logic. But it is plain,
d, priori, that there are other principles of suggestion besides
these two, and the following considerations lead me to think
that there are other exercises of the reasoning faculty than those
included in the two sciences here referred to.

In the first place, certain inferences not included in the ordi-
nary processes of conversion and syllogism were recognized as
exceptional cases by the old logicians. Leibnitz has mentioned
some with the remark that they do not depend on the dictum

* Now first published.

392 NOTES ON BOOLE'S

de omni et nullo, but on something of equivalent evidence. The
only question is whether we should be right in considering these
cases as exceptions, and, if they are so, to what they owe their
existence. One instance is the inversio relationis, e.g. Noah is
Shem's father, therefore Shem is Noah's son. Here we pass
from the idea of Shem to that of his father, and vice versa.
The movement of the mind is along a track distinct from that
which it follows, either in Algebra or what we commonly call
Logic. The perception of the truth of the inference depends on
a recognition of the correlation of the two ideas, father and son.
Again, take a similar instance : Prince Albert sat at the Em-
peror's right hand, therefore the Emperor sat at Prince Albert's
left, &c. &c. How shall we express such inferences sym-
bolically? Let S be Shem, N Noah, /father, s son ;

N=fS,
sf=l.

Eliminating /bet ween these two equations, we get

Nothing can be simpler than this : but the symbols s, / are of a
distinct nature from those employed in the Laws of Thought.
For/4 does not denote a species of -4, but an idea standing in
a different relation to it. The distinction between these two
kinds of symbols becomes more manifest when we reflect that
/2 is not identical with/ but denotes "father of father," or
grandfather. Now I do not see how these cases of inversion of
relation are to be dealt with symbolically without the introduc-
tion of such symbols. In the following examples I confine
myself to the cases afforded by relationship and the succession
of generations. Let A, -B, (7, denote three persons, s son, g grand-
son; then, if .5 is A* a son and C jB's, C is A's grandson, which
we may express symbolically by the following equations :

Eliminating B, we get

LAWS OF THOUGHT. 393

It would be more accurate in these examples to introduce
a symbol x or y to indicate that B is only one of the possible
sons of A, an individual ranged under the species sA. I shall
do that in the next example, in which the word son is replaced
by the more general term descendant denoted by d. The equa-
tions will now be

videlicet a descendant not of the first generation. The result of
eliminating B now is

C = ydxdA :
but by a principle about to be noticed

dx = x'd ;
.'. C = yx'zdA,
or C is included in the class of descendants of A.

The principle just used forms one of the recognized examples
of an inference not lying within the domain of Aristotelian logic.
It was called Transit io ex recto in obliquum. Whately, though
he says nothing of its nature, gives in his praxis of examples
one which depends upon it. A negro is a man, therefore he who
kills a negro kills a man ; let this derived notion killing be de-
noted by f, which may serve to indicate a general functional
dependence : then, M9 N, denoting man and negro respectively,
we have the following equations :

JV

.-. /Ar= x'fM,

or the killing of a negro is a kind of homicide. The evidence
of the truth of the equation

is the same as that in favour of the equation

xy = yx,
when x and y both belong to the kind of symbols used in the

394 NOTES ON BOOLE'S LAWS OF THOUGHT.

Laws of Thought. I shall not stop to inquire into the limita-
tions which it may perhaps require.
The general truth of the equations

a?2 = x and xy — yx

appears to suffer another exception in the case of relative terms,
that is, of adjectives of which the interpretation is functional of
the object to which they are applied. A small St Bernard dog
is not simpliciter a small dog ; the word meaning that which is
less than the medium size of the class of objects to which it is
applied. Here neither s2 = s nor sx = xs. If we say that in
order to save all these equations we may employ a different
symbol for every application of the adjective small, how can we
express the meaning which is common to them all, and in virtue
of which the word small exists as an element of language ?

Diffident as I am with respect to all these remarks on a
method in which I find so much to admire, I am yet more so
with respect to the following. But it seems to me that we
cannot say that

x (I - x) = 0

expresses proprio vigore, that is, in virtue of antecedent conven-
tions, what is called the principle of contradiction.

In ordinary language we have words which, independently
of this principle, express negation : we say red, not red, and the
like; but in the symbols employed in the Laws of Thought there
is no other means of expressing not red than by 1 — x, x de-
noting red. Now the interpretation of this symbol 1 — x seems
to me to be given by the principle of contradiction, and there-
fore I should rather say that the equation

is interpreted by that principle than that it expresses it. In
accordance with this view the equation

x* = x
would appear to be independent of the principle of contradiction.

BEMABKS ON CERTAIN WOBDS IN DIEZ'S ETY-
MOLOGISCHES WOBTEBBUCH DEB EOMANIS-
CHEN SPBACHEN (ETYMOLOGICAL DICTION-
ARY OF THE BOMANCE LANGUAGES)*

Addobbare. Is it not from adoperari? The word occurs so
early and is so much used that it can scarcely have been derived
from chivalry.

Camicia. As the b in cannabis becomes m in Spanish, I
would assume the following forms: canamisia, canmisia, ca-
misia ; the last being the form given by St Jerome, who speaks
of it as a linen garment : this explanation is confirmed by the
old French form chamse.

Laquais. Laquet, Naquet, NacJct, viz. Light-armed-men : the
word being plainly military and German. Compare the Greek
7u/^rJ9. The same class of people were also called pages and
servants or sergeants. Addison is right in supposing that ser-
geants-at-law represent the serving brothers of the Temple ; the
judges who called them " Brothers" being the Knights.

Albricias corresponds to the German botenbrod, and means
what is given to a messenger if you are pleased with him, as
when he brings you good news. I believe the true form is
albergias, or something of that kind, and the literal meaning
Tierberg e-money or gifts, that is, gifts for the entertainment of
the messenger. The German word in the Nibelungen Lied,
and the Spanish in the Cid.

Travar. Considering the French form entraver, it seems
more likely that the root of these words is trabea and not trabs.
From trabea came trabeare and trabeatio, the latter used in the
sense of Incarnation. The idea of entraver is simply wrapping
up in clothes, so as to hinder the free motion of the limbs.

* Now first published.

396 REMARKS ON CERTAIN WORDS IN

Donjon. Surely from dominatio, the highest or most com-
manding part of the place. The Proven9al form is Dompnon.
The French word donner in the phrase ' cette fenetre donne sur
la rue' is probably not from donare but from dominari, the
French word being thus a crasis of two Latin ones.

Treillis. There are two etymologies given of this word,
one in the first, the other in the fourth part of the dictionary :
the former from tri-licium, the latter from trichila: the second
is probably the right one.

Hanter. May it not come from hamitare for habitaref The
change of b into m is not very uncommon.

De balde. It does not seem necessary to refer to an Arabic
source for this phrase. The filiation of ideas may have been
something of this kind: In Lithuanian, balths means white;
the " Baltic Sea" is probably a " white sea," as well as that of
Archangel. Connected with this idea is that of brightness, as
we see in the name Baldur, and in many other northern and
German instances. Then comes the idea of doing a thing
openly, boldly and avafyavbav, balthaba, as in Ulphilas. Also
that of brightening up, making merry, &c. as in the Proven9al
Esbaudir. Again, between whiteness and vacuity there is an
obvious connexion, as in the word blank: and thus we get to
the English bald, &c. and to the Spanish phrases, de balde and
en balde.

Pantois. Paventols was perhaps an earlier form, the meaning
being to breathe as a person does when frightened. Compare
the Italian Spantare.

Assouvir. Surely from adsopire.

Jach&re. There must have been a Latin woid.jacerium from
jacere, in the sense of to lie neglected or fallow.

Charivari. Why the Lille Glossary translates Chativati
larnatium appears from the Sachsenspiegel, in which imme-
diately after sheep and geese kasten rait upgehavenen leden, &c.
are mentioned as part of the Morgengabe. This shows that the
word was used not in a ludicrous sense, but seriously.

Zorzal seems to be simply the German drossel. The English
form throstle seems to confirm this.

Massacre. The root is macella. In Proven9al " War against
the Albigeois," Mazel occurs in the sense of carnage. From Macella
comes Macellarius, a meat man, easily condensed in French into

DIEZ'S ETYMOLOGISCHES WORTERBUCH. 397

machecrier, which occurs in the Roman de Ron in the sense of
a butcher; and this would become massacrier, and thus give
rise to a concrete massacre. At Rouen the street where meat
was sold used to be called Rue de Massacre. Carnage, in the
same way, meant originally meat-eating, as opposed to manger
maigre.

Boucher. Surely Boucherie meant a place where you could
get munition de bouche, and a butcher is merely a purveyor.

Piloto. The connexion of ideas ^between Pilotis and Piloto
is illustrated by what I have seen on the Rhone, namely, two
men in the bows of the vessel taking soundings with long poles,
pilotis ; piloting the vessel by means of them.

Pimiento. As the name seems to have been given to the
spiced wine originally, and secondarily to the spice used in it,
the idea of calling it pigment seems to be an allusion to the
effect of spiced wine on the complexion ; a ludicrous or joking
name for a drink seems natural, and there are other instances
of it. Orpiment, by the way, is auri pigmentum.

Fregata. What is the origin of our English word to freight?
fregata, or frigate must be connected with this verb, as Carrick
with caricare and oneraria with onerare.

Orza. I should be inclined to suggest, though doubtingly,
that a lorza means on the left side, in consequence of the early
habit of mankind, which seems to recur at all times of speaking
of the cardinal points on the hypothesis of looking to the east.
Thus the phrase originally meant towards the Bear or North :
and secondarily to the left hand as the north is when one looks
east. The earliest trace of this way of thinking is in Genesis,
where it is said of Ishmael, that he shall dwell in the presence
of his brethren, that is, east of them.

Aubaine. Is not this word, like arban, heribannum, as auberge
is herberge f meaning the Lord's right, and especially his right
against a stranger. Why it should mean this particular right,
as arban means the Corvee, one cannot quite tell : but I imagine
that aubain in the sense of a stranger was a word formed in
consequence of a mistake of the meaning of the word aubaine.

Bis. Another instance of the use of this particle in com-
position to indicate something other than as it should be, is
Bistourner: the only church at Paris not towards the east is
popularly called St Benoit le Bistourne*.

398 REMARKS ON CERTAIN WORDS IN

Trovare. In confirmation of the derivation of this word from
turbare, it may be remarked, that invenio is literally to come in
upon. A person comes in upon you, finds you, or disturbs you;
the ideas manifestly being akin. So too the dog puts up the
game, disturbs it, or finds it.

Hurepe. This word with a little modification of the spelling
occurs in La lattaille de Karesme et de Charnage*, as the name
of a fish ; I suppose the Barrel.

Mora. In Dante this word is explained to mean a heap of
stones. "La grave mora" occurs in one of the early cantos of
the Purgatorio.

Estdble. It seems difficult to separate stall from stable: in
Marie de France (Purgatory of St Patrick) we find estaule in
the sense of stabilis, and in English the stall was until lately
spelt with a u.

Colmena. In the Vulgate the phrase * corner of the house-
top ' is translated angulus domatis ; and in the Yaudois trans-
lation the second word is rendered by colme. In the " Book of
Virtues" colme is replaced by meysoneta. So that there is little
doubt that colme was used for little dome or cupola rising above
the level of the roof. The transition from this sense to that of
lee-hive is from the form of the latter obvious, and the theme of
both colme and colmena must be culmen.

Cafre. Diez remarks in his grammar, that of the Arabic
words in Spanish hardly any refer to things of human feeling.
It seems very improbable that the word for Infidel should have
been taken from the Arabs, who did not call themselves so, and
with whom the Christians were not likely to come to an under-
standing as to its use. Considering that the word occurs also
in French, and the facility with which aspirates of different
organs pass into one another, I think the word must be Cathar,
the origin, as it is well known, of the German word Ketzer;
though Diez has remarked, he is not aware of any case in which
the Greek 0 becomes/, but the induction must be founded on a
small number of cases.

Indarno. It is so very unlikely that it should come from
the Sclavonic that I think it must be simply in danno, with the
first n changed into r. Compare the French phrase en pure
perte.

* In Fabliaux et Cvntes; edition of Barbazan, Vol. iv. p. 85.

DIEZ'S ETYMOLOGISCHES WORTERBUCH. 399

Trou. The word trabucar in Prove^al seems to be a mili-
tary one. It is constantly used in the " War against the^Albi-
genses" for battering or breaching with a trabs, or in the Italian
form trabocco. Trabuca is therefore probably the breach made
with this instrument, and the Italian buco is probably a cor-
ruption from hence and not a German word.

Nebli. As the Arabic derivation seems to be a failure, why
should it not come from milvius ? The change of m into n, of
v into b, and of the place of the liquid^ are all matters of frequent
occurrence in Spanish. Perhaps we ought to assume a diminu-
tive form milvillus, the final I being lost with the shifting back
of the accent.

Calibre. The word is principally used, speaking of the bore
of a cannon, or the diameter of a column ; but it has another
meaning which must be the original one. The word is used by
masons, carpenters and workers in metal, for the tool or model
which goes round anything and enables them to see if it be of
the right size; its form, &c. vary in different cases, but it is
always something that embraces or clips what it is applied to.
It therefore seems clear that calibre is simply clipper, the latter
word being borrowed from England or Holland, for I do not
know that it exists in modern German. If a Frenchman pro-
nounced clipper half a dozen times it would run into something
not to be distinguished from calibre. Compare the English
word caliper, and canif from knife; two things of the same size
are said to be of the same calibre, because they would fit the
same ; and hence the other sense of the word, which there is no
occasion to derive from the Arabic.

Caviar, l&avidpi, is certainly not a Greek word. I imagine
the Greeks intended the four vowels to represent the full sound
of the Italian or Provencal u, and that the word was originally
curee or curata, meaning simply cured roe. When it came to
the West from Greece, the perplexing number of vowels caused
the hardening of the u into v. Compare, for an analogous change
of a travelled word, the French word fashion derived from/a^ora.

SOME THOUGHTS ON THE FORMATION OF A
CHINESE DICTIONARY, AND ON THE BEST
MODE OF PRINTING CHINESE. IN A LETTER
TO THE REV. J. POWER, M.A., FELLOW OF
CLARE HALL, AND UNIVERSITY LIBRARIAN*.

Cantantes licet usque minus via Isedit eamus.

MY DEAR SIR,

THE study of Chinese is hindered by many difficulties,
the nature of which is not generally understood. The expense
of printing Chinese with our kind of types is one of them.
Much has been done in this matter by Breitkopf, of Leipsic,
who has been employed by the American mission.

Some of the results were exhibited at the Crystal Palace,
but I have not been able to learn anything of the details of
the analysis to which he subjects the Chinese characters. More
recently Professor Brockhaus has proposed, that in order to get
a complete and inexpensive Chinese dictionary, we should have
recourse to Lithography. Even so the undertaking would be a
great one, and it is very desirable that such a dictionary should
be arranged in the most convenient and useful manner.

Brockhaus proposes to follow the ordinary Chinese arrange-
ment ; according to which the characters are distributed under
214 radicals, remarking in favour of doing so, that it is conve-
nient to be in accordance with the Chinese practice f.

No doubt this is true; but the objections to this mode of
proceeding are considerable.

In the first place, it is essentially unscientific ; the radicals
are chosen on no definite principle. They seem to be, as Kant
somewhat too boldly asserted of Aristotle's categories, aufgerafft,
gathered up at random from all sorts of sources.

In the second place, the relation of the characters to the
radicals under which they stand, is arbitrary and uncertain.
In some cases the relation is one of mere resemblance, in others

* Previously printed for private circulation.

t The system of 214 radicals is not of high antiquity, and has not been always
followed since its first introduction.

THOUGHTS ON THE FORMATION, &<: 401

the radical is a component part of the character, while occa-
sionally it is very difficult to see what the connexion between
them is.

Again, there is no principle of arrangement under each ra-
dical, except according to the number of strokes of which the
character is composed; and in a complete Chinese dictionary
there would be on an average 200 characters under each radical,
and in some cases more than 1000. The result of this is, that
it is necessary to have a supplementary index of characters, of
which the radical is difficult to recognize, and in this there is no
principle of arrangement, except the number of strokes.

Gallery, of whose improvements I am about to speak, after
mentioning the way in which Chinese dictionaries are con-
structed, remarks, that it is not wonderful that so few persons
attain to a knowledge of the language, and they, only after
years of painful labour. In fact, to be able to use a dictionary,
is a great part of the whole business of learning to read Chinese.
The principle on which Gallery proceeds, he derived from his
instructor, Gon^alves, whom he speaks of as the ablest of
Chinese scholars, and who has published several works, in which
it is followed, though he has nowhere fully explained it.

Most of these works are in Portuguese, which is perhaps the
reason why they seem not much known. Gallery's own work
is in a mixture of Latin and French. It was printed at Macao,
and it is said that most of the copies were accidentally destroyed.
It consists of two parts; the first introductory, the second, a
dictionary of perhaps 13,000 characters, arranged according to
his own method, in which the principle of Gonc,alves is em-
ployed, in subordination to a phonetic classification. The facility
with which this dictionary is used seems to make it desirable
that no other should be constructed until it has been considered
whether the same method, improved, if possible, ought not to
be employed in it.

The principle of Go^alves is, as he states in his Arte China,
not wholly new. It is based on the almost necessary con-
nexion which there is between learning to read and learning to
write Chinese.

The early missionaries held that all Chinese characters con-
sisted only of six different kinds of strokes. Gon^alves increases
the number of elementary strokes to nine. These may be com-

26

402 THOUGHTS ON THE FORMATION OF

pared to the letters with which our words are spelled, in this
respect at least, that a Chinaman, in writing a given character,
will always employ them in a given order, just as when we write
man we write the m first, the a next, and so on.

Gallery affirms, that throughout China a uniform method of
forming the characters obtains, though little is said by Chinese
writers on the subject. Some directions, he remarks, have been
given, but too vague to be of much utility. I do not know if
he refers to a tract, of which Sir John Davis published an ac-
count in the Transactions of the Koyal Society of Literature.
A collection of characters arranged according to the order in
which the elementary strokes are employed will, as similar cases
continually recur, very soon enable any one, even without in-
struction, to acquire a knowledge of the Chinese method. This
arrangement is of course based upon giving an order of prece-
dence to the nine elementary strokes analogous to the order of
the letters of the alphabet. But this the Chinese themselves
do not seem to have done, and without this step, however de-
finite their method of forming the characters, no principle of
arrangement could be hence obtained. That they should never
have introduced this improvement, is one of the many instances
of the mixture of sterility and ingenuity by which they seem to
be distinguished.

When we have learned the order in which the strokes are
to be employed in forming characters, and have given an order
to the strokes themselves, we are in possession of a principle
in virtue of which all the characters of the language may be
arranged, so to speak, alphabetically. In practice, however, it
is desirable to combine this principle with others, and especially
with that which results from the analysis of which the great
majority of characters obviously admit. It is clearly more con-
venient to arrange the characters in the first instance according
to the number of strokes, than to set out with a purely alpha-
betical arrangement ; and this is the plan followed by both
Gon^alves and Gallery, in their vocabulary of primary charac-
ters. The arrangement therefore may be compared to that of
a spelling-book, in which lists of short words arranged alpha-
betically are followed by similar lists of longer ones.

Thus far there seems no reason to deviate from their method;
but the question becomes more difficult when we inquire what

A CHINESE DICTIONARY. 403

characters (and on what grounds) should be accounted primary.
My own impression is, that as the main object to be attended to
is facility of reference, we ought to be guided by the eye, and
whenever a character is clearly made up of others, to treat it as
a compound character, whatever the significance of its parts or
the relation between them. The great majority, it has been
already remarked, of characters, are in this sense compound;
and no one who is at all used to Chinese writing, can have
any difficulty, except in a few cases, in dividing them into their
elements.

Generally speaking, one or other element is in the Chinese
arrangement the radical under which the compound character
is placed, and this is the part of the method of radicals which
is in practice most convenient. Even so it is defective in giving
the compound characters mixed up with others, which on other
grounds are placed under the same radical. Still few compound
characters, considering their whole number, will be found in the
supplementary index.

Gallery's method differs from what is now suggested in this
respect; he does not avail himself* of the analysis of compound
characters into simple ones, in all cases in which such an
analysis is distinct and obvious, but only in a certain class,
though undoubtedly the largest and most important class of
cases.

More than five-sixths, according to one estimate, eleven-
twelfths according to another, of all Chinese characters are not
only compound, but made up on a uniform plan.

They admit of analysis into two parts, one of these simpler
parts having the same sound as the whole character or a modi-
fication of it, and the other having some reference to the mean-
ing. Thus the former indicates the pronunciation, and is there-
fore called by Gallery ' the phonetic element.' To the other he
gives the name ' classifica.' Gon^alves had called the latter
Generic, and the former Differential elements f.

The names are not satisfactory, for the phonetic element
may as well be made use of to constitute a class or genus as

* Or rather, he ought not.

t I do not apprehend that Gon9alves conceived it necessary that the differen-
tial element should always be phonetic. He would have given the name to what-
ever he found differentiating a generic character,

26—2

404 THOUGHTS ON THE FORMATION OF

the other, and then the latter must be considered the diffe-
rentia.

The nomenclature of both writers indicates the influence of
the old view, adopted by Remusat and quoted with apparent
censure by Gallery, that the class of characters of which we are
speaking was based on a scientific arrangement of objects, ac-
cording to genus and species.

The name phonetic is unobjectionable, and the other ele-
ment might be termed non-phonetic. But it would, I think,
be better to call it the logical element, inasmuch as it has rela-
tion to the word as such ; that is, as it has a meaning, and is
not a mere sound.

Gallery presents a list of 1039 characters, which enter into
others as phonetic elements. These he distributes into classes
according to the number of strokes, arranging those in each class
on what may be called the principle of graphic analysis, namely,
that of Goi^alves. Under each of these 1039 characters he
places those of the same or the kindred sound, which are formed
by the union of that character with one of those which he called
1 classifiers/ these being arranged subject to certain modifications
in the same way as the phonetic elements, under each of which
we find on an average about a dozen characters.

The plan, though in many respects very instructive, is not
all that one wishes for in a complete dictionary. His list of
primary phonetic characters contains many themselves made up
of a phonetic and a logical element.

The classifiers are for the most part phonetic elements also,
but when they are not, one has to look for them in a separate
list, given only in the first part of his work.

Moreover, there are certain characters, some of them common
and important, and not admitting of analysis into others, which
do not seem to have been used by the Chinese, in the formation
of compound characters, either as logical or phonetic elements,
and which therefore he does not notice at all, or if he does, only
incidentally.

But the great defect is, that he cuts himself off from the
advantage resulting from the analysis of characters in all cases
in which neither element is phonetic. Some indeed he seems
to introduce by mistake. Thus, the character for dog has no
relation in point of sound to that for bark, which is compounded

A CHINESE DICTIONARY. 405

of dog and mouth, and is pronounced as no other character is in
which dog is really the phonetic element. It is clear that both
elements are logical, we might say quite intelligibly to dogs-
mouth, instead of to baric.

It is no practical objection to Gallery's arrangement, though
it shows the difficulty of adhering, if we were required to do so,
to a definite plan, that in some cases both elements of the cha-
racters are phonetic, though in different ways, one indicating the
sound, and the other showing that the former does so*.

It is in this case a sort of signftim diacriticum, and the cha-
racter which is thus employed the most frequently is that for
mouth. For instance, Jive, and one of the words for /, are both
pronounced eu, and the character for the latter is the same as
that for the former, with the addition of that for mouth.

A similar instance has given rise to one of the innumerable
foolish things which are said and repeated about the Chinese.

Ho happens to mean corn, and concord or comfort, and the
character for it, in the latter sense, is the same as in the former,
with the same addition as in the preceding instance.

The mouth is here used diacritically, yet people have been
found to say, that the grossness of the Chinese is shown by their
having a character for happiness which indicates that they have
no higher idea of it than mere eating.

It so happens that the same sound ho also means a child's
crying, and in this sense also is represented by corn and a mouth
differently placed in relation to one another. What authority
can there be for an interpretation in the former case which is
obviously inapplicable in the latter? This, however, is a less
offensive error than those into which the early missionaries fell,
in seeking for the doctrines of Christianity in China. I can only
allude to their interpretation of the word yang.

If a Chinese were to say that the English are a particularly
selfish people, because the same symbol denotes unity and per-

* Cases might be pointed out in which both elements are representatively pho-
netic, so as to form a sort of reduplication. In some cases it may be said that
both elements are at once phonetic and logical. Thus assuming that the two ele-
ments of pi, to compare, are both pi spoons, the idea of comparison results from
their similarity. In other cases the same element is phonetic and logical. Thus
tsien, a small coin, consists of tsicn, small, and kin, metal or coin. Compare our
word 'groat.'

406 THOUGHTS ON THE FORMATION OF

sonality*, lie would only imitate the example of errors long
gravely maintained by European scholars. No more effectual
mode of getting rid of these errors presents itself than making
a complete analysis of all compound characters, in order after-
wards to recognize and classify the different principles which
have guided their formation. Eemusat's remark is perfectly
just, that the Chinese characters are formed in a variety of ways,
and that nothing but confusion can result from any attempt to
analyse all on the same principle. Chinese etymology consists,
as Humboldt has observed, of two parts, that of the characters,
and that of the spoken language. Both parts involve great
difficulties, and as yet neither has been treated scientifically.
The former part is particularly attractive: there is no more
amusing book than a Chinese dictionary. Perhaps my saying
so may remind you of the painter's reflection, ' che dolce cosa
e la perspettiva.' One instance may be enough to show you
the sort of interest I mean. The character for heart, which
expresses generally all mental operations, combined with that
which represents an enclosed and divided field, means to think
or consider : we have here a graphic representation of the Latin
1 contemplor,' formed, as there seems little reason to doubt, from
'templum,' in the sense in which the augurs used the wordf.

A very interesting part of the study of the Chinese character
would be the comparison of it with Egyptian hieroglyphics, not
in order to revive the old notion of an historical connexion
between them — between the flowery region and the lands of the
lotus and the papyrus — but in order to see how similar problems
have in the two cases been dealt with. It has been said that
* Thoth was wiser than Fo' (Fo-Hi), which may be true, but
still the comparison is worth making.

It is curious, that while in Egypt the feet or legs seem to be
the symbol of activity, so as to give a verbal signification to the
symbol with which they are associated, or, in some instances,
rather grotesquely joined, the hand should often serve the same
purpose in China. Other points of analogy might doubtless be

* Guesses at Truth.

+ Grimm's derivation of templum, from the same root as tepeo, making it refer
to the sacrificial fire, seems open to more than one objection : in the first place,
the augurial sense of the word appears to be the primary one, aiii this connects it
with T^nvfiv ; and in the second, it would then be particularly strange that the
house of Vesta should not have been a templum.

A CHINESE DICTIONARY. 407

indicated, though the less complicated forms of the Egyptian
characters can hardly admit of phenomena so various as those
which are presented by the Chinese. In the latter, for instance,
I believe we might trace that curious principle of language,
which, for want of a better name, may be called ' the principle
of intelligibility,' of which we cannot have a better instance
than the conversion of ' mandragore ' into ' main de gloire ;' I
mean, that in Chinese groups of strokes, originally forming only
a part of a complex picture, have probably, in some cases from
accidental suggestions, shaped themselves into the likeness of
other simpler characters.

With regard to the etymology of the spoken language, such
an arrangement as Gallery's is of the greatest value. When we
find different sounds associated together under the same phonetic
element, we may, special cases being set aside, conclude that
they are, with reference to the Chinese organs of speech, cognate
sounds, and thus establish the laws by which our investigations
are to be guided. The change of Ch into T, which we are
familiar with as the peculiarity of the Fokien dialect, is one of
the most obvious phenomena thus made manifest*.

To return from this digression, to the formation of an index
to a Chinese dictionary. I should propose to form a list of all
simple characters, and of all in which there could be any serious
doubt or uncertainty as to their analysis. The latter would not
be a very large addition to the number of the list. Gallery's
own estimate is that his 1039 phonetic elements result from
about 300 primary ones.

A small number, and those capable of further analysis, of
his classifying characters do not belong to his phonetic list;
but if we say, that all the elements he employs cannot much
exceed 300, we shall not perhaps be far from the truth. Taking
account of omissions, accidental and otherwise, we may perhaps
say, that 500 elements would appear in our list. I admit this
seems a small number; but Gallery speaks of having gone
through almost all the Chinese characters, and having omitted

* Compare with this the change of tr into cr, in craindre from tremere, veintre
(Hymn on Eulalia) from vinccre, &c. The same thing is seen in the English cor-
ruption of ask into ast. Is not triticum formally equivalent to Kpidri ? Compare
also the Greek and Latin names for Carthage ; Oxmantown and Ostraantown,
(Worsaae, Danes and Northmen in England.)

408 THOUGHTS ON THE FORMATION OF

only such primary characters (he calls them by perhaps a better
name, 'indivisible or fundamental characters') as were rarely
found, or useless. To allow 200 for such omissions seems suf-
ficient. However that may be, I should propose the formation
of such a primary list, and its being printed in a tabular form,
as a frontispiece to the index. Gallery has done this with his
1039 phonetic characters, and they are all visible (and of a suf-
ficient size for clearness) at one opening of the book. These
being arranged in the manner already mentioned, the remainder
of the index is to be placed under these, as keys or headings.
Under each I would place, in order, all with which it combines :
first, all the simple characters ; then, all the binary characters,
and so on. No doubt there would be a good deal of repetition
in this ; every compound character would be entered twice at
least; those consisting of three elements, three times; and so
on. But the advantage of being able to find any character you
want with comparatively little trouble, as soon as you have
recognized one of the elements it is composed of, seems to out-
weigh this disadvantage, and perhaps about 80 quarto pages
would be enough for the index to a dictionary of thirty or forty
thousand characters.

It is to be observed, and this I think a very important part
of the plan, that I do not propose to use compound characters
at all. The original 500, or whatever the number may be, would
be the whole number of characters used, and therefore of types
required.

You are reading, we will suppose, a Chinese book, and come
to a character you do not know. Seeing that it consists of
woman, mouth, and heart, you look for it under any one of these
three characters, and in a little while find the other two grouped
side by side with a number which enables you to refer to the
body of the dictionary.

There is but one character in the language made up of these
three elements, and therefore in order to recognize it without
ambiguity, it is not necessary that you should actually see it
before you in the index, or that you should even be told there
what you already know, that, to speak heraldically, woman oc-
cupies the dexter chief, and mouth the sinister. Some cases
undoubtedly there are— probably only few— in which the same
elements, differently arranged, form different characters; but

A CHINESE DICTIONARY. 409

nothing can be easier than to devise diacritical signs ; by which
the same group of characters should be made to refer, without
possibility of mistake, to the different compound characters.

Take a simple instance already noticed. Under corn you
find mouth, accompanied by an arrow pointing to the right, and
again a mouth with the arrow pointing upwards. The reference
to the first would be, to ho in the sense of comfort, and to the
second to ho in the sense of crying ; for in the first case mouth
stands to the right of corn, and in the second above it.

This principle once admitted, namely, that a character may
be as clearly recognized by means of its elements alone as if
a fac simile of it were given, may of course be applied much
more widely than merely to forming the index of a dictionary.
It seems to furnish the solution of the chief difficulty by which
the study of Chinese has hitherto been impeded. For we thus
get a mezzo termine between the unintelligibility of Chinese
written with Koman characters, and the impracticable expense
of a complete fount of Chinese type. Even if we had the 3000
elements*, which the ingenuity of Breitkopf has devised, where
should we, here in Cambridge at least, find a compositor suf-
ficiently learned to put them together ?

The reason why so many more elements are required to
imitate compound characters, than are necessary, if we content
ourselves with simply representing them, is of course the varia-
tion in size and shape, requisite in order to give uniformity in
these respects to the compound character. This uniformity of
size and contour is a matter of Chinese taste with which in
books intended for European use we need not trouble ourselves.
It is, by the way, an inconvenient taste even for the Chinese,
because in order to make complex characters distinct, the simpler
ones must be unnecessarily large. If it be said that the Chinese
would never become accustomed to characters of the proposed
kind, we may answer that even if this be so, the necessity of
printing in Europe books intended for Chinese use is not very
obvious.

Whatever may have been the case formerly, there can hence-
forth be probably no difficulty to hinder the printing whatever
is intended to be read in China, at presses (which, by the way,

* Am I right in thinking there are 3000 type elements ? or are there only 3000
punches ?

410 THOUGHTS ON THE FORMATION OF

is an incorrect phrase in speaking of Chinese printing) esta-
blished in the towns to which Europeans have now free access.
And it must be remembered that we could scarcely hope to pro-
duce in Europe what the Chinese would account a handsome
book. The softness of impressions from wood can hardly be
imitated with metallic type, and Chinese paper cannot, I believe,
be used in our printing presses.

A collateral advantage, resulting from what is now proposed,
would be the facility of learning to read Chinese. The difficulty
of analysing the characters would be removed, and when once
a student was able to read a book printed in the new method,
the transition to the usual characters would not cause more dif-
ficulty than Greek contractions, or than the ligatures in Sanscrit.
Another advantage would be that as the characters would follow
one another in regular order, accompanied only by brackets to
form them into groups, and by a few simple diacritical signs,
any ordinary compositor would be able to set them up. Pro-
bably it would not be found very difficult to distinguish the
phonetical elements by printing them with red ink, which to
beginners would be a great assistance. By similar means we
might distinguish the same character, according as in any sen-
tence it presented itself as a noun or as a verb.

Brockhaus has proposed a different way of printing Chinese
for European use, namely, in Roman letters with a numerical
reference under each word to its place in the dictionary he wishes
to see made.

There are two or three objections to this plan. In the first
place, difficult as it is to remember Chinese characters, it would
be found much more difficult to remember the meaning of a
number, even with the help of the pronunciation, because num-
bers give very little for the mind to fasten on, and can never be
exclusively associated with a single class of ideas. Imagine the
difficulty of remembering the plot of a story the persons in which
were denoted only by numbers.

Again, in all questions relating to what I think must, by
and bye, form an interesting part of comparative philology,
namely, the theory of the Chinese characters, such a plan would
be useless, even if we could suppose that all Chinese scholars
agreed to use the same dictionary.

Lastly, this method would form no introduction to the study

A CHINESE DICTIONARY. 411

of works printed in the Chinese character, a class to which the
great mass of Chinese literature must always belong. A man
might give years to the study of this Stratford atte Bowe Chi-
nese without being able to read the commonest characters.

A more radical reform has sometimes been proposed, namely,
simply to print Chinese in Eoman letters. Why are not the
absurd Chinese characters laid aside? has been asked in much
the same tone as the question one occasionally hears, of why
legal terms and forms should be used in conveyances ? Get rid
of these, it is said, and any deed might be written on a single
sheet of paper. The answer in the two cases is much alike.

If it had been possible during the last six hundred years to
enforce brevity in legal instruments, not only the practice but
the theory of conveyancing would be very unlike what they
now are. The complicated relations which have grown up
amongst us, the various subtle modifications of which the idea
of property has been found susceptible, could never have been
developed on such a system. Not only the outward form, but
that which the form represents, would have been different. Our
thoughts, and the mode in which we express and record them,
act and react on one another.

The influence of writing on the history of language, which
has been made the subject of an interesting essay by William
Humboldt, has been greater in China than anywhere else. The
hand and eye have, so to speak, brought into subjection the
voice and ear ; the reason of which is to be sought partly in the
original nature of the language, and partly in the general dif-
fusion of education.

The language of China, especially the written language, is
in many respects what it is, in virtue of ,the character, which
we cannot now give up without introducing ambiguity and con-
fusion*. To a certain extent the Koman letters may be used,
and this has already been done, as for instance by Morison and
Gon9alves. The dialogues of the latter are particularly valuable,
from giving both the Mandarin and the Canton pronunciation.

* The best plan by which indeed many of the difficulties would be removed,
would be to write all compound characters like fractions, I mean with the pronun-
ciation of the whole character above, and those of its elements below a horizontal

line. Lin, a wood, for instance, would be denoted by

mou, mou

412 THOUGHTS ON THE FORMATION OF

So far as the question, as to giving up the Chinese character,
can be decided by authority, it seems sufficiently settled. I may
refer particularly to what is said on this subject in Mr Kidd's
work on China. One kind of influence exerted by the character
is sufficiently peculiar to deserve mention. We have many
words whose meaning has been changed in consequence of a
mistake caused by accidental resemblances of sound. Such, for
instance, has been, at least in popular use, the case with demean.
Johnson even thought the secondary meaning had the authority
of Shakespeare. Again, there are words whose meaning has
been influenced by ' juxta-position,' by their occurring, so to
speak, in contact with others. Such are implicit, and buxom*.

But both kinds of influence may concur in Chinese. Not
only the sound of the word, but also the way in which the
sound is expressed, by bringing the character into constant
association with another, may influence the meaning. Take as
an instance the character already noticed, composed of heart,
woman, and mouth. The two latter characters alone form a
binary character, pronounced ju, and meaning even as, sicut.
This binary character is the phonetic element of the ternary
one, of which it forms the upper part. The latter is pronounced
shu, and means goodness or kindness. But it is related that
Confucius taught that this word is the summing up of all mo-
rality ; that it means the state of mind in which a man interests
himself in the happiness of others, even as in his own.

This development of the meaning of the word was, it is pro-
bable, merely the result of an acccidental coincidence of sound,
and of the selection of the one character to be the phonetic clement
of the other. But, error or not, this opinion as to the meaning
of the word has perpetuated itself; and what in this case is
referred to the authority of Confucius, has probably happened
tacitly in many others.

One of the difficulties in making a Chinese dictionary arises
from the number of compound words, that is, words each of
which means something separately, but when grouped together
express a single idea.

Remusat went so far as to say, that the compound word was
polysyllabic, and that each character merely represented a syl-
lable. This question is scarcely worth the attention which has
* So too in German Ehe.

A CHINESE DICTIONARY. 413

been given to it ; but the important point is, that the unwary-
scholar frequently endeavours to give separate translations to
each element of compound words. These are given in the best
dictionaries ; but there may be some difficulty in making a list
of them complete and easy of reference. In printing it would
be well, I think, to connect the elementary characters by a
hyphen.

The matter is so peculiar that you will not object to my
giving you an instance of the errors it is apt to produce. In
Kemusat's version of one of the 'Four Books,' as they are
called, of Confucius, it is said that Confucius lived in accordance
with the seasons and with the earth and water. The meaning
of this is certainly not clear; but 'water-earth' simply means
' climate.' The habits of Confucius were not in accordance with
the earth, whatever that may mean, nor with the water, but
simply, which is quite intelligible, with the climate.

The instances which in the Notes to Humboldt's letter to
him Eemusat quotes from other languages ('horseman' is his
English instance), are not quite parallel, for though no gram-
matical form indicates the relation between their parts, yet
ideally one of them is a substantive, and the other a modifying
adjective: whereas in Chinese the compound word is a new
formation, of which the meaning is suggested only by those of
its parts. 'Elementa quodammodo manent in composite*,' we
cannot define the matter more precisely. We see here, as in
the formation of the Chinese characters, and in the structure of
the language, the tendency to merely external union. There
is contact and combination, but no interpenetrating compound
growth, and the whole resembles not a picture but a mosaic.

The same remark might be made as to Chinese style, which
is all compact of set phrases and antitheses. 1 cannot enter on
all the matters of detail connected with the Index, of which I
have endeavoured to give you an outline. My ideas of them
are of course very imperfect. Gallery's merits, with respect to
Chinese lexicography, are doubtless great, both in his exposition
of the ultimate dissection of the characters, and in showing,
more clearly and fully than had been done before, the presence
of a phonetic element in the great majority of characters. His
assumption, that a set of phonetic elements were deliberately

* S. Thomas Aquinas de Prineipiis.

414 THOUGHTS ON THE FORMATION, &c.

and simultaneously invented, is unphilosophical, and seems to
Lave led him into liis principal error, that in all compound cha-
racters one element is phonetic.

This error is decidedly opposed to competent Chinese autho-
rities, and would place us, if we adopted it, in the dilemma of
either rejecting obviously correct analyses, or of setting aside
the laws by which the affinities of sounds are governed.

I must here conclude these remarks. They are the result
of your kindness, which has led to my seeing the works on sub-
jects of Chinese literature recently added to the Library, and
lias thus recalled my thoughts to matters which my increasing
illness had made me lay aside. You know the circumstances
in which I write, or, to speak more accurately, dictate. Vive
et vale.

Yours very truly,

E. L. ELLIS.

March 17, 1854.

VALUE OF ROMAN MONEY*.

GRONOVIUS'S estimate of the value of Roman money is
vitiated by two principal errors : his doctrine that 100 denarii
went to the pound weight of silver, *a doctrine connected with
his theory that the proper and direct meaning of sestertium is
two pounds and a half of silver, but which is contradicted both
by testimony and by the denarii, which like the bricks in
Richard II. are alive to this day to witness to the contrary ;
and his confounding the pound Troy with the Roman pound.
The errors tend to balance, one making the denarius too little
in value, and the other making our currency of too small value;
but his result is of course mere haphazard, to say nothing of his
neglecting the question of alloy.

The basis of the calculations in the Dictionary of Antiquities
is much more satisfactory, but the calculations themselves are
wrong. The articles Sestertius and Denarius do not take into
account that our shilling circulates as a counter above its in-
trinsic value. The value of the denarius is determined by com-
paring its weight of fine silver with that of the shilling. Now
as our coinage since 1816 is at the rate of 665. to the pound,
the result is the same as if the price of silver had been taken
to be 66^. per ounce standard, which certainly is not its real
price. The rate of coinage was purposely fixed above the
variations of the bullion market to prevent melting. Sixty-
two pence is the price commonly assumed in calculating the par
of exchange, and is rather a large average price. Taking the
data given in the article Denarius, and this price of silver, the
denarius of the end of the Republic is worth (not 8'6245<#.
as it is there made) but 8'099d., or in round numbers, not S^d.
but Sd.

The error will be nearly the same in the value of the later
denarius.

* Journal of Classical and Sacred Philology, Vol. i. p. 92.

416 VALUE OF ROMAN MONEY.

The value of the sestertium resulting from the value of the
denarius which I have quoted is £8. 19s. 8d., though by some
error of calculation it is reduced to £8. 17s. Id.; the real value
is £8. 8s. 8J</., so that the two mistakes, like Gronovius's, tell
against one another.

It is curious that the later value of the denarius gives the
sestertium £7. 7s. 7^d., a sum in 7 as the other in 8.

In the article Aureus, the writer says that the sovereign con-
tains 113'12 grains of fine gold. It really contains (neglecting
the third place of decimals) neither more nor less than 113
grains. The result is that lie gives the aureus as £1. Is. Id.,
and a little more than a half-penny, instead of as nearly as
possible £1. Is. 2d.

The following is an outline of my calculation :

Eequired the price of 60 grains of silver, ffths fine, at
per ounce, standard. (1 ounce = 480 gr.)

X = 6° 480 37 30 ' (Standard bein£ f$ tlls fine')
31 x 29

-D n .
Reducing,

31 x 29 = 302 - 1 = 899,

3 x 37 = 111,

x = 8*099^. = value of early denarius,
250 denarii = 1 sestertium,

240 pence =£1;

,,809-9 „ 101-23
value of sestertium = £ = £ - = £8*435,

y o ±2

= £8. 8s. Sd. 4 or £8. 8s. 8^d. nearly.

The later denarius is 52-5 gr. or 8'75 of the earlier, and the
sestertium is in the same proportion.

THE COURSE OF MATHEMATICAL
STUDIES*.

THE seventh query f, so far as it relates to the limits beyond
which it is not expedient that the undergraduate course of ma-
thematics should extend, seems naturally to form a part of a
more general question, namely, how the whole time given to the
study of mathematics may "be most advantageously employed.
In order to discuss this more general question, it is necessary to
consider on what grounds the study of mathematics is made to
form part of our system of education.

I. The grounds are two-fold : mathematics1 are studied as
ancillary to natural philosophy and as a means of training and
developing the mind. In the latter point of view they are
chiefly valuable, because they deal with necessary and not con-
tingent truth J. Of every necessarily true proposition which the

» Cambridge University Commission, 1852. Evidence on Mathematical Studies
and Examinations, p. wi.

1* The seventh query is : Would you be disposed to recommend the limitation
of some of the subjects included in the present range of the examinations, for
instance to omit such propositions and applications of the Calculus of Variations,
of the theories of Definite and Elliptic Integrals, of the Planetary and Lunar
theories, of the theories of Heat, Electricity, and Magnetism, of the undulatory
theory of Light, as require for their treatment a very refined and laborious analysis ?
Might such higher mode of treating these subjects be advantageously reserved for
examination for special prizes at periods subsequent to the Degree of B. A. ? Would
not the concentration of the attention of Students upon a smaller number of sub-
jects, and those restricted within narrower limits, tend to increase the accuracy
and raise the character of their knowledge, and to bring their instruction more
completely within the grasp of the public and recognized teaching of the Univer-
sity?

£ This applies to mixed as well as to pure mathematics ; the necessity of the
conclusion being, however, in the latter absolute, and in the former hypothetical,

27

418 THE COURSE OF MATHEMATICAL STUDIES.

mind distinctly apprehends as such, the contradictory is seen to
be inconceivable ; this inconceivableness of the contradictory
being ex parte mentis the criterion of necessary truth. Never-
theless, although when we think of any simple proposition in
arithmetic or geometry, we perceive not merely that it is true,
but that it must of necessity be so, this is nowise the case with
respect to all demonstrated or demonstrable results. The in-
tuition, so to speak, of the ablest mathematician is confined
within a narrower circle than that of the truths which he can
prove. He may satisfy himself of the cogency of each step of
the demonstration, and yet the essence of the conclusion — the
fundamental principle of its truth — remains unseen. The on is
manifest, but the S«m obscure ; and consequently a proposition
contradictory to that to which he has been led does not appear
to him an absurdity, but simply an untruth. It might, for what
he sees, have been true, though he knows that actually it is not,
and thus while he is aware that his conclusion is true neces-
sarily, yet still it seems as if it were so only contingently and
as a matter of fact, the demonstration appearing assensum con-
stringere, non rem. In a word, his conception of the matter is
still imperfect. But between this state of mind and that which
is produced by the contemplation of any elementary proposition,
there is no fixed or definite boundary. Every one who has
really studied mathematics must remember cases in which, after
long and patient thought, the reason of the truth of a propo-
sition, with the demonstration of which he may have been
acquainted for years, has seemed to dawn on him ; the propo-
sition thenceforth becoming, as it were, a part of his own mind
— a matter about which he is no more capable of doubting than
about the primary conceptions of form and magnitude. The
mind thus brought into nearer, if not immediate, contact with
necessary truth is conscious of its own development ; and herein,
I believe, resides the special benefit to be derived from the study
of mathematics, a benefit, that is, distinct from the exercise of
patience and attention which it undoubtedly requires, but which
is required also in other pursuits. The study of mathematics is
especially valuable, not because it gives the Student practice in
ratiocination but because it enlarges the sphere of his intuition,
by giving him distinct and conscious possession of truths which
lay hid in his conceptions of figure, number, and the like. But

THE COURSE OF MATHEMATICAL STUDIES. 419

in order to this kind of mental development, it is necessary not
only that the Student should master the successive steps of the
demonstrations set before him and retain them in his memory,
but that his mind should become imbued with their spirit and
essence. His real progress therefore is not to be measured
simply by the extent of ground over which he has passed : it
varies also according to the degree in which he has approached
towards a complete intuition into the results which he is able to
prove.

I believe that this principle ought to be our guide in ex-
amining the merits and defects of a course of mathematical study
intended to form part of a liberal education. But the connexion
of natural philosophy with mathematics must, to a greater or
less extent, modify the conclusions to which it would lead us.

II. It would be impossible to trace in detail the conse-
quences which appear to follow from this way of considering
the subject. They may be classed under two heads, the choice
of subjects, and the choice of methods. With respect to the
former, my impression has long been that a good deal might be
omitted which now enters into our course of reading, not only
without impairing its utility, but with positive advantage. A
more rigorous subordination of details to fundamental principles
would not only save the Student's time, but would make the
principles themselves be more clearly apprehended. Everything
received into our course ought to justify its admission there,
either by its own importance or by its connexion with something
more valuable than itself. Mere exercises of industry and in-
genuity, long numerical calculations, complicated processes of
algebraical reduction, tricks of transformation for the evaluation
of integrals and the solution of differential equations, and the
like, may all be accounted comparatively useless. These things
may be impressed on the memory but will hardly long remain
there, and meanwhile are felt to be rather a burden than an
acquisition. So in mixed mathematics, — many of the approxi-
mate formulae in optics, the less important astronomical cor-
rections, detailed descriptions of philosophical instruments, &c.,
might all be advantageously laid aside. Not that these things
are not worth knowing, but that they do not properly belong
to such a course of mathematics as we are considering — in which
the chief end proposed is a clear insight into fundamental priri-

27—2

420 THE COURSE OF MATHEMATICAL STUDIES.

ciples. In general it may be said that formulae of approxima-
tion are unsuited to the end we have in view ; they give little
or nothing on which the mind can rest : their value resides in
the practical application which is to be made of them, but which
the Student never makes. It is the predominance of approxi-
mate results which renders the lunar and planetary theories
unsatisfactory portions of the Student's course. They are, how-
ever, by no means to be omitted or curtailed, and with respect
to the latter, the evil might be lessened, though not without
some inconvenience, by giving more prominence to the general
theory of the variation of parameters as we find it in the Me-
canique Analytique and in some of Poisson's memoirs. Sir W.
Hamilton's essays in the Philosophical Transactions, and those
of Jacobi in Crelle's Journal, might, perhaps, give some ad-
ditional materials for the formation of a course of study on this
part of natural philosophy.

III. Secondly, as to the choice of methods, and especially
as to the preference to be given to geometry or to analysis.
Ever since the introduction of the modern analysis into Univer-
sity reading, there have been complaints of its having super-
seded the older methods and traditions of the Cambridge system.
Those who favoured its progress affirmed, and most truly, that
by its aid the Student advances faster, and goes farther, than
he could do without it ; he gains in fact more knowledge of the
subjects set before him. But this argument had little weight
with those who held that not the knowledge but the process of
acquiring it — the training and discipline of the mind was the
thing chiefly to be thought of.

It has been said that if information merely is the end in
view, mathematics have less claim on our attention than many
other things, and that most of the arguments in their favour
cease to be applicable if geometry is discarded or disparaged.
Of late these views seem to have gained ground in the Univer-
sity : their influence may be traced in the recent legislation on
the subject of mathematical honours. The principle on which
this re-action against the newer methods is chiefly based, namely,
that the mind of the Student ought to be as much as possible
conversant with fundamental conceptions is, I think, perfectly
correct. But it does not follow that analytical methods ought
to be discouraged. Demonstrations may be geometrical, arid

THE COURSE OF MATHEMATICAL STUDIES. 421

yet in a high degree artificial ; and first principles may be lost
sight of in a maze of triangles, no less than in a maze of equa-
tions. Though in mathematical investigations there is no royal
road, yet there is a natural one, that, namely, which enables the
Student, as far as possible, to grasp the natural relations which
exist among the objects of his contemplation. If this route be
followed, it matters but little whether the reasoning be expressed
by one set or kind of symbols or by another — in plain words —
in short hand — or algebraically. To -change the notation is
merely to translate from one language into another.

It is common to find persons in Cambridge and elsewhere
who insist upon it that geometry is geometry, and analysis
analysis ; but it may be doubted whether this notion of an
absolute separation between the two things is not the result of
a want of familiarity with either. It seems to be supposed that
if a mathematician treats a problem geometrically, he has to
think about it for himself, whereas if he treats it symbolically,
the symbols think for him. Perhaps it may be said that the
fact of there being any tendency towards so childish a notion is
in itself evidence of the mischief produced by the use of sym-
bols ; and certainly if symbols were never used, the notion could
not exist. But neither could it exist if they were rightly used
and rightly understood. The phrases which I believe may now
and then be heard from some of our younger analysts, such as
" the irrefragable a?," and " putting it into the mill," for ex-
pressing the conditions of a problem symbolically, show perhaps
that those who use them have but a half understanding of what
they are doing. But this evil is not to be remedied by discou-
raging the use of symbols. That our methods should be geo-
metrical is not by any means essential ; they ought to be natural,
and it has been too hastily supposed that they will necessarily
be so if symbols are excluded : whereas it is not by precise
adherence to any particular mode of expression that we are to
bring the Student to a familiar apprehension of the principles
of what he is engaged on. This is to be accomplished rather
by a " melange heureux de synthese et d'analyse," to use the
words of a great master in the art of which he speaks, than by
imposing either on teacher or students any unnecessary re-
straints. Let us consider the question more generally. When
the conditions of a problem have been stated, the solution may

422 THE COURSE OF MATHEMATICAL STUDIES.

be evolved from them by innumerable sets of combinations.
It does not often occur that even a practised mathematician
divines the simplest and the best. His choice among the routes
which he may follow is determined by an infinity of circum-
stances, and more especially by the way in which the conditions
have been expressed. " Words shoot back on the understanding
of the wisest," and so do symbols; and if the conditions of the
problem, whether geometrical or mechanical, or, if we will,
logical, are expressed by means of algebraical symbols, he will,
in all probability, not deal with them as he would have done
had they been expressed in common language : the reason of
which is, that the combinations and inferences which are the
most obvious when one mode of expression is employed, cease
to be so when it is replaced by another. Hence and from other
causes arises a variety of forms of demonstration, often, it is
true, perplexing, and yet, if attentively considered, full of in-
struction. For as to a mind which has attained to a perfect
mastery of the subject, and by which, therefore, the connexion
of thejlata of the problem, with its solution is perceived as by
intuition, all the demonstrations appear to be in their essence
identical, — different modes merely of presenting the same con-
ceptions,— so contrariwise the comparison of the different demon-
strations by which a given result has been established, tends to
make us recognize the grounds of their essential unity. It is
not by merely fixing in the memory the successive steps of a
single mode of demonstration, or even by studying several, if
we allow them to remain in the mind as distinct and hetero-
geneous processes of thought, that we are to acquire a complete
insight into the subject in hand, but by a more discursive
method, — by inquiring perpetually into the grounds and reason
of what we are doing, — by interpreting our symbols and follow-
ing the train of geometrical or physical conceptions to which
their interpretation leads, and again by retracing our steps and
passing from general considerations or purely geometrical rea-
soning to the technical language of symbols. Every change of
form should be suggestive of a new aspect of the subject, and
it is thus that the simplest way of considering it is to be dis-
covered.

In confirmation of some of the opinions which I have been
endeavouring to express, I may refer to Poinsot's admirable

THE COURSE OF MATHEMATICAL STUDIES. 423

tract on the motion of a rigid body. He lias there shown, with
great felicity both of thought and of expression, that the art of
combining symbols is by no means the whole of mathematical
analysis ; — that we must join to it the art of interpretation, and
that in many cases the essence and meaning of a result are
scarcely more obvious in the equations which express it than in
the original "mise en equation."

It did not belong to his purpose to point out that on the
other hand geometrical are not necessarily natural methods of
demonstration ; but that there is a real distinction cannot, I
think, be questioned. If it were not so, — if the Student felt
that by studying a subject geometrically he acquired more real
insight into it than he could else have got, geometry would be
more popular in the University than it now is. In truth, the
difficulty of remembering many geometrical demonstrations is
in itself a proof of their artificial character ; for that which the
mind has once completely grasped it does not easily forget.
What we want is the introduction of a freer and more liberal
method*, and especially the abandonment of the notion that
anything is gained by a rigorous separation of geometry and
analysis. It is this which for the most part makes our geometry
pedantic, and many of our analytical text books dry and sterile.
If it be asked how such a change could be brought about, I am
inclined to think it could only result from a change in the
opinions of those on whom the character of our studies chiefly
depends, — the Professors, Tutors, and Examiners. It could
hardly be made the subject of direct legislation.

IV. The same remark would apply to the subject more
especially suggested by the seventh query, namely, the proper
limits of an undergraduate course of mathematics. In every
branch of mathematics there are parts which from their abstruse-
ness ought not to be introduced into the degree examination ;

* What may be called the new geometry seems to be little studied in the Uni-
versity ; yet the method of which it makes so much use, namely, the generation
and transformation of figures by ideal motion, is more natural and philosophical
than the (so to speak) rigid geometry to which our attention has been confined. It
has been well said that the differential calculus is the symbolical expression of the
law of continuity, and probably the principles of the calculus would be better
understood if notions connected with this law were introduced at an earlier period
of our course. See, on the impossibility of severing our conceptions of space from
those of time and motion, Trendelenburg's Logische Untersuchungen.

424 THE COURSE OF MATHEMATICAL STUDIES.

but it would be found difficult to trace any precise and perma-
nent line of demarcation by which these might be separated
from the rest. In the progress of every science its methods tend
to become simpler ; and to refer especially to one of the subjects
mentioned in the query, namely, Electricity, I may remind the
Commissioners of the great simplification which the theory of
induced electricity has recently received. Professor William
Thomson's theory of electrical images has made, so to speak,
elementary, problems which previously required a " very refined
and laborious analysis." This theory, if electricity is to be
studied at all, would now almost of necessity form a portion of
the undergraduate course. It is, however, sometimes doubted
whether not only electricity but also the cognate theories of
heat, light, and magnetism ought not be excluded from the
degree examination. I confess to being unwilling that the ter-
minus of our mathematical studies should be made to recede,
and am disposed to believe that with the changes suggested in
the earlier part of these remarks, sufficient time would be found
for these subjects to be, up to a certain point, satisfactorily
studied. They now engage so much of the attention of scientific
men that it seems particularly desirable that the highest class
of Students should leave the University in a condition to follow
their progress and development. A young man will not wil-
lingly forget what he learned at Cambridge if he finds that it
enables him to understand the discoveries and researches which
are now going on; on the contrary, he will probably always
retain, at least, an interest in scientific matters. This advantage
would in many cases, perhaps in most, be lost if the subjects
in question were not studied until after the B.A. degree. Few
even of our best Students could then be induced to devote them-
selves to fresh mathematical studies, notwithstanding the influ-
ence of any system of prizes either for mere proficiency or for
original research. Of all such prizes* it may, I think, be said
that they could not be made to form a natural element of the
system of University education.

I do by no meana deny, or even doubt of, their utility ; but
we must remember that a University may be considered in two

* The remark does not apply to the Smith's Prizes, the examination for these
prizes being in effect a kind of sequel to the degree examination, and, therefore,
not requiring a distinct course of reading.

THE COURSE OF MATHEMATICAL STUDIES. 425

points of view, — as a seat of learning and as a place of education.
Much may be done by means of prizes to encourage learned men
in the pursuit of the kind of knowledge to which they have
especially dedicated themselves ; but as things are, and perhaps
as they ought to be, even a liberal education must end about
the age at which the Bachelor's degree is commonly taken.
And it may further be affirmed, with much show of reason, that
those whom after that epoch circumstances still permit "inter
silvas Academi quaerere verum," may with advantage, so far as
the symmetrical development of the mind is concerned, turn
from mathematical to other studies.

V. With respect to the subjects mentioned in the seventh
query, I have already made some remarks on the lunar and
planetary theories, as well as on electricity and the kindred
branches of physics. There remain, therefore, only the calculus
of variations and definite and elliptic integrals. Of these sub-
jects the first seems not unsuited for University reading. It
involves important principles and admits of important applica-
tions. From its connexion with the theory of conditions of
integrability it forms a natural sequel to the integral calculus ;
and, on the other hand, if the planetary theory were to be
studied in the manner which I have suggested, some previous
acquaintance with its principles would be indispensable. In
favour of definite integrals there is not so much to be said, many
of them depending for their evaluation on particular artifices,
which must uselessly burden the Student's memory ; and if in
an examination he attempts to determine the value of one with
which he is not already acquainted, he will often only waste
time and ingenuity to no purpose. Still certain definite inte-
grals must be known, and the theory of definite integrals of
periodic functions is especially important from its connexion
with Fourier's theorem and the development of discontinuous
functions. The difficulties of this theory have, I think, been
sufficiently removed to justify its introduction into our course ;
and it is not to be forgotten that no other step in the recent
progress of analysis has exercised so great an influence on ma-
thematical physics. The general theory of elliptic integrals is
too extensive and too abstruse for University reading, and the
study of isolated propositions almost useless. But Abel's theo-
rem ought to be studied as the natural development of the theory

426 THE COURSE OF MATHEMATICAL STUDIES.

of symmetrical functions, nor is there any difficulty in the de-
monstration by which an intelligent student can be embarrassed.
Likewise Abel's method for the division of the complete elliptic
function might with advantage replace Gauss's solution of the
binomial equation. It includes this solution as a particular case,
and from its generality is far more intelligible: Boscovich's
doctrine, that the more generally a subject is considered the
more easily is it understood, being for the most part true.
These instances, in which portions of the theory of the com-
parison of transcendents serve to complete and illustrate the
theory of equations, tend to show that a course of mathematical
study cannot well be made to adhere precisely to any definite
classification of the different branches of mathematics.

VI. One of the obstacles which hinder our mathematical
studies from being quite what they ought to be is touched on
in the tenth query*. It is there asked whether the number
of problems proposed in the Senate-House examination is not,
regard being had to the time allowed for solving them, greater
than it should be. It may be answered that it is necessary to
set before the candidates for high honours more problems than
any one is supposed capable of solving in the given time, in
order by the variety of subjects to provide sufficient employ-
ment for each, and at the same time to leave a certain freedom
of choice ; and, further, that if no more problems were proposed
than the ablest questionists might be presumed capable of solv-
ing, the number would still be too great for those of inferior
ability. All this is true, and it is therefore much less easy to
point out a remedy than to perceive the evils which result from
the present state of things, not only in the problem papers but
throughout the examination. He who for a season can remem-
ber a great deal, and who while he remembers it can reproduce

* The tenth query is : Is it your opinion, or the contrary, that the problems
proposed in the examinations bear too large a proportion to the questions derived
either immediately or by a very simple deduction from the books which are com-
monly read ? Are they for the most part so proposed, as to admit of their being
readily apprehended and their solution effected by a Student who thoroughly
understands the principles and applications of the branches of Mathematics upon
which they depend, or are they not unfrequently involved in such a form as to
require for their solution a peculiar tact distinct from accurate and philosophical
knowledge ? Are you not disposed to think the number of problems proposed to
be solved in the time allowed (as many, generally, as seven or eight in one hour of
time) is greater than the best prepared Student can be expected to complete 1

THE COURSE OF MATHEMATICAL STUDIES. 427

it rapidly and well, generally attains to more honour than he
quite deserves ; and though the undue influence of mere memory
is somewhat diminished by papers consisting of original pro-
blems, yet at any rate the student is trained to be quick and
ready rather than wise and thoughtful. In devising problems
it is difficult to avoid mere puzzles — things to be solved only
by some happy guess ; and if, on the other hand, examiners
were to confine themselves to tolerably obvious deductions from
known propositions, the problem papers would cease to be a
counterpoise to the rest of the examination. If the candidates
for high honours were, as in the Smith's prize examination,
examined apart from the rest, it might be possible, so far as
they were concerned, to diminish these difficulties by varying
the modes of examination. In a select examination there could
be no excessive inconvenience in giving almost unlimited time
for the consideration of the questions proposed, and in these
questions the candidate might be required to state accurately,
and in detail, the grounds of his views on fundamental prin-
ciples in analysis, geometry, or physics. Again, in such an
examination books might perhaps be introduced, and if so, in-
teresting questions might be proposed of a kind now inadmis-
sible; to mention one class only — the candidate might be re-
quired to form an opinion on any controverted point, to examine
for instance the correctness of Sir James Ivory's doctrine, that
in certain cases the ordinary condition of fluid equilibrium is
insufficient, and to state his reasons for adopting or rejecting it.

It must, however, be remembered that after all possible im-
provements the complaint that schools " lack profoundness and
dwell too much on seeming," will always be more or less just.
A course of study, of which the most obvious purpose is to pre-
pare the student for an &rt$6£Jft?, can never be quite what a
course of study ought to be, and might be made if higher ends
could always be kept in view.

CAMBRIDGE: PRINTED BY c. j. CLAY, M.A. AT THE UNIVERSITY PRESS.

RETURN TO:

CIRCULATION DEPARTMENT
198 Main Stacks

LOAN PERIOD 1
Home Use

2

3

4

5

6

ALL BOOKS MAY BE RECALLED AFTER 7 DAYS.

Renewals and Recharges may be made 4 days prior to the due date.
Books may be renewed by calling 642-3405.

DUE AS STAMPED BELOW.

FORM NO. DD6
50M 1-05

UNIVERSITY OF CALIFORNIA, BERKELEY
Berkeley, California 94720-6000

U.C. BERKELEY LIBRARIES

I