eee:
a

Sei ener ae
OCT pon

eee

a

en

Shel

Penne

Ax A
Anh AAA

* A Z
“aaa tt

Neier s,
PRT Nas pmeAnRacr Ait >

RAR AAO

4

NNN A

wR, J RARKE ‘al 53 eee a
J Ah \"§ Fie meteenennht
Pe Hal panannsnstenannrtr
aye rs ae “\h PN nan nm ah oh AAP ER ne pANARARA
eee sae het ie PARARRRR ADAG.”
~ ZR YP ZAR ppen ea 2
elke = el. Cee ta

ae . mame arn SAR Sha > =~
maar ea AR fog MARA pn’ h ee ey 4
aor ater ant AAA. ‘yf Whe oe Z
ie ae Ss => 7) 27
aeeenintnsane ame ere De
ee eee
VES Cp Be WW Be oe
io as A meth ne
AA AA RAR pF aS se >
ent ARAN RI oft? sp, ao UO SENRINIARR RM
4 > pa y??, : L
oy, Pp 2M. MM Spur esti pF
SN nee Ree BN yl nantinnnnnen ec eens, ~.. £4
‘ Nanna Aaa wy . z > pyre nan ananet RRR, x Amn 27
Sr i a EE a a aneeee™ YEAR AANA fin 4
ai Mee pen NT ce Me MEN,
= Ara a : —_ ‘ae An 2 co - = As
a Meena SS ee ON A re a a AA ~ Aaa
"95? ASAT Ae XN we An *y OE = fA 2A
Cee WY ALA YOM NSN RRA ANION AY >, 39 Bye SN yy,
3 ry 1) A RA aA w aK A ne Wi > I ¥y 2, | AA ; AAna,
3 3 yin AR AR oc af RAW N f zy 2 77. pA BY F 1M a’ ARRAS
2 a 5 Fe ann GONE eee ae 4 g >> 2, 9 ; a Aar
aeRO ean Re Re Ue
. CEMA ne WM Be Ao AANA
ATTEN ON, NM ap eee nnnnn
ne Braga nanneGer <= Sn" aM: OE oe A pane
annennn nh Gat nen A Ae = SN moma nnn axe
SRA EO NARA. a ar MAS NAAR ARR RAINY
DA Sp ee EN he “awh rie Danna We : An ee.
7 3 4 lat om
La >> 22. { “ fi A afar WA, ; Pg als
— KARA A Fh Baw Ops NN AN = A ARNIS OO BRAN NTR
WEAKARR AZ. Se Seer LN LPP, ARR ne NN An! r
TAN! aetacacas hy we 2-5) a pe AN BRE ERREKIKKR Ane rae
= yh PP A Sse oa OA CAN, opal ptaae pace

eee eee

“Sh ser A. rRAR CS RAAdA
XE AAA ann: fall pes = Sen :
N 2 al nn nap Ugo y)
Waesaa _ ANA
= WAaA P ao

»» Ar At anh
Vv i fe pete

AAR aR

~<Cerr pipes Retaeret 6 wn Oe ali BA

TAK Anne aan” re vv
ran ve ae ae AN :
pens ey |
af’ > ~ ; Aah A R
>» AR a Aa

a \A' .
Sh Wah AARO Race a
SAY fog AA

ww
Vv

Sy)
\ ’ %
Woy
SIN
VUS
=
_
>
3
>
>

=

=~
Tye

fale
CRARN

Leif
ae ‘ oy

RS

hr
oR yr a
SR eh
nee KAR
. WAR ar

aA pao ane RANE Me ae
ees Xs hs ee ae . Sow ws py
aT ENG. a
i : AARAR aan art
hh EA

AL
vt AYO WD ann

‘SS Das mm “Tar atom

Pa vwN 4

Eh a
‘
: =
. s
ia
* ad 7
~ i s m ‘ »
aves
: —
ae ‘
' onus -
nl >> “
- / ee _
- : ae

TRANSACTIONS

OF THE

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

ESTABLISHED NOVEMBER 15, 1819.

VOLUME THE EIGHTH.

«
CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS:
AND 80LD RY

JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. DEIGHTON; AND MACMILLAN & CO., CAMBRIDGE.

M.DCCC.XLIX.

tt
>

=i “

Ve ee Mee

N°.

Ill.

IV.

V.

VI.

VIL.

VIII.

CONTENTS OF THE EIGHTH VOLUME.

PART I.

Ow the Foundations of the Theory of Probabilities: by R. L. Euuis, Esq.
EFAs PL elloroMOp oD rinitimO Oleg elarteretayars/<lateloraiiia sieteleie bate \clebela Sais a s/a/e) <iaia vie «

On the Reflexion and Refraction of Light at the Surface of an Uncrystallized
Body: by the Rev. M. O’Brien, late Fellow of Caius College .........-.

On the Possibility of accounting for the Absorption of Light, by supposing it
due to the Motion of the Particles of Matter: by the Rev. M. O’Brien,
LP INATES NGA CHL ee COO ee OU band Obs GbbBe OAEHOS Opp anoSuasoK

On a new Fundamental Equation in Hydrodynamics: by the Rev, JAMES
Cuauuis, M.A., Plumian Professor of Astronomy and Experimental Philo-
sophy in the University of Cambridge... .. 1.0.1... 0eecerteecccececececes

Observations on the Nature of the Biliary Secretion ;—the object being to shen,
that the Bile is essentially composed of an Electro-negative body in chemical
combination with one or more inorganic bases: by GrorcE Kemp, M.B.,
Si Tea tail COG on ca G cep boboe iPebopopdan dns s05500c Kasse ODSrOREOUCe

On the Motion of Glaciers: by Witu1am Horxtns, M.A., and F.RS.,
Fellow of the Cambridge Philosophicul Society, of the Geological Society,
and of the Royal Astronomical Society.......... 0.202. 00ecee cee eceeeeees

On the Theory of Determinants: by A, CaytEey, Esq., Fellow of Trinity
(ONE 2. sneno an nneca nce Haan oonInOo ade ndosoaE AaBoDoEsC ABAoaseodeSmODsC
On Small Finite Oscillations: by the Rev. H. Hoxpircu, Fellow of Caius
College, and of the Cambridge Philosophical Society......... iataleteletetetersicte
On some Cases of Fluid Motion: by G. G. Sroxss, B.A., Fellow of Pembroke
(On gocon cEcoye cegb6 UaROOOr CODD ED HO JOU hbo cnPoccopperotandgtioncaond

Notice on the Occurrence of Land and Freshwater Shells with Bones of some
extinct Animals in the Gravel near Cambridge: by P. B. Bropiz, F.G.S.,
Op LOMA! (Chee paccennorndoudcnnnonocacnccbooepbapoeodecoonOc lor

62

PAGE

I

iss
~I

31

44

50

75

89

N°.

XII.

XIII.

XIV.

XVII.

CONTENTS.

PART II.

Ow the Foundation of Algebra, No. III.: by Aueusrus De Morean, Esq.,
V.P.R.A.S., F.C.P.S., of Trinity College; Professor of Mathematics in
University College, London . - -\-\strssstase new ilerean cic viosec= =e asl eeivir's es

On the Measure of the Force of Testimony in Cases of Legal Evidence: by
Joun Tozer, Esq., M.A., Barrister-at-Law ; Fellow of Gonville and Caius
(CHAE ABE Baa SCOT IB OGIS FHC bd doe od acne SSE OGRORRSCos Dene ae

On the Motion of Glaciers, [Second Memoir]: by Witu1am Hopkins, Esq.,
M.A., and F.R.S., Fellon of the Cambridge Philosophical Society, of the
Geological Society, and of the Royal Astronomical Society ..........-+++-+

On the Fundamental Antithesis of Philosophy: by W. Wurweut, D.D.,
Master of Trinity College, and Professor of Moral Philosophy..........--

On Divergent Series, and various Points of Analysis connected with them:
by Aucustus De Moraean, Esq., V.P.R.A.S., F.C.P.S., 4 ie ree
Professor of Mathematics in University College, London . .

On the Method of Least Squares: by R. L. Evuis, Esq., M.A., Fellow of
Printiy College . <0.» .siicc cove ver ieieisjele sinks sisieesislejecciccciceecascerececes

On the Transport of Erratic Blocks: by Wiiu1am Horxtns, Esq., M.A.,
and F.R.S., Fellow of the Cambridge Philosophical Society, of the Geological
Society, and of the Royal Astronomical Society ....-.+-+-+++0++ssseere ree

PAGE

139

143

204

N°. XVIII.

XXI.

XXII.

XXIII.

XXIV.

XXV.

XXVI.

XXVII.

XXVIII.

XXIX.

XXX.

CONTENTS. v

PART III.

PAGE
Ow the Foundation of Algebra, No. IV., on Triple Algebra: by Avcustus
Dre Moraan, Esq., V.P.R.A.S., F.C.P.S., of Trinity College; Professor

of Mathematics in University College, London.....-+-++++++++++00+ee00e- 241
On the Values of the Sine and Cosine of an Infinite Angle: by S. Earnsuaw,
M.A., of St John’s College, Cambridge........ .0..+--++02+e--seceeeees 255

On the Connexion between the Sciences of Mechanics and Geometry: by
the Rev. H. Goopwin, Fellow of Caius College, and of the Cambridge

TE PIS TI ATETU ROT RTL oe eae 9S nOE MOD. o CORLOD Ie Gt GOI oC OOO IDE SSC OSE OTe 269
On the Pure Science of Magnitude and Direction: by the Rev. H. Goopwin,
Fellow of Caius College, and of the Cambridge Philosophical Society...... 278

On the Theories of the Internal Friction of Fluids in Motion, and of the
Equilibrium and Motion of Elastic Solids: by G. G. Strokes, M.A.,
Fellow of Pembroke College, Cambridge .........2.++1+++++sceeseeceseeees 287

Calculations of the Heights of the Aurore Boreales, of the 17th September
and 12th October, 1833; with Observations upon the Locality of the
Meteor: by Ricuarp Porrer, M.A., late Fellow of Queens’ College,
Cambridge, and Professor of Natural Philosophy and Astronomy, Uni-

Opera) Ciao, ILORIG Peep son ancosnocsononasoaparuepmoo cnc hotobodD Maas 820
The Mathematical Theory of the two great Solitary Waves of the First
Order: by S. Earnsuaw, M.A., of St John’s College, Cambridge....... 326

On the Geometrical Representation of the Roots of Algebraic Equations:
by the Rev. H. Goopwin, late Fellow of Caius College, and Fellow

of the Cambridge Philosophical Society... 12.1.2: +++ 0.00+eeecceeceeeeeees 342
On a Change in the State of an Eye affected with a Mal-formation: by
em eA TREE SU ns AEN OTL OI RETMILOY Llermarataiaie ease rate ea aia relsistevelsiaveters erate cyers 361

A Theory of Luminous Rays on the Hypothesis of Undulations: by the
Rey. J. Cuauuis, M.A., Plumian Professor of A i ae and (nem
mental Philosophy in the University of Cambridge.. Sebo cocoos Elie

A Theory of the Polarization of Light on the ae of D Ondulations :
by the Rev. J. Cuanuts, M.A., Plumian Professor of te and
Experimental Philosophy in the University of Cambridge .. ocopoooag eat!

On the Structure of the Syllogism, and on the Application i the Theory
of Probabilities to Questions of Argument and Authority: by AuGustus
De Morean, Sec. R.A.S., of Trinity College ; Professor of Mathematics
UU MMV Er StmCOMeRe, -LONGOM cai on \na's\o\s!2\o/00)-\0\+1es/afela\ss\s/«)« a\o\clels)ee™ # oie.eie)<ia\e 3879

Supplement to a Memoir On some Cases of Fluid Motion: by Grorcr G,
Stokes, M.A., Fellow of Pembroke College, Cambridge .......+-+..+++1++- 409

vi

N°. XXXI.

XXXII.

XXXIII.

XXXIV.

XKXV.

XXXVI.

XXXVII.

X XXVIII.

XXXIX.

CONTENTS.

PART IV.

Own a New Notation for expressing various Conditions and Equations in
Geometry, Mechanics, and Astronomy: by the Rev. M. O’Brien, late
Fellow of Caius College, Professor of Natural Philosophy and Astronomy
ae oiorets Golleren SE OniOr stele reety ete eae tetateta le ete] Yale ale a= alee 2] inte ies (ele) bins

On the Principle of Continuity, in reference to certain Results of Analysis :
by J. R. Youne, Professor of Mathematics in Belfast College ........-..+

On the Theory of Oscillatory Waves : by G. G. Sroxss, M.A., Fellow of Pem-
GY DREN COM PIE ieee since rniaieinie late Soe eielelereletaelee Reece eminent isles

On the Internal Pressure to which Rock Masses may be subjected, and its
possible Influence in the Production of the Laminated Structure: by
AG JETS EE DOT eks VIG LOSES 0s Ana AB Bingn bodice 2 obnanasoooaeeAn

On the Partition of Numbers, and on Combinations and Permutations: by
Henry Warzurton, M.A., M.P., F.R.S., F.G.S., formerly of Trinity
College

On a Peculiar Defect of Vision: by Henry Goong, M.B., of Pembroke
(QU PIR o hoo Sa.45 goa FOO E BeOS SUDO RON e Sere -cOognnS One Dog enae aoe

Contributions towards a System of Symbolical Geometry and Mechanics : by the
Rev. M. O’Brien, Professor of Natural Philosophy and Astronomy in
King’s College, London, and late Fellow of Caius College, Cambridge ......

On the Symbolical Equation of Vibratory Motion of an Elastic Medium,
whether Crystallized or Uncrystallized: by the Rev. M, O’Brien, late
Fellow of Caius College, Professor of Natural Philosophy and Astronomy
Siem MS Eee MG OLE MLD UME eet olers ete ascic close vale nfe nist -iaeie leis /=[e(ere/</oi!0\<ieis@is ore

A Theory of the Transmission of Light through Transparent Media, and
of Double Refraction, on the Hypothesis of Undulations: by the Rev.
J. Cuartuis, M.A., Plumian Professor of Astronomy and Experimental
Philosophy in the University of Cambridge .......20++:+eeeceeeeeeeceee

PAGE

415

429

441

456

471

493

497

508

INES:

XLII.

9 UI ie

XLII.

XLII.

XLIV.

XLV.

XLVI.

XLVII.

XLVIII.

XLIX.

L.

LI.

LIL.

CONTENTS.

PART V.

Ow the Critical Values of the Sums of Periodic Series : by G. G. Stokes,
M.A., Fellow of Pembroke College, Cambridge ........--0--+-+0+ee+eeees
A Mathematical Theory of Luminous Vibrations : by the Rev. J. Cuauuis,
M.A., F.R.A.S., Plumian Professor of Astronomy and Experimental Phi-
losophy in the University of Cambridge ..........0+.000-eecscee een ceeces
Supplement to a Paper “ On the Intensity of Light in the neighbourhood of a
Caustic :” by Grorce Bippexu Airy, Esq., Astronomer Royal.........
Some Remarks on the Theory of Matter: by Rosert L. Euxis, M.A.,
Fellon of Trinity College, Cambridge... 00.00. .+0+s+sseenecese sesso.
Methods of Integrating Partial Differential Equations: by Augustus Dr
Morean, of Trinity College, Cambridge, Secretary of the Royal Astrono-
mical Society, and Professor of Mathematics in University College, London.
Second Memoir on the Fundamental Antithesis of Philosophy: by W.
Wuewe.L, D.D., Master of Trinity College, and Professor of Moral
PESOSOP/Yf lo eieinele cles sniai) aes si eais(eieis eee si ae al Aine ease eIaiee wale enioe 2
Observations of the Aurora Borealis of November 17, 1848, made at the
Cambridge Observatory: by the Rey. J. Cuauuis, M.A., F.R.S., F.R.A.S.,
Plumian Professor of Astronomy and Experimental Philosophy in the
Elnapen siti mapa OMINUTIUgE nw yoral toate e set-l el xieistatahes sisiclcls croc. iareraieelctesiae eis ee
On Clock Escapements: by Epmunp Brcxerr Denison, Esq., M.A., of
Bransiyn Coneze;, CAmMPItd ces Neel elec sicioie «281-1 saree eeneitatasiec cise uae:
(Suprrement.) On Turret-Clock Remontoirs: by E. B. Denison, M.A.,
Ofmme la rarest gs COME G; «53; ois. siakajnictelevoleia oeetetn “Poe isyascinior aero earaeson ne oleae oes
On the Formation of the Central Spot of Newton’s Rings beyond the Critical
Angle: by G. G. Sroxes, M.A., Fellow of Pembroke College........+...
Of the Intrinsic Equation of a Curve, and its Application: by W. WuEwE.u,
DED ry Mastersone Lrinityp Couece jem arerceiea te ciaceess ca thane coe att.
On the Variation of Gravity at the Surface of the Earth: by G. G. SvoxKes,
M.A., Fellow of Pembroke College ....--:..200c0ccesccnecseacstcvvecves
On Hegel's Criticism of Newton's Principia: by W. Wuewe-t, D.D.,
Masters of, Eninit ya Collepeta 1. acayaste Jorn atte ecg wocooeaeanicon:
Discussion of a Differential Equation relating to the breaking of Railway
Bridges: by G. G. Stokes, M.A., Fellow of Pembroke College.........

Vii

PAGE

584

595

600

606

614

633

ADVERTISEMENT.
WSS

THe Society as a body is not to be considered responsible for any
Jacts and opinions advanced in the several Papers, which must rest

entirely on the credit of their respective Authors.

TRANSACTIONS

OF THE

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

VouumeE VIII. Parr I.

CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS;
AND SOLD BY
JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. & J, J. DEIGHTON ; AND T. STEVENSON, CAMBRIDGE.

M.DCCC.XLIV.

a

Ke
bas
7
=
»

»
>

4

EEG LT

I. On the Foundations of the Theory of Probabilities. By R. L. Evwis, Esq.
M.A., Fellow of Trinity College.

[Read Feb. 14, 1842.]

Tue 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 Con-
dillac. It rejects all reference to ad 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.

I shall be satisfied if the present essay does no more than call attention to the in¢onsist-
ency 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 possible ways in which it may take
place, divided by the total number of such ways which may occur on the given trial.

Ut (ig cece on m,, denote equally possible cases which may occur on one trial, a.b...../. those
which may occur on a second trial, a3b;....p; those belonging to a third, &c.: then a,,
bez... A, debg...&c. &e. are all equally possible complex results.

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

Eee
1.2...p1.2...(& — 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 determined, 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 @ 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

Vou. VIII. Parr I. A

m? (1 — m)*-P

2 Mr. ELLIS, ON THE FOUNDATIONS OF THE THEORY OF PROBABILITIES.

another will occur more frequently; or a case in which he may be able 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 thinking 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 t and if all these 6 cases
tend to recur equally on the long run, the event must tend to occur @ times out of 6; 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 mag-
nitudes of the several terms of the binomial expansion.

[*]

The general term of {m+ (1 —m)t*, is —_+>=— m? (1-m)*-?, which is the probability
(p][* - P]
that an event whose simple probability is m will reeur p times on & 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. i

A coin is to be thrown 100 times: there are 2'° 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 2!° 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 2! possible sequences give the
difference less than 7, which is proportionally 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 constat, but the single case producing 100 heads may occur so much oftener than any
ease 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. Precisely 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
occurrence. 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,

Mr. ELLIS, ON THE FOUNDATIONS OF THE THEORY OF PROBABILITIES. 3

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 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 fre-
quency of recurrence of different possible results of similar trials. These judgments are founded,
not on the fortuitous and varying circumstances of each trial, but on those which are per-
manent—on what is called the nature of the case. They involve the fundamental 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 them-
selves—namely, the connection between the axiom just stated, and the inductive principle;
the real utility of Bernouilli’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 Probabilités.”

(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 Bernouilli’s
theorem afforded a demonstration of a general law of nature, extending even to the moral world.

At p. xuii. of the Essay, prefixed as an Introduction to the third edition of the Théorie des
Probabilités, after giving some account of the theorem of James Bernouilli, Laplace proceeds :
‘On peut tirer du théoréme précédent cette conséquence qui doit étre regardée comme une loi
générale, savoir que les rapports des effets de la nature, sont a fort peu pres constants, quand
ces effets sont considérés en grand nombre....Je n’excepte pas de la loi précédente, les effets
dus aux causes morales.”

It appears not to have occurred to Laplace, that this theorem is founded on the mental phe-
nomenon of expectation. But it is 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,,..‘*Il suit encore de ce théoreme que dans une série d’événemens inde-
finiment prolongée, l’action des causes régulieres et constantes doit Pemporter a la longue, sur
celle, des causes irréguliéres....Ainsi des chances favorables et nombreuses etant constamment
attachées 4 Vobservation des principes ¢ternels de raison de justice et d’humanité, qui fondent
et qui maintiennent les societés; il y a un grand avantage a se conformer a ces principes, et
de graves inconvéniens 4 s’en écarter. Que lon consulte les histoires, et sa propre expérience on
y verra tous les faits venir 4 Yappui de ce résultat du calcul.” Without disputing the truth of
the conclusion, we may doubt whether it is to be considered as a ‘“‘résultat 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 history were merely confirmations
of the “résultats du calcul.” We are tempted to say with Cicero—‘“‘hic 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 proportional 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 un-
certain.

A2

4 Mr. ELLIS, ON THE FOUNDATIONS OF THE THEORY OF PROBABILITIES.

Nevertheless it has been applied to a great variety of inductive 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 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 occasion. This presumption the theory of probabilities estimates at

m+1
m+2°
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

But here two questions arise ; What shall constitute a ‘next occasion?” What degree

eal F :
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

5 11 P ° . .
have one. Yet the theory gives ie the presumption in both cases. If right in one, it

11
cannot be right in the other. Again, let all the flags be red. Is it aa that the next

vessel will have a red flag, or a flag at all? If the same value be given to the pre-
sumption 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 presumption in favour of some common cause producing mo-
tion in that direction. But this presumption depends not merely upon the number of observed
movements, but also on the natural 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 movements. But in the admis-
sion thus by implication made, that regard must be had to the similarity of the move-
ments, too much is conceded for the interests of the theory. For are the retained move-
ments absolutely similar? The planets move in orbits of unequal eccentricity and in different
planes: they are themselves bodies of very 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 determining the direction of their movements would be strength-
ened ; 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,

Mr. ELLIS, ON THE FOUNDATIONS OF THE THEORY OF PROBABILITIES. 5

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 seem-
ingly 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 @ priori pro-
bability was greater than 4 is 3. Such at least is the received result. But in reality,
the a priort 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 ace is thrown. Here is a com-
plex 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 oceur. They are in fact related as genus
and differentia. Beside 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 circumstances 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 # is the probability that any observed event had on an @ priori
probability greater than 4, or that three out of four observed events had such an @ priori pro-
bability, seems totally to want precision. A priori probability to what mind? In relation to
what way of looking at them ?

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

4 . 1 > : .
in a determinate manner) was at let us take a second set of h trials for which the same quantity

1
and 1,

ne m —
is —: and so on to
m m

When the trials have taken place, we shall have approximately,

1 2 m—1
h(a tate t +1)
m m m

of the sought events. Of these

h (; te ~_ 1 =) iS 4
~+— ~+— etyes
2m) * (; m é
had a priori a probability greater than 4. Summing these series and dividing the second by
38m +2
ae i 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 3,
which is the received result.

the first, we get

6 Mr. ELLIS, ON THE FOUNDATIONS OF THE THEORY OF PROBABILITIES.

(17.) Thus, it appears this result is based upon some thing equivalent to the following as-
sumption :—There are an infinity of events whose simple probability @ priori is w, and another
infinite number for which it is a. These two infinities bear to one another the definite ratio of
equality, (w and a’ may represent any quantity from 0 to 1.) Now in reality, as we have seen,
these numbers are not only infinite, but in rerwm natura indeterminate, and therefore the assump-
tion 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 considered, 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

h 1 QP
(= t oerh Soap wes tis

WG a) + eel

a 1 5
would have belonged to trials where the simple @ priori probability was > ae the ratio of

of which

these two expressions is ultimately

fees, (ye

fy wda 2 ;
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 asswme 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,

aie Re : 1
that this simple probability common to all these allied phenomena is > Ay

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 regularity, 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.

II. On the Reflexion and Refraction of Light at the Surface of an Unerystallixed
Body. By the Rev. M. O’Brien, late Fellow of Caius College.

[Read Nov. 28, 1842.]

THe object of the present paper is to determine completely the Laws of Reflexion and Re-
fraction of Light, without introducing any empirical considerations, or omitting to take into aceount
the normal vibrations which are generated in cases of oblique incidence. Though several eminent
mathematicans have written upon this subject, I believe that most of what is here contained is new.
I must state, however, that I have not been able to procure a Memoir by M. Cauchy, which
he constantly refers to in his Ewercices d’Analyse et de Physique Mathématique (for 1840),
and in which he has given a general method of arriving at the equations of condition relative
to the limits of bodies. I can therefore only guess at the physical principles upon which he
obtains his equations of condition, which equations, in the form he has given them in the
Evxercices for 1840, are particular cases of those obtained in the present paper. As M. Cauchy
states that he has made use of some new principles in obtaining his equations of condition
(see the Nowveaua Evercices, Prague 1835, p. 203), I am justified in assuming that the
method employed in the present paper is different from his; for I have deduced my equations
of connection, not from any new physical principal, but from an old and obvious one, which
has been either directly used, or tacitly assumed by all the writers upon the reflexion and refraction
of Sound and Light, that I am acquainted with. This principle is very clearly stated by
Poisson, in the Mémoires de 1’ Institut, Tom. x. p. 320.

The following is a brief outline of the course pursued in the present paper.

In Section I. I have proved some very simple theorems by means of which I have after-
wards deduced the laws of reflexion and refraction, without assuming the integrals of the
equations of motion, or supposing the waves to be plane.

In Section II. I have deduced the equations of connection of the vibratory motion of two
media, separated by a plane, from the principle above alluded to. These equations of con-
nection are apparently the same as those given by Mr Green in the Cambridge Transactions,
Vol. vir. p. 11.3; but they differ from them very materially with respect to the constants in-
volved in them, and on that account they, and the results deduced from them, are perfectly
free from difficulties* which seem to me to be fatal to the correctness of Mr Green’s equations,
and which he appears to have felt himself. I shall not however enter into this subject now,
as I shall be obliged to do so on a future occasion.

I have shewn that these equations of connection are considerably simplified when we
suppose the ether to have the same constitution as ordinary gases, and neglect the variation of
temperature.

In Section III. I have applied these equations of connection to determine completely the
laws of reflexion and refraction of polarized light, both as regards direction, colour, and in-
tensity, taking fully into account the production of normal as well as transversal waves in the

" One difficulty I have mentioned a little farther on, Another difficulty is this, that there are just the same constants (4) and
(B) in Mr Green’s Equations of Connection, as those in his Equations of Motion: which arises, first, from an error in the form
of the function «, (Cambridge Transactions, Vol. vit. p. 7), and secondly, because qd, is not symmetrical round the axes of y and
# at the plane of separation, as Mr Green assumes it to be.

8 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT

case of oblique incidence, where the vibrations take place parallel to the plane of incidence.
The laws, which the directions of the normal rays obey, are curious, and have not been noticed
before, so far as I am aware; nor indeed can I perceive that these rays have been taken into
account in a satisfactory manner by writers upon this subject.

In this section I have shewn that if we take the equations of connection in their simplest
form, Fresnal’s formule will result from them on two suppositions; first, that normal waves
are propagated very slowly compared with transversal waves; and secondly, that normal waves
are propagated with the same, or nearly the same, velocity in vacuum and in transparent media.
The former hypothesis seems to me to be very improbable, for it is very difficult to conceive
a stable medium in which normal waves are propagated more slowly than transversal. I may
observe here, that M. Cauchy’s equations and results are obtained by assuming the truth of
this hypothesis, (see his Ewercices for 1840, p. 135), and appear, on this account, liable to
objection.

In Section IV. I have shewn that Fresnal’s formule may be applied, without making any
vague use of the symbol Wea to the case of Total Internal Reflexion, and that he was
fully justified in the very remarkable interpretation he put upon his formule in this case.

In Section V. I have shewn that normal waves will never produce any sensible effect on
the eye by producing transversal vibrations, provided the velocity of propagation of normal
waves be either very great, or very small, compared with that of transversal waves.

In Section VI. I have attempted to prove, from well established experimental laws, that
polarized light consists of vibrations at right angles to the plane of polarization.

In Section VII. I have briefly shewn how we must proceed when the equations of con-
nection are not taken in their simplest form, in which they are used in Section 111.

Lastly, in Section VIII. I have obtained expressions which apply to substances of high
refractive power, such as the diamond, and from which I have deduced results in exact
accordance with the experiments of Mr Airy. These expressions are different from those of

Mr Green, which certainly cannot be correct, since they give (see Cambridge Transactions,
2 2

1
Vol. vii. p. 23,) g = more than “ » for plate-glass; and a =more than 5° for diamond: which
a a

results are utterly at variance with experiment. The fact is, Mr Green’s original mistake
respecting the constants (4) and (B), mentioned above, obliges him to suppose that the index
of refraction is the same for normal and for transversal waves, and this makes his results true
only for substances of very low refractive power; for instance, they are quite at fault in the
case of common plate-glass, both as regards the intensity and the rotation of the plane of a
polarized ray. If y is put=y in my result, it agrees with Mr Green’s, which confirms the
correctness of what I have just stated,

SECTION I.
Preliminary Observations.

Berore we proceed to the direct investigation of the laws of Reflexion and Refraction, we
shall make a few observations, which will be found useful hereafter.

(1.) Let a, 8, y be the small displacements at any point (wyx) of a wave propagated
with a normal velocity (v); p, g, s the direction of the cosines of v, and V the actual velocity

of the vibrating particle, i.e. the resultant of the velocities = Zo ae

AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 9

Then we have the following relations, viz.

| da pda da qda da sda

and similar equations connecting the partial differential coefficients of B, y, and V,
or any function of these quantities.

It is easy to prove these equations, without assuming the integrals of the equations of
motion, in the following manner:

Let P be the point (wyz), AP the wave-surface which contains P at any time ¢, 4’P’ the
position of this wave-surface at the time ¢+d¢, PP’ the normal
to the wave at P, PQ’ a line parallel to the axis of #
meeting the wave 4’ P’ in Q’. Then assuming dw to represent
PQ’, we have

|

PP’ = PQ cos P’ PQ = pdx.
Also, since the space PP’ is described in the time dé
with the velocity v, PP’=vdt; hence we have

Now at the time-(¢ + dé), any point of the wave 4’P’ is in
the same phase of vibration as any point of the wave AP at
the time (¢); therefore a, B, y, V, or any function of these
quantities will not be altered by putting «+ dw, and ¢ + dé,
for 2 and ¢. We have therefore

da da : da pda
—d —_— =0 hich b 1) b —_—=--—.
da t+ ae dt=0, which by (1) becomes ae = eae
In the same way we may shew that
da qda da sda
Wie «sae kde eae.

and thus the truth of the equations (4) is proved.

(2.) Suppose the wave-surface to be a cylindrical surface perpendicular to the plane of xx,
the vibrations to take place parallel to that plane, and therefore @=0, g=0, and a and y

independent of y: then we have the following relations between V, v, and the partial differential
coefficients of a and vy, viz.

da dy

—=V7V —_—_— =

deed ag 72

d
OB) toe a =- +eeeeefor normal vibrations.

dw dz v

TAY

ds da

da d

dep? ag 77?

d 5 i

(0) yarns a ye 8 +es»»-for transversal vibrations.

dw dz

da dy V

Vor, VIII, Pazr I. B

10 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT

To prove these formule, let PP’ be the normal, and
@ the angle which the direction of V makes with PP’,
then V is equivalent to Vcos@ along PP’, and V sin 0
perpendicular to PP’; therefore, since s and p are the
sine and cosine of the angle which PP’ makes with the
axis of w, we have

= V cos@.p — Vsin@.8...+-.(1),

= = Vcos0.8 + Vain. p..----(2)s

p : 8 biay Pricceuneny t
Q) mck (2) Sa and Wes (2) 5? Bive (since p? + q? = 1),

pda sdy _V sda pdry gehe
midi 6 ae a ena 9s A aA a ae
which, by the equations (A) last article, reduce to
da dy da dy V.,
at ae ae BS eG ee

In these two equations, and in (1) and (2), put @=0, and we immediately obtain the formule

(B); again, put O=—, and we obtain the formule (C)*.

(3.) If ww, tw, Us, &e.,..u, be any functions of # and ¢, such that the equations
Wy Aan + Ue aba maceemer eon l0)ss scene eheatipsecesas \acecenen sss) (L))
du, du, du, duz du, du,
da 7 ape y Ea ae ae ee Cee (2)
@ dt da dt da dt
are true for all values of w and ¢3; 4a, do, dg,... &c. being any }constants; then must
y= Ga Ga cagucceans Sila

d(1) dl) is
For —— - a, ——
or ay di gives by (2)

d
yah duz dU, _
(a, - a, art (a, — a,) posses ti(@aay —/a,) a ¥=0,... (8):
_ (3) d(3) d(2)
A a SL hee BNE,
Bay Sei Bes by He?

Be Pu, & p
(4 a,) (a = @,-)) de + (a, =) a,) (a, = @n_}) = wosoue tT (@n-2 — ay) (d,-2 - a,-}) a ial ld

" The formule (B) and (C) are particular cases of the following, viz.
da da da

Vv
det ay t de yom
dg da\? (dy dB\? (da dy\? V2,
(78-32) + (%-%) + (72-72) = tag

+ The result i icle i
sult of this Seek ¥ also true when aj, ao, a3... &c. are variables, provided they vary very slowly compared with 7,
are . a a .
Yay Ua, &c.5 in which case = Ga &e will be extremely small compared with ee a » ke.
a

which may be easily proved.

AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 11

And by repeating this process, we find finally that
(a, — 4) (@ — G-1) (4 - Gyuz) (se0ee. ) (a - a.) = 0,

which shews that some one of the constants a,, a3, &c. must be equal to a,. Suppose that
a,=a,, and put u,t+u,=wu'; then we have w' instead of the first two terms of (1), and

1 1
— = oe instead of the first two equations of (2), and therefore, just as before, we may

x
shew that

(a, — a,) (@ — G,_1) (....--) (4, — a) = 0.

Therefore a, must be equal to some one of the quantities a3, a,, a5, &c.; let it be a3, then
proceeding as before, we may shew that a,=a,, and again, that a,;=a;, and so on. We
have therefore :

Gy = Ug = Ay capes =n.

(4.) The equations (4), (B), (C) in the preceding articles, may be very readily obtained
from the integrals of the equations of motion in the case of plane-waves of polarized light.
For when the wave-surface is a plane and the light polarized, we have

a=au, B=bu, y=cu,
where w=f(vt-pw—qy-—szx), and @, 6, c, any constants.

By differentiating these expressions with respect to #, y, x, and ¢, observing that p, q, s
are now constants, we have immediately the equations (4).

To obtain the equations (B) and (C), we must put g=0, 6=0, and then we have

r= (i) (2) = oro (f

Now a? +c? = (ap +cs)? + (as — cp)’,
du da du dy du da du dy
— =v — = — =v — c

Vv? da dvy\* da dry\*
h a= = [See a Se St
See vw (= a Z) % ($= 7)
: F da dy
Now for transverse vibrations, we have ap+ces=0, or FF ce ae =0, and for normal,
@ x

d d ° :
as—cp=0, or a —“Y 0: hence the truth of the equations (B) and (C) is manifest.

(5.) If any of the quantities p, q or s, be imaginary, (a case we shall have to consider
hereafter) the first method of proving the formule (4), (B), (C), fails, but the second method
does not. In such a case we call the vibrations transversal when the condition ap +cs=0
holds; and normal when the condition as-—cp=0 holds; and it follows easily from the
equations of motion, (see Cambridge Transactions, Vol. vit. p. 416) that transversal and normal
waves, thus defined, are in general propagated with different velocities; i, e. the constant v is
different for these two species of vibration.

(6.) It is important to observe that, in articles (1), (2), the wave is supposed to be pro-
pagated in the direction PP’, i. e. from P towards P’. If therefore p, q, s be positive
quantities, the motion of the disturbance along PP’ tends to increase a, y, and x; if p be
negative it tends to diminish w, if q negative y, and if s negative x.

B2

12 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT

SECTION II.

The general Equations of Connection of the Vibratory Motion of two Elastic
Media, separated by a Plane Surface.

(7.) Tue two media are supposed to consist of discrete particles symmetrically arranged,
and acting upon each other with forces varying according to any law which ensures stable
equilibrium. By the Surface of Separation, we simply mean an imaginary plane described
between the two media, the particles of one medium lying on one side of it, and those of
the other on the other side. In the immediate vicinity of this plane, the media are supposed
to exercise a mutual repulsion, so that no mixture takes place. We shall take the plane of
separation to be the plane of wy.

(8.) We shall obtain the general Equations of Connection of the vibratory motion of the
two media, by means of the following self-evident Principle.

When a very small vibratory motion is communicated to a stable system of particles, such
as the two media just described, we may assume that the vibratory motion will always remain
very small, and, at most, of the same order of magnitude as the original motion.

This principle is either tacitly assumed, or employed as self evident, by all the writers who
have treated of the problem of the transmission of waves from one medium into another. Poisson
states it very clearly in the Mémoires de Institut, Tom. x. p. 320, and makes use of it precisely
as we shall do in the present paper. It is evidently assumed in the Article Sound, Ency. Metrop.
p- 776; for by saying that the two media must have a common elasticity at their junction, and
that that elasticity is expressed by E(1 +s), and E’(1 + i's), the writer supposes that there
is the same slow variation of elasticity at the surface of junction as elsewhere, and therefore the
same slow variation of pressure, and consequently the same small vibratory motion.

(9.) To apply this principle to the case we are at present concerned with, let wyz (x = 0)
be the co-ordinates of the equilibrium position of any particle (P) of the lower medium in the
immediate vicinity of the plane of separation, a, B, vy its displacements at the time ¢, and let
a+da, y+ dy, 2+ 5x, at da, B+6R, y+ dy be the co-ordinates and displacements of any
other neighbouring particle (Q) of the lower medium; also let 7+ Aa, y+ Ay, + Ax, a+ Aa,
B+AB, y+ Ary be those of any particle (P’) of the upper medium.

Put r= da?+ dy?+ dx’, and r?= Aa®+ Ay’ + Az’,
and let mf(r), m'p(7’) be, respectively, the forces exercised by Q and P’ on P. Then, if X be
the whole force, parallel to the axis of a, brought into action upon P by the vibration, we have
(see Cambridge Transactions, Vol. v11. p. 403)

X=Em f(r) da + */' (r) Sa (Seda + SySB + Sz3y)}
lL , , 1 ‘ U
+ am {p(r') dat af (7) Aw (Avdat+ AyABR+ AzAy)},
= referring to the lower medium and >’ to the upper.
In this expression we shall substitute for Sa the series
da
da

Also, let a’, (', ry’ be the values which the displacements a + Aa, B+AB, y+ Ay, assume
when wv, y, * (=0) are substituted in them for 7+ Aa, y+ Ay, x + Az, then we have

,

, da da’ da’
a+ Aa=a+ ara ive (hie ee ek

Now the differences of the corresponding displacements of two contiguous particles at a distance
from the plane of separation must be indefinitely small, (supposing of course, as is always done,

da da
ow + agr! + qeo* + &C...000.

AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 13

that the interval between two contiguous particles is extremely small, compared with the length
of a wave); therefore, by the principle stated in Article 8, the same must be true of the displace-
ments in the immediate vicinity of the plane of separation, which cannot be the case unless we

have a =a. Hence
, ,

da da da’
Aa= ao” + Fy Ay+ a + &Ci saksee

Substituting these expressions for 5a, and Aa, and similar expressions for 6B, dy, AB, Ay,
and observing that all sums involving odd powers of da, dy, Aw or Ay, must vanish, in
consequence of the symmetrical arrangement of the system about the axis of x, but that sums
involving odd powers of ds or Ax do not vanish, since the particles are not arranged symme-
trically with respect to the plane of wz, we have

da dy F da dy
X=-(C+D) > Ts D aah (C’ + D’) sar D' aak higher differential coefficients,

where — C = 3mf(r) dz, 2 Sees (r) dx dz,
ad

C' = X'm' $(7’) Az, D' = Yn’ a P(r’) Aa? Az.

(We assume the two first constants in a negative form, because dz is negative, whereas Ax is
positive).

It is evident that -C+C’=0, is one of the conditions of previous equilibrium, therefore
we have C’=C in the expression for X.

Now since the length of the wave is extremely large compared with the sphere of action
of the molecular forces, the part of X involving first differential coefficients, has its several
terms extremely large compared with those of the part involving second and higher differential
coefficients (see Cambridge Transactions, Vol. v1. p. 408): therefore, unless the former terms
mutually destroy each other, X will be extremely large compared with the corresponding force
which acts upon a particle at a distance from the plane of separation (for this force involves
only second and higher differential coefficients, see Cambridge Transactions, Vol. vit. p. 408) ;
and if this be the case, the vibratory motion of the particles at the plane of separation will
be extremely large compared with that at a distance from it, contrary to the’ principle stated
in Article (8). Hence the terms of X involving first differential coefficients must destroy each
other, and we therefore have

da dy _ , da ,dy
(C+D) at D as (C + D’/) ae Ds rs aidaseansipieies Da se satetle sy (L).
In exactly the same way we may shew that ae B, and

Ov

(C+D) =e De ape ae a Pye ea a 2s ee SE EPRI: C)
Lastly, the force parallel to the axis of x is
1
Sm § f(r) dy + 7f@) ox (Svda + dy dB + dx dy)}
, 1 "40
+ Lm'§p(r’) Ay + aPC) Az (Aw Aat+ AyAB+ Az Ay)},
which, treated as above, gives y’=-y, and

(+m) Dev (F +2) =(C+E) 40 (52. +) wresentaye

14 Mr. O’BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT
1 , , 1 , , ,
Observing that =m af (r) dy%dx=-D, Ym 3 ¢(r') Ay Az=D,

1 - , 1 .
and putting — E and £’ to represent =m ef) os? and S’m 7 P(') A* respectively.

(10.) Hence it appears that, if a, B, y be the displacements at any point (wyz) of the
lower medium, and a’, (', y’ those at any point (a’y's’) of the upper, and if we put a’ = x,
y'=y, *=%=0, then the equations (1), (2), (8), and the equations, a=a’, B=, y=7’;
will hold for all values of @ and y. Now this being the case, we may differentiate these

d dy’ é
equations with respect to w or y; therefore = = as and therefore (1) may be put in the
form .

da dy ~ (da dy’
(C + D) (+2) EG + D') (+>) 5
and a similar alteration may be made in (2) and (3).

Hence, if we put C+D=M, C+D'= M’, C+E=N, C+ E'=N’, we have the follow-

ing equations:

a=a, B= yHy evn seat cwetiae aeaisaniss aesaree (D2);
da’ dv’
u (+2) = a aaa
dz dua ds da (E)
ene ap! éy'| Jebivatlebs Veceus oss bee ses ;
“ef (4+2) eae ee “i dy
Bi-y da df dry! da df
— _ — = es ==, ae eeeeee F .
wet (S45) ete (+a) )

These are the general equations of connection of the vibratory motion of the two media ;
they hold at all points of the plane of separation, i. e. they are true for all values of w and y,
x being put equal to zero.

(11.) We shall now compare with the last of these equations the equation of connection
furnished by the common law of elasticity, in the case of two ordinary elastic fluids separated
by plane surface.

Let p be the pressure at any point of the lower medium when at rest, considered as a
common elastic fluid; then the pressure when it is in a state of vibration, will (by the law of
elasticity) be (See Airy’s Tracts, note, p. 278, 2nd Ed.)

ba dy dz n , da dB dy }
Pass UD ETE e p{i-n (seas +2) :

n being a constant nearly equal to unity, depending upon the alteration of temperature during
the vibration.

Similarly, the pressure in the upper medium will be
: Pile EL sy
Lan (=— Sesh le
of Mg (e+ +S)

Now these two pressures ought to be equal at the plane of separation; also, by the con-
ditions of previous equilibrium, p =p’.

Hence, when z=0, we have

da dB dy , (da df dry’
n(ie+ dy +a) ar las? ay" Soe

AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 15

Comparing this with the equation (/'), we see that
Mien
N=M, N'=M’, and 7 = 77°
(12.) In our ignorance of the constitution of the liminiferous ether, it is natural to
assume that it is of the same nature as ordinary elastic fluids, and that it accordingly obeys
the common law of elasticity; we shall, in the first instance, make this assumption, and there-
fore put M=N, M'=N’, and M’=eM, where e=—, a quantity not differing much from
n
unity; and then the equations (Z) and (F) become

d d da dv’
oo ae = (ae tee)

Fee dz ae. oi te ee HS),
dp dy _ peal
dz dy = dy

da dp dy da’ dp’ =)
(G+o+ =) = (= + dy | dz Sekiseate ACL) s

Further, if we neglect the variation of temperature, and therefore put ¢ = 1, these equations,

in virtue of the equations (D) differentiated with respect to # and y, assume the simple forms

Bae NE eh 8 BE hea

ds de’? de dz’ ds. ds 707’

(13.) The equations of connection just obtained, along with the equations of motion given
in the Cambridge Transactions, Vol. vii. p. 409, are sufficient to solve all problems respecting
the propagation of waves from one medium into the other. We shall assume that these equations
of motion hold up to the very plane of separation: which of course is not accurately true,
since there will most probably be a variation of density in the media in the immediate vicinity
of that plane. If we describe two planes parallel to the plane of separation, one above it and
the other below it, including between them the slice of the two media in which this variation
of density is sensible, it is easy to see that, in consequence of the smallness of the sphere of
action of the molecular forces compared with the length of a wave, the thickness of this slice
will be extremely small compared with the length of a wave. Indeed, if one medium exercised
a sensible action only upon those particles of the other which are immediately contiguous to
the plane of separation, the thickness of this slice would be actually zero. We shall therefore
consider this slice to be of insensible thickness, and regard it as a physical plane. This being
assumed, we may, without sensible error, suppose that the equations of motion hold up to the
very plane of separation. All therefore that is proved of the propagation of waves in a sym-
metrical medium in the Cambridge Transactions, Vol. vit. p. 416, &c., we shall assume to be
true up to the very plane of separation.

We shall in the following Section, use the equations of connection in their simplest form,
viz. (D) and (J); and afterwards, in Section vir., shew how we must proceed when they are
taken in their most general form, viz. (D), (£), (F).

16 Mr. O’BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT

SECTION III.

Application of the Equations of Connection just obtained to the case of ordinary
Reflexion and Refraction.

(14.) We shall first consider the case of cylindrical or plane-waves perpendicular to the plane
of xx (which will therefore be the plane of incidence), the vibrations taking place at right angles
to that plane.

In this case a=0, y=0, a'=0, y’=0, and @ is independent of y: therefore the six equations
of connection, (D) and (JZ), Section 11, reduce to two, viz.

B=B; pe So ee

We shall suppose that the whole motion consists of three sets of waves (for we shall shew
presently that it cannot in general consist of only two), one set in the upper medium, and two
in the lower. Let ($+ , be the whole displacement at any point of the lower medium, the part B
arising from one of the sets of waves, and the part B, from the other; then we must write 8 + G,
instead of B in the two equations of connection, which therefore become

B+ B= Peccei g(B Yaa) ea (8):

Now, using the notation in Article (2), we have

Hence, and by the equations (4), Article (1), = and (2) immediately give us

&

V+
v

8 pee
2, Vimy V' venous (4).

‘

Vt Vi=V'ssseee (3),

Again, since by the equations (4), Article (1), we have
dV pdVv dV, p, dV, dV’ p dv’

da” Se: eee eee Cue ue ade c

and by (3), V+ V.- V’=0;

U
. P Pp < ‘
and since >? ar? ‘ are either constants (in the case of plane-waves), or vary very slowly

compared with V, V,, V/, on account of the extreme smallness of the length of a wave of light ;
we have by Article (3), (see Note),

Now v=», therefore

P= P vores (5); P = MP «2+... (6) {here A= =}.

Hence, observing that 9, 9,, 9’; are each zero, we have

z
sats, Peay an 2:
2

im

AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 17

@ s_

(3) gives —- = 7? which is in general inconsistent with (6); we must there-
v

fore take s=—s. We may suppose s’ to have either sign. Hence by (3) and (4), we have
V+V=V', s(V-—V)=uns8'V’, which give
_s—ps Q8

fl SAN (7); V’= ;
S+ us S+ps

If we take s,= 8, ——

Vowteoni (8):

(15.) We shall now interpret these results.
Supposing p and s positive, the normal propagation of the wave V tends to increase w and x
(see Article 6); and since p = Pr! 8,=— 8, that of the wave V, tends to increase w and diminish = ;

and since p’=pp, s=+ oy —~<,» that of the wave V’ tends to increase xv, and to increase or
oT

diminish x. Hence we have two cases according as we take the upper or lower sign of s’.

Fig. (1) represents the first case; X’4X and Z’AZ Z
are the co-ordinate axes; NA, NA, N'A are the normals (1) N
to the waves V, V_, V’ respectively, the arrows representing
the direction of normal propagation, N and N’ tending to
increase w and z, and N, to increase w and diminish x
Since p,=p, and p'=yp, we have 4 NAZ’= 4 N,AZ,
and sin N'AZ=, sin NAZ’. This is the ordinary case
of reflexion and refraction. If a, a,, a’ be the maximum
values of V, V,, and V’ respectively, the intensities of the
three rays N Nand N’ will be proportional to a”, a?, a”.
Now by (7) and (8), we have

28

,
&—ps
Balin, Sees Mi
S+ us S+ ps

=

These are Fresnel’s formule for the intensities of the reflected and refracted rays of a ray
polarized in the plane of incidence.

Fig. (2) represents the second case, in which s’ is negative ;
and therefore N’ tends to diminish x. This case may occur in
the following manner. An incident ray along NA will produce
a reflected ray along AN, and a refracted ray along AN”,
ZN” AZ being equal to 2. N’AZ; and another incident ray
along N’A will produce a reflected ray along AN”, and a re-
fracted ray along AN. Now let the intensities of the two rays
along AN” be equal, and let one of these rays be half a wave
behind the other; then they will interfere and destroy each
other, and we shall have remaining only a ray along NA, one
along N’A, and one along AN, (namely, the sum of the two
along AN). ‘This is exactly the second case.

(16.) It is evident, that, in the ordinary case where the rays N, and N’ are the effects
produced by the ray N, the normal propagation of N’ will be from and not towards the plane
of separation: therefore s’ must have its positive value, and consequently the second of the above
cases cannot occur.

(17.) If we suppose either V, or V’ equal to zero, the equations (7) and (8) give us
either s—s'=0, or s=0, neither of which equations can be generally true. Hence the
incident ray must, in general, be accompanied by a refracted and a reflected ray, or the equations
of connection cannot be satisfied.

Vor. VIII. Parr I. (0)

18 Mr. O’BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT

(18.) It appears from (7) and (8) that V, and V’ have the same periodic time as V; it
follows, therefore, that the colour of the reflected and refracted ray is the same as that of the
incident.

(19.) We shall now in the second place consider the case of cylindrical or plane-waves
perpendicular to the plane of wx, the vibrations taking place parallel to that plane.

In this case 3=0, and a and y are independent of y: therefore the six equations of con-
nection, (D) and (£) Section 11, reduce to four, viz.

If we attempt to satisfy these equations by three sets of waves, as in the preceding case,
we shall immediately arrive at the conclusion, »°=1; which shews that these equations cannot
be satisfied in this manner. The reason of this is obvious; for, in the case of vibrations per-
pendicular to the plane of incidence, it is clear that no normal waves will be produced by the
refraction and reflexion: but in the present case, supposing, as of course we do, that the in-
cident vibrations are transversal, we have every reason to suppose that normal vibrations will
be generated by the reflexion and refraction. Therefore, since normal waves are in general
propagated with a different velocity from that of transversal, we shall have to take into account
a set of normal waves in the lower medium, and one in the upper also, not coinciding with
the transversal waves.

Let at+a,;+a, and y+71+ 2, be the whole displacements at any point of the lower
medium, and a’+a”’, ++", at any point of the upper; the parts a, y2, a’, y’, arising
from the normal waves brought into existence by the reflexion and refraction. Then, the four
equations of connection become,

Gea =ia Pa csdeceicescearcsecal(l)> Yt yt yoay ty” cecceccecceccesees (2),
d de d Oe my
ae ttt a) = F(a’ + a’) ones (8); Se AY ae ais) oe Ly it Ys Bates (4).

From (1), or (2), by the equations (A), and by Article (3), we have, as in the preceding
case, (Article 14),

PoP RPE.
1 2
d(2 d(1
Also (3) - _, and (4) + — give us, by the equations (B) and (C), Article (2),
Ve ae Vv Veer V,,
a re (6), Far ite a (7)
d(i d(2
Also a) and 2, give us, by the equations (B) and (C),
Vs + LACH - Vi. ps = V's’ — V"p" eoeeee (8),
Vp + Vip, + Vos, = V'p'+ V's" ....0- (9).

These equations, namely (5), (6), (7), (8), (9), completely solve the problem as in the pre-
ceding case.

From (5) we get p,=p, and therefore s;=+s, As in the previous case we must take
the lower sign; for otherwise V and V, would enter into each of the equations, (6), (7), (8),
(9), in the form V+V,, and therefore we might eliminate altogether the quantities V, V;, V’,
V,, V”, from these equations, and so obtain an equation which would not be generally true,

AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 19

since it would contain no disposable quantity, p,, p’, p”, 85 8’, 8”, being all given in terms
of p by (5). We have therefore s, = — s.

For the reasons given in Article (16), we must take the positive values of s’ and s”, and
the negative value s,; i.e. we must put

sat ee = + ras - (5p) +0), $= — heh - (tp) . ).

Hence, if we take the arrows N, N,, N2, N’, N”, (3) £ a
to mark the directions of the rays, as before, Fig. (3)
will represent the circumstances of the case, and we
have

Z£N,AZ'= 2 NAZ, sin N’AZ =p sin NAZ,

”

sin N,AZ’ =~ sin NAZ’, sin N"AZ = — sin NAZ’.
©

Thus, both the reflected and the refracted normal

”

: seen ae

ray obey the law of refraction, putting — and — in-

v v
stead of «4. The transversal rays are circumstanced

just as in the preceding case.

(20.) We now proceed to compare the intensities of the rays N, N,, and N’, and we
shall do this, first, on the hypothesis that normal waves are propagated very slowly compared

with transversal.

”

On this hypothesis we may suppose that = and ~ are zero, and then, by (5), we have,

P2=0, p’=0. Hence, writing a, a, a’, for V, V,, V’, as before, we have by (6) and (8)5
G@+%=pa, s(a-—a)=s a,
,
8-8 } 2
and therefore a, =" 7a W= e ; a
ps+s pS+8
These are identical with Fresnel’s formule for light polarized at right angles to the plane

of incidence.
To determine a, and a”, we have, by (7) and (9), (observing that s,=-1, s”=1 by (10)
and (11) )

3 as @

p(a+a)-a=pa+a’, or a,+a’= (up—p’)a’ by (6);
V1 pe—1 2ps a Ue

9) es ae a=>— a,»
Uetv Bw pests Ve

(21.) We shall now make a different hypothesis, and suppose that v, is equal or very nearly
equal to v”.

On this hypothesis, we have by (5) p,=p", and by (10) and (11) s,=—s8”; therefore by
(6) and (8) we obtain

therefore a, =

a+a=pa', 8(a-a)=s'd,
which give us Fresnel’s formule just as before.
Also by (7) and (9) we have
a, =a", p(a+a)-s'a=p'a+s’a’;
we —1 2ps

therefore a, = a! = ——, 73a
us ps+s

C2

20 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT

(22.) Lastly, we shall try whether any other hypothesis leads to Fresnel’s formule in this case.
If Fresnel’s formule hold, it is easy to see that
s(a—a,)=s'a’, and therefore s(V—V,)=s'V';

therefore by (8) we have V,p,= Vp", and therefore by (7) and (5)

”
ve =v",

from which it is evident that no other hypothesis except those in Articles (20) and (21) will lead
to Fresnel’s formule in this case.

he second hypothesis here employed seems to me to be the only one we can adopt; for
it is extremely difficult to conceive how normal vibrations could be propagated more slowly than
transversal in a stable medium. .

If we do not suppose that v, is very nearly equal to v”, we may proceed to find V, and V’ in
terms of V in the following manner.

Substitute for V’ and V, their values got from (6) and (7) in the equations (8) and (9); then,

: V2
putting — =v, we have
v

(us —8') V—- (us+s') V,=V" (vp, —p"), (up —p’) (V+ V,) =V"(s" — vs,).

Hence, if for brevity we put Hae (up — p’) =n, we have
2

(us - 8’) V—-(us+8)V,=(V + V),

,
ss —
and therefore V,= ole Ail)

Mst+s +7
‘ : : ate alle a a4,
and from this expression we may easily find V’, since V' = :
Bb

Since p=yp’, and p,= vp", we have

n= @-1)@-1).7*

” =
8” — vs,

SECTION IV.
Explanation of the case of Total Internal Reflexion.

(23.) Since p = pp’, p’ will be >1 when p is >p (which it may be when y« is <1),
and then s’ will be impossible; and Fresnel’s formule become imaginary; which indicates that
the equations of connection cannot be satisfied by the three rays in Article (15), or the five rays
in Article (19). We shall now consider how the equations of connection may be satisfied under
such circumstances, and first in the case of vibrations perpendicular to the plane of incidence.

Let us suppose that the general value of V is

ae & ehcp =i Son (4)

It is allowable to give V this value, though it is imaginary, since it is an integral of the equa-
tions of motion, and is capable of satisfying, analytically, the equations of connection, and the
equations (4), (B), (C), Section 1, (see Article 5). Moreover, by superposing two such imaginary
values of V, viz. aek(t-P*-)V=1 and qe-*t-p*-s:)V=1, we obtain a real value, viz. a cos k (vt—pw—sz),
which will of course satisfy the same linear equations as the two expressions of which it is the sum,
i.e. the equations of motion and of connection.

AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 21

Assuming then this value of VY, we have at the surface of separation,

V= . eh(et—p2) Vai,

,
S—ps a = 2s a
and therefore V = pe — ek (ve-pey V=1, V’ = _ ek(vt=pr) V=1,
‘s+ ps 2 S+mus
Hence if 4 a e8 vt-P7+#) V=i, and 4 a’ ek \t-p2—s2) v=1, be the general values of V, and V’, we have
,
s—ps 28
a) = a, a — a,
S+ us S+ ps
u 2
v
and of course k’v’ = kv, p=-p= P , and s)=+ th as
v fh

Now let us assume

,
N/ 1 Ee ee (supposing p>p), and eae a Sin tan w,
then @ = eFV-Iq, a’ = 2 cos w er’ =a,
and the general values of V, and V’ become
V, = Lae thot-prtezmlV=1,,,,..(2), Vi = acoswewt-yarel/-+koz, |, (8).

Now let us superpose the system (1) (2) (3), taking the lower of the double signs, with
another system formed from (1) (2) (3), by putting —% for & and therefore —k’ for k’ and
taking the upper of the double signs. The result of this superposition will be the following
real system, viz.

V = acosk (vt — px — sz),
V,= acos $k (vt — pw+sx)+2w}, V'=2acoswe *%* cos {k' (v't — p'#) + wh.

These values of V, V,, and V’, since they are real, and satisfy the equations of motion
and of connection, represent a possible case of motion. The expression for V, shews that there
is a reflected ray, of the same intensity as the incident ray, but having its phase altered by the
quantity 2w. The expression for V’ gives 4a°cos’we ?*°* for the intensity of the refracted ray,
which quantity rapidly diminishes as z increases, since #’ = <, and )’ is extremely small. This
indicates a complete extinction of the refracted ray. If we had taken the upper signs instead of the
lower, and the lower instead of the upper, in the above process of superposition (as we might have
done), we should have obtained e**?*, instead of e~?*%*, in the expression for the intensity of
Vv’. Now this represents an intensity which increases rapidly with x, and therefore a vibratory
motion which becomes extremely large compared with that which gave rise to it, contrary to the
principle stated in Article (8). We must therefore take the signs as we have done above.

The alteration of the phase of the reflected ray is given by the equation

A del

8

ho
tanw = — =

This is exactly the first case of total internal reflexion considered in Airy’s T'racts, p. 361.
(second edition) *.
(24.) We shall, in the second place, apply the same method to the case of vibrations parallel

ROAR oat |
to the plane of incidence. To do this we have only to put tan w of MES

: > and
ps ws

1
* The uw in Airy’s Tracts is the same as the ji here,

| 22 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT

we arrive at exactly the same expressions as before, for V, and V’, with this difference, that

2a cosw

the coefficient of V’ is instead of 2a cosw.

This is exactly the second case considered in Airy’s J'racts, p. 361.

SECTION V.

Why Normal Waves never produce any sensible Effect on the Eye directly or
indirectly.

(25.) Wer must suppose of course that normal waves cannot produce vision directly, (i. e.)
that when such waves are incident on the retina they do not affect the optic nerve in such a manner
as to give rise to the sensation of light. But we have proved that when a transversal ray
undergoes oblique refraction it brings into existence normal rays, and it would be easy to shew
that, in the same manner, the oblique refraction of a normal ray will produce transversal rays.
Therefore, though normal waves cannot affect the retina directly, they may do so indirectly,
by giving rise to transversal waves. Now it is a matter of fact that they do not produce this
indirect effect, and it therefore becomes necessary to explain theoretically why they do not.

(26.) If we take the hypothesis in Article (20), it is easy to do this. For suppose the
normal ray, generated by the oblique refraction of a transversal ray at the first surface of a prism
or lens, to fall on the second surface at an angle of incidence sin~‘p, and let the transversal
ray produced by this oblique refraction emerge at an angle sin~'p’, then, as in Article (19), we
may shew that z, -7, and therefore p’= =P Now by our hypothesis = is very large,
therefore, unless p is very small (in which case the transversal ray will not be produced at
all), p’ will be >1, and s’ impossible; i.e. the transversal ray will be extinguished. (See
Article 23).

Thus the normal waves generated by the first refraction, will not produce transversal waves
at the second refraction.

Again, if we take the hypothesis in Article (21), and assume moreover that ov” and v,
are large compared with v and wv’, it is easy to see, by similar reasoning, that the normal rays
will be extinguished immediately after their production by the first refraction.

It is evident, therefore, that on either hypothesis (adding to the latter, that v, and v” are

large compared with v and v'), normal waves will produce no sensible effect on the eye, even
indirectly.

SECTION VI.

Whether Polarized Light consists of Vibrations at Right Angles to, or Parallel
to the Plane of Polarization.

(27.) Tere can be no doubt of the truth of Sir D. Brewster’s law of tangents, and
the laws of the rotation of the plane of polarization given by M. Arago, and Sir D. Brewster.
From these laws we shall attempt to prove, that polarized light consists of vibrations perpen-
dicular to the plane of polarization, in the following manner.

AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 23

If possible, let polarized light consist of vibrations parallel to the plane of polarization ;
then taking the equations of connection in their most general form, viz. (D), (£), (F),
Section 11., and proceeding as in Articles (14) and (15), we find, for light polarized perpen-
dicularly to the plane of incidence, the following formule :

s—eus’ 25
=, =~" _V;
S+teus S+eEeus
ee CD
where ¢€= 4
C+D

Now, by the law of tangents, V, ought to become zero when the tangent of the angle of
incidence is p, % e. when s’=ys. Therefore we have

1
s—ep’s=0, and therefore e=—,
Be

and this gives us
218s

ast+s’

ey re
MS+s
Now if U, U,, U’, Uz, U”, be the velocities, when the light is polarized im the plane of
incidence, we have, by the laws of the rotation of the plane of polarization,

U, s—ms pwst+s' U U' _wst+s U
Vi sie paos Vi WO sti Ve

Hence, by the above expressions for V, and V’, we have

’
S—pms 2s
C= ;U, U'=
S+ pus S+us

(i U,

and from these equations we find
e(04+0,)=0' ...,.-(1), 8(U0—,) =0U’.....<()-
Now from the equations of connection (D), we have, as in Article 19, equations (8) and (9),
s(U —U,) — U.p,= s'U'— U"p’,
p(U + U,) + U,8, = p'U' + U's",
which, by (1) and (2), and since p=pp’, become
Usp2= U"p", Use.= U"s".

”

Now since 2 Pa. and v”, v. are essentially positive, p” and p, have the same sign;
2
therefore U, and U”, and therefore s, and s” have the same sign. Now by Article (19),
equations (10) and (11), s, and s” have opposite signs, which is absurd. The only way to
get over this, is to suppose that U, and U” are zero, but then it will be impossible to satisfy
the equations of connection for all values of p. (See Article 19).

Hence it follows that, if we adopt the hypothesis which supposes polarized light to consist
of vibrations parallel to the plane of polarization, and take into account the experimental
laws above stated, we arrive at an absurd result.

We may therefore conclude, that Fresnel’s hypothesis is true.

24 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT

SECTION VII.
How we must proceed when the Equations of Connection are taken in their most
general Form.

(28.) THE equations (5), (8), and (9), Article (19), are still true, being deduced from
the equations (D), Section 11., without making any hypothesis respecting the constants. But
instead of (6) and (7), we shall arrive at two equations, somewhat more complicated in form,
as follows.

: or 5 d
The equations (Z) and (F'), Section 11., may, in virtue of the equations = =e—, —~=—2
be put in the forms

(c+ Dd) (SE - 2) = 4D) (SZ -9) sew - Do

U

dz d dx dea dx’
dy da\ _ , (dy da’ ; ; da
(C48) (+54) = (C+ 2B) (G+ SF) +W-£ D+E) =.

Now, in these equations, as in Article (19), we shall puta+a+a, yt pity, a+a’y
yl ty", for a, ys a’, y’> respectively, and then, as in Article (19), and by the equations
(A), (B), (C), Section 1., we obtain immediately the following equations,

V+,
v

”

“4 Vv orl mn Yi
(C + D) = (C+D) -2('- D) (sp G+" 5)

- Ll ne
(C+) 2- (C3 yk (D' - E'-D+E) ("= -p").
Us v v v .

From these equations, and the equations (8) and (9) Article (19), we may find V,, V’,
V,, and V” in terms of V. We shall not calculate these values, as they are rather complicated,
and not necessary to the object of the present paper. The last of the equations just obtained
considerably simplifies when we suppose the ether to obey the common law of elasticity, in which
case we have D'- E’-D+4E=0. (See Article 12.)

It is easy to see that Fresnel’s formule cannot be deduced from these equations, unless D=D’,
and E = £’, and therefore it will be useless to employ the equations of connection in their
most general form, as it is highly probable that Fresnel’s formule are experimentally true for
a great number of substances.

SECTION VIII.
Intensity and Phase of the Reflected Ray, in the case of highly Refractive Substances.

(29.) ‘THERE are some substances, such as the diamond and other bodies of high refractive
power, for which Fresnel’s formule do not appear to be accurately true. It is easy to account
for this in the following manner.

When we do not assume that v,=v", we have, by Article (22),

ps—s'—7 ‘pl
aerate where n = (v* — 1) (v? - 1) DEE

‘ = .
8” — v8

AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 25

Now, by Article (26), we must suppose that v, and v” are very large compared with
» and v’, and consequently that s, and s”, and therefore y, are impossible quantities. Let us
accordingly put

= tand/ =1......(1); WS = tania Vet rates (2),

ps—s wS+8

and we then have
ps — 8 cosw

= : ev te)va1 V.
‘as +8 cosy

Now this value of V, indicates, as in Section iv, that the intensity of the reflected ray is

“us —s' cos “ a (us—s')P- 7?
us +s! cos ‘ (us + sy? —7

a’, (by (1) and (2).

And that its phase differs from the phase of the incident ray by the quantity (Wy + w).

vy 2 vo” ; » "
To calculate y, we observe that s” = ai - (a) alas V/ —1 very nearly, since — is very
v v

”

Chale) v ° D : 4
large: and similarly, s,=—p—,./-1. We here give s” and s, opposite signs, because the ex-
v

pressions for V, and V” will contain the factors e'*¥-1 and els’:V-1*, Now one of these
(namely, V”) ought to diminish rapidly as x increases, and the other (V,) ought to do so as
decreases; but this cannot be if s” and s, have the same sign, therefore we must take these
quantities with different signs.

2 Be
Hence n= (? - 1) (Tm — 1) o . ~ ?
©.)

: Ve
and therefore, since vy =—,, we have
v

s v—1 p-1 }P
-7= pr.

v+lou

If we suppose the light to be incident at the polarizing angle, the expression for the intensity
of the reflected ray becomes (since at that angle us = s’= p)

———

ee -7 Poe \ +1 Op

CHa eee Sin
p-7 1+( p )

P+ Qu

=—1

3 5 + eel
For common plate-glass we may put #=-, and therefore (“ ) = nearly; and for

2m

s 12 we =—1\2? 2

diamond we may put p = 9 and therefore ( —) =1 very nearly. Hence, supposing that
Mo

v is the same for both, the intensity of the reflected ray at the polarizing angle is about six

times greater for diamond than for plate-glass. But we have every reason to suppose that py

(the index of refraction for normal waves) and » (that for transversal) will increase together.

* See the process of superposition in Section IV.

Vor. VII. Parr I. D

26 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT, &c.

wrat 4, ;
Let us suppose, at a venture, that the value of y is aa for glass, and . for diamond: then the

i 2 1 1 :
value of BS is about —— for glass, and — for diamond; therefore we have
P+) 100 15
2

2_" for glass
iti, cane

7

a
a?= S for diamond.

These expressions, if correct, would indicate that the reflected ray was scarcely visible for glass ;
and faint, though decidedly visible, for diamond: which, I believe, is the case. From this example

’ ; 11
it is clear that if we suppose the normal index of refraction to be less than about i when the

3 Pay . :
transversal index is less than ro the reflected ray at the polarizing angle will be scarcely visible
for plate-glass and substances of lower refractive power : and if we suppose the normal index not

less than about ; when the transversal is greater than 2, the reflected ray will be decidedly visible.

Supposing this to be true, — 7” will be very small for substances of moderate refractive power,
and therefore Fresnel’s formule will hold for such substances, at least the deviation from Fresnel’s
formulz will be insensible.

Hence, for substances of moderate refractive power w will be always small; but y, and there-
fore the phase (w+) will increase rapidly by very nearly 180° while the angle of incidence is
passing through the polarizing angle; this is evident from (1).

For substances of high refractive power, — ° will not be very small; therefore there will be
a sensible deviation from Fresnel’s formule. Moreover w will not be very small, and \, and
therefore the phase (w+) will increase by a quantity somewhat less than 180°, while the angle
of incidence is passing through the polarizing angle.

These results are in strict accordance with the experiments of Mr Airy; see the Cambridge
Transactions, Vol. tv, p. 422.

III. On the Possibility of accounting for the Absorption of Light, by supposing it due
to the Motion of the Particles of Matter. By the Rev. M. O’Brikn, late Fellow

of Caius College.

[Read Feb. 14, 1843.]

WueEn we take into account the motion of the particles of matter (see Cambridge Transactions,
Vol. vir. p. 421*), we arrive at the following equation for determining the velocity of propagation

(wv), viz.
_ mC mC

the disturbance being proportional to cos k (vt — w).
If we put kv = m, this equation becomes
n® (v° — mB) (v° — mB) = C {m, (v? — mB) + m (v? — mB) {v*......+0(1).

which is a quadratic equation for determining v? when 7 is given, i. e. when the colour is given
q gs given, ’

2

247, : Fi é
since — is the time of vibration.
n

This equation affords a complete explanation of the dispersion of light, and it may also be
applied to account, apparently in a satisfactory manner, for the absorption, as follows.
Suppose that the roots of the equation are impossible, then we shall obtain four values of v,

A F 1 ==
which we may put in the form, —- = £ ¢ £ 44/—1.
v

u my, . . . .
Now a = ae"('-i)¥= is an integral of the equations of motion; hence we have four integrals

included in the formula
a = aett-(FenV=IMVA, or gerne , en(t*eu)/=1,

From these imaginary integrals we obtain the real integrals
a= ae". cos n (¢ + ew).

Now we must not suppose a continually increasing intensity of vibration; and therefore the
upper sign of the exponential coefficient must be rejected, as is usually done in similar cases: we
have therefore

a = ae" cos n(t = eu).

This expression indicates a continually decreasing intensity of vibration different for different
colours (since is evidently a function of m), and thus the supposition that the roots of (1) are
impossible, leads to an explanation of the absorption of light.

It is easy to follow out this explanation into detail, and to shew that it agrees with experiment
so far as it goes; but the object of the present paper is to prove, very briefly, that there is a serious
objection against the supposition that the equation (1) has impossible roots, and therefore against
the explanation of absorption depending on the motion of the particles of matter. To do this, we

* Since the paper here referred to was printed, I have been | persion and Absorption of Light; but I am not aware that
informed that Professor Lloyd had previously read a paper on | his paper has been printed, for I have not been able to pro-
the same subject, in which he gave an explanation of the Dis- | cure it.

D2

28 Mr. O'BRIEN ON THE POSSIBILITY OF

must investigate the equations of connection of the vibratory motion of two media separated by a
plane (as in a previous paper in the present part of the Cambridge Transactions), supposing that
one of the media is composed of material as well as ethereal particles.

We shall make the same suppositions, and use the same notation, as in the paper just referred
to; assuming the upper medium to contain material particles, and a, 3, yy, to belong to any one of
them, a, 3, 7, and a’, 3’, ry’ belonging (as before) to the etherial particles.

Then the force parallel to the axis of w on any particle of ether at the plane of separation
will be ;

da dy . da dy :
(C+D) ae + Daas -(C'+D) ae D > + terms of superior order.

The terms of superior order here alluded to consist, in the first place, of higher differential
coefficients of a, B, y, a’, f', y’, and secondly of terms arising from the action of the material parti-
cles, the largest of which we have assumed, in obtaining the equation (1), to be of the same order
of magnitude as the second differential coefficients of a, 8, y. Hence we have at the plane of
separation

da dy . , da , dry’
(C+D) + DTH +D) 7+ Te cote et cere eerecccccces (2).
In the same way we obtain
dB dy ; meuyey dry’
— —_= D') FH DD oi ccsccscces VosVeascnces A
(C + D) ae 1D (C+D) We te ae (3)

dy da dB , 1 dy’ , (da dp
(C+ £) ae th (<+) =e +£)7 +0 ue eeacerult)s

We also find, just as before,
a =a, B= 8B, Cem SY tavivesteosnscsen(5)s
In addition to these six equations of connection, we obtain three others in the following

manner.
At the plane of separation the force acting on any particle of matter is

1
=m f(r) da, + = W'(r,) da, (Sa, da, + dy, 6B, + dx, d,)}
+a part arising from the action of the ethereal particles.

This may be reduced, as the force on an ethereal particle, to the form

d d ;
(C,+D) <4 DY 4 terms of superior order ;
dz ‘dx

observing that we include the part arising from the action of the ethereal particles among
the terms of superior order for the same reason as before. We have, therefore, at the surface
of separation, ;

da d
C. 4D) == Boa fi
(C,+ D) ag ts ae 0

and similarly (C,+ D) a +D, dy, =0 Esc Seavedcsecesbontes(O)t
x

d d
(Cr) “ep (F sa =a)

dz @

These nine equations, namely, (2), (3), (4), (5), and (6), are the complete equations of

ACCOUNTING FOR THE ABSORPTION OF LIGHT. 29

connection which apply to the case of reflexion and refraction, the motion of the material parti-
cles being taken into account. It is evidently not allowable to simplify these equations by
putting E = E’ and D=D?’, as we did in the previous paper. Moreover, instead of having
C=C’, we have C=C’ +C.

Our present purpose requires us to apply these equations only to the case of rays incident
directly on a refracting surface; we shall therefore suppose that the quantities a, ry, 4.7, a, 7,
are each zero, and that B, 8, 3’, are functions of x only: then the nine equations of connec-
tion reduce to three, viz.

dB, dp dp
=1Oeeeee 7)s —=h —......(8), —— = 0 cesses
B = B......(7) de gg ®) de 7 Ort):
; r
where ha Ste.

We shall assume v to be the velocity of propagation in the lower medium, and v’, v” the two
velocities in the upper, namely, the two roots of the equation (1). We shall suppose that the waves
in the upper medium are an incident and a reflected, and in the lower, two refracted waves, one
propagated with the velocity v’, and the other with the velocity v”, for it will be impossible to
satisfy the three equations (7), (8), (9), with only one refracted wave. Hence, using the same
notation as in the previous paper, we have from (7), (8), and (9),

VitNVe =Vi $V" ceateisicesu(lO)s

1 py’ 14
V—V) = = + yeeeeee0e.(11).
(V-V)-=S4+— (11)
Vv’ Ve

ee a OM Waraesestetastdan (12).

When V’ V” belong to the two refracted waves, and V’ V/’ to the corresponding waves of the
particles of matter, observing that the two latter waves are propagated respectively with the same
velocities as the two former (See Vol. vii. p. 421).

Hence, if we assume in general that

V= aer(t-j)V=1, V, = a, en4)VA, Ve a’ et(t-#) Val &e., &c.
We have from (10) (11) and (12), putting x=0,

at+a,=a+a" Soocsanse (GED)
v v
ian ie +s Gewese esteesn (LA),
vy”
i aptar Chas secesares (15).

Also by the two equations in the middle of page 423, Vol. v11.* we have (a’ and a! or a’ and a”
here, correspond to a and a, there)

ve? —-mB vw? — mB, v"

1 i ee —
v?—mB, v?—-mBv

Ge hoes es (16)s

* In the second of these a—a, is written by mistake instead of a,~a,

30 Mr. O'BRIEN ON THE ABSORPTION OF LIGHT.

Also (13) + (14) gives
2a= (2 +5) a’ + (: + >) (ie pe ee Cua)
v v

And from (16) and (17) we have
” hs see
ae ee ae hea deexees(h8)
"x v?—m B, v'*—-mB

Now if we suppose the roots of (1) to be impossible, we have

1 — 1
—~=etnV—1; = Se ay =>
v v0

Making these substitutions in (18), we evidently find an expression for a’ of the form

a’ =f (cosw+/—1sin w) a, or, fe®V-). a;
where f and w are real quantities, the former of which does not change sign when — nis abd ">
but the latter does. Now a’ becomes a’’ when v” is put in place of v’, and v' in place of v"; i.e.
when — 7 is put for 7. Hence we have

al" =fe-°V-1. a.

Hence the general expression for V'+ V” is

faten=. er FVAy e-ev>, en(t=7) V=1t .

or, fa feet gmt +a) oO Bale hr naa oa! Sameer neers (0) B
(19) therefore is the symbolical disturbance in the upper medium arising from the symbolical

disturbance ae"('-i)V— in the lower. By changing the signs of » and », and therefore of w, we
find that the symbolical disturbance a e-"(‘-=)V—= in the lower medium gives rise, in the upper, to

fa fem™.e- {mt—e)+0} J=i 4 em me, em {mt—e)-0} Vid,

Hence, superposing these two sets of disturbances, we find that the real disturbance

3
a@ cos n (¢-=)
v

in the lower medium gives rise to the real disturbance

fale.cos {n(t—ex) + wt + e7"™. cos fn(t — ex) - w} |
in the upper.

Now this latter expression indicates a continually increasing intensity, and therefore if the roots of
(1) were impossible, light after refraction would continually increase in intensity in passing through
the refracting substance; a result which is quite at variance with experiment. Hence we may
conclude that the roots of (1) cannot be impossible, and that the explanation of absorption given
above is not true. In fact, that explanation falls to the ground if we be not at liberty to reject the
integral .a@ e”” cos n(¢ — e #) and retain ae~””” cos m(¢ — ex), which we cannot do without violating
the equations of connection, as is evident from the process just gone through.

It appears, therefore, that though the action of the material upon the ethereal particles affords a

complete and satisfactory explanation of dispersion, we must look to some other source for an
explanation of absorption.

IV. On a new Fundamental Equation in Hydrodynamics. By the Rev. JaMEs
CuaLiis, MA., Plumian Professor of Astronomy and Experimental Philosophy in
the University of Cambridge.

[Read March 6, 1843.]

TuE object of this communication is to shew, that in addition to the two fundamental
equations of Hydrodynamics already recognised, a third is necessary to complete the analytical
principles of the science.

For the purpose of reference I shall call the two known equations, the dynamical equation, and,
the equation of continuity of the fluid. 'The same notation will be made use of as in my last
paper: p is the pressure and p the density of a particle whose co-ordinates at the time ¢ are w, y, x,
and the components of whose velocity V are w, v, w, in the directions of the axes of co-ordinates.
X, Y, Z are the impressed forces in the same directions. A differential coefficient is put in
brackets to indicate that the differentiation refers both to the co-ordinates and the time: a
differential in brackets means that the co-ordinates alone are differentiated. All differential co-
efficients not in brackets are partial.

1. It will be assumed that in any case of fluid motion an unlimited number of surfaces may
be drawn at each instant, cutting at right angles the directions of motion. In other words, it
is assumed that the directions of motion at every instant fulfil the condition of geometrical
continuity. In my last paper it was shewn that if

U v
dy, = da + dy + ds, sslanaseel a)

1 2 : ° ats .
the factor W being such that the right-hand side of the above equality is an exact differential,

the general differential equation of all these surfaces at all times isdyy=0. It is not necessary
that the surfaces should be continuous: that is, it is not necessary that the equation of a given
surface should be the same function of the co-ordinates through its whole extent. But that the
condition of the geometrical continuity of the directions of the motion may be maintained, each
surface must be made up of parts, either finite or indefinitely small, which are surfaces of continuous
curvature. Hence the quantity N has a real value for every part of the fluid in motion; at least,
motions for which this is not the case, if there are such, do not come under our consideration.

2. Let the integral of the equation dyy=0 be y(a,y,%, 7) =0, the arbitrary function
of the time being included in the function yy. The surfaces of which this is the general equation
I shall continue to call swrfaces of displacement. Since the equation y/(a, y, x, t) = 0 embraces
all the surfaces of displacement at all times, it will include the surfaces of displacement of a
given element of the fluid at two successive instants of its motion, if the path of the element
in the interval be continuous. It is not necessary that the path of an element through its whole
extent should be determined by the same equations, but it is necessary for the continuity of
the motion that it should be made up of parts, either finite or indefinitely small, which are
geometrically continuous, and that the directions of motion at two successive instants should not
make a finite angle with each other. The condition of the continuity of the motion of each element
is therefore expressed analytically by the equation d\/(q, y, x, t) = 0, the symbol 3 having reference,

32 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION

as in the Calculus of Variations, to the function yy, while the co-ordinates and the time vary with
the varying position of a given element. Hence,

d dw, d d
ney + Fe bas Eby + Groene.

But de =uédt, dy= vot, and dx= wot. Consequently,

ay dy, dy, a
y

WO iets cinwiatnteniewnieteie 2)
s

The main object of the arguments in this paper will be, to shew that the equation just
obtained is a necessary and fundamental equation of Hydrodynamics. I propose to call it, with
reference to the principle on which it was investigated, the equation of continuity of the motion,
to distinguish it from the equation of continuity of the fluid.

It may here be remarked, that in the place of the actual surface of displacement we might
have reasoned in the same manner on a surface having with it a contact of the second order
at the point ayx; for instance, the surface whose equation is,

@- a) Y= BY, @- 7 _

; 5 1=0
me n° p 4

the six parameters a, 3, y, m, ”, p, being functions in general, both of the co-ordinates and
the time. Writing F = 0 for this equation, it is clear, that when the co-ordinates and parameters
vary with the change of position of an element, we shall have oF = 0, provided there be no abrupt
change of the parameters, and consequently no abrupt change of the curvature of the surface
of displacement and of the directions of the lines of motion. This equation, therefore, to which
the equation dy (x, y, x, #) = 0 is equivalent, expresses the condition of continuity of the motion.

3. Before entering on the consideration of equation (2), it will be shewn by an example
that the two recognised fundamental equations are insufficient for the general determination
of fluid motion. One instance of contradictory results legitimately deduced from those equations
will suffice for this purpose. The example I have chosen is as simple as possible.

Let the fl uid pe inc 2 pr essi ble, and the mc ti on be parallel to the pl y Me : equation
du dv If 4 =
+ = 0. u=mez andv =—m that equation
] 1 Y q

is satisfied. These values make wdaw + vdy an exact differential. Hence the dynamical equation

of the continuity of the fluid for this case is

gives, p=C — 7 (x + y’), the arbitrary quantity being either constant or a function of the time.

: : ee ne. ‘ 2 ‘
By putting p =0, we obtain a + y? = —— for the equation of the free surface of the fluid, which
m
is therefore at all times cylindrical, and hence the velocity is every instant the same at all points

of the surface. But the differential equation of a line of motion is dy = = = -%. The lines
7

of motion are therefore rectangular hyperbolas having the axes of co-ordinates for asymptotes,

and the directions of motion are consequently different at different points of the cylindrical

boundary. Hence it is impossible that the boundary can be constantly cylindrical. This

contradiction proves that the equations on which the reasoning was founded are either erroneous or

insufficient. We have no reason to suspect any error in the principles from which they were derived,

and must therefore conclude that they are insufficient. It will appear afterwards that this instance
does not satisfy the conditions of continuity.

IN HYDRODYNAMICS, 33
vt , v= ek > and w= jyoaie we readily obtain from equation (2),
dy

dy dy ody dy _
+n (Fe oes +o) & Dachvd Beigoelcula):

which equation determines N,

A remark is important here, It appears from the equality (1), that wdw+vudy+wdzx
= Ndy,, and it might hence be supposed that when the left-hand side of this equality is integrable,
we are at liberty to assume N =1, and to consider yy identical with the quantity which is usually
called @ in Hydrodynamics, But it is clear from the reasoning by which equation (2) was
obtained, that N is a quantity of the same kind as the velocity, and that v is supposed to be
freed from any factors which do not verify the equation yy =0, whilst dd is merely a substitu-
tion for wda+vdy+wdz, and its integral @ is subject to no such operation. It is not,
therefore, allowable in any case to suppose the two quantities to be the same, on which account
I have here employed the letter y, in the place of the @ of my former paper, When uda + ody
+ wdz is integrable, in general N=f (). F (9).

5, For the purposes of the reasoning on which we shall presently enter, it is required
to shew, first, that when wdw+vdy+wdz is an exact differential (dp), the integral of the
dynamical equation may be taken from any one point of the fluid to any other, and that the
arbitrary quantity to be added is either a constant or a function of the time only, This will
appear as follows.

The general dynamical equation is equivalent to the three equations,

a2 -X+(Z)=0, 78 * (G)=% a - 2+ (2) =o, (6).

4. Since w=

in which P is substituted for a or for k* Nap. log p, according as the fluid is incompressible

or compressible. Assuming xu dx+Ydy+Zdz to be an exact differential, putting (d)) for
dP dP

(= —-X Jae a+ ( - ) a y+ (= - z) dz, and adding the above equations after multiplying

da
them respectively by da, dy, dz, it is known that we obtain for the case in question,

(dd) + (a <e) +3 (a. ( + ve + a5}) =10)Janwonbueyen(ii)s

ag*

2 2
which, if V* be substituted for aa a il aad jis equivalent to

dy?
dy @p dV an ap | dn a&e
(ae gan Vig) e+ (gt dydt* ro) ee (= + dedi ve) a We

But the quantities in brackets must be respectively identical with the quantities on the left-hand
sides of the equations ‘s a (6). Hence by reason of those equations,

dn dp ,av_, an dp av, ay ap A
dw * dwdt* i MON ayareloag® tragt Geant © oo re
+e - : ree fs d
Hence, dividing the foregoing equation by da, it will be seen that = and = may be of any

arbitrary values. The integral of that equation may consequently be taken from any one point
to any other of the fluid, and the arbitrary quantity to be added is independent of co-ordinates,

Vou, VII. Parr I. E

34 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION

6. It is next required to introduce into the equation of continuity of the fluid, by means
of equation (3), the condition of continuity of the motion. For this purpose the process must
be gone through which is given in my former paper (Camb. Phil. Trans. Vol. VII. Part III.
pp- 385, 386). The result there arrived at is, -

du dv dw_u dV vo dV w dV r(- *)
an do) dwar, avi dy a ee ,
where r, 7’ are the principal radii of curvature of the surface of displacement at the point vyx.
u_ du wo dy wi dx H -
Tama an ae ges gee ence if
dV be the increment of velocity along the line of motion corresponding to the increment ds, the
required equation becomes for incompressible fluids,
dV 1 1
+7 (- e *) Geet (ay.

When the fluid is compressible we have the equation,

If ds be the increment of the line of motion, we have

Now u = a = yy
du dv dw dV Le eel
en ar dy + as = rr +7 ( ~)
By substituting these values in the equation above, it will readily be found that

in which d.Vp is the increment of Vp along the line of motion corresponding to the increment
ds of the line of motion. I have obtained equation (9) in my former paper (pp. 387 and 388)
by elementary considerations, and equation (8) might clearly be obtained in a similar manner.
That method, being independent, may be adduced in confirmation of the reasoning here employed,
and of the general equation. (2), by means of which the reasoning has been conducted. It also
has the advantage of shewing distinctly that the increment d.Vp in (9) must be limited to
the direction of the line of motion, unless Vp has the same value at all points of a given
surface of displacement; and that dV in (8) must be similarly limited, unless the velocity be
the same at all points of a given surface of displacement.

The equations (8) and (9) may be called equations of absolute continuity. When they
are satisfied consistently with the respective dynamical equations, there can be no breach of

continuity and the motion is possible. Examples will hereafter be adduced to illustrate the
use of these equations.

7. I propose now to determine by means of equations (8) and (9) in what cases of possible

motion wdaw +vdy+wds is an exact differential. This important question has not yet received
a satisfactory answer.*

* Lagrange in the Mécanique Analytique argues that wda+ | and again, when the motion begins from rest. These theo-
vdy+wdz is an exact differential when the motion is so small | rems occur in the Edition of Poisson’s Traité de Mécanique
that powers of the velocity above the first may be neglected; | of 1811, but are omitted in that of 1833. Lagrange’s reasoning

IN HYDRODYNAMICS. 35

First, let the fluid be incompressible. It has been shewn in Art. 5 that when wda + vdy + wdx
is an exact differential, the dynamical equation may be integrated from any one to any other
point of the fluid. But the result obtained by integrating that equation in this manner,
does not give a possible motion unless the equation (8) be similarly integrable. Let this be
the case. Then the first condition that must be satisfied is, that each surface of displacement

; Ac 2 ear
be a surface of equal velocity. For on no other supposition can the differential coefficient ae
s

remain the same, in passing from a given point to another indefinitely near in an unlimited
number of directions. In the annexed figure P and Q are any

two points of the fluid; QR is an orthogonal trajectory to the p

surfaces of displacement situated at a given instant between

P and Q; PR is a line drawn on the surface of displacement

which passes through P, and intersecting QR in R. Now by

hypothesis the integral of equation (8) may be taken between

arbitrary limits. Therefore the integral from P to Q along PQ R
is the same as the integral along PR and RQ. But the integral

along PR is nothing, because PR is on a surface of equal velocity.

Therefore the integral from P to Q is the same as the integral

from R to Q. Supposing therefore the surface of displacement through P and the velocity in
this surface to be given at a given instant, the velocity at any point Q is a function of the line
QR. Let QR=s. Then Vds is a differential of a function of s and the time. Since, there-
fore, dp = Vds, is also a function of s and the time. But the equation @ = 0 is the equation
of a surface of displacement. Hence for a given surface of displacement s is constant. This
proves that the surfaces of displacement are parallel to each other, the orthogonal trajectories
are straight lines, and the motion is rectilinear.

Q

Again, let do be the increment of any line drawn arbitrarily on any surface of displacement.
Then since the direction of the variation of co-ordinates in the equation (7) may be any whatever,
we shall have,

dr op dV
ere
dines dened do

2 dr. ; : ; i ee 3
But since a is the effective accelerative force perpendicular to the direction of motion, and

o
: ‘ : oe : dx
since, as we have seen, the motion is rectilinear, it follows that om 0. Also as 0.
o
& ; : du
Consequently q a =0. This proves that the equations wda +vdy+wdz=0, and ai da
o
v dw , Collage : :
arr dy + FF dz = 0 are true at the same time. The latter equation is the former differentiated

with respect to ¢, on the supposition that dw, dy, dx do not vary with the time. It follows

with respect to the first is liable to this objection :—he concludes | tain ¢—A asa factor. We may therefore assume that wda +udy
that du_dv du _dw dv _dw fr mai +wdz =(t—h)*(Ude + Vdy+ Wdz), one at least of the

dy dw’ dz dx’ dz dy’ OT APE NBMSTOS 4 CANE, quantities U, V, W not vanishing when ¢=h. Since t—h is
tions, whence it follows that those equalities are approximate; | unaffected by the sign of differentiation, if the left-hand side of
whilst the inference that wda + vdy + wdz is a complete differ- | the equality be an exact differential, Udw + Vdy + Wdz must
ential, requires that they should be ewact. No reason is assigned | be an exact differential also. But the latter quantity is not
by Lagrange for the other Theorem. The following argument | necessarily an exact differential when t=h; therefore neither
shews it to be without foundation. If’each of the quantities w, | is the other.
v, w vanishes for a certain value / of ¢, they must each con-

E2

36 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION

that the direction of motion through a given point remains the same in successive instants.
This is rectilinear motion, and it thus appears that the rectilinearity of the motion is in accordance

with the dynamical equation.*
When the motion is perpendicular to a plane, r and r’ are each infinite, and equation (8) becomes

ein =0. This is true whether the motion be in parallel straight lines or in concentric circles about

ds

a fixed axis. But equation (8) does not enable us to determine whether in the latter of these two
kinds of motion, wda + vdy+wdsx can be an exact differential. This question will be con-
sidered further on.

Reserving then the case just mentioned, the following will be the conclusion to which the
foregoing reasoning conducts:—The only motions of an incompressible fluid which are possible
when udx + vdy + wdz is an ewact differential of a function of three independent variables,
are rectilinear motions,

8. Now let the fluid be compressible. For the same reason as that adduced in the case
of incompressible fluids, equation (9) cannot be integrated between limits entirely arbitrary
unless Vp is constant along a given surface of displacement. And again, as before, if s be drawn
at a given instant the orthogonal trajectory to surfaces of displacement from any point to
a given surface of displacement, then Vp at that point is a function of s. Hence, since
p (uda + vdy + wdz) = Vpds, it follows that the left-hand side of this equality is integrable.
But by hypothesis wdw + vdy + wdz is an exact differential dp. Hence, since pd = Vpds,
p is a function of @, and p and @ are each functions of s. But Vp is a function of s,
Therefore V is also a function of s. It is thus shewn that the surfaces of displacement are
surfaces both of equal yelocity and equal density. By reasoning precisely as in the case of
incompressible fluids a like conclusion is arrived at; viz. that the only motions of a com-
pressible fluid which are possible when udx + vdy + wdz is an ewact differential of a function
of three independent variables, are rectilinear motions.

The above result and the analogous one respecting incompressible fluids, are evidently
dependent on the fact that when wdw + vdy + wdx is an exact differential dp, both @ and V
are functions of the variable s, which is a line drawn at a given instant in the direction of
the motion of the particles through which it passes, commencing at an arbitrary origin and
terminating at the point wyx. And again, this fact is a direct consequence from the general
equation (3), as may be thus concisely shewn. That equation, on multiplying by N, becomes

d d
woe V? =0: or, since V=N. oye it becomes ay +N ¥ =0. Now when WN is a function
dt ds dt ds

of ¢ only, and consequently wdw +vdy+wds is integrable of itself, the last equation by
d

integration gives yy a function of s and ¢, Therefore also N a or V, is a function of s and ¢.

And since db =Vds, ¢ is also a function of s and ¢.

9. It remains to consider what are the forms of the surfaces of displacement which satisfy
the condition of rectilinear motion,

* If all the parts of the fluid have a common motion in | it must be under given circumstances, and their amount may
a common direction, the surfaces of displacement will partake | be calculated in the same manner as for a solid body. These
of this motion, and the motion of the particles in space will | motions may therefore always be considered to be eliminated
not be rectilinear. Such common motions are not the proper | by impressing equal motions on all the parts of the fluid in a
subject of consideration in Hydrodynamics. When they exist, | contrary direction,

IN HYDRODYNAMICS. 37

Since V and as are the same at all points of a given surface of displacement when the

; cea a c ths : 1 Gel Mit
motion is rectilinear, it follows from equation (8) that in incompressible fluids - + — is the
r fT

same at all points of the same surface of displacement. This is true also when the fluid is
compressible. For since p is given for a given surface of displacement, and since the equation

d
—da+ = dy + <7 ae = 0, obtained in Art. 7, proves that the surface of displacement through

a given point does not vary its position, it follows that = is the same at all points of the

ale and

same surface of displacement. It has been already shewn that this is the case with

. 1 Hoe 4 * ;
Vp. Hence equation (9) shews that ek Ie is the same at all points of a given surface of
id

1 1
displacement. Again, if dr be an indefinitely small constant quantity, ——— + ———— is con-
P ADS y 4 y +or r+o0r

1

s ‘ Sil 1
stant for the next contiguous surface of displacement. Hence if SoS ao arg) =a: —
r +or rr +o6r

=— =" aconstant, It follows that r and 7’ must each be constant
for a given surface, and consequently that not only is the curvature the same, but the principal
radii of curvature the same at all points of the surface. The only surfaces that possess this
property are the surface of a sphere and that of the common cylinder. Hence the only motions,
whether of incompressible or compressible fluids, that are possible when wda +vdy + wdz is
an exact differential, are in straight lines drawn from a jfived centre or perpendicular to a
Jined axis.

1
=c+ dc, we have = +
i

10. By reviewing the reasoning which has conducted to the above conclusion it will be
seen that after proving the dynamical equation to be integrable from any one point of the
fluid to any other whenever wdx+vdy+wds is an exact differential, the equations (8) and (9)
were assumed to be integrable in like manner. It is necessary therefore to inquire under what
circumstances the result obtained in the preceding Article is consistent with that assumption.

Let udw +vdy + wdzx be an exact differential, @ be a function of 7, and 1° = a + y? + 2%
Then the equation of continuity of a compressible fluid becomes,

2 2 2 2 2
(we - <2) fo £8 gee e@ Sb (as ts 4) = 0,

dr® , r rT

adr? dé dr'drdt dr

which does not agree in giving @ a function of r unless the impressed force either be nothing
or a function of 7. No such limitation is necessary with reference to incompressible fluids,
because the equation of continuity applicable to them becomes,

which gives @ a function of 7, whatever be the impressed force, It is, however, necessary
that Xda + Ydy + Zdz be integrable.

11. The investigation I have now gone through, shews that there are several defects in
the reasoning of my last paper, which I will endeavour to point out as distinctly as possible.
The first occurs in Art. 6 (p. 377), where it is asserted that “‘Vdr is not an exact differential,
unless the variation of V from one point of space to another at a given instant, depends only

38 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION

on the change of position in the direction normal to the surface of displacement.” This is
not true as a general proposition, but it is true with reference to fluid motion, solely in
consequence of the condition expressed by the general equation (2), as appears by the reasoning
in Arts. 7 and 8 of this paper. Hence the Proposition proved in the ‘Note’ added to the
former Paper fails in giving support to the above cited assertion, because it takes no account
of that equation. In fact, the proof neglects the curvature of the lines of motion, and therefore
only amounts to shewing that in rectilinear motion a surface of displacement is a surface of
equal velocity when wda+vdy+wdzx is an exact differential, or the converse—In the same
, a aa 4 het ea dv dw at 5 “
Article (p. 378) it is said incorrectly, tha ap i? + apy + Gp d® = 0 because for a surface
of displacement wda +vdy + wdx = 0.” This is true only when the position of the surface
of displacement through a given point is invariable, which should first have been shewn to be
the case. The correct reasoning is given in Art. 7 of the present paper.—At the beginning
of Art, 7 of the former paper (p. 379), it is supposed that in rectilinear motion the lines of
motion may pass through “fixed focal lines.” The more complete investigation of the present
Essay shews that they must be limited to passing through a fixed centre, or a fixed axis.—

“ce
The assertion (in p. 382) that a and V are constant for a given surface of displacement
°
at a given time, when wda + vdy + wdz is an exact differential,” is true, but, on account of
the defects already mentioned, does not follow from any previous reasoning.—It is not generally
true as asserted in p. 389, that “the variation of V at a given point is the same as if 7
and +r were constant,” and consequently the equation derived from that supposition is of no
value. I am not aware of any other points that require adverting to.

I proceed now to make some uses of equation (2) which will shew the importance and
necessity of it.

12. First, let it be required to determine on what hypotheses the general dynamical equation
is integrable. To do this it is necessary to introduce into the dynamical equation the condition
expressed by the equation (2), or by its equivalent equation (3). I have already gone through
the process for this purpose in Arts. 10 and 12 of my former paper. The result there obtained,
expressed in the notation of this paper, is

d.fNdwy V*_
Nit Sree ayaiiiain tees
It is supposed in this equation that Xdw + Vdy + Zdzx is an exact differential. This condition
being fulfilled by the impressed forces, the equation is integrable either if the second term vanishes,
or if Ndw be integrable. Since ee = fas, in the first case, se 0 and the motion
is steady; in the other, wiw + vdy+wds is an exact differential. These are the only cases
in which the general dynamical equation is integrable.

r F Li = c
13. Next let it be required to find the factor Wy in proposed instances of motion, and

to determine whether the motions are possible.
To make the equation (2), viz.

eg Oy. il gy

dt dx dy dz
convenient for this purpose, it will be transformed into another equivalent equation in the
manner following. The equation \,=0, being by hypothesis the equation of a curve surface,
may be supposed to contain besides the variables v, y, x explicitly, certain parameters a, b, c, &c.
which are functions of the co-ordinates and the time, and vary with the varying position of a

IN HYDRODYNAMICS. : 39

given element of the fluid, but which are constant in passing from point to point of a surface
of displacement through either a finite or an indefinitely small space. Let, therefore, the
equation \/ = 0 be equivalent to f(w, y, #, a,b,c, &e.) = 0. Then
dy df da df db_ df de
Wilda. dé) ah di’ deidt * 7a
dw SCAN df da dw df db d« df de da
“da “dv * da’ du'dt *db'dw dt *de'dw'di*

d d
and so for mA and eo
dy dz

oS) +2. (FG) +2. (3) + + &e. uo ew no,

Hence, substituting in equation (2), we have

Now differentiating the equation f(«, y, x, a, 6, c, &c.) =0, we obtain, since a, b, c, &c. are
constant for a given surface of displacement,

af df df _ 4 v w
ra da + = Boo geet =0= 7 dw+ = dy + de.
Hence, w= ap > v= a, w= a and consequently,
af af df af ap df\ |
teas a =) ag (FZ s) + qe’ (= a &e. +n (4 dy? + Fa = 0. 2+s0.004+(10).

We shall presently illustrate the use of this equation in finding N.
It is plain that the equation f(#, y, x, a, b, c, &c.) =0, may be that of a surface having a
contact of the second order with the surface of displacement at any point wyx, the parameters

in the equation of such a surface being a, b,c, &c. For instance, let the equation of the surface of
contact be :

(@- a), Wf), =~)

nae ae Pp —1=0.

ad

then we have for determining N the general equation,

Za (3 ae B ie ale ey Gos. (=) Gee (4) (22) #
m?> \dt ne \dt p dt m3 dt ns “\dt pa Noe
a 2 ~ 2 = ‘4
ev (© ») ow P) ats >)
m ni p
I proceed now to adduce some examples of finding NN, and of applications of the equations
(8), (9), and (10), to determine whether proposed instances of motion are possible,

Ex. 1. Let the case of motion be that considered in Art. 3. This instance gives w= ma,
d d
v= —my, and satisfies the equation ie + = =0. Also wde +vdy=m(ada —ydy), and
d -
a aA -%. Hence the general equation of the surfaces of displacement may be assumed to
u

be a — y’ — a’ = 0, and the general equation of the lines of motion, wy = c’.
q y

af da\ df df
Let f=a?-y-a. = Nhe re 20) eee
et f ye —a Then -- qa’ (5) = Qa ($5) am 2. ig Qy

40 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION

“(@)

Hence -a (3) +2N (2° +y*) =0, andN = ery
But by the equation a - y® — a =0, (since #, y, and @ vary simultaneously with the position
ad agi da) _ a(S) =u- ot) av=-
of the element,) we have a (F) 7] (=) a ($) 0, an 7 u= ma, (3 v my,
Hence a (=) = m(a* +), and consequently N=. This value makes o dw + dy sa
dt 2 N N
exact differential, and the equation (8) is therefore verified, We have now to see in what

: : : dV ds
manner equation (8) is verified. This equation for the instance before us becomes oe 0,

r being the radius of curvature of the curve of displacement at the point ay, and ds the increment

P e+ yi v+y ida
of the line of motion at the same point. Hence 7 = — Se » and ds oe - There-
ar, ms . :
fore are Bee Oe =0, This equation cannot be integrated unless y is eliminated by means

Vw+y @
of the equation o = c’; that is, it can be integrated only along a line of motion. The dynamical
equation must therefore be integrated in the same manner, and the arbitrary quantity to be
added is a function of co-ordinates as well as the time. The fluid must be conceived to be
included between two hyperbolic surfaces indefinitely near each other. This explains the contra-
diction met with in Art, 3,

Ex, 2. Let the equation of the surfaces of displacement be @ — tan-' =0@. Putting therefore

f for 6 - tan“12 » we have

df *)=(F qf sy geet & @
a5: ai z a) do w+y’ dy wt +y"

dé i de = WON. ee
Hence (Ge) + aan and N= ~ (T)(@ +y’).

Now since y = w tan 8, the motion is evidently parallel to the plane of wy in concentric circles about

d0\
a fixed axis, Hence at any distance x from the axis V = r (=) . Consequently V = — ~ (a? + y*)

=-Vr. Therefore if V weet we have N a function of ¢, and wdw + vdy an exact differential,

although the motion is curvilinear. This is the case of motion alluded to at the end of Art. 7,

Ex. 3. Let it be required to determine whether in an incompressible fluid the surfaces of
displacement can be concentric spherical surfaces, the centre of which is always on the axis of
x, and at the same time the motion be such that a given particle in successive instants is at
the same distance from the common centre,

Here if a = the co-ordinate of the centre, and @=the radius of any surface of displacement,
we have f= (~—a)? +y°+ 3° - a’,

df jd
Hence ue (3) = —2(@-a) (32) = — 2V_ (wv —a) suppose.
df da . (da
qa’ (F) = 0, because by hypothesis (5) =0,

IN HYDRODYNAMICS. 41

df af af
ag 2) ao ate
V (@—
“ —2V (w-a)+4N$(~—a)? + y+ 2°} =0, and N= — 2)
V (w — a)? d V (a -
Bee ee anh ees
dx a dy a
me V(w - i
Sy ie AES) Cd pele
a a

We have now to find V, by means of equation (8). Since the above value of N shews that
udx + vdy + wdx is not for this instance an exact differential, V must be differentiated along

Bes we have V = V,cos@, and dV =dV_cos@; so

2da
a

@
a line of motion. Hence putting cos @ for

dV
Equation (8) therefore becomes a +

‘

gle

1 1
that — =——. Alsods=daand-+-—=
Vis r

Hence V, = — and V= == cos@. Thus the motion is completely determined. It is plain

that this i would be arated by a smooth solid sphere moving in an arbitrary manner in the
fluid, with its centre always in a given straight line.

Ex. 4. Poisson’s determination of the simultaneous motions of a sphere and the surrounding
fluid (Memoirs of the Paris Academy, Tom. x1, and Connaissance des Tems, An. 1834) differs
from the foregoing. Let us therefore inquire, assuming the motion to be such as Poisson has
found, whether the conditions of continuity are satisfied.

For the sake of simplicity I shall consider the fluid to be incompressible. Poisson assumes
that wdw + vdy + wdz =d¢@, and finds values of the velocities which, if R* = (#— a)? + y° + x%
may be thus expressed :

3 ng aN 3 sic
pli”), » ore y (wa), Om pet (@ ia) s

T being an arbitrary function of the time, and c the radius of the ae These values make
Pag i gh ing
dy? dz?
integrating wdw + vdy + wdz =0, the equation of the surfaces of fac ae will be found
to be R*—h* (wv —a) =0 for the positive values of w —a, and R* + h? (w — a) =0 for the negative
values. his change of equation implies a breach of continuity*. If we put R cos@ for
w—a, we obtain R*—h’?cos@=0 for the polar equation of the curve which by its revolution
about the axis of w generates a surface of displacement. The lines of motion lie in planes
passing through the axis of w. The general polar equation of these lines will be found to be
R-esin®@=0. From the latter of these equations the value of ds is to be found, and from
the other the values of r and r’, for the purpose of ascertaining whether the equation (8),

“=

udue+vdy+wdz an exact differential, and satisfy the equation =

no MONE Loe : : : c -
viz. => + ds (- + =)=0, is satisfied by these values so as to allow of its being integrated

between arbitrary limits, the dynamical equation having been already integrated in this manner.

* According to this solution the fluid in contact with the | which the sphere moves forwards, a result, to say the least,
sphere and in a plane passing through its centre perpendicular | very improbable.
to the axis of 2, moves backwards with half the velocity with

Vou. VIII. Parr I. F

42 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION

>

dR (1 + 8.cos?@)}, 1 _ 6cos@ (1 gee PS am —s
2cos 0 . R(1 + 3cos? 6)? o) R(i + 3cos? @)2
dV iy fe dV 38dR 3+cos’@
Henve, + de(" +7) = 9 + oe * Ta peor ~

This equation cannot be integrated independently of the equation R —esin’@ = 0; that is, it
can be integrated only along a line of motion. Hence the conditions of continuity are not
satisfied. .

Ex. 5." Let it be supposed that the motion is in straight lines drawn from the vertex of
a cone, and let the fluid move in parallel slices so that the motion parallel to the axis of the
cone is the same at all points of any section perpendicular to this axis: it is required to
determine whether this motion is possible.

It will be found that ds =

The equation of the surfaces of displacement is 2° +y*+2°- R’=0, and the equation of
the lines of motion R-—wsec@=0. Hence ds=dR=dwsec@, and R=r=7'. Equation (8)

d 2d. : § df (dR
consequently becomes de oe’ =0. Now putting f for w+ y?+°-R’, we have a )

V oe dR\ dt
d. V
=-2RV, 2 = 20, ow ae , Lanes, Hence -2RV+4N(a2’?+y°+ 2) =0, and N=—7R
Therefore wu = wee = = (x) by hypothesis. Consequently v = a 5 = = He , and

y- 2b) _ 9) sec@. The above values of uw, v, w, do not make wdw+vdy+wdz an
wv
exact differential. Hence the dynamical equation must be integrated along a line of motion,

: Cans
and the equation (8) with the same limitation. Consequently “ar dp (#) sec@, and the above

, “ t
ae + = = 0, which by integration gives p(x) -10 - Hence yard sec @,
and the motion is completely determined. This solution agrees with the one I deduced from

particular considerations in the Cambridge Philosophical Transactions (Vol. V, Part 11. p. 186).

equation becomes

The preceding example is instructive as shewing that the motion may be rectilinear when
ude +vdy+wds is not an exact differential. Another inference may also be drawn from it.
Let the motion be steady, and let W be the velocity at a point of the axis distant by h from

2

Wh ;
the vertex. Then wt, whence f(#) =h* W, and V = GE see 0. If gravity (g@) be sup-

2
posed to act parallel to the axis of #, the dynamical equation gives for the pressure (p),
224

W*h
p=C-ga- ant sec’ 0,

2 7,4
and if p=0 where w=H, C=gH+ ie sec?@. If now H be assumed to be so large that
the second term of the expression for C may be neglected, we shall have, C=gH, and
2 Bt ..
= J — 2 .
p=e(H-2) oat Be 0

It might hence be argued that wd# + vdy + wdzs is an exact differential for this case, since C
is independent of sec@. But the objection to this inference is, that if it were true, the above
value of p might be differentiated supposing sec@ variable, which would manifestly be in-
correct, for the result would be at variance with the differential from which this value was
derived. The fact is, the neglected quantity has no effect on the numerical computation of
p, but as it contains sec@, we cannot regard C as independent of co-ordinates.

IN HYDRODYNAMICS. 43

Ex. 6. Let it be required to determine whether in a compressible fluid the surfaces
of displacement can be spherical surfaces, the centres of which are always on the axis of «,
and at the same time the motion be such that the radius of the surface of displacement of
a given particle remains the same in successive instants.

da

Let (vw - a)? + y? + 2° -R’?=0. Then by reasoning as in Ex, 3, V= (3) cos 8 = V_ cos 0.

By hypothesis (32) is the same for all points of the same surface of displacement. Hence V,

is the velocity at any point of the axis of w. The equation (9) may be put under the form,

pdt ds p ds R
subject to the limitation of integrating along the line of motion s. The dynamical equation,
subject to the same limitation, is

: dV ye
k® Nap. log p + [que es

2

Hence, carrying the approximation only to the first power of the velocity, we have

kd dV hed av

sie —, and ieee =
pds dt pdt dt
1 av dV

te ha alata

ds+f'(¢). Therefore

Rds RB? ds
The motion is symmetrical about the axis of w, which is plainly a line of motion. Hence the
above equation is true when V, is put for V and w for s. It thus becomes,

i
Vv em (gr lv =)

dV av dV, Vi dk
Lita Ces. se ae eet) | |

df Lo dx* ak (eae R? is)

Now in this equation R = #-—a, and — = “, for a is a function both of wand¢#. The
xv wv
d
differential coefficient = will in all cases be very small, if the velocity of the particles be small
d
compared to the velocity of propagation of the motion. Hence rae — Vy (2 -<*) = V_ nearly,
wv

ued : : -
regarding - a quantity of the same order as V. Also as V, may be considered a function of

dV, dV, dR av, @V, &V, dR &V, er cas!
R and ¢, a amas oan nearly. And = amt aa ae nearly. By substitution in

the foregoing equation, we have

PY meray {av V;

FE hot 1° aaa laa = #4) ;
This equation gives V, by integration, whence V is known from the equation V = V,cos@. Thus
the motion is completely determined consistently with equation (9), and this is the proof of
the possibility of the assumed kind of motion, so far, at least, as regards small motions. The
above solution is that which I have employed for finding the resistance of the air to the vibrations

of a ball-pendulum.

Camprinerk Onservatory.

March 2, 1843.
F2

V. Observations on the Nature of the Biliary Secretion ;—the object being to shew,
that the Bile is essentially composed of an Electro-negative body in chemical combi-
nation with one or more inorganic bases. By GEORGE Kemp, M.B. St. Peter's

College.

[Read March 6, 1843.]

Tue following observations on the nature of the Bile, form a portion of some researches
into the elementary composition of that secretion, commenced in the laboratory at Giessen.
Professor Liebig suggested the following mode of conducting the inquiry.

A portion of ox-bile as received from the gall-bladder was to be evaporated to dryness,
and then submitted to ultimate analysis, without any farther manipulation.

This plan was abandoned for the following reasons.

The gall-bladder of every animal yet examined contains, in addition to the bile, another
body, always varying in quantity, and possessing physical properties differing so essentially
from the biliary secretion, that I determined in the first place to separate and examine this
body, to which the name of Mucus of the gall-bladder has been given. The analysis proved
that this body contains 15°4 per cent of nitrogen, while the bile itself contains only 3°5 per
cent of that element, so that the results obtained in the manner originally proposed would
have been constantly varying, and always erroneous. The fats and fatty acids also, contained
in the bile, would have led us still farther astray; eventually, therefore, I determined on re-
moving the mucus and fatty acids before attempting the analysis of the fluid. Previously,
however, to entering on the manipulation employed, it will be proper to give a sketch of the
principal opinions which have been hitherto entertained on the nature of the bile.

The first proximate analysis of this fluid, of any importance, seems to have been made by
Thénard in the year 1806, with the results contained in the note*. According to his opinion
the bile is principally composed of biliary resin and picromel; the biliary resin he supposed
to be held in solution by the picromel. Berzelius in 1807 instituted an analysis of which the
table + below gives a summary view. He considered the biliary resin and picromel as one body,
altered by the manipulation of Thénard, who made use of nitric acid in his analysis. To
this body, composed of biliary resin and picromel, Berzelius applies the name biliary matter.
An analysis instituted by Dr. Prout about the same time confirms the analysis of Berzelius in
every essential point. At a subsequent period Gmelin undertook the investigation of this
secretion; his results induced him to imagine that the opinion of Thénard with reference to
the existence of biliary resin and picromel, was correct, although the substance described as
picromel by Gmelin differs very essentially from that body as described by Thénard, He,

Water vsessessesnesssesstesteneenesees i Se Water assvescsseusvactclerreteatsavtevas cevencotetiens ieesaa nana
Biliary resin “ Biliary matter with fat .... .
Picromel serseteeecgnancnne dave . Mucus of the gall-bladder
Yellow colouring matter........ L Osmazome, Chloride of Sodium,and Lactateof Soda 0:74
SOda.....sssesserersyenenenes wee ¥ Botha tes ison scedmteriaiate acest. eeiecee eat eee Ora
P hosphate of Soda . 25 Phosphate of Soda, Phosphate of Lime, and traces
Chloride of Sodium 40 of a substance insoluble in alcohol............. } Onl
Sulphate of Soda.... Pree Gs |}

Sulphate of Lime...........00. 15 aia

A trace of Oxide of Tron.

Dr. KEMP, ON THE NATURE OF THE BILIARY SECRETION. 45

moreover, found other substances denominated cholic acid and taurin, which have since been
proved to be products of manipulation. In the year 1826, Demar¢ay employed himself in the
laboratory of Liebig in preparing, and submitting to analysis, a substance obtained from the
bile when treated with diluted sulphuric acid, and to which he subsequently gave the name
choleic acid. This body was in all probability the picromel of Thénard, and the matter
which remained after removing the choleic acid, denominated choloidic acid by Demarcay, bears
a striking resemblance to the biliary resin of Thénard; as, however, no elementary analysis
was made by that chemist, the matter must remain in doubt. The choleic acid of Demarcay
is an important body, as Professor Liebig has acceded to the opinion that it is the essential
organic ingredient of the bile; a conclusion, however, which subsequent researches tend to
overthrow; indeed, the opinion of Demarcay was grounded on the following circumstance.
After he had prepared his choleic acid, and combined it with soda, the compound possessed a
considerable number of the physical characters of the Bile, and in estimating the quantity of
soda which combined with a given quantity of his choleic acid, he found the quantity of the
base almost precisely the same as that contained in the same quantity of the dried bile. One
unfortunate oversight, however, occasioned this erroneous inference of the identity of the two
bodies. The choleate was converted into the sulphate of soda, in order to estimate the
quantity of the base. On applying the same method to the bile, the chloride of sodium
contained in that fluid became converted into sulphate of soda, and thus the quantity of soda
combined with the organic body was supposed to be considerably greater than it really was;
for on looking over the analysis of Thénard it will be seen that the quantity of chloride of
sodium stands in the proportion of 4: 5 to the soda; a quantity much too large to be over-
looked, as it would occasion an error in the second whole number of the atomic weight. Indeed,
the circumstance of the similarity found in the dried bile and in the choleate of soda in this
one experiment, was evidently purely accidental, as the chloride of sodium is always present in
bile, and that in constantly varying proportions. In fact, the choleic acid of Demarcay seems
to be a product of decomposition of the bile effected by means of sulphuric acid; and the
errors in the late work of Professor Liebig on the subject have arisen from not taking into
account the other product of manipulation, the body which Demarcay has denominated
choloidic acid. The labours of Demarcay were however exceedingly valuable, as they directed
the attention from proximate to ultimate analysis, and were the means of inducing the illustrious
Berzelius to make one more effort towards effecting the solution of this difficult problem*.
A paper, which has recently appeared I believe in an English form, was in the year 1841
laid before the Royal Academy of Stockholm, purporting to be an analysis of the bile of the
ox, and the characteristic properties of its component parts. This claborate research was
conducted in the same manner as his former analysis, the object being to eliminate what he
considered the proximate principles of the bile; and the results confirmed all his former in-
vestigations on this subject. He concludes by stating the theory, that the bile in its healthy
and perfectly fresh state is essentially composed of bilin, a body agreeing in every physical
character with the biliary matter of his former analysis, and that this body is continually
undergoing a change into two acids, fellic acid and cholic acid; that at the same time these
two acids form binary compounds with bilin, to which compounds he has given the names
Bilifellic Acid and Bilicholic Acid. To this theory we shall have occasion again to advert.
These last researches of Berzelius seemed to discourage any farther attempt to elicit facts
by means of proximate analysis; and at the request of Professor Liebig, I commenced a series
of ultimate analyses, with the limitations alluded to above. It appeared desirable also to extend

* Ueber die Analyse der Ochsengalle, und die characterisi- | den Kongl. Vet. Acad, Handl, 1841. S. 1—64, iibersetzt von
renden Eigenshaften ihrer Bestandtheile; von J, Berzelius, (Aus | Dr. Wiggers.

46 Dr. KEMP, ON THE NATURE OF THE BILIARY SECRETION.

the inquiry, which had hitherto been confined almost exclusively to the bile of the ox; I
therefore proposed to examine the bile of that animal as the type of the graminivorous, the human
bile as the type of the omnivorous, and the bile of some decidedly carnivorous animal, the lion
or tiger for instance, as the type of that class of animals. It was further proposed to institute
an inquiry into the differences which exist in the bile of different species of fish. Thus I hoped
that some general character at least would be found to illustrate the nature of the secretion in its
relation to the researches of physiology and pathology. The results of the first investigation
which were made at Giessen have since been published in the Journal of Erdman and Marchand at
Liepsig, and in the London Medical Gazette; it will, therefore, merely be necessary to give a
general outline of the manner in which the investigation was pursued ; the subsequent portions
of the research have, by the kind permission of Professor Cumming, been carried on in the
laboratory of this University. At the onset of the inquiry it seemed most important to take a large
average, and the bile obtained from twelve oxen killed at the same time at Frankfort was evaporated
in a water-bath to dryness; the mass was reduced to powder and treated with alcohol sp. gr. -840,
in order to remove the mucus; the clear fluid obtained by filtration was again evaporated
to dryness, powdered, and treated with ether, in order to remove the fats and fatty acids in
combination with soda, and this treatment continued until the ether on evaporation gave no residue.
The substance was now dried at a temperature of 110° of the centigrade thermometer, reduced
to a powder and submitted to analysis. The solution of this substance was perfectly neutral ;
on burning it however in a platinum crucible; an alkaline ash was left, which consisted of
carbonate of soda, and chloride of sodium. The carbonic acid which was found combined with
soda was of course the result of the combination of the carbon of the organic portion of the
bile during combustion with the oxygen of the atmospheric air. In the bile therefore soda itself
was present in combination with organic matter, and as in the bile the alkaline property of the
soda is suspended, we have positive proof that the soda in the ox-bile is combined with an
electro-negative body; for in no other way can we account for the perfectly neutral character of
the bile. Those who are acquainted with the description of the bile in physiological works, will
remember that it has been described as an alkaline fluid; and Schulz has made the statement that
one ounce requires half a dram of acetic acid for its saturation. His account is, however, much
too vague to place any dependence upon, for what is usually called acetic acid is merely a
solution of acetic acid, and the strength of the solution has not been recorded by this author.
It is certain that a portion of a strong acid may be added to the bile without any acid reagency
taking place before the quantity of soda combined with the electro-negative body (naturally
contained in the secretion) has become saturated. And we must not be surprised that the electro-
negative body set at liberty produces no change on litmus paper, as bodies of very high atomic
weight seldom produce any visible reaction on test-paper. We may instance the new alkaloid
Berberin, which has an atomic weight of more than 4000 (O = 100) ;—the combining weight
of the body which we are about to examine is between 5000 and 6000. But to come to actual
experiment on the subject.

I have tested the fresh bile of more than forty oxen, the human bile, the bile from the tiger,
* the fox, the cat, * several kinds of monkey, the dog, * the wolf, * an Indian bull“, and the secretion
as found in the codfish; in all these cases, with two exceptions, the bile was perfectly neutral.
One of these exceptions was the bile from a child which had been burnt to death, and which
was not examined until three days after its removal from the body, in which state, it is needless
to remark, that decomposition had already commenced, and even in this case the alkaline reaction
was barely perceptible. The other was that of the bile obtained from an Indian bull, in which the
secretion was not only decomposed, but absolutely putrid. To return however to the ox-bile.

a The bile of the animals marked (*) was obtained through the kindness of Dr. Clark, Professor of Anatomy in this University.

Dr. KEMP, ON THE NATURE OF THE BILIARY SECRETION. 47

Having now decided that the bile of the twelve oxen under examination contained an alkaline
base, the physical properties of which had been suspended by combination with a body in an
opposite electrical condition, the next point was to determine the quantity of soda contained in a
given quantity, and thus estimate the combining weight of the organic body with which the
base was combined. On this being ascertained, an analysis was made to determine the quantity
of organic elements. For the sake of brevity the analyses will be given together, after the
description of the manipulations. A further portion of bile obtained from twelve more oxen was
now submitted to examination, and the results, both with respect to combining weight and proportion
of organic elements, were as nearly identical with the former portion as our present modes
of analysis, and the nature of the research, warrant us to expect. It may here be remarked,
that animal bodies in general present great obstacles to minute analysis, from the difficulty with
which they are burnt, and from the readiness with which they attract moisture. The body
contained in the bile is so hygroscopic, that even in the act of mixing and introducing it into
the tube a sufficient quantity of moisture is absorbed to render the estimation of hydrogen always
too high. Having now determined analytically that the bile of the ox contained an organic electro-
negative body in combination with soda, it seemed desirable to attempt a synthetical proof. Here
serious objections presented themselves. The bile is more or less influenced by every chemical
reagent yet tried, or, to use the words of Berzelius, “it has so great a tendency to undergo changes
in its composition, that the action of different reagents upon it converts it into different compounds,
which vary according to the processes employed to extract them; exactly as oils and fats are
converted into sugar and fatty acids by the action of the oxides of lead and zinc.” It appeared
probable, however, on consideration, that by extreme dilution of reagents, and carefully avoiding
a greater excess than necessary, we might succeed, if not in isolating the body for analysis, yet
in separating it from the soda with which it was originally combined; uniting it again with a
fresh portion of soda, and thus in forming the bile artificially. If the composition of the body
thus formed should by subsequent research furnish us with results identical with those obtained
from the bile in a natural state, I conceived that no candid person would reject the evidence either
as unsatisfactory or unsound. A portion of the dried extract of the bile freed from mucus and
fatty acids was dissolved in alcohol of as great a strength as could easily be obtained, and then
treated drop by drop with diluted sulphuric acid. The sulphate of soda thus formed being
insoluble in alcohol, could of course be separated by filtration, the organic elements previously
combined with the soda remaining in solution. The sulphuric acid was added in the slightest
possible excess, in order to ensure the complete separation of the soda, and the clear solution obtained
by filtration was now treated with an excess of carbonate of soda deprived of its water of crystal-
lization. The excess of sulphuric acid was precipitated in the form of sulphate of soda, while
a portion of the carbonate readily combined with the electro-negative body remaining in solution.
The solution obtained by filtration was now evaporated to dryness, and submitted to analysis.
As no change in the physical characters of the body had been made by this process, I was not
surprised to find that the combining weight and ratio of organic elements were found by analysis
to be identical with the bile in its natural state. But the question may be asked, Why not (having
separated the soda by means of sulphuric acid) have evaporated the solution, and then analyzed
the body thus isolated ? My reason for not doing so was, that it was necessary to add sulphuric
acid in slight excess, and this in proportion as the solution became concentrated by evaporation
would have rendered the result unsatisfactory, as we know that sulphuric acid of moderate
strength decomposes the bile, and converts it into the choleic and choloidic acids of Demargay.
Thus it would have been as much a matter of probable evidence whether the isolated body
was the matter contained in the bile, as whether the body separated as above and recombined with
a base, was the electro-negative substance, the composition of which we wished to determine.
It now only remains to give a summary of the general results obtained from the analysis of the
ox-bile, before passing on to the consideration of the bile of other animals.

48 Dr. KEMP, ON THE NATURE OF THE BILIARY SECRETION.

1. A portion of the substance was burnt in a platinum crucible, and an ash remained con-
sisting of
Carbonate of Soda =11°16 per cent.
Chloride of Sodium = 0°54 per cent.

2, Another portion treated in the same manner gave
Carbonate of Soda = 11°13 per cent.
Chloride of Sodium = 0°37 per cent.

The organic portion gave on combustion with chromate of lead :—
1 2

Carbon = 6460

Hydrogen = 9°62 9:40
Nitrogen = 3-40 .... .. 340
Oxygen = 22°88 oo... .eseeseeeneeee 22:35

100°00 100:00

The human bile, from the smallness of its quantity, presents us with still greater difficulties than
the ox-bile, the portion obtained from an adult under the most favourable circumstances being
barely sufficient for the necessary number of analyses. The first portion of human bile which
I examined was removed about eight hours after death from a man who died suddenly under an
attack of delirium tremens. Having separated the mucus and fat as above described, it was
submitted to analysis with the following results. On burning a portion in a platinum crucible
it was found to contain 6:6 per cent of soda and 1:87 per cent of chloride of sodium. ‘The organic
elements were in the following proportions :

Carbon = 68°80
Hydrogen = 10°40
Nitrogen = 3:44
Oxygen =17'36

100°00

The general conclusion from the above analysis is, that human bile, as well as the ox-bile,
is an electro-negative body in combination with soda. ‘Two other cases of bile obtained from
children who died in consequence of severe burns confirmed this conclusion. The next examination
was into the nature of the bile of fishes. I have not yet been able to obtain a sufficient quantity
of this secretion for anything more than a cursory examination, the results however, so far as
they went, were exceedingly satisfactory. The bile of four large codfish gave 2°61 per cent of
chloride of sodium, 1:8 per cent of lime, 4°3 per cent of soda with a trace of magnesia. I had
merely substance enough to estimate the quantity of carbon and hydrogen, which were 68-60
and 10°8 per cent respectively. From this analysis we see that this species of bile is also an
electro-negative body, but combined with three bases, lime, soda, and magnesia. I have recently
obtained the bile from a tiger, which was treated in the usual manner to remove the mucus and
fat. A portion burnt in a platinum crucible gave an alkaline ash, the nature of which I have
yet to determine. The solution of the bile itself was perfectly neutral; we therefore conclude
that its nature is similar to all the others which we have examined, and that in the carnivorous,
as well as the graminivorous and omnivorous animals, the bile is essentially composed of an electro-
negative body in combination with one or more inorganic bases. That the body is not the choleic
acid of Demargay, is evident from the difference of the elementary composition which exists
between them. ‘The bile of the ox contains nearly 65 per cent of carbon, human bile upwards
of 68 per cent, while the acid of Demarcay contains between 63 and 64; and the difference in
the quantity of hydrogen is so great that we cannot construct any formula under which bodies
differing so widely from each other can be included. The choleic acid also when combined

Dr. KEMP, ON THE NATURE OF THE BILIARY SECRETION. 49

with soda is precipitated by acetic acid; the body contained in the bile is not precipitated
by that reagent. We know also that ox-bile, treated in the manner directed by Demargay for
the preparation of choleic acid, is resolved into two bodies, the choleic acid and the chloidic,
the latter forming a very large proportion of the results, probably as much as one half; and it is
remarkable, that adding the quantity of elements found in the two, and taking the mean, we have
almost exactly the quantity as given by the analysis of human bile. The highest authority
on all subjects connected with chemical research is undoubtedly Berzelius, and he has lately
given it as his opinion that the bile is essentially composed of bilin, bilifellic acid, and bilicholic
acid. Considering, however, that these bodies were eliminated by means of reagents which he
himself has acknowledged as more likely to yield products than educts, we are perhaps justified
in supposing that these bodies were the results of manipulation; it is at any rate highly improbable
that in the very large number of elementary analyses made, we should in each case have accidentally
procured bile in which precisely the same point of transformation of bilin into the other products
should have been arrived at. One experiment was however made which proved that the body
described by Berzelius as bilin does not always exist at all in the bile. I obtained the biliary
secretion from an ox immediately as slaughtered, and while it was quite warm: the mucus and
fatty acids were removed with as great dispatch as practicable, the dried bile was then dissolved
in alcohol freed from water as thoroughly as possible, and through the solution a stream of
carbonic acid gas was transmitted for the space of three hours, without the slightest precipitate
or even opacity occurring. Now one of the principal characters of bilin, according to Berzelius,
is, that if combined with a base its tendency to combine is so slight that the combination is
destroyed by carbonic acid. In the above experiment, therefore, if bilin had been present, the
carbonic acid would have combined with the soda, forming the carbonate of soda, which is insoluble
in alcohol, while the bilin would have remained in solution.

Such are the principal facts which I beg to lay before you. It remains yet to be determined
whether the electro-negative body in the bile is the same in all animals. A certain analogy seems
to exist between the bile of the ox and that of man; but it would be premature to place on record
any reasonings which, however probable at the present stage of the investigation, more accumulated
evidence may not confirm. The subject is in progress, and bids fair to give decided and satisfactory
results.

G. KEMP.

Sr, Perer’s Couiece.

Vor. VIII. Parr I. G

VI. On the Motion of Glaciers. By Wriit1am Hopkins, M.A. and F.R.S.,
Fellow of the Cambridge Philosophical Society, of the Geological Society, and

of the Royal Astronomical Society.

[Read May 1, 1843.]

SECTION I.
On the Present State of Theories of Glacial Motion.

De Saussure appears to have been the first to examine with accuracy, and to describe in detail,
the various phenomena which the Alpine glaciers present to us. ‘The phenomena connected with the
motion of glaciers, constituting the class with which alone we are concerned in the present communi-
cation, engaged their share of his attention, though his observations did not aim at that degree of
exactness with which observers of the present day are conducting their researches. Nor did he
fail to speculate on the causes of glacial movements. He considered glaciers to slide along the sur-
faces over which they move, the motion being due to the inclination of those surfaces to the horizon,
and the action of gravity on the moving mass; and though he was not the first who adopted this
theory of glacial movement, it is now usually associated with his name, from his having been the
first to acquire any exact knowledge of such movements, or to form, perhaps, any very definite
conception of the mechanical causes to which they might be referrible. From his time to a recent
period the subject seems to have excited little comparative interest; but within the last few years
glacial phenomena have been investigated with great care, and attention has been again directed
to them, not only as forming an interesting branch of physical enquiry, but also as pregnant with
geological inferences of the first importance. We are especially indebted to M. Agassiz for his
active researches among the Alpine glaciers. The influence of his name has awakened an interest
in them which might otherwise have long slumbered; and whether some of the opinions he has
promulgated respecting the motion of glaciers be ultimately established or refuted, geology must
continue equally indebted to him for the manner in which he has directed our attention to the im-
portance of the subject in its geological bearings.

One of the consequences of these renewed researches has been to cast great doubt on the ade-
quacy of De Saussure’s theory to account for the motion of glaciers. "The inclination of the surface
over which some of the Alpine glaciers move is found to be so small as to render it apparently
inconceivable that such glaciers should not only descend, but overcome powerful obstacles to their
descent, if there be no other moving force than that of gravity. The mean inclination of the
surface of the Aar glacier is stated not to exceed 3° (and that of its bed must be still less), an incli-
nation much smaller than that at which a very smooth hard body will descend down an equally
smooth and hard plane*. Nor is the difficulty diminished by the consideration of the great

* The following results are given by Professor Whewell in his | be the angle of the plane down which sliding will just take place,
Mechanics of Engineering, on the authority of Morin. If 6 | and p the coefficient of friction, we have for

Values of | Values of
MB 6

Hard Limestone on Hard Limestone............
Brass on Brass
Brass on Tron ......se000-

Cast Iron on Cast Iron.......
Cast Iron on Cast Iron, greased..
Brass on Iron, greased .....sccesecseessenecerseeees

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 51

weight of the moving mass, or of the extent of its surface in contact with that over which it moves;
for, according to the observed laws of sliding bodies, the motion is independent of both these
circumstances. ‘This difficulty has been hitherto regarded, and with reason, as a most serious
if not an insuperable one to the sliding theory. Another has also been frequently urged, for which,
however, there is no real foundation. It has been contended that if a glacier moved by sliding
over its bed from the mere action of gravity, it ought to move with an accelerated motion, whereas
the motion is observed to be wnaccelerated. If the force retarding the motion were solely that
of ordinary friction of the surface over which it moves, the objection would be valid, because the
retarding force of friction is independent of the velocity acquired; but in the case of a glacier
moving down an irregular valley and over an irregular surface, all the retarding forces do not act
on the mass in the same manner as friction in the ordinary cases of sliding bodies. Besides the
friction, there will be retarding forces acting at an indefinite number of projecting points along
the sides or bottom of the glacial valley. Such forces will depend on the velocity of the glacier,
and therefore the whole accelerating force on the mass will be some function of the velocity, and
the motion will not necessarily be an accelerated motion*. The difficulty now spoken of, there-
fore, seems to have arisen from an imperfect conception of the problem; but the one first mentioned
is sufficient to shew that the solution afforded by De Saussure’s theory is far from being satis-
factory.

The rejection of the sliding theory has led to the adoption, by different persons, of two other
theories, which have been denominated respectively the dilatation and expansion theories. ° They
both rest on the same principle—the expansion of water in the act of freezing. The former has had
recently for its principal advocate M. Agassiz. It is found that a portion of the water arising
from the dissolution of the superficial ice of the glacier by the direct rays of the sun and the
warmth of the summer atmosphere, infiltrates into the minute pores and cavities of the ice, where,
it is contended, it is frozen by the cold of the glacier, and, in freezing, expands and produces a
dilatation and consequent onward motion of the whole mass. According to the expansion theory,
the motion is due to the freezing and consequent expansion of water collected, not in minute
pores and crevices, but in cavities or fissures of considerable dimensions. A repetition of these
processes is supposed to keep up the continuous motion of the glacier.

These theories appear to me to involve insuperable difficulties, both physical and mechanical.
Supposing the capillary cavities in the one case, and the large ones in the other, to become
full of water, and that water to be frozen, the cavities will be completely filled with solid ice.
How is another set of cavities to be formed for a repetition of the process? Such an effect cannot
be ascribed to an internal dissolution of the ice as a consequence of external temperature, for
though the internal temperature of the glacier might be depressed far below the freezing point
in winter, it cannot possibly be raised above that point, or even up to it, except at the extreme
surface, during the summer. That water does percolate through the pores of glacial ice with
extraordinary freedom, M. Agassiz has proved by making the percolation evident to the eye, but he
has not proved that it freezes there. The temperature of the upper portion of a glacier, where
the percolation has been observed, is, in fact, very little below that of freezing, and does not ap-
pear to be sufficiently low to convert water into ice while moving with the freedom with which
it descends through the glacier. Wherever congelation does take place the capillary pores must
necessarily, I conceive, be filled up, and where it does not, the percolating water must proceed till
it meets with the larger fissures, through which it will descend freely to the bottom of the
glacier. The existence of the larger internal cavities of the expansion theory is purely hypothetical;

* The descent of water along a river-course, or of ice floating down its current, is not necessarily with an accelerated motion, and
for a reason exactly similar to that assigned in the text.

G2

52 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

and a repetition of the process to which the motion is referred is perhaps still more inexpli-
cable than in the dilatation theory.

If, however, we chose to allow the alternations of congelation and dissolution required by
these theories, it might still be shewn (as I have done elsewhere*) that the effectiveness of the
causes of glacial motion assigned by them must probably be very much less than that of gravity
whenever the inclination of the bed of the glacier is not much less than that of any known
glacier. I think it unnecessary, however, to repeat such investigations in this communication,
or to insist on other difficulties involved in these theories, because there is an obvious and con-
clusive test to which they will doubtless be soon subjected. It is manifest, that, according to
either theory, the velocity with which any proposed point of a glacier will move must be approxi-
mately proportional to its distance from the upper and fixed extremity. If, therefore, it should
be found, on the contrary, that the motion near the two extremities of a glacier is nearly the same,
the refutation of both these theories will be complete. M. Agassiz has been engaged in the most
careful determination of all the circumstances connected with the motion of the glacier of the Aar,
and Professor Forbes has in like manner been occupied with the Mer de Glace of Mont Blane.
The results in the latter case are already partially known through Professor Forbes’s letters to
Professor Jameson}, and appear to be totally inconsistent with both the theories of which we
are now speaking. The full details of the surveys of these two glaciers will form most important
additions to our knowledge of glacial phenomena. In the mean time sufficient has been said to
indicate the great, and, as I believe, insuperable difficulties both of the expansion and dilatation
theories.

A conviction of the inadequacy of any of the three theories above mentioned to account for
the motion of glaciers, has led Professor Forbes to suggest another theory. In common with that
of De Saussure, it attributes the motion of a glacier to the action of gravity ; but whereas, according
to the sliding theory, gravity is enabled to act effectively in communicating motion to the glacial
mass in consequence of the facility with which the lower surface of the glacier moves over the
bed on which it rests, the theory now alluded to attributes the efficiency of gravity to the facility
with which contiguous particles of the ice itself may move with reference to each other. Such
at least is my conception of the theory, and it is only in this sense that I can understand it as a
mechanical theory: for if it be merely meant to assert that certain phenomena of glacial motion
are similar to those which would present themselves if the glacial mass were really a viscous fluid,
the assertion is only equivalent to a particular geometrical representation of the phenomena in ques-
tion. In this sense the theory asserts nothing respecting mechanical causes, and therefore cannot
be classed with the theories already mentioned.

Regarding this view of glacial motion, however, (in the absence of its more complete development)
according to my conception of it as a mechanical theory, it may be asked, what reason have we
to suppose that the adhesion of contiguous particles of glacial ice is much less than that of a
particle of ice in the lower surface of the mass to the contiguous particle of the bed of the glacier?
The general mass of glacial ice is extremely hard and compact, and has unquestionably a great
cohesive power, so that when we consider the probable effects of terrestrial heat and subglacial
currents in destroying the adhesion between the glacier and its bed, it would appear the more.
probable that this adhesion should be much less instead of being much greater than that between
contiguous portions of the ice itself. I do not insist on the absolute conclusiveness of this reasoning,
but on its sufficiency to shew the necessity of proving, by independent experimental evidence, that
glacial ice does possess this property of semi-fluidity or viscosity, if we would attribute to that
property the effectiveness of gravity in setting a glacier in motion.

* The investigations alluded to were printed and privately | of glacial motion assigned respectively by the three theories men-
cireulated among most persons interested in glacial researches. | tioned in the text.
The object was to compare the degrees of efficiency of the causes + Edinburgh Quarterly Journal of Science.

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 53

It may perhaps be answered, that the best way of making such experiments is by observing
the glaciers themselves, or in other words, that it is better to make our theory depend on obser-
vation than on direct experiment; and, undoubtedly, it is thus that we do arrive at the highest
order of evidence which the greatest problems of physical science admit of. We set out with
some determinate hypothesis, of which we calculate the consequences. These calculated results are
then compared with the results of observation, and the degree of accordance between them will
constitute the evidence in favour of our original hypothesis. The conclusiveness, however, of this
inductive process of reasoning must depend on the rigorousness with which we can calculate
our results, and the accuracy with which the phenomena to be accounted for can be observed. If
our methods possess, in both these particulars, the requisite degree of exactness, we shall be certain
of demonstrating the truth or detecting the fallacy of our original hypotheses, and of thus elimi-
nating, as it were, all but the true one. In the case before us, however, the required exactness
is not attainable, for it will appear, in the course of this paper that the particular phenomena to
which Professor Forbes would seem to appeal in evidence of the truth of his theory, are equally
consistent with that which I shall offer. Consequently, the necessity of direct experimental proof
of the viscosity of glacial ice assumed in this theory cannot be superseded, in the present state of
our knowledge of the motion of viscous fluids and of glacial movements, by an appeal to phenomena
which those movements themselves present to us.

This review of the existing state of glacial theories is sufficient to shew how imperfect a
solution of the problem of glacial motion has yet been offered. All the above theories repose
more or less on hypotheses unsupported by the direct evidence of experiment or observation. The
theory of De Saussure is apparently in opposition to the ascertained facts respecting the motion
of sliding bodies; in the theories of dilatation and expansion, the alternations of thawing and freez-
ing is an unsupported assumption, and the mechanical adequacy of the causes assigned by these
theories (supposing them to be real causes) a pure hypothesis; and in the last-mentioned of the
above theories, the viscosity of the glacial mass necessary to give effectiveness to the moving force
of gravity, seems to be opposed to the evidence of our senses. It would be difficult perhaps to
conceive the solution of any mechanical problem in a much more unsatisfactory state than the
one before us; for, of the different solutions which have been proposed, each involves some difficulty,
which, if not removed, must ensure its ultimate rejection.

In considering these difficulties it occurred to me, that the motion down an inclined plane
of a mass of ice having its lower surface in a state of disintegration, might take place according
to laws different from those observed in the sliding motion of rigid bodies, and, without forming
any very definite conception of the manner in which the motion might be modified under this new
condition, I determined to try the experiment. The results were such as to remove entirely,
I conceive, what appeared to be an insuperable objection to the sliding theory, by shewing that
ice, under the condition above stated, is capable of descending with a slow unaccelerated motion,
by the action of gravity alone, down planes of much smaller inclination than those over which
known glaciers are observed to move. In the next section I shall describe the experiments which
leave no doubt, in my estimation, as to the real cause of glacial motion.

SECTION II.
On the Cause of Glacial Motion.

1. Ewperiments.—A slab of sandstone was so arranged that the inclination of its surface
to the horizon could be slowly and continuously varied by the elevation of one edge, ‘The sur-
face was in the state in which it had been sent from the quarry, and in which such stones
are sometimes laid down as paving stones, retaining the marks of the pick with which the

54 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

quarry-man has shaped them, without any subsequent process for rendering the surface smooth.
The slab thus presented a grooved surface (the grooves running in very nearly parallel directions),
having some resemblance to those over which existing glaciers move, but having little of the
smoothness of roches poliés. 'The best measure, however, of the degree of its roughness is this
—when placed at an inclination of about 20°, a piece of polished marble would just rest upon it.

The slab was so placed that the direction of the grooves coincided with that of greatest
inclination. A frame of about 9 inches square and 6 inches in depth, without top or bottom,
was then placed on the slab and filled with lumps of ice from a neighbouring ice-house, in
such a manner that the ice, and not the frame (which merely served to keep the ice together
as one mass) was in contact with the slab. In the experiments in which the following results
were obtained, weights were placed on the ice such that the pressure on the slab was at the
rate of about 150lbs on the square foot.

inati : Mean S:
cae Spaces in decimals of an inch though which the loaded ice ee

: in inches
hey descended in successive intervals of 10 minutes. Coraline

3° 308 ,05 ,07 ,03 ,04 ,05 ,07 ,06 ,04

6 309 ,10 ,09 ,07 ,08
9° 317 514 419 420
534 327

341

mass descended with an accelerated motion.

When the inclination was 9° about two-thirds of the weight was removed; the velocity was
diminished by nearly one half.

When the inclination of the slab did not exceed one degree, there was a smali but very
appreciable motion.

On the surface a slab of the same kind of stone smooth but not polished, there was appreciable
motion at an angle of 40 minutes. Nor am I prepared to say that either in this, or the preceding
case, the angle was the least at which sensible motion would take place.

When the surface used was that of polished marble, there was sensible motion with the
smallest possible inclination. The motion, in fact, afforded almost as sensitive a test of deviation
from horizontality as the spirit level itself.

In all these experiments the ice melted continually but very slowly at its lower surface in
immediate contact with the slab. During the night the temperature descended below that of
freezing, and the motion entirely ceased.

The angle at which the accelerated motion just begins to take place is that at which the ice
would just rest upon the inclined plane, if the temperature of the slab and of the air were at or
below the freezing temperature, so that no disintegration of the ice should take place. This angle
appears to be nearly the same in the case of ice, on the grooved slab I made use of, as for that
in which polished marble was the sliding body, and is that whose tangent determines the coefficient
of friction between the slab in question and. solid ice. When the slab was of polished marble
this angle was very small.

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 55

2. In the experiment above detailed we have these results :—

(1). For all angles less than that just mentioned the motion was not an accelerated motion.
This result was verified in every experiment I made.

(2). For inclinations not exceeding 9 or 10 degrees, the velocity, ceteris paribus, was
approximately proportional to the inclination. This, I doubt not, would hold in all cases in
which the inclinations should be sufficiently small compared with the angle of accelerated motion.
It is manifestly equivalent to the assertion, that the velocity is proportional to the moving
force.

(3). The velocity of the mass was increased by an increase of weight.

3. It is not very difficult to give a general explanation of the mechanism of this motion. Con-
ceive a very thin slice of the sliding body in contact with the inclined plane on which the motion
takes place to become instantaneously fluid: an indefinitely small motion would necessarily take place,
by which the lower surface of the portion of the mass retaining its solidity would be brought in con-
tact with the plane. If the plane were horizontal, it is manifest that this indefinitely small motion
would be vertical; but it appears sufficiently evident, that if the plane be inclined the motion will
be compounded of a vertical motion by the action of gravity, with a motion parallel to the plane
arising from what may be termed a momentary floating of the solid body on the small portion
which has been supposed to become fluid or disintegrated, and depending partly on the inclina-
tion of the plane. The instant the solid portion of the body comes in contact with the plane,
the motion will be arrested. At that instant, suppose another thin slice of the body to become fluid ;
the same motion will be repeated, and so on. A discontinuous motion would be thus produced ;
but if the successive slices which become disintegrated be indefinitely thin, 7. e. if the liquefaction
or disintegration be continuous, the resulting motion will be continuous, and it will, moreover,
be uniform if the disintegration be so.

The fact that motion takes place down planes of such small inclination compared with that
necessary to make the ice slide independently of its disintegration at the lower surface, may simply
be stated as due to this circumstance—that, whereas the particles of ice in contact with the plane
are capable, so long as they remain a part of the solid mass, of exerting a considerable force to
prevent sliding, they are incapable of exerting any sensible force when they become detached
from the mass by the liquefaction or disintegration of its lower surface.

When the sliding mass is small (as in the experiments above described) the exact uniformity
of the motion will be destroyed by local irregularities in different parts of the inclined plane down
which it takes place, or temporary irregularities in the disintegration; but where the whole in-
clined surface on which the motion takes place is always the same (as in the case of a glacier),
and the mass is sufficiently large, all local or temporary irregularities will, in a great measure,
counteract each other, and will therefore not materially disturb the uniformity of the motion,
which will be preserved so long as the intensity of the causes of disintegration remains unaltered.

4. Temperature of the Lower Surface of a Glacier—The essential condition under which
gravity becomes effective in putting the loaded ice in motion in the experiments above described,
is that the lower surface of the ice shall be in a state of disintegration, or that its temperature
shall be that of zero of the centigrade thermometer. In order, therefore, that our results may
be applicable to any proposed glacier, we must shew that the temperature of its lower surface
must be zero. For this purpose, let us conceive the earth to be covered with a superficial crust
of ice, and, for the greater simplicity of explanation, let us suppose the conducting power for heat
within the icy shell, and in passing into it from the earthy nucleus, to be the same as in the
interior of the nucleus. The temperature of the ice, to a certain depth beneath the external
surface, would be subject to sensible annual variations of temperature, which would become
insensible at a certain depth (#,), where the temperature (%,) would be constant. The mathe-
matical determination of a, and % will be given in the concluding section. ‘The temperature zw,
would necessarily be less than zero (centigrade) and at greater depths than a, the increase of

56 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

temperature would be proportional to the increase of depth, the rate of increase (with our present
supposition respecting the conductive power of the ice) being exactly the same as in the actual
case of the earth, provided the ice should always remain solid, i. e. if the temperature, thus increasing
with the depth, should not rise to zero at the lower surface of the icy crust. Now, though more
accurate observations on the internal temperature of glaciers are wanting, it is probable from those
of M. Agassiz, that the internal temperature of glaciers in those regions in which their motions
have been observed, and at depths below the influence of external variations, is not less than
—1°(cent.). The least depth in the actual case of the earth at which the temperature is sensibly
constant may be stated generally at about 60 feet, below which the rate of increase of temperature
in descending may be taken at about 1° (cent.) for every 100 feet. Hence, supposing the same
to hold for ice, the internal temperature of our icy shell, where exposed to the same external tem-
perature as an actual glacier, would be below zero at every point, provided its depth were less
than 160 feet. If the thickness of the shell were greater than that quantity, the temperature of
its lower part would be higher than zero if ice were capable of receiving such higher tempera-
ture; but since that is impossible, the heat which would be employed in raising the temperature
of the lower portion of the shel] above zero if it could retain its solidity, would be actually
employed in converting into water its lower surface, which would thus be retained at the constant
temperature of zero, and in a state of perpetual disintegration.

If instead of supposing the icy shell to cover the whole surface of the earth, we suppose
it to be of comparatively small extent, the same conclusions will hold, provided its linear super-
ficial dimensions be sufficiently great with reference to the depth, which in the above case has
been estimated at 160 feet. Such is the case in all considerable glaciers. Hence, assuming
the truth of our data, if a glacier in those regions in which it is accessible to observation,
exceed 150 or 160 feet in thickness, its lower surface must be in a constant state of disintegration,
as a consequence of the internal heat of the earth. This result is liable to error, depending on
our imperfect knowledge of the internal temperature of glaciers, and the conductivity of glacial
ice; but in those parts at least, where the thickness of a glacier is considerably greater than 150
feet*, it leaves no reasonable doubt, I conceive, of the truth of our conclusion respecting the
state of slow perpetual disintegration of the lower surface.

5. Agency of Subglacial Currents.—The internal heat of the earth, however, is not the only
cause producing this constant disintegration. Another and probably very effective agency exists
in the subglacial currents, which, during the summer, are principally produced by the rapid
melting of the ice at the upper surface of the glacier, whence they descend through open
fissures, and afterwards force their way between the glacicr and the bed on which it rests. I
cannot appeal to any direct experiments to determine the effect of water at the temperature of
zero in dissolving ice at the same temperature, when running in contact with its surface, but
its efficiency in this respect is sufficiently proved by its action on the upper surface of a glacier
when the direct rays of the sun and the temperature of the atmosphere are sufficient to dissolve
the superficial ice, and thus to create innumerable rivulets running upon the surface till they
meet with a fissure into which the water is precipitated, and finds it way to the bed of the
glacier. hese little superficial streams shew their effect in disintegrating the ice by the manner
in which they cut out for themselves their own channels, thus assisting greatly in the degra-
dation of the surface. Its effect on the lower surface of the glacier is probably greater than
on the upper, on account of the hydrostatic pressure under which it must there act, The
descending water must reach the bed of the glacier at almost every point of it, and cannot

Si That such is the case throughout extensive portions of large | from his cabane on the glacier of the Aar, of which the depth
glaciers, there seems to be no doubt. M. Agassiz informed me | could not be much less than 780 feet.
that he had discovered a nearly vertical hole in the ice, not far

|
Mr. HOPKINS, ON THE MOTION OF GLACIERS. 57

afterwards collect and proceed in uninterrupted channels, because if such channels were once
formed they must necessarily be immediately destroyed, or at least impeded at numerous points
by the motion of the glacier. The existence of such impediments to the motion of the water,
and the consequent formation of subglacial reservoirs, is proved by the continued flow of the
streams which issue from the lower extremities of glaciers during the night, though the supply
from the upper surface is entirely stopped immediately after sunset, when the melting ceases,
and does not recommence till a considerable time after sunrise the next morning. During the
intervening ten or twelve hours the whole of the water beneath the glacier at sunset would
necessarily discharge itself if its course were unimpeded, even from the longest and least inclined
of the Alpine glaciers, before sunrise the next morning; whereas the volume of water issuing
from the glacier of the Aar is very little less in the morning than in the evening. This
equable supply can only arise from the discharge during the night from reservoirs formed
during the day. Hence it will follow that these subglacial currents, commencing from almost
every point of the glacier, will be forced under every part of it by hydrostatic pressure, by
which, as above asserted, its disintegrating action on the lower surface of the ice will doubtless

be increased.

SECTION III.
Phenomena depending upon the Motion of Glaciers.

6. Relative Velocities of the Central and Lateral Portions of a Glacier.—The central
part of a glacier moves considerably faster than its sides, but, according to Professor Forbes*, the
change of velocity takes place not far from the lateral boundaries, the whole central portion
moving with nearly the same velocity. In the month of August last summer, the central part
of the Aar glacier, near the cabane of M. Agassiz, was moving at about the rate of a foot a day,
while near the sides it was less by one third or one half. On the Mer de Glace the motion
appears to be generally greater, in the ratio of about 3: 2, but varying in different parts of the
glaciert. The difference between the central and lateral motions seems to be less than in the
former case.

On the Mer de Glace the velocity near the lower extremity appears to be somewhat greater
than near the upper one. On other glaciers no adequate observations on this point have yet been
made.

7. Crevasses or Fisswres—The fissures which traverse a glacier are among its most distinct
and striking phenomena. When the glacial valley contracts in descending, the following facts
appear to be established.

The fissures are transverse and curved, having their convexity turned towards the upper
extremity of the glacier.

Systems of fissures, preserving a certain identity of character with respect to number and
form, remain fixed in position, not with reference to the moving mass, but with reference to the
fixed objects around. It is not however to be understood that each fissure of a system remains
absolutely stationary, but that each system remains so in the same sense in which what may be
termed a system of breakers on the sea-shore may be said to be stationary, although every suc-
cessive wave-is in constant motion. In like manner every fissure must move through a certain
space with the glacier, and then disappear by closing, or be so modified as to lose its identity t ;

* Letters to Professor Jameson—Edinburgh Journal of ¢ I consider a fissure to remain identically the same so long
Science. as it continues to intervene between the same identical portions
+ Ibid. | of ice.

Vout. VIII. Panr I. H

58 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

and when, during this motion, it has passed forward a certain distance, a succeeding one originates
at the same point, moves forward in the same manner, and ultimately disappears at the same
point as those which have preceded it.

If the sides of the containing valley be divergent, the longitudinal fissures predominate, and
diverge from the axis of the glacier in a manner accordant with the divergency of the sides.

8. The continued convexity of a crevasse turned towards the upper extremity for a great
length of time would manifestly be inconsistent with the fact already stated, that the central
portion of a glacier moves considerably faster than its sides; for such relative motion must have
the effect of continually lessening and ultimately destroying the convexity. Let us examine how long
a time it might require to produce this effect. ;

Let PN, be a transverse fissure when first formed in a gla-
cier, of which NO is the axis. We may, for an approxima-
tion, suppose PN, to be the arc of a circle whose center is
O,. Since N, will move faster than P, the position and form = ss
of the fissure will change, but, as the change will depend only a
on the relative motions of different points of the PN,, we may \
here suppose P to remain at rest, and the other points of the \\
fissure to move only with their relative motion. It will be suf- NN
ficiently near for our purpose if we suppose this motion such is
that the fissure shall always retain the form of the arc of a ‘\
circle. Suppose it to come into the position PN after a time int
t, and let O then be its center of curvature. We may first fe
examine what change of curvature will take place in the fissure in x
the time ¢, the curvature being measured by the angle PON. oe

Let PON=0, PON, =90,, and PO=r, PO,=7,3 and nae

let v be the relative velocity of N. Then \ io
NN = ot, \
andr vers. 9@=7, vers. 8, — Dirctectesccersectee™(L)> \|

Also, if b= sin of the are PN,, He

r sin@ = 65,

and 1, sin@, = 6.
Hence (1) becomes

vers.@ vers.0, v

—— = —, -;z!
sin 0 sin 0, Rr

é
or tan = = tan 5

which gives the curvature at the time Zz.

If 6 and, therefore, 9 be not too large, we shall have approximately
2v
6 = 6 eee hs 3
or if @ and 6, be expressed in degrees,

=o, (2.2),
aw OO

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 59

To take a numerical example, let us suppose PN to be 2000 feet, where the relative velocity
(v) is ,3 feet*, the unit of time being one day ; we shall then have

ro0-(4)
u 58

nearly. Consequently it would in this case require nearly two months to diminish the angle
@ by one degree. If 6 =8" or 10°, the change of curvature during a whole summer will
scarcely. be sensible to the eye.

When the whole curvature is destroyed we must have 9=0, or

180 2v

nearly with the above values of v and b.

If @,=10°, ¢ = 580 days, supposing the relative motion to be ,3 feet each day.

If the central part of the glacier move through a foot each day, the curvature in the
above case would be destroyed after the highest point (N) of the fissure should have moved
through 580 feet.

9. Professor Forbes has shewn by his observations on the Mer de Glace, that there is
little variation of velocity except at points near the sides of the glacier. Consequently, if
we take a fissure of which the extremities do not approach too near the sides, the relative
velocities of different points of it may be much less than supposed in the above example, and
a fissure might remain for several years without losing its convexity. The period during
which these crevasses preserve their identity as open fissures, has not yet been made suf-
ciently a matter of observation. Whatever this period, however, may be, it is manifest, that
since it is too short for the convexity above described to be destroyed, the crevasses must
generally close after moving with the general mass of ice through a space extremely small com-
pared with the length of the glacier. After being closed they will form swrfaces of discon-
tinuity within the glacier, i.e. surfaces along which there is a discontinuity in the cohesive
power of the ice, There will, in fact, be no cohesion along such a surface when the crevasse
first closes, though it may be afterwards partially restored. As the existence of these surfaces
may exercise an important influence on the relative motions of different parts of the glacier,
it may be well to examine more particularly the positions they will assume in consequence of
the observed relative motions of the center and sides of a glacier.

10. Surfaces of Discontinuity—Let AA’ in the annexed diagram represent a fissure im-
mediately after its first formation. For the greater distinctness of explanation it is supposed to
extend from one side of the glacier to the other. Suppose it to move to BB’ before it closes.
During that time other fissures will have been successively formed at 44’, and will in like manner
have moved forward; so that between Ad’ where the fissures are formed and BB’ where they
finally close, there will be a system of open fissures as represented in the figure. Below BB’ the fis-
sures will no longer be open, but will form swrfaces of discontinwity, as above described. The

" If the central velocity be represented by unity, the velocity of the sides will probably be very frequently between ,6 and ,8, and
therefore the relative velocity of the center between ,2 and ,4.

H 2

60 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

successive lines in the diagram represent successive positions
of any one of these surfaces, or simultaneous positions
of successive surfaces originating in the same system of
fissures. If we suppose RUR’ to have been an open
fissure at AA’ when SVS’ was so at BB’, there will be
a number of surfaces of discontinuity between RUR’ and
SVS’ corresponding to the number of open fissures be-
tween 44’ and BB’; and the same would hold between
each consecutive pair of lines which at a previous epoch
coincided simultaneously with Ad’ and BB’.

If another system of fissures be formed at aa, they
will give rise to a corresponding system of surfaces of dis-
continuity, of which the dotted line rr’ may be taken as
the general type. These surfaces will intersect those of the
former system at angles more acute as they become more
remote from aa™*.

Hence then it follows, as a simple geometrical conse-
quence of the existence of transverse fissures and of the
more rapid movement of the central portion of the glacier,
that the whole mass must be traversed by numerous surfaces
of discontinuity; all those originating near the higher ex-
tremity of the glacier becoming very nearly longitudinal as
they descend, and others being less so, according as their
origin is more remote from that extremity, The whole
mass will thus be divided by these intersecting surfaces
into innumerable portions. Cohesion, as before intimated,
may be partially restored along the surfaces of discontinuity,
but the difference of velocity in the central and lateral por-
tions will have a constant tendency to give slightly different
motions to contiguous portions, and thus to prevent the
restoration of cohesion. ‘The whole glacier will thus be-
come a dislocated mass; and that it actually is so is indi-
cated by the facility with which it breaks up into vertical
masses whenever irregularity of motion is superinduced by
irregularities in the bottom or sides of the glacial valley.
I consider a glacier, therefore, as an aggregate of numerous
parts, cohering so imperfectly as to allow a much greater
facility of motion among themselves, than if the mass were
perfectly continuous. The glacier will thus derive a much
greater facility of adapting itself to the configuration of the
valley through which it descends, than if its power of adap-
tation depended merely on the plasticity and compressibility
of glacial ice—properties which it must doubtless possess,
though possibly in so small a degree that they may only

C
0

become sensible under the action of the enormous pressure to which I shall hereafter shew the
glacier must be subjected whenever its motion is considerably impeded.

* This exposition respecting the surfaces of discontinuity is | alternate layers of ice of different structure, which constitute the

similar to that given by Professor Forbes with reference to the | ribboned structure,

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 61

SECTION IV,
Explanation of Phenomena depending on the Motion of Glaciers.

1l. Relative Velocities of different parts of a Glacier.—According to our theory, the velocity
of any portion of a glacier will depend (1) on the inclination of its bed, (2) the disintegration
of its lower surface by the internal heat of the earth, (3) on subglacial currents, (4) the
depth of the mass, and (5) local and lateral obstacles. The first and second causes will generally
haye nearly the same effect both in the central and lateral portions; but the third cause will
manifestly produce in general the greatest acceleration in the central parts, and the fourth
cause will produce a similar effect, if the glacier be deeper in the center than at its sides
[Art. 2 (3)], while the greatest retardation will be produced on the lateral portions by the last
of the above-mentoned causes. These causes sufficiently account for the greater velocity of the
center of the glacier.

Again, the second of the above causes will probably act with approximate uniformity through-
out the whole length of the glacier, but the third cause will act with the greatest energy at
the lower extremity, because the subglacial currents will be increased by innumerable tributaries
as they descend. his cause, therefore, will tend to make the velocity greater, as we approach
the lower end of the glacier, while the greater depth of the mass at the upper extremity will
tend to give the greater velocity to that part of the glacier [Art.2 (3)]. In winter the effect
of the currents must be very inconsiderable, and we should consequently expect that there would
be a tendency in the portions of the glacier in the higher regions to move faster than those in
the lower, in which case there must be a longitudinal compression, and consequent closing up
of transverse fissures in a greater or less degree. During the summer, on the contrary, the sub-
glacial currents will be most efficient, and we should expect that they would give the greater
velocity to the lower extremity of the glacier, in which case the mass would be brought into
a state of longitudinal tension, by which new transverse fissures would be formed, or old ones
reopened.

12. Internal Tensions and Compressions arising from the unequable Motion of the Glacier.—
The mathematical determination of the internal state of tension or pressure of a solid, but
extensible and compressible body, acted on by external forces, presents difficulties which are
at present insuperable, except in the most simple cases; nor can demonstrable conclusions of
a less determinate character be arrived at except by an exact knowledge and careful application
of mechanical principles. The cases I shall consider are the simplest of the kind, and admit
of simple and conclusive reasoning. Let us first suppose a glacier to be a continuous mass, and
to descend down a gradually contracting valley, so that the mass may be everywhere sub-
jected to lateral compression; and let us also suppose that points near the upper extremity
of the glacier tend to move with a smaller velocity than those more remote from it, and the
central with a greater velocity than the lateral portions, from the causes above explained. Our first
object is to determine the direction of greatest tension at any proposed point.

Conceive the mass divided into two portions by an imaginary surface, which, for the greater
distinctness, may be supposed vertical or nearly so at every point. The mechanical action
between any two contiguous particles, situated on opposite sides of this geometrical surface, may
be resolved into two forces, the one normal and the other tangential to it. The normal force
may be either a pressure or tension; in the latter case there must be cohesion between the
particles. The tangential force may arise from cohesion, or may be of the nature of friction,
and independent of the existence of cohesive power. Now let us conceive the normal cohesion
at every point of our imaginary surface to be destroyed. Then, since the part of the mass
near the lower extremity tends to move faster than the other part, these two portions will

62 : Mr. HOPKINS, ON THE MOTION OF GLACIERS.

necessarily separate, if the surface intersect the lines of motion of the particles through which it
passes, and the internal state of pressure and tension will be altered. But, again, let us suppose
this surface to coincide with the line of motion of every particle situated in it, and while the
normal cohesion is destroyed, conceive the tangential force between contiguous particles to be
still maintained by friction. Since, by hypothesis, the mass is in a state of transverse com-
pression, it is manifest that the destruction of the cohesion along the internal surface in the
position now supposed will cause no separation of the two portions into which the mass is
thus divided, or any modification of the previous motion, or of the internal pressures and tensions
due to it. The same will be true if another such surface existed as near as we please to the
former. But in this case, it is manifest that the direction of greatest tension at any point be-
tween these two surfaces must be in the direction of these surfaces, i.e. in the direction of
motion; for since by hypothesis no cohesion exists between the portion of the mass included
by those surfaces and the contiguous portions, it is impossible that any tensions should be
impressed upon it in directions transverse to its bounding surfaces of no cohesion. Consequently,
the same must hold when the cohesion of the mass is unbroken, since it has been shewn that
the destruction of the cohesion would not affect the state of internal pressure or tension.

13. From the former part of the preceding paragraph, it follows that if any surface be
described within the mass, perpendicular at every point to the direction of motion, there will
be a maximum tendency to destroy the cohesion along such surfaces, so far as that tendency
depends on the relative motions of the portions of the mass near the upper and lower extremities
respectively.

14, Again, if our imaginary surface be longitudinal, and coincide with the direction of
motion of the particles through which it passes, it is manifest that the greater motion of the
central parts will cause an action of the particles on one side of the surface, on those on the
opposite side of it, and in directions tangential to it. This force will depend on the tendency
of the one set of particles to move faster than the other, and will evidently be greatest in the
direction in which that tendency is greatest, i.e. in the direction of the motion. If it be
sufficiently great the cohesion will be destroyed. There will be no tendency to produce open
fissures in the case we are considering, on account of the lateral compression to which the mass
is assumed to be subjected, but there will be a tendency to produce longitudinal surfaces of
discontinuity. To investigate the effects of the internal forces thus called into action, let the
following diagram represent a portion of a glacier bounded by the transverse sections 4d’ and
BB’, originally plane, but brought into the positions there represented by the relative motions
of the center and sides of the mass. Let ab, ed, &c. be any longitudinal surfaces along which
the tangential forces are called into action, and therefore in the direction of motion; and for the
greater simplicity suppose every thing approximately symmetrical with respect to the axis OO’.
Also let w, w.......W,......W, be the weights of these longitudinal portions into which the mass
is thus divided; V, Vo......V,......V, the velocities with which they would respectively move, in-
dependently of the action of adjoining portions on each other; they may be supposed to diminish
from the center to the side; v, v»......U,......U, their actual velocities; f, fo...---fueee--fo the tan=
gential longitudinal forces of contiguous portions on each other.

Now if W denote the weight of a mass of ice moving down an inclined plane, of which
the inclination = a, in the manner described in my experiments, the moving force of gravity
along the plane will be Wsina. Let a retarding force (f) be applied to the mass, and let
the velocity of descent then = v. Then, if V be the velocity when f does not act, we shall
have, by the second observed law of motion in such cases (Art. 2),

vo Wesina-f
V 6hWWsina ’°

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 63

and therefore,

The central portion abb’a' in the preceding diagram will be retarded by the force 2/,,
and therefore, since its weight =2w,, we shall have

f (: =) w, sin
= = = a
1 V; 1 >

a being the inclination of the bed of the glacier. The second portion, of which the weight = w,
will be retarded by f, —jf,, and, therefore,

f-fi=(1-¢) W, sin a.

2

We shall therefore have the following series of equations :

Bs) :
—-—],sIna
V, i

I, -o=(1

Ot Mr. HOPKINS, ON THE MOTION OF GLACIERS,

h—hi - (1-2) w, sin a,

1

Vn+1 5
Sasi -fi = (: — Fy | Mati SIN a,

Pray

Peet ees — eer vantessessesensves

v i
fie fe (1 - 2) w, sina,

Suppose @ to be the greatest value of the tangential forces (f) which one portion of the
mass, on account of its nature and structure, is capable of exerting on a contiguous por-
tion, without destroying their cohesion, and the one sliding past the other. The above equations
shew that f, f,, &c. are in ascending order of magnitude. Let f, = @; then will f, fi....--fi-r
be less than @, which will also be the limiting value of fis, fizoe-+++-foeeeee+) and will be their

actual values if we suppose @ to be the same for every two contiguous portions. In this case,
we shall have

and _ therefore

eee eee ee tree tees

Hence if ed, e'd' represent the boundaries of the mn‘ portions on each side of the axis, the por-
tions between these lines and the boundaries, 4B, A’B’ respectively of the glacier will move with
the same velocities as if they were not affected on the one hand by the lateral action of the central
portion, and on the other by that of the side of the containing valley, assuming this latter action
also = @. If each longitudinal portion of the mass were perfectly rigid, the central portion edd’c’
would remain unbroken (since f, f,.-.f,-1 are less, than @), and would move with a common velocity,
sliding past the adjoining portions, as these portions would again slide past those contiguous to
them. The central part would thus be brought into the position represented in the diagram
by the dotted lines at its upper and lower boundaries; but if the mass have some degree of
plasticity (as is doubtless the case with ice), it will be brought into the position defined by the dark
transverse lines ; for any such portion as cf will be acted on by a tangential longitudinal force
on one side in the direction ed, and on the other by an equal force in the direction fe; and these_
forces, while they counteract each other with respect to the progressive motion, will ¢wist the mass
from the form of a rectangular into that of an oblique-angled parallelopiped. The forces f, f., &e.
willin like manner twist the component portions of the central mass edd’c’ in a degree propor-
tional to their intensities, and therefore in the least degree those parts nearest the axis; so that
the central parts of ce’ and dd’ will have little curvature. A small additional motion, however, will
thus be given to the middle of the central portion, but with the degree of plasticity here supposed,
it may be considered as much less than that due to the sliding of one portion past another.

If @ be considerably smaller near the sides than at points more remote from them, the
width ec’ will be large, and there will be little variation in the velocity of the glacier except
at points near its edges, as stated by Professor Forbes to be generally the case.

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 65

15. If we add together the two sides respectively of the first » of the above equations,
we have
Vv, . vO, .
fa= (1 - ) W, SIN G@ + woeeee + (: - =, w,, sina.
But v, =v, =...... =v, nearly; and if we suppose 7, V,...... V,, not to differ much from each
other (as will probably be the case in most glaciers), we may substitute for each a mean value
V. Then we have

C= fr— (1 = *) (w, + Wo + «+20. W,) SID a,

where v is the common velocity of the central portion; or, if
Wy + Wa + cc.cee + W, = W,,

p= (: - 3) W,, sin a.

If the whole mass be very wide, like that of a glacier, and a be equal to the ordinary incli-
nation of a glacier (from 3° to 10° or 12°), and if the retardation V-wv be considerable, @ may
become a force of enormous magnitude. In order that the motion of the mass may be entirely
destroyed, cd must coincide with 4B, and we shall have
gp = Wsina,

where W = weight of the whole mass.

This explains the prodigious power which large glaciers are capable of exerting to overcome
local obstacles to their motion, arising from irregularities along the sides or bottoms of the valley
down which they move.

16. If the tangential action along gh, instead of being equal to @, be equal to @’ less
than @, the portion eh will be accelerated by the difference of the lateral actions, @ and Oe
and similarly for any other portion; but it will be observed that the portion cf, the nearest to the
center of those against which sliding takes place, will be neither accelerated nor retarded by
these lateral actions. Hence, if, in any proposed glacier, the velocity is nearly the same for the
central portion (dd’), but diminishes with considerable rapidity on approaching the sides, we
shall have two points (d, d’) which may be determined approximately, in any transverse section,
at which the velocity will be the same as that of a glacier whose thickness should be the same
as the depth at these points, and in which the conditions at its lower surface should be the
same as for the longitudinal portions through d and d’, but whose motion should be unimpeded
by any lateral obstacles. This conclusion is not unimportant as shewing that the slowness of
glacial motion does not result from lateral or local impediments, but is a necessary consequence
of the action of the bed of the glacier on the lower surface of the mass, as in the experiments
above detailed. It is this unimpeded or mean motion which ought in strictness to be compared
with the motion in these experiments.

17. In the preceding investigations the mass has been supposed to be continuous, but it
is easily seen that similar reasoning will apply if the mass be more or less dislocated. In
such case its cohesion will oppose comparatively little resistance to the formation of transverse
fissures; and the greatest tangential force (p) which can be exerted will be much less than
when the cohesion is continuous. The sliding of one longitudinal portion past another, and the
more rapid motion of the central portions, will thus, as already remarked, be much facilitated.

18. Formation of Crevasses.—It has been shewn (Art. 13), that if the mass of a glacier
were continuous, there would be the greatest tendency to form fissures in directions perpendicular
to those of motion, when the lower extremity of the glacier moves faster than the upper one.
Hence, if the tension becomes sufficient to overcome the cohesion, fissures would be formed in

Vor. VIII. Parr I. I

66 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

these directions, and would therefore be curved with their convexity towards the higher extre-
mity of the glacier, the glacial valley being convergent. The degree of curvature would de-
pend on the convergency of the lines of motion. If the mass be more or less dislocated, there
will still be a prevailing tendency to cause fissures to open in the same direction, though their
formation will necessarily be modified by the pre-existing dislocation. There will be the greatest
tendency to form these transverse fissures, or crevasses, where the change of velocity is most
rapid, or where lateral or other obstacles produce the greatest irregularity of motion. This
accounts for the permanent existence of systems of crevasses in particular localities, as already
noticed (Art. 7). Particular local causes may produce tensions which are not longitudinal, and,
therefore, crevasses which deviate from the general law of formation; but the general transversal
directions of these fissures proves beyond doubt the predominance of a general longitudinal
tension during the period of their formation.

This period, according to our theory, would be the summer, as already shewn (Art. 11).
In the winter, it has been also shewn, the motion must probably tend to produce in general an
internal longitudinal pressure, and therefore to close previously existing fissures. And here it
should be remarked, that it is not essential in order to produce these latter effects, that the
motion of the glacier near its upper extremity should be absolutely greater than that near its
lower end, but that the former of these motions should bear a greater proportion to the latter
during winter, than during summer.

19. In our previous reasoning the glacial valley has been sup-
posed to be convergent in descending. Let us now suppose its width
to increase, and its sides to become divergent below CC’. It is a very B
general law in such valleys, that where the valley expands its descent
becomes less rapid. Assuming such to be the case, the part of the
glacier below CC’ will tend to move more slowly than the part above
that line. Consequently, the former of these portions will be in a state
of longitudinal compression, which will prevent the general forma- c
tion of transverse fissures. Also the pressure along CC’, which will
be greatest in the centre, will push forward the mass below, so as
to make it tend to move along diverging lines of motion. Hence if the
mass remain continuous it will be in a state of transversal tension,
or if the continuity be broken, a system of longitudinal diverging cre-
vasses will be formed. Such systems have been recognized both by
M. Agassiz and Professor Forbes.

20. Passage of a Glacier through a narrow Strait.—Let us suppose the glacial valley to
contract suddenly at BB’ in the following diagram, and consider the motion of the glacier after
it arrives at that section. Conceive the mass divided into different portions by longitudinal planes
of discontinuity, as in the figure. The central portion cdd'c’ represents, as before, that in which
no sliding of one part of it past another takes place, the planes where this relative motion begins
being ed and e’d’. The more the central motion is impeded the greater will be the force f,, (Art. 14),
and the narrower will be the breadth dd’. The motion of the lateral portions will be much impeded
in such a case as that represented, and near to B and B’ may be entirely arrested, but there will be
no action which can destroy the motion of the part edd’c’. The central portion, bounded by the
planes of discontinuity through B and B’, will in fact move very much in the same manner as if
those planes were the immoveable boundaries of the glacial valley.

Unless BB’ be too narrow, therefore, the motion of the glacier will be only retarded and not
destroyed ; but even this retardation may be counteracted by other causes. The effectiveness of
the subglacial currents will be increased by the contraction of the valley, and very generally the
inclination of a valley increases as its width diminishes. These causes may compensate for the

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 67

retarding effects of the lateral action on the flanks of the glacier. The same explanation will
apply whether we suppose the cohesion along the planes of separation to be entirely destroyed or not.
It is only necessary that the tangential action between the central and contiguous portions should
not be sufficient to prevent the former from sliding past the latter.

21. Position of the Surface of a Glacier.—
Let P, Q be two points on the surface of a
glacier situated on the same line of motion.
C a point fixed in space in the vertical line
through Q. Draw Pp parallel to the bed
(AB) of the glacier. If the thickness BP of
the glacier at P remained constant while P
moved to the vertical line QC, P would come
to p, and the thickness of the glacier along AC
would be increased by Qp. Draw PM hori-
zontal, and let MPQ = a, the inclination of the
surface of the glacier, and MPp = 3 = that of
the bed of the glacier. Then if PM =a,

Qp= MQ —- Mp
= a (tan a — tan f3).

68 Mr. HOPKINS. ON THE MOTION OF GLACIERS.

If we suppose an upward expansion of the mass to take place in consequence of the freezing and
consequent expansion of infiltrated water, according to the theory of dilatation, this expansion will
also increase the thickness of the glacier above A. Let ¢ denote this increase for a unit of thick-
ness, while P moves through the horizontal space PM; then will eh be the whole increase, h being
the thickness 4Q of the glacier. On the contrary, the thickness will be diminished by the melting
of the superficial ice during summer, occasioned by external influences, and of the ice in contact
with the bed of the glacier, as the effect of internal heat and subglacial currents. Let A and 6
denote the depressions of the surface below the point C, due to these causes respectively, in the
time (¢) of moving through PM. Then if D denote the whole depression of the surface in the
vertical through C in the same time, we shall have
D=A+6-ech —a(tana — tan f).

Of the quantities involved in this equation D, A, a and a may be easily observed. For this
purpose conceive two vertical poles fixed firmly in the ice at P and Q in the same line of motion,
their upper extremities coinciding as nearly as possible with the mean level of the glacier at the
time. The inclination to the horizon of the line joining them would give the value of a; and the
height to which the poles should project above the surface of the glacier after the time (¢)
would give the value of A for that time. To determine the corresponding value of D, we
might observe the vertical distance of the surface of the glacier from the fixed point C when the
poles should be first fixed, and after the time ¢ of moving through PM, repeat the observation. The
difference between the observed vertical distances below C would give the required value of D.

The only attempts at the independent determination of « have been made, I believe, by observ-
ing the distances at different times of fixed points on the surface of the ice. Such determinations
I consider entirely valueless, on account of the impossibility of separating the effects of dilatation
from those of pressures and tensions depending on other and independent causes. If, however,
instead of horizontal we should make vertical admeasurements, the value of e for a given depth
of ice might, I conceive, be determined with great accuracy. If two short horizontal poles were
firmly fixed in a vertical line in the vertical wall of a crevasse, and an inextensible line or chain
were fixed to the lower one, any variation of the known distance between the two poles might
be ascertained with great accuracy by observations made at the upper one, and thence the
value of « might be accurately determined*. :

Supposing the quantities D, A, a, a and ¢ to be determined, our equation will still contain
three unknown quantities, 8, 6, and h, which cannot be determined by direct observation. I think
it probable, however, that « might be found to be inappreciable, or, at least, extremely small, so that
the term eh might either be neglected or expressed approximately by means of an assumed value
of h. We might also eliminate 6 from the above equation by making one of the observations for the
determination of D as late as possible in the autumn, and the other as early as possible in the follow-
ing spring, since the corresponding value of 6 would doubtless be very small on account of the absence
of subglacial currents during the winter. The value of D in this case would probably indicate
an elevation of the surface. Let this value therefore be denoted by - D. We should thus have

D = ch + a(tana — tan B) — A,
D,+A-eh D,+A
—__—_———_ = tan a — ———_
a a
nearly, the value of ef for the winter being small enough to be neglected. If tan 8 were thus deter-
mined, the value of 6 corresponding to any observed values of D and A, would be given by our
previous equation.

Also if 8 were known we should have immediately the difference of thickness at P and Q;
for this difference = Qp =a (tana — tan #3).

or, tan 3 = tana —

* It appears singular that those who insist so much on glacial dilatation should never have subjected their views to this simple test.

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 69

The determination of 3 would afford an obvious means of approximating to the thickness of
the glacier at any proposed point. For suppose 3 determined for all those different portions of
the glacier where a difference of inclination of the upper surface might indicate a corresponding
difference in that of the lower one. Let the length of the successive portions, beginning at the
lower extremity, be a, a,....-.@,3 and let a ap...... a,> Pi B2------ 3, be the corresponding
values of a and 3. Then if A, be the vertical thickness at the lower extremity, and /h the re-
quired thickness at a distance =a, + a,+......a@, from that extremity, we shall have

h=h, + a, (tan a, — tan B,) + ...... + a, (tan a, — tan f3,).

The chief practical difficulty in the application of this formula would be in the determina-
of 2, B., &c. with sufficient accuracy. It appears not improbable, however, that the limits of
error in determining B by the formula above given for tan 3, would be such as to render the deter-
mination a sure approximation to the real value; and, at all events, if it were found impracticable
to determine all the quantities 3, B,...... 3,, and therefore the complete thickness of the glacier,
such of them as should correspond to the more accessible and least irregular parts of the glacier,
might probably be determined with considerable accuracy, and thus the rate of increase of thick-
ness in these parts would be known.

SECTION V.

Internal Temperature of a Glacier.

22. In a previous section I have given the general reasoning by which we conclude that
the temperature at the lower surface of a glacier of considerable thickness cannot be higher than
zero of the centigrade thermometer. Since this conclusion, however, is of the first importance in
the theory which has now been offered of glacial motion, I shall give the mathematical investigation
of the problem. The case taken for direct investigation will be that of a large sphere, like the
earth, of which the temperature increases as we descend, coated with an external shell of ice, the
temperature of the shell being at every point below zero (cent.), that the ice may in every part
remain perfectly solid. We shall thus be able to deduce the limiting thickness of the icy crust
compatible with this condition of perfect solidity. If the thickness exceed this limit, then must
its lower surface be in a state of constant disintegration, as already explained (Art. 4).

We have no exact knowledge of the conductive power of ice, but there is no reason to doubt
its being very small. I shall suppose it (for the greater simplicity of investigation) to be the same
as that of the earthy matter supposed to constitute the nucleus of the sphere; and for the
same reason I shall also suppose the conductive power from the nucleus to the icy envelope to
be the same as in the interior of the nucleus, or in that of the icy crust. I shall also assume the

external temperature to be represented by V + C cos (2rt+2). So long as this is less than

‘

zero, the problem will present no peculiarity arising from the circumstance of the exterior crust
being composed of ice; but however much the external temperature may exceed zero, the superficial
temperature of the crust cannot, from the nature of ice, rise higher than zero. Hence while
the external temperature is below zero, we shall have the ordinary case of a solid body placed
in a medium of which the temperature varies according to a given law; but when the external
temperature rises above zero, the condition at the surface will be that the superficial temperature
of the mass shall be constantly at zero. Instead of this last condition, however, we may suppose
that, during the time it would hold, the ewtermal temperature shall be zero; for it is manifest that
the two condtions will in the case we are contemplating be very approximately the same. Hence,
then, the case for investigation will be that of a sphere of large dimensions cooling in a medium

70 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

Tv . . : .
of which the temperature is V + C cos (2 at +7) when this quantity is negative, and zero for
those values of ¢ which render the expression positive. If V =0 the first of these conditions will

1 3 1
be satisfied from t=0 to fage from ¢=1 to ioe &ec.; and the second from t=. to #=1,

oO

from {=~ to t=2, &c. If V do not =0, the former of these periods will be shortened and

2
the latter lengthened, or the converse, according as V is positive or negative ; if, however, V be
small compared with C, the periods will be approximately as above stated, and such, therefore,
we shall consider them. They will be semi-annual, if we take one year as the unit of time.

The theorems given by Poisson, in his Théorie de la Chaleur, Articles 194, 195 ‘and 196, will
enable us to obtain the required solution. e

23. If the external temperature be represented by the general formula

B+ Acos (mt +e) + A, cos (m,t + €,) + Ap (cos mat + €&) + &.......+-. (1),

and w denote that part of the internal temperature which depends on the external, we shall have,
at the depth « beneath the surface,

b =a
u=B+—dAe 2V'F cos Ce = aaah
D \ a 2
! “8V Fos ( Ls my 3)

+—A,e mt + 6 — — —
D, 1 1 €) 5 2

TRC i astietrecane

Where D cos ene Dsing-La/™,
a 2

b/2m m
Gea
a

a

. 2).

and therefore D=8+

with similar formule connecting D,m,0,, D.mz62, &c. Also a? = — 5 where & represents the
c

conductivity and ¢ the specific heat of the matter constituting the globe; and 6 is a quantity
depending on the conductivity and radiating power of the surface.

Now generally if ’ (#) denote any function whatever, continuous or discontinuous, whose values
recur whenever ¢ is increased by @, so that p (¢ + 0) = P(#), we have the general formule

() OG
o = fo @)at

,

UC) /
+ ah ¢ (t') cosa dt .cos = ct = ["¢ (#’) sin oat at «sin =
+ =f ee) cos a dt. cos ==" + = ol) sin =" at A sin
+ &e.
which will coincide with (1) when the following equations are satisfied,
Qar Aq 67
ey? 1 ai ig eh aay i &e.

=5 [pyar

Mr. HOPKINS, ON THE MOTION OF GLACIERS.

2 Qart’ 2 Qrt
A cose = af ot) cos = dt', A sine = 5 oO) sin a dt’,

2 ’ Aart ; 2 ether Bk 25
A, e038 «, = 5 J." (t) €08 = dt, Aysine = 5 f(t) sin dt,

2 2 1) rt! Q
A, cose, = 7 fc p(t) ry ad dt, A,sine, = =f op (#) sin

tee ee CE &@ie—s &e:

In the application of these formulae to the case before us we have

B= [i p(t)at+ [o@yat

2(n +1) rt
)

ll
—
Ker
NS
=f
Q
°
ro)
D
or
wo
FS]
oe
+
ro |
(pete
——
a
BR

Ve
B09 tars.
3 que as Ph sir
A cos ¢ = 2 2 [' {7+ Coos (ent +2) heos.2nt dt
0 2
Gg,
=2 ‘| Vcos 2rt' + —cos (ane + 7) dt
) 2 2
=i)
A sine = 2 25 74 C cos (2xe'+ Z)} sin ene a¢
24
j : a CYT nol Ge edi -
=2f [ Vsinane +$ {sin (4x2 +7) — sin aa dt
_ ey Cc
Beat ais
Hence, oe “aw we.
2 Tv 2

Also taking the general term,

Acoso, =2 [17+ C cos (20¢' + 5) cos 2(m + 1) rt'dt’
0 "

4 , Cc f f a fs
= 2f'{V cos 2 (n+ 1) wt ee [ cos (2@ +2) a0 +2) + cos (2m -) |}
L 2
= 0 when m is even;

aif
ot

. ———— when m is odd.
n(n + 2)

d,, sin e, = afr C cos (2 at’ + =) | sn 2(m + 1) rt’ dt’

=2 ({rsin eq +1) rt’ + 2 [sin (2 (n + 2) rt! + 5) + sin (2nne-7) |b ae

~ dé’,

71

72 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

Vaan A

= — —.—— when 7 1s even;
ar n+ti

=0 when 7 is odd.

Consequently when m is even

7 | Fares
==, 4,=-—-—;
2 7r7n+il
and when 7 is odd,
iC 1
co, A, =—.——_...
* : "3 n(n+2)

Hence, substituting in (2) we have

V Va me =
, c b (~-S)e ave could = — i/o —2)
= 2 2 a

u@=—--—+=

b Cc 2a a
tg ia 7s cos (4 mt — —\/2 4 — 6)
Oe UG = Tr 8 7
pe ee t+— --V/37-6.
5. ae" cos (6 + ae aN 1 — 6)

+ &e.

24. If vw denote the temperature which would exist at any point within the sphere at the
depth w beneath its surface, if the external temperature were always equal zero, we have
(« being small compared with the radius of the sphere)

V=%+ 9%
where v, is the superficial temperature of the sphere, and y the rate at which the temperature
depending on the original heat of the sphere, increases with the depth.

Let m denote the temperature of the sphere at the depth #, as depending both on the
original heat, and the variable external temperature; then

U=V+U,
or
vc Bye Prema mae
BS 32 oe (a = = )ie ge enn.(2 — a)

3 K+ m tye t 5 (= 5) cos ( mite V0 6)
b C ne a @
a ane ant ——\/27-58
Die Beem. eee eee

the complete expression for the temperature required.

25. I am not aware of any experiments for the determination of a and 6 for ice. Poisson
has given their values for the case of the earth, deduced from observations made at Paris on the
annual variations of temperature at different depths. They are

a@=5,11655 ),
in metres.
b = 1,05719
He also gives
Vo = 0°,0265 Kenderal
centigrade).
ry = 0°,0281 grade)

Substituting the above values in the expressions for D, D,, &c. we obtain

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 73

b

ag 37 nearly
p 63

ye ek ee
&e. = &c

A year is taken as the unit of time.

26. In the preceding investigation the sphere has been supposed to have a complete shell
of ice. The result will also be sensibly the same if, instead of the whole surface of the sphere
being covered with ice, a small portion only of it be so covered, provided the thickness of the
ice be small compared with its superficial dimensions. This is the actual case of a glacier, to
which therefore equation (3) will be approximately applicable. Let us proceed then to the inter-
pretation of that equation.

We observe that when #a=a few multiples of a, the value of the periodical terms becomes
insensible, on account of the exponential involved in them. Let x, be the least value of w for which
we may neglect these terms. Then, if w, be the temperature at that depth,

Vs
Mar eatit sep Carh Ya
a
TT eee bes: coc ccccccsccccs (4),

neglecting the small quantity v,. Consequently the temperature at a certain depth is independent
ooh yy : ; :
of annual variations, and lower by — — — than it would be if the exterior shell were composed of
T “

rock instead of ice; for, in that case the value of B (Art. 23) would be the mean external tem-

Cc
perature V, instead of ———.
2 Tw
If a, be the depth for which the temperature = 0, we shall have
V
0O'= a = SS Sp Y V2»
iG eV
m=-~(=-3), Pota(ete tet s atelaletete (5)
y\r 2

which, if we give to y the value above stated (Art. 25), will be the numerical value of a, in
metres.

If «w, be less than the thickness of the glacier, the formula (3), and therefore (5), will be no
longer applicable; for (3) would give the temperature of the ice at depths greater than x,, higher
than zero, which from the nature of ice is impossible. In such cases the lower surface of the
ice, at whatever depth it might be, would be necessarily at zero, because the heat which, if the
superficial crust were not ice, would elevate its temperature, will be employed in melting the ice
at its lower surface, which will thus be kept at the zero temperature.

With the value of y above given, equation (5) gives the value of x, supposing the ratio of
the conductive power of ice to its specific heat to be the same as for the rocky crust of the earth.
If this be not the case, the equation (5) will still give the depth at which the temperature = zero,
by assigning the proper value to ry as depending on the ratio just mentioned for ice.

As a numerical example, suppose V = 0, and C= 15° (cent.) | We shall have at the depth w

0

u, = — 5° nearly ;

5
3028
178 feet nearly.

Vou. VIII. Panr I. K

and a, = feet,

74 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

27. The temperature — 5° (cent.) appears, however, to be much lower than that observed
at different depths by M. Agassiz, and which did not exceed half a degree. The difference may,
I conceive, be easily accounted for. In our investigation the surface of the glacier has been sup-
posed to be exposed to the winter temperature, whereas, as soon probably as the mean temperature
of the twenty-four hours descends to zero, the surface is protected from the external cold by a
coating of snow, which increases as the temperature diminishes, and thus it is probable that the
temperature of the surface of the ice* may descend but little below zero during the whole winter.
If we suppose its lowest temperature to be about — 1°,5 (cent.) we shall have «, = — 0°,5, and w, = 54
feet nearly. If the conductive power of ice be less than that of common rock, the value of 2,
will be proportionally less.

Taking this last value of #,, it follows that if the thickness of a glacier should exceed 50
or 60 feet +, the temperature of its lower surface would necessarily be zero, as already explained.
Now the thickness of glaciers is doubtless much greater in general than 50 or 60 feet{, and
therefore we conclude, that generally the temperature of the lower surface of a glacier cannot
be less than zero, and must, consequently, be in a state of constant disintegration, unless the
conductive power of glacial ice be much greater than that of the ordinary matter forming the
crust of the globe.

28. From the conclusion of the last article it appears, that if we would investigate accurately
the internal temperature of a glacier of considerable thickness, we must take, besides the condition
given by the superficial temperature, the additional one that the temperature at the lower surface
shall always =zero. In this case, however, the resulting expression for the temperature would
become so complicated, that it would be useless, I think, to give it, especially with the uncer-
tainty which exists respecting the superficial temperature of the ice during winter. The conclu-
sion above enunciated, which is not invalidated by this uncertainty, is all that is requisite for the
theory of glacial motion which has now been offered.

W. HOPKINS.

* It appears to be established, that the snow which falls on all | thickness was estimated roughly at about 150 feet, that there might
but the higher regions of a glacier is again dissolved in the spring | be no doubt of its being an extreme value. The thickness of 50 or
or early summer, and does not contribute to any permanent increase | 60 feet as deduced above, is probably much nearer the truth,
of the glacier. + See Note, Art. 4.

+ As a deduction from the general reasoning of Art. 4, this :

CAMBRIDGE,
May 1, 1843.

VII. On the Theory of Determinants. By A. Cayuery, Esq. Fellow of Trinity
College.

[Read Feb. 20, 1843.]

Tue following Memoir is composed of two separate investigations, each of them having
a general reference to the Theory of Determinants, but otherwise perfectly unconnected. The
name of ‘ Determinants” or ‘‘ Resultants” has been given, as is well known, to the functions
which equated to zero express the result of the elimination of any number of variables from
as many linear equations, without constant terms. But the same functions occur in the re-
solution of a system of linear equations, in the general problem of elimination between algebraic
equations, and particular cases of them in algebraic geometry, in the theory of numbers, and,
in short, in almost every part of mathematics. They have accordingly been a subject of very
considerable attention with analysts. Occurring, apparently for the first time, in Crenner’s
Introduction a V Analyse des Lignes Conches, 1750. They are afterwards met with in a Memoir
On Elimination, by Bezont, Mémoires de lV’ Académie, 1764. In two Memoirs by Laplace and
Verndermonde in the same collection, 1772. In Bezont’s Theory of Equations, and in Memoirs
by Binet, Jowrnal Polytechnique, Vol. 1x.; by Cauchy, ditto, Vol. x.; by Jacobi, Crelies Journal,
Vol. xx11.; Lebesgue Liowville, Vol. v1. &. The Memoirs of Cauchy and Jacobi contain the
greatest part of their known properties, and may be considered as constituting the general
theory of the subject. In the first part of the present paper, I consider the properties of
certain derivational functions of a quantity U, linear in two separate sets of variables (by the
term ‘ Derivational Function,” I would propose to denote those functions, the nature of which
depends upon the form of the quantity to which they refer, with respect to the variables entering
into it, e.g.the differential coefficient of any quantity, is a derivational function. The theory
of derivational functions is apparently one that would admit of interesting developements.) The
particular functions of this class which are here considered, are closely connected with the
theory of the reciprocal polars of surfaces of the second order, which latter is indeed a par-
ticular case of the theory of these functions.

In the second part, I consider the notation and properties of certain functions resolvable
into a series of determinants, but the nature of which can hardly be explained independently
of the notation.

In the first section I have denoted a determinant, by simply writing down in the form of
a square the different quantities of which it is made up. This is not concise, but it is clearer
than any abridged notation, The ordinary properties of determinants, I have throughout taken
for granted; these may easily be learnt by referring to the Memoirs of Cauchy and Jacobi,
quoted above. It may however be convenient to write down the following fundamental pro-
perty, demonstrated by these authors, and by Binet.

Gyipner p> To = pe +oB.., pa t+oaf..,.. rrr dO))
a’, B p> i pa +oa'B.. : pia i o'B’..

K 2

76 Mr. CAYLEY, ON THE THEORY OF DETERMINANTS.

An equation, particular cases of which are of very frequent occurrence, e.g. in the investi-
gations on the forms of numbers in Gauss’ Disqzuisitiones Arithmetica, in Lagrange’s Determi-
nation of the Elements of a Comit’s Orbit, Sc. I have applied it in the Cambridge Mathe-
matical Journal to Carnot’s problem, of finding the relation between the distances of five points
in space, and to another geometrical problem. With respect to the notation of the second
section, this is so fully explained there, as to render it unnecessary to say any thing further
about it at present.

§ I. On the properties of certain determinants, considered as Derivational Functions.
Consider the function

U=a (aé + Bn +...) +.0.00. qa).
w(aE+ Bint...) +

(n lines, and terms in each line),

And suppose
KU= eviaers. a| | States (2).

(The single letter « being employed instead of KU, in cases where the quantity (KU), rather
than the functional symbol X, is being considered).

FU=- BEES Deets PIB MIRAE Ce «Stee? |i ines e's (3).
~ E+ Syt.e, a ; B aici
RE+s yn +..,5 a 5 Bp >
qU=- RY+R a +.., S@+S8@ +..5.. seven (4):
ab +By t+.-> a F B ‘vad
NEtB yt. a 5 B aes

The symbols K, F, J possess properties which it is the object of this section to investigate.
Let 4, B,.., 4’, B’,..,..be given by the equations :

= BY, y'.- eB = y's 8... eats £0. 0s (5).
[ee yy” me Nd
a at B'; of” as , nee Y"s oC by
ts y” cy. ys Oe.

(The upper or lower signs according as (mz) is odd or even).
These quantities satisfy the double series of equations,
Aa+BGB+..=k,...... (6).
Aa + BB’ +..=0,

Aa+BB+..=0,
ype BB+ coe

&e.

Mr. CAYLEY, ON THE THEORY OF DETERMINANTS. 17

Aa +A a +i. = Ky cecees (7).
AB+ A’ B'+.. = 0,

a FOR as = 0,
BB+B'P'+..=«,

&e.
The second side of each equation being (0), except for the r‘” equation of the 7" set of
equations in the systems.

Let A, w,--- represent the 7, »+1",... of the series a, B,..., L, M,... the corresponding
terms of the series 4, B... , + being any number less than (7), and consider the determinant

Flat BN banaee (8),

NS OW AY)
which may be expressed as a determinant of the mn order, in the form
Amen ee Letom ON Olean eee ees (Q)s

Aone) OO
0 on ig
0 Or Ont

Multiplying this by the two sides of the equation

K= Fe eal» cee ber (10),
he

and reducing the result by the equation (© ), and the equations (6), the second side becomes

RUC ee ore he ese ee (11),
Ok

xk O 0

0 re) yp”

0 ph td, prt)

which is equivalent to
kK" pw, Ose seevee (12).
pet, pero

Or we have the equation
FF eeceae & = 77) Bet ena le weaes + (13);

Aa ERY
which in the particular case of r=m, becomes

Aa Biel
A’, B’ at Kis a gases, (14)>

78 Mr. CAYLEY, ON THE THEORY OF DETERMINANTS.

which latter equation is given by M. Cauchy in the Memoirs already quoted; the proof in the
« Ewercises,” being nearly the same with the above one of the more general equation (13). The
equation (13) itself has been demonstrated by Jacobi, somewhat less directly. Consider now the

function FU, given by the equation (3). This may be expanded in the form
FU = (r&é + syn+--) [A.(av4 aa’ +..) + B(pvt wa’ +...) +..] 4+
(WE + s'n+.-) [4'. (awd a’a’+..) + B (pot v'a'+..)+..] +

which may be written
FU=a.(AE + Bunt...) + eevee (16).
aw. (A'E + Bly +...) +

By putting
A=a.(pd4+RA'+..)+B.(RB+ BB +..) ...... (17).
B =a.(sd+s/4'+..) +3. (sB + 8'B'+..)

A’=a'.(n4 4+ 2A’ +..) + BY. (RB + v'B’+..)
B =a’.(s4+84'+..) +3. (sB+s'B’+..)

Hence,
KFU= ‘A eeeei(l8) i:
IA’; B’
= A, B. aps se bee tan Ne Basen (19).
AGB" R, 8’ A’, Bl

Or observing the equation (14), and writing

re oe =) geedss (20).

ase!
Ey 8s - =P precageal Cid b

Be s’

This becomes

KEU =a S Lis yan” 1ss.cee (22):
Whence likewise

KPO =J.0 1(RO Ry iba 505\(28)-
Consider next the equation

IFU =- no+Ra'+.., sut+sa'+.. ee CL
Ag+3Byn+.., A » B
ANE +Bnt+.., A’ ; Bi
=- 1 1 fo ope Biers. (29):
AB ers R » A, B
( ey A , » RB’

Mr. CAYLEY, ON THE THEORY OF DETERMINANTS. 79

=-Jf ak eenteee (20)5
2 AB
a’ A’ B’

If the two sides of this equation are multiplied by the two sides of the equation (2),
written under the form

k= UP yet. Ol Beet ess (27)
-ap
.a B’
The second side is reduced to
—JSf 7 ab + Bn.. ’ aE + B'n.. seband \€43))6
x K :
a’ . K

== JE. («)*s%5 UO. <2... (29).
And hence
EMU O Hid HRA ED EE OR, wceocoo (30).
And similarly
PT Of — Pile (MU) eee se wesc (31).
Also combining these with the equations (22), (23),
LM HO SOLERO © LOS
KFU KIU KU

It may be remarked here that if U, V are functions connected by the equation
y M/ q

HU =OR V8 Ont. Ui = Cd Vid ccsccs (33).
Then in general
U= ort, VPs sce (34).
To prove this, observing that the first of the equations (33) may be written
FUP. (o=. V)y nik (35),
we have
© FU =F .F. (O73. V)y | saeces (36),
or
JL .(RUY"?.U = IL LK (1. VY? Ve cesses (87).

Or, if neither J, £ nor (KU) vanish, this equation is of the form

i= Ta, see oe (88).
whence substituting in (33),

1 NER CENCE (39).
which demonstrates the equation (34); and this equation might be proved in like manner from
the second of the equations (33). If however, J=0, or £=0, the above proof fails, and if
KU =0, the proof also fails, unless at the same line »=2. In all these cases probably, cer-
tainly in the case of KU =0, n +2, the equation (34) is not a necessary consequence of (33) ;
In fact, FU, or TU may be given, and yet U remain indeterminate.

80

Mr. CAYLEY, ON THE THEORY OF DETERMINANTS,
Let U|, a, B

y+ 4, B,, &e... be analogous to U, a, B..., 4, B, &c... and consider
the equation

K.(KU,FU +g.KU.FU,)

| nKAt+gKd, «B+ecB,..
| n,A'+gxA’, «B+ gx B,

Multiply the two sides by the two sides of the equation (2), the second side becomes, after
reduction,

kxt+gen-(Aat+BB+--),

PkaCAia BAG es)\os |||. seseee (41).
gx. (4a+BB+..), ««+gn.(A/a'+ B/B'+..) |
Multiplying by the two sides of the analogous equation
K,= Dy CHAE Soe dobaosnecacad (263)
(evry [dior |
and reducing, the second side becomes
ck,-(@,+8a), Kk,-(8, + gB)-.- | sesieien (43).
ck, (a) +ga), xk,-(B/+8P)
SY Holo 6 (COP aay Bf) Ae Gop (44).
whence
K.(KU,. FU + gKUFU,) = (KU)"*.(KU,)"".K(U,+gU). «2... (45).
and similarly
K.(KUAU + gKUTU,) = (KU)"'.(KU,)"".K.(U,+¢U). ...... (46).
Tn a similar manner is the following equation to be demonstrated,
4 .(KU,FU+ gKUFU,) =F (KUWU+ gKUTU,)= ...... (47).
ls CH Rae (CHOU) \eenise

av+ta/a.., Ba+B/a’..
a&+Bn.. a, + ga

» B, +8P
@E+Bn.. a) +ga 5 B/ +8f
Suppose ra
U=Z(pE+ont+...)(av+a a+...) ..00.. (48).
This expression being the abbreviation of
U=(p& +079 +..) a t+aa'+..)+ ...... (49).
(pétont+--)(aa+a/a'+..) 4

+
[( —1) lines, or a smaller number].

=0, peewee (50),

which follows from the equation (©).

Conversely, whenever KU = 0, U is of the above form.

Mr. CAYLEY, ON THE THEORY OF DETERMINANTS. 81

Also
Figs = Aga we) YBa Be ea. aerresen (Ol)
RE + Sy Zap ‘ Dac
nE+s'y Za'p ; Lac |
which may be transformed into
FU= Ag+a'a’...; BUY Ba’... RE+8y..., VE+8 7... , tee 4 (52),
j Pp ’ o a ‘ a
F 5

p OF sien Bey fa ‘ats ree

(for shortness, I omit the demonstration of this equation).

And similarly,

I= Wan iR Oates SH 4 Si@e soe AE+By.., AE+B'.., | coeeee (53),
P p) o a 3 a
, ,

p ; o b : b’

where it is obvious that if the sum = contain fewer than (7 —1) terms, FU=0, TU =0.

The equations (52), (53) express the theorem, that whenever reo = 0, the functions FU; LU
are each of them the product of two determinants.

If next ei!
U,=U+0.
Taking g=—1, in the formule
K.(K(U+U) FU — KUF (U+0)) = K.(K (U+U) TU - KUT (U+0)) ...... (56).

= (KU). (K(U4+U))*-!. KU.
Or observing the equation (50),
K .(K (U+U) FU — KU. F(U+0)) = K.(K(U+U) TU — KUT (U+U)) = 0. ...... 57.
Hence F.(K(U+U) ZU —- KUT (U+U)) = 7. (K(U+U) FU - KUF(U+U)) are each
of them the product of two determinants. But this result admits of a further reduction. We have
F.(K (U+U) TU — KUT (U+U)) = 7. (K(U+U) FU — KU. F(U+0)) ..... (58)
=- Jf (KU)*. (K (U+0))"* av+a/a.., Bw+B/a'..

Substituting a,=a+Zpa, &e... also observing that if the second line be multiplied by a,
the third by a’,.. and the sum subtracted from the first line, the value of the determinant is
not altered, and that the effect of this is simply to change a,, a/.. into a, a’.. in the first line,

and introduce into the corner place a quantity — U, which in the expansion of the determinant
is multiplied by zero. This may be written in the form

— Jf (KU)? (K (U + U))*"* BEA Hie sg GOP Be sr | senses (59),
ak + By Spa > Zea

dé 4 B'n pa’ i Sad’

Vou. VIII. Parr I. AD;

82 Mr. CAYLEY, ON THE THEORY OF DETERMINANTS.

which may be reduced to

Jf. (KUYPAKR (C+D) ee. (60),
av+da'+.., But+Pa'+.. af+Bn.., @E+B'n.... |
Pp > o a ; a

If each of these determinants were multiplied by the quantity (KU)"~', expressed under the
two forms ‘

ye ae ee ea a (61).
oe ee R, B'

They would become respectively

KU v : av Seales ga te
Ap+Bo ; A'p + Bo’

E , he ob 5s (62).
Aa+Aa.., Ba+Ba'..

So that finally

F.(K(U+ U)TU -KU.4.(U+ 0)) =7.(K(U+ 0) FU-KU.F(U+ 0)) =
Dh = 2s a, wv me {hare 2 : n rhe | w+. (63).

KU Ap+Bo.., A'p+B'o..,; us AGA i BORE GE 25
' | :
The second side of which may be written under the forms
K(U+ De
(“a

vor at
AV+A®.. ’ BLr+B@..

SBE
A.(4p+Bo..)+4'.(A'p+Bo..).., B.(Ap+ Bo..)+3.(4'p+B’o..) ..

| RE+Sn.. ; RE+S'n.. yor «+» (64).
R.(Aa+4'a'..) + S.(Ba+Ba..).., R'.(Aa+ A'a’..) + S’.(Ba+ Ba’..) |

And
moO" Ro+R'e'.. : Sv4S'a'
(a) R.(4p+Bo..)+R'.(A'pt+Rc..).., S.(Apt+Bo..)+S'.(A'p+B'o..)..;

.

KU

AE+Byn.. 5 AE + B'n.. shee +o (65).
A,(Aa+Aa’..)+B.(Ba+Ba..).., A. (Aa+ A'’..) + B’.(Bat+ Ba ..)

And again, by the equations (52) (53), in the new forms
TT)\ 2-2
(age) Side # Bo. (AE + Bye] (4a + Bios MEA em Bad «send
[(Aa+ d’a’..) (ra +r'a'..) + (Ba+ Ba’..)(sw+s'a'..)..it — ... (66),
K(U+U)\"
(“a) A=.§[(Ap+ Bo..) (RE +s7..) + (4’p' + Blo...) (WE + 8’ +..)..] x
[((Aa+ A’'a’..) (awt a’o’..) +(Ba+ Ba’..) (prsn'a’..).. J}. ... (67).

Comparing these latter forms with the two equivalent quantities forming the first side of (53),
and observing (33), (34). It would appear at first sight that

Mr. CAYLEY, ON THE THEORY OF DETERMINANTS. 83

K(U+U).TU-KU.9 (U+U) =
—2

pei eee

{=[dp + Bo..) (AE + By) + (A'p + Blao..) (VE + B'y..)..] x
[(4a+ A a’..) (px + r'e’..) + (Ba+ Ba’..) (sv+s‘a'..)..]}
K(0+0)FU-KU.F(U+U)=
_— n—-2
(SO 3 fp + Bo..) (RE +s8n..)+(4p4+Bo..) WE+5..) +..]
x[(da+ A'd’..) (av+ a’o’..) + (Ba+ Ba’..) (sx + w'a’..).. ]f,
which however are not true, except for 2=2, on account of the equation (57). In the case
of m =2, these equations become
me Uy tf — KU. (0 +0) =
[((Ap + Bo..) (aE + Bn...) + (A'p+ Bo...) (VE + 3y..) +--+] x
[(4a+ Aq ..)(nw+Ba’..) + (Ba+ Ba’..) (svt Sivoo es oo le xcaaces (68),
K(U+U)FU-KU.F(U+U0)=
[(Ap+ Bo..) (nE+sy..)+ (Ap + Bo..) (WE +8-.) +--] x
[(4a+ 4 a'..) (aa+ a’o'..) + (Ba+ Ba’..) (pvt ve't+..)+..] vee (69),
and it is remarkable that these equations ((68), (69)) are true whatever be the value of (m),
provided = contains a single term only. ‘The demonstration of this theorem is somewhat tedious,
but it may perhaps be as well to give it at full length. It is obvious that the equation (69)
alone need be proved, (68) following immediately when this is done.
I premise by noticing the following general property of determinants. The function
a+ Zpa, B+ 0a, Scone (WO,
a’ + pa’, B+ So'a,

(where Spa =p, + p.dz--. + psa), contains no term whose dimension in the quantities a, a’... ,
or in the other quantities p, o..., is higher than s. (Of course if the order of the determi-
nant be less than s or equal to it, this number becomes the limit of the dimension of any term
in a, a’...or p, o..., and the theorem is useless). This is easily proved by means of a well
known theorem,

Zpa, Doa..
Zp'a, So'a

whenever (s) is less than the number expressing the order of the determinant. Hence in the
formula (70), if = contain a single term only, the first side of the equation is linear inp, c...,
and also in a, a’..., i.e. it consists of a term independent of all these quantities, and a second
term linear in the products pa, pa’... ca, ca’.... This is therefore the form of K(U+U)...
Consider the several equations
k= KU = Aa+ BB +.. ove (72).
=A d+ BB+ +

It is easy to deduce
k,=K.(U+U0)=KU+ Apa + Boa + seoos. (73).
+ A'pa' + Bod +

L2

84 Mr. CAYLEY, ON THE THEORY OF DETERMINANTS.

To find the values of 4, B, &c... Corresponding to U + U, we must. write

A= MBN sy + cease. (74),
= mB + Ny” in
where
M’= Ys 6” ° 3 N’= =5 | Ng. e’ enscee (75),
mw yn | Sie
} : |
oe 28 "5 SZ AF ‘ pe one ae ae

wu gue Sue we
Re =

The order of each of these determinants being n—2, and the upper or lower signs being
used according as nm —1 is odd or even, i.e. as ” is even or odd. Hence
4 =A+ Mca’ + Nira + wwe. (76).
M”.¢@ +N’ .7a@'+..
And _ therefore
x4 —«A,= A*pat+ AB.ca+AC.ra+ vena (77).
AA'pa' + (AB'— xu')oa'+ (AC'- cn’) ra’ +..
AA" pa" + (AB —«M")oa" + (AC"- KN") ra" +
The additional quantities C, + having been introduced for greater clearness. Now the
equations
AB —«\M' =.4'B, AC’ -KN' = AC,.. «2.2. (78),
AB"-«M"= A"B, AC"-«N"= A'C,

written under the form
ABa AB =0MiG)) AG’ rtd Gracin @ us -+-daoves (79),
AB WRE UM, ACHE AVCE KN”

are particular cases of the equation (13), and are therefore identically true. Hence, substituting
in (77),
«,A—«A,= A’pa+ ABoa+ ACTa...+ = sevaee (80),
AA'pa' + A'Boa' + A’Cra’... +
A” Apa’ + A” Boa’ + A" Cra’... +

=(pA4+oB+...)(da+ dA'a +...).

Forming in a similar manner, the combinations «,B—«B,...«,d’-—«4/, «,B- x B/,..., mul-
tiplying by the products of the different quantities da + d’a’..., Bu+ B’a’...,...RE+Sn+...,
R'E + S’n,... and adding so as to form the function K(U+U).FU- KU.F(U + U), we
obtain the required formula, viz. that the value of this quantity is

[((pA+oB+..)(RE+sy+..)+ (Apt Bo...) (WEF 8'n..)+-.]% eee (81),

[(Aa+ A‘a’..) (aw+a’e’..) + (Ba+ Ba’..) (pa + v'a' +...) + -]J
with this theorem, I conclude the present section,—noticing only, as a problem worthy of in-

vestigation, the discovery of the forms of the second sides of the equations (68), (69), in the
case of = containing more than a single term.

Mr. CAYLEY, ON THE THEORY OF DETERMINANTS. 85

j Il. On the notation and properties of certain functions resolvable into a series of deter-
minants.

Let the letters

Ohip eee ro. Aacpeoc (4),
represent a permutation of the numbers
De Oh eaty lor mateo soi (2)

Then if in the series (1), if one of the letters succeeds mediately or immediately a letter
representing a higher number than its own, then for each line this happens there is said to be
a ‘“‘derangement” or “inversion” in the series (1). It is to be remarked that if any letter
succeed (s) letters representing higher numbers, this is reckoned for the same number (s) of
inversions.

Suppose next the symbol

emarsostes (3),
denotes the sign + or —, according as the number of inversions in the series (1) is even
or odd.

This being premised, consider the symbol

{"P an, waninee (4),
Pk TK-*
denoting the sum of all the different terms of the form
+, +,..Ap,,, Gs,9-»Ap,, Ts9 ++ coveee (5).
The letters
ihe. Ubieco Up Be SG EAnoc G6 Rieoacesac (6),

denoting any permutations whatever, the same or different, of the series of numbers (2). The
number of terms represented by the symbol (5) is evidently

(CUSEA EBT Oem Gonoten (7).

In some cases it will be necessary to leave a certain number of the vertical rows fi) Gisc
unpermuted. This will be represented by writing the mark (+) immediately above the rows in
question. So that for instance

ei
ea Adi - 4 sooose (8).
Pron + OnDy
The number of rows with the (+) being (w), denotes the sum of the
Gn2R2 k)iat Eas. (9)5

terms, of the form
+, +,.. Ap,,, Gs, see Ors —i ee eee Ap,, os, eee 6,5 Pi: © eseces (10).
This is obvious, that if all the rows have the mark (+) the notation (8) denotes a single

product only, and if the mank (+) be placed over all the rows but (1), the notation (8) be-
longs to a determinant. It is obvious also that we may write

{: -0,--0,.. mA = +, +,..|Apio,- ee rei) |e eesre (11),

PRT K+ Dy PAT K* + Pu, Po,

where = refers to the different permutations,

Uy Ug ert les A Orel Ugieea Uy 5, SECM aid odeee vse cae, G12)

86 Mr. CAYLEY, ON THE THEORY OF DETERMINANTS.

which can be formed out of the numbers (2). The equation (11) would still be true, if the
mark (+) were placed over any number of the columns p, c....

Suppose in this equation a single column only is left without the mark (+) on the second

side of the equation; the first side is then expressed as the sum of a number

(1.250. k)is eon generally (1 2s EPs Sot.U(1S)5
of determinants, according as we consider the symbol (4) or the more general one (8). And
this may be done in (7) or n—«a different ways respectively.

It may be remarked, that the symbol (8) is the same in form as if a single column only
had the mark (+) over it; the number (m) being at the same time reduced from (m) to (m — # + 1).
For, the marked columns of symbols may be replaced by a single marked column of new symbols.
Hence, without loss of generality, the theorems which follow may be stated with reference to a single
marked column only.

Suppose the letters

Pis Press PK Tis Toeee ORS Mice Abe. . (14)
denote certain permutations of
Gy Ap .-- ays (eh B» spores REL. bes (15),
in such a manner that
Pi= Gos p2=Gg.2° Praag)s o1=Br> o2 = Bys++ on = By5 5a) orOO? (16).
Then the two following theorems may be proved:
t +
Fe pe = shies Peay. scans: (17).
a; By

PRO
If () be even, but in the contrary case

eS =+4,.. i: ‘Biee | Fe (18).

PKOR ay B:

By means of these, and the equation (11), a fundamental property of the symbol (3) may
be demonstrated. We have

i“ Seas =24,. i By. ee (19),
ie cel te pes S(degigeD) eevee. (20)
cn By
a, By
oP Baye:

ay Bx PxPR
which when (m) is even, reduced itself by (17) to
ay By
uy
Bere es
But when (m) is odd, from the equation (18),
+
(* ae = a ae met (21).
Since the number of negative and positive values of +, are equal.

Mr. CAYLEY, ON THE THEORY OF DETERMINANTS. 87

From the equation (20), it follows that when (7) is even, the values of a symbol of the
form

ae y See (22),
a, Py

is the same, over whichever of the columns a, .. the mark (+) is placed. To denote this
indifference, the preceding quantity is better represented by

ce B, am 3.2638 (23),
a, By

this last form being never employed when (7) is odd, in which case the same property does
not hold. Hence also an ordinary determinant is represented by

a +
foe : res
aks By kk

the latter form being obviously equally general with the former one.

neseee (ELD

It is obvious from the equations (17), (18), that the expression (22) vanishes, in the case
of (7) even whenever any two of the symbols (a) are equivalent, or any two of the symbols
(8), &c.; but if (7) be odd, this property holds for the symbols (8), &c., but not for the
marked ones (a). In fact, the interchange of the two equal symbols, in each case, changes
the sign of the expression (22), but they evidently leave it unaltered, i. e. the quantity in
question must be zero.

Consider now the symbol

Giaes aoe (25),

eke
which, for shortness, may be denoted by
;
SACK 22p . e.-25, 6):

I proceed to prove a theorem, which may be expressed as follows :

t t
{4.k.2p}.{B.k.2q} = [AB] k.2p 429-2)... (27),
where

A eee Ss Si Aer r Sale net seecee (28).

The number of the symbols r,s... being obviously 2p—1, and that of w,y...2q—-1. The
summatory sign S' refers to 7, and denotes the sum of several terms corresponding to values of
1 from J=1 to l=k. Also the theorem would be equally true if Z had been placed in any
position whatever in the series r, s ...; and again, in any position whatever in the series a, y ... d,
instead of at the end of each of these. With a very slight modification this may be made to
suit the case of an odd number instead of one of the numbers 2p, 2q; (in fact, it is only to
place the mark (+) in { AB|..% over the column corresponding to the marked column in {4..},
{A4..} being the one for which the number is odd), but it is inapplicable where the two numbers
are odd. Consider the second side of (27). This may be expanded in the form

ha asta ph bec 2 Ato pe ge oe A CAR tok (ees (29),

where = refers to the different quantities s,.., w, y,.. as in (11).

2. 8g++XgYg++ *

88 Mr. CAYLEY, ON THE THEORY OF DETERMINANTS.

Substituting from (28), this becomes
=. Si, aie Si, a (+ #,.. +, +, AleneT 3 Aly co rb Lee pet aA Benya) POOLE (30).

Effecting the summation with respect to , y.- this becomes
t
=. S;,-. 9,22 + es ei Salis a Sxtie (31),

kk.. ly
S now referring to s,...only. The quantity under the sign = vanishes if any two of the
quantities 7 are equal, and in the contrary case, we have

t t
ie 1 “fy =+£,{B.k.2q}, ...... (82);

kk.. ly
which reduces the above to
t
$B. 2qh. Dt Hee. H,Ay gy Abie cg2 vee (33),

= referring to the quantities s..., and also to the quantities 7. And this is evidently equivalent
to

fd tkvepyh Blk. 2g}) \t).. (34),

the theorem to be proved. It is obvious that when p=1, q=1, the equation (27), coincides
with the theorem (©), quoted in the introduction to this paper.

VIII. On Small Finite Oscillations. By the Rev. H. Houivitcn, Fellow of Caius
College, and of the Cambridge Philosophical Society.

[Read May 15, 1843.]

THE system of bodies here considered is supposed to be such, that their position, and the forces
acting upon them in that position, depend upon a single variable; and the object is to find general
expressions, which may be applied to any particular case, without performing any integration, for
the length of the isochronous pendulum and the time of oscillation, rocking or sliding, when the
body or system of bodies is slightly disturbed from its position of equilibrium, the approximation
including the square of the variation of the independent variable.

By the principle of vis viva,

mv? +mvi +... =2m [Pdp + am, f P\dp, + ...
Let w be the independent variable, and a its value when the system is in equilibrium, and a + x its
value at the end of the time ¢; also let 8 be the value of x at the beginning of the motion, when
the system is disturbed and left to the action of the forces upon it.

Pdp
Let —— = =
chats SOS (aa),
Pi dp,
- “du 7 PM=G(+8),
indie eG: MEO Hrsg Loe ly (1),
du

. U,=mf (a) +m (a) + ---
U, = mf, (a) + m gd; (a) + ...
U, = mf, (a) + m 2 (a) + «.

2

aod WePdp + miPdpe+.. S (w+ e+ 0, + ...)dw a Shed brah a try (2);

but, when the system is in equilibrium, w~ = a and U, =0, or mf(a) + m, p(a) + ... = 0, which
determines its position when at rest, and as dw = dz, the integration of (2) will give

2? — 2 23 — 33 z! — 3

B + U,. ie + U;. B

2 2.3 2.3.4

m [ Pdp + Mm, [Pidp, SPB oe ey

} d
Again, let = = (uw) = (a +2),

ds,

da 8M) =E(a +2),

md s* Oe. + wee
and V = ee esaigeaiesives sasiectivasipearss CS)

Vor. VIII. Parr I. M

90 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS.

2

. mds? + mds, +... = (%+ Vis + ye + 1) dat

°\ dx Sp er Bee
and (% + Vi.2+ V,.—) 55 =U (#* - B) + Or. 3 + U;. ahd + cccccevece (4).

; dx° . 2 2
For a first approximation, Vo. FF Vee U, . (x* — B°), from which it appears that = vanishes when

; d
2 =+ 3, and since s= Iv (w)du = R (uw) = R(a + B) = By (a) + R, (a). B, = vanishes, when
s = R,(a) + R,(a)-; which shews that each body of the system vibrates to an equal distance on

each side of its position of equilibrium.
— 2
The time of oscillation = 7 “V/ oe TT. A me ey Sie GO <
mf, (a) +m, p; (a) + «.-

1
And L the length of the pendulum = — g. ee lait mn’ {a)" ten,
m f, (a) + m, p, (a) +...

- d
In the case of gravity, Pdp = —gdy, and f(w) = - =,

mds? + mds, +...
= Wa Seis da cdelevccccans Ceccvccceece 5
md-y + m,d°y, + ... (6),

and ..

the position of equilibrium being determined from mdy + m,dy,+... = 0.
If the body be rigid, and X, Y the co-ordinates of its centre of gravity, and Mk? the moment
of inertia about the centre, and @ the angle of rotation be made the independent variable; then

ds ds; dX*+dY*
eT oe LT a a AT. a
d X*
= Mk* —
+ MR >
dX*
2
¢ i de
and «, L= ayo SEEDODC DOSED COLO CORCO DEI (6)
de

k ad ‘ : ; _. mas + mids te

When a rigid body oscillates about a point of suspension, the expression ay sane
2 2
mem , the point of suspension being made the OTIPIN: .0.500.00sr2e0e (7)-
The equation (4) for the purpose of integration may be more conveniently put under the form _
dz
SF (p+ qzetrs) =a. (8 — 8°) +b.(B-) +¢.(Bi-#')
= (B -%). {(a + be + ex* + eB’). + B) + OR.
Assume (a+ bx + cx* + ef"). (z + B) + bB’ = (a+ bx tex + cB’) .(z +B + 5B’ + eP’2)-
w. (a + bz + cx + eB"). (0 + ex) = 5,
or, a0 + bdx + aex = 6, omitting the squares of B and zx;
-. a0=6, and bd +ae=0,

becomes L =

Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 91

be
> and e= — —

or, é= 2?

alo

b &
and therefore (a + bx + ex*° + cB’). (2+ B+—B -—=P*s),
a a

or (a+ bx + ex? + oft). (1 +2) .{e+e.(1-2-8)}

differs only from the factor (a + bs + cs* +c’) .(z+ 8) +5" by quantities of the fourth de-
gree of 3 and sz: or

Ga (p+ 4x4 ret) =(B-s). (1428) {a+ (1 ~ = .p)} (a+ be + es! +08")

is true to the fourth order of those quantities; and the limits of the oscillation of the system are,

B+e. (1-2.8) =(+9. (1-2-8), and

dz* i 5
o qe (P+ 4% +73") = (B-2)-(y+2)- ( -=,-B’) (a+ bx + ex* + ef),
and if a+cB’=a
dz 1 prqxtrs

—dt= :
V(B-2)-(y +2). vB a+6z +¢2z*
nes

The position of equilibrium must be a stable one, and therefore [mPdp + [m,P,dp, + vee

a maximum, or U, is negative, and .. a= a+ c(3? = — U, + cf is positive, and .-. also e
is positive.
Expanding therefore the last term of the above expression,
/P
~ dt= 5 ie (£2) (FE 2-2) I.)
J/ (B-2)-(y +2) v/s ve 2p 2a 2p 8p 2a 4pa 8a°/ J
a

to be integrated between x = 8 and x =—vy, excluding the powers of above the second.

2" ds

For this purpose, taking iReGh Ee » let (8B — #). (y+) =(B-2).2,

‘i pa al? | > and + en Laid = aoe 3

1+.2* V/(B-3).(y +2) l+a
"dx (Ba? -— vy)". 2da 2dau B+y\

ea eA 4 by 6m
‘s J (B= 8). Cy +2) (14+ a*y""" 1 + a (4 al

M2

92 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS.

Z wa Ye EE aa (Baty)
=. {8 ae ‘Trai’ ot Wydare = ee}

= +a
the other terms vanishing as m is not greater than 2.
The limits of = being 6 and —+y, those of w are o and 0, and
2n — 3).(2n — aa
pee between these limits = off bets ly TEs Ee
(1 + a*)” (2n —2).(2n —4)...4.2
a"dz { Bry n—-1 3
sep Aa a aN Pg EG ERE Sp coi
| Geos 2 : PEP ae

7
2°

= - 7m. {ft —nB".(B+ > .B)+n."—*. spy

--19- (2-1). 0-9 -98)

x
Hence Esa TT:
if dx mneioce
JVB-#)- Gyre) 24”
f ds ny Pp
V@-)-G+) 2
Pp

rag VE (8). +%)- VEL Ba) 9]

p : 3 ® qb 156"
PAE (fF 8- , J},

t=7 ae 2 2
a p @ 4p 2pa Aa

Vo
= NV Tet CB) ser eesseseeesnstetsniee pabayih Ciel weve (9)s

where C A (F Ess z YA MU; 2 =)
OANA ROR a ieee Vite 6 Gay L
eV
and L = cs v7," oa eee eee ee een eoeneseereer serves reeessees (10),

where L and # are expressed by quantities and their differentials.

The times of descent to the position of equilibrium and of ascent from it, will be found
by the integration of equation (8) between the limits, x = (3,0), and. x= (0, —-y); but as the
first powers of the are will appear, it will be sufficient to integrate

Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS.

dz

i JEL 3) aS
Time of descent = /P

q
pP a

3

but tan7? Jt = tan-? ai
*. time of descent = /2. fz + (4
a

2 2p
eetaistoaterae ascent = J? fe (4
a |2 2p

ars

Sai seas)

— 2tan7 w+ ch.

EA(t-9).8 ame hee}
eeeee ascent = s/t. fo tan? ieaceckal

b
a

Excess of time of ascent over the time of descent, or

ea Vy
Aon eg 2), g/l

a@X\a p B J U,
which is remarkable as not involving zr.

The excess of the are or angle of ascent = y — 3

2U, V,
om = 7) Gusts sates (13),
be U2 @
Pete o snpobe (14).

93

These results are on the supposition that the displacement of the system was by an increase
of the independent variable; in the opposite case, the odd differential coefficients of V and the

even ones of U must have their signs changed.

Example. Two bodies m and m,, moving in a circle and connected by a rod subtending
an angle 4a at the centre are acted upon by a repulsive force in the circumference, varying as

the n' power of the distance.

Let 26 be the angle at the centre between the radii passing through §' the centre of force

and m, .. 20+20,=4a,
P=k.(asin@), and
ae Pdp + P,dp, U

do °°” k.(@ay*
ee mds? + mds‘
de
If the bodies are equal, V = 8 ma’,
U, = 8ma’k.(2asina)""'.

= 4a" (m +m).

and L= &

p=2asing;

=m sin" @ cos @ — m, sin” 0, cos 0,

(7 cos* a — sin’ a),

k.(2asina)"~'. (sin*a — m cos’ a)’

and t=7 a) E Aaah ee In OO 2n-2 J};

256

where Aa is the whole angle of oscillation.

sin’ a nm cos? a — sin’ a

94 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS
xpressions may be put under more convenient forms: for

When the body is rigid, the general e
if the differential coefficients be taken with respect to the angle of rotation round the centre of
gravity, X and Y being its co-ordinates, and Mk* the moment of inertia round the centre,
V =Mi? + M(X>+ Y}),
V,=2M(X,X.+ MY),
V, =2M (X24 .X,X;,+ ¥2+ 1%).
And U =— MY,,

U, = — MY,
U, = — MY3,
U,= — MY,;
V,=M(e+X)), Vi =2MX,X,, and V,=2M(X;+X,4;+ Fi);
ke + X?
= Lee Ae ANE eon wa eeceaepietaeees eos e ra (15),
one ket Mik + FF xe) 19) AK}
4(k* +X?) 16Y, 4 te a (06)
XXY, AG aie Gate :

” 4Y, (+X) * 48° ¥?"

In the case of a particle, L =
Xx, Ne 1%, D.AE Pe
and C= 7x abe ORS ge ye OD:
Ewample. A rod oscillates upon two planes, inclined at the angles a and a, to the horizon;
he distances a and a, from the extremities of the rod.
Here X = A sin @ + B cos8,
Y = Msin@ + N cos0,

where @ is the inclination of the rod to the horizon, and
a sin a COs a, — @ CoSa sina,

the centre of gravity being at t

Pe Ee s
sin (a + a) 4 r sin (a + a) :
wet cos a sin a, — a, sina cos ay DM otitis (a+ a, -sina sina,
sin (a + a) sin (a + a)
«. ¥,=Mcos@—N sind =0,
M N

i andes 2 = —
/ M* + N? / M* + N®

a+a,).(asin’a, + a sin’a
(a+a,) - ( 1+% ieee:

Let M?+ N?= =
sin’ (a + a)

and dM RS (4+). Mae 2a, — a, sin 2a) Pp;
2 sin’ (a + a)

then =e
/Q

aa, |

aaah and p= Tn+ er

X&, =

al

Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 95

iE A Q-a’a’ 1 KPQ
and J =a Vz Lf +B. eras oan oe eae ale

If the planes include a right angle, and the centre of gravity be in the middle of the rod,

2

L=a+—,

a
B B
eet ES -)
Tv = (: + 16)?
both of which are independent of the inclination of the planes to the horizon.

If a particle move in a curve, by a constant force (g), making a given angle d@ with the
axis of x; then,

dp cos bda + sin pdy ; ds
U = = . = . — Sn
Fau * du SD 2) du
ds*
= aa?

where @ is the angle made by the normal with the axis of w; and making this the variable,
V = R®, where R is the radius of curvature ;
V; =2RR,,

V, =2R° +2RR,,
U=g.sin(d — 6). R,
U, = -—gcos(p — 6). R+g.sin(p - 6). R,,

U,=-—g.sin(p — 0). R - 2g cos (p — 9). +g. sin(p — A). R.,
U; = g cos (p — 0). R —3g sin(p — 0). R, — 3g cos (Pp — 0). R2 + g.sin(p —4).R;
In the position of equilibrium U=0, or p-6=0;
. U, =—-gRh,
U,=—2gh,,
U,=-gR-3gR..
Hence Baa R, and

‘Ta 1

R WIN. iy Hae ® eyes }
rar JE fis ae. (5 + Be ome) proce saan)?
Excess of time of ascent = 16,
3/2R
2R,.A0
Excess of angle of ascent = ae

If the are be made the independent variable,

E 1 R, RY )}
wi an J es ieee 19).
rr raft fisae (sat tak ash? (19)
2Rh,:As
E f time = - — ;
xcess O ime 3\/eR

Excess of are= = —————_.

96 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS.

This last result compared with the former, shews that an increase of the angle of vibration is
attended with a diminution of the arc, and vice versa.
3 b°
Ewample. In an ellipse, R = a seeing)”

3be _, A
.sin@.cos@.(1 —e sin’ @)-2,

3b’? nee ‘ : oe 2
Re = . (1 —é sin? @)-5. (cos’ @ + e* sin’ @ — sin®@ + 4° sin’ @ . cos* @),
a

and by substituting these values in (18),
R AG R\3
= —.<{1 ~-[/4—38.[— é
Tet Jt { “ 16 [ (=) }}
If the ellipse become a circle, R® = ab,

and ren JZ. (1+ +o).

If the axis of a cycloid be inclined at @ to the vertical,

R= 2acos0,
R, = — 2asin0@,
R, = — 2acos@.
L =2acos0,
ee ee a a.
g 12
Increase of angle of ascent = — ? tan. AG.

The time of oscillation in a cycloid therefore decreases, as the are increases, when the axis is not
vertical.

If a central foree kf (6), varying according to any law act on a particle in a given curve,
the co-ordinates of the centre of force being a, (3; then, taking the arc for the independent variable,

U=kf(s) > ann ee

Tet, es: wa £2). o(s);

*. U =k.o(s).%,=0, at the position of equilibrium,
=k. p(%). x,
U, =k. p(2) 23,
U; = 8k, (%). 23 + kp(®) . 3
but x =(w -a)?+(y— 8’);

d
= 2.(w — a) = +2.(y¥—- 8). o , and if @ be the angle made by the normal with the axis

da

of a, —=sin@ and =cos@;

&

d
ds
#,=2.(w-a.sinO +2. (y — B).cos8,

Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 97

® = 242. }(@ —a). cos — (y - B).sin@)} 5°

=2+42.{(w-a).cos@- (y—/).sin0}.R-’,
85= —2.$(w -a).sin@ + (y— 8). cos0{. R-* — 2. §{(w— a) .cos@ — (y —f).sin@).R~*. R,,
2, = —-2R*-2.$(#-a). cos — (y — B).sinO}.(R-° + R-*. R, - 2R“R))
omitting the terms which vanish; but since U = 0, (y — 8). cos@ + (a—a).sin@ =0,
and therefore (w — a).cos@ — (y — 3).sin@ = — 6.

&. 6

Hence Fogle
By R,
27 Re
x, o-R 6. (RR, - 2R})
A Te R'

s

Also V= =1, and V,=V,=0;
ieee ata Fe Fava ot oe adhere Soni el 3 (20).

-U, kf(d).(0-R)

: 2
mth se Se [2 - = - 3a,.d.-log (2) |}

3B_ Bo
iy eae =) 38 ~ B) LO ee
ee Ax. VE. pes 16 = letE ee ans Ry * RS OO a

If the force vary as the distance, d;.log ae = 0, and the force does not affect the cor-

rection of the expression for the time.

The ess of the time of ascent at, of. As
excess of the eo =- in =
3 °(8— RE. /KRFO)
ads - ey As
eee eee ewe eee eee eee wer eeeeee ALU eeseee Sah 72 Rk uot 3
26—3R Ae
. angle eos ty ia 3 .

If the force be constant and act in parallel lines, 6 is infinite, and f(6) constant, and formula
(21) becomes the same as (19).

(1). To find the time of oscillation of a particle in the centre
of a hollow sphere, the force varying as the n‘™ power of the dis-
tance, and the density being = 4.1”.

Let QOR=¢ and QCP = 6; then the volume of a particle
at Q=r'drsin@.d@.dq@, and its force on the particle at P,
where CP =a and QP=p, is

wer *dr.sinO.d0.dp.p";

ey og iene =n. rdr. sind. d0.dp pr? Seer

Vou. VIII. Parr I. N

98 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS.

=p.7"**dr.sin@.d@.dp.p"'.(w—7.cos@) + ,.. since p?= 17° +.a° — 2ra cos.

Let k=pr"**dr.sin 0d@.dq;

». U=kp"-'.(# —cos@) +...
U,=k.(m—1).p"*.(@— 7. cos0)? + kp""" 4+...

=u.r"*"*'drsin@.d@.d@ }(m —1).cos*@+1}, when a =0;
which, being integrated from 6=0 to = 7; from P=0 to P=27, and from r=7, to r=7%,

we have finally
U; & Amr. (n + 2) é Cosi oe rate )s
3.(m+n +2)

7 3a .(m+n +2)
and T = = a m+n+2 m+n+2 3
4u.(m+2).(73 -7 )

or if M be the mass of the hollow sphere,
Pan, «f__3.(mtn 42). GE =4) ,
(m +2). (m+ 3). M. (rettt? = petty

If the force varies as the distance, 7’ =e
VM

If the force varies inversely as the square, 7' is infinite.

If m+n+2=0, =——_+__ = log (=) ;
m+n+2

3. (n ~ 1). log =
And if 8=0, T= ——_
slik. Setlieanntes SCe ys mer a

(2). To find the correction for the time of oscillation, we have
U;=k.(m-1).(m—38).(m— 5). p""7. (w—1. cos 6) + 6h. (m — 1). (nm — 8). p"~>. (w — r cos 6)"
+ 3k. (n—1). p"-3,

or, making w= 0, there results for the attracting particle
Uz =. (m—1).7"*"~'dr.sin@, d0.d@[(m — 3). (nm — 5). cos'@ + 6. (m — 3) . cos’6 + 8],
which, integrated between @ = (0, 7), is 2u. aa,
and again between b= (0, 27) and r=(r,, 72) is for the hollow sphere
« 4u.a.n.(m —1).(n + 2) ymin _ ymeny,

U, ;
‘ 5. (m+n) z
Reet wi _3.(m +n +2). (727° — r**) : 3B n.(n—1).(m+n+2) ry —ri™
} (10 +2).(1m +8). MM. (rete t? — pm tnt2y* po 89 ° m+n Reeser 5 |S

If the sphere be solid and density uniform, and the force inversely as the square of the dis-

ance, J’ = on :
4p
If r equal forces 2k6. p(6'), be placed in the angles of a regular polygon, the time of oscilla-

on of a particle at the centre will be found to be

Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 99

eee tS ane Oe oe)
—2kr. P(@) + p:(a*). a*} >

no pa) + @ pi (a)

and as kr is the whole quantity of attracting matter, the time is the same while kr is, and
therefore if the matter be in the form of a ring, the above is still the time of oscillation.
n—1

If the force = k. 6", then p(e) i. » and

2

a ar/2 ———
/-rk.(n+1).a— 640° ’

On Rocking Bodies.

In the position of equilibrium, the centre G of the rocking
body, will be in the same vertical line as the point of support ;
that is, when 4 is at A,, AG will be vertical.

Let AG=a, AN=y, NP=z2a,
4N,=y,, N,P=%,,

and PO being a common normal,

let AOP = 0, and 4,0,P = 0,3;
then @ the angle rocked through = @ — 6;
and if X, Y be the co-ordinates of G, measured from 4,,
X=a,-—x.cosp+ (a-—y).sing

Y=y,+a.sno+(a Bee

i dy da
i aye aa x 3
also, sin@ re cos @ da and ds = ds,;

dX d: d
: Bee Ne tcc tea nituapy
dp

d d
= c05 0 « 75 — e058 cosh FE + ©. sin gh ~ sin Osing . T= + (a—y).conp

.sin b+ (a—y).cos

=a@.singd+(a—-y).cosd,
adY dy, dea dy

ae ip Lage Be OP ake ek ae
; d , d F
= sin 6, . ia + 008 8. sin. + 0.05 — sin 8. cos . 7 — (ay) sing
= vcosp — (a —y).sin p;
d
or — = vu]
ae ro (23)
dos
To find X, Y and their differential coefficients with respect to @; we have, from (22)
X =0,
Y = &%

N2

100 — Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS.

From (23), “= Y=a,
Y, = 0,
dy, dy,
=~ Gg
y. da, = da, i:
| oe dp 1 dp )
d*y, day, d*y,
Fa se ae Tap ag
dx, da, dy,
1 age agi * ag’
da, da, da, dy,
Y= ag ag apt" * ag
d it will b ssary to express dey ii
e —
and it will be nec y Pp dp’ dg
let R and » be their radii of curvature at P;
ds ds
R = a6, > and f= °T) 3
d*s d’s
let Ri = ag ry; = ae” &e.
d d
then ed sin 0, eek
dp dp
d*y, dé, ds, d’8,
age a aa a@ Oa?
? d
ani = cos 0,. Rey F
d do
d*x, d@, ds, 1
ig > (Sa apa oe
Ga, dé; ds, d°0, ds, ‘ dQ, d*s,
ioe cal peel Pa ae, 2 ieee
ag? cos 0, dg de in 6, ge dp sin 6, dg dg
Now dp = 40 - d0, = d9-d0.>;
jae) ox
: dp A eas
dé, we A?
dp R-r

Paes EAS Oe ia) y Nes.

dr ,aR dr a0, dR dd,

a ay epee) a pe aaa ee
ds "ag ag” “a0 ag)’ hE ae
aia CF) canis (R=)

+ cos @,.

—, &c. in terms derivable from the separate curves ;

d's,

Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 101

Rr, — 7 R,

Gir)
do dé, d0, do
3 pat ge 9 ett Fam Ln sai "he Wh
@s war ree WEEE a
dg (R-1) aes FET
d0, do
3 =~ ee Bind
ie Oe rR). (BGG a9)
G7),
_Rin-R |, (R?r, — 7? R,)?
(Gary 7 9)

If then, a be the angle, the common normal makes with the vertical in the position of equili-
brium; since s =s,, there results,

dy, Ge Rr
dp id hoor
d’y ie Rr, -rR
7 aes (merece To
da, nye Rr
Ppa saad ae
Pan, Rr, — PR,
dg pean 0) ala ek (R—71)
Ee cos ie sina. Rr gated 2sina.r cs ae
= a. a a
dd (R - 7)? (#7): Ci):
Rr, — r' R, (R?r, -— 7°? R,)?
peek ary: ape a + eet as

which values being substituted above, we have finally

A,=G@, XA,=—sina.

-r
Rr
Y, =0, Fasc as -a,
R -2r i Rr, — 7 R
eae” Op pea nt “RonF”
; R-2r Rr, — 7 R,
Mes URE Sc py OP Sn Cee pe
F R-2r a
Y,=a-cosa Ro + Rr’ cosa (R pp ie Ran’
+ 3r.sin B= tity co! ee tet Ee, 8 cos ad Nees NY
RR See ney LO, (R- ry
ss ke 2
Length of pendulum = == dnesebmecn oe) s
cosa -a@

102 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS.
T ‘ é ; nares
and T=7 —.(1+ C.A¢’), where C is to be determined by the substitution of X, Y, &e.
&
in (16).
If the pendulum be suspended by a point, 7 and its differential coefficients vanish, and

1
X,=-X,=-¥Y,=Y,=a, and... C= 16°

kr :
L=a+-—, measuring a downwards, and
a

penVE. (1+42).

1 Rr’ .(R -2r) R'r, — rR,
io mig BO = Pen (2)
Ot 16 16(R-r).§{Rr-(R-+r).at 16(R-717)'. {Rr-(R-71).a} 2)
—3(R’r, — 7° R,)’ R’r.(r - a) 5 (Rr, — 7° R,)?

16(R-7)'. {Rr—(R—r)-a) 4(R—-7)'.( +a) | 8° (R71). {Rr (R—7) ah?
If R and x be constant, or the curves be circles,

Ge Rr. (1% — acos a) 1 cos a. (27° R — 7° R*) ; @ sin’ « Rey
-a(R-7.(# +a) * 16 16(R-7). {Rreosa—(R—r).a} 4(R—r). (e+ ay
asin’ a. R*r’. (R - 27) 5 sin? a. (R?r — 2r°R)?

+4 @ioph [Rram= (ho fee Pals 6 Rat)! inea RG) a ee
If # and r be constant, and also a =03

Gs 1 R’r. (ry — a) Rr’. (2r — R)
“16 *a(R-7). (+0) | 16(R—r). [Rr -(R—1).a} clalaly pisleetateieie (27).

Ex. One sphere within another,

L=1.(R-1)
Taer/E. (1+ Paes).

Ex, If an ellipse whose semiaxis a is horizontal rock within another ellipse whose semiaxis
a, is also horizontal,

(2 +B). (a2b — aby)

by fe Sea Ne a,
aa, — a,b? + a°bb,
ab? a es
east ot 5? SE OE
C= a‘aie < 1 r a‘azbb, \ (a? aa a’)
L 4 (a:b — a°b,)®. (K+ 8) © 16° 16(a2b — a®b,)?. (bea? — a°bb, —a°a’)’
3a‘as ate?b, a ateb

16 * (a,b — a*b,)°. (a,b? - abd, — aaj)”
If.the bowl becomes a plane,
k? + 6°) .b
rie (H* + 6°)

ae een oR.
a — 6?

Mr, HOLDITCH, ON SMALL FINITE OSCILLATIONS. 103

tanVE. frag. (; Be & ata a}:

4b? , (24+ b) 7 16 b
If a body be suspended by an axis whose radius is r on a circular support, whose radius is R ;
and a, be the distance of the centre of gravity below the point of support,

+e
io “i '

cosa.

= tp

and, if the pendulum be suspended on another axis, the radius of which is 7,, and be isochro-
nous with respect to these axes;

a, — a
Rr, r q
cosa. — cosa.- +@-a
R-+1r, -
a; — a?
and if the axes are equal, L = =a+4a,
a, — a
Rr cosa
and kh? = (a + a) R — + GQ,

and therefore if Kater’s pendulum be supported on a concave or convex surface, the length is
independent of the curvature of the surface.

R
If A= == , and it rests on the first axis,

parE. t 6. (+ A A’ . (7 +a) _ ar lh

16 4rL. (A +a) (4 + a)
which is not independent a a, unless R is infinite, and therefore 4 = 7, in which case,

pan VEL (5+ az)}:

On Sliding Bodies.

When a body oscillates by sliding contact on a horizontal plane, X and its differential
coefficients vanish, and by (15),

L Tee, 5 Jie
T= JE. 1 2 (3 aos —. F) J
Ks g { abe 4k? 16Y, ° 48 Ye
The equation (2) becomes Y = asin @ + (a4 —y).cos@, since 0, =0, and y,=0, and .. p=0@;

nA = ©. cos (a ~y) «sind + sin. — cosa <4

vn. cos @—(a—y).sin@,

— wsin@ — (a—y).cos8 + cos@. ge + sin@.

dy
Hs do de’

d
Pe) a ee ee

104 Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS.

. Fy=-Y+%p

Y¥,=—-Fe+735
and taking the limits, Y= a,
YA 05

7 .
Y,=7r-—a,
YL

¥,="5

Y,=a-7r4+%,

T r-a (r-a)? 1 r, 5 r
and T'=—.4/ —. 1406. ( —+—= 2 y A :
6 2 { ak * 16 16.(r—a) \ 48 (r—a)¥)
Example. An ellipse with its axis major horizontal.
res et ah? a 8a' ee?
~ +a sin’. 2! 6 ae
2
=) +1 O + e+ + ceeeee 3
kb

= ae

4b! 168°

The same principles may also be applied, with great facility, to the oscillations of floating bodies.

4p A =
andere ee afi + ae. (55 <—-)}.

H. HOLDITCH.

ee ee

IX. On some. Cases of Fluid Motion. By G. G. Sroxes, B.A, Fellow of
Pembroke College.

[Read May 29, 1843.]

THe equations of Hydrostatics are founded on the principles that the mutual action of two adja-
cent elements of a fluid is normal to the surface which separates them, and that the pressure is equal
in all directions. The latter of these is a necessary consequence of the former, as has been shewn
by Mr. Airy*. An exactly similar proof may be employed in Hydrodynamics, by which it may
be shewn that, if the mutual action of two adjacent elements of a fluid in motion is normal to their
common surface, the pressure must be equal in all directions, in order that the accelerating force
which acts on the centre of gravity of an element may not become infinite, when we suppose the
dimensions of the element indefinitely diminished. In Hydrostatics, the accurate agreement of the
results of our calculations with experiments, (those phenomena which depend on capillary attraction
being excepted), fully justifies our fundamental assumption. The same assumption is made in
Hydrodynamics, and from it are deduced the fundamental equations of fluid motion. But the
verification of our fundamental law in the case of a fluid at rest, does not at all prove it to be
true in the case of a fluid in motion, except in the very limited case of a fluid moving as if it were
solid. Thus, oil is sufficiently fluid to obey the laws of fluid equilibrium, (at least to a great extent),
yet no one would suppose that oil in motion ought to be considered a perfect fluid. It would
appear from the following consideration, that the fluidity of water and other such fluids is not
quite perfect. When a mass of water contained in a vessel of the forin of a solid of revolution is
stirred round, and then left to itself, it presently comes to rest. This, no doubt, is owing to the
friction against the sides of the vessel. But if the fluidity of water were perfect, it does not
appear how the retardation due to this friction could be transmitted through the mass. It would
appear that in that case a thin film of fluid close to the sides of the vessel would remain at rest, the
remaining part of the fluid being unaffected by it. And in this respect, that part of Poisson’s
solution of the problem of an oscillating sphere, which relates to friction, appears to me in some
degree unsatisfactory. A term enters into the equation of motion of the sphere depending on the
friction of the fluid on the sphere, while no such term enters into the equations of motion of the
fluid, to express the equal and opposite friction of the sphere on the fluid. In fact, as long as we
regard the fluidity of the fluid as perfect, no such term can enter, The only way by which to
estimate the extent to which the imperfect fluidity of fluids may modify the laws of their motion,
without making any hypothesis as to the molecular constitution of fluids, appears to be, to calculate
according to the hypothesis of perfect fluidity some cases of fluid motion, which are of such a
nature as to be capable of being accurately compared with experiment. The cases of that nature
which have hitherto been calculated, are by no means numerous. My object in the present paper
which I have the honour to lay before the Society, has been partly to calculate some such cases
which may be useful in determining how far we are justified in regarding fluids as perfectly fluid,
and partly to give examples of the methods by which the solution of problems depending on partial
differential equations may be effected.

In the first seven articles, I have mentioned and explained some general principles, which are
afterwards applied. Some of these are not new, but it was convenient to state them for the sake
of reference. Others are I believe new, at least in their developement. In the remaining articles, I
have given different problems, of which I have succeeded in obtaining the solutions. As the pro-

* See also Professor Miller's Hydrostatics, page 2.
Vote .Yiil,; Pant I. oO

106 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

blem to be solved is usually stated at the head of each article, I shall here only mention some
of the results. Asa particular case of the problem given in Art. 8, I find that, when a cylinder
oscillates in an infinitely extended fluid, the effect of the inertia of the fluid is to increase the mass of
the cylinder by that of the fluid displaced. In part of Art. 9, I find that when a ball pendulum
oscillates in a concentric spherical envelope, the effect of the inertia of the fluid is to increase the
b+ 2a°
2(b'- a‘)
b that of the envelope. Poisson, in his solution of the problem of the sphere, arrives at the strange
result that the envelope does not at all retard the oscillating sphere. I have pointed out the errone-
ous step by which he was led to this conclusion, which I am clearly called upon to do, in venturing
to differ from so high an authority. Of the different cases of fluid motion which I have given, that
which appears to be capable of the most accurate and varied comparison with experiment, is the
motion of fluid in a rectangular box which is closed on all sides, given in Art. 13. The experiment
consists in comparing the calculated and observed times of oscillation. I find that when the motion
is small, the effect of the fluid on the motion of the box is the same as that of a solid having the
same mass, centre of gravity, and principal axes, but having different moments of inertia, these
moments being given by infinite series, which converge with great rapidity. I have also in Art. 11,
given some cases of progressive motion, deduced on the supposition that the same particles of fluid
remain in contact with the solid, which do not at all agree with experiment.

In almost all the cases given in this paper, the problem of finding the permanent state of tem-
perature in the several solids considered, supposing the surfaces of those solids kept up to constant
temperatures varying from point to point, may be solved by a similar analysis. I find that some of
these cases have been already solved by M. Duhamel in a paper inserted in the 22nd Cahier of the
Journal de V Ecole Polytechnique. The cases alluded to are those of the temperature in a solid
sphere, and in a rectangular parallelopiped. Since, however, the application of the formule in the
two cases of fluid motion and of the permanent state of temperature is different, as well as the
formule themselves to a certain extent, I thought it might be worth while to give them.

mass of the ball by times that of the fluid displaced, @ being the radius of the ball, and

1. The investigations in this paper apply directly to incompressible fluids, as the fluids spoken
of will be supposed to be, unless the contrary is stated. The motions of elastic fluids may in most
cases be divided into two classes, one consisting of those condensations on which sound depends, the
other, of those motions which the fluid takes in consequence of the motion of solid bodies in it.
Those motions of the fluid, which take place in consequence of very rapid motions of solids, (such
as those of bullets), form a connecting link between these two classes. The motions of the second
class are, it is true, accompanied by condensations, and propagated with the velocity of sound, but
if the motions of the solids are not great we may, without sensible error, suppose the motions of
the fluid propagated instantaneously to distances where they cease to be sensible, and may neglect the
condensation. The investigations in this paper will apply without sensible error to this kind of
motion of elastic fluids. ;

In all cases also the motion will be supposed to begin from rest, which allows us to suppose that
uda + vdy + wdzx is an exact differential dp, where wv, v and w are the components, parallel to the
axes of a, y, and x, of the whole velocity of any particle. In applying our investigations however
to fluids such as they exist in nature, this principle must not be strained too far. When a body is
made to revolve continually in a fluid, the parts of the fluid near the body will soon acquire a rota-
tory motion, in consequence, in all probability, of the mutual friction of the parts of the fluid; so
that after a time wda@ + vdy+wdzx could no longer be taken an exact differential. It is true that
in motion in two dimensions there is one sort of rotatory motion for which that quantity is an exact
differential; but if a close vessel, filled with fluid at first at rest, be made to revolve uniformly round
a fixed axis, the fluid will soon do so too, and therefore that quantity will cease to be an exact dif-

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 107

ferential. For the same reason, in the progressive motion of a solid in a fluid, the effect of friction
continually accumulating, the motion might at last be sensibly different from what it would be if
there were no friction, and that, even if the friction were very small. In the case of small oscillatory
motions however it would appear that the effect of friction in the forward oscillation, supposing that
friction small, would be counteracted by its effect in the backward oscillation, at least if the two
were symmetrical. In this case then we might expect our results to agree very nearly with experi-
ment, so far at least as the ¢ime of oscillation is concerned.

The forces which act on the fluid are supposed in the following investigations to be such that
Xdz+ Ydy + Zdzx is the exact differential of a function of z, y and x, where X, Y, Z, are the
components, parallel to the axes, of the accelerating force acting on the particle whose co-ordinates
are v, y, x. The only effect of such forces, in the case of a homogeneous, incompressible fluid, being
to add the quantity p /(Xdw + Ydy + Zdz) to the pressure, the forces, as well as the pressure due
to them, will for the future be omitted for the sake of simplicity.

2. It is a recognised principle, and one of great importance in these investigations, that when
a problem is determinate any solution which satisfies all the requisite conditions, no matter how ob-
tained, is the solution of the problem. In the case of fluid motion, when the initial circumstances
and the conditions with respect to the boundaries of the fluid are given, the problem is determinate.
If it were required to find what sort of steady motion could take place between given surfaces, the
problem would not be determinate, since different kinds of steady motion might result from different
initial circumstances.

It may be well here to enumerate the conditions which must be satisfied in the case of a homoge-
neous incompressible fluid without a free surface, the case which is considered in this paper. We
have first the equations,

aye 1d 1d
= - a eins Oe Z = = 2 ah a — W3> eee abn repenstaspiees(cA)s
pee pas p dz

du du du du

putting a, for vU— + wW—

dt’ da" dy dz’

and x, and omitting the forces,

and @,, @,, for the corresponding quantities for y

We have also the equation of continuity,

du dv dw _ B
ate Fy sar Pe Re cocsocos| Y#3})-
(A) and (B) hold at all times for all points of the fluid mass.

If o be the velocity of the point (w, y, x) of the surface of a solid in contact with the fluid
resolved along the normal, and y the velocity, resolved along the same normal, of the fluid particle,
which at the time ¢ is in contact with the above point of the solid, we must have

U =\ On spastedslon tactertn’ crise se sia Beano too ped mnalettey (@) 5
at all times and for all points of the fluid which are in contact with a solid.

If the fluid extend to infinity, and the motion at first be zero at an infinite distance, we must
have
Ui Ot.) aie0,eateany ininite, CIstanGem >.assctes «ariepauceite. (6).

An analagous condition is, that the motion shall not become infinitely great about a particular
point, as the origin.

* For greater clearness, those equations which must hold for all | values of the variables, or of some of them, are denoted by small
values of the variables within limits depending on the problem | letters. The latter class serve to determine the forms of the arbi-
are denoted by capitals, while those which hold only for certain | trary functions contained in the integrals of the former.

o2

108 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

Lastly, if 2%, %), Wo, be the initial velocities, subject of course to satisfy equations (B) and
(a), we must have
Qt 06,5 IVs ey = WEE —"O.nyes sa peeces bees ce scaeeeeeel Oe
In the most general case the equations which w, v and w are to satisfy at every point of the
mass and at every time are (B) and the three equations
dw, dw, da: _ do, da, _ day (Cc)
ij, dae a as ae :
These equations being satisfied, the quantity a,dw+a@,dy + @,d will be an exact differen-
tial, whence p may be determined by integrating the value of dp given by equations (4). Thus
the condition that these latter equations shall be satisfied is equivalent to the condition that the
equations (C) shall be satisfied.
In nearly all the cases considered in this paper, and in all those of which the complete solution
is given, the motion is such that wdw + vdy +wdz is an exact differential dp. This being the
case, the equations (C) are, as it is well known, always satisfied, the value of p being given by the

equation
P Env - = oe) + ()'+ (4)} hee eee. (D),

y (4) being an arbitrary Pa of ¢, which may if we please be included in @. In this case,
therefore, the sizgle condition which has to be satisfied at all times, and at every point of the mass
is (B), which becomes in this case

eho eG _

dai * dy? dz (4)

In the case of impulsive motion, if «,, v,, w,, be the velocities just before impact, wv, v, w,
the velocities just after, and q the impulsive pressure, the equations (A) are replaced by the equations

and in order that these equations may be satisfied it is necessary and sufficient that (w — u,)dv
+ (v — v,)dy + (w — w,)dz be an exact differential dp, which gives

q=C— p@.
The only equation which must be satisfied at every point of the mass is (B), which is equivalent to
(£), since by hypothesis «, v,, and w, satisfy (B). The conditions (a) and (b) remain the
same as before.

One observation however is necessary here. The values of wv, v and w are always supposed to
alter continuously from one point in the interior of a fluid mass to another. At the extreme boun-
daries of the fluid they may however alter abruptly. Suppose now values of w, v and w to have
been assigned, which do not alter abruptly, which satisfy equations (B) and (C) as well as the con-
ditions (a), (6) and (ce), or, to take a particular case, values which do not alter abruptly, which
satisfy the equation (B) and the same cares and which render wda + vdy + wdx an exact

; d
differential. Then the values of o. = and © = will alter continuously from one point to another,
y
but it does not follow that the value of p ar cannot alter abruptly. Similarly in impulsive

; d d ;
motion the value of g may alter abruptly, although those of et 5 = and = alter continuously.
av’ dy x
Such abrupt alterations are, however, inadmissible ; whence it follows as an additional condition to
be satisfied,
that the value of p or q, obtained by integrating equations (A) or (F), shall
not alter abruptly from one point of the fluid to another. er

ooeee(d).

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 109

An example will make this clearer. Suppose a mass of fluid to be at rest in a finite cylinder,
whose axis coincides with that of x, the cylinder being entirely filled, and closed at both ends. Sup-
pose the cylinder to be moved by impact with an initial velocity C in the direction of #; then shall

The (Oh DEOS iS

For these values render «dx + vdy + wdzx an exact differential dd, where @ satisfies (); they also
satisfy (a); and, lastly, the value of q obtained by integrating equations (F'), namely, C’ — Cpa,

does not alter abruptly. But if we had supposed that @ were equal to Cx + C'@, where 0 = tan!

?

the equation (£) and the condition (a) would still be satisfied, but the value of g would be
C” — p(Cw + C’8), in which the term pC’@ alters abruptly from 2rpC" to 0, as @ passes through
the value 27. The condition (d) then alone shews that the former and not the latter is the true
solution of the problem.

The fact that the analytical conditions of a problem in fluid motion, as far as those conditions
depend on the velocities, may be satisfied by values of those velocities, which notwithstanding corre-
spond to a pressure which alters abruptly, may be thus explained. Conceive two masses of the same
fluid contained in two similar and equal close vessels 4 and B. For more simplicity, suppose these
vessels and the fluid in them to be at first at rest. Conceive the fluid in B to be divided by an
infinitely thin lamina which is capable of assuming any form, and, at the same time, of sustaining
pressure. Suppose the vessels 4 and B to be moved in exactly the same manner, the lamina in
B being also moved in any arbitrary manner. It is clear that, except for one particular motion of
the lamina, the motion of the fluid in B will be different from that of the fluid in A. The velocities
U, V, W, will in general be different on opposite sides of the lamina in B. For particular motions of
the lamina however the velocities uw, v, w, may be the same on opposite sides of it, while the
pressures are different. The motion which takes place in B in this case might, only for the con-
dition (d), be supposed to take place in A.

It is true that equations (4) or (7), could not strictly speaking be said to hold good at those
surfaces where such a discontinuity should exsist. Still, to avoid the liability to error, it is well to
state the condition (d) distinctly.

When the motion begins from rest, not only must wdw + vdy + wdzx be an exact differential do,
and uw, v, w, not alter abruptly, but also gp must not alter abruptly, provided the particles in
contact with the several surfaces remain in contact with those surfaces; for if this condition be not
fulfilled, the surface for which it is not fulfilled will as it were cut the fluid into two. For it follows

dp

from the equation (D) that ae must not alter abruptly, since otherwise p would alter abruptly

from one point of the fluid to another; and “ neither altering abruptly nor becoming infinite, it

follows that @ will not alter abruptly. Should an impact occur at any period of the motion, it
follows from equations (7") that that cannot cause the value of @ to alter abruptly, since such an
abrupt alteration would give a corresponding abrupt alteration in the value of q.

3. Aresult which follows at once from the principle laid down in the beginning of the last article
is this, that when the motion of a fluid in a close vessel which is at rest, and is completely filled, is
of such a kind that wdw + vdy + wdzx is an exact differential, it will be steady. For let uv, v, w, be
the initial velocities, and let us sce if the velocity at the same point can remain uw, v, w. First,
uda+vdy+wdzx being an exact differential, equations (4) will be satisfied by a suitable value of p,
which value is given by equation (D). Also equation (B) is satisfied since it is so at first. The
condition (a) becomes v = 0, which is also satisfied since it is satisfied at first. Also the value of p

: , . d
given by equation (D) will not alter abruptly, for os = 0, or a function of ¢, and the velocities ae &e.,
c du

Ded

110 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

are supposed not to alter abruptly. Hence, all the requisite conditions ate satisfied ; and hence,
(Art. 2) the hypothesis of steady motion is correct.

4. In the case of an incompressible fluid, either of infinite extent, or confined, or interrupted in
any manner by any solid bodies, if the motion begin from rest, and if there be none of the cutting
motion mentioned in Art. 2, the motion at the time ¢ will be the same as if it were produced instan-
taneously by the impulsive motion of the several surfaces which bound the fluid, including among
these surfaces those of any solids which may be immersed in it. For let w, v, w, be the velocities at
the time ¢. Then by a known theorem wdw +vdy+wdz will be an exact differential d@, and

will not alter abruptly (Art. 2). @ must also satisfy the equation (), and the conditions
(a) and (b). Now if w’, x’, w’, be the velocities on the supposition of an impact, these quantities
must be determined by precisely the same conditions as w, v and w. But the problem of finding
uw, v' and w’, being evidently determinate, it follows that the identical problem of finding w, v
and w is also determinate, and therefore the two problems have the same solution; so that

u=u, v=v, w=w’,

This principle has been mentioned by M. Cauchy, in a memoir entitled Mémoire sur la Théorie
des Ondes, in the first volume of the memoirs presented to the French Institute, page 14. It
will be employed in this paper to simplify the requisite calculations by enabling us to dispense
with all consideration of the previous motion, in finding the motion of the fluid at any time
in terms of that of the bounding surfaces. One simple deduction from it is that, when all the
bounding surfaces come to rest, each element of the fluid will come to rest. Another is, that if the

velocities of the bounding surfaces are altered in any ratio the value of @ will be altered in the same
ratio.

5. Superposition of different motions.

Tn calculating the inital motion of a fluid, corresponding to given initial motions of the bounding
surfaces, we may resolve the latter into any number of systems of motions, which when compounded
give to each point of each bounding surface a velocity, which when resolved along the normal is
equal to the given velocity resolved along the same normal, provided that, if the fluid be enclosed
on all sides, each system be such as not to alter its volume. For let w’, v’, w’, v’, o’, be the values
of u, v, &e., corresponding to the first system of motions; w’, v”, &c., the values of those quantities
corresponding to the second system, and so on; so that

, ” , “” , ” Ld ” , ”
WWW Feoerg VEVHEY + ceeyn WHE WHEW t+ very VE VTV Feeen TH OHS Fee

Then since we have by hypothesis w/da + v'dy + w'dx an exact differential dd’, w’dax + v"dy
+ w’ds an exact differential dy’, and so on, it follows that wda+vdy+wds is an exact dif-
ferential. Again by hypothesis y’ =o’, y”"=o", &c., whence y=o. Also, if the fluid extend to an
infinite distance, «, v, and w must there vanish, since that is the case with each of the systems
u, v', w', &c. Lastly, the quantities @’, p”, &c., not altering abruptly, it follows that @, which
is equal to q' + p+ ...» Will not alter abruptly. Hence the compounded motion will satisfy all
the requisite conditions, and therefore, (Art. 2) it is the actual motion.

It will be observed that the pressure p will not be obtained by adding together the pressures
due to each of the above systems of velocities. To find p we must substitute the complete value of
gf in equation (D). If, however, the motion be very small, so that the square of the velocity is
neglected, it will be sufficient to add together the several pressures just mentioned.

In general the most convenient systems into which to decompose the motion of the bounding
surfaces are those formed by considering the motion of each surface, or of a certain portion of each
surface, separately. Such a portion may be either finite or infinitesimal. In fact, in some of the
cases of motion that will be presently given, where @ is expressed by a double integral with a
function under the integral sign expressing the motion of the bounding surfaces, it will be found
that each element of the integral gives a value of @ such that, except about the corresponding

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 111

element of the bounding surface, the motion of all particles in contact with those surfaces is
tangential.

A result which follows at once from this principle, and which appears to admit of comparison
with experiment, is the following. Conceive an ellipsoid, or any body which is symmetrical with
respect to three planes at right angles to each other, to be made to oscillate in a fluid in the
direction of each of its three axes in succession, the oscillations being very small. Then, in each
case, as may be shown by the same sort of reasoning as that employed in Art. 8, in the case
of a cylinder, the effect of the inertia of the fluid will be to increase the mass of the solid by
a mass having a certain unknown ratio to that of the fluid displaced. Let the axes of co-ordinates
be parallel to the axes of the solid; let w, y, x, be the co-ordinates of the centre of the solid,
and let M, M’, M”, be the imaginary masses which we must suppose added to that of the solid
when it oscillates in the direction of the axes of x, y, x, respectively. Let it now be made to oscillate
in the direction of a line making angles a, 3, y, with the axes, and let s be measured along
this line. Then the motions of the fluid due to the motions of the solid in the direction of the

three axes will be superimposed. The motion being supposed to be small, the resultant of the

Be a! 1 :
pressures of the fluid on the solid will be three forces, equal to Mcosa if , M’ cos B == 4
Lg at

ds 3 F oa 5 :
M” cosy ane respectively, in the directions of the three axes. The resultant of these in the

x - d’s
direction of the motion will be J, qP where

M,= Mcos*a + M’ cos* + M” cos*y.
Each of the quantities M, M’, M”’ and M, may be determined by observation, and we may
find whether the above relation holds between them. Other relations of the same nature may be
deduced from the principle explained in this article.

6. Reflection.
Conceive two solids, 4 and B, immersed in a fluid of infinite extent, the whole being at rest.

Suppose 4 to be moved in any manner by impulsive forces, while B is held at rest. Suppose
the solids A and B of such forms that, if either were removed, and the several points of the
surface of the other moved instantaneously in any given manner, the motion of the fluid could
be determined: then the actual motion can be approximated to in the following manner. Conceive
the place of B to be occupied by fluid, and A to receive its given motion; then by hypothesis
the initial motion of the fluid can be determined. Let the velocity with which the fluid in
contact with that which is supposed to occupy B’s place penetrates into the latter be found,
and then suppose that the several points of the surface of B are moved with normal velocities
equal and opposite to those just found, A’s place being supposed to be occupied by fluid. The
motion of the fluid corresponding to the velocities of the several points of the surface of B can
then be found, and A must now be treated as B has been, and so on. The system of velocities
of the particles of the fluid corresponding to the first system of velocities of the particles of the
surface of B, form what may be called the motion of A reflected from B; the motion of the
fluid arising from the second system of velocities of the particles of the surface of 4 may be
called the motion of A reflected from B and again from A, and so on. It must be remembered
that all these motions take place simultaneously. It is evident that these reflected motions will
rapidly decrease, at least if the distance between A and B is considerable compared with their
diameters, or rather with the diameter of either. In this case the calculation of one or two
reflections will give the motion of the fluid due to that of A with great accuracy. It is evident
that the principle of reflection will extend to any number of solid bodies immersed in a fluid;
or again, the body B may be supposed to be hollow, and to contain the fluid and A, or else
A to contain B. In some cases the series arising from the successive reflections can be summed,

112 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

in which case the motion will be determined exactly. The principle explained in this article has
been employed in other subjects, and appears likely to be of great use in this. It is the same for
instance as that of swecessive influences in Electricity.

4. Ifa mass of fluid be at rest or in motion in a close vessel which it entirely fills, the
vessel being either at rest or moving in any manner, any additional motion of translation com-
munieated to the vessel will not affect the relative motion of the fluid. For it is evident that
on the supposition that the relative motion is not affected the equation (B) and the condition
(a) will still be satisfied. Also, if 2,, mz, ws, be the components of the effective force of any
particle in the first case, and U, V, W, be the components of the velocity of translation, then

dV dW
nae CCA an ‘dt > @3t+ Che ?
will be the components of the effective force of the same particle in the second case. Now since
by hypothesis a,da + @.dy + @,dz is an exact differential, as follows from equations (C), and
U, V, W, are functions of ¢ only, it follows at once that

dU dV d
(=, + 7) dv + (w+) dy + (=. +7) dz

is an exact differential, where wv, y, x, are the co-ordinates of any particle referred to the old axes,

which are themselves moving in space with velocities U, V, W. But if a, y,, *;, be the co-

ordinates of the same particle referred to parallel axes fixed in space, we have
a=a+fUdt, y,=y+fVdt, x,=2+ fWdt,

whence, supposing the time constant, dv = da,, dy =dy,, dz = dx, and therefore

@,+

dU f dV dw
(=, + a) aa + (a. + ai) dy, + (=, + a} dz,
is an exact differential. Hence, equations (4) can be satisfied by a suitable value of p. Denoting
by p the pressure about the particle whose co-ordinates are w, y, #, in the first case, the pressure
about the same particle in the second case will be

r dU dV dw
PHO - 05, rage “ar *}?
none of the terms of which will alter abruptly, since by hypothesis p docs not.

Since then the present hypothesis satisfies all the requisite conditions, it follows from Art. 2
that that hypothesis is correct. If / be the additional effective force of any particle of the vessel
in consequence of the motion of translation, and we take new axes of a’, y’, 2’, of which the first
is in the direction of F', the additional term introduced into the value of the pressure will be
— pF a’, omitting the arbitrary function of the time. The resultant of the additional pressures on
the sides of the vessel will be equal to # multiplied by the mass of the fluid, and will pass
through the centre of gravity of the fluid, and act in the direction of — a’.

8. Motion between two cylindrical surfaces having a common avis.

Let us conceive a mass of fluid at rest, bounded by two cylindrical surfaces having a com-
mon axis, these surfaces being either infinite or bounded by two planes perpendicular to their
axis. Let us suppose the several generating lines of these cylindrical surfaces to be moved
parallel to themselves in any given manner consistent with the condition that the volume of the
fluid be not altered: it is required to determine the initial motion at any point of the mass,

Since the motion will take place in two dimensions, let the fluid be referred to polar
co-ordinates 7, 0, in a plane perpendicular to the axis, » being measured from the axis, Let
a be the radius of the inner surface, 6 that of the outer, f(@) the normal velocity of any
point of the inner surface, #'(@) the corresponding quantity for the outer.

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 113

Since for any particular radius vector between @ and b the value of @ is a periodic function
of @ which does not become infinite, (for the motion at each point of each bounding surface is
supposed to be finite), and which does not alter abruptly, it may be expanded in a converging
series of sines and cosines of @ and its multiples. Let then

p =P, + Zr (P, cosnO + Q, sin 20) ........0005 (1).

Substituting the above value in the equation
d d ad’
Yr G 2) + Bis Ulepodsdantosa . (2),
dr \ dr de
which p is to satisfy, and equating to zero the coefficients of corresponding sines and cosines,
which is allowable, since a given function can be expanded in only one series of the form (1),
we find that P, must satisfy the equation

of which the general integral is

P,=Alogr + B,
the base being «, and P, and Q, must both satisfy the same equation, viz.
d dP,
1 | ———)) P= 20.
dr ( 3 PE bs
of which the general integral is

(Pr Opn One

We have then, omitting the arbitrary constant in @, as will be done for the future, since we

haye oceasion to use only the differential coefficients of @,

p = A, logr + SY {(4,r-" + 4’,7") cos 20 + (B,r-” + B’,2”) sin MOE weve so (8)
with the conditions
dp
a ay 0) WEEE <3 (edhaocEncccpnadsbecobeeenn (4),

d
fF @) Maen. = Di pas ves daclgtenacmsbnre ee CO):

Let f (9) = C, + 2/(C, cosné + D, sin x),
F (0) =C, + SY (C',cosnO + D',sinn6) ;
7 1 2r 7 7 1 29 , ’ 1 2A, . , ‘
so that Cy = a S(O) de’, C,=— f S(@) cosn'dé', D, = =i FQ) sinnd' dé,
with similar expressions for C’,, &c. Then the condition (4) gives
1, na n Ya
= + By {(- 4,a-@+ + A’,a""}) cosn@ + (— B,a7 "+ + B’,a"-!) sin nt

= Cy, + 3) (C, cos nO + D, sin n6) ;
whence,
A,=aC),

1
A,a-“+) — 4) at = 2 Gi.

1
Bya-C+) ~ Bia} = —— Dy.

Vou. VIII. Parr I. P

114 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

Similarly, from the condition (5), we get
Ay =b C'os

Me
A,b-+9 = AB = —— C's,

B,b-0+) — Bib = - =D,
n

It will be observed that aC,=6C’,, by the condition that the volume of fluid remain unchanged,
which gives ;
a ff (0')d0' =b fF (6) de’.
From the above equations we easily get

2n B2n
a" ; fo-n+! C’, =a"! i

2
n n (b2" — a?”

and, changing the sign of m,

1
J a n (Ga — ae"
with similar expressions for B, and B’,, involving D in place of C.

We have then

/

fpr4 Ce — q't} Crt

p = aC, logr + Sy . ("= a®)=18G5"*! C", —a-"*"'C,) cosnd

+ (b-"*! D', — a-"*" D,) sinn@] a?"b?"7r-”
+ [(6"t! C’, — a"t! C,) cos nO + (6"*! D’, — a'*' D,) sin nO) 1°} «2... ree (65
which completely determines the motion.

It will be necessary however, (Art. 2), to shew that this value of @ does not alter abruptly
for points within the fluid, as may be easily done. For the quantities C,, D, cannot be greater

1 23 = . sas .
than -f +(0)d0@, where each element of the integral is taken positively; and since by
To

hypothesis f(@) is finite for all values of 6 from 0 to 27, it follows that neither C, nor D, can
be numerically greater than a constant quantity which is independent of . The same will be
true of C’, and D',. Remembering then that >a and <b, it can be easily shewn that the
series which occur in (6) have their terms numerically less than those of eight geometric series
respectively whose ratios are less than unity; and since moreover the terms of the former set
of series do not alter abruptly, it follows that @ cannot alter abruptly. The same may be
proved in a similar manner of the differential coefficients of @. The other infinite series ex-
pressing the value of @ which occur in this paper may be treated in the same way: and in
Art. 10, where @ is expressed by a definite integral, the value of @ and its differential coefficients
will alter continuously, since that is the case with each element of the integral. It will be
unnecessary therefore to refer again to the condition (d).

If the fluid be infinitely extended, we must suppose C’, and D’, to vanish in (6), since the
velocity vanishes at an infinite distance; we must then make 6 infinite, which reduces the above
equation to

atl

p =aC,logr -— =F $Cicos 8 ¥ Dy ci si wi onc venoes ces tvecne he

This value of @ may be put under the form of a definite integral: for, replacing C,, C, and
D,, by their values, it becomes

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 115

a Sah i ht. Ie lis NF) pBmiLye., N\ 70°
= ber f f@yag’ ~~ 37 ~ (2) f £(0’) cos n (0 - 6’) d6’,

which becomes on summing the series

»

a

a Qn , , a 2m , L , ,
zz esr f f(6')ae af log {1 ~ 2“ cos (@- 6+) + | f (8) dG ;

r

dp a p2fl ar cos (0 — 6’) — a ie om

whenee ‘dies, me A {; +? — 2ar cos (0-6) + a} F(@) a0 ‘
If we suppose to become equal to a the quantity under the integral-sign vanishes, except
for values of 6’, which are indefinitely near to @. The value of the integral itself becomes f (6)*.
Hence it appears, that to the disturbance of each element of the surface, there corresponds
a normal velocity of the particles in contact with the surface, which is zero, except just about
the disturbed element. The whole disturbance of the fluid will be the aggregate of the dis-
turbances due to those of the several elements of the surface. The case of the initial motion
of fluid within a cylinder, and the analogous cases of motion within and without a sphere, which

will be given in the next article, may be treated in the same manner.

The velocity in the direction of r given by equation (7), jc 7) A
a \

a

n+)
+=) (*) {C, cosn@ + D, sinn6},
r

\

aCy

a

and that perpendicular to 7, and reckoned positive in the same direction as 0, (- aa
pe

n+1
=e (<) {C, sin 0 — D,, cos nO}.
Yr

Conceive a mass of fluid comprised between two infinite parallel planes, and suppose that
a certain portion of this fluid contains solid bodies bounded by cylindrical surfaces perpendicular
to these planes. The whole being at first at rest, suppose that the surfaces of these solids are
moved in any manner, the motion being in two dimensions. Conceive a circular cylindrical
surface described perpendicular to the parallel planes, and with a radius so large that all the
solids are comprised within it. Then, (Art. 4), we may suppose the motion of the fluid at any
time to have been produced directly by impact, On this supposition the initial motion of the
part of the fluid without the above cylindrical surface will be determined in terms of the normal
motion of the fluid forming that surface, as has just been done. If C, be different from zero,

; : F : ; t ac ; 5
then, at a great distance in the fluid, the velocity will be ultimately ——°, and directed to or from
Te

the axis of the cylinder, and alike in all directions. Since the rate of increase of volume of a
length / of the cylinder is equal to la [°" f(0') a0! = 2rlaC,, it appears that the velocity at
a great distance is proportional to the expansion or contraction of a unit of length of the solids.
If however there should be no expansion or contraction, or if the expansion of some of the solids

should make up for the contraction of the rest, then in general the most important part of the

P : ; : EemGcosidyit.
motion at a great distance will consist of a velocity ———— directed to or from the centre, and
r

C’ sin @ : ; are ; P
another ——,— perpendicular to the radius vector, the value of C’ and the direction from which
7

@, is measured varying from one instant to another. The resultant of these velocities will vary
inversely as the square of the distance.

* Poisson, Théorie de la Chalewr, Chap. Vii.
P2

116 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

Resuming the value of @ given by equation (6), let us suppose that the interior cylindrical
surface is rigid, and moved with a velocity C in the direction from which @ is measured, the
outer Shh being at rest: then f (0) =Ccos@, F(@)=0; whence C,=C, and the other co-
efficients are each zero. We have then

2 432
a= oe (= +r) COS ieee sislcisie eels ts <ivemn (Oe

Suppose now that the inner cylinder has a small oscillatory motion about an axis parallel
to the axes of the cylinders, the cylinders having their axes coincident in the position of
equilibrium. Let y be the angle which a plane drawn through the axis of rotation, and that of the
solid cylinder at any time makes with a vertical plane drawn through the former. The motion of
translation of the axis of the cylinder will differ from a rectilinear motion by quantities depending
on W*: the motion of rotation about its axis will be of the order y,, but will have no effect on
the fluid. Therefore in considering the motion of the fluid we may, if we neglect squares of w,
consider the motion of the cylinder rectilinear. The expression given for @ by equation (8) will
be accurately true only for the instant when the axes of the cylinders coincide; but since the
whole resultant pressure on the solid eylinder in consequence of the motion is of the order w,
we may, if we neglect higher powers of y, than the first, employ the is tae value of @
given by equation (8). Neglecting the square of the velocity, we have

dp
ugg

d :
In finding the complete value of rf it would be necessary to express @ by co-ordinates re-
ferred to axes fixed in space, which after differentiation we might suppose to coincide with others
fixed in the body. But the additional terms so introduced depending on the square of the velocity,
which by hypothesis is neglected, we may differentiate the value of @ given by equation (8) as if
the axes were fixed in space. We have then, to the first order of approximation,

eit
dp _ dt [0
Fi -—t + r| cose.

If 2 be the length of the cylinder, the pressure on the element /ad@, resolved parallel to w
and reckoned positive when it acts in the direction of ,

1 ££ dC
‘< b?
=- pala (c s aj cos’ 6dé ;
P-a’ la
and integrating from @ =0 to 0 =27, we have the whole resultant pressure parallel to v

+a dC
Sr pa ele a

dt

aCe ; :
Since = is the effective force of the axis, parallel to w, and that parallel to y is of the order y/*,

we see that the effect of the inertia of the fluid is to increase the mass of the cylinder by

&
— au, Where » is the mass of the fluid displaced. ‘This imaginary additional mass must be
supposed to be collected at the axis of the cylinder.

If the cylinder oscillate in an infinitely extended fluid b = @, and the additional mass becomes
equal to that of the fluid displaced. This appears to be a result capable of being compared with
experiment, though not with very great accuracy. Two cylinders of the same material, and of the”

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 117

same radius, but whose lengths differ by several radii, might be made to oscillate in succession in a
fluid, at a depth sufficiently great to allow us to neglect the motion of the surface of the fluid. The
time of oscillation of each might then be calculated as if the cylinder oscillated in vacuum, acted on
by a moving force equal to its weight minus that of the fluid displaced, acting downwards through
its centre of gravity, and having its mass increased by an unknown mass collected in the axis.
Equating the time of oscillation so calculated to that given by observation, we should determine the
unknown mass. The difference of these masses would be very nearly equal to the mass which must
be added to that of a cylinder whose length is equal to the difference of the lengths of the first two,
when the motion is in two dimensions. This evidently comes to supposing that, at a distance from
the middle of the longer cylinder not greater than half the difference of the lengths of the two, the
motion may be taken as in two dimensions. The ends of the cylinders may be of any form, provided
that they are all of the same. They may be suspended by fine equal wires, in which case we should
have a compound pendulum, or attached to a rigid body oscillating above the fluid by means of
thin flat bars of metal, whose plane is in the plane of motion. Another way of getting rid of the
motion in three dimensions about the ends would be, to make those ends plane, and to fix two
rigid planes parallel to the plane of motion, which should be almost in contact with the ends of the
cylinder.

9. Motion between two concentric spherical surfaces.—Motion of a ball pendulum enclosed
in a spherical case. 4

Let a mass of fluid be at rest, comprised between two concentric spherical surfaces. Let the
several points of these surfaces be moved in any manner consistent with the condition that the
volume of the fluid be not changed: it is required to determine the initial motion at any point
of the mass.

Let a, b, be the radii of the inner and outer spherical surfaces respectively ; then employing the
co-ordinates 7, 0, w, where 7 is the distance from the centre, @ the angle which 7 makes with a fixed
line passing through the centre, w the angle which a plane passing through these two lines makes
with a fixed plane through the latter, the value of @ corresponding to any radius vector comprised
between a and 6 can be expanded in a converging series of Laplace’s coefficients. Let then

P = Vot Vy vovees #°Viy + oc 5
V,, being a Laplace’s coefficient of the x order.

Substituting in the equation,

rp 1 df, ,dp 1 &o
ae oad db (sine a) Tanke fa

which @ is to satisfy, employing the equation

a a (si oo) +a oe
A ~ — {sing —— = = eitlie «eet bays
sin@ dd do) * sin'@ da” (9),
and then equating to zero the Laplace’s coefficients of the several orders, we find
ar V,,
te 7 (a+ 1)V, = 0.
The general integral of this equation is
(oe
V,= Cr piel ’

where C and C’ are functions of 6 and w. Substituting in the equation (9), and equating coeffi-
cients of the two powers of r which enter into it separately to zero, we find that both C and
C’ satisfy it, and therefore are both Laplace’s coefficients of the m' order. We have then

im By (FeSO) asad vases.) (0);

118 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

where Y, and Z, are each Laplace’s coefficients of the 2 order, and do not contain 7. Let

f (0, w) be the normal velocity of the point of the inner surface corresponding to 6 and w,
F(@, w) the corresponding quantity for the outer; then the conditions which @ is to satisfy

are that
d
“ =f (0, w) when r=4,
d
c= F (0, w) when r=b,
dr
Let f(@, ~), expanded in a series of Laplace’s coefficients, be
yoy a Oe ar

which expansion may be performed by the usual formula, if not by inspection: then the first

condition gives
Sn a Se eae Ste Las

and equating Laplace’s coefficients of the same order, we get
MV 40° = (1 4-1) Zpa- > = Py gon oeesoesevece (11).
Let F(0, w), expanded in a series of Laplace’s coefficients, be
Per eateries act ase

then from the second condition, we get

7 Veh aN GU Agden at SaP raaectice clesiewn'seis ok (12).
From (11) and (12) we easily get
P,/b"+? — Pa’?
Ta ay’
Z, _ gent pent MP ae? oa! P,a-" - 3

(n +1) (24) — ae")
provided m be greater than 0. If m=0, we have
Log. =P, eee, = PY.
But the condition that the volume of the fluid be not altered, gives
as lig f(O, w) sin OdOdw = 0 [” [-" F(, w) sin OdOdw,
or 47a°P, = arb? P’,,
which reduces the two equations just given to one.

We have then, omitting the constant Yo,

Pia S32 Spent 2 I U 2 2
p=- = 35 48 —@a able ps (P’, "+? — P,a"**)r"
antl gZentl
5 SRE PbO) S Peareyeonh ae ete (13),

which determines the motion.
When the fluid is infinitely extended, we have P’,=0 since the velocity vanishes at an
infinite distance, and b = co, whence
noe Pia? ea aps
r a+r

It may be proved, precisely as was done, (Art. 8), for motion in two dimensions, that if
any portion of an infinitely extended fluid be disturbed by the motion of solid bodies, or other-

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 119

wise, if all the fluid beyond a certain distance from the part disturbed were at first at rest, the
velocity at a great distance will ultimately be directed to or from the disturbed part, and will

: : . 1 ‘5 i F
be the same in all directions, and will vary as 2 The coefficient of — will be proportional
3 ae

to the rate of gain or loss of volume of the part disturbed. If however this rate should be zero,
then the most important part of the velocity at a great distance will in general be that depend-

3
5 aP, , . 7
ing on the term — in @. Since the general form of P, is
2%"

A cos@ + Bsin@ cosw + C'sin@ sin w,

we easily find, by making use of rectangular co-ordinates, changing the direction of the axes,

D cos 6;
2

and then again adopting polar co-ordinates, that the above term in @ takes the form F

9, being measured from same line passing through the origin. The motion will therefore be the
same as that round a ball pendulum in an incompressible fluid, the centre of the ball being
in the origin ; a case of motion which will be considered immediately. In order to represent the
motion at different times, we must suppose the velocity and direction of motion of the ball to
change with the time.

The value of @ given by equation (13) is applicable to the determination of the motion of
a ball pendulum enclosed in a spherical case which is concentric with the ball in its position of
equilibrium. If C be the velocity of the centre of the ball at the instant when the centres of
the ball and case coincide, and if 6 be measured from the direction in which it is moving, we

shall have
f(@) = Ccos#, F(0) = 0;

* P,=0, P,=Ccos@, P,;=0, &e., P’,=0, &e.,

and the value of @ for this instant is accurately

Cae B ‘
—-- (: + } cos 0,
2

“Boe P
which, when b = ©, becomes :
Ca’ cos 0

a oe
which is the known expression for the value of @ for a sphere oscillating in an infinitely extended,
incompressible fluid.

It may be shewn, by precisely the same reasoning as was employed in the case of the cylin-

der, that in calculating the small oscillations of the sphere the value of ? to be employed is
a

“ae
dt el a

d
and from the equation p= — pas we easily find that the whole resultant pressure on the

sphere in the direction of its centre, and tending to retard it is

4 m™pa BN AG,
3h—a (a =) ‘dt’

d atl} Siiemety «2 : dc . :
and that perpendicular to this direction is zero. Since at the effective force of the centre

in the direction of the motion, and that perpendicular to this direction is of the second order,

120 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

the effect of the inertia of the fluid will be to increase the mass of the sphere by a mass
4 apa” ( iy ) B+ 2a

=- — a = =
3 BD - a B a (7

2a*

» being the mass of the fluid displaced; so that the effect of the case is, to increase the mass
which we must suppose added to that of the ball in the ratio of 6'+2a* to b'—a’.

Poisson, in his solution of the problem of the oscillating sphere given in the Mémoires de

U Institut, Tome x1. arrives at a different conclusion, viz. that the case does not at all affect
the motion of the sphere. When the elimination which he proposes at p. 563 is made, the last

oy &O aC’

2a*ced(l — dy) aa? dé
propagation of sound, and 6 the ratio of the density of air to that of the ball, ¢ and ¢’ being
functions derived from others which enter into the value of @ by putting »=c, where ¢ is the
radius of the ball. He then argues that this term may be neglected as insensible, since it involves
3 3 Ae

6 in the numerator and a® in the denominator, tacitly assuming that a: + bs
since @ is not large. Now for the disturbances of the air which have the same period as

dp

those of the pendulum QF is not large compared with @, as it is for those on which sound

term of equation (f) p. 550 becomes }s where a@ is the velocity of

is not large

depends. Let then Poisson’s solution of equation (@), p. 547 of the volume already mentioned,
be put under the form

o-3fe(e-2) AsDpratee-2-e ld}

f' and F’ denoting the derived functions, and all the Laplace’s coefficients except those of the
first order being omitted, the value of @ just given being supposed to be a Laplace’s coefficient
of that order. Then if we expand the above functions in series ascending according to powers

of = we find
$= 510+ FO} - So" + FO} + FO — FPO + os

and in order that when a@= © this equation may coincide with (10), when all the Laplace’s
coefficients except those of the first order are omitted in that equation, it will be seen that it is
necessary to suppose f’”’(#) — F’’(t), and therefore f(t) — F(¢), to be of the order a’, while
S( + F(A) is not large. Putting then

O=xO+ea,
FO= x(t) - ea),

co eall-9 exlied)ofe(-d)-#(3)}

will contain a term of the order a*, and the term which Poisson proposes to

we shall have

3 a
so that a+o)
dt’

leave out will be of the same order of magnitude as those retained.

In making the experiment of determining the resistance of the air to an oscillating sphere, it
would appear to be desirable to enclose the sphere in a concentric spherical case, which would at the
same time exclude currents of air, and facilitate in some measure the experiment by increasing the
small quantity which is the subject of observation. The radius of the case however ought not to be
nearly as small as that of the ball, for if it were, in the first place a small error in the position of the

Mr STOKES, ON SOME CASES OF FLUID MOTION. 121

centre of the ball when at rest might not be insensible, and in the second place the oscillations
would have to be inconveniently small, in order that the value of @ which has been given might be
sufficiently approximate. The effect of a small slit in the upper part of the case, sufficient to allow
the wire by which the ball is supported to oscillate, would evidently be insensible, for the conden-
sation being insensible in a vertical plane passing through the axis of rotation, since the alteration of
pressure in that plane is insensible, the air would not have a tendency alternately to rush in and out
at the slit.

10. Effect of a distant rigid plane on the motion of a ball pendulum.

Although this problem may be more easily solved by an artifice, it may be well to give the direct
solution of it by the method mentioned in Article 6. In order to calculate the motion reflected from
the plane, it will be necessary to solve the following problem :

To find the initial motion at any point of a mass of fluid infinitely extended, except where it
is bounded by an infinite solid but not rigid plane, the initial motion of each point of the solid
plane being given.

It is evident that motion directed to or from a centre situated in the plane, the velocity being
the same in all directions, and varying inversely as the square of the distance from that centre,
would satisfy the condition that wda +vdy+wdzx is an exact differential, and would give to
the particles in contact with the plane a velocity directed along the plane, except just about
the centre. Let us see if the required motion can be made up of an infinite number of such
motions directed to or from an infinite number of such centres.

Let a, y, =, be the co-ordinates of any particle of fluid, the plane vy coinciding with the
solid plane, and the axis of z being directed into the fluid. Let w’, y’, be the co-ordinates of
any point in the solid plane: then the part of @ corresponding to the motion of the element
da dy of the plane will be

Wy (a’, y') da! dy’
V (a — al + yy + a
and therefore the complete value of @ will be given by the equation 5
Be av’, y')da' dy’
p= sh i LA 2) se seseeeseseeeeeeses (14).
-at-a fw= al + (yo +

dp

The velocity parallel to = at any point = Fr

bi le. pe Vv’, y)xda' dy’ i
-2d-» f(a—a’)?+ (y-y)? + v8
Now when # vanishes the quantity under the integral signs vanishes, except for values of a’
and y/ indefinitely near to w and y respectively, the function y/(a’, y’) being supposed to vanish
when w’ or y’ is infinite. Let then w =w+£, y =y+y, then, & and », being as small as
we please, the value of the above expression when x =0 becomes

ch om wr (o+by +n) dédy
— the limit of UM ae ae are a
Now if y/(a’,y’) does not alter abruptly between the limits # - & and #+€, of a’, and y — »,
and y+, of y/, the above expression may be replaced by
xd&édy

. . 5 & 7
— (a, y) x the limit of bai Gaeta
which is = — 27 y(a, y).
Vor, VIII. Parr I. Q

when x =0.

122 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

If now f(a, y’) be the given normal velocity of any point (a’, y’) of the solid plane, the ex-
pression for @ given by equation (14) may be made to give the required normal velocity of
the fluid particles in contact with the solid plane by assuming

U , 1 ‘ ,
¥@,y)= ot et iMe ahd
whence
<a f(a’, y') da’ dy
Om mala Wisc —-xv# P+ y-y) + Bette
This expression will be true for any point at a finite distance from the plane vy even when f(a’, y’)
does alter abruptly; for we may first suppose it to alter continuously, but rapidly, and may then
suppose the rapidity of alteration indefinitely increased: this will not cause the value of @ just
given to become illusory for points situated without the plane ay.
If it be convenient to use polar co-ordinates in the plane wy, putting v=qcosw, y= q sina,
a =q' cosw, y' =q' sinw, and replacing f(a’, y’) by f(q', w), the equation just given becomes
pee 2 ft fh leeds
21%, 5 $q* + q° — 2qq/' cos (w —w') + 2°} 3
To apply this to the case of a sphere oscillating in a fluid perpendicularly to a fixed rigid
plane, let a be the radius of the sphere, and let its centre be moving towards the plane with
a velocity C at the time ¢. Then, (Art. 4), we may calculate the motion as if it were produced
directly by impact. Let h be the distance of the centre of the sphere from the fixed plane
at the time ¢, and let the line h be taken for the axis of z, and let 7, 0, be the polar co-or-
dinates of any point of the fluid, being the distance from the centre of the sphere, and @ the

angle between the lines x and h. Then if the fluid were infinitely extended around the sphere
we should have :

The velocity of any particle, resolved in a direction towards the plane,

3
= < {cos’ @ — 4 sin’ 6}.
For a particle in the plane wy we have
reosO=h, rsind=q’,
and the above velocity becomes
Ca’ (2h? -q*)
a(n? + 4)!
We must now, according to the method explained in (Art. 6), suppose the several points of
the plane wy moved with the above velocity parallel to x. We have then
, nn Cad (2h?-q?
Gs) ae a:
2(h? + q°)
whence, for the motion of the sphere reflected from the plane,
Cat alas (2h? - q*)q'dq‘dw'
0 *o (h? + 9g)? $q? +g? —2qq/ cos (w — w’) + aye
We must next find the velocity, corresponding to this value of @, with which the fluid pene-
trates the surface of the sphere. We have in general

x=h-reos0, g=rsing,

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 123
whence {q° + 9° — 2qq/ cos(w — w') +2°}-2= fh? + 7° +9" 2hr cos @ — 2q'r sin @ cos (w — w’)}~4.
Now supposing the ratio of a to h to be very small, and retaining the most important term, the value
of = when 7 =a will be equal to the coefficient of + when @ is expanded in a series ascend-

r

ing according to powers of r,

Ca} ff if (2h* — q”) fh cos +q' sin @ cos (w -— w')} dq dw’

4m vq %y (A + q'*)*
2 (2h? — q*)q'dq Ca’ cos @

=-1Ca SS eS Se crporcccmicnes (le

Ca feos f G+ a) Shs (17)
In order now to determine the motion reflected from the plane and again from the sphere,

3
we must suppose the several points of the sphere to be moved with a normal velocity cone,
h

or, which is the same, we must suppose the whole sphere to be moved towards the plane with
3

a velocity aE Hence the value of @ corresponding to this motion will be given by the equation
_ _ Ca’ cos as
7) —— 716h372 wee ewe wes eee eeeresere ).

For points at a great distance from the centre of the sphere, the motion which is twice
reflected will be very small compared with that which is but once reflected. For points close to
the sphere however, with which alone we are concerned, those motions will be of the same order
of magnitude, and if we take account of the one we must take account of the other.

Putting g=rsin@, x =h—rcos@ in (16), expanding, and retaining the two most important
terms, we have

p=Cc (x - 28") Se eee) ea)

& being a constant, the value of which is not required, and the second term being evidently found
by multiplying the quantity at the second side of (17) by r. Adding together the parts of @

: d :
given by equations (15), (18) and (19), putting » = a, replacing C by ae and taking for h the

value which it has in equilibrium, just as in the case of the oscillating cylinder in Article 8, we
have for the small motion of the sphere
dp dC af 38a\dC
Gan scar Ye (1+=5) “az 0089

The resultant of the part of the pressure due to the first term is zero: that due to the
3

F 3 , = 3a
second term is greater than if the plane were removed in the ratio of 1 +o to 1. Conse-

4
quently, if we neglect quantities of the order =p the effect of the inertia of the fluid is, to add

3 a’\ pw ; : F q :
a mass equal to {1+ ails to that of the sphere, without increasing the moment of inertia
v

of the latter about its diameter. The effect therefore of a large spherical case is eight times as
great as that of a tangent plane to the case, perpendicular to the direction of the motion of
the ball.

The effect of a distant rigid plane parallel to the direction of motion of an oscillating
sphere might be calculated in the same manner, but as the method is sufficiently explained by
the first case, it will be well to employ the artifice before alluded to, an artifice which is fre-

Q2

124 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

quently employed in this subject. It consists in supposing an exactly symmetrical motion to
take place on the opposite side of the rigid plane, by which means we may evidently conceive
the plane removed.

Let the sphere be oscillating in the direction of the axis of «, the oscillations in this case, as
in the last, being so small that they may be taken as rectilinear in calculating the motion of the
fluid; and instead of a rigid plane conceive an equal sphere to exist at an equal distance on the
opposite side of the plane ay, moving in the same direction and with the same velocity as the
actual sphere. Let 7, 0, w, be the polar co-ordinates of any particle measured from the centre of the
sphere, @ being the angle between 7 and a line drawn through the centre parallel to the axis of x,
and w the angle which the plane passing through these lines makes with the plane ws. Let 7”, 6, w’,
be the corresponding quantities symmetrically measured from the centre of the imaginary sphere.

If the fluid were infinite we should have for the motion corresponding to that of the given
sphere

Ca’ cos@
p=- eee ake eessrieee atte: (20).

The motion reflected from the plane is evidently the same as that corresponding to the motion
of the imaginary sphere in an infinite mass of fluid, for which we have

Ca’ cos 0
p=- gph ieeteseseeeseseees (21).

Now 7?’ cos @’ =7 cos@, 7 sin @’sinw =r sin@sinw, 7 sin @’ cosw +r sin @ cosw = 2h;
whence 7”? = 7? + 4h? — 4hr sin 0 cos w,

and equation (21) is reduced to

Ca'r cos 0
eo 2 fr? + 4h® — 4hr sin @ cos w}i*
aoe a'r r d a
Retaining only the terms of the order os or Be so as to get the value of — to the order Be’

the above equation is reduced to

p=- SEAR cetsene cee seeneemaa ren (22),

ag

and the value of ei when 7 =a is, to the required degree of approximation,
r
Ca’ cos 0
16 h°

For the value of @ corresponding to the motion of the imaginary sphere reflected from the real
sphere, we shall therefore have ©

Ca’ cos@
| Near rere eee eer eennns see eesverves (23).
Adding together the values of @ given by (20), (22) and (23), putting 7 =a, and replacing
dC “aA Bs Ais
C by apt have, to the requisite degree of approximation,

dp a 3 a*\ dC
SiS 5 (eS Sis ;
as ( * 16 z aoe

Hence in this case the motion of the sphere will be the same as if an additional mass equal to

8 a? , : ve aie
(1 + 16 = = were collected at its centre. The effect therefore of a distant rigid plane which is

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 125

parallel to the direction of the motion of a ball pendulum will be half that of a plane at the
same distance, and perpendicular to that direction. It would seem from Poisson’s words at page
562 of the eleventh volume of the Mémoires de l'Institut, that he supposed the effect in the
former case to depend on a higher order of small quantities than that in the latter.

If the ball oscillate in a direction inclined to the plane, the motion may be easily deduced
from that in the two cases just given, by means of the principle of superposition.

11. The values of @ which have been given for the motion of translation of a sphere and
cylinder, do not require us to suppose that either the velocity, or the distance to which the
centre of the sphere or axis of the cylinder has been moved is small, provided the same particles
remain in contact with the surface. The same indeed is true of the values corresponding to a
motion of translation combined with a motion of contraction or expansion which is the same in
all directions, but varies in any manner with the time. The value of # corresponding to a motion

* cos 0
of translation of the cylinder is — we C being the velocity of the axis, and @ being

measured from a line drawn in the direction of its motion. The whole resultant of the part of
the pressure due to the square of the velocity is zero, since the velocity at the point whose co-
ordinates are 7, 0, is the same as that at the point whose co-ordinates are r and 7-@. ‘To find the

d 1. 2
resultant of the part depending on = it will be necessary to express ¢ by means of co-ordinates

referred to axes fixed in space. Let Ow, Oy, be rectangular axes passing through the centre of
any section of the cylinder, @ the angle which the direction of motion of the axis makes with
Ow, 0 the inclination of any radius vector to Ow; then

Ca? i ae eae
= - (7 cos@ cosgm +7 sin 6 sin w)
Po)

_ @&(C’x+ Cy)

em a+ y? “4
putting C’ and C” for the resolved parts of the velocity C along the axes of w and y respectively.
Taking now axes Aa’, Ay’, parallel to the former and fixed in space, aa a and 3 for the
co-ordinates of O, differentiating @ with respect to ¢, and replacing = by C’, and —— = pay, GS

and then supposing a and # to vanish, we have

2 (tl dC’ |
dp aC? —2a®(C’n + C"y)? 7 (: Ben? ids
dt aw+y @w+y') ery

The resultant of the part of the pressure due to the first two terms is zero, since the pressure
at the point (w, y) depending on these terms is the same as that at the point (—a, —y). It
will be easily found that the resultant of the whole pressure parallel to w, and acting in the

: : - Cc
negative direction, on a length / of the cylinder, is equal to rpla’ sr and that parallel to y

equal to rpla® ere The resultant of these two will be rpla?F, where F is the effective force

of a point in the axis of the cylinder, and will act in a direction opposite to that of 7. Hence
the only effect of the motion of the fluid will be, to increase the mass of the cylinder by that of
the fluid displaced. In a similar manner it may be proved that, when a solid sphere moves in
any manner in an infinite fluid, the only effect of the motion of the fluid is to increase the mass
of the sphere by half that of the fluid displaced.

126 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

A similar result may be proved to be true for any solid symmetrical with respect to two
planes at right angles to each other, and moving in the direction of the line of their intersection
in an infinitely extended fluid, the solid and fluid having been at first at rest. Let the planes
of symmetry be taken for the planes of wy and ws, the origin being fixed in the body: then it
is evident that the resultant of the pressure on the solid due to the motion will be in the direction
of the axis of w, and that there will be no resultant couple. Let C be the velocity of the solid
at any time; then the value of @ at that time will be of the form Cy(a, y, =), where C alone
contains #, (Art. 4), and the velocity of the particle whose co-ordinates are , y, x, being pro-
portional to C, the vis viva of the solid and fluid together will be proportional to C*. Now if no
forces act on the fluid and solid, except the pressure of the fluid, this vis viva must be constant*;
therefore C must be constant; therefore the resultant of the fluid pressure on the solid must be
zero. If now C be a function of ¢ we shall have

don.
P= —p(%¥ 2) +P

p being the pressure when C is constant. Since therefore the resultant of the fluid pressure

: , : dC ‘ : x ;
varies for the same solid and fluid as on the effective force, and for different fluids varies as p,

the effect of the inertia of the fluid will be, to increase the mass of the solid by m times that of
the fluid displaced, m depending only on the particular solid considered.

Let us consider two such solids, similar to each other, and having the co-ordinates planes
similarly situated, and moving with the same velocities. Let the linear dimensions of the second
be greater than those of the first in the ratio of m to 1. Let w, v, w, be the velocities, parallel
to the axes, of the particle (w, y, x) in the fluid about the first; then shall the corresponding
velocities at the point (ma, my, ms) in the fluid about the second be also w, v, w. For

udma +vdmy +wdmz =m(uda + vdy + wdz) «1. sevcee (24),

and is therefore an exact differential, since wda +vdy+wdz is one: also the normal at the
point (#, y, x) in the first surface will be inclined to the axes at the same angles as the normal
at the point (mw, my, mz) of the second surface is inclined to its axes, and therefore the normal
velocities of the two surfaces at these points are the same; and the velocities of the fluid at these
two points parallel to the axes being also the same, it follows that the normal velocity of each point
of the second surface is equal to that of the fluid in contact with it. Lastly, the motion about
the first solid being supposed to vanish at an infinite distance from it, that about the second will
vanish also. Hence the supposition made with respect to the motion of the fluid about the second
surface is correct. Now putting @ for {(wdw +vdy +wdz) for the fluid in the first case, the
corresponding integral for the fluid in the second case will be m@, if the constant be properly
chosen, as follows from equation (24). Consequently the value of that part of the expression for
the pressure, on which the resistance depends, will be m times as great for any point in the

© If an incompressible fluid which is homogeneous or hetero-
geneous, and contains in it any number of rigid bodies, be in
motion, the rigid bodies being also in motion, if the rigid bodies
are perfectly smooth, and no contacts are formed or broken among
them, and if no forces act except the pressure of the fluid, the
principle of vis viva gives

>) 2
wee ABI pd Secstccous states tenets (a),

where v is the whole velocity of the mass m, and the sign = ex-
tends over the whole fluid and the rigid bodies spoken of, and
where d§ is an element of the surface which bounds the whole,
p, the pressure about the element dS, and v the normal velocity of

the particles in that element, reckoned positive when tending into
the fluid, and where the sign // extends to all points of the bound-
ing surface, To apply equation (a) to the case of motion at
present considered, let us first confine ourselves to a spherical
portion of the fluid, whose radius is 7, and whose centre is near
the solid, so that dS refers to the surface of this portion. Let us
now suppose 7 to become infinite; then the second side of (a) will
vanish, provided p, remain finite, and v decrease in a higher ratio

than = Both of these will be true, (Art. 9.); for » will vary

é z .
ultimately as za? Since there is no alteration of volume, Hence
if the sign = extend to infinity, we shall have 2mv? constant.

}
’

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 127

second case as it is for the corresponding point in the first. Also, each element of the surface
of the second solid will be m* times as great as the corresponding element of the surface of the
first. Hence the whole resistance on the second solid will be m* times as great as that on the
first, and therefore the quantity 2 depends only on the form, and not on the size of the solid.

When forces act on the fluid, it will only be necessary to add the corresponding pressure.
Hence when a sphere descends from rest in a fluid by the action of gravity, the motion will be
the same as if a moving force equal to that of the sphere minus that of the fluid displaced
acted on a mass equal to that of the sphere plus half that of the fluid displaced. For a cylinder
which is so long that we may suppose the length infinite, descending horizontally, every thing
will be the same, except that the mass to be moved will be equal to that of the cylinder plus
the whole of the fluid displaced. In these cases, as well as in that of any solid which is sym-
metrical with respect to two vertical planes at right angles to each other, the motion will be
uniformly accelerated, and similar solids of the same material will descend with equal velocities.
These results are utterly opposed even to the commonest observation, which shews that large
solids descend much more rapidly than small ones of the same shape and material, and that the
velocity of a body falling in a fluid, (such as water), does not sensibly increase after a little
time. It becomes then of importance in the theory of resistances to inquire what may be the
cause of this discrepancy between theory and observation. The following are the only ways of
accounting for it which suggest themselves to me.

First. It has been supposed that the same particles remain in contact with the solid through-
out the motion. It must be remembered that we suppose the ultimate molecules of fluids, (if such
exist), to be so close that their distance is quite insensible, a supposition of the truth of which
there can be hardly any doubt. Consequently we reason on a fluid as if it were infinitely divisible.
Now if the motion which takes place in the cases of the sphere and cylinder be examined, sup-
posing for simplicity their motions to be rectilinear, it will be found that a particle in contact
with the surface of either moves along that surface with a velocity which at last becomes in-
finitely small, and that it does not reach the end of the sphere or cylinder from which the whole
is moving until after an infinite time, while any particle not in contact with the surface is at
last left behind. It seems difficult to conceive of what other kind the motion can be, without
supposing a line, (or rather surface) of particles to make an abrupt turn. If it should be said
that the particles may come off in tangents, it must be remembered that this sort of motion is
included in the condition which has been assumed with respect to the surface.

Secondly. The discrepancy alluded to might be supposed to arise from the friction of the
fluid against the surface of the solid. But, for the reason mentioned in the beginning of this
paper, this explanation does not appear to me satisfactory.

Thirdly. It appears to me very probable that the spreading out motion of the fluid, which
is supposed to take place behind the middle of the sphere or cylinder, though dynamically possible,
nay, the only motion dynamically possible when the conditions which have been supposed are
accurately satisfied, is unstable; so that the slightest cause produces a disturbance in the fluid,
which accumulates as the solid moves on, till the motion is quite changed. Common observation
seems to shew that, when a solid moves rapidly through a fluid at some distance below the
surface, it leaves behind it a succession of eddies in the fluid. -When the solid has attained its
terminal velocity, the product of the resistance, or rather the mean resistance, and any space
through which the solid moves, will be equal to half the vis viva of the corresponding portion
of its tail of eddies, so that the resistance will be measured by the vis viva in the length of two
units of that tail. So far therefore as the resistance which a ship experiences depends on the
disturbance of the water, which is independent of its elevation or depression, that ship which
leaves the least wake ought, according to this view, to be ceteris paribus the best sailer. The
resistance on a ship differs from that on a solid in motion immersed in a fluid in the circumstance,
that part of the resistance is employed in producing a wave. -

128 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

Fourthly, the discrepancy alluded to may be due to the mutual friction, or imperfect fluidity
of the fluid.

12. Motion about an elliptic cylinder of small eccentricity.

The value of @, which has been deduced, (Art. 8), for the motion of the fluid about a circular
cylinder, is found on the supposition that for each value of r there exists, or may be supposed
to exist, a real and finite value of @. This will be true, in any case of motion in two dimensions
where wdw + vdy is an exact differential, for those values of v for which the fluid is not interrupted,
but will be true for values of 7 for which it is interrupted by solids only when it is possible to
replace those solids at any instant by masses of fluid, without affecting the motion of the fluid
exterior to them, those masses moving in such a manner that the motion of the whole fluid might
have been produced instantaneously by impact. In some cases such a substitution could be made,
while in others it probably could not. In any case however we may try whether the expansion
given by equation (3) will enable us to get a result, and if it will, we need be in no fear that it
is wrong, (Art. 2). The same remarks will apply to the question of the possibility of the ex-
pansion of @ in the series of Laplace’s coefficients given in equation (10), for values of 7 for
which the fluid is interrupted. They will also apply to such a question as that of finding the per-
manent temperature of the earth due to the solar heat, the earth being supposed to be a homogeneous
oblate spheroid, and the points of the surface being supposed to be kept up to constant temperatures,
given by observation, depending on the latitude.

In cases of fluid motion such as those mentioned, the motion may be determined by conceiving
the whole mass of fluid divided into two or more portions, taking the most general value of @
for each portion, this value being in general expressed in a different manner for the different
portions, then limiting the general value of @ for each portion so as to satisfy the conditions
with respect to the surfaces of solids belonging to that portion, and lastly introducing the con-
dition that the velocity and direction of motion of each pair of contiguous particles in any two
of the portions are the same. The question first proposed will afford an example of this method
of solution.

Let an elliptic cylinder be moving with a velocity C, in the direction of the major axis of a
section of it made by a plane perpendicular to its axis. The motion being supposed to be in two
dimensions, it will be sufficient to consider only this section. Let

r = c(1 + € cos 260)

be the approximate equation to the ellipse so formed, the centre being the pole, and powers of ¢
above the first being neglected. Let a circle be described about the same centre, and having a radius
y equal to (1 + k)c, k being {«, and being a small quantity of the order. Let the portions of
fluid within and without the radius y be considered separately, and putting

T=C+R;,
let the value of corresponding to the former portion be
P+Qz2+ Rx,

P, Qand R being functions of @, and the term in s* being retained, in order to get the value of ae
r

true to the order e, while the terms in z*, &c. are omitted. Substituting this value of fp in
equation (2), and equating to zero coefficients of different powers of x, we have

which is the only condition to be satisfied, since the other equations would only determine the co-
efficients-of x*, &c. in terms of the preceding ones. We haye then

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 129

1 PJ
p= P+ Qz- ~ (a + a) pele eentdetst (2) =
Now if & be the angle between the normal at any point of the ellipse, and the major axis, we have
— =0 + 2esin 20,
and the velocity of the ellipse resolved along the normal
= Ccos = C (1 - €) cosO + Ce cos 36.........(26).
The velocity of the fluid at the same point resolved along the normal is

db 5. inoo
—. 2
dp zea 90’

dp 2. dp
Of ats ee 856 (27)
Let P and Q be expanded in series of cosines of @ and its multiples, so that
P=?P,cosn@, Q=2;Q,cosn 6,

there being no sines in the expansions of P and Q, since the motion is symmetrical with respect to
the major axis ; then

2
p= Saves +Q,s- = (@, = ~ 1) 8° t COS NO... 60+e0000 (28) 5

d * 1 n®
ae => $Q, - : (Q, - = Pads} COS 2O ..ceeeeee eciceiteneete (29) ;
1 do i) ie Q, Ps j
a a (2: - 7) s}sinne ts oroiaanyadaisiaalcts npeooe (30).

For a point in the ellipse, x =cecos20, whence from (27), (29) and (30), we find that the
normal velocity of the fluid

= >> {Q, cos n 0 + <[n (n — 2) *. — Q,| cos (x —2)04+=[n (n a= Q,, | cos (m + 2) 0},

which is the same thing as

nD Pe n
35 {Eom (ou ~ 2) PF Qi] + Qs + Sf (w+ 2) 2 ~ Qf e088 oan (90s

if we suppose P and Q to be zero when affected with a negative suffix. This expression will
have to be equated to the value of C cos & given by equation (26).
For the part of the fluid without the radius y we have
A
p = A, logr + ZY — cosnO*,
Vn
since there will be no sines in the expression for , because the motion is symmetrical with

respect to the major axis, and no positive powers of 1, because the velocity vanishes at an infi-
nite distance.

From the aboye value of @ we have, for the points at a distance ry from the centre,

dp A sendy
dr = ¥ cri

* The first term of this expression is accurately equal to zero, | problem in the present article independent of the proposition
since there is no expansion or contraction of the solid, (Art, 8). | referred to.
I have however retained it, in order to render the solution of the

Vou, VIII; Parr I, R

130 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

Equating the above expressions to the velocities along and perpendicular to the radius vector
given by equations (29) and (30), when x is put = ke, and then equating coefficients of cor-
responding sines and cosines we have

te
Qa — HQ, + kent = et eee eee ee ee ee (32),
. P. A
Co Osea team ane Sc Banccenasbnedoc (33),

when 2>0, and equating constant terms we have
A
(1 -k)Q = is ;

from which equation with (32) and (33) we have, putting y = (1 + A)e,

nA, A,
Se aaa when 2>0, and sie

Substituting these values in the expression (31), it becomes
A —2 nA, Ass
= +

n
et} ent ce 3

€ € ; Ain ea
s{£@+)@-2) +5 (+1) (m+ 2) } cosa + ~ eos 20,
In the case of a circular cylinder the quantities 4,, 4., d3, &c. are each zero. In the present
case therefore they are small quantities depending on e. Hence, neglecting quantities of the order
e in the above expression, it becomes
A, 2A,

ce

n
na
cos 30 — 3} :

arr COs n0,

ce

which must be equal to C§(1 —«)cos@+ecos30}. Equating coefficients of corresponding
cosines, we have

4,=-C(1i-e)c’,

As. = =<C ee*s
and the other quantities 4,, 42, &c. are of an order higher than ¢. Hence, for the part of
the fluid which lies without the radius y, we have

p= -Cia- ¢) — cos 0 + < cos BO eaten caren COA):

and for the part which lies between that radius and the ellipse we have from (28)

gp = — Ce{(1 - ©) cos + ¢cos3 6} + C§{(1 — €) cos + 3 cos 30}

The value of @ given by equation (35) may be deduced from that given by equation (34)
by putting =e +s, and expanding as far as to x. In the case of the elliptic cylinder then it
appears that the same value of @ serves for the part of the fluid without, and the part within
the radius y. If the cylinder be moving with a velocity C’ in the direction of the minor axis of a
section, the value of @ will be found from that given by equation (34) by changing the sign of e,
putting C’ for C, and supposing @ to be measured from the minor axis.

a -

ie eee

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 131

If the cylinder revolve round its axis with an angular velocity w, the normal velocity of
the surface at any point will be 2wee sin2@. Since e* is neglected, we may suppose this normal
velocity to take place on the surface of a circular cylinder whose radius is e; whence, (Art. 8), the
corresponding value of @ will be

wec* ,
sin 20.

If we suppose all these motions to take place together, we have only, (Art. 5), to add together
the values of corresponding to each. If we suppose the motion very small, so as to neglect the

: 4 d dC CKO
square of the velocity, we need only retain the terms depending on ai 2 ae and Pry in the

de

value of ae? and we may calculate the pressure due to each separately. The resultant of the

dw. :
pressure due to the term ag will evidently be zero, on account of the symmetry of the corre-

sponding motion, while the resultant couple will be of the order ¢*, since the pressure on any
point of the surface, and the perpendicular from the centre on the normal at that point, are each of

CH OIMT : ? » 63
the order e. The pressure due to the term de will evidently have a resultant in the direction

of the major axis of a section of the cylinder; and it will be easily proved that the resultant

U

d dC
pressure on a length / of the cylinder is rpc*l (1 — 2e) < That due to the term ain will be

dC’
dt
its axis oscillates in a direction making an angle a with the major axis, and if C” be its velocity,
which is supposed to be very small, the resultant pressures along the major and minor axes will be

uw

» acting along the minor axis. If the cylinder be constrained to oscillate so that

apel(1 + 2e)

pw (1 —2e) cosa ae and p» (1 + 2e) sina Ty respectively, where » is the mass of the fluid displaced.
"

dem

Resolving these pressures in the direction of the motion, the resolved part will be u (1 —2e cos 2a)

2 uw

e : 23 iad Ses
or p (1 - rhe 2a) faa, being the eccentricity ; so that the effect of the inertia of the fluid will be,

2

é : e A
to increase the mass of the solid by a mass equal to p (1 — 28 2a), which must be supposed to be

collected at the axis.

A similar method of calculation would apply to any given solid differing little either from
a circular cylinder or from a sphere. In the latter case it would be necessary to use expansions
in series of Laplace’s coefficients, instead of expansions in series of sines and cosines.

13. Motion of fluid in a closed bow whose interior is of the form of a rectangular parallelopiped.

The motion being supposed to begin from rest, the motion at any time may be supposed
to have been produced by impact (Art. 4). The motion of the box at any instant may be
resolved into a motion of translation and three motions of rotation about three axes parallel to
the edges, and passing through the centre of gravity of the fluid, and the part of @ due to
each of these motions may be calculated separately. Considering any one of the motions of
rotation, we shall see that the normal velocity of each face in consequence of it will ultimately
be the same as if that face revolved round an axis passing through its centre, and that the
latter motion would not alter the volume of the fluid. Consequently, in calculating the part of

R2

132 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

¢@ due to any one of the angular velocities, we may calculate separately the part due to the motion
of each face. :

Let the origin be in a corner of the box, the axes coinciding with its edges, Let a, 6, ce,
be these edges, U, V, W, the velocities, parallel to the axes, of the centre of gravity of the interior
of the box, w’, w”, w”, the angular velocities of the box about axes through this point parallel
to those of x, y, x. Let us first consider the part of @ due to the motion of the face wz in
consequence of the angular velocity w’”.

The value of @ corresponding to this motion must satisfy the equation

’p do
—_— NO enieie ne tele misistgltcisleivisie;s ciate s=ie'e ssiee (SO)
dz dy’ (38)
with the conditions
dp
SeI= Os WHEDIM = JO MOLHL esters cdensiessjessives(S7)>
wv
‘e = Oye When 4/10) weletetalelaee aes ee'= Sonepocodd se (38),
d
ae = Bis (« -*) 3 when y= Olieamyerarteceieie cists (39),

within limits corresponding to those of the box.
Now, for a given value of y, the value of @ between # =0 and w =a can be expanded in a

convergent series of cosines of — and its multiples; and, since (37) is satisfied, the series by
a

which = will be expressed will also hold good for the limiting values of wv, and will be conver-

av
. Nav Pan ee.
gent. The general value of @ then will be of the form =? Y,, cos ——. Substituting in (36),
a
and equating coefficients of corresponding cosines, which may be done, since any function of v can
be expanded in but one such series of cosines between the limits 0 and a, we find that the

ary nTy

general value of Y, is Ce « +C’e @ , or, changing the constants,

nt (b—y) nm (b—y) n3y

Eg gn ry
Y, ='A,((e a +e a )+ B,(e« +E @ )s
when » > 0, and for 2 =0,
Y,=A,y + B.-
From the condition (38) we have

7 Anh ppenmb, Nov
A,+— 2; nB,(e* —e @) cos =0:
a
whence A, = 0, B,=0, and, omitting B,,
nw(b-y) _nm(b—y) mr x

p= i A, (e a +e @ )cos

From the condition (39), we have

7 amb amb Nv a
——D/nA,(e* —e€ 2) cos =" (»-).
a a

amb amy 1
=e fr - (=F (e# = 6 7) ;

—

= Oates 2s ae eee

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 133

whence
nm (b—y) am(b—y)
° mt ss ——————
4a° > hes +e a Nr v
p= = i) aa FED cos air?
E€4 —'¢€ a

putting =,, for shortness, to denote the sum corresponding to odd integral values of m from 1 to co .

It is evident that the value of @ corresponding to the motion of the opposite face in con-
sequence of the angular velocity w” will be found from that just given by putting b — y for y,
and changing the sign of w”; whence the value corresponding to the motion of these two faces
in consequence of w’”’ will be =

nth nTy nth nTy
ie = —_ —
4a a? 1 (e¢ —1l)e 4 +(€ * —1)e4 NTL
>, cos
r 0 n3 nth _amh a .
€ a —€E a

Let this expression be denoted by w’’y(a, a, y, b). It is evident that the part of @ due
to the motion of the two faces parallel to the plane yx will be got by interchanging w and y,
a and 6, and changing the sign of w” in the last expression, and will therefore be—w’” yy (y, b, w, a).
The parts of @ corresponding to the angular velocities w’, w’, will be got by interchanging the
requisite quantities. Also the part of @ due to the velocities U, V, W, will be Uw + Vy + We,
(Art. 7), and therefore we have for the complete value of

Ue+Vyt+ Weta iW(e, a, y, 6) — Wy, b, v, a)} + wo {Wy by 2 €) — W(x, 6 ys 6D}
+” $1 (z, c, @, a) — (a, a, x, €)t.

According to Art. 7 we may consider separately the motion of translation of the box and
fluid, and the motion of rotation about the centre of gravity of the latter; and the whole pressure
will be compounded of the pressures due to each. The pressures at the several points of the box
due to the motion of translation will have a single resultant, which will be the same as if the
mass of the fluid were collected at its centre of gravity. Those due to the motion of rotation
will have a single resultant couple, to calculate which we have

co) = wo” Wa, a, y, b) — NACE b, @, a)} + &e.

Since for the motion of rotation there is no resultant force, we may find the resultant couple
of the pressures round any origin, that for instance which has been chosen. If now we suppose

the motion very small, so as to neglect the square of the velocity, we may find a as if the
axes were fixed in space. We have then for the motion of rotation

dw”
ie Poa iW (a, a, y, b) — Wy, 6, w, a)} — &e.

“it d mw d , d ”
Hence we may calculate separately the couples due to each of the quantities = ; a and = ¢
dt

mr

~ é d : é
It is evident from the symmetry of the motion that that due to * will act round the axis

of z, and that the pressures on the two faces perpendicular to that axis will have resultants
which are equal and opposite. Also, since y(a, a, y, b) = — \/(0, a, y, 6) and yw (a, a, 6, b)
= — \ (a, a, 0, b), it will be seen that the couples due to the pressures on the faces perpen-
dicular to the axes of w and y will be twice as great respectively as those due to the pressures
on the planes yx and wy. The pressure on the element dydx of the plane y* will be p,_,dyds,

and the moment of this pressure round the axis of x, reckoned positive when it tends to turn
the box from w to y, will be

134 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

d wn
an py vO a, y, b) — Wy, b, 0, a)} dydz.

Substituting the values of the functions, integrating from y= 0 to y =}, and from x =0 to s=¢,
: 1 ; a 3‘ ee
replacing >, — by its value 96° and reducing the other terms, it will be found that the couple
n

due to the pressure on the plane yx is

3 mw 4 d mr anh 4 mw etal

pabedw 8pac dw 1 1l-e « _ 8pbic dw ll-e @
24 dt * dt on? Sand _ dt 9 > nta*
Tv Leben e T 14 oF

We shall get the couple due to the pressure on the plane wz by interchanging a and 3,
changing the sign of w”, and measuring the couple in the opposite direction, or, which is the same,
by merely interchanging a and 6. Adding together these two couples and doubling their sum

mat wr

we shall find that the couple due to Si C2 where
dt dt
nwb nwa
82pce _ 1 l-e @ l-e 4 abe
C= 2o Tle canbe = ee (ee Olen ee (40) :
l+e 4 I+e 4

, ,

d 5 i
Similarly, the couple due to an will be — A- tending to turn the box from y to x, and

" ”

d ;
that due to at will be — B ore)

derived from C by interchanging the requisite quantities. Hence, considering the motions both
of translation and rotation of the box, we see that the small motions of the box will take place
as if the fluid were replaced by a solid having the same mass, centre of gravity, and principal
axes, and having A, B and C for its principal moments. This will be true whether forces act
on the fluid or not, provided that if there are any they are of the kind mentioned in Art. 1.

Putting 4, B,, C,, for the principal moments of inertia of the solidified fluid we have

tending to turn the box from z to w, where dA and B are

C= pabe (a? + 0).

12
Taking the ratio of C to C,, replacing each term such as
_amb nz
1-e 4 Qe ‘ 384 1
ae by 1- nai» putting for a tr
l+e 4 lt+e @

3 ; 884 . ¢
its approximate value 1°260497, and for —~ its approximate value 1:254821, and employing sub-
Tv

sidiary angles, we have

ic = 1:260497 Bessy). 1°254821 bees ~ Ls versin 20
(Gh ab(a? +b’) b(a? +B) n? 4.
b§ aot ; ‘

+ a(a* +B) 2075 versin 20, -1,

ath nTa

where tan@,=e 2¢, tan@’,=e 2, so that

nb
L tan 8, = 10 — dies L tan 6’, = 10 — ane ; where i = “6821882.

Mr. STOKES, ON SOME CASES OF FLUID MOTION. 135

The numerical calculation of this ratio is very easy, on account of the great rapidity with
which the series contained in it converge, both on account of the coefficients, and on account of the

Y

rapid diminution of the angles 6, and 6’,. The values of and = will be derived from that of 5

by putting ¢ for a in the first case, and ¢ for b in the second. The calculation of the small motions
of the box will thus be reduced to a question of ordinary rigid dynamics.

These results appear capable of being accurately compared with experiment. For this purpose
it will only be necessary to attach a box, capable of containing fluid, to a rigid body oscillating
as a pendulum. The box may itself form the rigid body. The centre of gravity of the interior
of the box should be in a vertical plane passing through the axis of suspension, which will be known
by observing whether the position of equilibrium of the whole is affected by filling the box with
fluid. The mass, moment of inertia, and depth of the centre of gravity of the solid, including the
box, must first be found. The last of these may be found by loading the upper part of the
oscillating body till the equilibrium just becomes unstable: the moment of inertia will then be found
by means of the time of oscillation when the weight is removed; or else both may be determined by
the times of oscillation when the solid is loaded with another of known mass and form and placed in
a known position, and again when it is not loaded. The same must then be done when the box is
filled with fluid. We shall thus determine the moment of inertia and depth of the centre of
gravity of the fluid; and, subtracting the moment of inertia due to the motion of translation of the
fluid, we shall thus get that due to the motion of rotation of the box, and thus determine in
succession by observation the quantities 4, B and C, or any one of them. These quantities might
also be determined by making the box oscillate by torsion, and observing the time of oscillation. It
must be remembered that the moment of inertia due to the motion of translation of the centre
of gravity of the fluid, being capable of being derived from the general dynamical principle, that the
motion of the centre of gravity of any system is the same as if the whole mass were collected there,
and the external moving forces applied there, is of no use whatever in determining the question
of the equality of the pressure in all directions, or that of the amount of friction. It would seem to’
be most convenient to have the centre of gravity of the fluid im the axis of suspension. In this case
if M, M’, be the masses of the solid and fluid, u, »’, their moments of inertia, ¢, ¢’, the times
of oscillation, in seconds, when the box is empty and when it is full respectively, h the depth of the
centre of gravity of the solid, 7 the length of the second’s pendulum, we have

n=l? Mh,
wtp =lt? Mh;
whence p/ = 1(t'* — t®) Mh.

If the centre of gravity of the fluid be at a depth h’ below the axis of suspension, we shall have
pl =1(t? —#?) Mh +1t* MX’; in this case p’— M’h’* will be the moment of inertia due to the
motion of rotation of the box.

When one of the quantities a, b, becomes infinitely great compared with the other, the ratio

s becomes 1, as will be seen from equation (40). This result might have been expected. When

‘

a=b the value of € is 156537.

‘

The experiment of the box appears capable of great variety as well as accuracy. We may take
boxes in which the edges have various ratios to each other, and may make the same box oscillate in
various positions.

15. Initial motion in a rectangular bow, the several points of the surface of which are moved
with given velocities, consistent with the condition that the volume of the fluid is not altered.

136 Mr. STOKES, ON SOME CASES OF FLUID MOTION.

Employing the same notation as in the last case, let F (w, y) be the given normal velocity at any
point of the face in the plane vy. Let Jaye F (a, y) dwdy = Wab, and let

F (a, y) = f(a, 9) + W:
then, since the normal motion of the above face due to the function f (a, y) does not alter the
volume of the fluid, we may consider separately the part of @ due to this quantity. For this

part we have
’p dp adh
+ ay? * dz?

da dy
as =0, when
d

with the conditions

@ =/0 (OF @ ecwcwe=-+-ee-- (42);

aa =0, when y=0 or Db .......2.004+-- (43),

dy

d

se ==) 0, WHHL E Cress Seatenlebevsscess+- (44),
dp

ae =f(a, y), when s=0...... “pp GaG050 (45),

within limits corresponding to those of the box.

For a given value of x the value of @ from #a=0 to «=a and from y=0 toy=b may be
expanded in a series of the form

MTree
Don
> > Pr, n COS

4 A d
the sign = referring to m, and >’ to m: and since the values of p; ne and bl do not alter
y
abruptly, and equations (42) and (43) are satisfied, it follows that the series by which @,
d
a and = are expressed are convergent, and hold good for the limiting values of a and y.
an

Substituting the value of @ just given in (41), equating to zero coefficients of corresponding
cosines, and introducing the condition (44), we have, omitting the constant, or supposing A), =0,

pm(c~z) _pm(c—=)

P=B>B, Anale © +e © + cos —

nary
>

2 2
m n
where w= —
a

é M7
Determining the coefficients such as 4,,,,, from the condition (45) in the usual manner we have,
m and n being > 0,

" 4c pr SPT 5 (9 fon Mav ny
Ag,» = aaah (« =e) § J Fe, y) cos Sra dxdy,
2 wre Se b03 a pb nry
= —d d = a d *
Aon ara b—e tb) J i} Ff (@, y) cos dady*,

with a similar expression for 4,,,, whence the value of @ corresponding to f(w,y) is known. Ina
similar manner we may find the values corresponding to the similar functions belonging to each of
the other faces. If W’' be the quantity corresponding to W for the face opposite to the plane wy,

* The function f(., y) in these integrals may be replaced by F(a, 7), since fe ( W cos a cos da dy=0, unless m=n=0.
~ a

eo =

Mr STOKES, ON SOME CASES OF FLUID MOTION. 137

and U, U’, correspond to W, W’, for the faces perpendicular to the axis of #, and if V, V’, be the
corresponding quantities for y, there remains only to be found the part of @ due to these six
quantities. Since U, U’, are the velocities parallel to the axis of w of the faces perpendicular to that
axis, and so for V, V', &c., the motion corresponding to these six quantities may be resolved into
three motions of translation parallel to the three axes, the velocities being U, V and W, and
that motion which is due to the motions of the faces opposite to the planes yx, wz, wy, moving with
velocities U’ — U, V’— V, W' — W, parallel to the axes of a, y, x, respectively. The condition
that the volume of the fluid remains the same requires that

1 1 1
cay 3a SWB (= We):
7 U U)+5V AM W) =0
It will be found that the velocities
w==(U'-U), v=F(V'- Vy w= =(W'- W),

satisfy all the requisite conditions. Hence the part of @ due to the six quantities U, U’, V, V’,
W, W’, is

1 x , y? 3
Vy + We ye I) iG any:
Ux+Vy+ +(U Uae A Pyne Vee

This quantity, added to the six others which have already been given, gives the value of @ which
contains the complete solution of the problem.

The case of motion which has just been given seems at first sight to be an imaginary one,
capable of no practical application. It may however be applied to the determination of the small
motion of a ball pendulum oscillating in a case in the form of a rectangular parallelopiped, the
dimensions of the case being great compared with the radius of the ball. For this purpose it will be
necessary to calculate the motion of the ball reflected from the case, by means of the formule
just given, and then the motion again reflected from the sphere, exactly as has been done in the case
of a rigid plane Art. 10. In the present instance however the result contains definite integrals, the
numerical calculation of which would be very troublesome.

G. G. STOKES.

PremBroke CoLiEcE,
May, 1843.

Vou. VIII. Parr I. SS)

X. Notice on the Occurrence of Land and Freshwater Shells with Bones of some
extinct Animals in the Gravel near Cambridge. By P. B. Broniz, F.G.S., of
Emmanuel College.

[Read, April 30, 1838]

Tue discovery of recent shells associated with bones of some extinct mammalia, and other animals,
is a subject of considerable interest, especially as the same fact has also been noticed in several
other distant localities. The shells in question were found in a gravel pit at Barnwell, ad-
joining the river, in a bed of fine sandy gravel, about fourteen feet from the surface, the whole bed
consisting of alternating layers of fine white sand and pebbly gravel, resting upon a thin bed of
brown clay; altogether amounting to a thickness of about twenty feet. The stratum in which
most of the shells occur is composed of a thin bed of shelly gravel, abounding in many perfect
specimens, and comminuted fragments of the same fossils. 'To this succeeds an equally thin |
bed of fine white loam, containing shells far more perfect but less numerous. This gravel, though of
course derivative, appears to differ from the coarser beds of the same formation; for while the
latter chiefly consist of rolled fragments of older rocks, the former, on the other hand, contains
but a small proportion of such materials, and appears to be more immediately derived from a
finer sediment formed by local inundations. Indeed, many of the terrestrial and aquatic shells
are of so fragile and delicate a texture, that they must have been inevitably injured had they
been swept away by any violent aqueous action. In most of the specimens, the mouths of the
Univalves, and the hinges of the Bivalves, are in excellent preservation, whilst the associated
bones exemplify the same fact. The shells are also very abundant, and generally of small size ;
all the genera, and most of the species being identical with those now living, though one or
two species do not appear to be so. Among the terrestrial specimens the following genera and
species may be enumerated.

Helix hortensis. \ Bulimus clavulus. Pupa umbilicata.
Bie carthusiana. cit -oo- Sex-dentata.

The aquatic shells afford examples of the following genera:

Cyclas, a new species. |] Valvata obtusa. ae auricularis.

2 ET BS spivorbis. YY wees glutionosa.
Succinea amphibia.

ERR oblonga. Planorbis marginatus. re
Poco Le species undetermined. ssssss. and some others.

species undetermined.

Paludina, species undetermined. { Testacellus.
Shen oon Opercule of

The above undetermined species may not, perhaps, have any living representative. The Rev.
Leonard Jenyns has decided the Cyclas to be a new species. Seed-vessels of Chara or Gyrogonite,
and wood partly charred accompany them.

The bones discovered in the shelly gravel consist of the following specimens. A large tibia
and a small molar tooth of an elephant. ‘Tibia of the gigantic ox. Lower portion of the horn
of a stag. Tibia of a deer, with teeth and vertebra of the same animal. From the brown clay
forming the basis of the gravel, and overlying the chalk marl, was obtained the pelvis of a
small elephant ; but no shells occur in this bed. Some of the other localities, in which I have
also observed the same facts, are in the neighbourhood of Maidstone in Kent, and Salisbury in

Mr BRODIE, ON LAND AND FRESHWATER SHELLS, &c. 139

Wiltshire. In the former place a bed of brown clay fills up fissures in the lower green sand,
containing bones of mammalia and other animals. The shells accompanying them belong chiefly
to the genus Pupa. In the latter locality a thick bed of brown clay affords the bones and
teeth of elephants, with remains of the horse and deer, a jaw of a fox, and some others. The
only shells hitherto found associated with them belong to the genus Helix. Recent shells also
occur with bones of numerous quadrupeds in clay and gravel near Ilford, Essex, where several
of the shells appear to be identical with those above mentioned. (See Loudon’s Magazine, Vol. 1x.
p- 263, and Lyell, Vol. 111. p. 140.) Recent marine shells have also been discovered by Sir P.. Egerton
in a bed of gravel in Cheshire, which are described in the second Volume of the Geological Pro-
ceedings. From the occurrence then of the same facts in these distant localities, it may be
asked, whether any conclusions might be drawn with regard to the probable contemporaneous
origin of these respective deposits; and what argument might be founded on the excellent
preservation of many recent land and freshwater shells associated with bones of some extinct
animals, in strata, evidently of diluvial origin.

Since writing the above, I have observed that there are two distinct beds containing shells.
The uppermost, is the fine, white sandy stratum, containing Helix and Paludina in great abun-
dance, with other shells. While the lower one, is a hard white marl (resembling chalk), charged
with numerous Pupa, small Planorbes, some Clausilia, and a very few Seed-vessels of Chara.
Large and small fragments of wood abound. ‘This distinction of the two shell beds is necessary
to be observed, because they do not contain shells common to both. No Pupa, Chara or wood
occur in the upper sandy layer; indeed the general characters of each are very different; one
being a fine sandy shelly bed; the other, a hard white marl, and in this latter formation
the bones were found. These two beds however lie within a few inches of each other, so that
the distinction is chiefly necessary, with reference to the different Testacea and Mollusca which

they each contain.
P. B. BRODIE.

Emmanvet CoLencer,
April 28, 1838.

The following Nores to the above communication are added by Professor SEpGwick.

In a paper by J. Okes, Esq., published in the first Volume of the Cambridge Transactions (p. 175), there is
a description of some fossil remains of a beaver dug up from the bed of the Old West Water about three miles
south of Chatteris: and in a subsequent communication he described numerous fossil bones found in beds of
gravel which extend from Barnwell Abbey to Jesus Common. All the specimens were subsequently deposited
in the Woodwardian Museum: and, with those derived from the Barnwell gravel, were some species of land
and fresh water shells (Helix hortensis, &c.) well preserved and in a few instances retaining traces of their
original colours. Mr Okes considered these shells to belong to the period when the bones and gravel were
deposited. But the conclusion admitted of some doubt, as the pits from which the bones were derived gave
no clear sections ; and it was just possible that the shells might have fallen down among the bones (during the
progress of the excavations), from the superficial part of the gravel.

Similar phenomena fell under my own notice, a year or two afterwards, while workmen were employed in
excavating the foundations of the new houses at the west end of Barnwell. But there was still a difficulty ;
because the sections did not shew the exact position of the shells, so as to prove that they were strictly con-
temporaneous with the deposit of the bones. The observations of Mr Brodie have settled this question, and
there can now be no doubt that the shells above mentioned were as old as the period of the gravel.

140 Mr BRODIE, ON LAND AND FRESHWATER SHELLS, &c.

It is well known, that there are three kinds of diluvial drift in the country round Cambridge.

(1.) The great brown clay forming the table-land between Cambridgeshire and Bedfordshire—extending
in patches, sometimes of very great extent, through Norfolk, Suffolk and Essex, &c. &c.—often containing
rounded masses of stone of considerable size.

(2:) A coarse gravel generally occupying the crests of hills. e.g. Harston Hill, and the Gogmagogs.
Among the rolled masses in this gravel may be found specimens of most of the older formations of England.

(3.) The fine flint-gravel of the lower lands, the well known material for the repair of the roads. It is
a very extensive formation, and though often indicating the irregular action of water, is generally at a nearly
dead level. It appears to have been formed during some long period, while the water (by the elevation of
the land) was slowly subsiding to what is called a lower level. This gravel is the newest formation of the
three; and over it comes the bog earth, the deposits from land floods, and the vegetable soil.

Bones of the beaver (?), and certainly of the mammoth, rhinoceros &c. &c., have been found in the brown
clay (No 1.), but the organic remains of mammals are most abundant in the flint-gravel (No. 3.), though
generally in a very bad state of preservation, and much rubbed and comminuted. In a few spots, where the
coarser flint-gravel alternates with masses of fine siliceous sand, the bones and teeth &c. are found entire ;
but even then they are generally so brittle that they cannot be extracted or preserved without considerable
difficulty. If the above explanation be true, there can be no difficulty in understanding how remains of
mammals, and land or freshwater shells, may have been drifted into the flint-gravel during the period of its
formation: but this is a point, the discussion of which would lead me beyond the limits of this note.

Among the fossil remains of mammals in the flint-gravel (No. 3.), the bones of the following genera
deserve notice.

(1.) Mammoth, or fossil elephant. The remains of this species are most abundant—such as molars and
tusks, sometimes quite entire ; fragments of the pelvis and other strong portions of the skeleton.

(2.) Rhinoceros, also very abundant. In the Barnwell gravel I have found many molars, a humerus,
a femur; and bones of the extremities, such as metatarsals and metacarpals, &c.

(3.) Hippopotamus—one or two fragments of molar teeth, but rare.
(4.) Equus—teeth and other bones in considerable abundance.

(5.) Bos, very abundant—Remains, such as teeth, horns, fragments of the leg bones, jaws, portions of the
cranium, &c. &c. are scattered through the flint-gravel. The species, or variety, seems to have been like that
of Walton, and was of enormous size. A portion of a fine cranium was found by Mr Okes: a perfect ramus
of the lower jaw was found by Dr Clark in the gravel pits of Parker's Piece.

(6.) Cervus—bones, &c. of several species—among them are fragments of the horns of the gigantic cervus
(Irish Elk).
A. SEDGWICK.

Trinity Couiecr,
March 8, 1844.

N.B. The above communication of Mr Brodie was unfortunately mislaid; and, in consequence, its publication has
been delayed more than five years.

TRANSACTIONS

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

VotumeE VIII. Part II.

CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS;
AND BOLD BY
JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. & J. J, DEIGHTON ; AND T. STEVENSON, CAMBRIDGE.

M.DCCC.XLIV.

XI. On the Foundation of Algebra, No. UI. By Aucustus. De Morcay,
V.P.R.A.S., F.C.P.S., of Trinity College ; Professor of Mathematics in University
College, London.

[Read, Nov. 27, 1843.]

In two former papers (Vol. vit. pp. 173, and 287) I have described the view which I take
of the three fundamental symbols of Algebra, 4 +B, AB, and 4”. This third communication is
a generalization of the view taken in the second.

In establishing an independent definition of 4’, a step which was seen to be indispensable
if all its cases are to be explained from the commencement, a preliminary extension of the idea
of a logarithm is necessary. I proposed to call the extended logarithm by the name of the
logometer. In recapitulation it may be desirable to state, that the complete symbols of the new
algebra are all capitals, that the positive and negative quantities of the older algebra are small
letters, and that the equation R = (7, p) means that # represents a line of the length 7 inclined
to the unit-line at an angle p.

Any definition of 4? may be allowed which satisfies the conditions

Ap A AraC ANCH (AG), (Am) eA
or rather the proper definition is the most general of those which satisfy these conditions ; unless
it should happen that the results of the most general definition can themselves be conveniently
expressed in terms of the results of a less general definition. This does happen, as I am about
to shew in regard to 4”; a wider definition of it even than the one I gave in my last paper is
practicable, and must be considered: though it will turn out not to be necessary, because it is
capable of expression in terms of its more simple case.

The function which, whatever be its name (I call it the logometer), plays the part of the
logarithm in a complete system of algebra, is fully defined by the equation

AA+2°XB=X(AB), or A(a, a) +A(}, B) =X (ab, a + ),
and this function being settled, 4” can be no other than \~'(B) A). In my last paper, I pro-
posed as the definition of \ (7, p), the line which has the projections log r and p on the unit-line
and its perpendicular: so that X(r,p)=logr+p,/-1. But it would equally satisfy the
fundamental condition if we were to propose as the definition of \ (7, p) the following,

A (7, p) = log r (m+ m./-1) + p(utvy/-1),

where m, ”, », v are any constants. ‘The geometrical de- pay
finition is as follows. ‘Yo find the logometer of a line R ina
or (7,p); let there be two fixed lines OF and OG oe
(m+n ,/ —1 and »+v4/-1), which we may call the
base of length and the base of direction: accordingly OF wie
has ,/(m* +n) and OG has ,/(u* + v°) units. Let ees
OP: OF :: log r (the units in the logarithm of the
length of R) : 1, and let OQ : OG as p (the units in the
angle of R) : 1; then will OR, the diagonal of the paral-
lelogram on OP and OQ, be the logometer of R.

Vout. VIII. Pazr II.

140 Mr. DE MORGAN, ON THE FOUNDATION OF ALGEBRA.

In my last paper, the bases of length and direction were on the unit-line and its perpendicular,
and the lengths of these bases were units.
If the logometer be p+ 4 \/-1, its primitive, (7, p), 1s found from

lop SEE, a PERRY
mv—Np mv — np
We are now to express X”: it is convenient to express the radical letter XY by its length and
direction (v, &), and the exponent by its projections, v+w,/—1. If X*, which by definition
is \-'(YAX), be called Z or (x, ¢), we have
log x = (v — bw) log w—ecwk, (= (v + bw) & + aw log a,

2 2 2 2
m+n mp + Mv -4+y
fae, ec

where @ =

SN Seay = b) .
mv—-ne my — ne my—-Nue

If we prefer to express the bases of length and direction by their lengths and directions, as

m+n /-1= (87), w+ v/-1= (Kye),
we have
we" 1 _ €os («-y) k 1
~ ksin(k-—y)’ ~ sin (« —y)’ = sin (k—¥y)’
which are connected by ae — 6° = 1.

Some mode of expressing X* should be contrived, such as X},,,,; which may show its
dependence on the arbitrary constants in the bases; this will allow us to reserve X* for its
common signification, as an abridged form of Xin But, before proceeding further, I may
notice that the logometer of my last paper is not as general as it might be, even on the sup-
position that X* is to have no extended meaning. For if «—‘y be a right angle, and if k = g,
then a=1, b=0, c=1, logx =v logw—wé, (=vé +w loga, which two last equations simply
express that X has the ordinary meaning. That is to say, every result in the last paper remains
if, instead of the bases of length and direction being units, they be any equal lines, and if instead
of being on the unit-line and its perpendicular, they be on any lines which are at right angles
to one another, provided only that the base of direction be a right angle in advance of that of

length.
Returning to the most general definition, we have

a

xe or (a, 3 ha = (0 bw) log. x—cwk + [(w + bw) + aw log] Vv-1

= elt @- aN Nw) log e+ (0+ Ot eV—Nu]EV-1 = gr (b-aN-l)w elt +eV-ulev—1

Of the three fundamental equations 4°4° = A®*°, A®C®=(AC)® and (A*)°=A*, it is
instantly seen that the two first are satisfied by this new signification of the exponent; and that
they are satisfied independently of the relation between a, b, and c, or ac—b*=1. The third
is a little more intricate: the formation of (X’+’~')*+”%-! requires us to write (v — bw) loga
—cw€ for log @ and (v + bw) E + aw log a for &, v’ for v and w’ for w in the first or second of.
the preceding expressions for X*. This being done, it is found that in consequence of ac — b= 1,
the result is precisely the same as if vv’ — ww’ had been written for v and vw' + v'w for w, without
any substitutes being employed for w and —. But these last changes turn

v+wa/-1 into (v + w4/-1) (v' + w' 4/-1).

The theory of quantities once called real admits of no extension; for if € and w vanish,

@ 4, = € 8", or w. But the following deductions,
aes
-0+-0V-1
ov -1 -b0+a0V-1 aVv-1 a
Ennuy = E 4 se = €naan >

Mr. DE MORGAN, ON THE FOUNDATION OF ALGEBRA. 14]

show that the signification of ordinary exponentials involving 4/—1 is completely changed: thus
eee? ~’® inclined at the angle a@@ to the unit-line.

Without going further into details we may see that, as before remarked, it is not necessary to
retain this extended notion of 4”, since the consequences of the extension can be expressed by the
particular case in general use; which cannot be said of AB as compared with 4 + B, or of A®
as compared with 4B. This rejection is a generalization of the rejection of all logarithmic bases in
favor of e, and the extended definition of A® is itself a substitution of logarithmic bases in their most
general form, For whereas, in the common system, e€ and ¢¥~! are the logarithmic bases* employed
for ordinary and periodic magnitude, we have, in the system above described, employed

Aa Ne

signifies a line of the length e

and ¢@-#V-)",

Great care will be necessary, in verifying the conclusions, not to confound the meanings of 42

Niky
and A®, or the operations performed upon them. Thus the function whose mnu v-logometer fre
may be represented by
erg a ares
mney > Nexs 5)
AX

and Sones X. Without such care, the inquirer will infallibly be led to equations of con-
dition between m, m, », and vy, which he will find are satisfied by m=1, n=0, n»=0, v=1:
that is, he will imagine he has proved the system of my last paper to be necessary.

From the expression of X in terms of its logometer, we derive the following, e¢ meaning

(e, 0);

tvV-1 ore

loge + © & loge + (~+5V-1)é
= NE aa

Xx = (2, é) = Se sng Gey a Sane ‘

On this it is to be observed, that the notion formed from the ordinary modes of expression,
namely, that in €?+4%¥-! there is a peculiar reference to length in p, and to direction in q, is not
altogether correct. The imaginary part (it may perhaps be allowed to retain the nominal dis-
tinction of real and imaginary) determines the direction, but the length depends upon both parts.
The interpretation of (chee ae is, that it represents a line of the length €?~”! inclined to the unit-
line at an angle aq; or (€’~"!, aq). One case, and one only is indefinite, when (u + v4/—1) +
(m + n/ — 1) is real, that is, when m = 0, « = 0, or when m= 0, »=0, or when m: m :: pw: ¥,
which last includes the others. In this case the line takes the form (0, «© ) or (c, co) the inde-
finite character of the result arising from the coincidence of the bases of length and direction ;
it resembles the attempt in common algebra to form a system of logarithms to the base unity.
But when (4 + v4/-—1) + (m+n /-1) = —V-1, which gives b =0, a= —1, we find (a, &)
represented by atest eat . Here the bases of length and direction are at right angles to one another,
but that of length is in advance of that of direction. This case requires that p=”, v= —m,
and the logometer is (m+m,/-—1) (logw—£&,/-1).

There would be little use in entering into more detail than is necessary to illustrate the
general meaning of the symbol A”. But it must be considered necessary, in all future explana-
tions of the elements of algebra, to point out the complete meaning of this symbol, not only
to avoid defective reasoning, but to prevent the student from attaching an undue weight to the
connexion of 4/—1 with the representation of direction. It is a strong corroboration of what
seems to have been poiuted out by the course of the complete science up to the present time,
namely, that we must not expect any new imaginary or impossible quantities. I must own that

* As far as 1 know the bases actually employed are four, e and eV- in analysis as above described, 10 in the facilitation of com-
jz ° .
putations, and ,/2 in the numerical consideration of the musical scale,

12

142 Mr. DE MORGAN, ON THE FOUNDATION OF ALGEBRA.

I rather expected to find something of the sort in the present inquiry: remembering that the first
great difficulty arose from the inverse process to addition, the next from am inverse process to
multiplication, I should not have been surprised to have found a third in the most general direct
and inverse consideration of A”. But though we are not to look for any new inexplicables
from A+B, AB, or A®, it should be remembered that there is a scale of ascent in the funda-
mental mode of deriving them from one another which does stop anywhere. Addition being
obtained, and the general notion of operation, the solution of @(~+1)=a+e gives pa =ca,
and introduces multiplication. Next o(w+1)=c@w« gives @a=c', and introduces involution.
But (w+ 1) = c®’, the solution of which gives the next step, gives for @a a function which has
not been considered; though its particular cases

pl=a, p2 =c", p3= (i prac, &e.

are known. If a could be completely inverted, new inexplicables might, and perhaps would
arise, either from this or some succeeding case.

A. DE MORGAN.

University Cotiece, Lonnon,
October 7, 1843.

—— a

XII. On the Measure of the Force of Testimony in Cases of Legal Evidence. By
JoHn Tozer, Esq. M.A., Barrister-at-Law, Fellow of Gonville and Caius
College.

[Read Nov. 27, 1843.]

* Ow the question of the possibility or advantage of measuring numerically the force of tes-
timony, the opinions which pervade the legal literature of the English language differ almost
invariably from the conclusions of science. This paper contains an attempt to trace the effect of
those conclusions in their application to a practical example, and to shew that they afford the
best means of analysing the processes which are necessarily adopted in such examples. The mere
purpose of rendering demonstrated truths more accessible, might seem to assign to the observations
which follow a place in professional rather than in scientific literature: it must however be remem-
bered, that practical men are concerned with practical rules, and with principles no further than
may be sufficient to render those rules intelligible. The occasional devotion of time to higher pur-
suits can scarely be regarded by them as other than treasonable to their personal interests; the
assertion of the supremacy of science over art they must for the most part leave to the culti-
vators of science. :

The proposition that a moral certainty is a mathematical probability whose numerical measure
lies between unity and some definite numerical fraction, puts in issue either directly or indirectly
every question that can be raised on the subject treated of in this paper, though the subject itself
is of a much more limited extent than the proposition. The vague way in which the processes
by which this proposition, and those which must stand or fall with it, can alone be established
or disproyed, have been described by even the ablest of our legal authorities, removes every
feeling of diffidence in approaching the subject. Professor Starkie, in speaking of the mode of
estimating the weight of the united testimony of numbers, says, ‘If definite degrees of probability
could be attached to the testimony of each witness, the resulting probability in favour of their united
testimony would be obtained not by the mere addition of the numbers expressing the several pro-
babilities, but by a process of multiplication.” 1 Starkie, 3rd ed. 554. And in a work there cited
occurs this passage: ‘* On one side of the equation are mentally collected all the facts and circwm-
stances which have an affirmative value; and on the other, all those which either lead to an
opposite inference, or tend to diminish the weight or to shew the non-relevancy of all or any
of the circumstances which have been put into the opposite scale. The value of each sepa-
rate portion of the evidence is separately estimated, and, as in algebraic addition, the opposite
quantities, positive and negative, are united, and the balance of probabilities is what remains
as the ground of human belief and judgment.” Wills on Circumstantial Evidence, 14.

Symbolical language has given expression to no processes of greater refinement and beauty than
those employed in the investigations of the theory of probabilities. No elaborate ones are required
in this particular application of its principles; but the expression, ‘‘a process of multiplication,”
conveys to the mind no adequate idea of the simplest of them. Subjects which have’ been deemed
worthy of their attention by Laplace and Poisson cannot be thus dismissed.

“The notions of those who have supposed ‘that mere moral probabilities or relations could
ever be represented by numbers or space, and thus be subjected to arithmetical analysis, cannot
but be regarded as visionary and chimerical.” Starkie 571.

144 Mr. TOZER, ON THE FORCE OF TESTIMONY

“« Whenever the probability is of a definite and limited nature, (whether in the proportion of
one hundred to one, or of one thousand to one, or any other ratio, is immaterial), it cannot be
safely made the ground of conviction; for to act upon it in any case would be to decide, that
for the sake of convicting many criminals, the life of one innocent man might be sacrificed.” 574.

“The distinction between evidence of a conclusive tendency which is sufficient for the pur-
pose, and that which is inconclusive, appears to be this: the latter is limited and concluded by
some degree or other of finite probability, beyond which it cannot go; the former, though not
demonstrative, is attended with a degree of probability of an indefinite and unlimited nature” Ibid.

The above short passages are cited as containing a clear enunciation of the propositions dis-
sented from, and not as affording a complete exposition of the author’s views, for which the
work itself is referred to.

A passage from Lord Brougham’s Natwral Theology is also cited to by Mr. Wills, as includ-
ing the noble author among the advocates of the truth of the last of these propositions; it does
not however appear to do so. If the propositions are true, the conclusions here arrived at must
be erroneous.

The expression of the value of a probability numerically is a necessary consequence of any
attempt to express that value accurately: if a certain event has been observed to accompany a
certain set of appearances more frequently than the appearances have been observed to occur without
the occurrence of the event, we may say that a repetition of the appearances creates a probability
of the repetition of the event—we may even say that that probability is great or small; but if we
wish to say how great or how small, we are immediately forced on the enquiry, how many times
have the appearances to our knowledge occurred, and, out of these, how many times has the event
accompanied them. That the fraction which expresses the ratio of these numbers measures the pro-
bability of the occurrence of the event accompanying the appearances, is a consequence of the
definition of the term ‘ probability ;” and if the term ‘ moral probability” have any other definition,
that definition remains yet to be enunciated.

If the appearances are of ordinary occurrence, or capable of being resolved into others which are
so, the fact that the particular combination may never before have been presented to the senses of
the person deciding, is not material; the conceiving that if they were repeated a certain number of
times the event would accompany them a certain other number of times, is a process essential to the
conception of measuring the probability at all. If, again, the appearances afford some probability
of the event, but are so unusual that the judgment hesitates to assign the definite numbers it assigns
in the previous cases, the process is only varied to this extent: instead of assigning a numerical mea-
sure to the probability itself, we assign numerical values to the limits within which it lies. The
measure here then is indefinite, but it is so because, to the imperfect experience of the observer, the
probability is so; the indefiniteness has not been introduced in the process of measurement: the
least value also that the judgment assigns to the measure of the probability may be large enough
to measure a moral certainty, or the greatest so small that the probability must in ordinary occur-
rences be disregarded, without expanding or narrowing the limits through which indefiniteness may
range. If the probability be conclusive, its conclusiveness depends on the magnitude of the least
possible value of its measure; if it raise but a “light presumption,” it would do so if the measure
of the highest limit were that of the probability itself. Suppose, for example, a medical witness to
assert, that certain appearances had led him to the conclusion that a person had died from taking
hydrocyanic acid. To determine then whether the allegation possesses the degree of probability
which would warrant our treating the fact alleged as true, we estimate the ability generally of the
witness to judge, his opportunities of judging in the particular case, and his sincerity. The phe-
nomenon then that we witness is that of a man possessing the ability and the inclination to speak
correctly, which the values we assign to these would confer on this particular witness making this
particular allegation; if then in our opinion this phenomenon would in 997 cases out of 1000 be
produced by the fact asserted, and in three cases out of 1000 by some other cause, and if we have

i

Sa ae

IN CASES OF LEGAL EVIDENCE. 145

assured ourselves that to suspend our judgments from a fear of erring no more than three times out
: tees 7 Te
of 1000 would be to defeat the purposes for which laws were instituted ; a measures a probability

which we consider large enough to warrant decision ; and the testimony of this witness does therefore
warrant decision; and it would do no more if in our judgments the witness would be impelled to give
his evidence by any other cause than the presence of the acid no more than three times in a million ;
the actual value of the probability in such a case, perhaps cannot, and certainly need not, be assigned,
but the value of its inferior limit is definite, and its measure is a numerical fraction. The mind of
the person deciding may have done no more than perceive that the probability equalled or exceeded in
magnitude those on which he habitually decided in affairs of equal importance; but if called on to
assign measures to the probabilities he has employed, he must say that his decision would not be with-
held from a fear of erring three times in 1000, and that the chance of erring in the case before him
was within that limit : the employment of numbers is a consequence of the effort to be definite. If,
again, we wish to compare the effect of evidence on different minds, though each may say in a parti-
cular instance that enough has or has not been adduced to produce conviction, the answer to the
question, how much has been adduced ? or, how much will produce conviction? is, and is necessarily,
a numerical fraction. The conclusiveness or inconclusiveness of evidence is then altogether inde-
pendent of the definiteness or indefiniteness of the probability it raises; the only condition necessary
to conclusiveness is, that that probability should be measured by a numerical fraction which exceeds
some given definite magnitude. As regards criminal cases, the nature of the evidence does not admit
that demonstration can be obtained ; we cannot therefore ensure that out of some definite number of
persons punished one innocent person will not be punished as guilty; the only effect’of making
the standard of conviction indefinite, is to make the number of cases indefinite in which the wrongful
decision has occurred; but it leaves us in doubt as to whether the injustice is increased or diminished.

It is humiliating to intellectual pride to admit that our best exertions will not protect us from
inflicting wrong on others, but nothing can be gained by shrinking from measuring the extent of our
ability to do so. ‘Selon Condorcet, la chance d’étre condamné injustement pourrait étre équiva-
lente a celle d'un danger que nous jugeons assez petite pour ne pas méme chercher a nous y
soustraire dans les habitudes de la vie; car, dit il, la société a bien le droit, pour sa stireté, d’ex-
poser un de ses membres 4 un danger dont la chance lui est, pour ainsi dire, indifférente; mais cette
consideration est beaucoup trop subtile dans une question aussi grave. Laplace donne une définition,
bien plus propre a éclairer la question, de la chance d’erreur qu’on est forcé d’admettre dans les
jugements en matiére criminelle. Selon lui cette probabilité doit @tre telle qwil y ait plus de
danger pour Ja sureté publique, 4 Yacquittement d’un coupable, que de crainte de la condamnation
@un innocent.” Poisson swr la Probabilité des Jugements. 5.

Condorcet assumes that a man has no more fear of dying at 25 than at 20, and that he therefore

neglects a probability measured by , and infers that we may neglect this in our decisions.

1900
Condorcet, Probabilité des Décisions.

If in the term ‘danger to the public” we include the danger arising from a callousness or indif-
ference to the infliction of wrong, or from a diminution of respect for the laws, the definition of
Laplace seems unexceptionable.

In the example taken below, the formula first obtained applies to all facts the truth of which
may be established or disproved by experiment; it assumes that the witnesses giving their testimony
have no wish to deceive. The peculiarity in facts of this nature is, that the repeating and varying
of the experiments tends successively to eliminate the several causes by which the appearances could
have been produced, and to leave the’ fact attested as the only known cause by which they can be
accounted for. If the tribunal be competent to judge of the skill and success with which the
experiments have been conducted, their detail is submitted to its consideration; if it be not, the
conclusions are partly arrived at by the witnesses themselves, and taken on trust by the tribunal.

146 Mr. TOZER, ON THE FORCE OF TESTIMONY

The phenomenon then actually witnessed is that of a witness alleging that, from appearances
which his experiments have produced, he infers the existence of a certain fact, and the object is to
determine the probability of that fact being true. First then we consider, for each separate experi-
ment, on the hypothesis that the fact is true, what is the probability that it would have produced
appearances sufficient to convince the mind of the witness, and induce him to give the testi-
mony he has given. We then take each of the known possible causes of such appearances, and
similarly calculate the probabilities that each one of those severally would, if it existed, have pro-
duced them in such a way as to have impelled the giving of the testimony. And, lastly, the pro-
bability of some unknown cause having so acted. The probability of the hypothesis that the
alleged fact is the true cause, is then determined by the known processes of the science. If the
operations have been conducted in symbolical language, no step has thus far been taken without the
sanction of rigid demonstration ; the effect has been to resolve the probability whose value is sought
into the elementary probabilities of which it is composed. To the next step therefore, which is that
of assigning numerical values to the symbols in which the result is expressed, has been given all the
facility of which it is capable. In the particular case of persons accused of crime, the minimum
value of the probabilities which favour the accusation alone are required, the precise numerical
value of their measures never need therefore be assigned. ‘The values which in our judgments those
which favour the hypothesis cannot fall short of, and which those that favour any other hypothesis
cannot exceed, are all that are necessary to be decided; the result is a number which is not greater
than the numerical value of the measure of the probability whose value is sought: and as far as this
particular fact is concerned, conviction or acquittal must follow, as this measure does or does not
exceed the standard which justifies decision. The actual measure of the value of the probability
is left indefinite in magnitude; its least possible value alone is defined, but the assigning of accurate
values to the elementary probabilities, and thus defining the actual measure, will not in the slightest
degree affect the result.

The next formula applies to allegations of facts the truth of which cannot be tested by experi-
ment; the consideration of the credibility of the witness is also introduced ; the modification by
which it is made to differ slightly from that given by Poisson, does not affect the principle by which
it is obtained. The hypothesis that the fact alleged is true will account for its being alleged,
first, when the witness is neither deceived, nor intending to deceive; and secondly, when both the
one and the other, provided that among the various allegations which he may make for the purpose
of deceiving, he should chance to make that which is in fact true. The various ways in which he
may be deceived without intending to deceive, endeavour to deceive without being himself deceived,
and being himself deceived also endeavour to deceive without alleging the fact which did occur; all
suggest hypotheses which will or may, with some degree of probability, account for the testimony
being given, though the fact which it alleges is not true. The probability that the hypothesis
which assumes the fact alleged to be true is the correct one, is then as before given by the scientific
process, and this whether its truth be alleged by one or more witnesses, or alleged by some and denied
by others. The antecedent probability, of the fact alleged having occurred, is also taken account of
in the formula.

The same process applies to ascertaining the probability that a fact is true which is alleged,
but which is not material to the issue, of which an example also occurs. We then have a witness
alleging a fact the probability of whose truth we have measured ; and also other facts, the probability
of whose truth we wish to measure; and the former modifies the values of the probabilities, that
the witness deceives, or is deceived, which are involved in the equations which express the latter.

When the measures of the probabilities of those facts which must be proved to sustain the
accusation have been ascertained, their product will measure the probability of a series of facts
being true, from which the truth of the accusation is an inference; the probability of the accusa-
tion being true will therefore be this product, or this product multiplied by the fraction which
expresses the probability of the inference being true, on the assumption that all the facts of the

IN CASES OF LEGAL EVIDENCE. 147

Series are so, as the inference is or is not a necessary one: and the numerical values of the com-
ponent fractions, or of their limits being assigned a numerical measure of the probability of the
accusation being true, or of its inferior limit, will be obtained; and the evidence will or will not
warrant conviction as this number does or does not exceed the certain prescribed value; and
whether the precise value be or be not definitely assigned, that is, whether the probability be defi-
nite or indefinite, will be immaterial, so long as this condition is fulfilled.

In civil cases, the questions to be decided having been elicited by the parties in their pleadings,
the value of the evidence by which they are to be determined is estimated in the same way ;
but it will frequently be necessary to assign the numerical values with greater accuracy. The
following paragraph applies to such cases, and seems also to involye an admission of all that is
contended for in favour of scientific investigation, ‘In some instances, nevertheless, where from
paucity of circumstances the usual means of judging of the credit due to conflicting witnesses fail,
it is possible that the abstract principles adverted to may operate by way of approximation,
especially in those cases where the decision is to depend on a mere preponderance of evidence.”
Starkie 554. A paucity of circumstances or incompleteness of data is what distinguishes the evi-
dence in favour of events which are merely probable, from that which supports those which are
certain, and it is the business of the science to determine the probability of the truth of the event
from the data which are offered to support or disprove it, however limited in extent these data may
be. When the numerical measure of this probability is precisely 4, the data are insufficient for
decision, and in no other case; in criminal cases, this punetwm indifferens is claimed by the legal
presumption in favour of innocence.

If, therefore, in a case where the mere preponderance is to decide, we obtain a result by
assigning to the probabilities which favour the claim of 4 the least values of which in our judg-
ments they are capable, and to those which favour the claim of B the greatest values of which
in our judgments they are capable; and another result, from the greatest which favour 4’s claim,
and the least which favour B’s; then if each of these exceed 4, the decision is in favour of A,
and if each be less than 4, in favour of B; but if one be greater and the other less than q,
more accurate values must be assigned to the numerical limits, till both the limiting values of the
probability be made to exceed or fall short of 4, or till on assigning what in our judgments are
correct values, a result precisely equal to 4 is obtained; in which latter case no decision can be
arrived at. ‘The only peculiarity then in a case whose decision must depend on a mere prepon-
derance of evidence is this, that a more accurate estimate of the probabilities it involves must be
made.

A consideration of the investigations by which these remarks are illustrated, will shew that the
mode of estimating the force of evidence employed in a court, is a process which algebraic investi-
gation analyses, and of which it explains the theory ; and an approximation, (in most cases, scien-
tifically speaking, a rude one,) to a result which is obtained with accuracy by assigning numerical
values to the algebraic symbols. The complication which exhibits itself in the algebraic process is
in the nature of the subject, and is not in any degree introduced by the operation employed.
The difficulties are difficulties which belong to the act of in any way eliciting truth from a compli-
cated series of circumstances; the practical process, to a certain extent, evades, and necessarily evades,
these; the algebraic encounters them, and resolves them into their elements. The employment
of symbolical language facilitates the processes of deductive reasoning, but does not change them ;
the assigning of numerical measures to the probabilities involved defines with accuracy their mag-
nitudes, but in nowise modifies them.

Again, the analytical process does not exclude considerations other than those which result from
the bare probabilities. Presumptions of law may be adopted in its formula, and these may be
dictated by reasons of policy, or other motives, as well as by the necessity for substituting approxi-
mations in practice. They are inferences to which legislative enactment or judicial decision has
attached the legal consequences which properly belong to facts, and analysis therefore assigns to
them the measure of certainty. At the commencement of a criminal proceeding the law presumes

Vor. VIII. Parr II. U

148 Mr. TOZER, ON THE FORCE OF TESTIMONY

that the accused is innocent ; and the analyst therefore assigns unity as a measure of the probability
of his innocence, though it may merely represent the confidence which the state reposes in the
integrity of its individual members. The claim to the property in waste lands beside a road is
advanced by the owner of the adjoining freehold, with a probability in favour of its justice measured
by unity, which must be reduced below one half by any claimant who would deprive him of
the benefit of the presumption, The consequences attached by the statutes of limitation to the ex-
piration of the periods which they assign, cause 1 or 0 to be employed as the measures of pro-
babilities imperceptibly near in value to those to which by the non-expiration of the periods 0 or 1
would be immediately before assigned. Some legal presumptions have, however, the effect of
modifying the probability, that the inference which they establish is a just one; it is perhaps
immaterial whether that raised by the production of a subsequent receipt in favour of the payment
of previous rent be absolute or capable of being rebutted. If it were absolute, and generally
known to be such, the knowledge that a conclusive presumption existed would diminish the proba-
bility of such a receipt being given when any previous rent was unpaid. In the presumptions
raised in criminal cases against the innocence of a prisoner, the probability that the inference is
just can never be less than that which justifies conviction.

The presumption of guilt in the case of stolen property of which the possession by an accused
party is unaccounted for is defined by decisions on actual cases, and becomes more accurately so
as the number of these decisions is increased: the test of consistency among these appears to be simply
this, that for each case the probability that the guilt of the accused is the cause of the unaccounted-
for possession of the property should have the same numerical measure.

Proceeding to the investigation of the reasoning processes by the algebraic solution of an example.
In a case of alleged poisoning by arsenic, to determine from the testimony of the witnesses the
probability of the presence of the poison.

Let there be witnesses, who respectively allege, with a greater or less degree of confidence, that
they discovered As; and m others that they were unable to do so; and suppose there is no doubt
about the veracity of any of them.

An ordinary jury is not competent, from a detail of the processes of experiment, to decide on the
success with which they have been conducted. The phenomenon, therefore, which they witness is
the delivery of the testimony by a number of witnesses, whose respective abilities to judge it is a
part of their duty to estimate.

In the case where As is present:

Let p,.--p, be the probabilities that the first witnesses would elicit from its presence
such appearances as to induce them to allege its presence.
Qis+*Qm that the m latter ones would do so.
Then 1-—p, 1-—q would be the probabilities that they would not succeed in doing so.

Where As is not present :

Let 7;, 7...7, be the probabilities that some other substance has caused the appearances
in the case of the first witnesses.
81, 8.-.8, in the case of the latter m witnesses.

Then 1-7, 1 —s are the probabilities that appearances causing the testimony to be given would
not be exhibited by any other substance than As.

Then if P be the probability that As was present,
P= Pr+++Pn+ (1 = G1) -++00e(L = Om)
Pree+Py > (Ll — Gy) e081 — Ym) + Mreee?n (1 — 81) «(1 = 8)
If the jury were capable of judging of the evidence as furnished by the immediate result
of the experiments, selecting among the various causes of appearances which might be mis-
taken for those produced by As, would be performed by its members instead of by the witnesses,

wt?

IN CASES OF LEGAL EVIDENCE. 149

or as well as by them. In the particular case, antimony, the persalts of tin, and probably
some other substances, exhibit with some tests what to inferior skill would be such appearances.
Suppose there are m such causes of fallacious results, and let m several experiments be made,
Pi> Gi. 71, be the respective probabilities that As, if it were present, and each of the other sub-
stances severally, if it were present, would produce the appearances witnessed by
the application of any one test.
t, that some substance, other than those known and enumerated, would do so if it were
present.
Poy Yoerete + ..00++++D, J,+--f, the corresponding probabilities for the other tests.

P the probability that As is present.

Pi + Povee+++Py

Then P= .
ProeeDn + ree, + KC... 0-. +t, .t)...8,

If this substance exist in moderate quantity, and even an ordinary degree of skill be employed,
the experiments may be varied so as to produce appearances which could not have been produced
by some one or more of the causes other than the presence of As, and therefore a factor 0 will be
successively introduced into the terms qj.-.g,, 7--.7,, &c¢. of the denominator, and the expres-

sion is reduced to
1

P=
Bikes
1 + — eee —,
Pi Pr
ae i “ t 7
It may be observed generally, that it is the presence of this term —---—* in the denomi-

Pi n
nator representing possible hypotheses, yet unthought of, that distinguishes the proof of a physical
fact from a mathematical demonstration.

The successive elimination of the known causes of error is precisely that which common sense
employs in arriving at a moral certainty; when this cannot be effected, the previous expression
remains, and the probability of the fact alleged being true is arrived at by assigning numerical
values to its elementary probabilities.

The evidence which is the subject of the following formule is that, or nearly that which was
given in a case which occurred of a woman who was accused of having caused the death of her
husband, by administering As; it is merely used as an illustration, and therefore no particular pains
is taken to state the evidence very accurately. The death and its cause were not disputed, the
probabilities therefore of the presence of As, and of its having caused the death, are taken in the
investigation as measured by 1.

The first witness, whose evidence is here considered, alleged that she had seen the accused on
the morning of a particular day making some pills.

Consider, therefore, first, the probability of a fact being true which depends for its evidence on
the testimony of a single witness. In such a case the allegation may have been made, either because
the event alleged took place, and the witness saw and believed it to do so; or because the witness
believed it to have taken place, though it did not in fact do so; or because the witness was actuated
by a wish to deceive, and made the allegation without believing in its truth. Call the event
alleged £,, and as a convenient mode of expressing the probabilities involved in the investiga-
tion, let there be » —1 other events E,......£,, which include, first, all those by the belief in
the occurrence of which the disposition, on the part of the witness, to make the particular
allegation could be influenced, whether they might in fact have occurred or not: and secondly,
all those which the witness might be induced to allege on the particular subject, without believ-

ing them whether they could or could not have occurred.
u2

150 Mr. TOZER, ON THE FORCE OF TESTIMONY

Take then,

the respective probabilities of the happening of the events £,... E,, as derived from our knowledge
of the nature of the events themselves;

b, b, b,

Sb’ SB’ OD i ST es
the probabilities that they will be respectively believed, the witness being deceived ;

the probabilities that they will respectively be alleged, the witness not believing the one alleged
to have occurred ;

u the probability that the witness is not deceived ;

» that the testimony is not given with a knowledge that it is false;

p, that the occurrence of FE, would cause the allegation to be made;

p; that the occurrence of E; any one of the events E,... E, would do so;

a, that the fact alleged is true.

Now the occurrence of E, will cause the allegation to be made, if the witness be neither deceived
nor intending to deceive, of which the probability is wv; and also, if both the one and the other,
provided the event chosen for the purpose of deceiving be that which in fact occurred.

The probability of the hypothesis is (1 — w) (1 —v), and the probability of the particular mode
of being deceived being the believing in the occurrence of some one of the events F,, other than

E, is si 5? since FE, cannot be believed, but if H, be believed, the probability that EF, will
td |

, since EZ, will not be alleged; the probability therefore that £, will be

be alleged, is — a
: sy ae b,

believed, and E, alleged is ee Sn = a

And the whole probability that E, will be alleged on the hypothesis is

a, s b by }
3b-b,\" Sa-a Sa-a!
Hence,

a Sas b
pews (wo) pts fe t.
1

Again, the occurrence of E, will cause the allegation to be made, first when the witness is

deceived, and does not intend to deceive, but believes E, to have occurred.
The probability of the hypothesis is (1 — w)v.

that E, will be believed, and therefore alleged —

Secondly, when not deceived, but intending to deceive. ;

The probability of the hypothesis is w(1 — v);

ay

that ill be alleged :
at E, will be allege paca,

Thirdly, when both deceiving and deceived provided among the modes of deceiving the allega-
tion of the occurrence of 2, be not chosen.

ee

IN CASES OF LEGAL EVIDENCE. 151

The probability of the hypothesis is (1 — uw) (1 — v);

that the belief in the occurrence of E, is the mode of being deceived, = 2
that E, will be alleged, E, being believed, —“'— ,

La-a,

a, b,

that E£, will be believed and £, alleged,

Sh. bi Se ase
the whole probability, on the hypotheses, that 2, will be alleged, is

i - Bi \.
Te TS ee see

£, and E; being excluded in the several values of E,, because one is alleged and the other happened.

Hence,
b, a, a { b ay i
-=(1- i im a Se a iT ; ,
P; = ( ES aay al sea apt 20 *) Sys, eps La-a Al
ne
ee
And a = ~
Phy PN;
eal | =p;h
Sh =a Sh, Pil

The value of =p;h,; being

h h h h
Sph,= 1 - > ee 1 a Z
bie — 4)? Spree sina} te Na fe Bat

po DG-e) [t= == i“ sole iI iz ses} -{= SHES , =|

It is here supposed that the occurrence of an event, which is not believed to have occurred, will
not affect the disposition to believe in the occurrence of any one event which did not occur in
preference to any other; and that the disposition to allege the occurrence of any event which
is not believed to have occurred in preference to any other, is independent of the event which is
believed ; if this assumption be not made, the values of a@ will be different for different values of +,
and the values of b different for the different values of i, but the process will be the same.

The expression for 2p,h; is adapted to the case of all the circumstances by which the belief
or veracity of the witness can be influenced being known; when the data are less complete the
expression becomes much simplified, the result of course becoming less accurate, as from the insuf-
ficiency of the data it must do.

Taking

1 h by b, ha a,
so, = (ame, So 7) eae
fa, Ph = ( wo h, 3b-6 sion} t ee ”4 Thy, SG, Sa

b b, h ay, a \
+(l-u)(1-») (Ps o-s ts} (2: Tb-b Tb-+,

{2 h a,.b a,b ]
h, (3b-b)(Sa-a) (Sa-a,) (3b -b)ft-
If the data be so incomplete, that among the events by the occurrence of which, either the
belief of the witness, or his disposition to allege one fact in preference to another, is influenced, no

152 Mr. TOZER, ON THE FORCE OF TESTIMONY

reason be afforded for thinking that one rather than another has occurred, h = h,; and therefore

= disappears from the formula.

; If among the events, the belief in which may have prompted the allegation, no reason be
shewn why one rather than another should be believed, 6 =6,; and the multiplier of (1 — w)v be-

Sb-b
comes Se b en
Similarly, if no reason be shewn why the witness in attempting to deceive should make any

ae eee =
Sa-a Za-a,

And lastly, if in the case of a witness both deceived and intending to deceive, if there be no
reason why the probability that he would allege any one fact should differ from the probability that

particular allegation rather than another, =

he would believe any other, a = 5; and the multiplier of (1 — ~) (1 — v) becomes (: =5 2 ) 4
a-a

With these hypotheses we therefore have

5, Pia (i - u)v+ (1-v)u+ (1 — wu) (1-2) G- =").

And with the same assumptions,

a
p= 4o+ (1-4) A=) > —

a

And
1 1 1
+ syp, ee bo) * ben) Go) Ose) ae (4).
va (<-1) Gye,

The next material allegation was made by another witness to the effect, that she saw the accused
exchange some pills which she had procured for others: the evidence of this fact, as of the former,
is contained in the testimony of a single witness; but the antecedent probability of its occurrence
is different as we do or do not believe that previously alleged. If then z, be the probability

that this allegation is true,
1

1
+ ———____,_—__—
Pi shim +h — ™)}
the previous notation being preserved and adapted as regards the value of its symbols to this
particular allegation,

T2>

: =p, fhyr, + hy - m)}>

hy h,,
DA+thy? NN Shs)
being the probabilities of the occurrence of E,... E,,, on the supposition that the previous fact is
true, and ;
hy hy
em Sth):
on the supposition that it is not true.
Among the means of assigning numerical values to the probabilities of the accuracy and

sincerity of a witness, the comparison of the allegations of different witnesses as to immaterial
facts is one of the most important. This witness also alleged, that she saw the accused procure

eam « *
= a os

OO EN,

IN CASES OF LEGAL EVIDENCE. 153

the pills for which she substituted others from a surgeon, the surgeon however alleged, that if
she had done so, an entry should appear in a book that he produced, which entry did not appear.
Let p, p; be determined as in the value of z,, and let
e be the probability that the surgeon omitted the entry from design ;
f that he did so from neglect ;
Q that the fact alleged is true.
Then the truth of the allegation is consistent with the non-appearance of the entry ;
firstly: if he designed the omission, of which the probability is e;
secondly: if he did not design the omission, but neglected to make the entry of this, the pro-
bability is (1 - e) f.
Again, whatever may have caused the allegation, any hypothesis which excludes its truth
may be taken also to exclude the possibility of an entry being made; and therefore 1 will measure
the probability of its not appearing, on every hypothesis but that of the fact having occurred.

Hence Q = :

1
1 +———_—___ Shp,
inp je+fa-eop

Referring to the expression for 7, the probability of a fact being true, which has no other sup-
port than the testimony of one witness, we see that the values of w and v which satisfy the equation

Q-——__—_,

1+ = ph;

Prh,
are those which they would possess if the probability Q were raised by the testimony of this
witness alone, and this equation therefore affords the means of correcting the values of those
quantities.

The next independent fact was, that the accused bought As; it depended for its evidence
on the testimony of a single witness; if therefore +; measure the probability that she did so,
a, will be determined by the formula for 7,; the numerical values being adapted to the particular
allegation.

The witness, who spake to the making of pills, also alleged that she saw some given by the
accused to the deceased ; and that she herself took one of them which produced effects similar to
those produced by As: as far as this testimony is concerned, the probability that poisonous pills
were administered is compounded of the probabilities that any were administered, and that those
given, if any were so, were poisonous. In this example it is assumed, that the probability that
poisonous pills were given by the accused to the deceased is, as far as the testimony of this
particular witness is concerned, the same as the probability that her previous allegation is true,
or 7m.

Let then P, be the probability that the accused knowingly possessed poison,

P, that she administered it,
P, the probability of guilt.

9
Then, as far as these facts are concerned,
P, = P,.P,.

Now each of the facts, whose probabilities are measured by 7, m2, 73 and 7, afford some
probability that each of the facts whose probabilities are measured by P,, P, is true, and the
falsehood of any number of them less than the whole does not render either P, or P, 0. The
complete expression for each of these quantities will therefore be,

154 Mr. TOZER, ON THE FORCE OF TESTIMONY

WT + kya, Te (1-75) + kya; (1-72) Ts + k;(1—m) mim 73
4 ky Q—m)2 aes + kyr (l—m) (1-2) 3 + kee (l=) 7 (1-3) + By mw, (1—22) (1-73)
+ k,(1—m)? m2 (1-m3) +h =m)? (1-72) 13 + Ayo (1-7) (1-72) (1-715).

Both the facts being certain when all the circumstances concur, the factor of the first term
is 1; K,, h,, 4%; are the probabilities that the inferences are true when three only out of the
four elementary facts concur, and the remaining one is false; k,, k;, ks, k,, when two are true
and two false; and kg, ho, #) when one only is true; 4,, &c. are not or not necessarily the same
in the value of P, as in that of Py.

The evidence in this case is so far complete, and would or would not warrant the conviction
of the accused, as P, did or did not exceed or equal the standard of conviction; there was
however in the particular case a subsequent chain of facts spoken to.

First, a witness alleged that he sold the accused As after the administering first spoken to:
if p, be the probability that this allegation is true, ra will be determined by the formula for 7,.

Let Q; be the probability of the possession of 4s by the accused, with knowledge, after this
allegation.

Then Q,=1-(1 -P,) (0 — pi):

With regard to the administering subsequent to the second purchase, three witnesses severally
alleged, that they saw the accused administer a white powder, whose appearance, from their de-
scription, corresponded with that of As.

For the probability that this fact is true, let q,, 7,, s, be the respective values of p in
the formula for 7, for each of these witnesses respectively, and let p, be the probability sought.
Then, preserving the remainder of the notation,

1751 h,
But the fact of possession, with knowledge, of which the probability is Q,, concurring with the
admitted cause of death, affords, independently of the last fact, some probability of the second

administering.
Hence, if Q, be this latter probability,

Q,= $1, po +7, (1 — p.)}-Q + 4(1 — Q) + Pe?

i,, 1,, and 1, being the respective probabilities that As was administered, when it was possessed ;
and something like it was administered, when it was simply possessed, and simply when some-
thing like it was administered ; the cause of the death being in all the cases assumed.

After this second series of facts, we get

P,=1-(1- P,P.) - QQ) = Pi P. + QQ, (1 — Pi P,).

The numerical values below are assigned for the purpose of completing the illustration, and not
with a view to obtain the actual numerical result in the particular case, the assigning of those values
is no part of the scientific process, but is determined by a consideration of the situation and character
of the witnesses, and of the manner in which they give their testimony.

The operation is also completed for the purpose of shewing, by the attempt to assign numerical
values, that the practical approximation to a correct result must necessarily be a rude one. Though the
elementary probabilities are expressed by low numbers, the resulting numbers rapidly become very
large; and to assign at once the value of the resulting probability, without the assistance of the
processes of calculation, would be necessarily to assign them very inaccurately; and the process of-at
once determining the consequences of that value must be affected with at least as great a chance of
error. We may, perhaps, in criminal cases, make as small as we please the chance of an innocent

i

eo

IN CASES OF LEGAL EVIDENCE. 155

person being convicted ; but it must be either by increasing the chance of a guilty person escaping,
or by rendering the practical process more perfect; if we were to conceive the elementary proba-
bilities to be elicited by the skill of science, and presented to the jury as separate issues, they would
then have to decide on simple facts instead of on complex series of facts, and the remainder of the
process would be logical deduction, and therefore would exclude the possibility of error. A result
so obtained would possess all the accuracy of which the subject is capable: by as much as the prac-
tical process differs from this, by so much is it, as far as mere accuracy is concerned, inferior : the
difference is the price of practicability.
In the value of 7,

10 100
u — NA oe) — 1 0.
Sat Sin ; Sa a7 B00
111
Then — >p, —
mm iF 000°
1000

For determining the value of 7.

1 1 1 1 1
: Zips T- VEZ £03 (6-)F +0 > —

. 200 100 Za-—a™ 100
hy p; fe +f (1 —e)}

The values of w and v being assigned independently of this particular allegation.

First, Q =

1 301
Th ——— ee
oe hy py ‘Pi? 20000”

and e¢ $1, ft5 Hl,

1 43 2500
ar yes See ety
npile+fa-ey ier +

2500” 2543
Since Q differs so little from 1, the values of w and v are not materially diminished by the
evidence as to this collateral fact.

and

a 1

Sa—a_ 100

1 1
Assuming, therefore, -- 1=——, and -~1l=a
8> aT) 200’ v 3 ?

43 100 + # (20000 + 99)
2500 20000 + w

°
h ee
whence @& 1
82

1 : °
Employing then this value of aol in the expression for 7,, which is

1

1
Lh Ds

hy }
™m™+—(1- h
{ 1 P, 7) ¢ Pi hy

and putting Seo gr ot Pt i
a 1

Vou. VIII. Parr II. x

156 Mr. TOZER, ON THE FORCE OF TESTIMONY

1 = + 314413
tft tT 16418204’

d ¢ 1641
and 7, ¢—
> "1673
83
For 7; I See Te
1 1
hp iPiP 5?
3
and Ts; ane
1000 1641 3
For the values of P,, P,, then 7, < —— —— =,
1) 429 sear? ™:t rs? 3 ei
And for P, alone, substituting the values
99 18 8 49 11
kyt—, ke=1, ky=2, Kye=1, hy =2, kg=—, ky=—, ky =—, ky =1, Bey = —
ee 00? 2 *) 3 > 4 > 5 ) 6 10° 7 10° 8 BO 7 9 ? 10 100°
and observing that each of the coefficients k;, ks, k;, and ‘,, multiplies the sum of two terms,
121
e get P. —
we get Pict 95
Ron P:;
199 98 195 19 if 1 101
1 > 2 ’ 3 100’ 4 100’ 5 100° 6 10’ 7 > 8 100’ 9 100° 10 100
122 14762
Whence P, t —, and P,P.
» +755” ; 2 + T5129
Again, fi E i< Zee 1=0
ain, for = —; --1=0;
gems Pu 40° 0
1 40
—— maar: y 4c _
ply phP>—, pot a3
5041
and —
Q + 5043

Also in the value of p, for each of the quantities q, 7, s,

1 haa 1 -
a | es | ae d is
ae ane Sara i
q 459
“q 4501"
Whence results
1 h 96702579
Ns a hats 51282466080 °
gil
and pez + 912 .

a

-_

a

IN CASES OF LEGAL EVIDENCE. 157

And for Q. = {l,p.+ &(1 — p2)} Q + bes - Q),

99 4 9
Jip sere a oP Sie Sera and Q, +

5041
5043”

4.5500 2293 r

45992 , and QQ + 2319”

which give Q, {

3508
. Py =P, P.+ QQ (1 — P,P.) £ ea
In the following cases, Professor Starkie has assumed that a probability to be conclusive
must be indefinite; they are inserted here for the purpose of shewing that conclusiveness is in-
dependent of this property.

Ex. Two pieces of cloth, on being compared, correspond with each other at the junction; to
determine the probability that they were originally one piece. St. 570. n.

Let the edges be divided into » corresponding portions, and let p,, ps...p, be the probabilities
that any cause, other than the pieces having been originally one, would have produced the corre-
spondence of the several portions; then, it being certain that if they were originally one piece, the
edges would correspond. The probability that this is the true cause of the corresponding, is

uN

1 + P+ Po-e-Pn
P,> Ps++-Pn, and not on the question of its value being or not being accurately determinable.

For example, if the breadth be 18 inches, and this be divided into as many equal portions, and

, and the conclusiveness of the evidence depends on the smallness of the fraction

1
if the values of p,, p....p, can be accurately assigned, and are each = OGG} then the probability

that the pieces were originally one, is , which is a definite measure. But if, as is prac-

a 10°
tically true, it would be difficult or impossible to assign these measures with accuracy, and we can

only with certainty define their limits, let p,...p, be each > =: and ¢ a , the probability will

1 1
+
10% = + 5 j0%

dence is conclusive ; but the probability whose measure is definite, is many thousand times as great
as the other.

then be +

, and the measure will be indefinite. In either case the evi-
let

Ex. A is robbed of 1 penny, 2 sixpences, 3 shillings, 4 half-crowns, 5 crowns, 6 half-sovereigns
and 7 sovereigns ; B is found in the same fair or market in possession of the same combination of
coins. No part of the coin can be identified, and no other circumstances operate against B.

*« Although the circumstances raise a high probability of identity, it is still one of a definite
and inconclusive nature.” St. ib.

The hypothesis that B is innocent of the theft is opposed by the extraordinary coincidence of the
coins in number and value: the hypothesis that he is guilty, by the fact, scarcely less extraordinary
that there should be guilt which did not afford any other circumstances of suspicion. It is submitted,
that the want of conclusiveness is a consequence of the probability that guilt, if it existed, would
have left some other evidence of its existence, being as great, or nearly as great, as the probability
that the concurring of the coins in number and value was due to their identity. It would further

x2

158 Mr. TOZER, ON THE FORCE OF TESTIMONY, &c.

appear, that extreme indefiniteness is the distinguishing character of this problem. All the data, by
which the probabilities that either 4 or B would possess this particular combination of coins could
be determined, are wanting. The algebraic solution of the problem must therefore involve a sum-
mation through every possible hypothesis for each datum, and no judgment could venture to assign
limits to the resulting probability which did not leave a very wide interval to indefiniteness. It seems
impossible to conceive any addition to the data which would render the evidence of guilt conclusive
which would not also diminish this interval, although therefore the conclusiveness would not be a
consequence of the greater degree of definiteness, the progress towards the former would neces-
sarily be accompanied by a corresponding progress towards the latter.

JOHN TOZER.

Temety, March, 1844.

XIII. On the Motion of Glaciers. By Wittiam Hopkins, M.A., Fellow of
the Cambridge Philosophical Society, of the Geological Society, and of the
Royal Astronomical Society. [Second Memoir. |

[Read Dee. 11, 1843.]

1. In my former Memoir on the Theory of Glacial Motion, I have given a full develop-
ment of the sliding theory as supported by my own experiments. According to the views there
advocated, a glacier is a dislocated mass, all the planes of dislocation, or of discontinuity in the
cohesive power being vertical or nearly so, and thus facilitating the more rapid motion of the center of
the glacier with reference to its flanks, but not that of its superficial with reference to its inferior
portion. It was shewn that the lower surface must be in a constant state of disintegration, and it
was thence inferred, that the adhesion between the glacier and its bed must be almost indefinitely
less than that between contiguous particles of the solid ice, and that, consequently, the velocities of the
superficial and inferior portions of the mass must be equal, or differ from each other by quantities
small compared with that of either portion. In my present communication, I propose to investigate
the nature of the motion on certain other hypotheses respecting the constitution of the glacial mass,
that we may compare such motion, or the effects of it, with observed phenomena, and thus be enabled
to judge of the admissibility of our hypotheses. I shall not include amongst these hypotheses those
which belong to the dilatation or expansion theories, because, after the facts observed by Professor
Forbes respecting the relative velocities at different distances from the origin of a glacier, and the con-
tinuance of glacial motion during the winter*, it appears to me impossible not to recognise the total
fallacy of those theories. I shall only therefore consider hypotheses appertaining to views of the
subject which, in common with those developed in my former memoir, agree in assigning gravity
as the immediate cause of glacial motion, but differ as to the circumstances which render it effective in
producing that motion down planes of such small inclination. The hypotheses I shall take are as
follows.

(1.) A glacier may be conceived to be divided into strata, of which the surfaces are approxi-
mately parallel to the upper or lower surface of the mass. In such case, each stratum might slide
over that immediately subjacent to it, while the lowest stratum should slide in a similar manner over
the bed of the glacier, or remain firmly attached to it. In this motion each stratum must be sup-
posed to preserve its form and continuity as a solid mass, while between two contiguous strata there
is discontinuity, in the sense in which I shall here use the term, i. e. particles originally in contact
along the common surface of two contiguous strata do not remain in contact during the motion.

(2.) While the upper part of the mass retains its solidity the inferior portions may be conceived
to become disintegrated, so that while the component particles retain their solidity they shall lose their
cohesion ; the disintegrated portion thus assuming a character similar to that of a mass of sand. In
this case, we may conceive the motion of the disintegrated portion to take place by a sliding of the
elementary component particles past each other, each particle or element of the mass retaining its
original form, like the hard grains of sand during the motion of a mass of that substance.

* Travels through the Alps of Savoy, &c.,by Professor Foibes, | who wish to obtain a knowledge of glacial phenomena, or who feel
p. 861.—This work is full of admirable and well-digested details, | interested in the many objects of beauty and sublimity which the
founded on the most careful observations and admeasurements, and | Alpine regions present to the traveller.
eaunot be too strongly recommended for the perusal of all persons

160 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

(3.) The glacial mass may be conceived to have the property of great plasticity, and to move
by a change of form in the elementry particles composing it, the continuity of the mass, in the sense
above defined, being strictly preserved. It is in this sense that the continuity of a fluid is assumed
to be preserved in those cases of fluid motion which have been subjected to mathematical analysis.

(4.) The mass may be supposed to be viscous, and the motion to take place partly by a change
of form in the elementary portions of the mass, and partly by the destruction of the continuity sup-
posed to be strictly preserved in the preceding hypothesis. ,

My immediate purpose in this communication is to investigate certain properties of the motion
which would exist in glaciers constituted according to these several hypotheses, and to examine
somewhat more in detail than in my former memoir, the state of internal pressure and tension super-
induced by the unequal velocities of the central and lateral portions of a glacier. The explanation
of this inequality of motion given in my former memoir, will apply with little alteration if we should
adopt any of the preceding hypotheses ; it will not therefore be necessary to recur to that part of
our problem. We shall have to examine more especially the relative motions of the superficial and
inferior portions of the mass, to ascertain how far they may be consistent with observed phenomena,
and thus to test the truth of our hypotheses.

2. Let us first suppose the glacier stratified as in (1).

Fig. 1-
c
D
‘ae N
a7
B
A

Let ABCD represent the vertical section of a mass reposing on the inclined plane AB,
making an angle a with the horizon; and let MN represent the surface of one of the strata of
which we here suppose the mass to consist. We have first to consider under what condition the
upper part CDMN would slide over that on which it is superincumbent. Assuming the absence
of all cohesion between contiguous strata, the only force opposing the sliding motion will be
the friction along the plane MN. Now so long as the original texture of the lower surface of
the sliding body and that of the surface on which the motion takes place, remain unaltered by the
weight of the sliding mass or other cause, it is well known that the inclination at which the
sliding will begin is independent of the weight of the sliding body, and that, if the inclination
be a, we must have

tana=np,

where p is the constant ratio which friction bears to the normal pressure. If tan a were greater
than p, the whole mass would begin to move; and (supposing the friction the same throughout)
in such a manner that the relative motion of each stratum to the one immediately subjacent to
it would be the same for all the strata. Consequently, if we could ascertain from observation
that no such relative motion existed in the upper strata, we should be certain that none existed
among the inferior strata, unless at depths at which the assumed condition that the texture of
the sliding surfaces shall remain unaltered, may be no longer satisfied.

Now judging from the observations I have made on the descent of ice down inclined planes,
I much doubt whether it be possible that two surfaces of ice at a temperature below that of
freezing could, under any circumstances, be so smooth as to admit of the sliding motion above
contemplated at an inclination so small as that of some observed glaciers; and therefore I believe

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 161

that no such motion would take place in such a glacier, for instance as that of the Lower Aar,
even if it were perfectly stratified, and there were no adhesion whatever between the strata. Much
more then is such motion impossible in the actual case of a glacier in which there is little or no
indication of stratification, and none whatever of the want of powerful cohesion between two
contiguous portions separated by any nearly horizontal plane, such as MN. If, then, any mo-
tion of the upper portion take place by its sliding over a lower portion, it must be at a depth at
which the hard and crystalline structure of the ice is destroyed*. This brings us to the second of
the cases above specified, as possible modes in which glacial motion may take place.

3. There are three causes, I conceive, which may tend to destroy the crystalline structure
of the mass—temperature, moisture, and pressure. With respect to the first we may observe,
that during the summer the interior temperature, except at points very near the lower surface,
must necessarily be lower than that immediately beneath the upper surface, where however, there
is no such disintegration of the ice as we are now contemplating. Consequently there can be no
such disintegration, as the result of temperature, in the interior of the glacier. Similarly with
respect to moisture, if no sensible disintegration result from it near the upper surface where it is
most completely disseminated by immediate infiltration, it is not to be supposed that any such
effect will be produced in the interior of the mass, except at points so near its lower surface as
to be within the influence of the sub-glacial reservoirs and currents.

It would seem then that the only cause to which we can refer any disintegration of the mass,
except at points very near the lower surface, must be the pressure of the superincumbent portion.
And this must be allowed to be a possible cause of such an effect, for it cannot be doubted that
if ice formed under a small pressure were exposed to a very great pressure, its crystalline structure
would be effectively destroyed. Still it does not follow that we can assert it to be probable
that such is actually the case in existing glaciers; for the hard crystalline structure of glacial
ice is doubtless acquired gradually, and probably, in its ultimate degree, under a pressure which
bears a considerable ratio to the greatest pressure to which it afterwards becomes subjected ; and
on this account I should deem it the more probable hypothesis that no part of a glacier becomes
disintegrated merely by the pressure which it sustains. Without dwelling, however, on the
assertion of probabilities, we may, to a certain extent, appeal to observation. M. Agassiz has
descended a vertical fissure to the depth of nearly 200 feet, but we hear of no appearance of a
change of structure in the ice, such as here supposed, and which, had it existed, could hardly
have escaped his observation. But more conclusive evidence is found in the bore which
M. Agassiz sunk to the depth of nearly 200 feet. At the bottom of it the ice was found to be
excessively hard, and so little had its structure yielded to the pressure which it sustained, that
its specific gravity could scarcely tave exceeded that of the superficial ice, as proved by the
facility with which the broken fragments rose from the bottom of the bore to the surface when
the bore was filled with water. At the depth of the bore, then, we may assert the absence of
even the smallest tendency to disintegration, and therefore we are justified in concluding by in-
duction, that no very sensible effect of that kind existed at considerably greater depths, as for
instance, at the depth of 300 feet or upwards.

4. Nor does it appear to me possible that glacial ice, retaining its crystalline structure,
should possess a degree of plasticity sufficient to admit of a motion of the kind above specified in
(8). It may be conceived to be possible that the elementary particles of a fluid mass should change
their form indefinitely, and that a continuous motion might result from such change; but solid
bodies are susceptible of a relative motion of their parts, from this change of form, by the action

* It was stated by M. Agassiz, in his Etudes sur les Glaciers, | over the lower portion, so as to indicate the relative motion above
on the authority of M. Hugi, that the upper portion of glaciers | described. It is now well known that there is a remarkable absence
may be observed in deep transverse fissures to project in strata | of such appearances.

162 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

of external forces, only to a very limited extent, especially when in large masses. Thus if a force
be applied to lengthen a given solid mass, a small extension will be the immediate effect; but
however long that force may be continued, or however slowly it may be increased, we know of no
hard solid substance capable of more than very small extension, so long as it retains that structure
on which its hardness and solidity depend. Metals, for instance, with a hard crystalline structure,
are susceptible of very small extension, until that structure is destroyed by a sufficiently elevated
temperature, when their ductility may become indefinitely great, till it becomes fluidity. In the
same manner it would seem impossible to believe that glacial ice, a substance of very hard and
highly crystalline structure, can have more than an extremely small degree of extensibility ; nor
when it approaches that temperature which dissolves it, does it appear to acquire the property
of ductility above mentioned in metals, but to pass almost immediately from a hard crystalline
texture with powerful cohesion, to a state of dissolution in which the cohesion is entirely destroyed.
Reasoning thus from the known properties of ice, and from the analogies furnished by other sub-
stances, it would seem extremely improbable that a glacier should be susceptible of a continuous
motion due to a change of form in its component particles, independently of all sliding of one par-
ticle past a contiguous one, and of the sliding of the whole mass over the bed on which it reposes.
Though the two causes of motion considered in this and the preceding article are, when strictly
analysed, essentially distinct, the motions resulting from them, so far as such motions can be
subjected to observation in glaciers, would be nearly the same. Disintegration of the mass would
seem to be essential for the effectiveness of either cause. No evidence whatever of such disintegra-
tion has been obtained from observations made at accessible depths in glaciers; but supposing it to
exist at greater depths, it would seem to me far the more probable that it should reduce the
mass to a state more analogous to that of an aggregation of sand, than to that of an extremely
plastic or semifluid substance. But whether we adopt either of these hypotheses, or that of (4)
(Art. 1), which may be regarded as a combination of the other two, it is easy to shew, as I shall
proceed to do, that the whole mass must necessarily, during its motion, be in a state of longitudinal
compression ; a conclusion which I conceive to be inconsistent with observed appearances.

5. Let the annexed diagram represent a longitudinal section of a glacier, BDH being that

Fig. 2.

ae ap
& aie oy —
amen wee
vh wart b« D
a rt ee

of the bed on which it reposes. Let MQP be a line of particles vertical at any proposed instant.
In the motion we are now contemplating each particle will have a velocity infinitesimally greater
than that of the particle immediately below it, the lowest particle at M having no motion if there
be no sliding, as I am now supposing, along the bed BH. Thus the physical line MQP will, at
successive times, form the continuous physical lines Mqp, Mq'p'. These lines, to a certain depth,
will sensibly retain their vertical position ; for it has been shewn that to a depth of about 300 feet at
least the texture of the ice is such as to admit of scarcely a sensible change of form, or, conse-
quently, of a sensible difference of velocity in different particles to that depth. In fact, the almost
invariable and continued verticality of all transverse fissures to depths not unfrequently of from

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 163

100 to 200 feet establishes this fact beyond doubt. Hence if we draw CD parallel to AEG at the
depth of about 300 feet, the portion of the mass above that line will have no motion except
that which arises from the motion of the subjacent portions CD B. But the cause of motion we are
now examining is greatest at BC where the glacier is thickest, and diminishes as we approach
to D, where it vanishes. Consequently, the tendency to move will be greatest at the upper
extremity of the glacier, and therefore the whole mass must necessarily be throughout the greater
part of its extent in a state of longitudinal compression. In fact, a large portion DH of the
glacier towards its lower extremity could have no sensible motion from the cause under considera-
tion, (since its depth is less than PQ), except that produced by the pushing force exerted upon
it by the other portion.

Now this state of longitudinal compression appears to be quite inconsistent with observed facts,
at least during the summer-months, when the motion is probably always greatest. During that
season, the velocity on the Mer de Glace of Mont Blanc appears to be considerably greatest near the
lower extremity, and all observed glaciers, as already stated, are traversed by numerous transverse
fissures—facts which indicate unequivocally a state of longitudinal extension, and not of compression.
M. Elie de Beaumont has remarked the obyious appearances of extension which glaciers present, and
Professor Forbes has borne testimony to the truth of the remark. In winter, it is probable that
there may be a tendency to more rapid motion near the upper extremity of the glacier, as explained
in my former memoir (Art. 11), and a consequent tendency to produce compression ; but if the prin-
cipal part of the motion were due to the particular constitution of the mass above supposed, the
tendency to compression would be most obvious during summer, when the motion is greatest; a
conclusion totally at variance with the results of observation.

Hence, then, it appears that any theory resting on any of the four hypotheses respecting the
constitution of a glacier above stated (Art. 1), is not only raised on a foundation unsupported by
direct experiment, but leads to results opposed to those of direct observation. The theory which
assigns the viscosity of the mass as the principal cause of glacial motion necessarily involves these
difficulties, so far as it pretends to any distinctive character which may separate it from other
theories, which, in common with it, assign gravity as the primary cause of the motion to be accounted
for. The absence of longitudinal compression in a glacier is equally opposed also to the theories of
dilatation and ewpansion.

Formation of Transverse Fissures, Since the publication of my former memoir, I have
discovered that the explanation there given of the origin of transverse fissures, and of the fact of
the convexity of the curves which they form being towards the upper extremity of the glacier,
is imperfect. I shall now offer what appears to me to render the explanation complete.

In this investigation we shall only be concerned with the difference of the velocities of the central
and lateral portions, for, at least to the depth to which observed fissures extend, there is certainly
no difference of velocity for particles in the same vertical line. We may therefore consider the
glacier independently of its thickness, or as a lamina of ice. The explanation will thus, in some
degree, be simplified.

6. When a plain solid lamina having a small degree of compressibility and extensibility, is
brought into a position of constraint by forces acting in the plane of the lamina, the particles on one
side of a geometrical line will exert certain forces on the contiguous particles on the opposite
side of the line. If the lamina were formed of fluid particles the resultant action at each point
of this line of separation would be normal to it; but when the lamina is solid this will not
be generally the case, and therefore the force at any point of the line may be resolved into
two forces, one being normal and the other tangential to the line of separation; all forces being
supposed to act in the plane of the lamina. Suppose the line of separation to be a straight line
AA parallel to the axis of w, and let pq be a portion of it so small that the action on each
point of pq may be considered equal. Let Y,.pq denote the normal force exerted by the

Vor. VIII. Parr II. ve

164 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

particles immediately above pq in the annexed figure, Fig. 3.
on those immediately below it, estimated in the direction

qB; and let f.pq represent the tangential action on pq.

Again, let the line of separation coincide with BB, paral- ye
lel to the axis of y, and perpendicular to 4’4; and let o
X,.qs denote the normal force exerted by the particles
immediately on the right of gs on the contiguous particles
immediately on the left of it, and f’.qs the tangential
action. Join p and s, and let a perpendicular to ps
make an angle @ with 4’A or the axis of a Then if
X.ps and Y.ps be the resolved parts of the forces
which the particles on one side of ps exert on those on
the opposite side, estimated in the direction gd and qB
respectively, we shall have

X = X,cos 0 + fsin 0,
Y = Y,sin0 +f" cos@.

To prove these formule we have only to observe that the forces acting on the sides pq and
qs of the triangular element pqs must be in equilibrium with the forces -.X and — Y acting
externally on the side ps, neglecting small quantities of the third order. Hence we have

—-X.ps+X,.qs+f-pq = 09,
—Y.ps+Y,.pqt+f .qs=0,
which, since it =sin@, and a cos @, prove the above formuls’*.

We have also the relation
ui = f-

To prove this equation, complete the rectangular element pqsr. A tangential force will
act on the element along the side rs in a direction opposite to that of the tangential force (/)
acting along pq, the intensity of which will not differ from f by any finite quantity; and
similarly, a force (f’) will act on the side pr in the direction opposite to that on gs. The
moments of these forces with respect to the middle point of the rectangular element, will be

3f-pq-qs, and 4f’.pq.qs.

The direction of the resultant of the normal forces on qs will pass at a distance from the
middle point of the element small compared with gs; that distance will therefore not exceed
a quantity of the second order; and consequently the moment of the force X,.qs about the
middle point of the element will not exceed a quantity of the third order, and may be neglected
in comparison with the moments of the tangential forces f and jf’, which are of the second order.
Hence, the equilibrium of the element requires that we. should have

BF +PY-98 =3F-PY- Gs,
or fs:

With this condition we have

X = X, cos +f sin,
Y = ¥, sin 8 +f cos.

If a line be drawn through gq parallel and equal to ps, the distance between the two lines

will be a small quantity of the first order, and therefore the action on the line through q may

“See Poisson’s memoir ‘¢ Sur le Mouvement des Corps élastiques,”’ in the Mémoires de I’ Institut, Vol, 111. p. 383.

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 165

be considered to have for its resolyed parts the forces XY and Y, from which they cannot differ
by quantities exceeding infinitesimals of the first order.

7- Let the length of ps, or of an equal and parallel line through g, =; the resolved
parts of the forces upon it will be XX and XY. Let AR be the force on A estimated in a
direction making an angle @ with the axis of w, then shall we have

AR=AX.cosp+rY.sin gd,
or R= Xcosp + Ysing;

R is therefore a function of the two independent variables @ and @; and I shall now proceed
to find the values of 6 and @ which render R a maximum or a minimum. Differentiating with

respect to @, we have
0=X sing — Ycos@,

which shews that for any assigned value of @, or position of the line of separation, the max-
imum value of R will be that of the resultant of X and Y, and the corresponding value
of @, that of the angle which the direction of that resultant makes with the axis of w.
Differentiating with respect to 6, we have

O eatecnh + Bear
do dé
Substituting for X and Y in these two equations, we obtain
(X, cos 6 + fsin @) sin — (Y, sin 8 + f cos @) cosh = 0,
(X, sin 8 —f cos 0) cos p — (Y, cos @ — fsin 8) sin = 0.
Eliminating @, we have
(X, cos # + f sin 0) (X, sin @ — f cos 0) - (Y, sin@ +f cos @) (Y, cos@ — f sin @) = 0,
. (Xf + Yi f) (sin’ 6 — cos’ @) + (Xt — Y7) sin 6 cos @ = 0;

2
«parr ace eR tec ee
Again, from the two preceding equations containing @ and @, we have
(X, + f tan 6) tan @ — (Y, tan@ + f) =9,
(X, tan 0 - f) — (Y, — f tan 6) tan d = 0,

or

Xtand — Y, tand +f tan @ tan d — f = 0,
X, tan@ — Y, tang +f tan@ tang — f=0.

@ and @ enter exactly in the same manner in these two equations, and must therefore be equal.
Hence

ei :
x, -Y, nat tote a co ora peoes CE

Equation (1) shews that there are two positions of the line of separation through any proposed
point, at right angles to each other, for one of which the resultant action between the particles on
opposite sides of the line at the proposed point is a maximum, and for the other a minimum; and
since determines the direction of the resultant action, equation (2) proves that direction to coincide
with the normal to the line of separation, whenever that line is in a position for which the

Y2

tan 2h =

166 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

resultant action is a maximum or minimum. ‘These conclusions may also be arrived at by some-
what different though equivalent reasoning, as follows.

8. First, to find the value of @ which gives R a maximum or minimum, we have

R? = X?+ ¥%,
and therefore
dX dy
0= X — —
mp hae?

which by substitution and reduction gives
(X\f + Y,f) (sin? @ — cos? 0) + (X} — Y}) sin @ cos @ = 0,

or
af

t 26 = ——_—_ ..

a ee,
And, secondly, taking @ as the angle which the resultant of XY and Y makes with the axis
of z, we have

aoe Es y Y, sind Bones
X X,cos@+fsind
and if we put @ =, we shall determine that position of the line of separation for which the direction
of the resultant action at any proposed point of it coincides with the normal. We thus obtain
sin 8 {.X, cos @ + f sin 0} = cos @ {Y, sin# + f cos 6},
or
(X, -— Yj) sin @ cos 0 = f (cos? 6 — sin’ @) ;
af

Xx,-Y,
This equation shews that that position of the line of separation for which =, is that which
corresponds to the maximum or minimum action between the contiguous particles on opposite sides
of the line, as before proved.

-. tan20 =

9. The maximum action here spoken of is the maximum tension at the proposed point, and
since it is perpendicular to the corresponding line of separation, there will manifestly be the
greatest tendency to form a fissure along that line, and a fisswre will be formed along it if the
maximum tension be greater than the cohesive power at the proposed point.

10. To apply the investigation to the case of a glacier, let PQ (fig. 4) be a portion of the
mass contained between two parallel vertical planes perpendicular Fig. 4.
to the axis of the glacier and indefinitely near to each other.
By the more rapid motion of the central part, the element PQ P a
will be brought into the position P’Q’; and if pqrs be an infi- P : Q
nitesimal rectangular portion of PQ, it will be brought into the '
position p’q’7’s’. Let the longitudinal axis of the glacier be that
of v. The tangential force f will arise from the greater velocity
of the central portion of the mass. It will be of the same in-
tensity, as above proved, for each side of the element, and will Pie ee
manifestly act on the sides respectively in the directions q's’
and r’p’, q‘p' and r’s’. It is this force which distorts the element
from its rectangular form. The longitudinal force X, will
generally be a tension arising from the greater velocity near the lower extremity. The transverse
force Y, in actual glaciers, in which the sides have so generally some degree of convergency, may be

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 167

frequently large, and in the preceding formulz must be made negative, because it will generally be a
pressure, and not a tension. Equation (1) thus becomes

af
Gera

If X, = 0, and Y, =0, 0 = 45° or 135°, In the case before us it is easily seen that the former
of these values corresponds to the maximum and the latter to the minimum value of R; and, there-
fore, the direction of greatest tension at p’ will bisect the angle r’p’q’, that angle being supposed,
as in the previous reasoning, to differ but little from a right angle. Consequently the greatest
tendency to form a fissure will be along a line bisecting the exterior angle rp’q’. If the value of
X,+ Y, be finite, that of @ will be less than 45°, and the direction of the fissure will so deviate
from the above-mentioned position as to approximate more nearly to perpendicularity to the sides
and axis of the glacier.

If the angle q’p’r’ should deviate from a right angle by a finite quantity before the fissure
should be formed, it would not be difficult, to shew that the line of greatest tension might be still
considered to bisect that angle. This would cause a still further deviation in the direction of the
fissure towards perpendicularity to the sides,

Since the relative motion of particles situated in a transverse line varies most rapidly in the
lateral portions, the value of f will be greatest near the sides, and vanish along the axis of the
glacier ; while the value of X, + Y, will be approximately the same at the sides and center. Conse-
quently, the value of @ will diminish as the distance from the sides increases, and the fissures will
be curved; the curvature being most rapid near the sides of the glacier, and the convexity being
turned towards the upper extremity of the glacier. The force f will probably be much more
effective than .X, in producing the fissures near the sides of the glacier, while X, will possibly be
the more effective in the central portion. The incompleteness of my former explanation consisted
in ascribing the phenomena to the latter force only, to which alone the reasoning there applied
is applicable. The above investigation appears to me to offer the complete solution of the
problem.

tan 20 —

11. Riband or Laminated Structure—I have made no attempt to account for this curious
structure in glacial ice; but I would observe that it appears to me impossible that it should be
due, as some persons, I think, have supposed, to internal tensions or pressures, producing, as their
direct and immediate effect, an almost infinite number of parallel fissures, into which water percolates,
and forms, when frozen, the bands of blue ice. It is conceivable, as an abstract hypothesis, that
a mass should be accurately homogeneous, and that the external and internal forces should be such
as to have exactly the same tendency to produce a fissure at one point of the mass as at another ;
but practically, this state is no more possible than that a body should rest in a position of unstable
equilibrium—that a cone should rest permanently on its vertex, or.a needle on its point. Allowing
the nearest practical approximation to this state of the mass, fissures would necessarily begin to be
formed, first at particular points, after which the uniformity of condition throughout would be
instantly destroyed, and irregular fissures at intervals, large, compared with those between con-
secutive bands of blue ice in the riband structure, would be the necessary consequence. I repeat,
that the formation of a system of parallel fissures, of sensible or insensible width, at distances not
exceeding a few inches, in the mass of a glacier, is no more possible than that the mass should
permanently maintain a position of unstable equilibrium.

The internal pressures and tensions here spoken of are the consequences of external forces acting
on the mass, such as gravity and the resistances of the rocks with which the glacial mass may be in
contact. ‘There is, however, another class of internal forces, the molecular forces, the existence and
nature of which may be considered independent of the external conditions to which the mass is
subjected, though their action and effects may very probably be modified by those conditions. I
have investigated the effects of the first kind of forces, and have explained how transverse and

168 Mr. HOPKINS, ON THE MOTION OF GLACIERS.

longitudinal fissures may result from their action; it is to the molecular forces that I am disposed
to attribute the veined or riband structure, their action being modified in some unknown manner
by the general conditions under which the glacier exists. In expressing this opinion I am offer-
ing no theory of the curious structure in question, but only meeting the theoretical difficulty which
it presents to us by a confession of profound ignorance of the nature and action of those forces, to
which the peculiarity of crystalline structure is generally due. The mechanical solution of the
problem I conceive to be utterly hopeless, till we shall have arrived at some solution of the general
problem which crystallization presents to us*.

12. In conclusion, I will state the principal objections which have been urged against the
sliding theory, and indicate the answers which the preceding investigations afford. In doing this,
I shall refer principally to the work of Professor Forbes, already mentioned, as that in which those
objections are most systematically stated.

(1.) The enormous friction when a glacier moves over a bed of rock, is spoken of by all
opponents of this theory as an insurmountable objection to it. My experiments shew that the
friction, or rather the force analogous to friction, is extremely small.

(2.) Professor Forbes remarks (p. 362), ‘As I understand the Gravitation theory, it supposes
the mass of the glacier to be a rigid one, sliding over its trough or bed in the manner of solid
bodies.”—I am not aware that any advocate of this theory has fallen into the absurdity of con-
sidering a glacier as a rigid, when he has spoken of it as a solid mass. I have considered it as
a dislocated mass, glacial ice itself having some degree of plasticity.

(3.) When a glacier passes out of a wider into a more contracted channel, Professor Forbes
says that “the idea of sliding, in the common legitimate sense of the word, is wholly out of the
question." —The term “sliding” is certainly not restricted to the motion of a rigid body; it is
applicable to any solid body, in the sense in which a glacier is considered to be such, and on this
hypothesis I have distinctly explained, in my former memoir, how it may pass from a wider into a
narrower channel. In objections of this nature the distinction between solidity and rigidity would
seem to be forgotten.

(4.) ‘* The inclination of the bed is seldom such as to render the overcoming of such obstacles
as the elbows and prominences, contractions and irregularities of the beds of glaciers, even conceiv-
able, being, on an average of the entire Mer de Glace, only 9°, a slope practicable to loaded carts ;
but the greater part of the surface inclines less than 5°,” (p. 363.) This difficulty has arisen in an
imperfect conception of the enormous pressure which, according to our theory, must be thrown on
abrupt local obstacles+.

(5.) Another objection is founded on the fact that changes in the rapidity of glacial movements
are found to be simultaneous with changes of external temperature. ‘*In order to reconcile this to

* At the last meeting of the British Association at Cork, | facts mentioned by Professor Forbes might admit ofa similar <=
Mr. Phillips mentioned a curious fact, which seems calculated | terpretation.
to throw some light on one of the modes in which external con- I may here mention a curious effect of crystallization in the
ditions may modify the action of molecular forces, assuming | structure of hailstones, which may possibly have some bearing on
the lamellar structure of rocks to be due to such forces, It | the question before us. I had an opportunity of witnessing it at
appeared that certain Trilobites were frequently found in some | Cambridge, on the 9th of August, 1843, during one of the most
of the older rocks in South Wales, so deformed as to their | desolating hail-storms ever known in this country. Many of the
general proportions as to present, to a casual observer, the ap- | hailstones were of the form of rather flat double convex lenses,
pearance of different species. On comparing, however, a number | nearly as large as the palm of the hand, and consisted of white
of cases, it became evident that the specimens had been com- | opaque ice in the center, surrounded by a ring of dark transparent
pressed in a direction perpendicular to the planes of structure, | ice, with an exterior ring of ice like that in the center. In some
from which it was justly inferred, that the general mass in which | cases there were two or three dark rings, the central part and the
these remains were imbedded had probably been subjected to | exterior ring being always opaque. These successive rings (with
a great pressure in the direction above mentioned. It would | the exception of their circular form) exactly resembled the alternate
seem to be a legitimate inference from this fact, that the posi- | opaque and transparent bands in glacial ice, where the riband
tion of the planes of structure had probably been mainly deter- | structure is best developed.
mined by the direction of greatest pressure. Perhaps some of the + Art. 15. of my former memoir,

Mr. HOPKINS, ON THE MOTION OF GLACIERS. 169

the sliding theory, it should be shewn that the disengagement of the glacier from its bed depends on
the kind of weather which affects its surface and temperature.” The action of the subglacial
currents does fully account, I conceive, for the phenomenon in question.

(6.) It has been contended that, according to the sliding theory, the glacier ought to descend
with an accelerated motion. This objection never had any real foundation, but only arose, as I
have shewn in my former memoir, from an erroneous conception of the nature of the retarding forces
which must act on the glacier during its sliding motion, whatever might be the cause of such
motion. My experiments, however, afford the most complete answer to the objection.

(7.) It is said that the flow of heat from the earth is not sufficient to produce the effect which
this theory ascribes to it:—I reply, that all which the theory requires is, that the lower surface
of the glacier should be constantly kept at the temperature at which the disintegration of ice com-
mences. The tangential action of the bed on the bottom of the glacier will in such case be so
modified as to render it impossible for that action to prevent all motion.

(8.) Another objection has been founded on the existence of glaciers of the secondary order,
which are observed to rest on surfaces of great inclination. Professor Forbes remarks, ‘‘ M. de
Charpentier has very justly quoted several examples as proving, that if glaciers really slide over
the soil, as De Saussure supposed, these could not for a moment sustain their position at an angle
of 30° or more,” (p. 79). M. de Charpentier, I presume, would contend that if gravity were the
primary cause of glacial motion, such a glacier would descend with the rapidity of an avalanche.
But it appeared from my experiments, that a mass of ice might be placed on a surface as smooth
as that of a paving slab at an angle of nearly 20°, without descending with an accelerated motion,
even when the lower surface of the ice was lubricated by its being in a state of dissolution. Now
these secondary glaciers are generally at a great elevation, and of no great thickness, so that it
is highly probable that a considerable portion of their lower surfaces may be frozen to the rocks
on which they rest. This circumstance, together with the probable inequalities of the surface of
those rocks, leaves no difficulty in accounting for the want of accelerated and precipitous move-
ments in such glaciers as those above spoken of, nor even in those of still greater inclination.
They will descend down their highly-inclined beds with an unaccelerated motion, and will then
be precipitated, as avalanches, down the precipices which usually form their lower boundaries.

In another part of his work, Professor Forbes appears to give an opposite phase to the
objection derived from secondary glaciers, and to make it rest on the assumed fact of these
secondary glaciers being frozen to the rocks throughout the whole of their lower surfaces. That
these glaciers are partly frozen to their beds, I have above stated to be probable; that they are
entirely so, no proof has been or can be offered. We possess no knowledge of them which does
not justify the application of the sliding theory to them, as well as to other glaciers.

W. HOPKINS.

CamBrincE,
Dec. 11, 1843.

XIV. On the Fundamental Aatithesis of Philosophy. By W.Wwuewe.t, D.D.,
Master of Trinity College, and Professor of Moral Philosophy.

[Read Feb. 5, 1844.]

I nave upon former occasions laid before the Society dissertations on certain questions which
may be termed metaphysical :—on the nature of the truth of the laws of motion:—on the ques-
tion whether all matter is heavy :—and on the question whether cause and effect are successive or
simultaneous, As these dissertations have not failed to excite some interest, I hope that I shall
have the indulgence of the Society in making a few remarks on another question of the same
kind. In doing this, as my object is to throw some light if possible on a matter of consider-
able obscurity and difficulty, I shall not attempt to avoid the occasional repetition of a sentence or
two which I may have, in substance, delivered elsewhere.

1. All persons who have attended in any degree to the views generally current of the nature
of reasoning are familiar with the distinction of necessary truths and truths of experience ; and few
such persons, or at least few students of mathematics, require to have this distinction explained
or enforced. All geometricians are satisfied that the geometrical truths with which they are con-
versant are necessarily true: they not only are true, but they must be true. ‘The meaning of the
terms being understood, and the proof being gone through, the truth of the proposition must be
assented to. That parallelograms upon the same base and between the same parallels are equal ;—
that angles in the same segment are equal ;—these are propositions which we learn to be true
by demonstrations deduced from definitions and axioms; and which, when we have thus learnt them,
we see could not be otherwise. On the other hand, there are other truths which we learn from
experience ; as for instance, that the stars revolve round the pole in one day; and that the moon
goes through her phases from full to full again in thirty days. These truths we see to be true ;
but we know them only by experience. Men never could have discovered them without looking
at the stars and the moon; and having so learnt them, still no one will pretend to say that they
are necessarily true. For aught we can see, things might have been otherwise ; and if we had been
placed in another part of the solar system, then, according to the opinions of astronomers, experience
would have presented them otherwise.

2. I take the astronomical truths of experience to contrast with the geometrical necessary truths,
as being both of a familiar definite sort; we may easily find other examples of both kinds of truth.
The truths which regard numbers are necessary truths. It is a necessary truth, that 27 and 38
are equal to 65; that half the sum of two numbers added to half their difference is equal to
the greater number. On the other hand, that sugar will dissolve in water; that plants cannot live
without light ; and in short, the whole body of our knowledge in chemistry, physiology, and the other
inductive sciences, consists of truths of experience. If there be any science which offer to us truths
of an ambiguous kind, with regard to which we may for a moment doubt whether they are neces-
sary or experiential, we will defer the consideration of them till we have marked the distinction of
the two kinds more clearly,

3. One mode in which we may express the difference of necessary truths and truths of expe-
rience, is, that necessary truths are those of which we cannot distinctly conceive the contrary.
We can very readily conceive the contrary of experiential truths. We can conceive the stars moving
about the pole or across the sky in any kind of curves with any velocities; we can conceive the
moon always appearing during the whole month as a luminous disk, as she might do if her

Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 171

light were inherent and not borrowed. But we cannot conceive one of the parallelograms on the
same base and between the same parallels larger than the other; for we find that, if we attempt to
do this, when we separate the parallelograms into parts, we have to conceive one triangle larger than
another, both having all their parts equal; which we cannot conceive at all, if we conceive the
triangles distinctly. We make this impossibility more clear by conceiving the triangles to be placed
so that two sides of the one coincide with two sides of the other; and it is then seen, that in
order to conceive the triangles unequal, we must conceive the two bases which have the same
extremities both ways, to be different lines, though both straight lines. This it is impossible to
conceive: we assent to the impossibility as an axiom, when it is expressed by saying, that two
straight lines cannot inclose a space; and thus we cannot distinctly conceive the contrary of the pro-
position just mentioned respecting parallelograms.

4. But it is necessary, in applying this distinction, to bear in mind the terms of it;—that we
cannot distinctly conceive the contrary of a necessary truth. For in a certain loose, indistinct way,
persons conceive the contrary of necessary geometrical truths, when they erroneously conceive false
propositions to be true. Thus, Hobbes erroneously held that he had discovered a means of geome-
trically doubling the cube, as it is called, that is, finding two mean proportionals between two given
lines; a problem which cannot be solved by plane geometry. Hobbes not only proposed a construction
for this purpose, but obstinately maintained that it was right, when it had been proved to be
wrong. But then, the discussion showed how indistinct the geometrical conceptions of Hobbes
were; for when his critics had proved that one of the lines in his diagram would not meet the other
in the point which his reasoning supposed, but in another point near to it; he maintained, in reply,
that one of these points was large enough to include the other, so that they might be considered as
the same point. Such a mode of conceiving the opposite of a geometrical truth, forms no excep-
tion to the assertion, that this opposite cannot be distinctly conceived.

5. In like manner, the indistinct conceptions of children and of rude savages do not invalidate
the distinction of necessary and experiential truths. Children and savages make mistakes even with
regard to numbers; and might easily happen to assert that 27 and 38 are equal to 63 or 64,
But such mistakes cannot make such arithmetical truths cease to be necessary truths. When
any person conceives these numbers and their addition distinctly, by resolving them into parts, or in
any other way, he sees that their sum is necessarily 65. If, on the ground of the possibility of
children and savages conceiving something different, it be held that this is not a necessary truth, it
must be held on the same ground, that it is not a necessary truth that 7 and 4 are equal to 11; for
children and savages might be found so unfamiliar with numbers as not to reject the assertion that
7 and 4 are 10, or even that 4 and 3 are 6, or 8. But I suppose that no persons would on such
grounds hold that these arithmetical truths are truths known only by experience.

6. Necessary truths are established, as has already been said, by demonstration, proceeding from
definitions and axioms, according to exact and rigorous inferences of reason. Truths of experience
are collected from what we see, also according to inferences of reason, but proceeding in a less exact
and rigorous mode of proof. The former depend upon the relations of the ideas which we have
in our minds: the latter depend upon the appearances or phenomena, which present themselves to
our senses. Necessary truths are formed from our thoughts, the elements of the world within us;
experiential truths are collected from things, the elements of the world without us. The truths of
experience, as they appear to us in the external world, we call Facts; and when we are able to find
among our ideas a train which will conform themselves to the apparent facts, we call this a Theory.

7. This distinction and opposition, thus expressed in various forms; as Necessary and
Experiential Truth, Ideas and Senses, Thoughts and Things, Theory and Fact, may be termed
the Fundamental Antithesis of Philosophy; for almost all the discussions of philosophers have
been employed in asserting or denying, explaining or obscuring this antithesis. It may be ex.

Vor, VIII. Parr II. Z

172 Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY.

pressed in many other ways; but is not difficult, under all these different. forms, to recognize the
same opposition: and the same remarks apply to it under its various forms, with corresponding
modifications. Thus, as we have already seen, the antithesis agrees with that of Reasoning and
Observation: again, it is identical with the opposition of Reflection and Sensation: again, sensation
deals with Objects; facts involve Objects, and generally all things without us are Objects :—
Objects of sensation, of observation, On the other hand, we ourselves who thus observe objects,
and in whom sensation is, may be called the Subjects of sensation and observation. And this
distinction of Subject and Object is one of the most general ways of expressing the fundamental
antithesis, although not yet perhaps quite familiar in English. I shall not scruple however to
speak of the Subjective and Objective element of this antithesis, where the expressions are con-
venient.

8. All these forms of antithesis, and the familiar references to them which men make in all
discussions, shew the fundamental and necessary character of the antithesis. We can have no
knowledge without the union, no philosophy without the separation, of the two elements. We can
have no knowledge, except we have both impressions on our senses from the world without, and
thoughts from our minds within:—except we attend to things, and to our ideas;—except we
are passive to receive impressions, and active to compare, combine, and mould them. But on the
other hand, philosophy seeks to distinguish the impressions of our senses from the thoughts of
our minds ;—to point out the difference of ideas and things ;—to separate the active from the
passive faculties of our being. The two elements, sensations and ideas, are both requisite to the
existence of our knowledge, as both matter and form are requisite to the existence of a body.
But philosophy considers the matter and the form separately. The properties of the form are the
subject of geometry, the properties of the matter are the subject of chemistry or mechanics.

9. But though philosophy considers these elements of knowledge separately, they cannot really
be separated, any more than can matter and form. We cannot exhibit matter without form, or
form without matter; and just as little can we exhibit sensations without ideas, or ideas without
sensations ;—the passive or the active faculties of the mind detached from each other.

In every act of my knowledge, there must be concerned the things whereof I know, and thoughts
of me who know: I must both passively receive or have received impressions, and I must actively
combine them and reason on them. No apprehension of things is purely ideal: no experience of
external things is purely sensational. If they be conceived as things, the mind must have been
awoke to the conviction of things by sensation: if they be conceived as things, the expressions of
the senses must have been bound together by conceptions. If we think of any thing, we must
recognize the existence both of thoughts and of things. The fundamental antithesis of philo-
sophy is an antithesis of inseparable elements.

10. Not only cannot these elements be separately exhibited, but they cannot be separately con-
ceived and described. The description of them must always imply their relation; and the names
by which they are denoted will consequently always bear a relative significance. And thus the
terms which denote the fundamental antithesis of philosophy cannot be applied absolutely and
exclusively in any case. We may illustrate this by a consideration of some of the common modes
of expressing the antithesis of which we speak. ‘The terms Theory and Fact are often emphatically
used as opposed to each other: and they are rightly so used. But yet it is impossible to say
absolutely in any case, This is a Fact and not a Theory; this is a Theory and not a Fact,
meaning by Theory, true Theory. Is it a fact or a theory that the stars appear to revolve round
the pole? Is it a fact or a theory that the earth is a globe revolving round its axis? Is it a
fact or a theory that the earth revolves round the sun? Is it a fact or a theory that the sun
attracts the earth? Is it a fact or a theory that a loadstone attracts a needle? In all these
cases, some persons would answer one way and some persons another. A person who has never

Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 173

watched the stars, and has only seen them from time to time, considers their circular motion round
the pole as a theory, just as he considers the motion of the sun in the ecliptic as a theory, or
the apparent motion of the inferior planets round the sun in the zodiac. A person who has
compared the measures of different parts of the earth, and who knows that these measures cannot
be conceived distinctly without supposing the earth a globe, considers its globular form a fact,
just as much as the square form of his chamber. A person to whom the grounds of believing
the earth to revolve round its axis and round the sun, are as familiar as the grounds for believing
the movements of the mail-coaches in this country, conceives the former events to be facts, just
as steadily as the latter. And a person who, believing the fact of the earth’s annual motion,
refers it distinctly to its mechanical course, conceives the sun’s attraction as a fact, just as he
conceives as a fact the action of the wind which turns the sails of a mill. We see then, that
in these cases we cannot apply absolutely and exclusively either of the terms, Fact or Theory.
Theory and Fact are the elements which correspond to our Ideas and our Senses. The Facts are
Facts so far as the Ideas have been combined with the sensations and absorbed in them: the Theories
are Theories so far as the Ideas are kept distinct from the sensations, and so far as it is considered
as still a question whether they can be made to agree with them. A true Theory is a fact,
a Fact is a familiar theory.

In like manner, if we take the terms Reasoning and Observation ; at first sight they appear to
be very distinct. Our observation of the world without us, our reasonings in our own minds, appear
to be clearly separated and opposed. But yet we shall find that we cannot apply these terms abso-
lutely and exclusively. I see a book lying a few feet from me: is this a matter of observation ?
At first, perhaps, we might be inclined to say that it clearly is so. But yet, all of us, who have
paid any attention to the process of vision, and to the mode in which we are enabled to judge
of the distance of objects, and to judge them to be distant objects at all, know that this judg-
ment involves inferences drawn from various sensations ;—from the impressions on our two eyes ;—
from our muscular sensations; and the like. These inferences are of the nature of reasoning, as much
as when we judge of the distance of an object on the other side of a river by looking at it from
different points, and stepping the distance between them. Or again: we observe the setting sun
illuminate a gilded weathercock ; but this is as much a matter of reasoning as when we observe the
phases of the moon, and infer that she is illuminated by the sun. All observation involves inferences,
and inference is reasoning.

11. Even the simplest terms by which the antithesis is expressed cannot be applied: ideas
and sensations, thoughts and things, subject and object, cannot in any case be applied absolutely and
exclusively. Our sensations require ideas to bind them together, namely, ideas of space, time, num-
ber, and the like. If not so bound together, sensations do not give us any apprehension of things
or objects. All things, all objects, must exist in space and in time—must be one or many. Now
space, time, number, are not sensations or things. They are something different from, and opposed
to sensations and things. We have termed them ideas. It may be said they are relations of
things, or of sensations. But granting this form of expression, still a relation is not a thing or a
sensation ; and therefore we must still have another and opposite element, along with our sensations.
And yet, though we have thus these two elements in every act of perception, we cannot designate
any portion of the act as absolutely and exclusively belonging to one of the elements. Perception
involves sensation, along with ideas of time, space, and the like; or, if any one prefers the expression,
involves sensations along with the apprehension of relations. Perception is sensation, along with
such ideas as make sensation into an apprehension of things or objects.

12. And as perception of objects implies ideas, as observation implies reasoning; so, on the
other hand, ideas cannot exist where sensation has not been: reasoning cannot go on when there
has not been previous observation. This is evident from the necessary order of development of
the human faculties. Sensation necessarily exists from the first moments of our existence, and is

Z2

174 Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY.

constantly at work, Observation begins before we can suppose the existence of any reasoning which
is not involved in observation. Hence, at whatever period we consider our ideas, we must consider
them as having been already engaged in connecting our sensations, and as modified by this employ-
ment. By being so employed, our ideas are unfolded and defined, and such development and
definition cannot be separated from the ideas themselves. We cannot conceive space without bound-
aries or forms; now forms involve sensations. We cannot conceive time without events which mark
the course of time; but events involve sensations. We cannot conceive number without conceiving
things which are numbered ; and things imply sensations. And the forms, things, events, which are
thus implied in our ideas, having been the objects of sensation constantly in every part of our life,
have modified, unfolded and fixed our ideas, to an extent which we cannot estimate, but which we
must suppose to be essential to the processes which at present go on in our minds. We cannot say
that objects create ideas; for to perceive objects we must already have ideas. But we may say,
that objects and the constant perception of objects have so far modified our ideas, that we cannot,
even in thought, separate our ideas from the perception of objects.

We cannot say of any ideas, as of the idea of space, or time, or number, that they are absolutely
and exclusively ideas. We cannot conceive what space, or time, or number would be in our minds,
if we had never perceived any thing or things in space or time. We cannot conceive ourselves
in such a condition as never to have perceived any thing or things in space or time. But, on the other
hand, just as little can we conceive ourselves becoming acquainted with space and time or numbers
as objects of sensation. We cannot reason without having the operations of our minds affected by
previous sensations ; but we cannot conceive reasoning to be merely a series of sensations. In order
to be used in reasoning, sensation must become observation; and, as we have seen, observation
already involves reasoning. In order to be connected by our ideas, sensations must be things or
objects, and things or objects already include ideas. And thus, as we have said, none of the terms
by which the fundamental antithesis is expressed can be absolutely and exclusively applied.

13. I now proceed to make one or two remarks suggested by the views which have thus
been presented. And first I remark, that since, as we have just seen, none of the terms which
express the fundamental antithesis can be applied absolutely and exclusively, the absolute application
of the antithesis in any particular case can never be a conclusive or immoveable principle. This
remark is the more necessary to be borne in mind, as the terms of this antithesis are often used in a
vehement and peremptory manner. ‘Thus we are often told that such a thing is a Fact and not a
Theory, with all the emphasis which, in speaking or writing, tone or italics or capitals can give.
We see from what has been said, that when this is urged, before we can estimate the truth, or the
value of the assertion, we must ask to whom is it a fact? what habits of thought, what previous
information, what ideas does it imply, to conceive the fact as a fact? Does not the apprehension of
the fact imply assumptions which may with equal justice be called theory, and which are perhaps false
theory ? in which case, the fact is no fact. Did not the ancients assert it as a fact, that the earth
stood still, and the stars moved? and can any fact have stronger apparent evidence to justify per-
sons in asserting it emphatically than this had? These remarks are by no means urged in order to
shew that no fact can be certainly known to be true; but only to shew that no fact can be certainly
shown to be a fact merely by calling it a fact, however emphatically. There is by no means any
ground of general skepticism with regard to truth involved in the doctrine of the necessary com-
bination of two elements in all our knowledge. On the contrary, ideas are requisite to the essence,
and things to the reality of our knowledge in every case. The proportions of geometry and arith-
metic are examples of knowledge respecting our ideas of space and number, with regard to which
there is no room for doubt. The doctrines of astronomy are examples of truths not less certain
respecting the external world.

14. I remark further, that since in every act of knowledge, observation or perception, both the
elements of the fundamental antithesis are involved, and involved in a manner inseparable even

Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 175

in our conceptions, it must always be possible to derive one of these elements from the other, if we
are satisfied to accept, as proof of such derivation, that one always co-exists with and implies the
other. Thus an opponent may say, that our ideas of space, time, and number, are derived from
our sensations or perceptions, because we never were in a condition in which we had the ideas of
space and time, and had not sensations or perceptions. But then, we may reply to this, that we no
sooner perceive objects than we perceive them as existing in space and time, and therefore the ideas
of space and time are not derived from the perceptions. In the same manner, an opponent may say,
that all knowledge which is involved in our reasonings is the result of experience ; for instance, our
knowledge of geometry. For every geometrical principle is presented to us by experience as true ;
beginning with the simplest, from which all others are derived by processes of exact reasoning.
But to this we reply, that experience cannot be the origin of such knowledge; for though experience
shows that such principles are true, it cannot show that they must be true, which we also know. We
never have seen, as a matter of observation, two straight lines inclosing a space; but we venture
to say further, without the smallest hesitation, that we never shall see it; and if any one were to
tell us that, according to his experience, such a form was often seen, we should only suppose that he
did not know what he was talking of. No number of acts of experience can add to the certainty of
our knowledge in this respect; which shows that our knowledge is not made up of acts of experience.
We cannot test such knowledge by experience; for if we were to try to do so, we must first know
that the lines with which we make the trial are straight; and we have no test of straightness
better than this, that two such lines cannot inclose a space. Since then, experience can neither
destroy, add to, nor test our axiomatic knowledge, such knowledge cannot be derived from expe-
rience. Since no one act of experience can affect our knowledge, no numbers of acts of experience
can make it.

15. To this a reply has been offered, that it is a characteristic property of geometric forms that
the ideas of them exactly resemble the sensations ; so that these ideas are as fit subjects of experi-
mentation as the realities themselves; and that by such experimentation we learn the truth of the
axioms of geometry. I might very reasonably ask those who use this language to explain how a
particular class of ideas can be said to resemble sensations; how, if they do, we can know it to be
so; how we can prove this resemblance to belong to geometrical ideas and sensations; and how
it comes to be an especial characteristic of those. But I will put the argument in another way.
Experiment can only show what is, not what must be. If experimentation on ideas shows what
must be, it is different from what is commonly called experience.

I may add, that not only the mere use of our senses cannot show that the axioms of geometry
must be true, but that, without the light of our ideas, it cannot even show that they are true. If we
had a segment of a circle a mile long and an inch wide, we should have two lines inclosing a space;
but we could not, by seeing or touching any part of either of them, discover that it was a bent line.

16. hat mathematical truths are not derived from experience is perhaps still more evident,
if greater evidence be possible, in the case of numbers. We assert that 7 and 8 are 15. We find it
so, if we try with counters, or in any other way. But we do not, on that account, say that the
knowledge is derived from experience. We refer to our conceptions of seven, of eight, and of addi-
tion, and as soon as we possess these conceptions distinctly, we see that the sum must be fifteen.
We cannot be said to make a trial, for we should not believe the apparent result of the trial if it
were different. If any one were to say that the multiplication table is a table of the results of experi-
ence, we should know that he could not be able to go along with us in our researches into the founda-
tions of human knowledge ; nor, indeed, to pursue with success any speculations on the subject.

17. Attempts have also been made to explain the origin of axiomatic truths by referring
them to the association of ideas. But this is one of the cases in which the word association has
been applied so widely and loosely, that no sense can be attached to it. Those who have written
with any degree of distinctness on the subject, have truly taught, that the habitual association of the

176 Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY.

ideas leads us to believe a connexion of the things: but they have never told us that this association
gave us the power of forming the ideas. Association may determine belief, but it cannot determine the
possibility of our conceptions. The African king did not believe that water could become solid, because
he had never seen it in that state. But that accident did not make it impossible to conceive it so,
any more than it is impossible for us to conceive frozen quicksilver, or melted diamond, or liquefied
air; which we may never have seen, but have no difficulty in conceiving. If there were a tropical
philosopher really incapable of conceiving water solidified, he must have been brought into that
mental condition by abstruse speculations on the necessary relations of solidity and fluidity, not by
the association of ideas.

18. To return to the results of the nature of the Fundamental Antithesis. As by assuming
universal and indissoluble connexion of ideas with perceptions, of knowledge with experience, as an
evidence of derivation, we may assert the former to be derived from the latter, so might we, on the
same ground, assert the latter to be derived from the former. We see all forms in space; and we
might hence assert all forms to be mere modifications of our idea of space. We see all events
happen in time ; and we might hence assert all events to be merely limitations and boundary-marks
of our idea of time. We conceive all collections of things as two or three, or some other number :
it might hence be asserted that we have an original idea of number, which is reflected in external
things. In this case, as in the other, we are met at once by the impossibility of this being a complete
account of our knowledge. Our ideas of space, of time, of number, however distinctly reflected to
us with limitations and modifications, must be reflected, limited and modified by-something different
from themselves. We must have visible or tangible forms to limit space, perceived events to mark
time, distinguishable objects to exemplify number. But still, in forms, and events, and objects, we
have a knowledge which they themselves cannot give us. For we know, without attending to them,
that whatever they are, they will conform and must conform to the truths of geometry and arith-
metic. There is an ideal portion in all our knowledge of the external world; and if we were
resolved to reduce all our knowledge to one of its two antithetical elements, we might say that all
our knowledge consists in the relation of our ideas. Wherever there is necessary truth, there must
be something more than sensation can supply: and the necessary truths of geometry and arithmetic
show us that our knowledge of objects in space and time depends upon necessary relations of ideas,
whatever other element it may involve.

19. This remark may be carried much further than the domain of geometry and arithmetic.
Our knowledge of matter may at first sight appear to be altogether derived from the senses. Yet
we cannot derive from the senses our knowledge of a truth which we accept as universally certain ;—
namely, that we cannot by any process add to or diminish the quantity of matter in the world.
This truth neither is nor can be derived from experience; for the experiments which we make to
verify it pre-suppose its truth. When the philosopher was asked what was the weight of smoke,
he bade the inquirer subtract the weight of the ashes from the weight of the fuel. Every one who
thinks clearly of the changes which take place in matter, assents to the justice of this reply : and
this, not because any one had found by trial that such was the weight of the smoke produced in-
combustion, but because the weight lost was assumed to have gone into some other form of matter,
not to have been destroyed. When men began to use the balance in chemical analysis, they did not
prove by trial, but took for granted, as self-evident, that the weight of the whole must be found in
the aggregate weight of the elements. Thus it is involved in the idea of matter that its amount
continues unchanged in all changes which takes place in its consistence. This is a necessary truth: and
thus our knowledge of matter, as collected from chemical experiments, is also a modification of our
idea of matter as the material of the world incapable of addition or diminution.

20. A similar remark may be made with regard to the mechanical properties of matter. Our
knowledge of these is reduced, in our reasonings, to principles which we call the laws of motion.

Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 177

These laws of motion, as I have endeavoured to shew in a paper already printed by the Society,
depend upon the idea of Cause, and involve necessary truths, which are necessarily implied in the
idea of cause ;—namely, that every change of motion must have a cause—that the effect is measured
by the cause ;—that re-action is equal and opposite to action. These principles are not derived from
experience. No one, I suppose, would derive from experience the principle, that every event must
have a cause. Every attempt to see the traces of cause in the world assumes this principle. I do
not say that these principles are anterior to experience; for I have already, I hope, shewn, that
neither of the two elements of our knowledge is, or can be, anterior to the other. But the two ele-
ments are co-ordinate in the development of the human mind ; and the ideal element may be said to
_ be the origin of our knowledge with the more propriety of the two, inasmuch as our knowledge is
the relation of ideas. The other element of knowledge, in which sensation is concerned, and which
embodies, limits, and defines the necessary truths which express the relations of our ideas, may be
properly termed experience ; and I have, in the Memoir just quoted, endeavoured to shew how the
principles concerning mechanical causation, which I have just stated, are, by observation and experi-
ment, limited and defined, so that they become the laws of motion. And thus we see that such
knowledge is derived from ideas, in a sense quite as general and rigorous, to say the least, as that in
which it is derived from experience.

21. I will take another example of this; although it is one less familiar, and the consideration
of it perhaps a little more difficult and obscure. ‘The objects which we find in the world, for
instance, minerals and plants, are of different kinds; and according to their kinds, they are called by
various names, by means of which we know what we mean when we speak of them. The discrimi-
nation of these kind of objects, according to their different forms and other properties, is the business
of chemistry and botany. And this business of discrimination, and of consequent classification,
has been carried on from the first periods of the development of the human mind, by an industrious
and comprehensive series of observations and experiments; the only way in which any portion of
the task could have been effected. But as the foundation of all this labour, and as a necessary
assumption during every part of its progress, there has been in men’s minds the principle, that
objects are so distinguishable by resemblances and differences, that they may be named, and known
by their names. This principle is involved in the idea of a Name; and without it no progress could
have been made. The principle may be briefly stated thus:—IJntelligible Names of kinds are
possible. If we suppose this not to be so, language can no longer exist, nor could the business of
human life go on. If instead of having certain definite kinds of minerals, gold, iron, copper and
the like, of which the external forms and characters are constantly connected with the same properties
and qualities, there were no connexion between the appearance and the properties of the object ;—
if what seemed externally iron might turn out to resemble lead in its hardness; and what seemed to
be gold during many trials, might at the next trial be found to be like copper; not only all the
uses of these minerals would fail, but they would not be distinguishable kinds of things, and the
names would be unmeaning. And if this entire uncertainty as to kind and properties prevailed
for all objects, the world would no longer be a world to which language was applicable. To man,
thus unable to distinguish objects into kinds, and call them by names, all knowledge would be impos-
sible, and all definite apprehension of external objects would fade away into an inconceivable
confusion. In the very apprehension of objects as intelligibly sorted, there is involved a principle
which springs within us, contemporaneous, in its efficacy, with our first intelligent perception of the
kinds of things of which the world consists. We assume, as a necessary basis of our knowledge,
that things are of definite kinds; and the aim of chemistry, botany, and other sciences is, to find
marks of these kinds; and along with these, to learn their definitely-distinguished properties. Even
here, therefore, where so large a portion of our knowledge comes from experience and observation, we
cannot proceed without a necessary truth derived from our ideas, as our fundamental principle

of knowledge.

178 Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY.

22. What the marks are, which distinguish the constant differences of kinds of things (definite
marks, selected from among many unessential appearances), and what their definite properties are,
when they are so distinguished, are parts of our knowledge to be learnt from observation, by
various processes; for instance, among others, by chemical analysis. We find the differences of
bodies, as shown by such analysis, to be of this nature:—that there are various elementary
bodies, which, combining in different definite proportions, form kinds of bodies definitely different.
But, in arriving at this conclusion, we introduce a new idea, that of Elementary Composition,
which is not extracted from the phenomena, but supplied by the mind, and introduced in order
to make the phenomena intelligible. That this notion of elementary composition is not supplied
by the chemical phenomena of cumbustion, mixture, &c. as merely an observed fact, we see from
this; that men had in ancient times performed many experiments in which elementary composition
was concerned, and had not seen the fact. It never was truly seen till modern times; and when
seen, it gave a new aspect to the whole body of known facts. This idea of elementary composition,
then, is supplied by the mind, in order to make the facts of chemical analysis and synthesis
intelligible as analysis and synthesis. And this idea being so supplied, there enters into our
knowledge along with it a corresponding necessary principle ;—-That the elementary composition of
a body determines its kind and proportions. This is, I. say, a principle assumed, as a con-
sequence of the idea of composition, not a result of experience; for when bodies have been divided
into their kinds, we take for granted that the analysis of a single specimen may serve to determine
the analysis of all bodies of the same kind: and without this assumption, chemical knowledge
with regard to the kinds of bodies would not be possible. It has been said that we take only
one experiment to determine the composition of any particular kind of body, because we have
a thousand experiments to determine that bodies of the same kind have the same composition.
But this is not so. Our belief in the principle that bodies of the same kind have the same com-
position is not established by experiments, but is assumed as a necessary consequence of the ideas of
Kind and of Composition. If, in our experiments, we found that bodies supposed to be of the same
kind had not the same composition, we should not at all doubt of the principle just stated, but
conclude at once that the bodies were mot of the same kind ;—that the marks by which the kinds
are distinguished had been wrongly stated. This is what has very frequently happened in the
course of the investigations of chemists and mineralogists. And thus we have it, not as an
experiential fact, but as a necessary principle of chemical philosophy, that the Elementary Com-
position of a body determines its Kind and Properties.

23. How bodies differ in their elementary composition, experiment must teach us, as we have
already said that experiment has taught us. But as we have also said, whatever be the nature
of this difference, kinds must be definite, in order that language may be possible: and hence,
whatever be the terms in which we are taught by experiment to express the elementary com-
position of bodies, the result must be conformable to this principle, That the differences of elementary
composition are definite. The law to which we are led by experiment is, that the elements of
bodies continue in definite proportions according to weight. Experiments add other laws; as for
instance, that of multiple proportions in different kinds of bodies composed of the same elements ;
but of these we do not here speak.

24. We are thus led to see that in our knowledge of mechanics, chemistry, and the like,
there are involved certain necessary principles, derived from our ideas, and not from experience.
But to this it may be objected, that the parts of our knowledge in which these principles are in-
volved has, in historical fact, all been acquired by experience. The laws of motion, the doctrine
of definite proportions, and the like, have all become known by experiment and observation; and
so far from being seen as necessary truths, have been discovered by long-continued labours and
trials, and through innumerable vicissitudes of confusion, error, and imperfect truth. This is
perfectly true: but does not at all disprove what has been said. Perception of external objects

Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 179

and experience, experiment and observation are needed, not only, as we have said, to supply the
objective element of all knowledge—to embody, limit, define, and modify our ideas; but this
intercourse with objects is also requisite to unfold and fix our ideas themselves. As we have already
said, ideas and facts can never be separated. Our ideas cannot be exercised and developed in any
other form than in their combination with facts, and therefore the trials, corrections, controversies,
by which the matter of our knowledge is collected, is also the only way in which the form of
it can be rightly fashioned. Experience is requisite to the clearness and distinctness of our ideas,
not because they are derived from experience, but because they can only be exercised upon ex-
perience. And this consideration sufficiently explains how it is that experiment and observation
have been the means, and the only means, by which men have been led to a knowledge of the
laws of nature, In reality, however, the necessary principles which flow from our ideas, and
which are the basis of such knowledge, have not only been inevitably assumed in the course of such
investigations, but have been often expressly promulgated in words by clear-minded philosophers,
long before their true interpretation was assigned by experiment. This has happened with regard
to such principles as those above mentioned ; That every event must have a cause; That reaction
is equal and opposite to action; That the quantity of matter in the world cannot be increased or
diminished: and there would be no difficulty in finding similar enunciations of the other principles
above mentioned ;—That the kinds of things have definite differences, and that these differences
depend upon their elementary composition. In general, however, it may be allowed, that the
necessary principles which are involved in those laws of nature of which we have a knowledge
become then only clearly known, when the laws of nature are discovered which thus involve the
necessary ideal element.

25. But since this is allowed, it may be further asked, how we are to distinguish between the
necessary principle which is derived from our ideas, and the law of nature which is learnt by expe-
rience. And to this we reply, that the necessary principle may be known by the condition which we
have already mentioned as belonging to such principles :—that it is impossible distinctly to conceive
the contrary. We cannot conceive an eyent without a cause, except we abandon all distinct idea of
cause; we cannot distinctly conceive two straight lines inclosing space; and if we seem to con-
ceive this, it is only because we conceive indistinctly. We cannot conceive 5 and 3 making 7 or 9;
if a person were to say that he could conceive this, we should know that he was a person of imma-
ture or rude or bewildered ideas, whose conceptions had no distinctness. And thus we may take it
as the mark of a necessary truth, that we cannot conceive the contrary distinctly.

26. If it be asked what is the test of distinct conception (since it is upon the distinctness
of conception that the matter depends), we may consider what answer we should give to this question
if it were asked with regard to the truths of geometry. If we doubted whether any one had
these distinct conceptions which enable him to see the necessary nature of geometrical truth,
we should inquire if he could understand the axioms as axioms, and could follow, as demon-
strative, the reasonings which are founded upon them. If this were so, we should be ready to
pronounce that he had distinct ideas of space, in the sense now supposed. And the same answer
may be given in any other case. That reasoner has distinct conceptions of mechanical causes who
can see the axioms of mechanics as axioms, and can follow the demonstrations derived from them as
demonstrations. If it be said that the science, as presented to him, may be erroneously constructed ;
that the axioms may not be axioms, and therefore the demonstrations may be futile, we still reply,
that the same might be said with regard to geometry: and yet that the possibility of this does
not lead us to doubt either of the truth or of the necessary nature of the propositions contained in
Euclid’s Elements. We may add further, that although, no doubt, the authors of elementary
books may be persons of confused minds, who present as axioms what are not axiomatic truths ;
yet that in general, what is presented as an axiom by a thoughtful man, though it may include
some false interpretation or application of our ideas, will also generally include some principle
which really is necessarily true, and which would still be involved in the axiom, if it were cor-

Vou. VIII. Parr II. Aa

180 Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY.

rected so as to be true instead of false. And thus we still say, that if in any department of
science a man can conceive distinctly at all, there are principles the contrary of which he cannot
distinctly conceive, and which are therefore necessary truths.

27. But on this it may be asked, whether truth can thus depend upon the particular state of
mind of the person who contemplates it; and whether that can be a necessary truth which is not
so to all men. And to this we again reply, by referring to geometry and arithmetic. It is plain
that truths may be necessary truths which are not so to all men, when we include men of confused
and perplexed intellects; for to such men it is not a necessary truth that two straight lines cannot
inclose a space, or that 14 and 17 are 31. It need not be wondered at, therefore, if to such
men it does not appear a necessary truth that reaction is equal and opposite to action, or that the
quantity of matter in the world cannot be increased or diminished. And this view of knowledge and
truth does not make it depend upon the state of mind of the student, any more than geometrical
knowledge and geometrical truth, by the confession of all, depend upon that state. We know that
a man cannot have any knowledge of geometry without so much of attention to the matter of
the science, and so much of care in the management of his own thoughts, as is requisite to keep his
ideas distinct and clear. But we do not, on that account, think of maintaining that geometrical
truth depends merely upon the state of the student’s mind. We conceive that he knows it because
it is true, not that it is true because he knows it. We are not surprized that attention and care and
repeated thought should be requisite to the clear apprehension of truth. For such care and such
repetition are requisite to the distinctness and clearness of our ideas: and yet the relations of these
ideas, and their consequences, are not produced by the efforts of attention or repetition which we
exert. They are in themselves something which we may discover, but cannot make or change. The
idea of space, for instance, which is the basis of geometry, cannot give rise to any doubtful proposi-
tions. What is inconsistent with the idea of space cannot be truly obtained from our ideas by any
efforts of thought or curiosity ; if we blunder into any conclusion inconsistent with the idea of space,
our knowledge, so far as this goes, is no knowledge: any more than our observation of the external
world would be knowledge, if, from haste or inattention, or imperfection of sense, we were to
mistake the object which we see before us.

28. But further: not only has truth this reality, which makes it independent of our mistakes,
that it must be what is really consistent with our ideas; but also, a further reality, to which the
term is more obviously applicable, arising from the principle already explained, that ideas and
perceptions are inseparable. For since, when we contemplate our ideas, they have been frequently
embodied and exemplified in objects, and thus have been fixed and modified; and since this compound
aspect is that under which we constantly have them before us, and free from which they cannot be
exhibited; our attempts to make our ideas clear and distinct will constantly lead us to contem-
plate them as they are manifested in those external forms in which they are involved. Thus in
studying geometrical truth, we shall be led to contemplate it as exhibited in visible and tangible
figures ;—not as if these could be sources of truth, but as enabling us more readily to compare the
aspects which our ideas, applied to the world of objects, may assume, And thus we have an addi-
tional indication of the reality of geometrical truth, in the necessary possibility of its being capable
of being exhibited in a visible or tangible form. And yet even this test by no means supersedes
the necessity of distinct ideas, in order to a knowledge of geometrical truth. For in the case of
the duplication of the cube by Hobbes, mentioned above, the diagram which he drew made two
points appear to coincide, which did not really, and by the nature of our idea of space, coincide ;
and thus confirmed him in his error.

Thus the inseparable nature of the Fundamental Antithesis of Ideas and Things gives
reality to our knowledge, and makes objective reality a corrective of our subjective imperfec-
tions in the pursuit of knowledge. But this objective exhibition of knowledge can by no means
supersede a complete development of the subjective condition, namely, distinctness of ideas.
And that there is a subjective condition, by no means makes knowledge altogether subjective,

Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 181

and thus deprives it of reality; because, as we have said, the subjective and the objective
elements are inseparably bound together in the fundamental antithesis.

29. It would be easy to apply these remarks to other cases, for instance, to the case of the
principle we have just mentioned, that the differences of elementary composition of different kinds of
bodies must be definite. We have stated that this principle is necessarily true;—that the contrary
proposition cannot be distinctly conceived. But by whom? Evidently, according to the preceding
reasoning, by a person who distinctly conceives Kinds, as marked by intelligible names, and Composi-
tion, as determining the kinds of bodies. Persons new to chemical and classificatory science may not
possess these ideas distinctly ; or rather, cannot possess them distinctly ; and therefore cannot appre-
hend the impossibility of conceiving the opposite of the above principle ; just as the schoolboy cannot
apprehend the impossibility of the numbers in his multiplication table being other than they are.
But this inaptitude to conceive, in either case, does not alter the necessary character of the truth:
although, in one case, the truth is obvious to all except schoolboys and the like, and the other is pro-
bably not clear to any except those who have attentively studied the philosophy of elementary com-
positions. At the same time, this difference of apprehension of the truth in different persons does
not make the truth doubtful or dependent upon personal qualifications; for in proportion as persons
attain to distinct ideas, they will see the truth; and cannot, with such ideas, see anything as truth
which is not truth. When the relations of elements in a compound become as familiar to a person
as the relations of factors in a multiplication table, he will then see what are the necessary axioms
of chemistry, as he now sees the necessary axioms of arithmetic.

30. There is also one other remark which I will here make. In the progress of science, both
the elements of our knowledge are constantly expanded and augmented. By the exercise of observa-
tion and experiment, we have a perpetual accumulation of facts, the materials of knowledge, the
objective element. By thought and discussion, we have a perpetual development of man’s ideas
going on: theories are framed, the materials of knowledge are shaped into form; the subjective
element is evolved; and by the necessary coincidence of the objective and subjective elements, the
matter and the form, the theory and the facts, each of these processes furthers and corrects the
other: each element moulds and unfolds the other. Now it follows, from this constant develop-
ment of the ideal portion of our knowledge, that we shall constantly be brought in view of new
Necessary Principles, the expression of the conditions belonging to the Ideas which enter into our
expanding knowledge. These principles, at first dimly seen and hesitatingly asserted, at last be-
come clearly and plainly self-evident. Such is the case with the principles which are the basis of the
laws of motion. Such may soon be the case with the principles which are the basis of the philosophy
of chemistry. Such may hereafter be the case with the principles which are to be the basis of the
philosophy of the connected and related polarities of chemistry, electricity, galvanism, magnetism.
That knowledge is possible in these cases, we know ; that our knowledge may be reduced to prin-
ciples gradually more simple, we also know; that we have reached the last stage of simplicity of our
principles, few cultivators of the subject will be disposed to maintain ; and that the additional steps
which lead toward very simple and general principles will also lead to principles which recommend
themselves by a kind of axiomatic character, those who judge from the analogy of the past history
of science will hardly doubt. That the principles thus axiomatic in their form, do also express
some relation of our ideas, of which experiment and observation have given the true and real interpre-
tation, is the doctrine which I have here attempted to establish and illustrate in the most clear and
undoubted of the existing sciences; and the evidence of this doctrine in those cases seems to be
unexceptionable, and to leave no room to doubt that such is the universal type of the progress of
science. Such a doctrine, as we have now seen, is closely connected with the views here presented of
the nature of the Fundamental Antithesis of Philosophy, which I have endeavoured to illustrate.

W. WHEWELL.
AA

XV. On Divergent Series, and various Points of Analysis connected with them. By
Avucustus Dre Moreay, V.P.R.A.S., F.C.P.S., of Trinity College; Professor
of Mathematics in University College, London.

[Read, March 4, 1844.]

I sereveit will be generally admitted that the heading of this paper describes the only subject
yet remaining, of an elementary character, on which a serious schism exists among mathematicians
as to absolute correctness or incorrectness of results. When such a question arises upon a method
of pure mathematics, there can be little doubt that it must be one which is likely to lead to error
if not cautiously used; and it is probable that the contending parties have not made any close
agreement upon the use of terms. A review of the leading points of the controversy may be useful,
accompanied by an examination of the maxims which have been adopted, but I think not very
plainly stated, in the rejection of the series called divergent. The manner in which the rejection
just alluded to has been made will require that, instead of dividing series into convergent and
divergent, we should make a more general division, say into convergent and non-convergent.
Non-convergent series may be divided into those of infinite and finite divergence: the former of
which, as in the cases of a+a+a+t.,.. and 1—-24+3-—4+,., can be made, by summation of
terms, to differ from a given quantity to any extent; the latter, as in the cases 1-1+1-..,
and cos @ + cos2@+... cannot be made to differ from a finite quantity by more than an amount
which can be ascertained. It is obvious that only the converging series can, properly speaking, be
the objects of arithmetical calculation, in which they occur early, of which 4+ ='33333..; is a
sufficient instance, All others, whether of finite or infinite divergence, are equally out of the pale
of arithmetic to those who do not acknowledge different degrees of impossibility. I do not here
argue with those who reject everything which is not within the province of arithmetic, but only
with those others who abandon the use of infinitely diverging series, and yet appear to employ
finitely diverging series with confidence. Such appears to be the practice of those analysts
who object to diverging series, both at home and abroad. They seem perfectly reconciled to
1-1+1-1+...=4, but cannot admit 1+2+4+8+...=-1.

Many of an earlier school took an opposite position; they freely used infinitely diverging
series, but, with Euler, considered finitely diverging forms as indeterminate. To use a common
phrase, they spoke as they found: they could actually obtain by rules of algebra, finite expres-
sions from which they could evolve infinitely diverging series: but they were not able to find, or
to satisfy themselves they had found, similar equivalents for most cases, particularly the trigono- _
metrical ones, of the remaining species. They made an unguarded use of the word ‘ indeterminate >

F E 2 f 10) H
sometimes it meant wndeterminable, in the same manner as 5 when looked at as the solution of an

identical equation ; sometimes only wndetermined, either with reference to the state of science at
the time the word was used, or to the state of a particular question at some one particular stage of
the solution (as in the method called that of indeterminate coefficients). The moderns seem to me
to have made a similar confusion in regard to their rejection of divergent series: meaning sometimes
that they cannot be safely used under existing ideas as to their meaning and origin, sometimes
that the mere idea of any one applying them at all, under any circumstances, is an absurdity.

Mr. DE MORGAN, ON DIVERGENT SERIES, 183

We must admit that many series are such as we cannot at present safely use, except as means
of discovery, the results of which are to be subsequently verified: and the most determined rejector
of all divergent series doubtless makes this use of them in his closet. But to say that what we
cannot use no others ever can, to refuse that faith in the future prospects of algebra which has
already realised so brilliant a harvest, and to train the future promoter of analysis in a notion
which will necessarily prevent him from turning his steps to quarters from whence his predecessors
have never returned empty-handed, seems to me a departure from all rules of prudence. The
motto which I should adopt against a course which seems to me calculated to stop the progress of
discovery would be contained in a word and a symbol—remember ,/—1.

I do not pretend to have that confidence in series which, to judge from elementary writers on
algebra, is common among mathematicians: not even in convergent series. A few great forms,
which have had substantive and finite expressions assigned to represent the remnants after any
given term may, no doubt, be perfectly trustworthy. But as for the rest, I cannot bring myself
to that positive assurance with respect to any general class of series which the writers to whom I
shall presently allude appear to have with respect to such divergency as they do admit. The
main object of this paper is to show that they have underrated the character of most of what
they reject, and overrated that of all they receive.

I shall now proceed to the different points of discussion in order.

SECTION I.

All Divergent Series, whether their divergence be finite or infinite, stand upon the same
basis, and ought to be accepted or rejected together, as far as any grounds of con-
jidence are concerned which are not directly derived from experience.

I s#auu first examine the general arguments on which Poisson supports the contradictory of
the preceding assertion. This great analyst was at the head of the school in which definite integration
had been made in a great measure to take the place of expansion into algebraical series. A definite
integral is a particular kind of series, and has its converging and diverging cases, the latter being
either of infinite or of finite divergence. Thus a be dz is convergent, fede is infinitely
divergent, and Pr cos ada is finitely divergent. Perhaps in the natural bias derived from a
continual contemplation of integration under the form of summation, not of inverse differentiation,
may be seen the reason for the opinion of divergent series adopted by the definite integrators.
Let it only be granted that integration is as fully defined and as generally understood, as any of
the fundamental operations of arithmetic, and the question on diverging series seems to be settled
at once, and by a much easier argument than any of those usually proposed against them. To
take an instance ;— |, 2da cannot be other than [eda +t [Pode + fp Fda +...: but the
first is (on the above assumption) infinite, and the second is (log2)-'(1+2+4+...) which is
therefore infinite. Consequently 1+ 2+ 4... cannot, as usually held in algebra, represent —1. It
must certainly be charged upon those who have hitherto used divergent series, that they have
never reflected upon and explained, perhaps have never perceived, the singular apparent in-
consistency which they were every day committing; namely, treating those very forms as repre-
sentatives of infinity when they were consequences of integration, which they accepted as finite,
when they were results of algebraical development, Referring further discussion of this point
to a subsequent section, I now make two citations from memoirs by Poisson in the Journal
de (Ecole Polytechnique, Cahier 19, pp. 408, 409, 501.

Page 501 “On enseigne dans les élémens, qu'une série divergente ne peut servir 4 calculer
Ja valeur approchée de la fonction dont elle résulte par le développement: mais quelquefois on

184 Mr. DE MORGAN, ON DIVERGENT SERIES.

a paru croire qu’une telle série peut tre employée dans les calculs analytiques a la place de
la fonction; et quwoique cette erreur soit loin Wétre générale parmi les géométres, il west
cependant pas inutile de la signaler, car les résultats auxquels on parvient par lintermédiaire
des séries divergentes, sont toujowrs incertains et le plus sowvent inexacts.”

Pages 408, 409. ‘‘On peut voir dans les Mémoires-de Pétersbourg (Novi Commentarii, tom.
xvi1 et xvitt) la discussion qui s'est élevée autrefois entre Euler et D. Bernouilli au sujet des
séries de sinus ou de cosinus prolongées 4 V'infini. Les détails dans lesquels nous venons d’entrer,
ne semblent devoir laisser aucune obscurité sur ce point d’analyse: nous admettrons avec Euler
que les sommes de ces séries considérées en elle-mémes, n’ont pas des valeurs déterminées; mais
nous ajouterons que chacune d’elles a une valeur unique et qu’on peut employer dans analyse,
lorsqu’on les regarde comme les limites des séries convergentes, c’est-a-dire, quand on suppose
implicitement leurs termes successifs multipliés par les puissances d’une fraction infiniment peu
différente de Dunité.”

I hardly know which of the passages in my Italics ought to excite most surprise. Divergent
series, at the time Poisson wrote, had been nearly universally adopted for more than a century,
and it was only here and there that a difficulty occurred in using them. As to the second
passage, we may clear Poisson of absolute mistatement by remembering that he had. both head
and hands full of a subject which had tasked his great powers to their utmost, namely, the
substitution of definite integrals for series in questions of mathematical physics. As far as in-
tegration is concerned, I admit, and even think I shall presently show, that he was fully justified
in what he said: in the meantime I attend to his argument in favour of finitely diverging series.

Let us take the series 1—1+1—1+..., a remarkable specific case of both algebraical and
trigonometrical series. I collect from what I have quoted, and from numerous other parts of
his writings, that Poisson is content to equate } to1—1+..., considering the latter as a mere
form indicative of 1—g+g°—..., where g is a fraction infinitely near to unity, but less. He
will consent to use the limiting form of convergency, to walk on the line which separates con-
vergency from divergency, but not to cross that line, even by an infinitely small quantity.

In using the language of infinitely small quantities, I do not intend to direct any part of
my argument against the ideas connected with the phraseology, because both Poisson’s statements
and my comment on them might easily be translated into the language of the theory of limits.
Let us then take 1-1+1-.,. as indicating 1-g+g*—... where 1 —g is infinitely small and
positive. How can 1-g+g°-... be called convergent? Because the terms diminish without
limit, and g", if n be infinitely great, becomes infinitely small. The departure from finite
divergence, and commencement of real convergence, is infinitely distant. Now all that is
wanted to make 1+2+4+,.. equal to —1 is the presence of the infinitely great negative re-
mainder, which might be considered as not destroyed, but only removed, when the second side
of (1 —2)"1=14+24+2°+... + 2"+2""' (1 —2)7' is made an infinite series by » = @. If sup-
positions which only take effect at an infinite distance from the beginning of the series are
allowed to be made with regard to series of finite divergence, why may not the same be conceded
in the case of infinite divergence? Both 1-—1+... and 1+2+... are equally irreducible to
their finite equivalents by the arithmetical computer; both are equally creatures of algebra: ‘if
a reason can be shown for the distinction between them, those who adhere to infinitely diver-
gent series have a right to ask for it; but if, as I suspect, that reason be ewperience, I am
prepared to contend that, when integration is not employed, there has not been produced one
single instance in which divergency, properly treated, has led to error.

That experience is the guide may be safely inferred in all cases of rejection, when those
who reject do it to different extents. Poisson would admit 1° — 2* + 3° — 4° +... =0, since there
is no question that, g being less than unity, the mere arithmetical computer might establish,
to any number of decimal places, the identity of 1°-2°g + 3*g*-.., and (1 —g)(1+g)~*. But

Mr. DE MORGAN, ON DIVERGENT SERIES. 185

on this equation, 1*— 2°+ .., = 0, Abel, another rejector, remarks (Works, 11. 266), ‘¢ Peut-on rien
imaginer de plus horrible?”

Poisson’s mode of allowing }=1-—1+... is clearly equivalent to an adoption of the maxim
that whatever is true wp to the limit is true at the limit. When relations of pure magnitude
are in question, there is no doubt of the truth of this principle. But the words wp to must not
be understood inclusively, since then the principle would merely assert that what is true at the
limit and elsewhere, is true at the limit. With this caution, it is impossible to prove that a relation
of magnitude is true at the limit, if at the limit we have no longer calculable magnitude. We may
not say that what is calculable up to the limit is calculable at the limit, nor that what is complicated
up to the limit is complicated at the limit, &c.: but only that relations which are quantitatively true
up to the limits are so at the limits, if the limits be quantities. Assume 1-1 + ,., to be quantity,
determinate quantity, and that quantity may possibly be shown to be 4 and no other: but it
may not be assumed that 1—1+.,.. is a quantity, because 1-g+g°—... is a quantity, up to
its limit; or at least if such assumption may be made, no reason has been given for confining
it to any one class of limiting forms.

Again, it is clear enough from the manner in which Fourier, Poisson, Cauchy, &c. use the
limiting form 1—1+..., that they intend it to signify } in an absolute manner. The whole
fabric of periodic series and integrals, which all have had so much share in erecting, would
fall instantly if it were shown to be possible that 1-1+... might be one quantity as a limiting
form of A,—4,+... and another as a limiting form of B,- B,+.... Fourier’s celebrated
expression of a function by means of a definite integral, that of Poisson by means of a series
of periodic integrals, &c., are all stated as absolute truths, and used as such, though they are proved
only as limiting forms of one particular class of convergent series. A person who is much versed in
the writings of the above-mentioned analysts must feel to his finger’s ends that one well-established
instance in which 1—1-+... means other than 4 would throw doubt upon all they have written.
Now we have Poisson’s assurance that these series, though indeterminate, have each a unique value,
which can be employed in analysis when the series are considered as the limits of convergent series.
Here the word ‘indeterminate’ is loosely used, in the sense of not determinable by actual summation :
a unique value, which can be employed (and therefore of course first found) is not indeterminate in
any correct sense. But who is to assure us of this uniqueness of value? How could Poisson
undertake to make the assertion? By an induction—an extensive one I grant—but still an induction.
From (1+ @)~!'=1-—a+.., to*¥

1 ae e “dv 4 a pet

Sallie cumne= eee, c Fe Sr s
it is always observed that where the series-side of an attainable developement gives 1—1+... the
finite side gives 4. But this induction may be overturned: and if the stability of form which really
has hitherto characterized series of finite divergency should be found not to belong equally to those
of infinite divergency, it should teach us rather to suspect the former than to content ourselves with
merely empirical rejection. There are two ways of considering a series: absolutely, as a given
algebraical expression, and relatively, as the development of a given function, from which it
actually was produced. I do not defend the former mode of considering either convergent or non-
convergent series; and I fully believe that analysts have been led into error, as to both classes, by
incautiously reasoning on series of which the invelopments were unknown, I do not dispute that
the arithmetical value of a specific case of a series may, when that particular case is convergent, be
calculated : but, speaking of general series, it seems to me that it is dangerous to reason upon them

* This instance is very good for the purpose, since one side or the other must have all the difficulties of divergency: either the
integral or the series is divergent.

186 Mr, DE MORGAN, ON DIVERGENT SERIES.

until as general an inyelopment is found; after which, I incline to think that all conclusions upon
the series should be upon them considered as the developments of those particular functions which
produce them. My reasons are as follows.

1, Discontinuity of form is not perceptible in the series itself, though it may very possibly
exist ; to reason upon a series as a continuous function, without knowing from its invelopment that
it is so, is pure assumption. This remark applies particularly to series which are always conver-
gent, and most of all to series which are convergent and also begin to diminish from the first term.
If we spoke of mathematical results in the same sort of language as of physical phenomena, we
should say that there is inaptitude in developments to be the permanent arithmetical representatives
of finite continuous functions, and that series which must of necessity be always convergent, shew
this inaptitude by discontinuity, while the others escape from arithmetic altogether by divergency.

2, When divergent series are employed independently of their invelopments, it is impossible to
distinguish the cases in which they really represent infinity from those in which they are developed
forms of finite quantity. No one can actually calculate with the symbol = , even when its sign is
determinate: for even if © + o’ and o x ©’ would not puzzle him, it is certain that © — @'
and c + ©’ would require reference to the producing functions. As soon as © is attained, we
must stop for examination: this cannot be done if, when attained, it is seen under the divergent
form which equally belongs to finite quantities, that is, is not seen at all.

3. It cannot be questioned that series which are infinitely divergent, at least, may appear
as very different things in different cases. For instance, an algebraist would be inclined almost to
assert that 1+2+44+... must be —1; for he would say, if it be the object of algebra at all, it
must satisfy the equation x =1 +22, But now let us consider the series 1 + 2a-" + 22a-™ + 23 a7”
+... which is certainly convergent, if a and m be both greater than unity, and as certainly increases
without limit, as a — 1 diminishes without limit. When 2 =1, the limiting form 1+2+4+,,.. is
clearly the representation, not of — 1, but of ©. The series e~? + ve~'" + we-'™ + ,., satisfies the
equation

@2U xndU 8

ee pe Tie a? bY? 948 dO,

Or Tat ee () PO. a? bY’ ni" dé
where @@ and W/@ are arbitrary, and a and # are any constants independent of a, 6, and m. In
taking this form for U, I follow the example of Poisson, Cauchy, &c., who are always content with
such a form, provided only that it contain the requisite* number of arbitrary functions. To make
the form of U an algebraical equivalent of the series, we must determine 0 and y@ from
é€

=H
— [° nonh¥°de -
sere pO’ ddd ;

1 B
— = [ pOa* or'nvdd,

1-w@ 1

a useless attempt, even when w<1, unless discontinuous forms of y,@ be introduced. Here is a
clear case in which 1 +2 +4 +... represents co : are we then really to abandon the assertion that it
satisfies the equation 1+2s=%? If so, the opponents of divergent series have gained their point,
for those developments are not even to be trusted as to their symbolical properties. But I rather
argue that it is not so, in the following manner. Every equation, it is very well known, has as
many roots as units of dimension, only on the supposition that its problem is absolutely of
that dimension, and not a degenerate case of a higher dimension. Plenty of simple problems may be
proposed which illustrate this known result of common algebraical reasoning. Now the equation
which stands related to the series in question in the same manner as1+2s=% tol1+2+4+... 1s
px = we" +(x+1). If this last could be generally solyed, then p0O would be the series

* They assume that = pe, or 0; e*® + py bev? +... can | happen to be false, ‘ toujours incertains’ may be applied to many
always be represented by jp0e?9d 6, which I believe to be true if | of their results, and ‘le plus souvent inexacts,’ may follow,
p may be discontinuous, But it has not been proved; should it |

Mr. DE MORGAN, ON DIVERGENT SERIES. 187

required: if, after solution, 6 were made = 0, we should see that 1 + 22 =~, the result for w = 2,
would be only a degenerate form of a more complicated form.

This remark will illustrate my opinion that a series is to be considered strictly in relation to the
function from which it is developed. If a* + .a*+'+... be absolutely under consideration, the
equation @z = a*(1—.«)~' may be strictly obtained, and thence (1 —#)~' for 1+a+... But
there is no saying what further degeneracy of form may be seen in passing from px = a*e~'™ +
p(# + 1) to 1 +2z=2, which is not seen in passing from px = a* + p(x + 1) to the same.

My conclusion is, that a divergent series may have for its proper value either that which is
usually so considered, or infinity, according to the nature of the function from which it is expanded.
And since every equation has as many roots as it has algebraical dimensions, so many of them being
infinite as there are vanishing coefficients which precede the first finite coefficient, there can be
no right to say that the symbolical character of divergent series is forfeited, until either the symbol
co takes the place of the ordinary value in a case in which there is no degeneracy, or until some
Jinite value, different from the ordinary one, is shown, in some one particular case, to be the proper
representative of the series. Let 1+2+4+... be shown to be any thing but a root of either
1+2zx =, or of another equation which has degenerated into 1+2z =x; that is, let it come out
any thing but — 1 or ©, and as a result of any process which does not involve integration performed
on a divergent series—and I shall then be obliged to confess that divergent series must be aban-
doned, or rather, that the generalizations frequently made on the subject must be much curtailed.
But nevertheless, there is nothing to lead us to doubt that divergent*series of all classes, whether of
finite or infinite divergence, must be treated alike. If any one say that such a difficulty as the
preceding cannot occur in series of finite divergence, he must prove it.

It might perhaps be supposed that, in every doubt which has been raised in the preceding
remarks, the finitely diverging series have been much less hardly borne upon than the others—to an
extent which may make it seem to be almost admitted by myself that the foreign analysts, if not
justified in their dogmatical rejection of infinitely diverging series, have nevertheless chosen, and
judiciously chosen, to confine themselves to the safer of two paths. But it is to be remembered
that I have been obliged, as yet, to mention only their practical division, which really consists in
the separation of all finitely diverging series from the rest. Had I had to make my own division
of series, I should have admitted that there was one of two paths which was much safer than the
other: but I should have asserted that the labors of the writers in question did not extend over the
whole of that path. From the sort of appeal to induction which unfortunately must, in the present
state of our knowledge, help us to a part of our results on series, backed by considerably more of
demonstration than has been applied to the remaining cases, it seems to me pretty clear that the
proper line of demarcation does not separate series of finite and infinite divergence, but series having
all their signs alike from those of terms alternately positive and negative, or consisting of parcels o1
terms alternately positive and negative. This will be the subject of a subsequent section.

SECTION II.

The Operation of Integration as at present understood, is one of Arithmetic, as distin-
guished from Algebra, and must not be applied unreservedly to Divergent Series.

Accorpincé to elementary notions, we differentiate when we find the value of }p(w+h)-—path—'
in a calculable form when 4 =0. Integration is usually defined as the inverse question, which
must be, required pu when the calculable form of ip (w7 +h) - pu} h-' is given for h = 0. This
demands the solution of a functional equation, and it is easy to say, Let this equation be considered

MOI VL. Parr II. Ba

188 Mr. DE MORGAN, ON DIVERGENT SERIES.

as solved, and let the process of solution have a name. But the state of our knowledge makes it of
no use whatever to express a conventional solution, since our power of translating our convention
into ordinary language is confined to a small number of cases, all rendered backwards from the
direct process. Common integration is only the memory of differentiation: and the process of
parts, and the few other artifices by which it is effected, are changes, not from the unknown to the
known, but from the forms in which memory will not serve us to those in which it will. We may
assume that any function has an integral, and we may write down [cos wv dx or fe"da; we
may also have recourse to series, and by assuming an unlimited use of divergency, we may procure
abundance of nominal answers to any question. But we cannot be so much as sure of the fact that
every continuous function has an integral, except by recourse to the summatory definition, namely,

[godon [pas 9 (a+ ®=2) +9 (e+e 2=4) +49 (we noes

n n

in which m is made infinite. This definition, as is well known, never fails, nor can fail, to give one
value for every value of a and a, applied to one branch of the function, except only when pu
becomes infinite at or between v =a and v=a@. In this last case, we have not even the means of
universally defining {pudv: all the difficulties of divergent series meet us again.

In confining ourselves to this arithmetical definition of an integral, when one of the limits is
infinite, we must, as to a large number of cases, act precisely as if we separated a class of divergent
series from the rest, and insisted upon their retaining for their values the idea which the attempt at
arithmetical summation gives, infinity. The early problems by which the nature and use of
integration is suggested, being problems on concrete (mostly on space) magnitude, cannot afford the
means of generalizing our definition. No doubt the area of the curve y=e", represented by
i e’da, is greater than any surface which can be assigned: no doubt also that the series of inscribed
rectangles 1 +¢ + e+... is the same. When we shall have obtained the definition of an integral
by which we can state such a value for jf, ede as is the true correlative to (1—e)~' considered as
the value of 1+ 6+... then, and not till then, shall we be entitled to claim integration as an
instrument of algebra in the widest sense. Some of the objections raised against divergent series,
indeed most of those which are very plausible, are grounded upon the supposition that integration
may be as unreservedly applied to divergent as to convergent series, if the former are to be used
at all. That this cannot be done may be satisfactorily shown by instances, as follows :

1—acosav
Let pv = ——_—_ = 1 + wcosav +a’ cos2av+......
1—2Hcosav+a”
which never becomes infinite for any value of v, except only when # = +1; and the series is con-
vergent when @ lies between —1 and +1. Multiply both sides by e~“dv, and integrate from
» =0tov=o , in which case there cannot be any doubt about the purely arithmetical (or convergent)

a?

character of every integration. This gives us, ¢ being « +

a

Ee e“prdv=ltat+ et + ef att ti...

Teo

This resulting series is convergent for all values of w: for, since ¢ is less than 1, wv" must
become less than unity after a certain value of , and thenceforward S(@t")" must be more con-
vergent than any series of simple powers. If # lie between —1 and +1, the whole of this process
is purely arithmetical, and the identity of the two sides of the last equation might be approximately
verified by actual computation: if not, the original series, though divergent, is changed into
a convergent one by the process. Change w into w~', and let mv then become @,v; we find
pv+iv=1, and ;

Mr. DE MORGAN, ON DIVERGENT SERIES. 189

2 On 2 oa) een 2 oO a
was pode + fe giedv=—— fe dv=1.

Accordingly, if all that precedes be correct, we have

1 pt Mil 1
2+ («+-) t+ (+5) ie ae (#+5)e+ seec=els

a

which is certainly false, unless a convergent series can represent less than half the sum of its terms.
This last series is always convergent, except only when a=0, or ¢=1, in which case the last
equation is found to be algebraically true. If for w we write — a, the equation is found to be
true when ¢ is equal to the least of # and #~', but is certainly not universally true.
Apply the same process to
(1 — x) cosav ce
Te = COSEY + & COS BAU + A COSSAV + seccercoeeee
1—2wvcos2av+a@

and the result is

1 1)\ 5 2 1 25
= (1 +=) t+ (e+) ¢ + (a+ =) Grobe stiiems
a x a

on which precisely the same remarks might be made. I might multiply instances of this kind to
any extent; but the following consideration will render them needless, as showing that what we
have seen is precisely what we ought to have expected.

Integration, though only capable of an arithmetical definition, is the most decided changer of
form which we ever use. A change of value in a constant may introduce a totally different form
into an integral; and in particular, the assumption of infinite value for a constant has this effect
almost without exception. And in regard to definite integrals, there is hardly any end to the
known instances in which complete and apparently arbitrary changes of form (such as cannot pass

: 0 ; ° . d
one into another through io the like) arise from alteration of the specific value of a constant.

If then V be expanded into the series P, + P, + P, +... and if the sum of 7 terms,
P,+P,+...+ P,_, be called Q,; we obviously have

a a a ya
[Vdv =f" P,dv + J, Pidv +...+ J, (V- Q)dv
where 7 is made infinite after integration. When the series P, + P, +... is convergent, then, even
granting that /(V — Q,) dv may have circumstances peculiar to m= ©, it is of no consequence,
since considerations of form are rendered useless by evanescence of value: the elements of /(V—Q,,) du
must, by the hypothesis of convergency, diminish without limit as compared with the corresponding
elements of {P,dv, /P,dv, &c. Even if integration converted the convergent series into a diver-
gent one, this would still be the case. But if P,+P,+... be divergent, we have no longer any
right to draw any conclusion about /(V — Q,) dv from observing what takes place with S Pode,
aA P, dv, &c. Applying this to our first example above, we have
| —wcosav cos (v + 1) av — wcosnav

Ma aachan a =1+4+#cosav +...+ a cosnav + w"*!
— 2@cOs av + a

1 —2ucosav +a

1

change wv into a’, and add; which gives

1 gl
(a a =a cos (n + 1l)av— (a + 5) cosnau

1 1
1=2+(0+-) cos av took (att =) cosmav + are
ri) iB 1—2uvcosav+a”

aw

This equation is identically true, the only restriction being that m must be a positive integer
(0 included). Consequently, we have, as specimens of legitimate inference from integrating a
divergent series,

BB2

190 Mr. DE MORGAN, ON DIVERGENT SERIES.

@ 1 1
2 ih (w+) cos nav — (a+ =i) cos (z +1) av
—_ a” as} -

-2
VT 0 2 € dv
1—2HMcosav+2@

€

q\ 2a TT Aes Nee
=1+ (+2) “+ (at+5) 2 ti.t (a'4<) :
& wv

e “dv

1 fi ee
Se (a + av) —2cosav

cosnave “dv.

1 tod (att 4 a OF) —4(a" + a") cosav
eee. 4 (@ + a”) -—cosav

The series continued ad infinitum is expressed by the value of the integral just found, ‘in
which m is made infinite, being the very remainder which is called nothing in the original and
fallacious process. Many interesting forms might be derived from the preceding and similar
cases, but having no reference to the subject of this paper.

When the terms of a divergent series separately vanish, the series having remained divergent
up to the time of evanescence, it is customary, in elementary works, to assume that the series
itself vanishes: or 0+0+0+... is taken to represent 0. Very frequently, no doubt, the in-
velopment shows that this is correct; and I think I shall be able to show that if the function
be perfectly continuous on both sides of the epoch of evanescent form, a reason can be given why
it must be so. But so far as the series itself is concerned, we have no right to come to such
a conclusion, unless we can shew that as the evanescent form is approached, the inyelopment
diminishes without limit. The following instance will show the necessity of this caution.

« e~*' cos bt dt
1+2
tainly is not comminuent with a, but approaches the limit dre

fF cos bt(1+#)-'dt. Expand the first side into

is convergent for all values of a, however small, and cer-

The integral if
0

*, the well-known value of

a p2nt2
-at — 1 gee — 1)" 7 Phy Cat es ee
i ; cos b¢ (1 Pose (- 1 4+ (-2) <a)
which, from
T (2m + 1) cos {(2% +1) tan-* (ha-*)t

ff e7 “cos bt. dt = (84 ayer ’

0

gives, making tan~*(ba~*)=0
(b° + a*)-3 cos @ — 1.2 (b? + a*)-4 cos 30 + 1.2.3.4 (B° + a*)-? cos 50-12...
aah So lcosbiee 4 ae
+ (—1)"1.2.3,..2n (6? +a‘) 2 cos (2n +1)0 + (-1)"*? i bes eb

f Lee
If we neglect the last term, or suppose » infinite, we have expanded the given integral into a
divergent series of which all the terms are comminuent with a: for a=0 gives @=42. When we
have the remainder, we may, by retaining its proper value, allow the preceding form 0+0+0+...
to stand for 0: but otherwise the appearance of that form must be a warning, when it arises from
the value of a divergent series, that there may be some finite equivalent which is not to be neglected.
It is worth noting that immediately before the terms of the preceding series vanish, they are all
of one sign, or cos, cos 36, &c. are of alternate signs. This is one out of the constantly recurring
cases in which it happens that the difficulties of series are mostly incident to the divergent case in
which all the signs are the same: the illustration of which is-the subject of the next section but one.

Mr. DE MORGAN, ON DIVERGENT SERIES. 191

SECTION III.

It gencrally happens that the real analytical equivalent of the different values of an
indeterminate expression, is the mean of those different values.

Tuis principle must rest at present upon induction. When Leibnitz pronounced 4 to be the
value of 1 — 1 +... because there was no apparent reason why either 1 or 0 should be preferred, he
was not only right in his conclusion, but had a glimpse (though not in solid reasons) of a principle
which admits of such frequent confirmation that it may be suspected to be general.

In the first place, if we take any algebraical series, such as a + bw — cw? +aa* + ba*—ca°+,..
in which ce = a +b, so that when w = 1, the successive results of summation are a, a + 6, 0, a, a+b,
0, &c. we find by common processes that the analytical equivalent is the mean of a, a + 6,0, or
4(2a+6). The same thing happens if we take other forms which produce the same limiting form, as
a +bcos@ — ccos20 +...

Secondly, if we take a series 4, + 4,cos@ + dA, cos 20+.., or Fourier’s integral yea eS cos
w (« — v) pv dw dv, in such manner that it may represent the ordinate of a discontinuous curve,
the branches of which do not join at the common ordinate, it is found that for the abscissa of the
common ordinate the series and the integral represent in both cases, not either or both of the
ordinates, but the mean between them.

Thirdly, the indeterminate symbols sin ¢ and cos ¢9 are found in numberless cases to represent,
each of them, 0, the mean value of both sinw and cosv. ‘The mean value of any function pe,

between a and 4, is Jvpade divided by b—a.
Fourthly, if z lie between —7/ and + /, Poisson has shewn that

1 +1 1 ny; mr (wv —v)
= 5 = an = =
pe se pudv + i if cos j pear (from m= 1 to m= 0),

the second side of which is not changed in value, by changing the sign of /. And this second side
is the same whether we make w = — /, or 2 = +/; consequently it is wholly undecided whether it
is then to represent p(—/) or @(/). Poisson has shown that in either case it represents the mean
of p(l) and p(- J).

Fifthly, if we extend the term mean value, and, in cases in which the function becomes infinite,
define it as {%ada~ (b — a), the same principle applies, in a very peculiar manner, to the remaining
trigonometrical functions; if the part of the integral at which @a becomes infinite, be examined in
the manner which occurs so frequently in the writings of M. Cauchy. Let us take for instance,
tana. In ie tan vd, the finite parts destroy one another: and to obtain the expression for it we
must examine the integral from 4a —pto 4 +p, and from 37-4 to 3a +4, « being infinitely
small. Now the indefinite integral is — log cos aw, so that we have to examine

cos (37 — 1)

1
cos (4a —
aN eal log
cos (4a + u) cos (37 + )

. . ‘20 . . nite.
each of which is log (— 1) or 7/— 1, when » = 0. Hence jf tan ada is 2r4/— 1, which divided
by 27, gives \/ -1, the proper representation of tano, if this principle be true. Now if we

ps ¥ tan wv + tany - 3 .
examine the equation tan(w+y) = eee and make @ infinite, presuming that c and
1 — tan@# tany
2 +y are the same angles, we find tan o =+,/-—1. In the same manner cot @ is + \/-1.
It cannot be argued that since the values of tan a, from x = 0 to a = 7, have signs contrary to those
from w = 3 to w = 27, therefore if r\/—1 be taken for the first, — r+/— 1 should be taken for
the second: the reason being that the signs in the second semicircle are really repetitions of those in

192 Mr. DE MORGAN, ON DIVERGENT SERIES.

the first, and only contrary in an inverted order. And it must be remembered that, 4 being the
mean value of X, dA is not therefore that of ¢X: thus sin*w has the mean value 4, not 0%. Also
that, when a quantity is, at one or more epochs, infinite, its mean value is not necessarily positive
because all its values are positive. Thus tan?# has —1 for its mean value. The mean value of
secv or cosecw is 0. This remarkable coincidence of two modes so remote from each other of
determining the analytical meaning of tan © and cot © , depends at last upon e**=¥-! =0, an equa-
tion which more writers have virtually used than have openly dared to state it. The apparent dis-
turbance of the law of continuity when # = ©, as in cos* © + sin® oo = 0, &c. is perhaps what has
prevented the formal recognition of these relations: nevertheless they will, it may confidently be
asserted, not only obtain universal reception, but finally a rational and consistent explanation.

The following is a glimpse, perhaps, of the explanation, as applied to series. In every conver-
gent series, the limit of the sum of all its terms is the mean value obtained from all the summations:
the mean of partial summations 4,, (4, + 4), ...... (4; + 4, +... + A,)

: n—-1 n—2 1
is 4, + ——4,+ ——4A,+ ... + — Any
n n n

which, as 7 is increased without limit, has 4, + 4, + ... ad inf. for its limit. Hence, by Poisson’s
principle, by which I mean the assumption of the right to apply the maxim, ‘‘ that which is quanti-
tatively true up to the limit, is true in the same sense at the limit, when the limit presents an
incalculable form”—we may assert most positively, that 1— 1+ 1 —... must be 4 whenever it is the
limiting form of convergency: not on the metaphysical doctrine (probably suggested by the known
result) of Leibnitz, namely, that we can see no reason to prefer 0 to 1, or 1 to 0, and must therefore

. : : 1 n 1
take a mean; but because ” partial summations give the mean — x = le Se
U7) n 5

is even or odd, and the limit of both is. At the same time it is easily proved that whenever
the partial summation gives recurrences in which 0 occurs at stated equal intervals, the limit of the
means must be the mean of one period.

As in other cases, the diverging series whose terms are all of one sign is not elucidated by this
process, which nevertheless, provided we adhere to our principle, brings out the true algebraical
result for series which have terms alternately positive and negative. The mean of 1, 1 — a,
1-—a+a’, &c. (n summations), is

n+l
Dc aea pe ris ics Sane
l+a n(i+a)* n(1 +a)°
if, when m is infinite, we take (— a)”*’ as 0, the mean of the values between which we cannot then
choose, we have (1 + @)' as the limit.

according as #

SECTION IV.

Series of alternately positive and negative signs stand upon a much safer basis than those
in which all the terms have the same signs, aud that whether their divergence be finite
or infinite.

Art the very outset, namely, in the mode of finding whether the series is convergent or divergent,
there is every possible difference between the two species above-named, which we may term

progressing and alternating. The progressing series p(1) + (2) +... is convergent when the
first of the set

Pao P,=log#(P,-1), P,=loglog#(P,-1)... P,= (log)"« (P,_,—1)..-

Mr. DE MORGAN, ON DIVERGENT SERIES. 193

which is not equal to unity is greater than unity; and divergent when the first which is not
equal to unity is less than unity. But o(1)-—@(2)+... is necessarily convergent, provided
only that @(c ) = 0 continuously, or that the terms ultimately diminish without limit.

A progressing series must be either convergent or infinitely divergent ; an alternating series may
be convergent, or either finitely or infinitely divergent: but the infinite divergence of the latter
is of a different character from that of the former. I very much doubt whether it is quite
correct to apply the same phrases to both kinds of series.

It is easy to apply Poisson’s principle to alternating series, even when they are of infinitely
diverging form. We can always contrive to find positive quantities B,, B,, &c. in such a
manner that 4,B,—- A,B, + A,B,— ... is convergent, up to a certain value of a variable contained
in B, &c., which makes them become severally =1. Thus 1—a@+a’- is a limiting form of
1—ae-"+a@a-"— a@a-" +... which, m being > 1 is certainly convergent down to z = 1, exclusive;
and this whatever the value of a may be. Whether this limiting form is always (1 + a)! may be
a question; but, as I think is sufficiently shown in various parts of this paper, the question may
also be asked about the finitely diverging series which have been so confidently allowed.

When an alternating series is convergent, and a certain number of its terms are taken as
an approximation, the first term neglected is a superior limit of the error of approximation. This
very useful property was observed to belong to large classes of alternating series, when finitely or
even infinitely divergent: I do not remember that any one has denied that it is universally true,
while many have implicitly asserted it. When the series is convergent for a certain number
of terms, particularly if the terms become very small before they begin to increase again, it obviously
makes the divergent alternating series practically as useful as the converging series, perhaps even
more so, for it is very frequent that the greater the ultimate divergence, the greater also is the
primitive tendency towards convergence.

In any series P,- P,+ P,—... this theorem is obviously true as long as the remnant
P,, — P,,,, +... has the same sign as P,, or the positive sign. Thus, if P, — J eepgy a rire bee, WiG
have for the series P,— P, + Q, and P,—- P,+ P,— Q;: if Q, and Q, be positive, the series is
greater than P, — P, and less than P, — P, + P,; which is a case of the theorem. It is also clear
that if either Q, or Q, be negative this case is not true.

That the theorem is not universally true will appear in the following instances :

3 = cos’ a — cos*2a + cos*3a—.

1 -— 3t
1-f

It is not true that } always lies between cos*@ and cos* a — cos*2a, or that (1 — 3f) (1 — E\m
always lies between 1 and 1 — 3¢, whenever ¢ is positive. The following investigations, though they
will fully explain why it is that the theorem is so often true, are insufficient to distinguish accurately
between those in which it is and is not true.

When gw can be expanded into 4 —- Ba+Ca*—... (4, B, &c. being positive), we take
the known form

SS pe ate = Se

n

, ny ® (n) Z (n+1) ”
p0+po.a+ gt t® Mr am eg (Oa)

a
a" 1

ZeS cae Mt

in which @<1. If then Po, y’0, &c. be negative, and 0, "0, &c. positive, and if pa, p'a, &e.
each preserve, up to w = a, the sign it starts with when w=0, there is no question that the theorem is
true from # =0tow=a. ‘Thus common differentiation with respect to w will prove the theorem
for the case of

2 @~"dv ' '
i). =1—H + 2a* —2.3v'+...
o L+av

194 Mr. DE MORGAN, ON DIVERGENT SERIES.

For any particular series 4, — 4, +... it is enough that 4, — 4,#+ ... should be a continuous
function of 2 whose differential coefficients preserve their initial signs from «=0tow=1. But
though some of them should change sign, the theorem obviously remains unaffected as to summation
stopping at parts of the series in which no change takes place. It is then no wonder that the
theorem should be so frequently true.

Whatever value a function may have when w =0, it is obvious that if the commencing series of
signs, namely, those of 0, po, &c. be + — + —+-— &c. ad infinitum, the function itself, and all
its differential coefficients, are at the first instant in a state of mwmerical diminution. The reason
is that those which begin negative are algebraically increasing, while those which begin positive are
algebraically diminishing: this follows from the well-known (but much too scantily used) theorem
that a function is in a state of algebraical increase or decrease according as its differential coefficient
is for the moment positive or negative. Adopting for convenience the mechanical idea of the differ-
ential coefficient representing the velocity of the function, and supposing a to be the time elapsed,
say in seconds, let pa =A,- 4,a + 4,a°—,.. bea function of w, A,, 4,, &e. all being positive,
And first let A,, 4,, 4,, &c. present an unbroken series of diminutions, or 4,—A,, 4,— 4,, &e.
an unbroken series of positive signs, Then @a begins = A,, with retardation at the rate of
— A, per second. But 4, is less than 4,; therefore this rate of retardation cannot change the sign
of pa in one second, unless it receive an increase. But this there is no symptom of at the com-
mencement, since a) is positive, and the retardation begins by being checked. Hence, if a func-
tion start with a differential coefficient of a sign different from its own, and numerically less, it cannot
change sign within the next unit of increase of the variable, without the second differential coefficient

F ‘ Aten rrr

first changing sign. Nor can it even change sign before w becomes —* without a change of sign in
1

pa previously occurring. For if the velocity had continued uniform, it would then have been

A é : A : ;
A, - ZA or 0, and would not have changed sign till after w Se at least; but since the velocity

1 1

: on 2 3 , A
of retardation begins by being diminished (gv being positive), it must make this up before # = —

1
if a change of sign be to take place; that is, increase of retardation must come on, or gp’ a must

become negative. All this will be very plainly pictured in the curve y = pa.

Again, if @,« = 4, — 4,w+... and if A, > A, similar reasoning shows that @, v cannot change
sign before w = 1, unless @,"a first change sign. If neither p”a nor @," xv change sign from wv = 0
to w=1, then it is easily collected that 4,—- A, +... lies between A, and 4,— 4,. And if we
suppose 4,, 4,, &c. to diminish until we come to 4,, then if p, «= A, —d,,,@+ ... we see that if
neither p,-2t nor 0,18 vanish before # =1, we are sure that 4,_, and 4,_,—A,_, contain
A,_,—A,-, ++. between them; from which it may readily be proved that the theorem is true up
to the last but one of the converging terms, under the preceding pair of conditions.

The useful part of this theorem in calculation, is undoubtedly its wswal truth for all the
apparently converging terms of the series. And we see from the above that if these converging
terms last up to 4,, then m not being >, the theorem is true up to 4,,,, inclusive, if neither
P~,,-2@ nor Gay w vanish before w=1. But the theorem is not wmniversally true even for
converging terms. Let pa =3-2a + 2 — 20a*+20a*—20a°+ ... which has three terms con-
verging, and is of finite divergence; so that Poisson would admit - 8 =3-2+1-—20+4+20-.,,..
as the limiting form of the above when a=1. But —8 does not lie between 3 and 3—2. This
series is the development of (3 + # — a*— 19°) (1+ .)~’ and its second differential coefficient will
be found to change sign before # = 1.

We will now look at the theorem in another point of view. Every alternating series may be
reduced to a case of px -p(w~+1)+(w+2)—... in which @v is a positive function from
v=@ to v=o. If this be the proper developement of ya, then Wax + (wv +1) = a; con-

Mr. DE MORGAN, ON DIVERGENT SERIES. 195

sequently Wu+.(v+1) must be always positive from v=a# to v=o. Hence Wv cannot
change from + to — when v =a, without changing again from — to + before v=a+i1. Now
the theorem can only be disturbed by y,v becoming negative: for w= pw — (a +1), or,
W (a +1) being positive, pa>ga; again a= pa — d(a+1) + (w+ 2), or, (w +2) being
positive, yw <a — d(x +1), and so on.

Hence 1. No function y,x can be expanded as above unless it be one in which its changes of
sign go in pairs, the — + change following the + — change before the variable has received an
additional unit : 2. except at those epochs at which \, (# +) happens to be negative, the theorem
must be true. As long as px, @ (w +1), &e. continue diminishing, the theorem must either be
true, or there must be a minimum value of ya within a unit-change of the variable, reckoning
from the last change of sign. When ya changes from + to —, wW’a is negative, and when from
= reyes Ne is positive: there must then be a minimum value of yw between the two changes.
Now as long as @w diminishes, or pa is negative, yw +) (a +1) is also negative. After the
minimum is past, then, y/’# cannot continue positive until a has increased by a whole unit, or there
must be a maximum value within a unit-change of the variable, reckoning from the minimum. If
then the terms continue diminishing as far as @ (w + 7), it may be collected from the above that the
theorem is true for the several summations up to @ (w+ —1), except for those in the neighbourhood
of the last terms of which are found two roots of yy for values of not differing by a unit, followed
by a maximum value of wa, for a value of # not a unit in advance of that which gives the inter-
mediate minimum of the roots. And if \ # can ever become infinite, pa being finite, then ...
We (# +2), We (a +1), Wa, \y (w — 1), ... are all infinite, with alternate signs. From thisit will readily
be seen that in the greater number of cases the theorem must be strictly true.

Again, it is now known that every function pw can be expressed in the form = Ae", provided that
integration be included under the sign =, and also the finite summation of terms in which J is
infinitely great, and a infinitely small, and which give a finite sum by difference of sign.
Whether many cases of this reduction do not involve much greater difficulty than those of divergent
series, may be a question. However this may be, it is clear that in whatsoever manner ma may
be represented by =4e, in the same manner pa — p(w+1)+... may be represented by

ar
s Ae
1 +”

In all cases, then, in which the several terms of S4e™ are severally positive, and, if

an

infinite in number, can be arithmetically summed, it follows that Wa or =o , is also positive.

€
Thus for all cases in which pa can be expressed by [rev dv, xv being always positive between
the limits, it follows that the theorem is true.

We find then that this theorem must be true in the great majority of cases: as far as
observation goes it is not known to have failed in any one of the instances in which its use is of
importance. It is enough, without any thing else, to draw a great distinction between the pro-
gressing and alternating series. But this is not all: it is also matter of observation that there is
great difficulty in finding alternating series* which become infinite for one or more values of their
variable, without having recourse to those in which the law of the coefficients is discontinuous. It
is most easy, both to make the above theorem fail, and to procure a case in which infinity of value
can be obtained, by means of the development of common algebraic functions, presenting discon-
tinuous coefficients; but it is not easy with coefficients following a continuous law.

It cannot of course be proved that 4, — 4, + A, —... is necessarily a finite quantity, since cases of
exception may be procured: but some illustrations may be given of the tendency of this form to
represent only finite quantity. Probably nothing but the collection of such tendencies will ever
lead to a rigorous criterion for ascertaining in what cases it can represent infinite quantity.

In a great many cases, a large majority of those usually considered, the complete alternating
series A, — 4,4 + A,a* — .., diminishes without limit, as w increases without limit: and the faster

Vou. VIII. Parr Il. Ce

196 Mr. DE MORGAN, ON DIVERGENT SERIES.

A, A, &c. increase, the more rapidly does this diminution take place. We shall see, in the next
section, that this comminuence of 4, — d,w + ... and a’ is to be looked for as the rule, its failure
being the exception.

Let the series be transformed into
2

1 x a
=i oh = a gy heen
aerate agri! qd Teale nei Orr egal
3
= (A, = 5 Lay es i A le
(4, aA, +3a°A, 4) ay

which is easily done. Let @ be taken so small that the series just obtained shall still be alternating,
which can generally be done, though not always, and then, on account of the factor (1 + a#)~', it
is clear that the original series and 2’ are comminuent except only when the second series and wv
increase without limit together: that is, instead of supposing, as @ priori we should do, that the
alternating form with terms increasing without limit has an equal facility of approaching any given
limit, we are rather to look upon it that its facility of approaching any other limit except 0, as
increases without limit, is only equal to that of its approaching ©, or increasing without limit. I
am not, of course, disposed to attach much weight to reasoning which rather resembles that of the
theory of probabilities than of pure mathematics: but I do say that it must be better to take such
considerations at their proper value, as suggestions for the conversion of results of observation into
demonstrated theorems, than to allow isolated facts which evidently point at something, to remain in
their state of separation.

This inaptitude to represent infinity, and this tendency to comminuence with a-' are both cir-
cumstances which render the operation of integration much safer as applied to alternating than to
progressing series. But the principal distinction between the two kinds of series seems to me to
depend upon our present knowledge of the meaning of integration, as explained in a previous
section, being imperfect. The progressing series cannot be expressed differentially without the
operation of what we may call progressing integration; the alternating series can. This is exem-
plified in the two following remarkable theorems, given by Poisson :

PO + pl + G24... =LH0+ fy bedzx +2dm=* § \” cosammz pe dx},
p9- pl + P2-...=b PO + 2BKP ne (cos max — cos2mmrz) pxdzt.

We may now examine the sort of proof which we can obtain of the usual values of divergent
series, with the view of comparing finite and infinite divergence. Let V = P, —- P, + P,—... and
let P,, = 1 when w# = 1, independently of m. Also, before vw = 1, let the series be convergent ; after-
wards divergent. Let P, = P,_, - p,_,, whence p,_,=0 whenw=1. And

V = P.- (Po- po) + (P, - p) - --. or V=2P.+(po- Pi + Pe — ---)-

Again, let W= Q,+Q,+Q.+... and let Q,=2" when vw=1. Let Q,=2Q,_,;- 13

whence q,=0 when a=1. And W=Q,+(2Q,-q@)+(2Q-—M) +.--- or W=-Q-(Q+%
+. +.---). When w =1, we have :

V=tP,+(0-0+0-...); W=-1-(04+04+0+...)
and on the proper equivalents of the two evanescent forms it depends whether 1-1+1—-...=4
and1+2+4-+...=-—1 are true or not. Now instances enough may be produced in which

0+0+0-+.., is not an equivalent of 0: though, by instances merely, it would be found exceed-
ingly difficult to overturn 0 — 0 +0... = 0, as long as the common operations of algebra only are
used. But here again, when the forms of the integral calculus are employed, instances may be
produced in which, though the form 0-0 +... may still be called 0, it is only by means of a
discontinuity which, occurring as it does at the limiting form of an alternating series of finite

Mr. DE MORGAN, ON DIVERGENT SERIES. 197

divergency, has a tendency to destroy the exclusive confidence which many modern analysts have
placed in them.
The very foundation of this confidence is, as we have seen from the expressions of Poisson,
a full belief in the maxim that whatever is numerically true up to the limit is true at the limit.
To this principle, reasonable and convincing as it is, let us join the remembrance of a fact so
well ascertained, were it merely as a matter of observation, that alternating series are more safe
and more easily calculated than progressing series, and also the simplest of all theorems on
convergency, namely, that an alternating series is rendered convergent by mere diminution, if
-* sin aa
- dw. I

unlimited, of its terms. With these premises let us consider the integral
che

believe that this one integral might be made to throw a case of exception in the way of those
who have claimed privileges for the finitely diverging series over other non-arithmetical forms,
in every particular as to which their superiority has been asserted.
Se, We 2 : 4 2sinaw :
Poisson, agreeing in this point with all other analysts, asserts that of apy AP is 47, 0,
wee

or —tr, according as a@ is positive, nothing, or negative: any computer using the method of
quadratures would confirm this result in all its parts. But this integral is clearly the same as

3r

wT Qr
asinaw @ sinaw @sinaw
f ae fi a da +...
wv = v = «@

a 7 a -

which is an alternating series, since the second, fourth, &c. integrals are composed entirely
of negative elements. Moreover the terms diminish without limit, since the numerators of the
elements are recurrent, but the denominators constantly increasing, and without limit. However
small a may be, if it be positive, 4a is the real value of the series, obtainable by the computer:
and yet if a be absolutely = 0, each of the terms is also absolutely =0. But if 1-1+41-... is
to be taken as having the wnique value 4, which may be employed in analysis (the Italics are

Poisson’s expressions) because 1-g+g°—... is certainly (1+g)~', however little g may fall short
: ‘ 7 ap | fe
of unity, then surely 0-0+0-—0+... may here represent either — gq Of + G> Since, however

small a may be, when negative it gives the first, and when positive the second: notwithstanding
which, it is certain that 0-0 +0-—... is in this case = 0.

Here then we have O—0+0-—..., a limiting form, and that which is true up to the limit is
not true at the limit. But why is this principle abandoned, being, as it is, the very point on the
assumed clearness of which the line is drawn between the accepted and the rejected cases of non-
convergency? Is it that an infinite series of zeros must represent zero? I think I have shown
sufficient cause against that assumption. Is it by the principle of mean value discussed in the
last section? No one that I know of, except Leibnitz on grounds purely metaphysical, has ever
used this principle, and no one has hitherto stated it in general terms: and moreover the modern
analysts appear to require strictly arithmetical foundations, and would acknowledge no identity
of principle between their methods and one which produces tan o = We 1; they seem also to
suppose that they are quite free of the use of principles established by induction. Either then
the principle that whatever is numerically true up to the limit is to be held true at the limit
must be abandoned, or exceptions of discontinuity, in questions involving integration, must be
admitted to be possible in a manner which renders the cases to which Poisson and others have
confined themselves subject to as great difficulties as those which they have abandoned.

In a preceding part of this paper I spoke of it as a strong presumption that A, + 4,0 + A,w*+...
should represent 4, when w= 0, or that the form 0+04+0+... which follows 4, should =o.
If A, A,, &c. be all positive, and if the series be always divergent, however small w may be,
it is obvious that where the preceding represents a function of complete continuity, we may

cc2

198 Mr. DE MORGAN, ON DIVERGENT SERIES.

look for its value at #=0, from the limit of A,- A, + A,a°-—... as well as from that of
A,+A,v+... Accordingly, when there is continuity, all the presumptions of superior safety
which the alternating series presents may be applied to this intermediate case.

SECTION V.

On Double Infinite Series, in which the Terms are infinitely continued in both
Directions.

One look at the series

we + p(w — 8) + p(w - 2) +p(w-1l)t+ pat p(e+1)+p(w+2)+P(w+3)+...
will show that, whenever it can represent a definite function of «, which preserves its properties for
different values of w, it must be a solution of the equation (#+1)=.)«. Various modes of
proof, applicable however only to functions and processes of complete continuity, show that, in
all cases to which those proofs apply, the representation of the above is simply 0, or rather

, , 0 y Fi 5
0 , either 0, or, in particular cases, a And certainly, in all cases, it can be reduced to the

xv 3 A . *
limiting form 0+0+0+..., so that, if not always =0, the warning given in another part of this

paper is confirmed. Throughout this section, let Spw stand for a double series of the above form.
For pw write pa.a* and divide by a” which gives ...+ @(w~—1)a7"'+ Gw+G(w+l1)a+...
Now

mr

1 a at+a@ dv a+4@04aq'ax

put ——, pa + ue + . eae
a (1 - a)? (l1-a)® 2 (j-a)' 2.3
in which it need only be noted here that the numerators of the functions of @ all read backwards
and forwards the same in their coefficients. Now by the same rule

1

p(-2) +p(-#-1).44+..= $(-0)- GaP (- 2) +

1l-a

put p(w+1).at+..=

change w into —a in the last, and a into a~', add the result to the preceding equation, and

substract pa, which gives S(pa.a") =a" (0+0+0+...). Again, taking the calculus of

operations, let Epx=o (w +1), then, of all perfectly continuous answers, E~'dw must mean

p(#—-1). The whole operation performed upon da in S@wis...+E-'+ E+ E' +... or
0

1 x ° A cq 5 aigithe a
err i+ ENTE or 0. But it must not be forgotten that, in cases in which discontinuity is
ee ee a te

possible, it does not follow that E~" a always signifies @(w—m). For if we were to assume,
for instance

E-°pa=9(«-3)+ (t+ fA" ae) yo

we should be justified by the result E°E~*pa=wx, whenever a — (a +3) is negative, though
when a — w is positive, the preceding would not be the same as ¢ (w — 3).

This is an important point, not only in reference to the calculus of operations, but to every
case in which inverse operations are employed. There is, I am well aware, among mathematicians,
something like a disinclination to provide beforehand for discontinuity, which first showed itself in
the struggle against admitting discontinuous functions into the solution of partial differential
equations. But it should be remembered that, in our time, trigonometrical series of the most
continuous form have been shown to represent functions of the most capricious discontinuity. A

—

Mr. DE MORGAN, ON DIVERGENT SERIES. 199

mathematician has lately amused himself with preserving the first part of the air of ‘God save the
King’ for posterity by means of a case of Fourier’s integral ; and any one who has studied the pro-
perties of the series 4 cosw + Bcos2@ +... knows that a sturdy computer, who is not afraid of the
method of quadratures, might hand down the means of recovering the profile of his own face from
its equation: and that in a form which no analyst could tell at sight from the equation of a circle,
a parabola, or some other continuous curve. Nor is such discontinuity a mere possibility : it is
constantly occurring in the higher branches of mathematics, and its detection and treatment forms
the most distinctive feature of the most recent school. Surely then it is time to pay attention at the
outset of every plan of investigation to the possibility of the occurrence of discontinuity in inverse
operations.

I do not see how absolute error is to be avoided without such a precaution. Defining
Eu as p(v+1), nothing is clearer than the right to use the symbol EZ, and those derived from it,
algebraically : all the fundamental symbolic definitions are satisfied by it. If we are to assume, as
of necessity, that E~"g@a can be nothing but gp (w—n), the symbol Sa must represent 0, as
shown: and experience points out that it actually does so in every case in which there is no
discontinuity. But in certain cases, as I shall show, S'pw does not represent 0, but another
solution of y,(# +1) =a: there is then some flaw in the demonstration, which I take to be the
assumption without reserve of E~"pw = p(#—n).

I might give other ways of expressing Sa, all ending in the same result, that, unless some
special mode of introducing and allowing for discontinuity be adopted, it represents 0. But this
paper is already too long, and I therefore pass on to some cases in which it does mot represent 0.

Let us consider the series,

1 1 1 1 1
T+ (=e) 14-0 148 ‘ie@+ePe 1+ @+ 208°"
which is both ways convergent. We have the two following results,
ieee Sei EON he estas Savy, Hpac ae
K 1+ (b+ ke)’ ‘ 1 + (6 — ke)’
that one being taken in which e” is raised to a negative power. Let 6b lie between me and
(m+ 1)c: then we have

—=—= ii ets ie ae pag $e O“Mmle ---) sinvdv
1 1
++ (b—m +20)" 14 (b—m+ ic)
in which integration is performed on convergent series only. Hence,

og tek =
i (e- PFA He (mt Be— yo .--) sinvdv,
0

wo exam Ee Al OTE) o eo= bln ee (me —b)v

il
— = wi sin vdv= [ —_—_________ sin vodv,
4 lee (6 + pe) p az Bac k eo = en ?

where m’=m-+4. Now m'c —6 is numerically less than 4c; and Legendre has shown (see my
‘Differential Calculus,’ page 669) that if g be not greater than h,

us us
pees d 7 ek—e h
——___——— 81n 9a) = — >
a el? — e-v 2h = = Te
e+e *-+ 2cos —
h
wi te Ooty gcee™
7 ecn— a2 7 gee es. 4
(ioe rk a 2b an at 2rb
eo¢+e © +2cC08 (2m +1 ——) efutele — 2cos——
c c

is the value of the series, which is, as it should be, a solution of y, (b + c) = Wb.

200 Mr. DE MORGAN, ON DIVERGENT SERIES.

Before showing some consequences of this and similar results which will be interesting as
extensions of known theorems, I proceed to verify my assertion that this series, being double,
and not =0, will show signs of discontinuity. Let us consider the series

ie AGEs et) r

= +
od 148 i+(b+0) 1+ (b+ 2c)°

+ woes

which is convergent when ca is 0 or negative. This series is a solution of

Py eo
ap Chas ?
y Gee
4 + coswe’ da zsinwe’ da
whence y = sin a [ Sg es ne a f Fig a gers
BE ob

There is nothing in this result, as long as the final value of @ is negative, to hinder the
computer from finding the value of y by the method of quadratures, and comparing it with the
result of the convergent series. And even when a =0, the part of the first integral which comes
from between # =—a and # =O, a being infinitely small; is rendered evanescent by the factor

sin #, as special examination will show. If then we make w = 0, and if we venture to change the

-—1
sign of c, and put the two results together, we have, remembering that the term (1 + 8°) occurs
twice,

a! 1 0 1 1 >
— -f ( =k —>:) sin ve!"da
1+(b+pey 14+8 re Pl eae

-—@
1

Fa Geeer °C

ye 1

= -f sina e* da =——_;
re 1+8

a false result, but agreeing with the theorem already discussed, and which I think may now

be described as follows, The double series Sa is, if its two sides be perfectly continuous, = 0:

and any method which proceeds by neglecting discontinuity will end in S@w = 0, true or false.

But perhaps it may not be evident at once why I say we have neglected discontinuity in the
preceding process: if so, the following explanation will be necessary.

A continuous equation is one in which the two sides are algebraical equivalents, that is,
in which the right to use the sign of equality is independent of the value of any letter or letters.
If this right be destroyed by the passage of any one letter over a given limit, there is obviously
discontinuity. Now if a= pa + p(w+1)+... bea continuous equation, or if ya = pa + (w+1)
be universally true, we may convert it into

Wa=—-(#-1)+W(#-1), or pa =- p(#- 1) - p(a- 2)-...:
if this be granted, then Sq@w=0. Conversely, if Sw be not =0, then Yw= dat... being
true, Ww= — d(w—1)-... is false. Also, if the assumption of the permanence of any equation
make Sdw=0, then, whenever this last is not true, it follows that such assumption of per-
manence is erroneous. In the preceding result, we have assumed the permanence of the equation

o sinwe’" da 1 1

3 ia pe ipa
for all values of ce. The error of our result is manifest: this permanence then has no existence.
And the warning is that when c is made negative, we integrate over a diverging series: in
fact, our process assumes the ordinary development of (1 — e**)~! when ¢ and « are both negative,
or e*>1, and integrates that development.
There have been two discontinuities occurring in the preceding; the first dependent upon
the introduction of m, the second that just considered. The first may be treated as merely

~Mr. DE MORGAN, ON DIVERGENT SERIES. 201

incidental to one particular process; we were not bound to Legendre’s integral ; and this dis-
continuity disappears in the result. But the second is essential to the problem; the series
satisfies a certain differential equation, the complete solution of that equation is ascertained, and
therefore the series must be represented by its equivalent solution of the equation. No other
equivalent could have been anything but the one we found, or the same in a different form.
As matters stand, then, we cannot have a continuous relation between the series and its invelop-
ment: and this, I will venture to prognosticate, will continue until the definition of integration is
extended.

i earner sl a SP eee ee eee
= — .-. whic :
1+(b-c)? 14+ 1+ (6+c)’ 1 + (b+ pe)?
Proceeding just as before, and, b lying between me and (m + 1)e, we shall find, m’ being m+4,
as before,

Let us now try...

(mic—byv fe —(m'c—b)v

1 "¢€
iS’ —_—____——— = (— 1)” TGS,
1+ (6+ pe)’ ( yay Ci brew es paki

But Legendre has also shown the following, g being not greater than h:

us T
moe a F
(e?4— e 4) sin Re:

ee ald 7 2h
FSS IVI Se ’
hv —hv

+ iis Es 7
S 5 eh +e +2008 “=

whence the series in question is

2 ee i) a aie 6
(e¢—e °)sin (m+h-<)n (ef-—e ©) cos =
c c

2a(- 1)” Qa
——_ - 5 OT —————— NT 5
ec ae 2b Choma aee Qa
ec +e +2008 (2m +1 -—)m ef +76) — 2icos
ec
which is, as it ought to be, a solution of \/(b+¢)=—-—vy,b. Now consider the series,
Ae gto" eb t20)2
= - +———_—__ -...
Ite 1+(b+c)? 14+(6+2c)?
dy e*?

which is a solution of y + ——=
y Che setae

: * cosa edu > sinwe’"da
whence y = sina [ a cosa [ — —
ate ete? 1l+e
on which may be repeated all the remarks on the preceding case. But in this case, when ¢ = 0,
the value of y gives, as it should do, 41 + B)7.

The danger of integrating over a diverging series is thus shown to be incident to alternating
as well as progressing series. It cannot be denied that Poisson has separated the only case in
which integration can be used with some freedom and safety on non-arithmetical series: namely,
the finitely diverging series which lies between the convergent and divergent cases. | Whether
the freedom is entire and the safety absolute is more than can be determined at present :
unacquainted as we are with all the varieties of the discontinuity which appears in limiting cases
of integration, as now understood. On this point, I must refer to the preceding part of this
peper.

With regard to the alternating double series + A_,w-°— A_,v~'+ A,— A, w + A,w*—-... we
now learn that, whenever complete continuity exists, 4,- A,a+..., v being infinite, must have
the same value as 4_,w~'— A_,w~*+... when w-' is nothing; that is, must vanish, generally
speaking. This observed tendency of 4,- 4,a +... has been already noticed.

202 Mr. DE MORGAN, ON DIVERGENT SERIES.

I now take some results of the two series here discussed which are interesting in the way of

verification and extension, though not illustrative of the points on which I am specially writing. If
for b and e¢ we write b : a and c: a, we have
Ta Ta
= eo = eo) cos —
1 TT i SN See ! 1 Qa c
S =——_—_—3 = — ee A a ee ee
ewe Sake = == Qarb a@+(b+pe (Coe ee ah
a + (b+pe) Ge salle Hae eee ) A tiene 1 asin eee
c c

Make a =0, ¢ = 1, and we have
ao \* 1 ie 1 - 1 4 1 - peste we
ler = meee G=2yp" Clip ee GTi” Grey

1 1 1 1 1

2
= SSS eet a te
= =) Coe eS ct paaye wELI) so Gd)e, (4-38)
(Ca ee a \? 1 1 Nee os 1 1 rn
Pn do" \\sin =) ee (ae ie bly 4 2)he sas
(-1y @} ( 7 ) A 2 1 1 ye 1 ~ 1
Tn de*—")\\snnb)f “" G—2)") G—1" &* +1)? +2)"
Tv 1 J 1 1 1
Sah bee bo BA Spe

If we had commenced with §(6+pc)?-—1{ ', and had used the formula fe sin vdv
= (1 — m?)-', which Poisson would have admitted as a limiting form of hn ef ktm —-l” sin vdv, we
should have seen in the final result a right to substitute Thi i= 1 for a in the preceding formule ;

giving

. 27a . wa wb

sin —— sin — cos —

1 Tv c ai age 1 2or c c
“(b+pey?-@ ca 2ra Qrb° (6+pe)?-a* ac 2ra Sab"
cos —— — cos —— cos Spy!

Various formule might be obtained by differentiating these with respect to a, b, or c; and
various others by integration, one set of which is remarkable. Multiply the two first equations
severally by ada, and it will be seen that the second sides become integrable : integrate from a = 0,
and make the antilogarithmic change, which gives the following continued products, of which the
well-known formulz for the sine and cosine are particular cases.

Qra Qra Qa
a a a y(ee +e B age CO
Fay (alls 1+ Ue $$$) cr
( a ( +a) ( ae, (1 +Gay) aonb
1 — cos ——
ec

(1+) (4%) (1 uae aly ry petra mh ab
es (b — 2c)? b ea $(ecte Page Oe a + co

a” a’ ae ae wb ab
.(1+—— sa 1 +-———; |} .-. 1 (ec ~e Rae = ees
( + - x) ( + G+ =) d(ee +e °) + cos ; 1 — cos :
the second of which is readily deducible from the first.
arising from the substitution of a \/ —1 for a.

PNG EMI NA CNG AEE tt I Mi

It is needless to write down the forms

oe

aa ee

NOS

Mr. DE MORGAN, ON DIVERGENT SERIES. 203

From what precedes, we are warned to expect some discontinuity arising in the treatment of
any series f(v)+P(w+1)+... if h(v+n)=p(w~-—n), unless that series be an analytical

equivalent of 0. And even in this latter case, it is to be remembered that os is the real form, and
that when yw = 0, there may arise cases of exception in which the series represents a finite quantity,
and even infinity. This particular point has been so beautifully illustrated by Poisson, in his
treatment of the series $ + cos @ + cos2@ + ... that nothing is left for any one else to say, at
present.

In mentioning once more the name of this distinguished analyst, I may state that the point in
which I have freely ventured to question his judgment is not as to the wisdom of the course he
took, in rejecting divergency from the integral calculus as he found it, but as to the grounds on
which he asserted a final and fundamental difference between what he adopted and what he rejected.

A. DE MORGAN.

University Coniece, Lonpon,
January 15, 1843.

ADDITIONS.

Page 192, line 8. Ir is not asserted that cos* co + sin* co = 0, for. the mean value of each of the
terms is 4, and cos’ © + sin*co =1. Many errors may be made by forgetting that @ sinw (# = & )

or @ (0) is not the mean value of @ sin a, but ['¢ (sin w) da + 27.
0

Page 201, last four lines. If it should seem for a moment that this reasoning would apply
equally to 4,+ 4,~+... and — A_,#~!— A_,a~* — .,., remember that the theorem in Section IV
(to which the exceptions are only occasional) shows that 4_,a~-!— 4_,#-?+,,. lies between
A_,a" and A_,#-'!— A_,2~° when @ is great: but that we have no such argument from which to
infer the comminuence of — 4_, wa’ — A_,w~*—... and a. Still however, the equality of this
last to 4, +4, +..., when there is no discontinuity, would enable us to predict the very large
number of cases in which 4, + 4, v +... is infinite and negative when 2 is infinite and positive.

Vor. VIII. Parr II. Dob

XVI. On the Method of Least Squares. By R. L. Etuis, Esa., M.A., Fellow of
the Cambridge Philosophical Society.

(Read March 4, 1844.]

Tur 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 T’heoria
Motis, 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 investigation gave his second demonstration in the Theoria Combinationis
Observationum.

The subject has been also discussed by Poisson in the Connaissance des Tems 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 produce 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 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 reduced 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 Laplace’s, and which proves in the most general manner the superiority of the rule of least
squares, whatever be the law of probability 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 pro-
babilities. In this conclusion I cannot coincide; nor do I think Mr. Ivory’s reasoning at all
satisfactory.

Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 205

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 T’heoria Motis, 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 theorem, having been first given by him in the Théorie de la Chaleur.
It will be seen that the mathematical difficulty i is greatly diminished by the change.

GAUSS’S FIRST 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 observations,
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 @ priori. Undoubtedly the
conviction that the effect of fortuitous causes will disappear on a long series of trials, is an imme-
diate 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,
@ the observed value, e the error, then we have

@—-a=e,

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

= (w#,- a) =0;
1
or, b= Sis

which expresses the rule of the arithmetical mean, and which is thus seen to be absolutely true
ultimately when m 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 not only is Ze = 0 ultimately, but [fe =0, where fe is any function such that fe = — f(—e);
and therefore we should have

=f (wv - a) =0,

as an equation which ultimately would give the true value of # when the number of observations in-

creases 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
ppg

206 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES.

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 >f(#—- 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 @ priori probability of the observed phe-
nomena a maximum. That is to say, if a be the true value sought, 2, being the value observed
at the first observation, x, the corresponding quantity for the second, and so on, the errors at
the first, second, &c. observation must be a,— a, #,—a, &c., respectively; and if @e.de be the
probability of an error ¢ in any observation of the series, the quantity which is to be made a

maximum for a is proportional to
p (m- 4) Pp (%- 4)... P(w,— 4).

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

¢ @%- 4) P' (@%- 2)
+ &c. + =.
co) (a i a) p (z,.- a)
as the equation for determining @ in x Let a YW, then it becomes
iW (@ - a) =0.
Now we have assumed that the most probable value of a is given by the equation
>i (@ - a) =0:

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

- We =me, m being any constant ;
,
€
hence ae = Meé,
€

and de = Ceime,

Now as the error ¢ is necessarily included in the limits — o +0, we must have

Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 207

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 observations the law of proba-
bility 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 arith-
metical mean rests on a mere illusory prejudice, or, if it has a valid foundation, the reasoning
now stated must be incorrect. Hither 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 @ posteriori.

Suppose there were m different magnitudes a, a... a@,, and that each of these were observed
m 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 observations each containing » observations.

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

ky + he +...k, = K:

where # is that portion of K which is derived from observations of a,..

Then, ultimately, the most probable value which the given series of observations leads us to
assign toa, is (supposing @ is susceptible only of the values a, a,...a,) equal to a,, 7 being
such that the corresponding quantity k, 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, .-. @, includes all possible magnitudes from — «© to +o.

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.

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
a priori necessarily implies a result, viz. the universal existence of a special law of error, not only
not true @ 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
error already obtained, the method of least squares is to be deduced.

Let

6 =av+by+ &e.—- V,
6 = a0 + by + &e. — Vz
Pp eoDassOOOORE (a)
&c. = &e.
a,v + b,y + &e. — V,,

A
tl

be the system of equations of condition, which are to be combined together so as to give the
values of w, y, &c. he error committed at the first observation is ¢,, at the second ¢,, and
s0 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) proportional to

@ 7 (lax +b, y + &e. — Vi)? + (azw + bey + — V2)? + &e.)
; >

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

208 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES.

(q,a + by + &e. — Vi)? + (ow + boy +... — Vo)? + oo

a minimum: thas 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:

av da’ +y dab + &c. = SaV
oDab + yb? + &c.=SbV }..........+-(8),
&e. = &c.

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 # from the equations of condition stated in the last paragraph, we
multiply the first by 4, the second by p,, &e., and add: (4, u., &e. fulfilling the conditions

Sua=1, Su4b=0, &e. =0)
we find w= Suv — ne;

and if we assume that Spe is equal to zero, then the resulting value of a is SV: the error
of this determination being the quantity Ze, which we have assumed to be equal to zero,
without knowing whether it really is so or not.

Now supposing there are equations of condition, and p quantities to be determined, and
that m is greater than p, then we see that there are m factors 4, Ms--.“,, and p conditions for
them to fulfil. ‘They may therefore be subjected to —~p additional conditions.

This being premised, let us consider the probability that the quantity =e will not be less
than a, or greater than 8, a and # being any quantities whatever. The law of probability of
error at each observation being given, the question is evidently analogous 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= —/ and B=/, / being any positive quantity.

Then P is a function of /, and of w,...n,-

Let us now so determine p,...“,, (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 in our
determination of w, viz. Spe, lies within the limits + 7, 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 « 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, &e. *

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 + J, than if we 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 Due will be precisely equal to vw, bepdu. Then manifestly

P= Joy pdus
and we have therefore only to determine p.

Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 209

Let e, €...¢, be the errors which occur at the first second &c. observation ; @, ¢, de, 2 e2d €2.--
Pn.€nde, be the probabilities of their occurrence: the form of the function @ 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

Pies Pres tee Pn€, de eee de, Poe eee ewe wee rere ee ses reeaeseeseeeses aoa (lil)

and the first principles of the theory of probabilities show that the value of pdzw will be obtained by
integrating (1), e.-.¢, being subjected to the condition Spe = wu.

Thus
pdu= Ih ETDs Cae l y Ga LE ifsa Dex. alan cisie Selec tinsineciseclescieve« csi asy(2)

with the relation

~ My €; + My €y-6- + M, €, = UL
Consequently
U — hy Ey.02— fh, _1 € 3
pdu = de, [ prer---Pnrey-1Pr Ba deed ey toes seenes (3).
n
Now by Fourier’s theorem
+o
UW — €, — -8* — My_1 ©, - 1 © Uy — hes = lev = —
?, eS di da f Pe, cos (« ma Ma-1€n=-1 ste dew
Pn Teo, — BK,

which, replacing SE by a, becomes
Bo

o +o
Mh
—/ d -= de,,
El. a he de, cos a (wu ue) de,

Therefore

+2 +o

d --]
pau = “A [ da [ dey. f dey rer +++ Gye, 005 a (6 — Spe) sereseees (4).
0 —o

Now if wu and e, are to vary together
du = p,de,, and therefore
© +

p=~fodaf dey. f_

And finally,

=~ [au f da f € fe de, Pnén COS a (u — Tye) 6

= deness € €} «02 DE, ania soreveiee eee
Ss hs 1 = Fie gh nal U € ( )

Tw —o

de, Pier --- Py Eq COS a (06 — Dre) op. eee renee (5).

Now let us suppose that equal positive and negative errors are equally probable. In this case
pe = p(-«), and consequently,

es de sin anede = 0.
Hence (6) will become
= = f'au ik cos auda jive 6, COS apje, de)... te Pn€n COS ap, €, Le, weeeee vee (7)-
The next step is to find an approximate value of this expression.
When a=0 ye Pecos anede = [*? pede = 1;

as the error « must have some value lying between +.

210 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES.

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

y fess 1 & COS mae, de, ... eae Pn€n COS p,a€, de,
decreases sine limite, (being the product of m factors each less than unity) except for values of a
differing infinitesimally from zero.
Let Ak? = Ibe -ede, c= to pe-etde,
and develope each of the cosines in the above-written continued product. It is thus seen to be
equal to

i
1-—a Sk? + a! (= Sate! + Dee ashe? ig) - —&e,

Again, m being very large and ultimately infinite, it is evident that PT «' is of the same
order of magnitude as 7, while Sp? we kh? k3 is of the order of n*, the former term of the coefficient
of a‘ may therefore be neglected in comparison with the latter, which again may be replaced by
3 (2? k*)’, from which it differs by a quantity of the order of m. Similar remarks apply with
respect to the higher powers of a.

Thus the continued product may be replaced by

1 5 1 RSE
1— a Sk? + = anyway: ere tak (Sp kh’) + &e.

or by e~**"""; a function which is coincident with it when a is infinitesimal. When a is finite
both are, as we have seen, infinitesimal.
Consequently,
7 wo 9
--f du [ COSA eat se yacatalsesoviesa(8)s
m9 0

or
U

1 iy See 2 .
ase J, e HF dy Sa are dy,....(9),

Vado

uw = 2(Sp7k?)*v.
It is evident, that whatever 7 may -be, this expression for P is a maximum when

>2k? is a minimum.
Hence we get the following remarkable conclusion: When the number of observations increases
sine limite the most advantageous system of factors are those which make

Sy’ k? a minimum,
It remains to determine » from the condition of the minimum taken in connexion with those
already stated, viz. Sua=1, Spb=0, &c.=0. We have

where we have supposed

Wkuee
=a 1 igi lai ae
Sbdp a)
&e. =0

Let Aj, Az--. Ap be indeterminate factors, then we may put
Kim = ad, + dry + &e.
Hols = GoAir+ beds + &C. \ .eceeccecers (B)>
= &c.

Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 211

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

- a a .
we multiply equations (B) by = 5 = » &c. respectively, and add the results. For the second, we
fi, ce

&c. and then proceed as before. And

; a b,
employ instead of the factors we , &e., the factors re $ ie?
1

similarly for the others.
In consequence of the relations
Dua=1, Fu4b=0, &c. =0,

the new system of equations will be

a

ab
1=N25 pmnse |

=D>az+ttAX — + &e. |
Ie k*
0= &c.

These p equations determine ),, Az... A», and thus in virtue of (B) the values of m,, fs... My
become known. Finally as

= py Vi+ bs Vo+ eee + mee ’
w will be completely determined.
Now let us recur to the original equations of condition stated in the last paragraph.
e, =a,0+bhy+ &e. -V,
€, =d,0 +bh,y+ &e. — Vy
soongnangace (a).
&e. = &c.
6, =4,0+b,y + &e. - V,
From this system we deduce a new one, containing p equations. ‘The first of these is got by
multiplying equations (a) by

by by
Ke? 2?

i P z , &c., and adding the results: the second by using the factors

&e.: and so on as before, The resulting system will be, neglecting all errors,

03 5+ y3% 7+&e=35 ma

b?

ab Peace (cays
+yBoy + &= BT act (6)

2
&e. = &e.

The system ((3’) contains as many equations as there are unknown quantities w, y, &c. I pro-
ceed to show that if w be determined from this system, its value will be the same as if it had
been obtained from the most advantageous system of factors, namely, that which is determined
by means of (B) and (C). In order to prove this, we multiply equations (') by ,, Ay, &e.,
and add the results. ‘Then, in virtue of (C)

|
2 = XB eV t= oN + be.

Vor, VIII. Pazxr II. Er

212 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES.

V, V,
> wz=(, a, + Ao dy + Be.) + Ay ae + Ae be + Ke.) 7G + Ke.
1 2

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

@ = phy Vy + os Vo eee + Mn Vas
as before; which proves that the system (B') gives the same value for # as the most advantageous
system of factors. Moreover, as (/3’) is symmetrical in aw and a, y and b, &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 k, = k, = &c. and (@')
reduces itself to (3) given at p. 208 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
> (ax + by + &c. — V)? a minimum with respect towy, &c., we must substitute for it the

function 2a (aw + by + &c. — V)*, and 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
less inviting than the fourth chapter of the Théorie des Probabilités, which is that in which the
method of least squares is proved.

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

a1 fia [aa f dey. [dengue 1 €, COS a (U — Ze).

It is certain that Se lies between the limits + ¢. Therefore when? = ©, P should be
equal to unity. I proceed to show that this is the case.

wo 2 = +a +@
em dy if da 7 de... f de, Pi & «++ Pn &, COS a (% — Zpe)
0 —o -@

9
when m= 0.
Effecting the integration for w,
1 fa >
P =

cy a)
¥ Gyed.2 mda f de... [ de, Pie ++» Pr€, COS a Spe ... (10)

—o —-2

when m= 0,
since

wmtut Wee vee
[ e cosaudu = ——e~ i
m

and if e-™ sin audu = 0.
Integrating for a, we see that when m =0
+e +a
B= f dé, ».. if de, Pi, & «++ peje 7 (11).
Or,
+o +o
Praf giede,... f eid eres ee

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

Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 213

I proceed to show that in a particular case in which the value of P can be accurately
determined, Laplace’s approximation 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 pe =4e*, the upper
sign to be taken when € is positive.

Let p, = ws = &c. = 1, then

1 «© + +@
p= = da f ba 2 ONY ICS chips as (e**) de, cosa (u — Se),
To ee: 2
or,
1 * costa : ee :
p=-— H) ———.— da, since ih e-“cosaede = =°
aw J. (1+a°) D l+a

The value of p is thus given by a known definite integral, which has been discussed by M. Catalan
in the fifth volume of Liowville’s Journal.
It may be developed in a series of powers of wu. Up to w*~" no odd power of w can appear

2 2p
in this development, for i ee da is finite while p is less than m, and therefore the integral
0

may be developed by Maclaurin’s theorem. For higher powers the method ceases to be applicable,
and we must complete the development by other means. But as we suppose 7 to increase s. /. the
integral tends to become developable in a series of even powers only of w. Thus

2 cos ua e da © a
——.—_da= ae ——.d 3
f (1 + a’)" re if @a)s) 2 i sp (1 + a?)" peut ee

0

2 da
Let f Gta =f (n).
TI » ada —a* it:
nen i (4a) = f(n = is

and generally

2 2p
if _a Pda = + Af(n — p).

9 (1 4.0%)’
Now
hepa
Seer ee
TiS Nis eran ad

and generally,

T 1.3.,.2m2 —3—2p
SREP pioneer given ta

1 vee
MO a are

214 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES.

Thus
*cosuada 1 i 1 1.3
if =fnii--. u? + ————— _u‘~ &c.
F (i +a)" aye 2 2n-—3 2.3.4 2n—5.2n-—383
The coefficient of 2”? is
1355.-2)) — 2 1
= fn. z —
2.3...2p 2n—-1—2p...2n —3
or,
1 1
af oersgp

ip oy oko ep ahs

Let m become infinite, this becomes
1

SO) pane
and we have only to determine what f a then becomes.
Now by Wallis’s theorem

(2) - 1.3,..(2n — 1)3 Pre

= 2.4,,.2n—2

f 1 Qa 3 nas J
Therefore fn = — when 7 is infinite,
2\2n-1
1 fr 3
or nm=-|-
id ae (=)
Consequently,
2cosuada 1 /7\3 a. tt
ee a(n 1— — + —-.——. — &,
(l+a°)" 2\n 4n 2 (4n)?
1 if ted
= = ( ) e *” when m is infinite
Therefore,
1 ase

Now the value given for P at p. 210 is

1 2

To amano ot

In the present case mw = 1.

Bat [reede =1: and consequently Zp*k*=n.
Thus

ut
Bi 4" du, as before.
V nr >

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

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

The function p having been developed in a series of powers of w, 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

—

Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 915

: 1 : ae
neglect for instance every such term as —;,u"’, because we have a term in u”” divided by x’.
x ,

ue? 2(p—8)
Thus we retain —~ and neglect
nt =

, although, unless w be large, the former term is of the same

or a lower order of magnitude than the latter. That Laplace’s method does in a very general
manner give an approximation of this kind cannot, I 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

1 1 4 : Fev yy 5 9 ren
pe=- ae and the case in which «,, p9--., decrease in infinitum sine limite, I shall only refer
7 €

to p. 10 of Poisson’s paper in the Connaissance des Tems for 1827. Neither affects the general
argument. We now come to Gauss’s second method, which is given in the T’heoria Combinationis
Observationum.

GAUSS’S SECOND DEMONSTRATION.

The connexion between the method of Laplace, and that which Gauss followed in the T'heoria
Combinationis Observationum, will be readily understood from the following remarks.

After determining ,...u, by the condition that P should be a minimum, Laplace remarked
that the same result would have been obtained (viz., that Zu?A° must be a minimum), if the
assumed condition had been that the mean error of the result, z.e, the mean arithmetical value
of Sue 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 notation, &c.) It is in fact easy to see

that the mean value in question is equal to
a

updu 2

fy updu or to 2, updu;

hi pdu

and as
1 n ue
P= @SeRy
2 (Sp7k*)'

2 f” updu = =

which is of course a minimum when Sp?k? is so.

Gauss, adopting this. way of considering the subject, pointed out that it involved the
postulate that the importance of the error Dye, 7.e. the detriment of which it is the cause, is
proportional 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 asswme 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 repre-
sented by (Zpe)*. That is, that (ZSwe)* is the function whose mean value is to be made a
minimum. I now proceed to find it.

Due)? = Dye? + 2 Byj My €; €y ve+0-e've» (13).

216 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES.

+e
The mean value of & is [ €& pede = 2k’.
a
Hence, that of Sp2e? is 2 Sp? kh’.
The mean value of Spim2e,¢, is zero, positive and negative errors of the same magnitude
occurring with equal frequency on the long run,

Consequently,
mean of (Spe)? = 2 Dp?k? ......... (14) 5

and, therefore, as before, Su?k* is to be made a minimum. The rest of the investigation is of
course the same as that of Laplace.

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 m*, and then making m = 0, we find

cd ~ +o
[prt du = [ perder... [" penden Spe)? = 2 Bak;

and as the first member of this equation is evidently the mean value of w* or of (Zpe)’, this is a
new verification of our analysis.

As an illustration of Gauss’s principle, let the fourth power of the error be taken as the
nieasure of its importance ;

(Spe)* = Spte! + 6 Spy’ ps” e, eg + terms involving odd powers of e.
Therefore,
mean of (Spe)' = 2 Spits + 24 SuPurohy hg? ...sese0e (15)

and m4, --. #, 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.

Recurring to the equations of condition, as given in p. 208, we see that the values Laplace
assigns to the factors y, m, &e., are independent of V, V, &e. They depend merely on the
coefficients @ 6 &c., which are quantities known a priori, i.e. before observation has assigned
certain more or less accurate values to the magnitudes V, V, &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 a b &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 conclusion is wholly irrespective of the values of V, V, &c., and consequently
of those which we are led in each particular case to assign to a y &c. The comparison is one
of methods, and not at all one of resu/ts. But when V, V, &c. are known, another way of
considering any particular case presents itself. We can then compare 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 a 6 &c. equal, as in the former case, but also V, V, &c.) The true values of
the elements x y &c. may be different in each. But in affirming that & &c., are the most
probable values of w y &c., we affirm that the true values of w y &c. are more frequently equal
to & » &c. than to any other quantities whatever. Here we have no concern with the method
by which the values — 7 &c. were obtained. The comparison is merely one of results.

As for one particular law of error (that considered in p. 206), the results of the method of
least squares are the most probable possible; and as the function by which this law of error is

Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 217

expressed occurs in Laplace’s demonstration of that method, it has been thought that his ap-
proximation involved an undue assumption, 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 shew 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 examination
of the process by which he was led to them, I may just mention that in the whole of Mr.
Ivory’s reasoning, the probability that Zue 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.

ie Ke
e tk a Sm

aig © oR 12
2a/ x z Sm

without being aware that in Laplace’s notation 7 and a are infinite, and that consequently the
expression is infinitesimal. (Vide Tilloch’s Magazine, txv. 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 Tilloch’s Magazine.

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

he remarks that the influence of the error e on the value of w increases as a decreases, and
versa 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 inversely 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

Sae=0, whence S(aw — V)a=0,

the result given by the method of least squares.

218 Mr. ELLIS, ON THE METHOD OF LEAST SQUARES.

But, granting that the influence of an error e, ought to be greater when a@ is less, and
versa 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 @ 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 deter-

mine, by introducing the extraneous idea of equilibrium on a lever, that — is the function re-
a
quired, seems to be little else than a petitio principii, concealed by a metaphor*.

The second demonstration may be thus briefly stated.

The values of different sets of observations might be compared 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 v, y, &c. involved in a system
of equations of condition, 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 observations, 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 actwally 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 ealeulated errors, namely, those which are determined
from the equations of condition when particular values have been assigned to a, 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 resulting values
_ of the unknown quantities may be liable to the smallest average of error.

This problem Laplace and Gauss have both solved. ‘heir 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 squares is applicable, are such, Mr. Ivory
observes, that there exists no bias tending regularly to produce error in one direction, 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 natura 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.

* Ihave 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.

Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 219

Mr. Ivory states the following equations of condition:
e=au—m
e =ax2—m
&e. = &e.
He thence deduces the following value of a:

Sae Lam

a” = —— + ——., and those of e e’ are
ra’ os Sane

axtam adae
=
=a Za’

=

e=—-m+

" jena Sam aeae

e=-m his cn nS . &c. = &e.

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
Sae=0, which of course leads to the method of least squares, and that in this case, as we
shall have

axtam
=-—-m+ SS c= &e.

each error is determined by ‘‘the quantities of its own experiment.” But this reasoning is
perfectly inconclusive. In the case supposed, ee’ &e. are as much connected together as in

any other, as may be shown by eliminating Zam between the equations
alam ; ne a >am
7 e=-m™ eae

Sa Za?

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,
whether we use the method of least squares or any other.

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

e=—-m+ &e. = &c.;

La.pa ¥) La.gpa—

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 pa, pe, &c. whatever be
the form of @. As it thus would serve to establish, at least apparently, an infinity of contra-
dictory 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 pre-
sent communication may tend to make a very curious subject more accessible than it has hitherto

been.
Vou. VIII. Pant II. Fr

XVII. On the Transport of Erratic Blocks. By Wituiam Hopkins, M.A.
and F.RS., Fellow of the Cambridge Philosophical Society, of the Royal
Astronomical Society, and of the Geological Society.

[Read April 29, 1844.)

Tue instability of opinion which usually, and perhaps necessarily, characterizes the earlier
researches into any new and extended branch of philosophical enquiry, is strongly exemplified
in the different views which have been entertained respecting the causes to which the transport
of erratic blocks is to be referred. In the first stages of the enquiry rapid currents of water
were generally recognized as the most probable agents in these phenomena. No attempt, however,
was made to calculate the power of this agency, and the theory was associated with hypotheses
far too extravagant to bear the test of careful investigation. ‘The natural consequence was the
very general abandonment of the theory on the suggestion of another possible cause of the
phenomena in question. It was represented that floating ice might have acted as vehicles of
transport, and many facts were collected, from the reports of those who had visited the colder
latitudes, confirmative of this opinion. Again, this latter theory has been lately endangered by
the recognition, on the part of some geologists, of a third theory, which attributes the transport
of blocks to the sole action of glaciers; a view of the subject which has arisen out of the curious
and interesting observations recently made on the movements of existing glaciers, and the phe-
nomena indicating their far greater extension at some preceding geological epoch.

The entire rejection of any one of these theories would imply a forgetfulness of the fact,
that geology is, in a peculiar sense, a mixed science, not merely as involving investigations which
properly belong to widely different branches of physical and natural science, but also as treating
in some instances of phenomena, (as in the cases of erratic blocks of different kinds, or in different
localities) which, while they possess a great community of character, may be referrible to totally
dissimilar causes. Both glaciers and floating ice are manifestly adequate with respect to their motive
powers, to produce the phenomena in question. In the following communication I shall investigate the
transporting powers of currents of water, and shall shew that, under certain conditions, such currents
would be generated of sufficient velocity for the transport of boulders, and consequently that this
cause is also adequate to produce the removal of at least a large portion of the boulders which have
travelled from their original sites; and that, therefore, the theory is not to be rejected on account
of any apparent inefficiency in the cause of transport assigned by it, or the extravagances which
have been formerly associated with it. We shall thus, I conceive, be constrained to recognize the
general adequacy of each of the three causes of transport above mentioned; and in the further
examination of the problem it will only remain for the geologist to ascertain, as far as possible, the
share which each cause has had in producing the actual phenomena of transport by a careful
comparison of observed facts with the probable results of each mode of transport. Each group
of erratic blocks, or each mass of transported materials, may present in this respect a separate
problem; in the present communication I shall only offer on this branch of the subject a few
general observations, without entering into any discussion of particular examples, beyond what may
be necessary for the elucidation of general views.

Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS. 221

SECTION I.
Transporting Currents.

1. Tuese currents may be divided into River Currents, Tidal Currents, Ocean Currents,
and Elevation Currents. By the latter, I mean those currents which would be produced by the
more or less sudden elevations of determinate portions of that part of the surface of the earth
which is covered with water. They are the only currents among those above mentioned of which
it will here be necessary for me to speak. Currents of this kind are always accompanied by a
corresponding temporary elevation of the surface of the water, constituting a wave. We are
indebted to Mr. Russell for all the experimental knowledge we possess of the nature and properties
of this wave, of the laws of its motion, and of the current which attends it. He has denominated
it the great wave of translation. The details of his experiments will be found in the Proceedings
of the British Association. It will only be necessary for me here to state his general results.

2. Suppose a long canal to be filled with water, and, for the greater simplicity, let it be sup-
posed to be of uniform width and depth. ‘There are various ways in which a wave of translation
might be produced in this canal. One of the simplest, and most appropriate for our immediate
object, would be the sudden elevation of a determinate portion of the bottom of the canal, which
portion, for distinctness, may be conceived to be about its middle point, and of small extent as
compared with the length of the canal. Two waves will thus be sent off in opposite directions.
Each wave will move with uniform velocity, preserving very approximately the same form. Its
length will depend, in great measure, on that of the portion of the bottom elevated to produce the
wave. Each particle of water begins to move when the front of the wave reaches the vertical
transverse section in which the particle is situated, and continues in motion till the wave has passed
over it, when it is again left at rest. Its motion therefore is not oscillatory, but one of translation
in the direction of the wave’s motion. Mr. Russell has established experimentally the following
law of this motion :

(1) Every particle in the same vertical transverse section of the canal has the same motion.

He has also established the following law respecting the propagation of the wave:

(2) The velocity with which the wave is propagated is equal to that due to half the height of
the crest or highest point of the wave above the bottom of the canal*.

3. From these laws we easily deduce the expression for the velocity of each particle, i.e. for the
velocity of the current which accompanies the wave. Let LPN represent the position of a
longitudinal section of the wave, at the time ¢, and L’P’N’ at the time ¢ + d¢, AB being the bottom

Py

M M M, B

of the channel, and CFD the level of the general surface of the water. Let P, be the crest of
the wave, QP,=h,; P any other point on the surface of the wave at time ¢, P’ the corresponding

* It should be stated that the experiments and observations by | that the same laws hold, at least approximately, for much greater
which these laws were established, were made on canals not many depth, as I have assumed them to do in the application of these
feet in depth. There appears, however, to be no reasonable doubt investigations to the transport of erratic blocks,

FF 2

222 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.

point at time ¢+ d¢, and mP=m'P’ =h, and M,Q=Mm = H, the depth of the canal, supposed
uniform. Also let V be the velocity of propagation of the wave; then will LL’=nn'=mm’
= NN’ = V58t; and let v be the velocity of the current at P, and therefore also (by the first law)
at every point of the vertical transverse section through P. Also let 6 be the breadth of the
canal; the area of the transverse section through P will = (H +h) b.

Now it is manifest that a volume equal to that whose vertical longitudinal section is Z Pr L’ (or,
in the limit LP P’L’) and breadth b, must have passed through the transverse section MP in the
time o¢. ;

Let this volume = U; then if np=y,

dU =b.areaqp’
=b.pp’.dy
= bVOt. dy;
. U=bVAdt,—
integrating from y=0 to y=mP=h.

But by the first law we must have ‘

U =vb(H +h) dt;
and therefore equating these values of U, we have

Also by the second law

VeVg(H+h);
eran |
. va=V/e(H+ h) ah
_ Ve (+h) a(i : ay
F H ; H
If v, be the velocity of the current in the transverse section through the crest of the wave,

h,
v= ar ear [by (1)]

4. Let us now suppose the wave to diverge from a center; then assuming the breadth
of the wave to remain constant, and therefore the velocity of propagation (V) to be the same for
every part of the wave, we shall have

OU =2rp.pp’. dy
=2rVot poy,
where p= Cn, C representing the point from which the wave is diverging. U cannot be found
generally without knowing the relation between p and y, i.e. without knowing the form of the wave ;

but if we suppose the space CZ (r) through which the wave has diverged to be much greater than
the breadth (/) of the wave, we shall have approximately p = r, and therefore

6U = 2rVot. roy,
and integrating from y=0 to y=mP=h,
U=2rVrhot.

—

Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS. 223

Again, since U is now the volume which passes through the cylindrical surface whose radius
= Cm (p’) and height = MP (H + h), in time dt, we must have by the first law
U = 2rp' (H +h) vot
=2rr(H +h) voé nearly.

Equating these values of U we obtain

he
di da 2

hy
Lav pt HT

These approximate expressions for » and v, are of the same form as the accurate expressions
obtained in the preceding case, but h and h, are not here independent of the distance through which
the wave has travelled; they are functions of r, To determine them let us assume the vis viva of
each wave to remain constant during its motion. The element (dm) of the mass in motion at the
time ¢, will be the portion of the fluid included between the two cylindrical surfaces whose radii are
p and p' + op’ and height H+h (MP). Therefore

om = Qa p’ (7 +h) op’
= 2arr Hop’ nearly,
if r be much greater than 7, and H than h. Also

h
Mis sre
h
= Var nearly,
Hence
V2 r+l i,
Sv bm =2r = rf hop’.
Now let

h=(p'—7)
be the equation to the curve LPN when CL =a, a particular value of 7; then assuming the

form of the curve so to change that each ordinate shall be diminished in the same ratio, we
shall have generally when CL =r,

were,
and ,
Sv2dm = 20 ee r Wy ()| [%¢ (p'— )} dp’;
or putting i p-Tr= p> and x’ = (p)’,
Sv'2dm = 20 ie r {y (“) pee x’ (p) dp,

which will be independent of r, if

Y 2
x4 (—)? =c=a constant,
a

224 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.
cy
: Sie

Hence NE TOF

or, for any assigned value of p

Consequently,

v and v, « Ws

5. A diverging wave, such as above described, would manifestly be produced in the midst
of the ocean by the elevation of a portion of its bottom. The height and breadth of the
wave will depend on the area of the elevated portion, the height through which it is raised, and
the time occupied in the process of elevation. Suppose this area to be circular, and its radius = R;
and first suppose the elevation to be instantaneous, and the height =h,. The resulting wave
will have a steep front, like that of the tidal wave called a bore, the height of its crest
being at first equal to that of the elevated surface of the water above the level of the general
surface = h, in the case before us; and the breadth of the wave will be the space through
which its front shall have diverged from the boundary of the original disturbance, when that
boundary shall have been reached by the inner circular boundary of the wave.

6. Let us next suppose the elevation to take place gradually, its amount being still = h,.
The surface of the water above the elevated area will be raised to a height less than h,, and
consequently the height of the crest of the wave will be less than h,, and the velocity of the
current produced by it will be proportionally less than in the former case. If R be small,
a small increase in the time occupied by the elevatory movement may make a great difference
in h,, and consequently in the velocity and transporting power of the current; but if R be
large, the corresponding diminution in h, will be much smaller*.

4. If the elevated area be a parallelogram, of which the length is much greater than the
breadth, two waves will proceed in directions perpendicular to the longer sides of the area, to which
sides the fronts of the wave (except near to its extremities) will be parallel. The breadth of the
wave will depend on that of the elevated area, It is important to remark that the diminution
in the height of the wave, and consequently in the velocity of the attendant current, will be
much less rapid than in the case above considered of the circular wave. Instances of circular
waves would necessarily present themselves in the elevatory movements of such a district as
that of the Cumbrian mountains, while wholly or partially beneath the sea; and examples of
the other kind, in the simultaneous elevation of the whole of such a range as the great mountain
limestone ridge of the northern part of this kingdom.

8. In the case first considered the wave was supposed to be propagated along a canal of
uniform width and depth. If, on the contrary, the depth or width decrease, the velocity of
the current will be increased, as appears from the expression for v,, (Arts. 3 and 4). Thus, if
a portion of a great wave pass into the mouth of a channel which gradually contracts, the velocity

* For example, let R=20 miles, and let the elevation be instan- ) minutes without reducing h, very materially. But if, on the
taneous. The depth of the ocean might be such that it should | contrary, A did not exceed a mile or two, then, under the
require 15 or 20 minutes for the surface of the water above the | same conditions, h, would be reduced to a very small quan-
elevated area to be reduced again to the level of the general surface. | tity, and the transporting power of the wave would be almost
In such cases, the elevatory movement might occupy several | annihilated.

ee I ee te

Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS. 225

of the attending current may become much greater than in the uncontracted wave. Such must
have been the case with respect to the portion of a wave diverging from the district of the
Cumbrian mountains, and received into the strait which must have been formed by the
pass of Stainmoor previously to its emergence from the ocean, but subsequently to that of the
higher mountains to the north and south of it.

We may now proceed to investigate the transporting power of currents originating in the
manner above explained.

SECTION II.
Transporting Power of Currents.

9. WuatEver be the specific gravity of a body, if its dimensions be sufficiently small, it
can never acquire more than a small velocity in descending by gravity in any fluid of which
the density is not extremely small. Such a body may therefore be held in suspension in water
for a considerable time, and when placed in running water, soon acquires a horizontal velocity
indefinitely nearly equal to that of the current. It may therefore be transported to considerable
distances before it descend to the bottom ; or when once deposited on the bed of the stream, it may
easily be again disturbed, and carried onward as before. When the body is not however of very
small dimensions it can only be transported along the bottom by the impelling force of the current,
its motion being retarded by friction, or the resistance of solid obstacles. In this latter case it is
necessary to ascertain the relation between the velocity of the current and the dimensions
and weight of the largest mass it is capable of moving. This relation depends not only on the
volume and specific gravity of the mass, but also on its form; and therefore, in order to ascertain
whether certain given bodies could be moved by a given current, a separate investigation would,
in strictness, be necessary for each, supposing their forms to be different, though they might in all
other respects be the same. ‘To render our results immediately applicable however, with sufficient
accuracy for our general purpose, it will be sufficient to investigate the above-mentioned relation
for a few certain forms, and then to refer any proposed mass to that particular form to which it
most nearly approximates, among those for which the above investigation has been made.

10. A body acted on by a current may be moved by sliding or by rolling. In the former
case, a very uncertain element, the friction of the surface on which the body rests, is necessarily
introduced into our calculations. It will depend on the nature of the surface over which the
transport takes place, and on the force with which the body presses on that surface, and this force
will depend very much on the extent of that portion of the surface of the body which may be
in such close contact with the surface on which the body reposes as to exclude the water from
penetrating between them, and exercising there its upward pressure. In those cases, however,
in which the motion takes place by rolling, the uncertainty depending on friction is entirely
removed, for such motion is independent of the magnitude of the friction. Also, during a rolling
motion the body must be revolving round one edge as an instantaneous axis, so that the fluid
pressure will act on all points of the surface except those very near to that axis. I'he abstraction,
therefore, of the pressures on these latter points will have no material effect on the body’s rolling
motion, and may be neglected in our calculations. When the body passes from one. edge to
another, as a new instantaneous axis, the whole intervening surface might come in close contact
with that over which the body moves; but if these edges be not too far apart (as will generally
be the case in those bodies which tend to move by rolling rather than sliding) the body will
necessarily begin, by its momentum to move round the second axis, and will consequently admit
the fluid to exert its pressure on the lower surface of the body, after it has passed to a new
axis of instantaneous rotation.

226 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.

11. Hence if a body once begin to roll, and we would calculate the force of the current just
sufficient to keep it in motion, we may consider the fluid pressures as acting on every part of its
surface, and our results will be very approximately true, independently of the nature of the surface
over which the motion takes place, provided that surface be sufficiently firm to give the requisite
support to the rolling body. The force, however, thus determined might be insufficient to make
the body begin to move, since it might rest in such a position as to exclude the fluid action from its
lower surface. But here it should be carefully observed, that a current is ne¢.to be deemed ineffi-
cient in moving blocks of given weight and form, unless it be capable of moving all such blocks;
on the contrary, it is to be considered efficient for that purpose, if it be sufficient to move such of
them as may exist under conditions most favorable for transport. In many cases the incipient
motion might be due to accidental causes, as, for example, an impulsive blow from another mass
already in motion; and, moreover, it is probable that all blocks which may have been transported
by this agency to considerable distances, have been carried on by currents of considerably greater
force than that just sufficient to keep them in motion, and which may have been sufficient without
accidental causes to move them from rest, even under conditions not the most favorable for their
movement.

The preceding remarks are of the first importance as removing all doubt and uncertainty with
respect to the applicability of our calculated results to actual cases of transport by the agency of
currents, whenever those results involve the hypothesis of the rolling motion of the transported
mass. ‘Transported bodies may have moved by rolling or by sliding; but in the latter case, the
retarding action of friction and local obstacles introduces so uncertain an element as to render
calculation comparatively useless; but if in calculating the force necessary to move a block of
considerable magnitude, we assume it to have moved by rolling, we avoid in a great degree the
uncertainty arising from the above causes, and are in no danger of assigning its transport to a force
much less than that which has been actually required for that purpose.

We may now proceed to investigate the force which a current is capable of exerting on bodies
of particular forms, It will be sufficient for our purpose to take a few prismatic bodies, of which
the sections perpendicular to their axes are triangles, rectangular parallelograms, pentagons or
hexagons. These cases will shew how the transporting power of a current, as estimated by the
mass it is capable of moving, depends on the form of the mass; and will enable us to estimate, to a
sufficient degree of approximation, the velocity of a current capable of moving any proposed erratic
block.

12. Ifa plain surface, whose area = S' be placed at rest in a fluid, whose density is p,, moving
with a velocity v, in a direction making an angle @ with the plane, we shall have

vy
R = $(0) -— p,Sssin’9,

R being the moving force of the current on the plane estimated in the direction perpendicular to
the plane; and if R’ be the resolved part of this force in the direction of the current,

5 v?
R’ = $ (0) tes pi S'sin®@ ,

which will be the whole force in this direction, if we neglect the friction between the fluid and the
plane.
When 0=

Tv ie 3
>: numerous experiments, made by different persons, shew that

9

, x)
R = oh

yery approximately. The experiments have been made with different velocities up to 11 or 12

Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS. 227

miles an hour, and we are justified in concluding by induction, that the expression will hold for
nit 7 5
still greater velocities. Hence p (=) =1. It also results from experiment that the value of (0)

is very nearly unity for all values of @ not exceeding 45"*, and therefore, since the applications
I shall make of the above expressions are in cases where @ is less than that value, we may assume
generally

v

R ae pi S'sin’@,

ll a ae
and R = 5 pi Sein °0.

13. Let us first take the case of a prism, of which the axis is perpendicular to the current,
and the section a triangle ABC.

If this section bisect the prism, it is manifest that the resultant
of the whole pressure upon it produced by the current will pass
through the middle point of AC. If therefore a perpendicular to
AC through this middle point meet 4B in B, or between A and B,
it is manifest that the force of the current can have no tendency to Wie
make the prism turn over about the edge through B. Suppose the LO
triangle equilateral; then on whichever side the prism may rest, the 4
above perpendicular will pass through the opposite angular point,
and the prism will not roll; and if the triangle be not equilateral, it is easily seen that there must
necessarily be one side which, when the prism rests on it, will be met by the perpendicular. Con-
sequently no triangular prism can continue to roll by the force of a current round each edge in
succession.

To find under what conditions the prism will slide, I shall assume, as the most favorable
condition for such motion, that the water has access to the lower side of the prism. In this
case, taking p for the specific gravity of the prism, and p, for that of water, we shall have the
weight of the body in water

B

= (p > pi) gU,
U = volume of the prism, and g= accelerating force of gravity. Let d4B=a, AC =e, the length
of the prism =6, and CAB=6. Thenif R = the normal force on the side of which AC ds the
section due to the current, and FR’ the horizontal force, we have (supposing @ not much less
than 45°)

R =~ p,S'sin® 6, r

wy?
R= — pi Ssin°;

or, since S' = be,
2

a= < pibe sin® 0,

* The most detailed experiments I have seen on this point are | its length it was deemed better to publish it separately. When
contained in a work, entitled Nouvelles Experiences sur la | 0=45°, these experiments give (0)=1,08, and values approxi-
Resistance des Fluides, par MM. D’Alembert, le Marquis de | mating to unity as their limit, for smaller values of 0. For greater
Condorcet, et Abbé Bossut, Membre de Academie des Sciences, | values of 0, unity is no longer a near approximation to the value of
§c. &c. Par M. Bossut, Rapporteur, 1777. It was intended to | (0).
appear in the Transactions of the Academy; but, on account of

Vor. VIII. Parr II. Go

228 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.

vw
Fa = Pi besin® 0.

Therefore, the vertical pressure on the base

= (p-—pi) gU + Roos 8

2

=4(p - pi) gabesin + = pibe sin’ 8 cos 8

= Locsin @ $(p — p:) ga + v"p, sin 8 cos}.

If we suppose the force (F') opposing the body’s sliding to follow the ordinary law of friction,
we shall have
F =). vertical pressure on the bottom,
=n {(p -— pi) U + Roos 6},
(where « = coefficient of friction); and the condition of the prism being on the point of moving
will be
B= Ey,

Hence we obtain

ye? 2
er sin’@ = » {(p - plate pi sin @ cos 0},

basa late
v 2 \pi

or, —.eCeCGF—___________"___ .
2g sin@(sin@ — cos @)

This shews that a triangular prism with its axis perpendicular to the current cannot be moved
by sliding unless tan @ be >, whatever be the velocity of the current.

If the section ABC be equilateral, 9 = 60°, and we shall have

+ m(e3
de s/s)

If we take & = 2,5, which may be assumed as a mean value of that ratio, we shall have
Pi
2

fea:
a=({1-—~) —.
VA ) 28
14. Let us now take the rectangular parallelopiped, of which ABCD is the transverse
section. Let AB =a, AD=c, and the length=6b. Then

2 D c
ud) ——$—$$$$ $$
R= — pr. bes

and in order that the body may be on the point of rolling
round the edge perpendicular to the plane of the paper
through B, we must have

, AD AB
Bia = (pa peu A B

2

v c a
g Pibe.-> = (p — pi) gabe—,

Mr. HOPKINS, ON THE

and

Let c=na, then

If the section be square, »=1, and

TRANSPORT OF ERRATIC BLOCKS.

1 v

C= -_—-
1,5 2g

If the body be on the point of sliding, we must have
R= pu (p - pygu,

2

v
or, 7 pi be=m (p— pigabe,
1 vw
= ==
eae
eee
Pi
This value of a is less than the former one if
1
(Le So
n

If m=1, or the section be a square, the condition becomes

a>.

If this hold, the body will roll rather than slide.

15. Let us next take a prism of which the transverse section is a pentagon.

SRN Cimon ye Daas

Let the side of the pentagon =a, and the length of the
prism = b; ACQ=0, CQ being horizontal; and let R,R, be
the normal forces due to the current on AC, CD respec-

tively.

To find the tendency of these forces to turn the body

round an axis through B and perpendicular to the section, we
observe that a perpendicular to CD through its middle point
would pass through B, and consequently the moment of R,
round the proposed axis will=0. Also, since the direction of a c
R, will bisect the angle at HZ, we shall have, when the prism is on the point of turning about

the edge through B,
R, asin

or, substituting for R, and U,

'D ; a
ae Oe a?

2 5 8
; pi @b sin’ @ cos 36° = > (p-pd& = cot 36°.

GG2

B

229

230 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.

~

Therefore (since 9 = 72’)
1 z -
on ae ee 36°”

sin? 72°sin 36° v

ES) 8g
8 pi

a B
utting — = 2,5.
p s i ?

If we suppose the body on the point of sliding we find the value of @ nearly equal to that
just given, supposing p = 1.

16. Again, let the section of the prism be hexagonal. Let 4B =a, and R’ be the horizontal
force of the current on the side AC = that on the side CD. Then
when the body is on the point of turning about the side through
B, we shall have

D

j a

But 5 \
li = <p ab sin’ 60, A
and U=3a.H0.b; te
v 3 7.
25a ab sin’ 60 =—=(p - pi)ga’d B
4 sin? 60° v?
and Po ak F Be
ae
F VJ/3 v
TET
Pi

I
.
ea
ae |
|

nearly.

It will be observed that in all the preceding cases the results are independent of the lengths

of the prisms, as they manifestly ought to be, since by changing the length of a prismatic body |

situated as above supposed, the mass and the force upon it are changed in the same ratio.
The tendency to roll as compared with that to slide is easily shewn to be greater in this than in
the preceding cases; and if we take a prism of which the section should be a regular polygon
of a still greater number of sides, the tendency to roll would be still greater. It is unnecessary
to increase the number of examples of this kind; but there is another case somewhat different
from the above which is deserving of notice.
17. Many of the erratic blocks which may be referred to the agency of currents are so

rounded as to approximate more or less to the spherical form. Let APB represent the locus of
those points on the surface of the body which come consecutively in contact with the ground in the

Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS. 231

rolling movement, and which may be considered, for simplicity, as forming a plain curve. Moreover,
since the greater and smaller axes of this curve will not

differ much in magnitude, we may suppose it approxi-

mately to be an ellipse. Let its semi-major axis AC 4

=a,and BC =b; CP=r, CY=p, CY being vertical,

and therefore perpendicular to the horizontal tangent

at P, the point of contact. The horizontal force of c

the current (R’) will be approximately equal to that

on a sphere whose radius =a, and its direction will

pass nearly through C, which will also approximately

coincide with the centre of gravity of the body. Hence

when the body is in equilibrium in the position above B

represented, we shall have

R'p =(p-p)gUV/r — p?;

2
or, R= (p- peo - 15

and in order that R’ may be just sufficient to make the body roll over, this equation must hold

when the angle PCY is a maximum, i.e. when P is a minimum. Now
ip

SS =F = min.
yr (a? +b?) 2 — 7"

which gives r

and p

r (a a b?)?
po. 4a
(a? — b)
~ aee

4
Also U= : ma* nearly,
vy
and R=t7ra’. rua nearly,
as determined by experiment on the resistance on a sphere.
Hence the above equation becomes
1 v 4 a—b

grt gma (p— pie sme. aan >
2 _
‘ v2 (£~1)a. St 9C-9
2g pi ab

= 4 (¢ - 1) (a—b) nearly ;
Pi

232 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.

a
or if a—b=-,
n
n vw
a= ba
nae
Pi
n v
Si a

18. To estimate numerically the dimensions and weights of the blocks which may be moved in
2

each of the preceding cases by a current of given velocity, let us take == 4, which gives a
g&

velocity of about 16 feet per second, or about 10} miles an hour.
The triangular prism will never roll; when it is just on the point of sliding we shall have
(the section being equilateral)

This involves the unknown quantity ~. Suppose the friction such that the body would just
slide down the surface on which it rests, if that surface were inclined at an angle of 45° to the
horizon; then « = 1, and

a = 1,68 feet.
If w= ,5
a = 2,84 feet.
In the parallelopiped of which the section is a square, we have, when it is on the point of rolling,
a = 22 feet.

If the length =a the body will be a cube containing nearly 19 cubic feet, and weighing
nearly 13 ton.

For the pentagonal prism on the point of rolling,
a = 2,268 feet.

If the length of the prism be 2a, the volume will be about 40 cubic feet, and the weight nearly
3 tons.

In the hewagonal prism on the point of rolling,
a = 2,28 feet.
If the length of the prism = 2a, and therefore do not differ much from its height, its volume
will be upwards of 60 cubic feet, and its weight between 4 and 5 tons.
When the body is approximately spherical, let m = 4; then
a= & feet.

If we estimate the volume as equal to that of a sphere whose radius a’ is a mean between
a and 6, we find

a= $ feet,
and the volume about 52 cubic feet. The weight will be about 4 tons.
If n= 6,
a = 4 feet,
a=1ta,

and the volume may be taken at nearly 200 cubic feet. Its weight will be 14 or 15 tons.

Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS. 233

19. It will be observed in the expressions above given that the lines denoted by a vary, in
every case, as v*, and consequently the weight of the mass in each case, which varies as a@*, varies as
v®. Therefore the moving force of a current estimated by the volume or weight of the mass of any
proposed form which it is just capable of moving, varies as the sixth power of the velocity.

This proposition may be easily proved independently of induction from particular cases.

Let a denote the length of any parameter in a proposed body of given form. Then, when v is
given, the force (/’) of the current, estimated as above, varies as the surface of the body, varies as a’;
and when the surface is given, the force varies as v°. Therefore

F«a’v’,
and the moment of F' to make the body roll
a aU
= Cav’ (C =constant).
Also, the weight of the body < a*, and its moment tending to keep the body at rest

Hence, when the body is on the point of moving, we must have

Cav? = Ca’;

oc v,
and the weight « a* « v"; which proves the proposition.

This result shews how excessively erroneous an opinion we might form of the transporting power
of rapid currents from that of the ordinary currents subjected to our observation. Thus if a stream
of 10 miles an hour would just move a block of a certain form of 5 tons weight, a current of
15 miles an hour would move a block of similar form of upwards of 55 tons; and a current of 20
miles an hour would, according to the same law, move a block of 320 tons.

Again, according to the same law, a current of two miles an hour would move a pebble of
similar form of only a few ounces in weight. And here it should also be remarked, that minute
inequalities, or a want of perfect hardness in the bed of a current, which would produce little effect on
the motion of a large block, would entirely destroy that of a small pebble; so that the circumstance
of the transporting power of a stream of 2 or 3 miles an hour being inappreciable is perfectly
consistent with the enormous power of rapid currents.

20. Let us now investigate the space through which a block might be conveyed by the current
attending a single wave of elevation.

Let V be the velocity with which the wave is propagated.

A t M F M, B

®, the greatest velocity of the current, or its velocity in the transverse section through the crest
or highest point of the wave, which will be very near the front of the wave, assuming it to have the
character of a bore, as will necessarily be the case if the elevation producing it be paroxismal.

v the velocity of the current in any other section of the wave;

v, the velocity of a current just sufficient to move the block.

234 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.

Let AB represent the surface on which the block rests, CD the general surface of the water and
LPP,N the wave, M the block at the time ¢, and P the point in which a vertical through M meets
the surface of the wave.

MP and M,P, are the sections in which the velocities are v and v, respectively.
Let dM =, Al =CL =a’; then will

= = vel. of the block,

and 24 = vel. of the wave = V.

Also let v, be the velocity of a current just sufficient to move the block. Then, when the
velocity of the current at the point where the block is situated becomes = v,, the block will begin to
move; and as the velocity of the current increases, that of the block will always very nearly
= difference between the velocity of the current and that just necessary to move the block; so that
we may consider the instantaneous velocity of the block as approximately =v —v,. We shall

da
then have rigs rly
or, substituting for v its value given by equation (1), (Art. 3.)
daz h
—— = =. V Ug eeeccscccccccenccsscevercercncsesvcsver ves .
Gi ea = (1)
da’ -
Also aa Poa geeee Geader avatet ne tcicpaorsweneadienserscuvuneust qoate(oyla

dau S h Vo 3
we Forest eetee te es eeesneenneenenessennenanens(3)

h will be a function of # — x’ depending on the form of the wave, This form is not known, but as
an approximation we may assume LP, to be a straight line ; we shall then have

h mP Lm «@-—a2'

h, mP, Lm 1
1 being the length of the wave to which Lm is very nearly equal. Therefore
l ,
RES SSCS ESSE ae tate Ue ead
Sa dw _1l dh Fi
PER ee a i
Hence, by substitution in (3) and reduction, we obtain
hy dal Megs | ea Vf
1° dh Wy Vt, Uy %% Pet
V+,
and integrating,
hy , V i v
+a =C--h+— 7H. log.. 7, )
7 %, ae og. = h+H).

Let a = the original distance of the block, and 7 = the length of the wave; then when a= a —/,
we shall have h =h,. Therefore

Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.

Vo
h,+ H
a —-a+l V(h—-h) VA Vienne
— ii -—-— - log, eecvceves (5).
1 V5 hy v hy Vo hae
V+

2

235

Equations (4) and (5) give the two relations between , a’ and h. Our object is to determine

. . aa j
the value of a when the motion of the block ceases, when we have the condition ago which

gives from (1)
h
H+h

VE it =| Olwerslennteianiviscietocriscsbecicacherencerterel (0):

From (4) (5) and (6) the required value of w can be determined, and thence w —a, the space

through which the block will be transported, will be known.
Equation (6) gives

V2
== .A;
V-v,
and we have from Art. 3,
VY
ha poy
Also from (4),
w=xex-—.l,

1
Substituting this value of a’ in (5) we obtain

1 2) hy
a-a ZN et) elite OW Se

— — —.— lo¢
2 Se -
v, hy Vy h

l Vg
ae Viv, H

1

Since 2 will always be less than unity and 3 will generally be a small fraction, we shall obtain

a-a@

a near approximate value of

terms of the second order,

m-a_ Vg (: ee hs
it gece h,) 2° (V+e,)" Hh,
+h h—-h

POAT ; | fm ih des
i ( 2 a Teh
Omitting 33 and substituting the above values of h and h,, we obtain finally
a ee Ce
s=a ar TSie See aenete aaa tics sie geeae scales (7),

which gives the space through which the block will be transported.

If we put v, = 0 we have

Vy

Vito,

1
age — am SOc COO DED OCREE LER AaCeosnccocoo | t))5

if we expand the logarithm. We shall thus have, preserving

which gives the whole space through which each particle of the fluid is carried by the wave from its

original position.
Vou. VIII. Parr II. Hu

236 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.

v, :
If 7 be sufficiently small,

1 (v, — ¥,)?
Sr pmmeyar oc tare etree sea)
and Seen MEATY, eee caconersnaneaae- (10)
2V
= z ; a Au approximately.
2 H

I have supposed the section (Z P,) of the surface of the wave to bea straight line. It will
generally be some curved line having its convexity turned upwards or downwards according to the
nature of the disturbance in which the wave originates. In the former case, the value of s would
be greater, and in the latter less than that here determined, which may therefore be considered as
an approximation to the mean of the values of s for different waves, in which v, v, V and 7 should
be the same, but the original mode of disturbance, and therefore the form of the wave, different.

21. The following table exhibits numerical values of the velocity (V) with which the wave is
propagated, of the maximum velocity (v,) of the attendant current, and of the space (s) through which
a block may be transported, for certain values of the original depth (#2) of the water, of the height
(h,) of the wave, and of the velocity (v,) of the current just sufficient to move the block. The
values of H and h, are given in feet, those of V, v, and v, in miles, the velocities being estimated by
the number of miles described in an hour; s is given in terms of / the breadth of the wave. The
values of s are calculated from equation (9). The last column contains sy calculated from (10).

Us

Miles.

nearly

ol™

—_"~— —a——_—

I~ Sl~ wes

_
_

Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS. 237

SECTION III.
Application of the preceding Theory.—Comparison of different Modes of Transport.

22. Iw estimating the magnitude of a block which may be moved by a given current, the
transport has been supposed to take place over a horizontal surface, sufficiently hard and even
for the block to roll upon it without impediment. If the surface be otherwise constituted,
the motion may be impeded or destroyed. The softness of a clayey surface would probably
be most unfavourable to the motion; while the want of cohesion of a sandy bottom, from its
opposing a less effective resistance to a motion rather by sliding than rolling, might be highly
favourable to the transport of the block. In any case a constant action of denuding causes
will be highly favourable to it, by the successive removal of temporary and local impediments.
Abrupt inequalities, such for instance as those presented by ravines and steep escarpments,
would present insuperable impediments to this mode of transport. It is important however
to observe, that regular ascents, without rugged inequalities of surface, would offer no such
serious impediment.

The difficulty in this theory arising from the presumed inequalities of the surface over
which the blocks must have been transported, has» been, I conceive, in many instances, far too
much insisted on; for it has been made to rest on the assumption that the inequalities of surface
between the present and original sites of erratic blocks were the same, or nearly so, at the
time of transport as at present; an assumption which I regard as totally untenable. There
are three obvious causes of inequality of surface—elevation and disruption, denudation during
gradual emergence from beneath the ocean, and erosion after emergence. So far as sudden, abrupt
inequalities can be traced to the first cause operating previously to the transport, the difficulty
alluded to must be admitted; but in many cases existing inequalities have been produced by
post-tertiary elevations, which we have no right to assume to have been entirely anterior to
the transport of erratic blocks. Again, such great inequalities as those presented by the oolitic
and chalk escarpments, have doubtless been due in a great measure to denudation, during the
period of gradual emergence of the land, the higher levels being raised above the sphere of
denuding action, while the lower levels remained exposed to it. Minor local irregularities of
surface are also due in a great degree to erosion. All superficial inequalities, therefore, which
are referrible to these causes, must have been posterior to the removal of erratic blocks trans-
ported by currents, and form no objection to that mode of transport. The only other causes
which can materially affect the configuration of the terrestrial surface, are the deposition of
new sedimentary beds, and denudation produced by ocean currents previously to any partial
emergence of the surface. But it is manifest that both these causes, instead of creating those
abrupt superficial inequalities, which alone would form a serious impediment to the transport we
are considering, must constantly tend to destroy them wherever they may exist from other causes.
For these reasons, I believe that there is no validity in the objection above stated to the theory
of transporting currents. Those greater superficial inequalities which now exist, and are obvi-
ously referrible to denuding agencies, could not, I repeat, be the consequences of superficial
denudation, while the whole surface was submerged beneath the ocean; and minor abrupt inequa-
lities could not then have continued to exist, even if they had originally existed, for they would
have been destroyed by the action of transporting currents themselves, though no other cause
should have operated to produce that effect.

23. ‘These currents, in addition to their transport of larger blocks, must manifestly tend to
spread out the smaller detritus in a layer over the bottom of the ocean, supposed, for the reasons
above stated, to form an even surface*. As the bottom rises in the process of slow elevation,

* In the sense for instance, in which the bottom of the German Ocean or English Channel is an even surface.
HH 2

238 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS.

it will become exposed to all the action of denuding agents, which however will, in many
instances, make less impression on those parts where the covering of detritus is thickest, or is
composed of the coarsest materials. Such parts will therefore, ceteris paribus, emerge first
from beneath the surface of the ocean; and thus, in the first instance, will form islands, and
subsequently, when the whole shall have risen above the level of the sea, the summits of hills.
Such summits may consequently be expected to be capped with transported materials, of which
all traces may have been destroyed by denuding agents in the surrounding valleys. This pheno-
menon, of such frequent occurrence, is thus simply accounted for according to this theory.

24. It appears by the table above given, (Art. 21) that a wave of between 50 and 100 feet
in height, (in an ocean of the original depth there supposed), would be accompanied with a
current of which the velocity would be from 10 to 20 miles an hour; and it is demonstrated
in the first section, that (under conditions which I conceive to be entirely admissible) currents
of that velocity would possess a motive power abundantly sufficient to move the largest blocks,
the transport of which it would be deemed necessary to refer to this cause. But I would
particularly direct the attention of the reader to the fact, as exhibited in the values of s, in
the table just referred to, that the space through which a block may be transported by a
single wave, is equal only to a small fraction of the breadth of the wave. Consequently, a
great number of waves might be necessary for the transport of blocks to distances to which
they frequently have been transported. It must also be recollected, that sudden or paroxismal
elevations only will produce waves of elevation of considerable transporting power. Hence it
follows that this theory of transport is essentially and necessarily associated with that theory
which regards the phenomena of elevation as the consequences of a series of paroxismal move-
ments, the movements by which, in my opinion, those phenomena can be most satisfactorily
accounted for. The instantaneous elevation of a determinate portion of the bottom of the sea
would produce a wave whose height would be equal to that of the elevation itself, so that
it may be asserted in general terms, that the theory of transport by elevation currents, in its
application to existing phenomena of transport, involves the hypothesis of a succession of paro-
xismal movements beneath the ocean, the height of many of which must have varied from 50 to
100 feet at least.

25. If we allow the efficiency of each of the three recognized means of transport of
erratic blocks—glaciers, floating ice, and currents—the difficulty which remains is that of sepa-
rating the effects produced by these causes respectively. In some cases it is probable that
doubt will always remain from insufficiency of evidence, but in others, I conceive, our conclu-
sions may involve but little uncertainty. The distinctive characters in the transported materials
must be sought in the magnitude and form of the blocks, the state of their surfaces, and the
distribution of the general mass of the transported materials. The magnitude of a block can
hardly be considered to increase the difficulty of its transport by ice, while it increases in a great
degree the difficulty of transport by water. Again, blocks cannot generally be rounded by
attrition when floated on icebergs or carried on the upper surface of a glacier. A small portion of
those brought down by glaciers are rounded by being rolled between the ice and sides or bottom
of the glacial valley; but this is a rough grinding, and all the specimens I recollect to have
examined immediately at the termination of a glacier, wanted that more perfect smoothness of
surface which distinguishes a water-worn boulder. It might be contended that blocks floated
on icebergs might be rounded and polished before being taken up by the ice or after being
deposited by it. If such were the case, the effects must be produced either on beaches by the
action of breakers, or at the bottom of the sea by that of currents. The action of breakers,
on large blocks, however, as far as my observation has extended, rarely tends to give to them
a rounded form, but, on the contrary, to wear them into very irregular shapes, till they are
so reduced in magnitude as to be rolled about by the force of the waves; the most prominent

Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS. 239

points then become subject to the greatest attrition, and the surface afterwards assumes that
form and polish which distinguish a water-worn boulder. I do not recollect, however, to have
observed on any beach instances of this perfect rounding and polishing except in pebdles, much
too small to afford any explanation of the cases of many of the erratic blocks which have been
subjected to some similar and equally effective process of that kind. Moreover, should the
efficiency of this cause be allowed, it must be recollected that the sphere of its operation is
limited to the comparatively small area over which the waves break, for it is there alone that
they can exert any effective power. How then shall we thus account for the water-worn appear-
ance of innumerable blocks existing in the detritus spread out over a wide area, or in cases
where the transported materials exist in layers of great thickness? If it should be contended
that the water-worn appearance may be due to the other cause above alluded to—the action
of water remote from shallow coasts—it must be replied, that that force which is capable of
rolling a block is unquestionably sufficient to transport it, and therefore, that the solution does, in
fact, admit the existence of transporting currents.

There is also another important point to be remarked with respect to the transport by ice,
whether on land or by water—it affords no reason why the transported blocks should diminish
in size, and become more generally rounded and polished, the more distant they are from
their original localities. Such would necessarily be the consequences of transport by currents,
but it must be a matter of indifference whether a block has been floated on an iceberg or
carried by a glacier one mile or one hundred miles, so far as regards the form and dimensions
of the block when ultimately deposited by the ice which conveyed it. If the great majority of
the blocks transported from a given locality be rounded and polished, there is a strong presumption
that water has been the transporting agent; if, moreover, the blocks do not exceed a weight
of a few tons, the probability of that mode of transport is increased; and, finally, if we find
that the magnitude of the blocks generally diminishes as their distance from their original
site increases, till at length they degenerate into rounded pebbles, the previous probability
appears to me to approximate as nearly to certainty as we can reasonably expect.

On the other hand, when erratic blocks are extremely large, the presumption is in favour of
their having been transported by ice; and if, moreover, they retain sharp angular points and
edges on their apparently unworn surfaces, and their magnitude bears no relation to the distance
of transport, we may confidently conclude that the transporting agent has been ice, assuming
always that the transport is attributable to one of the causes we have mentioned.

The main distinction between the cases of transport by glaciers and by floating ice, must be
sought for, I conceive, in the distance which the blocks have travelled, and the nature of the
surface over which the transport has taken place, and not in the character of the blocks them-
selves. If the motion of glaciers be due to gravity, as I have endeavoured to shew in a recent
memoir, it would be an absurdity to attribute to their agency the transport of the blocks dis-
seminated over the extensive flat plains of northern Germany and Russia. In such cases I
should not hesitate to refer the removal of large angular blocks to the agency of floating ice. On
the other hand, the transport of numerous blocks on the flanks of the Alpine chain can hardly
be referred to any agency but that of glaciers of greater extent than those now existing. In
other cases the transport may have been effected by a combination of these means. Blocks may
have been brought down by glaciers from the mountains, and then floated on icebergs to distant
localities. This process has been recently observed, on a magnificent scale, in a high northern
latitude, and appears to me the simplest mode of accounting, in certain cases, for the transport
of blocks now far above the level of the sea. If Switzerland were depressed 1600 or 1700 feet
below its present level, the enormous angular block of Pierre a bot above Neuchatél would be
on the margin of an arm of the sea, occupying the present valley of Switzerland, while on the
Opposite margin there would be rocks bearing the strongest marks of glacial action. Under
this hypothesis, and without assuming any material change in the general configuration of the

240 Mr. HOPKINS, ON THE TRANSPORT OF ERRATIC BLOCKS,

surface, there remains no difficulty in accounting for the transport of the prodigious block above
mentioned from the Alps to the Jura; a fact which on any other hypothesis hitherto made,
presents, in my opinion, mechanical difficulties totally insurmountable. The supposition of an
elevation of 1600 or 1700.feet since the period of transport offers, as I conceive, no a priori
difficulty, when we recollect the evidences of recent elevation in other places. With conclusive
evidence that Snowdon has been elevated 1200 or 1300 feet within a period which we have no
reason for supposing more remote than that of the transport of erratic blocks, there can be
little hesitation in admitting the elevation above supposed in the region of the Alps within the
same period, as an hypothesis as probable at least as any other which might be adopted.

W. HOPKINS.

CaMBRIDGE,
April 29, 1844.

oR A NoSseA C-TLONS

OF THE ;

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

VouumeE VIII. Part III.

CAMBRIDGE:
PRINTED AV THE UNIVERSITY PRESS;
AND OLD Ly
JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. & J. J. DEIGHTON ; AND MACMILLAN AND CO., CAMBRIDGE.

M.DCCC.XLVII.

~ fog EM A ert,

YTEIOOE AL AOTILG © 0. Lands

sala hide ARO.

a + ; | . ' 4 ane

XVIII. On the Foundation of Algebra, No. 1V., on Triple Algebra. By Aucustus
De Morean, V.P.R.A.S., F.C.P.S., of Trinity College; Professor of Mathe-
matics in University College, London.

[Read, October 28, 1844.]

In the Philosophical Magazine for July 1844, Sir William Rowan Hamilton has published the
first part of a paper read before the Royal Irish Academy in November 1843, headed ‘On Qua-
ternions, or on a new System of Imaginaries in Algebra.’ To this paper I am indebted for the idea
of inventing a distinct system of unit-symbols, and investigating or assigning relations which define
their mode of action on each other. ‘The systems which Ff shall examine differ entirely from that of
Sir William Hamilton, both as being triple instead of quadruple, and as preserving, in their laws
of operation, a greater resemblance to those of ordinary Algebra,

$1. Description of triple systems. A system of Algebra of the n™ character is one in
which there are m distinct symbols, &,, &, ... &,, each of which is a unit of its kind, of a difference
from all other kinds such that @,£, + a.& +... cannot be equivalent to b,£, + b.f+... unless
a,=6,, 4 =6,, &c. This condition however is connected with the interpretation: a perfect sym-
bolical system might very well exist without it. Having assumed a system, and also the ordinary
laws of addition and subtraction, the introduction of the operation of multiplication requires that
meanings should be assigned to ££, ££, &c., so that each of them may be regarded as coincident
with such a form as a,£, + a,€ + .... On the manner of assigning this form the properties of the
system entirely depend; and if we are to preserve the ordinary rule of the convertibility of multipli-
cations and divisions, we must not only provide that &, & = &£,, &c., but also that eT ES && ... shall
give the same result in whatever order the operations are performed. his rule relative to mul-
tiplication may be reduced to two simple rules, 4B = BA, and A(BC)=(AB)C. It is exactly
the same thing as to additions, the convertibility of which is contained in the rules d+ B= B+ A
and (4+ B)+C=A4A+(B+C). This second rule is generally concealed in the common rule
of signs, according to which 4 +(B+C) or 4+(0+ B+C) is, by the assumed distributive
character of the sign +, allowed to be transformed into A +(+ B)+(+C) which again by the
rule of like signs, becomes d + B+ C, a symbol identical in meaning with (4+ B)+C. We
might also use the signs x and — in the same absolute manner, and assume a corresponding dis-
tributive character, and rule of like and unlike signs: considering x @ and ~ a as abbreviations of
1xa@and 1+ a. But it will be enough for my present purpose to note that the complete conver-
tibility of multiplications will be secured if every triple combination, as £££, E°&, &c. has a
meaning which is independent of the order of the operations.

Having settled the system, it must next be inquired, for the sake of the interpretation, what is
the modulus of multiplication, namely, what function of a,, a,, &e. is it which, in the product,
has the same value as the product of the functions of the factors. If, agreeably to the laws of
the system, the product of a,£, + a,& +... and a',€, + a@.€+ ... be A), + Ask + ..., Ai, Ao, &e.
being definite functions of a,, a’;, a,, @,, &c., the modulus is to be found from the solution of
the functional equation

P(A, Az...) X P(A), @z,...) = P(A, Az, -+-);
Vou. VIII. Parr III. Ir

249 Mr. DE MORGAN, ON TRIPLE ALGEBRA.

on which it is only to be observed that any powers, or products, or products of powers, of solutions,
are themselves solutions. 'The most convenient modulus is that which, in one or more definite
cases, reduces the system to the simple single or double Algebra already in use. In this common
Algebra, in its widest form, there are two unit-symbols, say € and », usually (not necessarily)
representing units of length taken off on the rectangular axes of w and y; and the laws of com-
bination are & = &, x? =—& En=n& =n, which give &y° = —E = (En)y, &c. The modulus of
multiplication of a&+by is 4/(a?+ 6°). Sir William Hamilton seems to have passed over
triple Algebra altogether on the supposition that the modulus, if any, of a& + by +c¢ must be
f(a +0?+c*). It is certain* that there cannot be a system of triple Algebra with such a
modulus; but it is by no means requisite that the modulus should be a symmetrical function of
a,b, and ec. I should also notice that in Sir W. Hamilton’s quadruple Algebra there is a complete
departure from the ordinary symbolical rules: AB and BA have different meanings.

§2. One mode of derivation of systems of triple Algebra. Let a&, by, eC, represent lines of
a, b, and ec units measured on the axes of a, y and x. Let it be a condition that b= 0, c=0,
reduces the Algebra to the common single system; which might be worded thus: let the Algebra
of the axis of w be the common single Algebra of positive and negative quantities. Also let 7 and ¢
be interchangeable, and related in the same manner to £ We have then, for the forms which
define the actions of the unit-symbols on each other,

& means &, n€ means pE + qn +qG
Mb irsmwaces a&+byn+eQ, GE serene LE +my+ ng,
Gases ab +en + be. Enh casts ss lE + nn +m;

and it will be found upon examination that the equations £9 = E(E), En? = (En), 1° ¢ = n(n,
n@=C(nQ, CE=E(CE), CP = EGE), ECnQ) = (CE) = C (En), will be satisfied by the following
eonditions ; in using which care must be taken not to form new ones by introduction of subse-
quently vanishing factors without recurring to the original forms. Some of these conditions are
included in the others, but it is nevertheless desirable to be reminded of them.

GQ.) a(q-¢)+ pq -—b)=l(a—-p). (4.) U(m+n) = 0.
(2.) + mp+na=a+ (b+e)l. (5.) 2mn =m.
(3.) P+ma+np=p + 2ql. (6.) m+n? =n.
(7, 8.) ln =(q-—b)m=(ce-q)m. Ql.) (¢ +e) (q-¢) =am-—pn.
(9, 10.) Im = (q—c)m = (b—q)m. (12) (q+ ©) (q- 0) = an - pm.

From (5.) and (6.) we have either

m=0, =0; m=0, n=1; m=4, n=t; me-4, net.

2? “

Proceeding by analogy, we might expect the triple Algebra which is the proper extension of
the common double one to give »' = — &, € = —£, the necessary conditions of which are
3.) al+ab+ep=-1. (i4.) an+b? + eq=0. (15.) am+be+ceq=0.

* Any one who will try to get three squares in which accented | three points of a sphere, each of which is opposite to both of the
and unaccented letters enter symmetrically, and of which the sum | other two; also three other points each distant by a quadrant from
is equal to the product of a?4 6%+4c? and a’+5'+c" is engaged, | each of the first three.
whether he know it or not, upon the following problem ;—To find

Mr. DE MORGAN, ON TRIPLE ALGEBRA. 243

But at the same time it is desirable to examine the case of y°=—&, ¢?=-€&, the conditions
of which are a=-—1, b=0, e=0. These two systems may be called the simple cubic and
quadratic systems, both being ¢riple. I now proceed to a mere enumeration of cases to be pre-
sently discussed.

Case A. Let m=0, m=0; which gives either of the following

(4,) m=0, n=0, q=-¢, (4) m=0, n=0, gq=c=b,
—2ac—p(c+b) =l(a—- p), a=p,
P= a+(b+e)l, P=a+2bl.
P= p-—2cl.
Neither gives a simple quadratic form, unless = —1, which is inadmissible.

Simple cubic forms are only such as are contained in
b=c=-q, m=0, n=0,
al+b(a+p)=—-1, 2bh(a+p)=-Jl(a—-p),
P=a+2bl=p-2bl,
which give p=a=1, J=-1, 6=0.

Case B. Let m=0, m=1. We have then

=, nC=-(P -e)E+9(n+0),
W =(q +e) (q-b)E+bn + €G CEé= %
C=(¢ +6) (T-D)E + en + BE. En= 4.

This is the case, and the only one, in which the action of € upon both of the others is imper-
ceptible. The following cases will be considered, the first of which is a species of simple quadratic
form, the second a simple cubic, the only one which the case yields.

Sees nt = & oe n€ =,
ee HE eis 8 SE = G w= —% =%
C= -E+n +l En =n. C=-7n. En = 7.

Case C. Let m=4, n=}. This gives 1=0, g=b=c, a=p. The only simple
quadratic and cubic forms are as follows:

P=F, nQ=-&, & = 8, nC =-14+4n+4G

wee-& adn thy Pa-EthntdG Go dnt hh
wai | Sadat 5 C= -E+ dnt he Ena an +3
Case D. Let m=-—43, n=4. The equations of condition are reducible to
g=t, yaaaas (b + 8c) (b—c) = - 2(a +p).

The simple quadratic and cubic forms are
P=é, nt =, B= £, n¢ =,
ee-E E=- dnt ¥G WBE +R“ CE mBE- dn 4 BG
ete n= anak C= -hE-tnt+dl En=hb + 40-26

I12

244. Mr. DE MORGAN, ON TRIPLE ALGEBRA.

§ 3. Simple and perfect cubic form. I now proceed to consider the simple cubic form in -
case B. The equations of signification* are (dropping the distinctive symbol & which is in-

operative),
Fe a ate 0
And the product of a+6y+e¢ and a'+b'n+ ¢¢ is
be’ + cb’ + aa + (ab' + ba'—cc’) n + (ac + ca’ — bb’) ¢.

If the equations of signification be also consistently algebraical, and if y = and ¢ =» satisfy
them, then a +bu+ev is a modulus of multiplication. Accordingly in the present instance, it
is sufficient that » and y should be severally equal to — 1, or else that they should be the
imaginary cube roots of —1. Let them be the latter: then a—b-—c, a+ub+ve, a+vb+pe,
are moduli, and since any product of roots of moduli is a modulus, we have, taking such roots
as are required by the condition that the Algebra is to become single if 6 and ¢ always vanish,
the following possible moduli,

a-b-e,

VJ (a +8? + 0+ ab +a — be),
a/ (a - 6’ — c’— sabe).

These expressions are connected with the third degree in the same manner as a*+ 6° with the second.
Changing the signs of b and ¢ &c., their modular character gives the following equations. Let

A=le'+ch'+aa, B=abl'+ba+ece, C=ac+eca+by.
Then (a+b+o0)(d+04+c)=A+B+C
(a? +B? + c?— ab — be — ca) (a? + 6? + c?-ab' — We — ca’) = A? + B+ C’- AB- BC-CA
(a? + B+ c8— 3abc) (a + b°+.c2— 8a't'c) = A+ B+ C— 8 ABC.

These might, I think, be made of the same sort of use in the theory of numbers with the equation
(a?+ b’) (a+ 6”) = (aa’— bb')?+ (ab'+ ba’), which is the modular equation of the common Algebra.
Thus of either of the forms a*+ b’+ c’— ab — be —ca and a*+b*+ c’— 3abe we may say that the
product of two instances must be a third instance.

It appears that this cubic form of triple algebra may involve three cases, according to
the modulus which we employ. Now we know that in common Algebra, a+ 6,/ —1 is made to
depend upon a length and an angle, in such a manner that the length is represented by the modulus,
and the product of two expressions has the product of the lengths for a length, and the sum of
the angles for an angle. Suppose that we make a + by +c to depend upon the modulus and
two angles, each having the same property as the angle of the former case: it is required to
express a + by + €¢ by [, 0, | in such manner that the following equation may be identically true,

[/, 0, p]-[l, 0, p'] =[W7, 0+0,6+¢')-
Without as yet specifying which modulus we are to take, we must examine into the conditions :
of a species of triple trigonometry, in which two angles form the base of every expression.

Looking at the form of the product of a+by+c¢ and a+b +c, it is obvious that the
problem is solved if we can assign

a

a c
7? Beat ze Co =z

* In this sense it ought to be remembered that they more resemble — x —=+ than ab=c.

Mr. DE MORGAN, ON TRIPLE ALGEBRA. 245

in such manner as to satisfy
Agu gry = Bog Cu + Bi, Cog + AagAuy>
Bo.ugsv = Bog, + Buy dos — Co gen vs
Coxuprv = Cop Auy + Ci v4o4 = By Buy

Here A,, is a species of cosine of (9, @), and B,, and C, , are two different species of sines.
The second sides of (MM) must admit the interchange of @ and u, and also of d and y. That this
and all other conditions of self-consistence are satisfied, will appear as follows. We have

Ay, = By Co, ar By, Coo a A, Ay,>
B,, = By Ao» te By, Ag, = CoCo,»
Cy, = Co 04g, 2 C, Ae. = By Boy:

Again, Ag.vgiv= Box 4,0Cog+v + Be. Cas ao + Agu 4og4
a (Bao A, aF B, Ao 7 Coo Cu) (Cop 4o» at Civ 4o¢ a? By Boy)
+ (Bog Aor + Bor dog — Cog Cor) (Coo 4uo + Cuo40. - Boo By 0)
+ (Boo Cuo + By ooo + 0 04u,0) (Bop Cor + Bor Coe + 4og4or)-

Develope these products, and the results will be seen to be identical with

(Bo. 404 ot By Ao = CoC o4) (CoA a Copz ban = B,,B,.)

48 (B, 0 4oy + By Avo - n,0Lo ») (Co ,4o4 ar Cy 4A o.0 ar Bo Bo)

0,u-~" 4,0
ar (Bo .Cos oF By Coo cts A, A, 4) (Bn Gor cf Cita ae A, >Ao,)>

which is BoC.» + By Cog + 4054ur-
The other equations may be treated in the same way.

I am able to find the solutions of all three varieties of this system by means of that
which the modulus is ,/(a* +b? + c? +ab+ac— bc); in which case the equation answering to
sin’?@ + cos*@ =1 in common trigonometry is

0.6 + Boy + Cig + Ap4 Bog + 4o5 Cog — Bog log =1-

in

We have
2 me Ne
?+hP+e+ab+ac—be= (a+"*) +3(5 “) ‘
Assume A,,+4(Bag+ Co 4) = cosd, 4(Bs4— Coa) = snioy
,o * 2 Lb .$ 20.4 9, f3
Then we must have equations of the following form
A, , =cos8+ Ly 4,
sin 0
Boy = wre Ly 45
sin 0
— Ly 4: -

Cay=- /8

246 Mr. DE MORGAN, ON TRIPLE ALGEBRA.

Substitute these. values in the first of equations (7), and we have
Lox udev = 3L94L,, + cosOL,, + cosp Lo 4 + dsin @ sin p.

Assume L,, = 4.(Py4 — cos 8) which gives Po,44,,= Po4P,, the only solution of which
is) P;, =e" *, giving
A, 4 = $cos 0 + te%**F?,

1
Boy = ¥ 00s 0 + —- sin® — de?*F9,

V3
Cy 4 = 00s 0 ~—_ sind — Leattee
06= 3 3 ze -
This gives Ags - Boy x fyi AE
3 3 a
Ap, — Boy — — 3A, , By ,Co4 = °° *F%.
We can now get solutions on the supposition that the other moduli are used. If we take

l=a-—b-—c, we have
Ay, = $ cos 8. e th) + 1,

Oa ar —aoted)
3 3°

seery sin i _n (e046) 1
i.

But if we use 4/(a? —b*—¢* — 3abe), we have
Ag = $0088. SOMthe) 4, 1 eerAd),

3

cos —— sin ss)

3
3S sin 7a) eT NaOtBd) _ J gfa048d),

e RO Bd) _ ZetaorAy,

We must remember that, of any two solutions of (J), either must be the other multiplied by a
solution of Po,,,4,.=Po4 P,,3 and any solution of (M) multiplied by one of the last is also a
solution of (M). And the fovtn of the solutions might be generalized, but in appearance only,
by writing cos (a’@ + 6’) and sin (a’@ + B'p) for cos@ and sin@. But by the same consideration
it appears that the system is not less complete if we write @ for a0 +. Adopting this simplifi-
cation, the equations of connexion between a@ &c. and / &c., are at full length as follows :

L=/ (2+0'+ c+ ab + ac — be),
a=1{2cos0+te%}, a+4(b+c) =lcos@,

1
Venema 7g! — kerk» VE G0) =Isin 8,

Lier
lh al yee ee ea a—(b+c) =le?.
From these premises it follows that the product of a + by + c¢ and a’+ b'n + eG or of [7, 0, ¢]

and [U', 0’, p’] is [17,0 +6, p+¢q’]. And it is certain that this is the only simple cubic system,
except that noted under case A, which as will afterwards be seen, is deceptive: also that this is the

—T

Mr. DE MORGAN, ON TRIPLE ALGEBRA. 247

only case of that system in which /= 4/(a°+ &c.), the equations (M) admitting no other solution
with that modulus.

We now come to the question of geometrical interpretation, the most difficult part of the question
in one sense, the easiest in another. Every system of Algebra admits of an infinite number of
geometrical interpretations. Take the common one, and instead of supposing a + y 4/ — 1 to stand
for a line r=4/ (a+) inclined to the axis of wv at an angle @=tan—'(y: x), let it stand for
any line r, inclined at an angle 0,, where 7, and 6, are unambiguous functions of r and 06.
Then the sign +-in [7,, 0,]+[7";, 6] must be defined in such a way that the preceding symbol
may stand for the line determined by r= 1/ }(w + a’)’+ (y+y’)’} and tan@ = (y+y’): (w+a’);
and similarly with the other signs. There is no question about the superior convenience and
primary character of the usual interpretation: but others are not therefore absolutely excluded.

Analogy would lead us to infer that a, 6, ¢ should represent lines on the axes of a, Yn B3
and even if we took them to represent areas on the planes of yx, xv, and wy, we should be
able to determine an area on the plane of yx (its form not being in question) by a line on the
axis of w. Again, the same analogy would lead us to take / for the absolute length of a + bn + ne
but all that is necessary is that /, 0, and @ should be sufficient determinants of that length. For
instance, we may say, let a + by + c¢ represent a length r = ,/(a*+ b’ +c’) inclined to the axes at
angles having cosines ), , v, proportional to a, b, ¢: but then we give up the convenient property
of the modulus of multiplication, and must form (#, A, M, N) the product of (7, 2, u, v) and
(7, X', »’, v’) from the conditions

Roos A =rr (uv + vp +X2),
ReosM =rr' (Ap'+ Qu — vv’),
Roos N =rr (Av +X — pp),

so that & must depend on the angles of the factors as well as on their lengths. The systems
I have given are the only ones in which the moduli represent the absolute magnitude of the
symbols.

I am not able to present any striking geometrical interpretation. The symbols of the triple
trigonometry on which it must be founded are mixed functions of circular and hyperbolic sines and
cosines. If we take the equilateral hyperbola a*— y’=1, and let w and y be called the hyperbolic
sine and cosine of @, the double of the sectorial area included between the axis of w, the radius vector,
and the curve (for analogy, the angle must be replaced by the double of the area of a circular
sector of radius unity), we have e?=COS@+ SIN @, using capital letters for distinction. We
might very easily invent interpretations: but I see none which I think worth presenting. The
transformation

= E & Se sin @ = £ cos (60° + 0)
will of course not be forgotten by any one who makes an attempt. ‘This entrance of both species
of sines and cosines is, both in this and other cases, the consequence of the determination to have
what may be called a doubly logarithmic system, or one in which both angles, or magnitudes
corresponding to them, have their sums in the product.
We may, if we like, consider the system as one in which there is a double modulus of mul-
tiplication ; let 7. e? = m, and we have

la=f(a+b' +c? +ab+ac—be), m=a-—b—ce,
a = %lcos 0 + 3m, a+4(b +e) =lcos6,
b = 41 cos (60° — 6) -4m, a4/3.(b-c) =lsind.

e = $1 cos (60° + 6) — 4m.
The product of [7, m, @) and [J', m’, 6’) is now [ll', mm’, 0 + 6’).

248 Mr. DE MORGAN, ON TRIPLE ALGEBRA.

The three axes on which a, 6, c, are laid down, ought not to be rectangular axes, but those
of y and x should be each inclined at 60° to the axis of w, so that units laid down on them may
be cube roots of — 1. The planes of wy and w= being at right angles, and A being the diagonal
of the parallelepiped on a, 6, ¢, we have ’ = A= 2be.

Should a simple interpretation be obtained, the ancient difficulty of the imaginary quantity will
immediately occur; for 4/m must take the place of m in 4/{J, m, 6], and m may be negative.
This system therefore will never be completely explained until it is interpreted on the supposition
that a, b, &c. have the forms a+a4/—-1, b+b./-1, and also 6, 1, &c. By analogy we
might have expected this, in the following manner. As soon as pure arithmetic is converted into
single Algebra by the extended definitions of + and —, and the new symbol ,/-—1 occurs, it occurs
in conjunction with both the forms +1 and —1; and at the same time the vehicle of explanation
takes two dimensions. If new distinct symbols be added, such as will require space of three
dimensions, it is therefore natural to suppose that each of those new symbols will combine with the
complete system of the double Algebra. By this, since @ + a@,,4/—1 may mean any line in the
plane of wy, it is reasonable to suppose that two new symbols will be required, to express removal
into the planes of yz and za, and that

(a+ ar/—- 1) 4+ (6+ b/-1)n t+ (C+ eVf- DG

will signify some line in space, determined by three lines in the three co-ordinates planes.

§ 4. Redundant biquadratic form. The last remark suggests an examination of the method
by which systems have hitherto proceeded, with a view to ascertain whether the hints which analogy
might give are exhausted. If we look at the series +1, —1, 4/—1, we see that one new
unit-symbol is introduced at each step, represented by a square root of the preceding. What then is
the system in which ove more unit-symbol is introduced, whose action resembles that of /- 1,
the combination with preceding symbols being of the complete character just described.

Let the fundamental symbol be

[4 pb gl=atpy—-1tlb+av-1D¥,

where (? means ,/—1. Accordingly, the product of [a, p, 6, q| and [a’, p’, 0’, q'] is [4, P, B, Q]
where
A=ad —pp' — bq -b'q, B=ab'+ab-pq -p'q,

P=ap+ap+bb'-qq, Q =aq + a'q+ bp’ + pl’.

The modulus of multiplication is found to be

2 \ 2
l= J {(a+54) + (p+ 49) .
f2 /2/

Now it is evident that, a line in space being determined by three data, we have here one to spare,.
since a, b, p and q must all be given before the fundamental symbol is completely determined. It
would be in our power for instance, to consider the symbol as meaning a line of given length drawn
from the origin in a given direction at a given time; or as determining a point which has a given
position at a given instant. Let a +p./-—1 represent in the usual manner a line in the plane
of wy, and let ¢ represent a unit somewhere in the plane of wx; we may easily see that it must
be at 45° to the positive axis of a, if the rule of angles in multiplication is to be preserved. To
satisfy this last condition, let [a, p, 6, g| represent a length 7 making an angle with the axis of
determined by

b- : b
ben0= as lsin@=p + aad

/2 V2

Mr. DE MORGAN, ON TRIPLE ALGEBRA. 249

Let ¢ signify revolution through 45° in the plane of wx, so that if a+p/1 =re*¥"',
b+q1/—-1=seV"!, we have (6+q/-1)¢ signifying a line s at an angle 8+47 in the
plane of wz. Moreover we have

Lcos@ =rcosa+scos(8+47), dsinOd=rsina+ssin(@+47),

so that the way to find 7 and @ geometrically is as follows. In any plane, say that of ay,
set off r and s at angles a and 3 +47: the diagonal of the parallelogram on these lines represents
the length 7 inclined at the angle @ to the positive axis of x. In various systems I find that
when /sin @ has the form M + N, one of the simplest interpretations consists in making N = M tan,
where w is the angle which the plane of the line and the axis of # makes with the positive side of
the plane of wy. In the present instance, this will give

, +q Lene oad Zsin@ lsin@tanw b+q
an = —— = - sy i
w prf2’ / 2 > rey ian P> ey ee emer V2

Here p, b, q can be found so as to give [/, 0, w] for any given value of a. The system is now
complete, all the rules of Algebra are true of it, and it only remains to give the results their
easiest geometrical form. The most natural mode of proceeding is to examine the mode of escaping
redundancy, which consists in assigning one relation between a, b, p, and g. The case of b = q will
appear exceedingly remarkable, when viewed in connexion with the imperfect system which I shall
describe in the next section.

According to our conventions, a+ p4/-1+6(1 +4/-1)¢ represents a line of the length
L=/ {a°+ (p+b/2)*} inclined at an angle having a : land (p+6,/2) : J for its cosine and
sine, with a projection on the plane of yx which makes the angle tan-*Sb4/2 : p} with the positive
axis of y. But the relation B = Q does not obtain in the product; and if we bring it about by a
proper use of our redundant letters, so as to represent the product [L, 8, Q] under the form
V+W./-1+X(1+4/-1)G we shall find that we have sacrificed the equation 4A(BC)
= (AB)C, which is no longer a formula of the Algebra. Owing to the redundant letter, two lines
may be identical in position, but must not therefore be considered as identical. Now the introduc-
tion of an equation of condition between a, b, p,q, and the alteration of the product in such a
manner as to satisfy this same condition, is, in point of fact, the substitution for the product of a
line equivalent in position only.

I shall resume this subject in the next section: but in the first place, observe that the modulus
admits of resolution into the square root of the sum of two other squares, namely

= A (eoteal (oteal}

Take another angle « such that

pta . 1
V2 ’ lsink =q + ya

This angle « is not a new directing angle, being in fact 0 -i; and ¢ is -f/- 1

leosxk =b +

The modes of interpretation will be better seen, so far as they are easily practicable, in the next
section.

§ 5. Imperfect form, derived from the preceding. The first system of triple Algebra which I
obtained was that in which P = a + by + eG, where @, 7’, and n¢ severally represent — 1. I did not
at first see that though this will give PP’= P’P, it will not give P’(P’P)=(P"P’)P, except in
particular cases; though it should have been obvious that 7’(, for instance, is not the same thing
as (y(€)n. Now this is precisely the case of the redundant system already noticed, in which 6 = q.
If we multiply together a+ p4/-14+b(1+/%-1)¢ and a+ p'/-14+0 (lL + f- 1G

Vou. VIII. Parr III. Kk

250 Mr. DE MORGAN, ON TRIPLE ALGEBRA.

under the condition that ¢° means 4/—1, and if we then reduce the result to a line of the
same value of J, 8, w, in which also b= q, we have

aad’ — pp’ —2bb' — (pv’ + p'b)/2 + (ap' +a'p)\/-1+ (ab +a’b) (1+ -/- 1)¢.

Now for b 4/2 write b, and let (1 + 4/—1)¢ + 4/2 be an independent unit symbol (it will be
found by our conventions to be a unit on the axis of 2), and for it write ¢; also for py si hey
unit on the axis of y, write 7. Then it appears that the product of a +by+c¢ (write ¢ for p
and then interchange it with 6 in the preceding), and a’ + b'y + c'¢ is

aa’ — (b+ c) (+e) + (ab +ba’)n + (ac + ca’)%
which is here produced, and can only be produced, from y?=-1, (?=-1, 7€=-1.

I shall give the interpretation of this synthetically, and with some minuteness, since the leading
features of it belong to most of the other imperfect quadratic systems which I have tried.

Let every line drawn through the origin be considered as having for its plane that plane which
also passes through the axis of #; and let the line in which that plane cuts the plane of yz
be called the émaginary awis of that plane and of all lines in it (except the axis of w itself).
Let a line (x = — y) which bisects the second and fourth right angles in the plane of yx be called
the neutral axis, and one perpendicular to it, which therefore bisects the first and third right angles,
the primary avis. Let every imaginary axis have for its sign the sign of the parts of y and
which lie on the same side of the neutral axis as itself: and let angles be measured positively
in every plane by revolution from the positive axis of # towards the positive imaginary axis.

Let a + by + ¢¢ represent a line of the length 7 = 4/ {a+ (6 +¢)"{ in a plane whose imaginary
axis make with the positive axis of y the angle =tan~‘(e : b) having for projections on the real
axis (the axis of w), and its own imaginary axis severally a@ and +c; or making with the
axis of a an angle @ whose sine is b+ : J and whose cosine is a : J.

For addition, subtraction, multiplication and division, of two lines, make them both revolve
round the axis of w into, say the plane of wy, taking care to bring the positive part of each
imaginary axis into contact with the positive part of y. Then add, subtract, multiply and divide
as in common double Algebra, and find the plane into which the results are to be finally
transferred by the following rules.

In addition, set off on the primary axis lines equal to the projections of the given lines on their
imaginary axes; or transfer the imaginary projections by revolution to their proper sides of the
primary axis. From the extremities of the lines so drawn, draw lines perpendicular to the
primary axis, meeting the imaginary axes of the two lines, so as to cut off two hypothenuses,
On these hypothenuses describe a parallelogram; its diagonal from the origin is in the imaginary
axis of the sum. And similarly for the subtraction, or the addition of the equal and opposite line.

In multiplication, first lay down on the primary axis lines proportional to the tangents of
the angles which the factors make with the axis of w, and then proceed (exactly as in addition)
to determine the imaginary axis of the product from the diagonal of the hypothenuses. And
similarly for division. ;

In every plane, as long as lines are taken in that plane only, there is one complete system
of double Algebra, admitting every rule of ordinary Algebra to its full extent. When lines
from another plane are introduced, we lose the equation 4(BC) = (AB)C, unless 4 and B be
in one plane.

The theory of powers and roots is absolutely identical with that of common double Algebra for
every line which is not on the axis of w, the plane of each line being the locus of all its powers.
And +1 has only two square roots, as usual; but — 1 has an infinite number of square roots, every
imaginary axis of a unit in length being one of them. Also both +1 and —1 have an infinite
number of third, fourth, &c. roots, one set of three, four, &c. in every plane.

Mr. DE MORGAN, ON TRIPLE ALGEBRA. 251

For a +bn+c@, or [/, 0, w], we may write

Zsin@ Zsin@, tan w
Zcos 0 + n+ ——.¢,
1 + tanw 1 + tanw

and if 4/,, — 1 denote the square root of — 1 which is at an angle w to the axis of y, we have

Ree pr iratan a? Ft tae sy VTL? me Mtge
Call these last 4/—1 and 4/,-—1; we have then

[Z, 0, w| =1(cos@ + sinO,/, -—1) = (cose + a N/ 24 sin 8 EB ay eo t).

The product of any two positive square roots of -1 is —1, and the product of a positive
and negative square root is +1.

The Algebra of the neutral plane, which passes through the neutral axis and the axis of @ is of
a very peculiar character. In the first place, neither side of the neutral axis is necessarily positive
or negative by our conventions, and the signs of this axis must be determined (like that of
tan3a) by the manner in which we come upon it. But this is not the chief peculiarity. If
we call the point whose co-ordinates are a, b, c, the subsidiary point of L or [1, 0, w], the
point and its subsidiary point are always in the same plane: but if the subsidiary point be on
the neutral plane (6 +c¢=0), the angle @ is 0 or zw, and ZL is on the axis of 2 But if on the
other hand LZ be on the neutral plane, but not on the axis of a, then 6 and ¢ are infinite (with
contrary signs): and in this case, whatever line A may be, L+ 4, A+£L, AxL, A+L, L~A,
are all on the neutral plane.

Hence ‘a unit, situated on the positive side of the axis of w’, is not a complete description of
any line: for under that description comes every case of 1+m(y-—() in which m is finite.
The fundamental unit 1 or 1+0y+0¢ is the line which requires that the preceding should be
augmented by ‘having its subsidiary point at its extremity.’ It is true that no alteration could,
in any case, be produced in / or 0, by substituting one case of 1+ m(y-—() for another; but
the effect would be seen in the value of w. The rules of addition and multiplication, as above
given, fail when one of the lines is of the form @+mn—m(€; we must replace them by others
drawn from the use of the projections themselves.

I look upon the preceding system, as the one which has most general resemblance to the
common system, from which I derived it, before I considered the subject generally.

It is demonstrably impossible that any system can give the convertibility of three factors, in
which a line of a unit in length is represented by cos@+sin@.P,, where P,P,,=-1. Calling
this A, it will be found that 4”4’A and 4’A’”A are not identical unless sin@. sin 6’. sin 0”. P,,
= sin @.sin 6’. sin 9”. P,,, which, to be universal, requires P,, = Py.

1 tan w

1 + tanw

§ 6. Second imperfect system deduced from the redundant system. It is natural to examine
that particular mode of getting rid of redundancy, which consists in reducing the modulus of
multiplication to the form ,/(a* + p*+ 6+ q’). This is obviously

a(b—q)+p(b+q)=0, or b(p+a)+q(p-a)=0.
Now if we examine the corresponding function in the product, we find *
A(B-Q)+(B+Q)
= fa(b-q) + p(b+9)} fa? +b? +p? + qt + fa WU —q) +p +q)} fat + pi +q'},

* Most easily seen thus: since whence follows the equation in the text, and also
A? + B?+ P24 Q?+/2.{A(B-Q)+ P( B+ Q)} 24 Re 2 24. $94. 72 4+ g2)(q'24. B24 py’ + 9/2
f ; ps = + 094 p® + 92) (a+ b+ p+ 4
is identical with the product of the corresponding functions of a Er, Dal Ihe a ERR a2 oweh
4, b, &¢. and a’, b’, &c., the parts affected with ./2 are identical; :

KK2

252 Mr. DE MORGAN, ON TRIPLE ALGEBRA.

So that if this condition be true of the factors, it is true of the product. Now if as before,
a+ pa/-1l=reY", b+ prs/—1=s86°%"", we have, for the expression of the condition,
tan(@-—a)=1. This gives either

Bt+inv=athn, lcos@ =r cosa — s sina, lsin@ =r sina +s cosa,
or Btin=at+3r, lcos@ = rcosa+ssina, Zsin@ = rsina — s cosa.
_ The first will be the most convenient.

But though this condition may be satisfied for the product, when it is so for the factors, the same
is not true of the components and the sum, unlessa: a :: p: p’.. This system then would be
perfect for multiplication, division, and all its consequences, as the former one is for addition and
subtraction.

If we endeavour to find the system in which the sum of two lines is the diagonal of the
parallelogram formed on them as they stand, at angles a and 8 + 4 to the axis of w in the two
planes; we find the condition to be p(b+q)=0. Now b= —gq satisfies this for additions, and
p=0 and b= —q for both additions and multiplications: but an examination of this last case
will shew that it gives nothing more than the common double Algebra; no line lying out of
the plane of ay.

If there can be a perfect non-redundant system formed out of the redundant system, there must
be some function f(a, b, p, q) such that f(4, B, P, Q) and f(a+a,b4+',p+p,q+¢7) both
vanish when f(a, 6, p,q) and f(a’, b’, p’, q’) both vanish. The second condition cannot be satis-
fied unless f (a, b, p, q) be of the first degree with respect to the letters specified, in which case
the first condition cannot be satisfied.

§ 7. Imperfect system, independent of all that precede. Let the laws of combination of
the symbols, £, 4, ¢ in the expression a& + by + c(, be

En =nt, &e., P=§, w=-, C= -6&,
n€=-& C&=m En=¢.

The product of a& + by + c{ and @E+b'n + J is
faa’ —(b +c) (B+ e)RE+ fac +ca'}n + fab’ + ba’ &

In this system, the properties of the neutral and primary axes, the conventions of sign connected
with them, the modulus of multiplication, the rule of addition and subtraction, and the meaning
of the angles @ and w, are precisely as in the system described in § 5. But the product of
two lines in this system differs from that in the preceding one as follows; the angle made by
its imaginary axis with the axis of y is the complement of that made in § 5. Or, signifying
by @ the angle made by [/, 6, @] or af +by+c¢ with the primary awis, then if [/, 0, p] and
[v, 4, ¢'] have the product [Z, 0, @] in § 5, their product is [Z, ©, — ®] in the present system.”
Let two imaginary axes be called opposite which are equally inclined to the primary axis on
opposite sides of it, and let the planes passing through them and the axis of w be called opposite
planes. Then A*” is in the plane opposite to that of A”; A‘, A™, A™, &c. are in the plane of
A; A’, A’, A®, &c. are in the opposite plane. Generally speaking 4’"+! is in a new plane for
every new value of m. But the character of the square roots of — £ resembles that in § 5, and we

have
[4 0, p] =l{cosO.& + sinO./,- FE}

5 1-
=10038.£ + ising. ~~, 4 Ising 4

Mr. DE MORGAN, ON TRIPLE ALGEBRA. 253

The imperfection of this system, as in the former case, consists in the want of the equation
A(BC) = (AB)C.

There is a remarkable new consideration, which presents itself in these systems of inverted
multiplication, as we might call them. When € is an inoperative symbol, that is, when £7 means
n and €¢ means (, the abstract number of common arithmetic, m, may be represented by a line
E+0n4+ 0¢. But, in the case before us, the multiplier m and the multiplier m€ are very distinct
things. The former has only the effect of multiplying the length by m, without altering angles.
But there is still a line which has the effect of the abstract multiplier m, upon aé& + by + c@:
it is
b-e

@

The product of these two lines is mat + mbyn+mce¢. Now the second line represents a line of
the length m, on the axis of w; not having its subsidiary point at its extremity, but at a finite
distance on the neutral plane. And thus it appears that every such line of the form € + py — p&
plays the part of the abstract multiplier 1 to every line of the form a& + by + (b + ap)¢.

e-b
me+m——n+m

§ 8. On looking back to §2, we see under case A, a perfect cubic form with the equations
of signification

icy Seah ee a Gr tamer Gos eee C1 Se oy RY fore 2

Accordingly every product is of the form mé&, or according to our usual interpretation, must
be laid down on the axis of w Look at the quadratic and cubic cases that come under C and D,
and it will be equally apparent that all products take the form m& +m(y +&) or mE +n(y- Q),
according to the system: consequently all products come into one plane. It would be easy enough
to make any number of triple systems, under such a condition.

The perfect quadratic system under B may be readily developed. Its modulus of multiplica-
tion is 4/ {a* + (6 —c)’} which will require that, in an explanation resembling that of § 5, the
neutral and primary axes should change places. The line m(y + Q) is one of no length in such a
system, and if (y+) be added to a&+6y + c¢, nothing is changed except the position of
the imaginary axis. Let all the explanations be as in § 5, after interchanging the neutral and
primary axis: then the system before us is complete when we add to the explanations in $5,
thus altered, the condition that the product of a& +by+c¢ and @&+b'n + '¢ is to have the
addition (bb’ + cc’) (y + @), giving a certain alteration in its imaginary plane.

I should have liked to have delayed the present communication until I could have examined
these and other cases in more detail. But as, owing to the approach of other occupations, any such
delay must have lasted a year, I determined to send my thoughts just as they are, in the hope that
others may be induced to pursue the subject. One great point of the interest which attaches to it,
is the hope that the generalized notions of interpretation which it gives, will be found applicable to
the common double Algebra, which is at present restricted to systems of linear co-ordinates: and as
to which, though the restriction is clearly unnecessary, the proper direction of generalization is
not seen,

A. DE MORGAN.

University Cotitece, Lonpon,
October 9, 1844.

254 Mr. DE MORGAN, ON TRIPLE ALGEBRA.

ADDITION.

In single Algebra, we use no angles, and, so far as geometrical interpretation is concerned,
only one dimension of space. In double Algebra, we use two dimensions of space, and the rec-
tilinear angle. It might be supposed that in triple Algebra we should use three dimensions of
space, and solid angles, considered as proportional to the areas of their subtending equi-radial
spherical triangles. I can make no use of these solid angles; but others may be inclined to try
them: I accordingly give the following results, connecting the solid angles of a system of co-ordi-
nates with the plane ones.

Let the positive sides of the rectangular axis of a, y, x, meet the sphere in X, Y, Z; let
P be any point on the sphere, and let the cosines of the angles PX, PY, PZ, be d, p, v. Let
the spherical excesses of the triangles PYZ, PZX, PXY, be a, 8, y: their signs being taken
so that the equation a + 8 + y=47, which obviously exists when P is inside the triangle XYZ,
may be permanent. We then easily obtain

Cas oS oe
(+p) +0)’ Q+p +r)’
SER l+u l+pv (1+A) (+2) +)

I+sina 1+snB 1+siny (1 +r) (+p) (+ v) — Aw’
20+) 0 +4)0+»)
(1+A+mu4+)°
AG +A) (1 +A 4p +r)
G+aA +p +4) ’
2(1 + sin a)
(1 — cosa + sina) + (1 — cos B + sin 3) + (1 — cosy + siny)’

cosa =1

which, since \? +p? +? =1

Also, 1 —cosa+sina=

1+A= &e.

Having since I read this paper in proof, examined Sir W. Hamilton’s system of quaternions, I
may state that, in my view of the subject, it is not quadruple, but triple, since every symbol is
explicable by a line drawn in space. His object has been, to secure interpretation, though it
should cost the loss of some of the symbolic forms of Algebra; and his success has been of a
most remarkable character. My object has been to detect systems in which the symbolic forms
of common Algebra are true, without making any sacrifice to interpretation. The redundant
biquadratic system in § 4 of this paper has a resemblance to Sir W. Hamilton’s quaternion-
system in some of its formula, and a still greater one in its redundant character. It yet remains

to be seen what systems exist in which the axes of y and x are mot symmetrically related to
that of .

December 17, 1844.

XIX. On the Values of the Sine and Cosine of an Infinite Angle. By
S. Earnsuaw, M.A., of St. John’s College, Cambridge.

[Read December 9, 1844.]

Tue usage of Mathematicians in reference to the symbols Sin ¢& and Cos ¢9 does not seem to be
in accordance with their expressed opinions. It does not appear to be questioned either by English
or Foreign writers, that when w becomes infinite Sin w and Cos w cannot be said to be in one part
of their periodicity rather than another. If this mean any thing, it must be understood to signify
that Sin co and Cos o are indefinite. Yet this is not borne out in the usuage of these symbols
which we find in the writings of any author. Indeed, an opinion has been expressed that their
indeterminateness is only apparent, and therefore not real: and that analysis has furnished definite
equivalents for them by legitimate processes of investigation on principles which are allowed: and
though some writers on Definite Integrals have abstained from stating in direct terms what are the
values which analysis assigns to Sin ¢ and Cos os, all agree in practically affirming “that both the
Sine and Cosine of an infinite angle are equal to zero.” But while we find these values used where-
ever Sin co and Cos occur in investigations, we do occasionally meet with expressions of doubtful-
ness respecting their universal truth. This seems to indicate that in the opinion of such writers
the values of Sin c¢> and Cos e depend on the circumstances under which they occur; but what those
circumstances are which have this power over Sin co and Cos I do not find any where pointed
out. In fact, upon tracing the origin of this doubt respecting the universal truth of the equations
Sin c = 0, Cos ~ =0, I find that it has arisen from the occurrence of certain results of a character
so obviously suspicious, perhaps I might say, erroneous and contradictory of evident truths, as to
create a reasonable doubt of the propriety of writing zero for Sin o and Cos © in those cases.
But though results have thus forced some writers to doubt respecting the general truth of the
equations Sin e& =0 and Cos © =0, it does not appear that they have any where admitted the
demonstrations of the truth of these equations to be defective. We find ourselves then in this
difficult position ;—-we have certain investigations presented to us in which there occur no doubted
steps, and these investigations present us with certain absolute results ;—but the certainty of these
results thus established by a process of mathematical reasoning, the accuracy of which is no where
called in question, we are afterwards required to look upon with suspicion;—and that sort of
suspicion which while it throws doubt upon every thing affords us no clue for ascertaining what are
the cases to which alone it ought to be attached. It is obviously desirable that some effort should

be made to remove this uncertainty. Now some light may be thrown upon this difficulty by
fo)

considering that Sinmw and Cosn go through a whole period of values while w increases by hale
n

- pe sue - : : 4 :
As long as is finite — is finite, and all the values included in a period are therefore consecutive.
n

But what happens when 7 increases in value? We easily see that as m increases the whole period
becomes condensed so as to occupy a shorter and shorter portion of the current variable; and
that when 7 approaches ¢, the values are no longer consecutive but simultaneous :—hence as n
increases towards ¢ a whole period of values of Sin w or Cosa tends to become simultaneous, and
in the limits are simultaneous; i.e., Sin co has at once all values from —1 to +1; and the same

256 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE

property belongs to Cos ¢. Consequently according to this view it is not true that Sin ¢ and Cos ¢
have each a single value, or any finite set of values definitely; but they each have all possible
values from —1 to +1 in such a manner and sense that not one of these values is pre-eminent
above another, and no one has a claim to be put forward above its fellows, but all stand in exactly
the same relation to the function Sin ¢ (or Cos e) so that at one and the same moment Sin
(or Cos co) is equal to every one of them but not more properly equal to any one than any other
of them. From this reasoning and kindred reasons of an equally general character, I satisfied
myself that Sin ¢ and Cos ¢ cannot be replaced by zero, unless under some special hypothesis,
and that when taken in a general sense they cannot justly be supposed to have definite values at
all. I shall now proceed to some considerations which are preliminary to a more formal proof that
they have not the value zero, even when considered as the limits of more general forms.

In conducting my inquiry into the values of the symbols Sin ¢, Cos c, I am unavoidably
brought upon the confines of the much controverted subject of divergent series. In a certain sense
which will be explained, I agree with Professor De Morgan that all non-convergent series stand on
the same basis, though I cannot subscribe to the train of reasoning by which this is usually main-
tained, involving as it appears to me some disputable positions. Much of the obscurity which
attaches itself to the subject of divergent series may be traced to the discordant and strange
significations applied to the symbol =, when used in connection with infinite series. The pre-
sumption is that when this symbol stands between two quantities it indicates, that either may be
used for the other in algebraical processes. A very eminent author states that it “‘ may be rendered
by the phrase gives as its result, when it is placed between two expressions, one of which is the
result of an operation which in the other is indicated and not performed ;’—an explanation which
agrees exactly with what Woodhouse states in his Principles of Analytical Calculations, who insists
upon this definition of it at intervals through his work with an earnestness which indicates the
confidence with which he regarded it as true. Now if this definition be closely examined it cannot
be understood to denote that the expressions connected by = differ in any thing but form; for
one side denotes that an operation is to be performed, and the other is the result of the actual
operation: if then the operation has been eorrectly and completely performed, there is no difference
except in form between quantities connected by =. But an examination of the Principles of Ana-
lytical Calculations, will not fail to satisfy us, that in giving this definition the author must have
understood it in some modified sense which he has not expressed in the definition itself. For when
it is said that “‘ = is a symbol which serves merely to connect an involved expression and the result
of an operation,” it is evident that “numerical equality” could not then be, what the author affirms
it is, a contingent result. But whatever was the sense which the author mentally attached to the
symbol, it involved a principle which necessitated the making distinctions where by ordinary minds
the difference cannot easily be grasped : for it was found impossible to be consistent without demand-

c : 1
ing a license to consider 4 and —— (as also
1+1

) as essentially distinct. Now
v+i1

what difference is there between 2 and 1 +1, except inform? Is not 1+1 an expression in which
an operation is to be performed the result of which is rightly denoted by 2? and if so, then by his
own definition 2 and 1+ 1 are algebraically equivalent. I must confess that I cannot consent to.
such distinctions as are here demanded without being satisfied that there is no means of avoiding
them; and I cannot but suspect that in the present case there is no other necessity for them, than
what arises from a misapplication of the definition which the author has given of the symbol =.
For if this symbol serve merely to connect an involved expression and the result of an opera-
tion, it is clearly a misuse of it to employ it in connecting an involved expression and a part
only of the result of an operation. Let me explain by an example. Professor Woodhouse writes
1

i+@

=1-—w+a°—...... Now the operation denoted on the left-hand is the division of 1 by 1+ 2,

OF AN INFINITE ANGLE. 257

and according to the definition of =, the other member is or ought to be the result of that opera-
tion. But we observe that 1 — «+ a°—...... is a series of terms following the same law through-
out, and shewing no indication of any terms which are not included in this law; yet it may
be asked, have we any just ground for knowing that all the terms resulting from the division
of 1 by 1+ do follow the same law throughout? Let us examine; if we stop after one term

of the quotient we find (eae 1- ie ek if we pursue the division a step further we find
& d
1 a : i
a | — ; another step gives =l-wt+a’- , and so on. In all these
1+a 1+a@ 14+ 1

partial operations we observe that one term of the quotient is an exception to the law followed by
the others. It is true, by continuing the process, we may push this anomalous term to any con-
ceivable distance from the beginning of the series, but there is not the slightest indication that
by so pushing it it will at length cease to be, or become zero: on the contrary, as Professor De
Morgan justly remarks, by the prolongation of the operation it is removed farther off but not

is of a character which can never

destroyed. Consequently, the operation represented by as
a

be completely comprehended in any series of terms which follow one law: and therefore, strictly
speaking, there is no such quantity as the definition requires which can be joined with it by the
symbol =. Shall we then join it with as much of the quotient as does follow a fixed law?
It is clear we cannot without violating the terms of the definition. When therefore we find

1

=1-—a#+4a°—... ad inf. without an implied remainder, we are at a loss to understand in what

1+a@
way this use of it is reconciled with the meaning attached to the sign = in the definition. Yet
it is certain, that most eminent writers do use the symbol = to connect a function with a series

every term of which is supposed to follow a fixed law, as though the operation denoted by the
function were capable of being represented by such a series of terms. Still, though it is thus
rendered evident that the usage has not been sanctioned by the definition, the discrepancy is
not very important in itself, seeing that an alteration may be admitted into the definition which
shall make it agree with usage. The definition may then stand thus ;—the sign = is used to
connect an involved expression with the result of an operation as far as it is eupressible in
terms which follow a fixed law. ‘The really important point now to be examined is, whether
that portion of a result herein included will in all cases represent, for algebraical purposes, the
properties of the expression from which it was derived. If it will so represent the expression,
then for algebraical purposes series of all kinds, whether convergent, periodic, or divergent, will
stand on the same basis, and their use in all cases be equally safe. I need hardly say that
this isa much disputed point, which has been warmly attacked and defended. I am induced to
venture into the field on the side of the assailants from having observed that its advocates have
defended the use of non-convergent series on grounds some of which are capable of being easily
shewn to be fallacious: and though I cannot bind myself to the justness of all the arguments
which have been opposed to them even by the most eminent and skilful analysts, I yet think there
are sufficient reasons left to justify us in rejecting non-convergent series when in accordance with
the above definition their remainders are thrown away.

Now according to the definition above proposed, it is evident that an invelopment and its series
are not equal, (they differ by the remainder) the question is, are they equivalent? does the series
embody all the algebraical properties of the invelopment, and no more? The discussions which
have been so earnestly carried on with the view of arriving at a satisfactory settlement of this
difficulty have not yet elicited any unanswerable arguments on either side: at any rate they have
not been of such a character as to set the question at rest. Though I do not presume to hope that
what is here brought forward will have the effect of satisfying those who entertain the opposite

Vou. VIII. Panr III. Lu

258 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE

views, yet something may perchance be said which will in abler hands be made useful in settling
some of the difficulties which beset the consideration of this perplexing subject.

1. The ground on which I would reject the use of non-convergent series is a conviction
that such series may have some algebraical properties which their invelopments possess not, and
may lack others which the invelopments have. For series of ordinary forms I think I shall be able
to prove the truth of this as satisfactorily as such an intractable subject as an infinite non-converg-
ing series admits of.

2. Let us notice first, that there is a presumptive ground of suspicion of the truth of this (viz.,
that the algebraical properties of a non-convergent series are identical with those of its invelop-
ment) in the rejection of the anomalous term (the remainder) which if preserved would certainly
render their (numerical as well as) algebraical properties identical. Has the remainder mo alge-
braical properties? If it has, then it will hardly be believed without proof, that in throwing
3t (and with it its properties) away we have not destroyed the algebraical equivalence which by its
means existed between the invelopment and the series. I will endeavour to illustrate my meaning
by instances.

8. It admits of no doubt that including the remainder the equation }=1-1+1-1+.....
ad inf. is strictly true. We are to examine whether this is algebraically true if the series be
taken without its remainder. Denote the sum of m terms of the series by S,; then it will be found
that for all values of », S,= S?. This equation being strictly true may be made use of in any
algebraical operation: and as it is true however large be the value of , it is impossible to refuse to
admit that S',,= S is a property of the infinite series. Hence 4, not being a root of this equation,
does not enjoy this property which the sum of the infinite series does enjoy, viz., that it is not
altered in value by being squared. 3 is the sum of the series inclusive of the remainder, and 5S‘, is
the sum of the same series exclusive of the remainder. Hence the rejection of the remainder has
altered the algebraical properties of the symbol by which the series is represented,

4, But the algebraical importance of the remainder may be rendered still more striking, and the
On . ‘ . : . @ :
impropriety of rejecting it put in a stronger view. For if any proper fraction 5 be put in the form

1+1+1+... to a terms

sei ate oe it will be found by the ordinary process of algebraical division that

1+1+1+-... toa terms
oe Shy he geen
1+1+1+... tod terms

Now many persons have found it difficult to reject 4 as the algebraical equivalent of 1-1+1-...
because by ordinary algebraical development this series ad infinitum can be obtained from 4.
It is here shewn however that the very same process which elicits the series from 4 would serve to
elicit it from any proper fraction whatever: and this being so, by what distinguishing property
are we to be guided, so as to be able to select amongst all proper fractions some one particular
value as the equivalent, the wnique equivalent of the infinite series? If 4 be selected as embody-
ing all the algebraical properties of the series, surely we must admit that for as good a reason

A : : : a :
; embodies the whole of its properties; and thence we cannot avoid allowing that 4 and p are in an
algebraical sense equivalent fractions.

5. But it is said in special favor of 4 that from whatsoever more general series 1-1+1-—...
be deduced the symbolical equivalent is always found to be 4. If deduced, for example, from

OF AN INFINITE ANGLE. 259

ay . - ~~ l+eo+ar+... ot}
1-av+a°-... by writing 1 for x the sum is 4. Now let us turn the fraction —————"—_
1+a+a°+... a

into a series by the ordinary process of division ; the result is, (b > a)

1+0+...a@ terms

a SS SE TS Sete
1+a+...b terms

This series differs from 1 — v + #*—... only in being more general, for it includes it as a particular
case (viz. when a=1, and b=2). If then it be lawful to write 1 for w in 1—@+a°-... it

: j , a
is equally lawful to do so in the more general case: which being done we have a 1=1+1-... ad

infinitum. Here then is ‘a well-established instance in which 1 — 1 + 1 — ... means other than 3;”
shall we say with Professor De Morgan, one such instance throws ‘* doubt on all that Poisson and

Fourier have written ?”

6. It will hardly be considered necessary to defend a system which requires us to receive as a
legitimate consequent that all proper fractions are algebraical equivalents. I apprehend therefore
the last article but one will be sufficient to shew that in numerical forms of series the ability of an
expression to furnish by legitimate expansion a proposed series is no presumption that the two
are algebraically equivalent. Weve then is fair ground for suspecting the existence of some grievous
violation of just reasoning in depriving an infinite series of its remainder, i.e. in supposing that by
pushing an expansion in infinitum the anomalous terms may be disregarded. In converting the
expression SBE Aa ata a into a series we observe that for all values of a and b (a <b) the

+1+1+... 6 terms
series of quotients are the same, and the various cases are distinguishable only by their remainders.
The distinctive properties then of these proper fractions by the process of development are not
thrown into the quotients, but are preserved in the remainders. How then shall we reject the
remainders in any equation which professes to exhibit the equivalence of its members ?

But there is yet another proof, which I shall now offer, that neither 2 nor even any proper
fraction whatever can be the proper equivalent of the series 1 — 1+1—......

7. In perusing what has been written upon this series, we cannot but perceive that some authors,
setting out with Tog 717 @ + 2 sees. as an equation admitted on all hands to be true when & is
w

less than 1, have argued that, being true when @ is less than 1, however small 1 — a may be, it must
needs be allowed in the limit. If the premises are true, I do not see how we can refuse to allow the
conclusion. But it is obvious the premises assume that the series is convergent towards 1-1+1-...
when 1 — w is indefinitely small’; is this true? If it is, I admit that 2. is the equivalent of the series

‘ j c F 5 a
1-—1+1-—... in as good a sense as ; is the equivalent of the converging series 1 —w + a...

wv
Mr. De Morgan questions this; but I see no objection in it which would not, if admitted here,
overturn the whole fabric of the Differential Calculus. But we have to answer the question asked
above, is it true that the series 1 -aw+a*—a' +... is convergent towards 1-1+1-—... as its
limiting form when 1 — w is indefinitely small?

y

8. Let y be any finite quantity, and assume 1 — # = +=: then when » approaches infinity,
n

P : ‘ hale a. y\"
1 - w will be indefinitely small; but then limit of w” = limit of (1 + *) = e*”, the upper or lower
mn
sign being used according as w approaches 1 from inferior or superior values. Here then is a proof
LL2

260 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE

that the terms of the series 1 -w+a°—... at an infinite distance from the beginning do not
converge towards 1 as their limit, but to one of the indeterminate quantities e~’ or e*”; the values
of these depending upon the law under which # approaches unity. Who shall prescribe this law ?
Surely it is (and must be left) arbitrary in the fullest sense of the word. It is not true then that
the converging and diverging forms of 1 — # + # —... approach the same form, viz. 1 -1+1—1+...
ad infinitum, as their common limit. For the limiting forms both of convergency and diver-
gency are arbitrary, yet so restricted that they never can mutually approach so near as to be
separable by only a single form: for e*” never can approach so near to e~” that only unity lies
between them, because y is necessarily finite, i.e. neither indefinitely large nor indefinitely small.

9. The unavoidable inference from the last article is that1 —1+4+1-—... is an isolated form
of 1 — @ + a@ —...... and separated from the limits of continuity on either side by a finite interval.
For the same reason it is an isolated form of 1 — v7 + a@°—a"*>4 ~?8— .., Let it now be admitted

1
that
i

is the equivalent of 1 —# + a*— ... ad infinitum, then it will follow that
+2

limit of = limit of (1 - 7+ a -...)

1+2
But limit of (1 -v7+a°—...) isnot =1-—1+1-......

Pave To
«. limit of isnot =1—-1+1-...
1+2

This then is the proof that 3 is not, (and in a similar way it would follow that = is not)

the proper equivalent of 1-1+1-—...,., even assuming an to be the proper equivalent of
a
a”

: F ‘s 1
1-—w+a°-—...... It is easily shewn, since

=1-—a+a°-... m terms, that 3 (1 +e) is

d

= limit (1 — w + a — a+... ad infinitum), which is therefore indefinite.

10. In a paper “ On Divergent Series” by Mr. De Morgan, there is a remark which shews the
important bearing of the results obtained in the preceding articles. ‘It is clear enough,” writes the
Professor, ‘“‘from the manner in which Fourier, Poisson, Cauchy, &c. use the limiting form
1—1+1-... that they intend it to signify 4 in an absolute manner. The whole fabric of periodic
series and integrals, which all have had so much share in erecting, would fall instantly if it were
shewn to be possible that 1 — 1 + 1 —... might be one quantity as a limiting form of 4,—4,+4,-...
and another as a limiting form of B, — B,+ B,—...”. I object, of course, to the assumption that
1—1+1-—... is a limiting form of the series alluded to; but passing over that, it is shewn above
that 1 — 1 +1-—... when taken as a form of 1 — a* + a’ — ..., which it certainly is, may be one
thing or another, according to the values arbitrarily assigned to a and b. Indeed it is stated
( > as well ES But

1+1+1 i+
Woodhouse either did not observe this evident contradiction, or must have got over it by the -

in Woodhouse’s Anal. Calc. p. 61, that 1-1+4+1-1+...

mystical maxim that

is not = 4, and is not = 4; which is perhaps the case, for in a

1+1+41 1+1

note he considers that Euler, Leibnitz, and Waring had fallen into a mistake by making cr
1 A e A .
&e. =4, 4, &c. However, passing by this doctrine, it serves the purpose for which I

?

1+1+41
quote it, for it exhibits Woodhouse as testifying to the propriety of taking 1-1+1-—-... to be

s- In fact

a form of the series 1 — + vw — a+... which arises from the expansion of ees

OF AN INFINITE ANGLE. 261

to this also Mr. De Morgan has given assent where he assumes that 1 —-1+1-—... is a form of

1-2 +a'—a+4+a-—...... I have brought forward these testimonies, because it is not very

unusual to cast a mantle of mystery over this subject, by introducing zeros into the expansion of
1

ee 1”

head: for Nes
1+27

But such a device, however much it may serve to satisfy the eye, cannot satisfy the

=a gives 1—w+°—w'+..., there being no terms between & and 2’, w* and a’*,

&e., in this, which is the general form of the series; and consequently it is not allowable to write
1
1+1+1
any difference in the sum of the infinite series: and if they make no difference, why introduce them ?

=1-1+4+0+1-1+0+..., if it be intended to insinuate thereby that the zeros make

11. On principles therefore which are allowed, and used by the writers quoted, it is established
that 1-1+1-—... has no definite equivalent, in the sense in which this word is generally under-
1

stood. I think also it is proved, that is in no proper sense the equivalent of 1-v+a°-...,

+@
except when this series is convergent. For that the two expressions may be equivalent to each other,
it is essential that each should exhibit the same degree of indeterminateness of value in particular
cases, and the same kind of discontinuity: but, as we have seen, there is no such agreement: on the

contrary, while it is admitted that, as w converges towards 1, approaches towards 3 as its

1+2
unique limit, it is here shewn that the other member of the assumed equivalence approaches towards
an indeterminate form of an ambiguous character, and absolutely refuses to approach in any case to
1-1+1-—... asa limit of continuity.

12. It is not the purpose of this paper to treat of Diverging Series in general, but only of the
recurring form 1—1+1-—..., and of this only because it has been connected with the values
of Sin co and Cos @, yet as the method above employed is applicable to the general form
Pxr=au+ a,x +... +a,0"+ ... I may state that the same mode of reasoning when applied to
this, shews that pw does not embody the algebraical properties of the series, unless the value of «,
and the form of the coefficients, be such as to make a,” tend to zero as its limit when » and »
approach <. Series which satisfy this test I call convergent series, whether the arithmetical sum
thereof be finite or infinite: and all such series are distinguished by this property, that their invelopes
may be safely used as equivalent to them in every sense both algebraical and arithmetical.

13. From this it is evident, that the operation of integration performed upon a series will often
(not always) have the effect of removing its discontinuity, and establishing a real equivalence though
none existed before. And so the operation of differentiation will not unfrequently have the effect of
introducing discontinuity, and destroying equivalence.

, 2a
Hence we see why we may put 1 for w in log, (1+a)=a—— + nee though we may not

write 1 for w in om: =1-—a+°—... from which it was derived by integration.
@

14. But, in pursuance of the object of this memoir, it is time now to turn to the series
1 — Cos + Cos29 — Cos 30+... which has been assumed to be a form which can approach
1-1+1-...... as a limit by diminishing @ towards zero. Now assume y to be any arbitrary

finite angle, and put 0 = + f which will be indefinitely small for the terms where m is infinite.

Hence in such terms Cosn@ = Cos+y= Cosy= a fimite quantity, not equal to unity, because y

262 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE

cannot be equal to zero. Hence the terms of this series at an infinite distance from the beginning
are subject to discontinuity, and cannot be made to approach 1 as their limit; because if @ differ
ever so little from zero there will always be a term so distant from the beginning as that 76 is
Jinite; that term and all following ones will not approach 1 as their limit. Consequently 1 -1+1-...
is an isolated form of 1 — Cos@ + Cus 29 -...

15. It is not necessary to repeat, in reference to this series, what has been already said upon
1-—«v+a°>—...5 it is sufficient to remark that all results are nugatory which have been obtained
upon the supposition of 1— Cos @+Cos2@-—- ... approaching 1-1+1-—... as its limit as
6 changes continuously towards 0. I might here add remarks in reference to the series
a, + a,Cos 0 + a,Cos20+4... +a,Cosv@ +... parallel to the remarks in (12) and (13).

16. Since it often happens that by integration as remarked in (13) a real equivalency is
established, it is not unusual to find such series cited as confirmations and verifications of the
propriety of the equivalency assumed to exist before integration. From what has been proved
above however it is evident that such verifications are of no value, and do not in any degree justify
the inference sought to be drawn from them.

17. I come now to examine the limiting values (if such there be) of Sin w and Cos w when
@ approaches co. As a preliminary step it is proper to remark, that oo is an indefinite symbol : and
when it is said that # approaches © as its limit, we are not to understand that a approaches towards
some definite value, but merely that it approaches to a value of which we have no other property
than this, that it is greater than any finite quantity. Yet there is such a thing as a restricted co.
Thus, if w be an odd multiple of x by the nature of its definition, this restriction will not hinder its
becoming infinite ; yet then the symbol © will be specific ; and accordingly it is possible that under
such a condition definite results in certain cases may be obtainable.

18. The above remarks respecting the essential indefiniteness of the symbol ¢o will enable us
at once to reply to some questions which have been found perplexing. The question has been
asked, is the series P,— P,+ P;-—P,+...... ad infinitum equivalent to the series (P, + A)
— (P, + A) + (P; + B) — (P, +B) +... ad infinitum? This has been rightly answered in the
negative ; but on erroneous grounds. ‘The true reason is this: the terms 4, B, C'... are introduced
in such a manner as necessarily involves the notion that © is an even number, and therefore it
creates an error unless it have been stipulated that co is an even number. As from the nature of
an infinite series no stipulation of this kind can be allowed, we are justified in saying that the two
series are not equivalent.

19. If a be defined to be a term of the series 0, 2, 4, 6 ..., then Cos v7 = Cos 0° when v= 0;
but if @ be a term of the series 1, 3, 5, 7..., then Cos am = Cos az when # = ©; but if w be defined
to be a term of the series 0, 1, 2, 3, 4..., then it cannot be affirmed that a is an odd number, nor yet
that it is an even number. To say only that a is a whole number, is to express oneself in a way that
requires the result to leave the question as to whether @# is odd or even undecided. Hence in this
case we cannot say that Cos co = Cos 0°, nor yet that it = Cos; but we must express the result in
such terms as leave undecided which of these two is the value of Cos © ; for to select one of them
and reject the other would narrow the restriction laid upon « by its definition, by deciding that it is
not only an integer, but that it is a specific integer. Hence then in this case Cosco = Cos0° or Cos +
indeterminably.

This mode of reasoning can be extended without difficulty to the case where # is a continuous
variable, and it leads us to this result, that on this hypothesis respecting the nature of a, Cos ©
(derived from Cosa by supposing w to approach towards © ) is equal to the Cosine of any angle
from 0° to 2a indeterminably. When I say indeterminably, I mean to say that we cannot fix on
one of these angles and reject the others without violating the generality of the hypothesis: should

OF AN INFINITE ANGLE. 263

we for instance say that Cos o =0, the selection of this particular value would be equivalent
to narrowing the hypothesis respecting wx, as it would restrict w to be an odd multiple of = » and con-

i nee Po em or cies =
fine its variation to the terms of the series Br puriaig, 5 -maehsG similar observations may be made

respecting Sino.

20. It is also very important to remark that Sinaw and Cos aa do not cease to be functions of
a when « approaches © .

For since Cosa = + » it appears that Cos@ has an ambiguity of value of which

(" + Cos =o

Cos 2a does not partake. We may follow out this mode of reasoning to shew that Cos # and Cos aw
have not the same number of corresponding values, and that if the value of one of these were given
the other would not be determinable from it except in an ambiguous form. Whatever indetermin-
ableness attaches itself then to Cosa when w approaches ©, the same, and also another kind of,
indeterminableness belongs to Cosaw at the limit. We are then particularly to take notice that
Cos derived from Cos may not be written for Cos cc derived from Cosax. Much error has
arisen from want of attention to this caution, Also Cos aw cannot be considered independent of a at
the limit # =o , inasmuch as it is subject to two causes of indeterminateness which are distinct from
each other.

21. Having thus given my reasons for considering that Cos c¢ and Sin have not definite
values, it may be proper to examine the proofs which have been brought forward by those who have
used definite values for Sin co and Cose. The following is the most direct proof I know of:

wf? Sinadx=(f" + "+ 7+... ad infinitum) Sinw da;
-. 1-Cosem =2—-242-... ad infinitum =1;
.. Cos eo =0.

To this proof there are two objections, either of which is fatal to it. In the very first step it is
assumed that co is an integer multiple of x. For this assumption there is certainly no autho-
rity, neither is it compatible with the indeterminate nature of the symbol in the left-hand
member of the equation. The next error is made in the summation of the infinite periodic series
2-2+42-..., which I have shewn in the previous articles of this memoir cannot be equal to 1.

22. As the reader may wish to have a further proof of the error of principle involved in
the first step of the above investigation, let him see the effect of a different distribution of © into
parts in the following process of reasoning, in which the question of summation of series is avoided.

2m 4a 2n
"Sina da =(f* + fz + far + seeene ad infinitum) Sin w da
) 3 3

1+0-14+1+0-1+4......)

2
3 iJ 3m Qn
"> (f+ fe + fre th vescee ad infinitum) Sin w dw
0 2 >

3 D
= sf Sinwda;
2 0

_—

“. f, Sinwdw=1-Cos@=0, .. Cos =1.

264 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE

It is for those mathematicians to reconcile these conflicting results, who maintain that providing
the last limit of # be ¢ it is no matter whether it be a specific co or a general «. The dis-
tinction is of first-rate importance in periodic functions. I think I am fairly entitled to affirm that
specific values for Sin co and Cos 9 are obtained by such processes as that in (21), only becauise
those very processes assume at the outset a specific form of co.

23. The next proof which I shall examine depends upon the principle of continuity ‘that
what is true up to the limit is true at the limit.” It is as follows:

-—ar

: P e :
Since fe-** Sinada = — cae (Cos a -+ a Sin @) ;
+a

. fe Sinada = ——_.
1+a’

This being true for all positive values of a, no matter how small, is taken to be true in the

limit when a@=0, which gives (since e~*” then =1 for all values of 2)

f° e~ Sinadaz = f° Sinwde =1;
. Cos @ = 0.

24. To this investigation I have two objections to bring forward. The step which assumes
that e~°* = 1 for all values of wis not true at the limit #= o, for however small a become aw
will be finite and arbitrary or infinite when a= ©. Hence as we diminish a towards zero e~*”
approaches, not to 1 as its limit when w= @, but to e” an arbitrary value depending upon
the relative laws with which w approaches © , and a zero. Now it is absolutely necessary in the
above proof that for ald values of w between zero and «0, e “" should be equal to 1; and as
this is not a true hypothesis, the proof fails.

-ae

e

should be the value of —

Again, it is essential to the above investigation that ; 3

l+a 1+a
(Cos# + aSinw) between the limits ~=0, e=c. But this will not be the case unless e~“”
vanish when a=. Now I have just shewn that when @ is made to approach zero e~*”
become e~” at the limit a=. ‘This step therefore of the investigation is erroneous, and the
proof fails.

Let us look at the first written equation in (23), and endeavour to answer these questions;
can e~“* in the left-hand member be always =1, and yet in the right-hand member = 0, when
w= co ? If #=co make e“*=0 in the right-hand side, what can prevent the same being true in
the left-hand side, seeing that the values of w are simultaneous in both members? Here is a
plain contradiction of hypothesis in the two members of the fundamental equation the consequences
of which no explanation can remoye: and as both hypotheses are required to be true together
to enable us to obtain the final result Cosco = 0, I conclude that this result is not proved to
be true. I think upon examination of the steps of the proof in (23) the reader will admit,
that it is conducted upon the supposition that, as v varies from zero to ©, e"” remains constant -
on the left-hand, and decreases from 1 to 0 on the right-hand.

25. These are the usual proofs that Cos «@ = 0; and it is not necessary for me to examine
more, as all that I have met with involve erroneous reasoning of a character similar to that
noticed in the two above given. Before concluding I wish however to notice one or two other
cases in which great caution is necessary in managing the symbols Sin c& and Cos o.

o Sinawx

26. The first which I shall notice is f dx, which has been said to exhibit some

. Ce

OF AN INFINITE ANGLE. 265

singular anomalies. It has been asserted to be equal to = a result which is manifestly sym-

bolically erroneous, seeing that it does not change sign with a, a property which the expression
to be integrated shews must belong to the true integral. Such an objection as this would be held
to be fatal to a result in other branches of analysis, and I am at loss to conceive why it has
not been allowed the same force in this. It is true a proof has been offered that the integral ought
to be independent of a; but if any thing can be inferred from that proof it is that the
integral ought to be imdefinite in every case. The proof alluded to is as follows:

Si :
ee In Cate ae Sin aw COM Sins Fe
@ x
Sin Ba, Sins « Sina
. = dx
| LETS ace re

whence it is stated that the value of the integral is in every case the same as when a=1: yet
as I have said before, this inference is evidently erroneous when —a@ is written for a. The
probability is that the true integral is such a function of a as is constant for ordinary values of a,
and changes sign with a; I say ordinary values, because it is easy to shew that the transformation
fails as a@ approaches zero. For since the equation av=x must be respected, by means of
which the transformation is effected, this shews that were a to become indefinitely small, x would
not be co when # approached oo; but in that case the limits for x would be 0 and y (y being
an arbitrary finite quantity). Consequently as a approaches towards zero, the integral approaches
towards an indeterminate form as its limit.

The value of the integral when a= 0, would therefore seem to be isolated: and cannot be
inferred from the above transformation. Expressed in a series the required expression for the
integral is
(a «)°

eee
Eien Gree iets

which confirms the preceding re be in the case when a approaches zero.

27. The next case which I shall consider is aise Cosba

>, dw, which has been stated to be
0 +2

oie ‘ i 1 F p A F
equal to af *@ when 6 is positive, and to ma when 6 is negative. As in the preceding case,
a

so here, the symbolical inaccuracy of the integral brought forward is sufficiently indicated by
the acknowledged necessity of empirically changing the form of it. As the erroneous principle
by which this result is obtained has found its way into a great number of other integrals which,
as well as this, are vitiated and rendered erroneous by it, I shall endeavour fully to expose it.

28. Denoting the required integral by P, we find
Sin(6.0) Sin(b.o)

i he
In the usual process, the last member of this equation is assumed to be zero: and with regard
to the first term of it that assumption may be allowed; but the last term of it, it has been the
object of this paper to prove, is indeterminate. It is also to be remarked, that this term forbids us
to make b approach towards zero, because when 6 is indefinitely small the right-hand member
approximates to ¢. Yet regardless of these cautions the right-hand member has been put equal
to zero, and the value of P has been then found by integration to be

P=Ce*+Ce™,
Vou, VIII. Parr III. Mm

d,P —a°P= — f,”Cosbadwa =

266 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE

The first term of this integral has been put equal to zero on the ground that b= would
make P= co were this term allowed to remain. (I shall shew presently that it is not allowable
to put b= o@). The value of C’ is then found by putting b = 0, the very supposition which must
necessarily render the result erroneous, seeing ae d,P —a°P is then equal to ©. I infer there-

fore that there is no certain ground for writing > — for C’; as little indeed as there is for rejecting
2a

the term Ce’, In fact, the given function being unchanged when — 6 is written for b, the inte-
gral must posees the same property, which gives C = C’, and therefore we ought rather to write

P=C(e" +e”).
29. TI shall now endeavour to shew that we may not put b= in the value of P.
It is easy to shew that
dam denn peron aaa
For all jinite values of b the right-hand member of this equation vanishes: but when b= ©
the term = Sin (6.0) cannot be put equal to zero; this term corresponds to the limit v=0. Also

© Cosba

ta da forbid us to
put 6=0. Hence if we put the right-hand member of the equation patel ‘ zero, we are to keep
in mind that that step involves a prohibition against putting 6 either equal to zero or «9. Exclusive
then of these values of b, we have

the intermediate steps by which this equation is obtained from P = ipso

&(aP) - BaP) =0;
and .«. aP = Be” + B’e~™.

For the same reason as before, B’= B; and by comparison of this with the value of P (admit-
ting that value to be correct for the present), found in the last article we learn that B is inde-
pendent both of a@ and },

B ab —ab
P(e Huenas)s

How B is to be determined, I know not, seeing that it is not allowable to put b=0, which is
the usual plan.

30. There is great advantage in forming two distinct differential equations for P, as we may
learn from one of them something which may assist us in managing the other. In Art. 29, we
have seen that, subject to the condition of b being finite, we have strictly dj(aP) — b°(aP) =0:
but this condition will not allow us to strike out the right-hand member of the equation in (28).
This shews that B and B’ in (29) are functions of b; and (29) shews that C, C’ in (28) are
functions of a.

In strictness then we ought to integrate the equation

d?(aP) - B(aP) = - ; Sin (bee ).

a> ne" Sin(beo db e* re Sin(beo )db
ab -ab piers
<¢ polo al eer ~ b

C being an absolute constant, the value of which I know no means of determining.

OF AN INFINITE ANGLE. 267

31. It is not necessary to examine other instances of definite integrals the values of which, as
they have hitherto been obtained, I believe are not to be relied upon. They involve either the
notions that Sino = Sin(a@.c)=0, Cos = Cos(a.c) =0; or else depend upon the sum of
the series 1-1+1-—...... being =}. The classes of definite integrals free from one or other
of these errors are very few in number, not including some of those which analysts have evidently re-
garded with especial favor. It will be evident, if what has been written in the preceding pages
be allowed, that nothing could be more troublesome than the very general adoption of 0 and
as limits of integration when trigonometrical quantities are involved. The expansion also of
functions in the form of series of multiple angles seems in very many instances to be attended
with much uncertainty, on account of the fact that Sinmw and Cosmw become discontinuous
when 7 is ©: and Fourier’s celebrated theorem, that any function whatever can be developed in
a series of Sines and Cosines of multiple arcs, I regard as being fallacious in all cases where
the coefficients do not converge to zero as 2 becomes ¢. As an instance, I have no doubt that
4 is not equal to1 + Cos # + Cos2w + Cos3v+...... for any value of w whatever. But this is
too wide a field to enter upon in this paper, the object of which is to shew that Sin ¢ and Cos
are not definite quantities, and that Sin(a@o), Cos(ae) are functions of a.

32. Perhaps it may be proper to add something in explanation of what is said in (26),

Sinazx Par ; : :
da, that it is such a function of a@ as is constant for ordinary

o
respecting the integral Hh =
0

values of a, and changes sign with a. This requires that a distinction should be allowed between
arithmetical values and symbolical forms; and such a distinction must be allowed, if any operation
© Sinaw

with respect to a@ is to be performed on the expression NR dw. An example will best

1)
explain what is meant.

In Fourier’s Theory of Heat, we find the equation

T

; = Cosy —3Cos 8y + +. Cos5y — ....0.

This equality is established (pp. 167—174) by a method which is remarkable for its exhibiting
no symptoms of the existence of failing cases: and hence it is with surprise we read soon after, that
the left-hand member changes its value when y is comprised between certain limits. Guided
by the investigation which Fourier gives of the sum of the series Cos y—4Cos3y + ...... we
could have had no suspicion that the result is erroneous in any case; yet it is manifestly erroneous

ef Tv 3a . . . *
when y lies between zi and a Hence the inference is plain, that the value . is not sym-

bolically correct, because it does not contain y, of which the proper form is obviously a function.
The author, at page 208, proves that

_, (2Cosy fs aed ae
4, tan (3) = e~* Cosy — Le-** Cos 3y + Le-** Cos Sy — verses

And consequently, admitting the propriety of putting #=0, we obtain

3, tan~' (200 Cosy) = Cosy — 4 Cos3y + 1 Cos 5y = ».00
Now from this it is obvious that 4 tan-’(2¢ Cosy) is numerically = . for non-critical values of

: Hs € ; . :
y, Whenever Cosy is positive; and equal to — rs numerically, whenever Cosy is negative.

MM2

268 Mr. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE, ETC.

It appears from this example that, as has been before remarked, a co, which for distinction I call
a symbolical ¢, is not to be confounded with co, a mere arithmetical infinity: for the former
ceases not to be a function of a.

tay f ® Sinawx ‘ 4 © Sine
In (Art. 26.) then when it is said that i - dx is not symbolically equal to ee dx,
0 et F x
the assertion is grounded upon this distinction between ac and o; and it is manifest that in

© Sinagr

this case, supported as it is by the example quoted from Fourier, ¢

0

dx is symboli-

« Sina

cally a function of a, while if dz is not a function of a. This distinction between a

0
and oo is of great importance in all definite integrals where the results are understood to be
symbolically exact; as they are always supposed to be when they are made use of in obtaining from
them other definite integrals by differentiation or integration with regard to parameters. It will be
very obyious to any one who examines the definite integrals which have been published, that many
of them have been obtained without sufficiently observing this caution with respect to symbolical
exactness.

S. EARNSHAW.

CaMBRIDGE,
November 9, 1844,

XX. On the Connexion between the Sciences of Mechanics and Geometry. By the
Rev. H. Goopwin, Fellow of Caius College, and of the Cambridge Philo-
sophical Society.

[Read February 10, 1845.]

1. IT is well known, that the first step in proving the elementary propositions of Mechanics
is usually to explain that for the purposes of demonstration forces are represented by straight
lines, and so simple a step does this appear to be, that it has been complained that students
frequently do not perceive that they have passed a distinct boundary-line in their transition from
Geometry to Mechanics. It becomes therefore a matter of interesting inquiry, what is the ground
of the connexion between the two sciences? is it merely conventional? or only partly so? or not
at all? Is the substitution of lines for forces to be looked upon as a mere ingenious device,
or has it such a natural basis in the reality of things, as to force itself in one form or another
on the mind of every one capable of appreciating the subject? This is the question which
I propose to examine.

2. Let it be observed then, that an indefinite straight line is merely the expression of the
idea of direction: the idea of direction is a pure idea capable of no simpler expression, and, as
I think, obviously not acquired from experience: no child ever walked from one point to another
by a roundabout path, until it discovered that one path was shorter than any other; there might
be a difficulty about understanding what was meant by a straight line lying evenly between its
two extreme points, but about the fact that you would go in one determinate direction from one
point if you wished to go to the other, there could be no doubt at all. I hold, therefore, that
the idea of direction is a pure idea, independent of all experience, and that all definitions of a
straight line are attempts, accompanied with more or less success, to give verbal expression to
this idea*.

And so when I draw a mark on paper which I call a straight line, this is a method of re-
presenting rudely to the eye a certain direction, it enables me to speak of that direction
intelligibly and to reason about it, the reasoning of course referring not to the mark on the
paper, but to the ideal line or direction of which that mark is the visible memorandum.

When we speak of a finite straight line, we limit the idea of mere direction by introducing
the new one of magnitude. The idea of magnitude is merely that of comparison of one
quantity with another, and a straight line of certain magnitude is represented by taking two points
on a given indefinite straight line, such that the distance between them is so many times greater
than the distance between two standard points.

Thus a finite straight line given in position is the expression of the combined pure ideas
of direction and magnitude ; and a mark on paper standing for such a line is the exhibition to the
eye of these two ideas.

And hence, further, we may say that all propositions concerning indefinite straight lines are
deductions from the pure idea of direction; all propositions concerning finite straight lines not
given in position are deductions from the pure idea of magnitude; and all those concerning finite
straight lines given in position are deductions from these two pure ideas combined.

* See Note (A).

270 Mr. GOODWIN, ON THE CONNEXION BETWEEN THE SCIENCES

We may put what has been said in other words by asserting that all properties of straight
lines are functions either of their direction, or their magnitude, or both; a straight line has no
other elements than these, and therefore every thing which is predicated of a straight line is
predicated simply in consequence of that straight line having a certain direction and a certain
magnitude.

3. Now from this point, I think we can see a simple road into Mechanical Science; for if
there be anything physical which depends on no other elements, than those of direction and
magnitude, there is no reason why a mark on a piece of paper should not stand for this physical
embodiment of the two ideas as well as for the geometrical: and further, if there be anything
physical, of which it can be predicated that it has no other elements than direction and magni-
tude, then all propositions which haye been proved for straight lines will have their corresponding
propositions, in fact will be true with a change of phraseology, in physics.

In devising a method therefore for representing to the eye the forces on which we reason in
Statics, the question’is not whether a force can be conveniently represented by an ideal straight
line, but whether a force has such qualities that the same representation which serves for demon-
strations respecting straight lines, will also serve for demonstrations respecting forces.

4, Now when we come to examine a Statical force, we find that it does involve, or rather it
is a physical expression of, those two ideas of direction and magnitude, and of no others. For
we measure a Statical force by the pressure which will counteract it; and what are the questions
as to the counteracting force? these two—in what direction it must be applied, and with what
intensity; it is clear that neither of these is sufficient without the other; for a particle left to
itself under the action of a force will move off in a certain determinate direction, and it is a truth
which requires no proof, but is purely axiomatic, that a force, however great, applied in any
other but the exactly reverse direction will not prevent motion ; and so likewise it is a self-evident
fact, that the counteracting pressure must be of a certain determinate magnitude and no other.
Thus, to a person who understands what I mean by the term Force, it will be apparent that the
only ideas involved, are those of direction and magnitude; any cause tending to produce motion
which involves any other element for its complete determination is not a Force, it may be called
so popularly, but jt is not included in the mathematical definition.

And it may be observed here, that as in Euclid, the definition given of a straight line, viz..
‘that it lies evenly between its extreme points,” is virtually superseded by the axiom, that “two
straight lines cannot inclose a space,” so in elementary books of Mechanics, although the definition
is given of a force that it is “any cause which produces or tends to produce motion,” yet the
fundamental proposition is usually made to depend on the axiom or fact (or whatever it is to be
called) that a force may be supposed to act at any point in its direction, which is the same thing
as saying, that if the magnitude be given the force depends on direction only.

When the science of Mechanics was first studied, the simple view of force which I have given
would, of course, not be immediately taken; the effect of force would probably be supposed to
depend on other circumstances; but this is a matter of no consequence: the question is merely.
what we mean by force now, and what it is supposed to mean in all mechanical treatises; and
it signifies not whether we start with the idea of a cause of tendency to motion involving the ideas
of direction and magnitude only, and call the embodiment of that idea force by definition, or
whether we examine the world we live in, and shew that such are the elements and the only
elements of force.

5. Let it be granted then that the only ideas involved in that of force, are those of direction
and magnitude, and we come to the case (already spoken of by anticipation) of a thing physical,
involving exactly the same ideas as the straight line in Geometry; and we therefore lay down
this proposition, that every theorem regarding straight lines will have its fellow in Mechanics, that

OF MECHANICS AND GEOMETRY. 271

the theorems of the one science can be translated into the language of the other, and that the
demonstration belonging to figures in which the marks represent straight lines will apply pre-
cisely as well to similar figures in which the marks represent forces; for in both cases the
representation must be conventional: no inkmark can be a straight line, and no proposition
concerning straight lines can be true of the inkmarks which represent them, and though it
requires a greater abstraction of the mind to speak of an inkmark as a force, yet the speaking
of it as a straight line is certainly as really conventional, and the proper utility of the figures
in both cases is that they assist the mind artificially in drawing deductions from the pure
ideas of direction and magnitude.

Velocity is another instance of a thing physical involving the ideas of direction and magnitude
only, and of which therefore it may at once be predicated that the propositions respecting the
straight line refer to it mutatis mutandis.

6. When it is said that every proposition respecting the straight line will have its fellow
respecting force, it is of course equally true that each proposition in Mechanics will have its
fellow in Geometry, and it will be asked, what proposition in Geometry corresponds to the parallelo-
gram, or rather the triangle of forces: to which I reply, that when two
lines AB, BC are given in position and magnitude, the straight line
joining the points A and C will be as strictly their geometrical
resultant, as the force represented by AC will be the resultant of
the forces represented by AB, BC: for by speaking of the resultant
of two lines we necessarily imply that the two lines are given to 4 ————¥y
determine some third object, and that object must be a straight
line, since the resultant of two things of the same kind must be of the same kind with those
which produce it, and if there be any line which is to be considered as the resultant of AB, BC
it must be AC, since this is the only new line whose position and magnitude is in any way
whatever determined by the positions and magnitudes of 4B and BC. If therefore we mean
by the resultant of two straight lines given in position the straight line which is determined
in magnitude and position by those straight lines, and this seems the most obvious meaning to
give to the term resultant, then AC is the resultant of AB and BC.

The proposition of finding the resultant of two straight lines given in position may be
generalized into that of finding the resultant of any number of straight lines forming an imperfect
polygon. For if all the sides of a polygon be given except one, then that one will be the
resultant of all the rest, inasmuch as it is the only new line whose position and magnitude
becomes determinate in virtue of the other sides being given. It may be said that the extremity
of one of the last sides may be joined with one of the angular points, and that thus some
other line will be determined, but the obvious answer is, that this will not employ ali the data,
and that the line so determined will be the resultant of all those which are really made use of.
In fact, a straight line may be given just as really, though not so directly, by giving in position
all the other sides of a polygon of which this straight line forms the last; to give those other
sides is, I say, precisely the same thing in fact as to give the line itself.

Conversely, a straight line may be considered as the resultant of any system of straight lines
which with it form a polygon; and also in such polygon any one side may be called the
resultant of all the rest; if two be missing, they cannot be replaced; but if one only, then is
that missing one just as fixed and determinate as if it were represented as part of the polygon.

In speaking of the direction of lines, it is of course necessary to distinguish between a line
AB, and a line BA, the direction of the one being considered

exactly the reverse of that of the other. Thus, in the pre- <4
ceding investigation AC is the resultant of AB, BC, not of ;
BA, BC: the resultant of those latter lines would be found by

—

taking AD parallel and equal to BC: then BD would be the
resultant of BA and BC.

2c

Cc

B

272 Mr. GOODWIN, ON THE CONNEXION BETWEEN THE SCIENCES

7. The principle of the third side of a triangle being the resultant of the other two may
be applied to demonstrate certain propositions in plane Geometry, which I here introduce for
illustration’s sake.

It may be shewn from this principle, that the lines drawn from the bisections of the sides
of a triangle perpendicular to the sides will pass through the same point. For suppose we
bisect two of the sides, and draw lines perpendicular to them, (it is of course necessary to
bisect the sides, because the middle point of a line is the only one which is similarly related to
the two extremities), then these indefinite lines determine a new point, viz. the point of inter-
section; now if we perform the same operation on the third side, the result must be such that
no new geometrical element is determined, since everything which is functional of the third side
is already implicitly involved in the knowledge of the other two; therefore this third line must
pass through the point of intersection of the other two, since if it did not it would determine
two new points, which, by what has just been said, is impossible.

The same reasoning applies to the propositions that the lines bisecting the angles of a triangle
pass through the same point; and that the lines joining the angular points with the bisections
of the sides pass through the same point,

And, I may remark, that we have here the explanation of the fact, that some propositions
in pure Geometry admit of simpler proof by referring to mechanical considerations than by the
ordinary geometrical methods; as for example the last proposition of those first cited finds its
solution at once in the property of the center of gravity of a plane triangle.

8. Taking the view which I have endeavoured to explain of these resultants, it will be
obvious how close the analogy is between this case and that of forces ;
for if AB and BC represent two forces, then AC we know represents
their resultant, and in general if two sides of a triangle represent
two forces their resultant is given by the third, and still more generally
if the sides of an imperfect polygon represent forces their resultant
is given by the last side. Now the same thing holds in this case
which was true in the case of Geometry, viz. that if AB, BC be given in position and
magnitude, the only third term determined is AC; and therefore if AB, BC represent two
forces, the magnitude and direction of the force AC is at once determined, but this can be
asserted of no other. Now I do not say that this could be considered as a proper proof of
the triangle of forces; but I do think that it is a way of considering the subject which, by
careful thought, will lead to the intuitive perception of the truth of the proposition. It would
be impossible to admit this as the only proof that the force AC would balance the two AB, BC,
but at least it shews that AC is related to 4B and BC in a manner in which no other force
is related, that it is at once determined by them, so that to give them is to give it, and that
this can be predicated in the same sense of no other force; and from this it seems possible by
degrees to arrive at an intuitive perception of the truth that AC is in fact the resultant of
AB, BC. And after all this is the point at which we should endeavour to arrive; the funda-

c

mental proposition in mechanics ought not to have a merely artificial basis, and to be such that

the mind rather concedes it because it cannot deny it, than sees it to be true; and I cannot feel
a doubt but that there must be some method of viewing the subject, which if we adopt, the
fundamental propositions of Mechanics will gradually grow into as perfect axiomatic clearness as
do the simple propositions of Geometry *.

9. ‘To illustrate this point by contrast, let us for a moment consider the proof which is
frequently given in elementary treatises of the triangle of forces, I mean that which is due to

* See Note (B).

OF MECHANICS AND GEOMETRY. 273

Duchayla. Now this proof is certainly convincing; that is to say, it is not possible to point
out any flaw in the steps of demonstration, but for perswading the intellect it seems to have no
kind of fitness. The proof is essentially artificial, and is based on a simple case of composition
of forces which seems very insufficient to suggest, as it is pretended that it does, the result
sought. The character of the proof seems, if I may so express myself, to be that of cunning
rather than honest argument; and yet I think that, however unsatisfactory the proof may appear
in this light, we must feel convinced that, supposing it accurate as we do, there must be a meaning
and principle about it at bottom, and that these are only smothered and obscured by the artificial
contrivances of the demonstration, This I think we shall find to be really the case if we
examine the proof in the light of the preceding observations. The first part of the proof seems
to involve very faintly the idea of force; the only principle introduced being this—that a force
may be applied at any point in its direction; and thus the distinctness of the proposition as a
mechanical one seems rather obscured, but this difficulty of course vanishes if this first part of
the proposition be what I should call a proposition in the science of Pure Direction; the proof
involves the idea of force only indirectly, and this is exactly what ought to be the case if the
proposition be true of several things, of which force is one: it is equally true of velocity, for
example: force is an embodiment of the pure idea of direction, and therefore all theorems of
pure direction will belong to force, not singly, but to it in common with all other embodiments
of the same idea. In fact, the first portion of Duchayla’s proof appears to be simply this, given
two straight lines in position to ascertain the direction which will be determined by them.

But direction is not the only idea involved in force: there is magnitude as well, and there-
fore there is a second portion of the proof we are considering, in which it is shewn that, allowing the
triangle of forces so far as direction is concerned, that part which regards magnitude necessarily
follows; the extreme simplicity of this part of the proof shews how intimate the connexion must be
between the two parts of the proposition, a connexion which I think we should not have been led to
expect from anything occurring in the proof itself, for, although the fact that the direction of
the resultant of two equal forces will bisect the angle between them is taken as suggestive of
the general law of direction, there is not a shadow of a hint that in this simple case the law will hold
as respects magnitude: so that a very remarkable proposition is proved by a mere artifice without
apparently the least reason in the nature of things why we should anticipate the result. But if we
consider the proposition from the same point of view as that from which we regarded the question
of the resultant of two straight lines, we shall see that there is a necessary connexion between
the two propositions, I mean those respecting direction and magnitude; for when we had two
lines 4B, BC given, the resultant AC became at once known both in direction and magnitude ; the
two things were co-ordinate, in fact, as this word suggests, they were merely two new co-ordinates
of C which became known from the two given co-ordinates 4B, and BC.

10. On the whole, therefore, I would urge that the proposition which we call the triangle of
forces is a result of the combination of the pure ideas of direction and magnitude, and will therefore
be true in some sense of all concrete existences which are embodiments of these two ideas and
no other: and therefore I explain the fact of the unmechanical character of the proof we have been
considering by observing, that the proposition is more general than the merely mechanical one,
includes in fact the triangle of forces, the triangle of lines, the triangle of velocities, the triangle
of couples, and perhaps other cognate propositions.

11. This subject will, I think, receive further elucidation as follows;

If / represent any quantity in magnitude only ; then if the quantity depend on direction also, it
will be necessary to assign the direction in which / is to be measured; but if this be done, it is
possible to affect 1 by a symbol or sign of affection, which shall indicate for itself the direction
in which it is measured, This symbol it is well known is e?¥—, which is such that if 7 represent a

You, VIII, Parr III, Nw

274 Mr. GOODWIN, ON THE CONNEXION BETWEEN THE SCIENCES

line as to magnitude only, then Je®¥-1 will represent the same line measured in a direction
making an angle @ with some fixed line.

Now if ABC be a right-angled triangle and BAC=0, and AB=/ in

magnitude, 4 °
then AB = le®\= =1c0s0 +\/ —11sin8 ime
= A
SIGE as Te net |
or, if we omit \/ —1 which is a sign of affection, Cc |
AB = AC + BC.

We may therefore say, that regard being had to direction as well as magnitude, AB the
hypothenuse is the swm of the two sides AC and BC, or perhaps it would be more distinct
to say that the hypothenuse is the equivalent of the sides, that is to say, that considered as
embodiments of the ideas of direction and magnitude one is equivalent to the other; if the
direction be disregarded it would be absurd to say that 4B = AC + BC, and in like manner, if
direction only be considered, there is no equivalence between the hypothenuse and sides, but com-
bining the two there is an equivalence, and one may be substituted for the other in all sciences which
are developements of these two fundamental ideas.

I may remark further, that we may consider the symbol Je®¥- as the type of the sciences
depending on these ideas, or rather one may say, that the symbol Ze®¥=! is the germ from which
may be evolved the fundamental principles of these sciences.

12. One more observation may be made on this symbolical representation. The symbol
le ¥-1 is as we know equivalent to the expression 7 cos 0 + V/ —11sin@, and therefore if this symbol
were given to a person as the representation of force, it must at once strike him that the fundamental
property of force was that of being made up of two other forces, which we will call as usual
its resolved parts. Now what I would wish to ‘observe, is, that this connexion supposed to be
suggested by the symbolical formula is precisely that which would probably be suggested to the
mind when it first began to engage itself with mechanical studies.

For suppose we have a force tending to draw a particle P in any direction OP; then if we wish
to examine the nature of this force, and determine its laws, the obvious = s
artifice would seem to be to constrain the particle in various ways, and 7
reason as to the result. Suppose, for instance, a plane drawn perpendi- /
cular to OP and indefinitely near the particle P, then it is manifest that vat
the particle will not move at all, this is a point which no one will
question, and therefore we arrive at one property of force, namely, that
it produces no effect in a direction perpendicular to its own. But,
suppose we incline the plane at some angle @ to OP, then motion
will ensue if not checked, and the question is, what force acting along the plane will be just
sufficient to check motion? To determine this, take any point O in the direction OP and
draw OP’ perpendicular to the constraining plane, then it is easy to see that whatever relation
the line OP has to the original force, the same relation has P’P to the resolved part in the direction
PP’; to make this apparent, I shall call a plane drawn through a point in the direction in which a
particle has a tendency to move and perpendicular to that direction the impossible plane, and
then the definition of OP will be, that it is the distance between the impossible planes corresponding
to P and O: now suppose any two other parallel planes to be drawn through P and Q, and let
them be perpendicular to PP’, then P’P is the distance between these impossible planes, as OP was
between the two former. This being the case, it will be allowed that if OP represents the original
force, P’P will represent the resolved part in the direction PP’, that is, the resolved part will

a)

OF MECHANICS AND GEOMETRY. 275

be the original force reduced in the proportion of PO: PP’ or of 1: cos 9. The question remaining
would be, whether a force could properly be represented by the distance between two impossible
planes, a question which might perhaps be answered satisfactorily and without much difficulty if
we consider that a finite line taken in the direction of a force will represent the two fundamental
properties of force, namely, magnitude and direction. But if it appear in this way that PP’
represents the effective part of the force acting on P it will be seen that in like manner P’O repre-
sents the ineffective or destroyed part. And therefore the result of the artifice of constraining
the particle P would be that when a force 7 acts on a particle, which is constrained by a
plane inclined to the direction of the force at an angle 0, the force is equivalent to a force
Zcos@ which is effective and a force /sin@ which is destroyed, or a force 1cos@ in the direction
in which motion is possible and a force /sin@ in the impossible plane. And this is exactly what
would result from considering foree under the light of the formula

Ze®9V> = Jcos 9 +4/ —1Isin 0.

13. This symbolical representation*, though depending on refined principles, is nevertheless,
I apprehend, valuable in the discussion of the question before us, because it is generally admitted
as a complete method of geometrical representation, and those who study the question must
perceive that its complete character is founded on something much deeper than a mere symbolical
artifice, inasmuch as it expresses the equivalence between a line, considered in its direction and
magnitude, and the two rectangular projections of that line. Now it has been the intention of
what immediately precedes to point out the corresponding necessary connexion between a force
and its resolved parts, and the perfect applicability of the same symbolical method to the two
cases tends, it is presumed, to strengthen the characteristic view of this paper, viz., the essential
identity of the Geometrical and Mechanical Sciences, considered as developements of the same
combined fundamental ideas.

14, The preceding remarks have been wholly devoted to the consideration of force as acting
on a single particle; it was my intention to have attempted a discussion of the case of a system of
forces acting on a rigid body, and to have shewn how the science of Mechanics diverges from that
of Geometry, by the introduction of this new idea of Rigidity; but perhaps what has been already
said will be sufficient to put in a clear light the fundamental views which it is my desire to
explain: my belief is, that these views contain the shadows at least of important truth, and
that they will be seen to do so by any one who will devote attention to the subject. The
great question is, what are the fundamental ideas of Elementary Mechanics, and what of Geometry ?
Are they the same, or are they cognate, or are they altogether distinct? If the last, then
the resemblance between certain demonstrations and propositions in the two sciences is a curious
and unexplained fact; but if the second or the first, then the explanation is obvious. And if
the relation of the two sciences be such as I have represented it, then it seems to me to be most
important that it should be recognized, and that for more reasons than one; first, this view
connects two streams of truth, usually I believe considered distinct, and traces them to one
fountain head, and this is an important simplification, in the same sense and for the same
reason that it is an important simplification to trace two phenomena to the same physical cause ;
but, again, the foundation of geometrical truth is a matter of less question in general than that
of mechanical; it is I suppose universally allowed that the propositions in pure Geometry are
as they are, because they could not be otherwise, that they are necessary truths in every sense
in which truths can be necessary, but there is not, I apprehend, such clearness of thought prevalent
respecting mechanical truth, it is difficult to make out from the ordinary books on the subject

* See Note (C),
NN2

276 Mr. GOODWIN, ON THE CONNEXION BETWEEN THE SCIENCES

what the writer’s belief is respecting the nature of the truths which he is developing; now this
point is entirely resolved if it is shewn that the principles of Mechanics are identical with those
of Geometry, that the two sciences not only have certain analogies, but are in essence identical
as being two developements of the selfsame ideas, and hence, if this be true, we see at once
the necessary character of the truths of Mechanics, or at least we shew them to stand on the
same ground as others which are supposed to be admitted as necessary. I will close this paper
by saying, that although I am well aware that what I have said in favour of the views propounded
may not with many appear to amount to demonstration, and indeed perhaps demonstration in
such a subject is not altogether possible, yet I am persuaded of their fundamental correctness
by this consideration as much as by any, viz., they do seem to point out the road to absolute
intuition of truth, they seem to mark out a method of thought according to which the elementary
truths of Mechanics will present themselves gradually with axiomatic clearness. And certainly,
whether this method be true or not, it cannot I think be doubted by any one who has reflected
on the foundations of truth, that this is the natural course, viz., that all demonstrations gradually
merge in intuition, and that all human knowledge converges towards that absolute intuition which is
the attribute of the divine mind.

NOTES.

Note (A), page 269.

Tue word direction appears to be the best abstract word for expressing the idea which is intended to
be embodied in the concrete form of a straight line; the evil of concrete terms is that they appear to
assign physical existence to that which can have none, and by thus leading away the mind from the
true idea tend to prevent the intuition of geometrical truth. If the idea intended to be embodied in the
terms point, straight line, and angle be conceived in their abstract form, the simple propositions respecting
them at once assume the character of axiomatic truth. I will here put down what appear to me to be the
best abstract terms for expressing these three geometrical elements—

1. A point = Position.
2. A straight line = Direction. 4
3. An angle = Inclination of directions.
I will illustrate the intuitizing force of these terms by applying them to the 7
doctrine of parallels. The idea of parallelism is that of identity of direction with- ~
out identity of position; and from this it is evident that a straight line CD falling on
=

two parallel lines 4B. A’B’ makes the alternate angles equal; for since the question 4° 7
is one of direction only, whatever is predicated of the line AB may be predicated /
of the line A’B’, since they differ in position only and not in direction. D

Nore (B), page 272.

The proposition in pure Geometry which seems more than any other connected with Mechanics is
Euclid 1. 32, and it will be worth while to point out the self-evidence of this proposition both for its
own sake, and also from the assistance it will afford in the intuition of the cognate mechanical proposition.

OF MECHANICS AND GEOMETRY. 277

I observe first, that Euclid’s last corollary seems to be the easiest proposition to grasp, and will be
admitted without formal demonstration as soon as its meaning is apprehended. For if we consider the
changes of direction which a straight line undergoes by successively coinciding with the sides of the polygon
it is clear that when it has been made to coincide with all in succession, it will at last come into its original
position, but a line which has revolved and come into its original position must have described four right
angles; whence the proposition is manifest. From this of course the first corollary and the proposition
itself immediately follow.

It appears therefore that Euclid 1. 32. is only a form of the self-evident proposition, that a straight line
being made to deviate from its original direction cannot assume it again until it has deviated through four
right angles.

Now the condition of forces being such as will produce equilibrium, is simply that the lines respecting
them shall form a polygon. And this proposition is I believe only an expression of the fact, that two forces
cannot counteract each other unless they act in the same straight line, or, to express myself more in con-
formity with the geometrical proposition, that if a force has been made to change its direction it cannot
produce the same effect as before unless its deviation has been through four right angles; but this thought
T have not yet fully developed.

Norte (C), page 275.

It may be well to remark here that the symbol reV-1 being the complete expression of magnitude and
direction is also the complete expression of linear and angular distance, if r be measured from a fixed point,

and 6 from a fixed straight line: consequently re?V-! may be taken as the complete position-index of a
point, or of a physical particle, in a given plane.

“ d
If we consider a particle whose position is changing with the time (t), then the symbol qr will
express the complete variation of the position-index, and will therefore give the magnitude and direction of
its velocity.

In like manner the symbol = .ré’-1 will give the complete variation of the velocity, and will therefore

be the symbol for the accelerating force in both magnitude and direction.
Now suppose the particle to be in motion under the action of any forces, the complete expressions for
which are P.c?¥-1, P’.e¥¥-1, &c.: then if M be the mass of the particle we shall have for its equation

of motion
MA. reN-1 = PANAS Pets...
This equation I have given merely in illustration of the principles of the preceding memoir, but in the
Cambridge Mathematical Journal, (Vol. 1v. p. 177.) I have shewn that the symbol re9V=1 may be applied

to mechanical investigations with considerable practical convenience.

H. GOODWIN.

XXI. On the Pure Science of Magnitude and Direction. By the Rev. H. Goovwin,
Fellow of Caius College, and of the Cambridge Philosophical Society.

[Read May 12, 1845.]

In a former Memoir, I have endeavoured to point out the @ priori and necessary character of the
fundamental proposition in Mechanics, by connecting it with the propositions of Geometry, and so
bringing the demonstrative character of the two sciences into one and the same point of view. I
there pointed out that the only elements of Force are Magnitude and Direction, and therefore that
the only simple ideas of which the term Force is the expression are those of Magnitude and
Direction, and hence, that all propositions respecting Force ought to follow demonstratively and
perhaps intuitively from the possession of those two ideas combined, even as the propositions
respecting straight lines arise necessarily from the same two ideas. In the course of that Memoir,
I spoke of such a Science as that of pwre direction, which should include within itself the Sciences
of Geometry or rather of Position, of Kinematics, of Mechanics, and possibly other Sciences ;
it is the design of the present Memoir, to attempt to establish the fundamental Proposition of such
a Science, or, as perhaps it may be more properly called, the pure Science of Magnitude and
Direction.

1. The fundamental problem will be, to determine the combined effect of any number of
causes, the magnitude and direction of each of which is given. It will be seen that this statement
is perfectly general; for a line given in a certain direction may be looked upon as cawse, the point
in space determined by its extremity as effect, or if two lines be given, having an extremity in
common, the line joining the other extremities which is thus determined may be regarded as effect;
so likewise, if a particle be animated by two simultaneous velocities, they may be looked upon as
causes, the resultant velocity as effect; and if a particle be acted upon by two forces, the resultant
pressure will be the effect which results from the two given forces as causes; and hence it will
appear, that the fundamental problem is to find the combined effect of any number of given causes.

2. Now, if the direction in which all the causes acted were the same, it is clear that the
combined effect would be found by mere addition of the quantitative symbols which measure
their respective effects; the only postulate here involved is, that two causes do not modify each
other’s effects, a postulate which is of the nature of an axiom, and which merely expresses such
truths as these, that if a point be taken in a straight line at a distance (a) from a fixed origin,
and another point at a distance (b) from the former, then the distance of the point last deter-
mined from the origin will be a+6, or again, that if there be two forces, one of which can
lift a weight P, and another a weight Q, then the two together can lift a weight P + Q.

8. Hence, when the direction of a number of causes is the same, the process of addition
serves to determine their combined effects, but when the directions are different, it will be necessary
to determine according to what law variation of direction modifies the effect of a cause; in other
words, suppose we take P as the quantitative symbol of the effect of a certain cause in a given
direction, what will be the symbol for the effect of a cause of equal intensity whose direction
makes an angle @ with the given direction ?

Mr. GOODWIN, ON THE PURE SCIENCE OF MAGNITUDE AND DIRECTION. 279

Now, it may be assumed, that the effect of the oblique cause can be measured by a symbol
of the form Pf(9), where f(@) is a modifying function, which would be 1 if @ were zero
and whose general form must be determined. This assumption appears admissible, because if
there be a symbolical expression for the effect of an oblique cause, it can be of no form more
general than that assumed, and if there be no such symbol, this will appear by the impossibility
of determining the form of f(@).

4. To determine the form of the function f, I observe, that the fundamental law of such
causes as we are considering is, that the exact reverse of any cause whose magnitude and direc-
tion are given is one of equal magnitude and exactly opposite direction; so that, if we denote
opposite affections by + and —, then +P must change into — P, while @ increases from 0 to 7:
moreover, the change of P, for rather of Pf(@)}, as @ varies continuously, must manifestly be
continuous, and not only continuous but wniform, that is, the rate at which the affection changes
from + to — must be the same at all stages of the change, since there is no reason why the
change should be more rapid for one value of @ than for another :—speaking symbolically, it
f(@+a)-f(@)

f()
This law, to which f(@) is subject, and which flows at once from the pure idea of direction,
is sufficient to determine the form of f. For suppose the angle m to be divided into » equal
parts; then, if the direction vary through one of them, the symbol representing the effect of

may be said that must be the same for all values of @ if a be a given quantity.

the oblique cause will be pf (=); if it vary through two such divisions, the symbol becomes
7

1\ Te alls Qa eee
pf (=\7(Z), or pir (z)\ ; (it also becomes Pf. P= ), and so on; and when the direction has

\n
varied through m angles each equal to -, the symbol becomes Pl (2) y. but by what has just

been said this symbol must represent the exact reverse of + P, and must therefore be = — P;
hence we have

P\s(=\\' = - P,
bey.

1
v = Tr Tr
r'—) =(— 1)” =cos— — 1)/ sin —
ft ( ) cos = ag 1) sin

Tv
if Fant;
and if we put 5 6

8
f(8) = (- 1)? = cos 8 + (= 1)3 sin O...... 10.000 0e0eee (A).
It will be observed, that ” may be made as large as we please, and therefore, the condition will

be satisfied of @ varying continuously from 0 to 7; also the change of f is not only continuous

é (0 + a) — f(b)

but uniform, for it is clear that the expression (— 1)" satisfies the condition that “-~———~ *~

f(9)
shall be the same for all values of @; and this follows necessarily from the mode of determining
J, without assuming that f(a) = — 1, for the fundamental law of variation of f is expressed by the

equation

r(E)l 10),

n

or, (8) = $f (B)}?,

280 Mr. GOODWIN, ON THE PURE SCIENCE

where 3 is any constant angle, the value of which is indifferent so far as the wniformity of
variation of f(@) is concerned; and it is clear that the form of f as thus determined satisfies
the condition of uniform variation corresponding to the continuous variation of 0.

Hence, therefore, the formula (4) may be regarded as an expression for the law, according
to which change of direction affects any cause, which depends on magnitude and direction only,
and which varies uniformly to its direct opposite while its direction varies continuously to the

exactly opposite direction.
5. I shall now interpret the symbolical equation (4); we have
F(0) = cos 0 + (— 1) sin 0;
hence, F(0) =1;

f(F) = 9s

2

and therefore the equation
Pf(0) = Pcos@ + (— 1)! Psin@

may be written thus,

P.f (0) = Pcos@.f (0) + Psin.f (J) ovsssesseane oeenn (BD:

From this form it may be concluded, that the equation expresses this fact, that the effect of an
oblique cause is measured by that of two whose directions are at right angles to each other,
the direction of one being the original direction that of the other the perpendicular to it, and the
intensity of the former being measured by P cos@ the intensity of the latter by P sin@; and
this comes to the same thing as saying that any cause is equivalent to two other causes, the
directions of whose action are perpendicular to each other, and whose intensities are measured
by Pcos@ and Psin@, @ being the angle which the direction of Pcos@ makes with the original
direction.

6. The proposition which I have just been endeavouring to establish is the fundamental
one of the pure science of Magnitude and Direction, and may be called the principle of the
resolution of causes; from it may be deduced with ease rules for the calculation of the resultant
effect of any number of causes, since we have only to resolve them in the same directions and
then add the effects; and if the demonstration given be free from solid objection, we shall have
established a proposition which contains within itself the theorems of position, the theorems of
the composition of forces, of moments, and of velocity both linear and angular.

7. As it may appear improbable that so general a proposition as that which I am con- °

sidering should be capable of proof without reference to particular cases, I am anxious to examine
the objections which may be made to the proof just given. But before doing so, I would observe,
that it does not seem unreasonable to suppose @ priori that it would be possible to establish
a formula indicating the variation of intensity of a number of causes, which have been all brought
under the same definition by saying that they are such as depend solely on intensity and direction ;
we may consider it as certain that a law which applies to one will apply to all, and the only question
is, whether we can establish such a law by reasoning which shall apply equally to all causes
comprehended in one definition. Now it is allowed on all hands that the symbols + and — are
proper symbols for exactly opposite affections, and therefore it cannot be objected to, that we
should consider + P and — P as symbols of two causes of equal intensity but exactly opposite in
direction; the question simply is, according to what law can + P vary continuously and uniformly

_—-

OF MAGNITUDE AND DIRECTION. 281

into — P while the angle of direction varies from 0 to two right angles, that is, from the original
direction to the exact opposite. The answer given by the preceding investigation is, that the symbol

@
(- 1)* is that by which P is to be affected ; the equivalence of this symbol to cos 6 + (— 1)? sin 6
is a proposition of pure analysis, and it must therefore be allowed that cos 9 + (— 1)! sin @ is the

0
true sign of affection for P, if it be granted that (— 1)* is the correct sign of affection; in fact,
8

the symbol (- 1)* expresses the continuous passage of a quantity from + to —, or of an affection
to the exact opposite, it is an algebraical fact that the symbol cos @ + (- 1)! sin @ will express
the same. If therefore there be any flaw in the reasoning it.must be in that part by which the

D

form of the symbol (— 1)7 is established ; but on reviewing that part it will be seen, that the only
assumption is that an oblique cause can be symbolized by an expression of the form Pf (6), where
P is the magnitude of the direct cause; for if it be granted that Pf(@) is the expression for
an angle @, it seems evident that P{f(6)}? will be the expression when the angle is doubled,
since P{ f (@)}* is derived from Pf (6) according to the same rule by which Pf (@) is derived from
P; the only question therefore is as to the legitimacy of the assumption of the symbol Pf (6),
and this is, I think, justified by the consideration that the expression for the oblique cause, if
there be one, can involve no other quantity of the same kind as P, and therefore P must occur
as a factor of the expression; the form could not, for instance, be Pf(@) + p(@), since (0)
would be a quantity incommensurable with P. But if this step be granted, the conclusion that

@

f(@) = (- 1)* seems inevitable, and the equation f(@) = cos@ + (—1)!sin@ is consequently true,
being algebraically deducible from the former.

8. Supposing then the demonstration free from real objection, we may regard the formula
@

Sf (0) = (- 1) = cos 6 + (—1)3 sin @......(A)

as expressing the law, according to which a cause depending solely on magnitude and direction
will vary to its exact opposite, a law true in the nature of things and which will only require
to be interpreted in the case of different causes which come under the definition; the question
will arise, can the formula be interpreted in a manner free from objection, at least, can an
interpretation which can be depended upon, be put on a symbolical formula from @ priori
considerations? I have already transformed the equation (A) into the equation (B), and have
made use of that form of the equation to give this interpretation, that the effect of an oblique cause
is measured by that of two whose directions are at right angles to each other, the direction of
one being the original direction that of the other the perpendicular to it, and the intensity of
the former being measured by P cos @ that of the latter by P sin 8. Now perhaps this inter-
pretation may appear doubtful, when we consider that the sign + connects two quantities which
express causes acting not in the same direction, and whose effect cannot therefore be summed
according to the usual rule; but I think this difficulty will disappear when we consider that the
possibility of resolution of a cause into two components is an obvious truth which may be seen
independently, that is, it is in general possible to find two causes in given directions, whose
effect shall be the same as that of a given cause, and the only peculiarity of the case in which
the directions of these causes are at right angles, is that the effects are entirely independent of
each other; for it will be observed that the continuous change of a cause with its exact opposite
necessarily introduces the idea of an impossible plane, or a plane in which the cause produces
no effect whatever; for it is clear that a plane equally inclined to the primitive direction and
the exact opposite of that direction will be a plane of indifference, or one in which the cause
produces no effect ; it may be seen therefore, without reference to the equation (B), that an oblique
cause may be supposed to be the resultant of two components, one in the original direction,

Vou. VIII. Parr III. Oo

282 Mr. GOODWIN, ON THE PURE SCIENCE

the other in the plane perpendicular to it or the impossible plane, and this being the case, all
that is done by equation (B) is to assign the relative magnitudes of the two components. We
have, in fact, these two things known respecting the oblique cause which we denote by Pf (6),
first, that

P.f(0) = Pcos6.f(0) + Psind.f (=) ee

and, secondly, that the oblique cause may be supposed to result from two component causes,
for one of which @ = 0 and for the other 0 = = and putting these two things together, there

can, I conceive, be no doubt as to the conclusion that these components are represented by P cos @
and P sin @ respectively. I am not saying that the auxiliary consideration just used is really
necessary for the interpretation of the equation (B), for I am inclined to believe that the generality
of symbolical interpretation would justify us at once in construing the equation thus:—the effect
of P acting at an angle 6 = the effect of P cos @ acting directly, combined with the effect of
P sin @ acting at right angles to the original direction; but at least the objection, if there be
one, seems removed by the consideration adduced, and that is my reason for adducing it.

9. On the whole, I would submit that the preceding investigation not only is free from
solid objection, but is in fact the true mode of viewing the subject; because it rests upon the
leading idea of a uniform continuous passage of a cause from + to —, while its direction varies
continuously. And if it be objected, that physical laws cannot be conceived of as the results
of symbolical equations, it is to be answered that this is exactly the advantage of this mode
of viewing the subject, that it shews that such laws as that of the composition of forces are
not physical laws, in the sense of being laws known by experience or by induction from
observation, but are necessary laws in the most exact sense of the word: there is nothing more
incredible in the fact of Demoivre’s formula containing the laws of Mechanics, than in that of
its containing the laws of Space, and it is as credible that it should be capable of proof from
that formula, that three forces are in equilibrium when they are each proportional to the sine
of the angle contained between the other two, as that the sides of a triangle are proportional
to the opposite sides. ‘The fact of our making these conclusions depend on the interpretation
of symbols is in the present state of analysis no objection at all, and it may well be supposed
that some such method would be necessary in order to bring into one investigation subjects
at first sight apparently so distinct as the laws of space and the laws of equilibrium of forces.

10. And I think it cannot be said that the method adopted in this paper is so artificial as those
which are sometimes applied to what are called fwnctional proofs of the rule for the composition
of forces; for although the quantity (- 1) or 4/1 enters into the investigation, still it is to
be remembered, (and I wish to lay great stress upon this,) that this quantity is not introduced
by any principles peculiar to this paper, it enters by mathematical necessity and must be inter-

preted; the only equation which it is incumbent upon me to prove is the equation f(@) = (- 1s,
and if this be established, the remainder necessarily follows. Indeed, so far from this method
being of an artificial and therefore incomplete kind, I would venture to question whether the
unsatisfactory character of some functional proofs of the law of composition of forces, and the
extremely complicated nature of all, may not arise from the oversight of the fact that a function
exists, which will express an oblique force in its totality and not only so far as it is effective.
On this subject, however, I will not enlarge, but only remark that it seems to me a point of
great beauty that a symbol, of such peculiar form as cos 0 + (-—1)} sin 0, which meets us at
every turn in analysis, should be the complete expression of a law by which all nature is
governed.

OF MAGNITUDE AND DIRECTION. 2853

11. I have now concluded all that I wish to say on the principal subject of this paper,
but before bringing it to an end I am desirous of making some observations on the general
question of the transition of quantities from the + to the — affection, which will, I believe, illus-
trate my general design.

That design may be stated to have been, to shew that there is in the nature of things
one law according to which causes, which depend solely on magnitude and direction, vary with
their obliquity from a given direction to the exact opposite, and according to which the cause
P varies till it becomes — P. Now in considering the general case of the passage of a quantity
from the + to the — affection, it is to be observed, that if the quantity be necessarily of one
dimension, as time for example, then future time being denoted by + ¢, past time will be denoted
by —7#, and ¢ will vary from + to — by simple diminution and passage through zero; in this
case the sign »/—1 can have no place as a sign of affection; I believe there is no conceivable
interpretation to be put upon #\/-—1. And in like manner if distance be measured along a
fixed axis the variation from + to — is by simple diminution; but, if space be considered in

two dimensions so that a line may assume all oblique positions in varying from + to —, then
a
the symbol (— 1)” indicates the law according to which the change from + to — takes place.

Here then are two laws according to which the affection of a quantity may be reversed, and
there may possibly be others, and probably instances might be found in which such change is
abrupt not continuous; for instance, Dr. Peacock illustrates the properties of 4/—1 by saying*,
that supposing +a to represent property possessed, and —a to represent debt, then \/—1.a
may represent property deposited “ which admits of similar relations when considered as property
possessed and property owed by another person;” it must however, I think, be admitted, that
the use of the symbol pf ail in this case is conventional in a sense in which it is not when
applied to lines or forces, and it may be doubted whether the symbol so used can be applied
to the practical purposes of investigation; and indeed, if I might hazard an opinion, I should
hold it probable that the symbol \/—1 can only then be successfully used when it expresses
a particular stage in the continuous change of affection from + to —.

12. It is not difficult to devise laws, according to which a quantity may change from +
to —, other than those which have been specified. Suppose for example f(@) to represent
the sign of affection for a quantity P, and suppose

8

Pf (0) = P(- 1)";
this form satisfies the condition that f(@)=1 and f(r) =-—1, and therefore so far agrees with
the actual law of lines, forces, &c., as that the affection of P passes from + to — while @ varies

from 0 to 7. But if we examine this case, we find that it is widely diverse from the actual
law just mentioned; for we have

Ah een? (= sin 4 + (-1)} sin (= sin :):
Now if @=0 f(@)=1

= 5 f()=(-0)

* Algebra, 1st Edition, page 366.
002

284 Mr. GOODWIN, ON THE PURE SCIENCE

BOYS (= ain *) f (0) + sin (= sin s)r(3).

2 3

and Pf (0) = P cos (= sin °) f (0) + Psin (= sin ) i (=) :

3

In this case, therefore, it would seem that the oblique quantity P would be equivalent. to two
components, one in the original direction, the other inclined at an angle of 60°, the magnitude

of the former being P cos (= sin =); that of the latter P sin (= sin *). But there will be a dis-

tinction between this and that of the ordinary formula cos 9 + (— 1)! sin @ which is to be observed ;
for in this latter case the impossible plane determined by {(@) =(-1)} coincides with that
determined by f(@) = (— 1)!, but in the present hypothetical case, we have two impossible direc-

tions, one corresponding to 0 = = or f (9) = (— 1)4, the other to @ = 2a -= or f (0) = — (- 1)3.

And therefore that which is analogous to the impossible plane in the ordinary case is an im
possible cone whose semi-vertical angle is an angle of 60°.
There will be two impossible cones in like manner belonging to the formula
62 2 °

F (0) = (— 1)" = cos% + (= ph sin

which, together with the formula (4), are particular instances of the general form,

e seuous badeetge (GY.

7

OD = k— 1 (e ne cos i + (- 1)3sin

13. The preceding cases are examples of the general formula f(9) = (-1)®, where © is some
function of @ which = 0 when @=0, and =1 when @=7, the direction or directions for which

f (9) = (- 1)! are given by 0 = lk i

It may be observed, that all examples must have this

property in common ; that if we suppose a quantity P to be composed of two others whose direc-
tions are the line for which @=0, and that for which f{(@) =(-1)!, and if we call these
components wv and y respectively, then w and y satisfy the condition

e+ = P*;
and therefore if # and y be regarded as oblique co-ordinates of a point, the locus of that point
@

is an ellipse; in the case of f(@) = (—1)" this locus becomes a circle.

14, It is evidently possible to vary indefinitely the law according to which f(6) shall vary
from +1 to —1, while @ increases from 0 to 7, even though we confine ourselves to the form
(8) =(-1)°; and all these laws will express modes in which the affection of a quantity may
be diametrically reversed ; I am disposed to look upon most of them as fictitious generalizations
which can have no type in the nature of things, just as we might construct a system of geometry
of four dimensions which could have nothing real corresponding to it. It may be possible,
however, to find some which have not this fictitious character, and which express physical laws.
We shall obtain a distinet conception of the manner in which the law expressed by the formula

0
f (0) = (—1)* differs from all others, by observing that if (—1)° expresses the law of change from
+ to —, the gradual change of affection, as compared with a change in the value of @, will be

OF MAGNITUDE AND DIRECTION. 285

31a

de 1 , F
—~ = — a constant quantity, which can be true of no other

expressed by ae now if O= -, oT

2

law satisfying the conditions f(0) =1 and f(7) = —1. If, for instance, @ = ue a form which satis-
Tv

d
fies the conditions f(0) =1 and f(7) = —1, we have a =2 & , and therefore the intensity of
Tv

the minus affection which is measured by © would increase more rapidly as the angle @ approached
the value 7. And this also shews distinctly what is meant by saying that the change of affec-
tion, in such causes as we have been considering, is wniform, for this is, in fact, saying that
d98 s . ; 5 :

7) must be constant, a condition which must manifestly be satisfied when the case is one of pure
direction, and when, therefore, there is no reason why © should increase more rapidly for one value
of @ than another. Whether there be real cases of change of affection coming under the general
type represented by (- 1)°, in which this condition of uniformity is not satisfied, remains to
be seen.

15. The condition of wniform change of affection is satisfied by the function 0 = = where
a

a is some constant angle, which, in the actual case of pure direction, is a right angle. If ©
have this value, we have

f(0) = age + (-1)} sin #8,
2a 2a

and the impossible direction is given by

@6=a.

For example, let a = —, then

ela

F (6) = cos 20 + (— 1)? sin 20,

a formula which represent the variation of a cause which changes uniformly, and produces exactly
opposite effects in directions at right angles to each other. It seems not improbable that this
formula may be found to represent something real: may it not represent the following case ?
Suppose a disturbing cause in an elastic medium which propagates simultaneously a condensed
wave in two opposite directions, and a rarified wave in the direction perpendicular to them;
then if @ be the condensation which would exist at a given time and a given distance from
the centre, on the supposition of the condensing cause only acting, may not the complete expres-
sion for the condensation in the direction determined by the angle 6 be
p cos 20 + (—1)! psin 26?
A rough approximation to this case would be that of a tuning-fork.

16. A more general law than that expressed by the formula f(@) = (-1)° is given by
Ff (6) = m(—1)? + m'(— 1)” + &e.
There is only one example of this formula which I shall notice :
Suppose we have a quantity P determined by the equation
Pf (0) = acos 0 +(—1)b sin @,.....(D);

which comes under the above form of f(@) for the equation, may be written

6
7.

b Z -b
PF (0) =" (=r + (eye

286 Mr. GOODWIN, ON THE PURE SCIENCE OF MAGNITUDE AND DIRECTION.

We have here an effect which results from two causes, which separately vary uniformly.
Considering equation (D), we may observe that the difference between it and the equation

Pf (0)= P cos 6 + (—1)! P sin @...... (EB)

is this, that, although both give @ = 0 and 0== as the directions of the components, yet the

values of Pf (0) and Pf (=) which are the same in the latter case (omitting the sign of affec-
tion) are different in the former: for in equation (D)

2 Pf (0) =a,
Pf(=) Bes)

In fact, considering the equations (D) and (E) geometrically, the former represents an ellipse,
the latter a cirele: the angle @ will manifestly be the eccentric anomaly.

My reason for introducing the formula (D) is that I may remark that, whereas the formula
(Z) represents force considered in the light of pure direction, the formula (D) corresponds to
a case of polarity. Pure force must, of course, be free from any polarity, that is to say, its
absolute magnitude must be the same in whatever direction it acts, the direction will modify the
effect, not diminish or increase it; but there are complicated instances of force in which this is
not the case, but in which there is polarity ; for example, in the case of an elastic medium under
constraint from the action of the particles of a crystallized body which contains it. Now the
formula (D) appears to be exactly calculated to express this kind of force ; to fix our conceptions
let an elastic medium have the same properties in all sections parallel to the plane of wy, and
have polarity in that plane; consider any one section, and let the properties of this section
of the medium be symmetrical about the axes of # and y, then the origin will be a position
of rest for a particle, and if it be disturbed, the force of restitution may be represented by such
a formula as (D).

In examining that formula we find that there are two directions perpendicular to each other,
for which the force of restitution is in the direction of displace-
ment; for all other displacements the force of restitution is not
in the same direction, but will have to be determined thus; let
AP be the direction of displacement: take APB proportional to
a+b, and AD, making the same angle with Aw as AB, propor-
tional to a — 6; complete the parallelogram ABCD, and draw
through P a line parallel to the diagonal 4C; this will be the
direction of the force of restitution. _ Hence then it appears, that
the formula (D) will represent the kind of law which determines
the force of restitution on a disturbed particle in the case of uniaxal crystals.

H. GOODWIN.

XXII. On the Theories of the Internal Friction of Fluids in Motion, and of the
Equilibrium and Motion of Elastic Solids. By G. G. Stoxes, M.A., Fel-
low of Pembroke College.

[Read April 14, 1845.]

Tue equations of Fluid Motion commonly employed depend upon the fundamental hypothesis
that the mutual action of two adjacent elements of the fluid is normal to the surface which
separates them. From this assumption the equality of pressure in all directions is easily deduced,
and then the equations of motion are formed according to D’Alembert’s principle. This appears
to me the most natural light in which to view the subject; for the two principles of the absence
of tangential action, and of the equality of pressure in all directions ought not to be assumed
as independent hypotheses, as is sometimes done, inasmuch as the latter is a necessary consequence
of the former*. The equations of motion so formed are very complicated, but yet they admit
of solution in some instances, especially in the case of small oscillations. The results of the theory
agree on the whole with observation, so far as the time of oscillation is concerned. But there
is a whole class of motions of which the common theory takes no cognizance whatever, namely, those
which depend on the tangential action called into play by the sliding of one portion of a fluid along
another, or of a fluid along the surface of a solid, or of a different fluid, that action in fact which
performs the same part with fluids that friction does with solids.

Thus, when a ball pendulum oscillates in an indefinitely extended fluid, the common theory
gives the arc of oscillation constant, Observation however shows that it diminishes very rapidly
in the case of a liquid, and diminishes, but less rapidly, in the case of an elastic fluid. It has
indeed been attempted to explain this diminution by supposing a friction to act on the ball,
and this hypothesis may be approximately true, but the imperfection of the theory is shown
from the circumstance that no account is taken of the equal and opposite friction of the ball on
the fluid.

Again, suppose that water is flowing down a straight aqueduct of uniform slope, what will be
the discharge corresponding to a given slope, and a given form of the bed? Of what magnitude
must an aqueduct be, in order to supply a given place with a given quantity of water? Of what
form must it be, in order to ensure a given supply of water with the least expense of materials
in the construction? ‘These, and similar questions are wholly out of the reach of the common
theory of Fluid Motion, since they entirely depend on the laws of the transmission of that
tangential action which in it is wholly neglected. In fact, according to the common theory
the water ought to flow on with uniformly accelerated velocity; for even the supposition of
a certain friction against the bed would be of no avail, for such friction could not be transmitted
through the mass. The practical importance of such questions as those above mentioned has
made them the object of numerous experiments, from which empirical formula have been con-
structed. But such formule, although fulfilling well enough the purposes for which they were

* This may be easily shown by the consideration of a tetrahedron of the fluid, as in Art. 4.

288 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

constructed, can hardly be considered as affording us any material insight into the laws of nature ;
nor will they enable us to pass from the consideration of the phenomena from which they were
derived to that of others of a different class, although depending on the same causes.

In reflecting on the principles according to which the motion of a fluid ought to be calculated
when account is taken of the tangential force, and consequently the pressure not supposed the
same in all directions, I was led to construct the theory explained in the first section of this
paper, or at least the main part of it, which consists of equations (13), and of the principles
on which they are formed. I afterwards found that Poisson had written a memoir on the same
subject, and on referring to it I found that he had arrived at the same equations. The method
which he employed was however so different from mine that I feel justified in laying the latter
before this Society*. The leading principles of my theory will be found in the hypotheses of
Art. 1, and in Art. 3.

The second section forms a digression from the main object of this paper, and at first sight
may appear to have little connexion with it. In this section I have, I think, succeeded in shewing
that Lagrange’s proof of an important theorem in the ordinary theory of Hydrodynamics is
untenable. The theorem to which I refer is the one of which the object is to show that
udax +vdy + wdz, (using the common notation,) is always an exact differential when it is so
at one instant. I have mentioned the principles of M. Cauchy’s proof, a proof, I think, liable
to no sort of objection. I have also given a new proof of the theorem, which would have served to
establish it had M. Cauchy not been so fortunate as to obtain three first integrals of the general
equations of motion. As it is, this proof may possibly be not altogether useless,

Poisson, in the memoir to which I have referred, begins with establishing, according to
his theory, the equations of equilibrium and motion of elastic solids, and makes the equations of
motion of fluids depend on this theory. On reading his memoir, I was led to apply to the theory
of elastic solids principles precisely analogous to those which I have employed in the case of
fluids. The formation of the equations, according to these principles, forms the subject of
Sect. 111.

The equations at which I have thus arrived contain two arbitrary constants, whereas Poisson’s
equations contain but one. In Sect. rv. I have explained the principles of Poisson’s theories of
elastic solids, and of the motion of fluids, and pointed out what appear to me serious objections
against the truth of one of the hypotheses which he employs in the former. This theory seems
to be very generally received, and in consequence it is usual to deduce the measure of the cubical
compressibility of elastic solids from that of their extensibility, when formed into rods or wires,
or from some quantity of the same nature. If the views which I have explained in this section
be correct, the cubical compressibility deduced in this manner is too great, much too great in
the case of the softer substances, and even the softer metals. The equations of Sect. 111. have,
I find, been already obtained by M. Cauchy in his Ewercises Mathématiques, except that he
has not considered the effect of the heat developed by sudden compression. The method which
I have employed is different from his, although in some respects it much resembles it.

The equations of motion of elastic solids given in Sect. 111. are the same as those to which
different authors have been led, as being the equations of motion of the luminiferous ether in
yacuum. It may seem strange that the same equations should have been arrived at for cases
so different; and I believe this has appeared to some a serious objection to the employment of
those equations in the case of light. I think the reflections which I have made at the end of
Sect. 1v., where I have examined the consequences of the law of continuity, a law which seems
to pervade nature, may tend to remove the difficulty.

* The same equations have also been obtained by Navier | T. vr.) but his principles differ from mine still more than do
in the case of an incompressible fluid, (Mém. de U' Institut, | Poisson's.

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 289

SECTION I.

Explanation of the Theory of Fluid Motion proposed. Formation of the Differential
Equations. Application of these Equations to a few simple cases.

1. Berore entering on the explanation of this theory, it will be necessary to define, or fix
the precise meaning of a few terms which I shall have occasion to employ.

In the first place, the expression ‘the velocity of a fluid at any particular point” will require
some notice. If we suppose a fluid to be made up of ultimate molecules, it is easy to see that these
molecules must, in general, move among one another in an irregular manner, through spaces
comparable with the distances between them, when the fluid is in motion. But since there
is no doubt that the distance between two adjacent molecules is quite insensible, we may neglect ‘the
irregular part of the velocity, compared with the common velocity with which all the molecules
in the neighbourhood of the one considered are moving. Or, we may consider the mean velocity
of the molecules in the neighbourhood of the one considered, apart from the velocity due to
the irregular motion. It is this regular velocity which I shall understand by the velocity of
a fluid at any point, and I shall accordingly regard it as varying continuously with the
co-ordinates of the point.

Let P be any material point in the fluid, and consider the instantaneous motion of a very
small element £ of the fluid about P. This motion is compounded of a motion of translation,
the same as that of P, and of the motion of the several points of E relatively to P. If we
conceive a velocity equal and opposite to that of P impressed on the whole element, the remaining
velocities form what I shall call the relative velocities of the points of the fluid about P; and
the motion expressed by these velocities is what I shall call the relative motion in the neigh-
bourhood of P.

It is an undoubted result of observation that the molecular forces, whether in solids, liquids,
or gases, are forces of enormous intensity, but which are sensible at only insensible distances.
Let E’ be a very small element of the fluid circumscribing E, and of a thickness greater than
the distance to which the molecular forces are sensible. 'The forces acting on the element E
are the external forces, and the pressures arising from the molecular action of E’. If the
molecules of £ were in positions in which they could remain at rest if H were acted on by no
external force and the molecules of E’ were held in their actual positions, they would be in
what I shall call a state of relative equilibriwm. Of course they may be far from being in a
state of actual equilibrium. Thus, an element of fluid at the top of a wave may be sensibly in
a state of relative equilibrium, although far removed from its position of equilibrium. Now, in
consequence of the intensity of the molecular forces, the pressures arising from the molecular action
on £ will be very great compared with the external moving forces acting on E. Consequently
the state of relative equilibrium, or of relative motion, of the molecules of E will not be sensibly
affected by the external forces acting on EH. But the pressures in different directions about
the point P depend on that state of relative equilibrium or motion, and consequently will not
be sensibly affected by the external moving forces acting on EZ, For the same reason they will not
be sensibly affected by any motion of rotation common to all the points of E; and it is a direct
consequence of the second law of motion, that they will not be affected by any motion of translation
common to the whole element. If the molecules of EH were in a state of relative equilibrium,
the pressure would be equal in all directions about P, as in the case of fluids at rest. Hence
I shall assume the following principle :—

That the difference between the pressure on a plane in a given direction passing through
any point P of a fluid in motion and the pressure which would ewist in all directions
about P if the fluid in its neighbourhood were in a state of relative equilibrium depends
only on the relative motion of the fluid immediately about P; and that the relative motion

Vou. VIII. Parr III. Pe

290 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

due to any motion of rotation may be eliminated without affecting the differences of the pressures
above mentioned.
Let us see how far this principle will lead us when it is carried out.

2. It will be necessary now to examine the nature of the most general instantaneous motion
of an element of a fluid. The proposition in this article is however purely geometrical, and
may be thus enunciated : —‘‘ Supposing space, or any portion of space, to be filled with an
infinite number of points which move in any continuous manner, retaining their identity, to
examine the nature of the instantaneous motion of any elementary portion of these points.”

Let w, v, w be the resolved parts, parallel to the rectangular axes Ow, Oy, Ox, of the
velocity of the point P, whose co-ordinates at the instant considered are wv, y, x. Then the
relative velocities at the point P’, whose co-ordinates are +a’, y+y', x +2, will be

du, du, du, llel t

— wg +—¥y +—s parallel to a

dau dy Up aE P :

dv GD. Guid BOs,
— epneee seeae a>

neglecting squares and products of a’, y', %. Let these velocities be compounded of those due
to the angular velocities w’, w’, w” about the axes of x, y, x, and of the velocities U, V, W
parallel to a, y, x. The linear velocities due to the angular velocities being w”2’ — wy,
wv’ — wz’, wy — wv parallel to the axes of 2, y, x, we shall therefore have

du , du mw / du ” ,
U=— a +(F +o )y + (= -6 J,
da dy z

Vv dv Pa) ap dv , A (5 ) ;
= (— - — a =
er 3 : y y deel

d
dw ae dw Way i ee
We (Fre )a + (Gal) +

Since w', w’, w’” are arbitrary, let them be so assumed that

dU dV dV dW dW dU

which gives

dw dv du dw dv du
ee toa ar eee tht PARE TE| sels eds
One ie =); w =) (= Sy w 2 ($ aah sdaigewescly

du du dv du dw
webb sa Ee Mea ttf
ie * dy ' da)? *2 as ag)

.

dv du dv dv dw
= i a os | i coclgaadeeeeuen vannce ic
V 4 (+a) "tay +3 ( \ x (2)

de * dy

dw du dw dv dw
moa(Ge ovale dey ode
2\da tds)’ +? dy ds)" tax”

The quantities w’, w’, w’” are what I shall call the angular velocities of the fluid at the
point considered. This is evidently an allowable definition, since, in the particular case in which

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 291

the element considered moves as a solid might do, these quantities coincide with the angular
velocities considered in rigid dynamics. A further reason for this definition will appear in Sect, 111.

Let us now investigate whether it is possible to determine a’, y', zx’ so that, considering only
the relative velocities U, V, W, the line joining the points P, P’ shall have no angular motion.
The conditions to be satisfied, in order that this may be the case, are evidently that the incre-
ments of the relative co-ordinates a’, y', x’ of the second point shall be ultimately proportional
to those co-ordinates. If e be the rate of extension of the line joining the two points considered,
we shall therefore have

Fa +hy + gx =er',

,

he + Gy +fx =ey', gon sya

gu +fy + Hz =e’;

du dv dw dv dw dw du du dv
x = =— =— Qf = — —— 22 => — 2h= —.
Beate sy dx’ so dy’ Hi dz’ f ds dy te Me eg dy + da

If we eliminate from equations (8) the two ratios which exist between the three quantities
a’, y', x, we get the well known cubic equation

(e — F) (e — G) (e- H) -f’ (e- F) -g (e-G) -h (e- A) - 2fgh=0,..... -(4)

which occurs in the investigation of the principal axes of a rigid body, and in various others.
As in these investigations, it may be shewn that there are in general three directions, at right
angles to each other, in which the point P’ may be situated so as to satisfy the required conditions.
If two of the roots of (4) are equal, there is one such direction corresponding to the third root, and
an infinite number of others situated in a plane perpendicular to the former; and if the three
roots of (4) are equal, a line drawn in any direction will satisfy the required conditions.

The three directions which have just been determined I shall call aves of ewtension. They
will in general vary from one point to another, and from one instant of time to another. If we
denote the three roots of (4) by e’, e”, e”’, and if we take new rectangular axes Ow, Oy, Ox,
parallel to the axes of extension, and denote by uw, U,, &c. the quantities referred to these
axes corresponding to u, U, &c., equations (3) must be satisfied by y’ = 0, z/ = 0, e=e’, by @/=0,
z/=0, e=e", and by w/=0, y/=0, e=e"”, which requires that f/=0, g=0, h,=0, and
we have

é=F =—, “=G=—, e=-H=—+.

The values of 1, V,, W,, which correspond to the residual motion after the elimination of

the motion of rotation corresponding to w’, w’ and w’”, are

U,= e' an’, V= ey |. W = eine.

The angular velocity of which w’, w’, w”’ are the components is independent of the arbitrary
directions of the co-ordinate axes: the same is true of the directions of the axes of extension,
and of the values of the roots of equation (4). This might be proved in various ways; perhaps
the following is the simplest. The conditions by which w’, w’, w” are determined are those which
express that the relative velocities U, V, W, which remain after eliminating a certain angular
velocity, are such that Uda’ + Vdy' + Wdz' is ultimately an exact differential, that is to say
when squares and products of a’, y' and x’ are neglected. It appears moreover from the solution
that there is only one way in which these conditions can be satisfied for a given system of
co-ordinate axes, Let us take new rectangular axes Ox, Oy, Oz, and let U, V, W be the resolved
parts along these axes of the velocities U, V, W, and x’, y’, z’, the relative co-ordinates of P’; then

PP

292 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

U = Ucosax + V cosay + W cos #2,
dz' = cos vxdx' + cos wydy’ + cos xzdz’, &e.;

whence, taking account of the well known relations between the cosines involved in these equations,
we easily find
Uda’ + Vdy' + Wdx' = Udx' + Vdy' + Wdz’.

It appears therefore that the relative velocities U, V, W, which remain after eliminating a certain
angular velocity, are such that Udx' + Vdy' + Wdz' is ultimately an exact differential. Hence
the values of U, V, W are the same as would have been obtained from equations (2) applied
directly to the new axes, whence the truth of the proposition enunciated at the head of this
paragraph is manifest.

The motion corresponding to the velocities U, V,, W, may be further decomposed into a
motion of dilatation, positive or negative, which is alike in all directions, and two motions which I
shall call motions of shifting, each of the latter being in two dimensions, and not affecting the
density. For let 5 be the rate of linear extension corresponding to a uniform dilatation; let o2,,
—oy, be the velocities parallel to «, y,, corresponding to a motion of shifting parallel to the
the plane a,y,, and let c/a’, —o'x/ be the velocities parallel to w, x, corresponding to a similar
motion of shifting parallel to the plane ws, The velocities parallel to #, y, x, respectively
corresponding to the quantities 6, ¢ and o’ will be (0+a+0)a/,(0-c)y/, (0—o')x;, and
equating these to U, V, W, we shall get

Ne 4 (e a Bu + é”), ae 4 (CA as es 2 e”), we 4 (e+ a — 2e”),
Hence the most general instantaneous motion of an elementary portion of a fluid is compounded

of a motion of translation, a motion of rotation, a motion of uniform dilatation, and two motions of
shifting of the kind just mentioned.

3. Having determined the nature of the most general instantaneous motion of an element
of a fluid, we are now prepared to consider the normal pressures and tangential forces called
into play by the relative displacements of the particles. Let p be the pressure which would exist
about the point P if the neighbouring molecules were in a state of relative equilibrium: let p + p,
be the normal pressure, and ¢, the tangential action, both referred to a unit of surface, on a ‘ieee
passing through P and having a given direction. By the hypotheses of Art. 1. the quantities p, ¢,
will be independent of the angular velocities ww”, w” ’, depending ol on the residual relative
velocities U,V,W, or, which comes to the same, on e’, e Casid e’”’, or on g,o’ and 6. Since this residual
motion is symmetrical with respect to the axes of extension, it follows that if the plane considered
is perpendicular to any one of these axes the tangential action on it is zero, since there is no more
reason why it should act in one direction rather than in the opposite; for by the hypotheses
of Art. 1. the change of density and temperature about the point P is to be neglected, the
constitution of the fluid being ultimately uniform about that point. Denoting then by p+p’,
p+p',p+p’ the pressures on planes perpendicular to the axes of w,, y,, %, we must have

p=pere’ie”), p= ple',e’,e), p’=p(e",e,e"),
¢ (¢, e", e”) denoting a function of e’, e” and e” which is symmetrical with respect to the two
latter quantities. The question is now to determine, on whatever may seem the most probable
hypothesis, the form of the function ¢.

Let us first take the simpler case in which there is no dilatation, and only one motion of
shifting, or in which e” = —e', e”” =0, and let us consider what would take place if the
fluid consisted of smooth molecules acting on each other by actual contact. On this supposition,
it is clear, considering the magnitude of the pressures acting on the molecules compared with
their masses, that they would be sensibly in a position of relative equilibrium, except when
the equilibrium of any one of them became impossible from the displacement of the adjoining

my

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 293

ones, in which case the molecule in question would start into a new position of equilibrium. This
start would cause a corresponding displacement in the molecules immediately about the one which
started, and this disturbance would be propagated immediately in all directions, the nature of the
displacement however being different in different directions, and would soon become insensible.
During the continuance of this disturbance, the pressure on a small plane drawn through the
element considered would not be the same in all directions, nor normal to the plane: or, which
comes to the same, we may suppose a uniform normal pressure p to act, together with a normal
pressure p, and a tangential force ¢,, p,, and t, being forces of great intensity and short duration,
that is being of the nature of impulsive forces. As the number of molecules comprised in the
element considered has been supposed extremely great, we may take a time 7 so short that all
summations with respect to such intervals of time may be replaced without sensible error by
integrations, and yet so long that a very great number of starts shall take place in it.
Consequently we have only to consider the average effect of such starts, and moreover we may
without sensible error replace the impulsive forces such as p, and ¢,, which succeed one another
with great rapidity, by continuous forces. For planes perpendicular to the axes of extension
these continuous forces will be the normal pressures p’, p’, p’”

Let us now consider a motion of shifting differing from the former only in having e’ increased
in the ratio of m to 1. Then, if we suppose each start completed before the starts which would
be sensibly affected by it are begun, it is clear that the same series of starts will take place in the
second case as in the first, but at intervals of time which are less in the ratio of 1 to m.
Consequently the continuous pressures by which the impulsive actions due to these starts may be
replaced must be increased in the ratio of m to 1. Hence the pressures p’, p’, p”” must be
proportional to e’, or we must have

p=Ce, p’=Ceée, p”=C"'e.

It is natural to suppose that these formule held good for negative as well as positive values
of e’. Assuming this to be true, let the sign of e’ be changed. This comes to interchanging
az and y, and consequently p’” must remain the same, and p’ and p” must be interchanged. We
must therefore have C”=0, C'’=—C. Putting then C = — 2u we have

p=-2ne, p’=2pne, pl’ =0.

It has hitherto been supposed that the molecules of a fluid are in actual contact. We
have every reason to suppose that this is not the case. But precisely the same reasoning will apply
if they are separated by intervals as great as we please compared with their magnitudes, provided
only we suppose the force of restitution called into play by a small displacement of any one
molecule to be very great.

Let us now take the case of two motions of shifting which coexist, and let us suppose
eé=ata, e’'=-—a, e”=—a, Let the small time + be divided into 2” equal portions, and
let us suppose that in the first interval a shifting motion corresponding to ¢ = 2c, e” = — 20 takes
place parallel to the plane x, y, and that in the second interval a shifting motion corresponding
to = 20’, e’’ = — 2o' takes place parallel to the plane w, x, and so on alternately. On this

se Sine * 5 ie .
supposition it is clear that if we suppose the time a to be extremely small, the continuous forces
n

by which the effect of the starts may be replaced will be p’= — 2u(¢ + 0’), p"=2pua, p= 20’. By
supposing indefinitely increased, we may make the motion considered approach as near as we
please to that in which the two motions of shifting coexist; but we are not at liberty to do so,

‘ : : im
for in order to apply the above reasoning we must suppose the time — to be so large that the
: n

average effect of the starts which occur in it may be taken. Consequently it must be taken as an
additional assumption, and not a matter of absolute demonstration, that the effects of the two
motions of shifting are superimposed.

294 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

Hence if §=0, i.e. if e +e” + e””=0, we shall have in general

p=- Que, p = — Qne”, p” =- DO gndinn naa saaesases(D)

It was by this hypothesis of starts that I first arrived at these equations, and the differential
equations of motion which result from them. On reading Poisson’s memoir however, to which
I shall have occasion to refer in Section 1v., I was led to reflect that however intense we may
suppose the molecular forces to be, and however near we may suppose the molecules to be to their
positions of relative equilibrium, we are not therefore at liberty to suppose them in those positions,
and consequently not at liberty to suppose the pressure equal in all directions in the intervals of
time ‘between the starts. In fact, by supposing the molecular forces indefinitely increased,
retaining the same ratios to each other, we may suppose the displacements of the molecules from
their positions of relative equilibrium indefinitely diminished, but on the other hand the force of
restitution: called into action by a given displacement is indefinitely increased in the same proportion.
But be these displacements what they may, we know that the forces of restitution make equilibrium
with forces equal and opposite to the effective forces; and in calculating the effective forces we
may neglect the above displacements, or suppose the molecules to move in the paths in which they
would move if ‘the shifting motion took place with indefinite slowness. Let us first consider a
single motion of shifting, or one for which e” = —¢’, e”’ =0, and let p, and ¢ denote the same
quantities as before. If we now suppose e’ increased in the ratio of m to 1, all the effective forces
will be increased in that ratio, and consequently p, and ¢, will be increased in the same ratio, We
may deduce the values of p’, p” and p’” just as before, and then pass by the same reasoning to
the case of two motions of shifting which coexist, only that in this case the reasoning will be demon-

strative, since we may suppose the time ap indefinitely diminished. If we suppose the state of
n

things considered in this paragraph to exist along with the motions of starting already considered,
it is easy to see that the expressions for p’, p” and p’” will still retain the same form.

There remains yet to be considered the effect of the dilatation. Let us first suppose the
dilatation to exist without any shifting: then it is easily seen that the relative motion of the
fluid at the point considered is the same in all directions. Consequently the only effect which
such a dilatation could have would be to introduce a normal pressure p,, alike in all directions, in
addition to that due to the action of the molecules supposed to be in a state of relative equilibrium.
Now the pressure p, could only arise from the aggregate of the molecular actions called into play
by the displacements of the molecules from their positions of relative equilibrium ; but since these
displacements take place, on an average, indifferently in all directions, it follows that the actions
of which p, is composed neutralize each other, so that p,= 0. The same conclusion might be
drawn from the hypothesis of starts, supposing, as it is natural to do, that each start calls into
action as much increase of pressure in some directions as diminution of pressure in others.

If the motion of uniform dilatation coexists with two motions of shifting, I shall suppose,
for the same reason as before, that the effects of these different motions are superimposed. Hence
subtracting d from each of the three quantities e’, e’ and e”, and putting the remainders in the
place of e’, e” and e” in equations (5), we have

mr

p os Zu(e” ar, e” ah 2e'), p’ = gu(e + ao ws 2e"), p" = gule’ ae e” ae 26"). .n. ae (6)

,

If we had started with assuming @(e’, e”, e””) to be a linear function of e’, e” and e”,
avoiding all speculation as to the molecular constitution of a fluid, we should have had at once
p' =Ce' + C'(e’ +e”), since p’ is symmetrical with respect to e’ and e”; or, changing the
constants, p’ = Sy(e” +e” —2e') + «(e' +e” +e”). The expressions for p” and p”” would be
obtained by interchanging the requisite quantities. Of course we may at once put « =0 if we
assume that in the case of a uniform motion of dilatation the pressure at any instant depends
only on the actual density and temperature at that instant, and not on the rate at which the

AND THE EQUILIBRIUM AND. MOTION OF ELASTIC SOLIDS. 295.

former changes with the time. In most cases to which it would be interesting to apply the
theory of the friction of fluids the density of the fluid is either constant, or may. without sensible
error be regarded as constant, or else changes slowly with the time. In the first two cases the
results peer be the same, and in the third case nearly the same, whether « were equal to zero or
not. Consequently, if theory and experiment should in such cases agree, the experiments must
not be regarded as confirming that part of the theory which relates to supposing « to be
equal to zero.

4. It will be easy now to determine the oblique pressure, or resultant of the normal pressure
and tangential action, on any plane. Let us first consider a plane drawn through the point P
parallel to the plane yx. Let Ow, make with the axes of w, y, x angles whose cosines are 1’, m’, n’ ;
Jet 7’, m", n” be the same for Oy, and 1”, m’”, n’” the same for Ox, Let P, be the pressure,
and (aty), (atx) the resolved parts, parallel to y, x respectively, of the tangential force on the
plane considered, all referred to a unit of surface, (vty) being reckoned positive when the part
of the fluid foweede —w urges that towards + in the positive direction of y, and similarly
for (wtz). Consider the portion of the fluid comprised within a tetrahedron having its. vertex
in the point P, its base parallel to the plane yz, and its three sides parallel to the planes wy, y,~,
za, respectively. Let A be the area of the base, and therefore /' A, 1 4, I” A the areas of the fee
perpendicular to the axes of w, y, x, By D’Alembert’s principle, the pressures and tangential
actions on the faces of this tetrahedron, the moving forces arising from the external attractions,
not including the molecular forces, and forces equal and opposite to the effective moving forces will
be in equilibrium, and therefore the sums of the resolved parts of these forces in the directions
of w, y and x will each be zero. Suppose now the dimensions of the tetrahedron indefinitely
diminished, then the resolved parts of the external, and of the effective moving forces will vary
ultimately as the cubes, and those of the pressures and tangential forces on the sides as the
squares of homologous lines. Dividing therefore the three equations arising from equating to zero
the resolved parts of the above forces by A, and taking the limit, we have

P\= 31" (p+ p'), (wty) = EU'm' (p + p'), (ats) = Zin’ (p +p’),

the sign = denoting the sum obtained by taking the quantities corresponding to the three axes
of extension in succession. Putting for p’, p” and p”’ their values given by (6), putting e’+e’+e”
= 30, and observing that 3/’*=1, Z/’m'=0, S/'n'=0, the above equations become

Pi= p —2pSl?e' +2u0, (vty) =-2pdI'm'e’, (atz) = -—2nSI'n’e’.

The method of determining the pressure on any plane from the pressures on three planes
at right angles to each other, which has just been given, has already been employed by MM. Cauchy
and Poisson.

The most direct way of obtaining the values of Je’ &c. would be to express J’, m’ and
n in terms of e’ by any two of equations (3), in which a’, y’, 2’ are proportional to /’, m’, n’,
together with the equation /'* + m® + m= 1, and then to express the resulting symmetrical function
of the roots of the cubic equation (4) in terms of the coefficients. But this method would
be excessively laborious, and need not be resorted to. For after eliminating the angular motion of
the element of fluid considered the remaining velocities are e'w,, e’y/, e”’%/, parallel to the axes of
YY,» The sum of the resolved parts of these parallel to the axis of is V’e’w'+ le" y/4 Ue" x’.
Putting for w/, y/, x/ their values U'a'+ my’ + n'x’ &c., the above sum becomes

40%

ow Sl*e +y SI m'e + 2S ne’;

but this sum is the same thing as the velocity U in equation (2), and therefore we have

Sida ae ale Pee (i+ ae SI'n'e'= -4 (s+ et

dx px

296 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

It may also be very easily proved directly that the value of 36, the rate of cubical dilatation,
satisfies the equation

Let P,, (ytz), (ytw) be the quantities referring to the axis of y, and P,, (xtw), (xty) those
referring to the axis of x, which correspond to P, &c. referring to the axis of 2 Then we see
that (ytx) = (sty), (sta) = (wtz), (wty) = (ytw). Denoting these three quantities by 7), T',, 7'5,
and making the requisite substitutions and interchanges, we have

delet soe aenn(S)

dv dw dw du du dv
nao(lte), maul r88), nant 8)

It may also be useful to know the components, parallel to v, y, x, of the oblique pressure on a
plane passing through the point P, and having a given direction. Let /, m, n be the cosines of the
angles which a normal to the given plane makes with the axes of w, y, x; let P,Q, R be the
components, referred to a unit of surface, of the oblique pressure on this plane, P, Q, R being
reckoned positive when the part of the fluid in which is situated the normal to which J, m and n
refer is urged by the other part in the positive directions of x, y, x, when J, m and m are positive.
Then considering as before a tetrahedron of which the base is parallel to the given plane, the
vertex in the point P, and the sides parallel to the co-ordinate planes, we shall have

P=/1P,+m7,+7T:,
Q= LT; + m P, oe nT, eee Cece cc seen te soesecenseccter senses (9)
R=1T,+mT, +nP,|

In the simple case of a sliding motion for which wu =0, v=f(a#), w=0, the only forces,
besides the pressure p, which act on planes parallel to the co-ordinate planes are the two tangential

See pk . dv : ves
forces T',, the value of which in this case is — 1 a In this case it is easy to show that the axes of
a

extension are, one of them parallel to Ox, and the two others in a plane parallel to wy, and inclined
at angles of 45° to Ow. We see also that it is necessary to suppose « to be positive, since
otherwise the tendency of the forces would be to increase the relative motion of the parts of the
fluid, and the equilibrium of the fluid would be unstable.

5. Having found the pressures about the point P on planes parallel to the co-ordinate planes,
it will be easy to form the equations of motion. Let X, Y, Z be the resolved parts, parallel
to the axes, of the external force, not including the molecular force; let p be the density, ¢ the
time. Consider an elementary parallelepiped of the fluid, formed by planes parallel to the
co-ordinate planes, and drawn through the point (a, y, x) and the point (w+ Aw, y+ Ay, z + Az).
The mass of this element will be ultimately p Aw Ay Az, and the moving force parallel to arising
from the external forces will be ultimately pX Aw Ay Ax; the effective moving force parallel

to w will be ultimately p ot ne Ay Az, where D is used, as it will be in the rest of this paper,

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 297

to denote differentiation in which the independent variables are ¢ and three parameters -of the

particle considered, (such for instance as its initial co-ordinates,) and not t, x, y, x. It is easy also

to show that the moving force acting on the element considered arising from the oblique pressures

: ; adP aT, af;

on the faces is ultimately (— + Fi are
we have by D’Alembert’s principle

; ‘ x) ape any an,

rag gt ae

} Aw Ay &2, acting in the negative direction. Hence

05) Ress seatoa, slags cle -itocia dette (10)

in which équati aperitif egy lx VEE ain ee epg!”
= S value — UU — vo + SS 5 liar =
in whic equa 10ns we mus pu or Dt dt + FE oa dy Ww i an simula iy or d

and — . In considering the general equations of motion it will be needless to write down more

Dt
than one, since the other two may be at once derived from it by interchanging the requisite
quantities. The equations (10), the ordinary equation of continuity, as it is called,
d dpu dpv dpw
SL ad lig tes Win ala ee All ihe ah (11)
dt da dy dz

which expresses the condition that there is no generation or destruction of mass in the interior
of a fluid, the equation connecting p and p, or in the case of an incompressible fluid the equivalent
equation wins 0, and the equation for the propagation of heat, if we choose to take account
of that propagation, are the only equations to be satisfied at every point of the interior of
the fluid mass.

As it is quite useless to consider cases of the utmost degree of generality, I shall suppose
the fluid to be homogeneous, and of a uniform temperature throughout, except in so far as the
temperature may be raised by sudden compression in the case of small vibrations. Hence in
equations (10) ~ may be supposed to be constant as far as regards the temperature; for, in the
case of small vibrations, the terms introduced by supposing it to vary with the temperature
would involve the square of the velocity, which is supposed to be neglected. If we suppose
p to be independent of the pressure also, and substitute in (10) the values of P, &c. given by (8),
the former equations become

Du dp Gu d@u du um ad (du dv dw

e( Nias wana, = tae) pay lap Hage ac)

Let us now consider in what cases it is allowable to suppose « to be independent of the
pressure. It has been concluded by Dubuat, from his experiments on the motion of water in
pipes and canals, that the total retardation of the velocity due to friction is not increased by
increasing the pressure. The total retardation depends, partly on the friction of the water
against the sides of the pipe or canal, and partly on the mutual friction, or tangential action,
of the different portions of the water. Now if these two parts of the whole retardation were
separately variable with p, it is very unlikely that they should when combined give a result
independent of p. The amount of the internal friction of the water depends on the value of x.
I shall therefore suppose that for water, and by analogy for other incompressible fluids, « is
independent of the pressure. On this supposition, we have from equations (11) and (12)

SS G5 aoeee (12)

Du dp du du d*u
= edgy = taal | eee eR em Op Bind oti teacs 18
e(a; BR) + a “ (Garage sr pee sae
du dv dw
—_ + ——«
dw dy * dz

Vou. VIII. Parr III. Qe

298 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

These equations are applicable to the determination of the motion of water in pipes and canals,
to the calculation of the effect of friction on the motions of tides and waves, and such questions.

If the motion is very small, so that we may neglect the square of the velocity, we may

Du _ du
ee: Sal.
determination of the motion of a pendulum oscillating in water, or of that of a vessel filled with
water and made to oscillate. They are also applicable to the determination of the motion of
a pendulum oscillating in air, for in this case we may, with hardly any error, neglect the
compressibility of the air.

The case of the small vibrations by which sound is propagated in a fluid, whether a

, &c. in equations (13). The equations thus simplified are applicable to the

d
liquid or a gas, is another in which = may be neglected. For in the case of a liquid reasons

have been shown for supposing » to be independent of p, and in the case of a gas we may neglect

d : oy ping
ay if we neglect the small change in the value of yw, arising from the small variation of
Pp

pressure due to the forces X, Y, Z.

6. Besides the equations which must hold good at any point in the interior of the mass,
it will be necessary to form also the equations which must be satisfied at its boundaries. Let
M be a point in the boundary of the fluid. Let a normal to the surface at M, drawn on the
outside of the fluid, make with the axes angles whose cosines are l, m,n. Let P’, Q’, R’ be
the components of the pressure of the fluid about M on the solid or fluid with which it is in
contact, these quantities being reckoned positive when the fluid considered presses the solid or fluid
outside it in the positive directions of w, y, s, supposing 1, m and m positive. Let § be a
very small element of the surface about M, which will be ultimately plane, S” a plane parallel
and equal to §, and directly opposite to it, taken within the fluid. Let the distance between S
and jS” be supposed to vanish in the limit compared with the breadth of .S, a supposition which
may be made if we neglect the effect of the curvature of the surface at M; and let us consider the
forces acting on the element of fluid comprised between § and S’, and the motion of this
element, If we suppose equations (8) to hold good to within an insensible distance from the
surface of the fluid, we shall evidently have forces ultimately equal to PS, QS, RS, (P,Q and R
being given by equations (9),) acting on the inner side of the element in the positive directions of
the axes, and forces ultimately equal to P’S, Q'S, R’S acting on the outer side in the negative
directions. The moving forces arising from the external forces acting on the element, and the
effective moving forces will vanish in the limit compared with the forces PS, &c.: the same
will be true of the pressures acting about the edge of the element, if we neglect capillary
attraction, and all forces of the same nature. Hence, taking the limit, we shall have

P=P, Y=Q, R=R.
The method of proceeding will be different according as the bounding surface considered is a
free surface, the surface of a solid, on the surface of separation of two fluids, and it will be
hecessary to consider these cases separately, Of course the surface of a liquid exposed to the
air is really the surface of separation of two fluids, but it may in many cases be regarded as
a free surface if we neglect the inertia of the air: it may always be so regarded if we neglect
the friction of the air as well as its inertia.

Let us first take the case of a free surface exposed to a pressure [], which is supposed to
be the same at all points, but may vary with the time; and let L=0 be the equation to the
surface. In this case we shall have P’ =/II, Q’=mlIl, R’'=mlI1; and putting for P, Q, R their
values given by (9), and for P, &c. their values given by (8), and observing that in this case
5 =0, we shall have

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 299
du du dv du dw \
i(1 - + 21— + (= 7) (= =) = 0, Sgn coaeees Ole!
( P) | dx uf dy da ist Pra | Ba (14)
Auk, Ge dL
da’ dy’ dz

d

in which equations /, m, 2 will have to be replaced by , to which they are pro-

portional.
If we choose to take account of capillary attraction, we have only to diminish the pressure I

1 1 : ays : ‘
by the quantity H (= as —), where # is a positive constant depending on the nature of the fluid,
ers
and 7,, 7 are the principal radii of curvature at the point considered, reckoned positive when
the fluid is concave outwards. Equations (14) with the ordinary equation

dL dL dL dL

—+u-——+v—+F

dt du dy dz
are the conditions to be satisfied for points at the free surface. Equations (14) are for such
points what the three equations of motion are for internal points, and (15) is for the former
what (11) is for the latter, expressing in fact that there is no generation or destruction of fluid
at the free surface.

The equations (14) admit of being differently expressed, in a way which may sometimes
be useful. If we suppose the origin to be in the point considered, and the axis of x to be the
external normal to the surface, we have /=m=0, n =1, and the equations become

dw du dw dv

—+—=0, —+—= 1 +2 te rf 16
pee aiden Se Pag rasaiepenesecsesees(LO)

The relative velocity parallel to x of a point (2’, y’, 0) in the free surface, indefinitely near
the origin, is Uw + Ber: hence we see that ie a are the angular velocities, reckoned
dx dy da dy
from # to x and from y to * respectively, of an element of the free surfaee. Subtracting the
linear velocities due to these angular velocities from the relative velocities of the point (a’, y/, 2’),
and calling the remaining relative velocities U, V, W, we shall have

du, du, (= =) ,

= — — x
da * dy” * \dz* de
ye dv Y dv , = P. a
- da tay! s) dz dy)’
dw
W =—?’
dz ;
Hence we see that the first two of equations (16) express the conditions that p Pll

dV , , sid : :
and lhe, which are evidently the conditions to be satisfied in order that there may be no
#

sliding motion in a direction parallel to the free surface. It would be easy to prove that these
are the conditions to be satisfied in order that the axis of x may be an axis of extension.

The next case to consider is that of a fluid in contact with a solid. The condition which first
occurred to me to assume for this case was, that the film of fluid immediately in contact with the
solid did not move relatively to the surface of the solid. I was led to try this condition from the
following considerations. According to the hypotheses adopted, if there was a very large relative

aa2

300 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

motion of the fluid particles immediately about any imaginary surface dividing the fluid, the
tangential forces called into action would be very large, so that the amount of relative motion
would be rapidly diminished. Passing to the limit, we might suppose that if at any instant the
velocities altered discontinuously in passing across any imaginary surface, the tangential force
called into action would immediately destroy the finite relative motion of particles indefinitely close
to each other, so as to render the motion continuous; and from analogy the same might be
supposed to be true for the surface of junction of a fluid and solid. But having calculated,
according to the conditions which I have mentioned, the discharge of long straight circular pipes
and rectangular canals, and compared the resulting formule with some of the experiments of
Bossut and Dubuat, I found that the formule did not at all agree with experiment. I then
tried Poisson’s conditions in the case of a circular pipe, but with no better success. In fact, it
appears from experiment that the tangential force varies nearly as the square of the velocity with
which the fluid flows past the surface of a solid, at least when the velocity is not very small. It
appears however from experiments on pendulums that the total friction varies as the first power
of the velocity, and consequently we may suppose that Poisson’s conditions, which include as a
particular case those which I first tried, hold good for very small velocities, I proceed therefore
to deduce these conditions in a manner conformable with the views explained in this paper,

First, suppose the solid at rest, and let Z =0 be the equation to its surface. Let M' bea
point within the fluid, at an insensible distance h from M, Let @ be the pressure which would
exist about M if there were no motion of the particles in its neighbourhood, and let p, be the
additional normal pressure, and ¢, the tangential force, due to the relative velocities of the
particles, both with respect to one another and with respect to the surface of the solid. If the
motion is so slow that the starts take place independently of each other, on the hypothesis of starts,
or that the molecules are very nearly in their positions of relative equilibrium, and if we suppose
as before that the effects of different relative velocities are superimposed, it is easy to show that
p, and ¢, are linear functions of w, v, w and their differential coefficients with respect to w, y, and zx;
u, v, &e. denoting here the velocities of the fluid about the point M’, in the expressions for which
however the co-ordinates of M may be used for those of M’, since h is neglected. Now the

F , ‘ F d |, du
relative velocities about the points M and MM’ depending on = &e, are comparable with — A,

a@

while those depending on wu, v and w are comparable with these quantities, and therefore in
considering the action of the fluid on the solid it is only necessary to consider the quantities
u, v and w. Now since, neglecting /, the velocity at M’ is tangential to the surface at M,
uw, v, and w are the components of a certain velocity V tangential to the surface. The pressure p,
must be zero; for changing the signs of w, v, and w the circumstances concerned in its production
remain the same, whereas its analytical expression changes sign. ‘The tangential force at M will
be in the direction of V, and proportional to it, and consequently its components along the axes
of w, y, x will be proportional to wu, v, w. Reckoning the tangential force positive when,
1, m, and n being positive, the solid is urged in the positive directions of w, y, x, the resolved
parts of the tangential force will therefore be vw, vv, vw, where y must evidently be positive,
since the effect of the forces must be to check the relative motion of the fluid and solid. The normal
pressure of the fluid on the solid being equal to g, its components will be evidently la, ma, nw.

Suppose now the solid to be in motion, and let w’, v’, w’ be the resolved parts of the velocity
of the point M of the solid, and w’, w’, w” the angular velocities of the solid. By hypothesis,
the forces by which the pressure at any point differs from the normal pressure due to the action of
the molecules supposed to be in a state of relative equilibrium about that point are independent of
any velocity of translation or rotation. Supposing then linear and angular velocities equal and
opposite to those of the solid impressed both on the solid and on the fluid, the former will be for
an instant at rest, and we have only to treat the resulting velocities of the fluid as in the first case.

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 301

Hence P' =la@ + v(u—w’), &c.; and in the equations (8) we may employ the actual velocities
u, v, Ww, since the pressures P, Q, R are independent of any motion of translation and rotation
common to the whole fluid. Hence the equations P’ = P, &c. give us

du du dv du dw
= ‘ esr yi oT == (= =) pase rp te
ae a Rawat Wag “| (aa ‘} ea dy * dx abe * dx

which three equations with (15) are those which must be satisfied at the surface of a solid, together
with the equation L=0. It will be observed that in the case of a free surface the pressures
P’, Q’, R’ are given, whereas in the case of the surface of a solid they are known only by the
solution of the problem. But on the other hand the form of the surface of the solid is given,
whereas the form of the free surface is known only by the solution of the problem.

Dubuat found by experiment that when the mean velocity of water flowing through a pipe is
less than about one inch in a second, the water near the inner surface of the pipe is at rest.
If these experiments may be trusted, the conditions to be satisfied in the case of small velocities
are those which first occurred to me, and which are included in those just given by supposing py =o .

I have said that when the velocity is not very small the tangential force called into action by
the sliding of water over the inner surface of a pipe varies nearly as the square of the velocity.
This fact appears to admit of a natural explanation. When a current of water flows past an
obstacle, it produces a resistance varying nearly as the square of the velocity. Now even if the
inner surface of a pipe is polished we may suppose that little irregularities exist, forming so many
obstacles to the current. Each little protuberance will experience a resistance varying nearly as
the square of the velocity, from whence there will result a tangential action of the fluid on the
surface of the pipe, which will vary nearly as the square of the velocity ; and the same will be true
of the equal and opposite reaction of the pipe on the fluid. The tangential force due to this cause
will be combined with that by which the fluid close to the pipe is kept at rest when the velocity
is sufficiently small.

There remains to be considered the case of two fluids having a common surface. Let
uw, v, w, », © denote the quantities belonging to the second fluid corresponding to uw, &c.
belonging to the first. Together with the two equations ZL = 0 and (15) we shall have in this
ease the equation derived from (15) by putting w’, vo’, w’ for uw, v, w; or, which comes to the
same, we shall have the two former equations with

)}=o, ea eAeccon Ul)

l(u—w') +m(v—-v) +n (w—w’) = 0....0.... eee (18)
If we consider the principles on which equations (17) were formed to be applicable to the
present case, we shall have six more equations to be satisfied, namely (17), and the three
equations derived from (17) by interchanging the quantities referring to the two fluids, and
changing the signs of /, m, m. These equations give the value of a, and leave five equations
of condition. If we must suppose y= co, as appears most probable, the six equations above
mentioned must be replaced by the six w’ =u, v' =v, w’ = w, and

Ip — wf (u,v, w) = lp’ -n'f(w’, v', w'), &e.,

f (u,v, w) denoting the coefficient of » in the first of equations (17). We have here six equations
of condition instead of five, but then the equation (18) becomes identical.

7. The most interesting questions connected with this subject require for their solution a
knowledge of the conditions which must be satisfied at the surface of a solid in contact with
the fluid, which, except perhaps in case of very small motions, are unknown. It may be
well however to give some applications of the preceding equations which are independent of
these conditions. Let us then in the first place consider in what manner the transmission of
sound in a fluid is affected by the tangential action, ‘To take the simplest case, suppose that
no forces act on the fluid, so that the pressure and density are constant in the state of

302 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

equilibrium, and conceive a series of plane waves to be propagated in the direction of the
axis of a, so that w=f(a,¢), v=0, w=0. Let p be the pressure, and p, the density of
the fluid when it is in equilibrium, and put p=p,+p'. Then we have from equations (11)
and (12), omitting the square of the disturbance,

dp d d i @
pol cle t age! 2 re Re eT)
a

= + = — a

pat dn a aE. da d
Let AAp be the increment of pressure due to a very small increment Ap of density, the
temperature being unaltered, and let m be the ratio of the specific heat of the fluid when
the pressure is constant to its specific heat when the volume is constant; then the relation
between p’ and p will be

4
3

p= Wid (pip) MARA 0s re (20)
Eliminating p’ and p from (19) and (20) we get

d@ ~ "* da 3p, dtd#
2

xv . 27e
To obtain a particular solution of this equation, let «w= (é) cos : + w (é) sin = Sub-

stituting in the above equation, we see that @(¢) and (4) must satisfy the same equation,
namely,

4or* 167°'n .,
p(t) + eo m Ag (t) + 3n"p Pp (4) = 0,
the integral of which is
2Qrb , . 2rbt
$ () = 6" (Coos cies +C sin") -
8a 167° 2° ; ‘ 2
where c= an'p,” b=mA — Date?” C and C’ being arbitrary constants. Taking the same
/ P,

expression with different arbitrary constants for y(t), replacing products of sines and cosines
by sums and differences, and combining the resulting sines and cosines two and two, we see
that the resulting value of « represents two series of waves propagated in opposite directions.
Considering only those waves which are propagated in the positive direction of x, we have

u = C,e~“ cos {= (bt -— aw) +C, Ete et tee

We see then that the effect of the tangential force is to make the intensity of the sound
diminish as the time increases, and to render the velocity of propagation less than what it
would otherwise be. Both effects are greater for high, than for low notes; but the former
depends on the first power of », while the latter depends only on ». It appears from the
experiments of M. Biot, made on empty water pipes in Paris, that the velocity of propagation
of sound is sensibly the same whatever be its pitch. Hence it is necessary to suppose that for air

a - is insensible compared with 4 or PI am not aware of any similar experiments made
p:

‘ ‘

2
on water, but the ratio of (4) to A would probably be insensible for water also. The

P,
diminution of intensity as the time increases is, in the case of plane waves, due entirely to
friction ; but as we do not possess any means of measuring the intensity of sound the theory
cannot be tested, nor the numerical value of u determined, in this way.

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 303

The velocity of sound in air, deduced from the note given by a known tube, is sensibly
less than that determined by direct observation. Poisson thought that this might be due to the
retardation of the air by friction against the sides of the tube. But from the above investigation
it seems unlikely that the effect produced by that cause would be sensible.

The equation (2!) may be considered as expressing in all cases the effect of friction; for
we may represent an arbitrary disturbance of the medium as the aggregate of series of plane
waves propagated in all directions.

8. Let us now consider the motion of a mass of uniform inelastic fluid comprised
between two cylinders having a common axis, the cylinders revolving uniformly about their
axis, and the fluid being supposed to have attained its permanent state of motion. Let the
axis of the cylinders be taken for that of x, and let q be the actual velocity of any particle,

so that w= —qsin@, v =qcos 0, w =0, r and @ being polar co-ordinates in a plane parallel to wy.
af d@f df idf 1¢f
Observing that da af as + oat ae’ where f is any function of x and y, and

d : 2 :
that 5 = 0, we have from equations (13), supposing after differentiation that the axis of w

coincides with the radius vector of the point considered, and omitting the forces, and the part
of the pressure due to them,

dp gq
—-p-—=0,
dr PS
d 1d
= Ae oe SUNOS) omrincs stapcclesinvans cam Cle)

and the equation of continuity is satisfied identically.
: Cc
The integral of (22) is digs naa C'r.

If a is the radius of the inner, and 6 that of the outer cylinder, and if g,, gq, are the
velocities of points close to these cylinders respectively, we must have g=q, when r=a, and
q=q when r=b, whence

1 ab
q -5+| (bq, — @q:) Salt Giger aig) pds ebieatevis~'os «ies (23)

If the fluid is infinitely extended, b = «, and

These cases of motion were considered by Newton, (Principia, Lib. 11. Prop. 51.) The
hypothesis which I have made agrees in this case with his, but he arrives at the result that
the velocity is constant, not, that it varies inversely as the distance. This arises from his having
taken, as the condition of there being no acceleration or retardation of the motion of an annulus,
that the force tending to turn it in one direction must be equal to that tending to turn it in
the opposite, whereas the true condition is that the moment of the force tending to turn it
one way must be equal to the moment of the force tending to turn it the other. Of course,
making this alteration, it is easy to arrive at the above result by Newton’s reasoning. The
error just mentioned vitiates the result of Prop. 52. It may be shown from the general equations

304 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

that in this case a permanent motion in annuli is impossible, and that, whatever may be the
law of friction between the solid sphere and the fluid. Hence it appears that it is necessary to
suppose that the particles move in planes passing through the axis of rotation, while they at
the same time move round it. In fact, it is easy to see that from the excess of centrifugal
force in the neighbourhood of the equator of the revolving sphere the particles in that part
will recede from the sphere, and approach it again in the neighbourhood of the poles, and this
circulating motion will be combined with a motion about the axis. If however we leave the
centrifugal force out of consideration, as Newton has done, the motion in annuli becomes
possible, but the solution is different from Newton’s, as might have been expected.

The case of motion considered in this article may perhaps admit of being compared with
experiment, without knowing the conditions which must be satisfied at the surface of a solid.
A hollow, and a solid cylinder might be so mounted as to admit of being turned with different
uniform angular velocities round their common axis, which is supposed to be vertical. If both
cylinders are turned, they ought to be turned in opposite directions, if only one, it ought to
be the outer one; for if the inner were made to revolve too fast, the fluid near it would have
a tendency to fly outwards in consequence of the centrifugal force, and eddies would be produced.
As long as the angular velocities are not great, so that the surface of the liquid is very nearly
plane, it is not of much importance that the fluid is there terminated; for the conditions which
must -be satisfied at a free surface are satisfied for any section of the fluid made by a horizontal
plane, so long as the motion about that section is supposed to be the same as it would be
if the cylinders were infinite. The principal difficulty would probably be to measure accurately
the time of revolution, and distance from the axis, of the different annuli. This would probably
be best done by observing motes in the fluid. It might be possible also to discover in this
way the conditions to be satisfied at the surface of the cylinders; or at least a law might be
suggested, which could be afterwards compared more accurately with experiment by means of
the discharge of pipes and canals.

If the rotations of the cylinders are in opposite directions, there will be a certain distance from
the axis at which the fluid will not revolve at all. Writing — q, for q, in equation (23), we have

b L 2
for this distance a are 24) -
2F an

9. Although the discharge of a liquid through a long straight pipe or canal, under given
circumstances, cannot be calculated without knowing the conditions to be satisfied at the surface of
contact of the fluid and solid, it may be well to go a certain way towards the solution.

Let the axis of x be parallel to the generating lines of the pipe or canal, and inclined at
an angle a to the horizon; let the plane yx be vertical, and let y and x be measured downwards.
The motion being uniform, we shall have ~=0, v=0, w =f («,y), and we have from equations (13)
d d dw dw
=o Te = EP OS & 7-8 sinatyp ( + ae

d
In the case of a canal - = 0; and the calculation of the motion in a pipe may always be reduced
z

o1e Oe : dp.
to that of the motion in the same pipe when 2 is supposed to be zero, as may be shown by
z
reasoning similar to Dubuat’s. Moreover the motion in a canal is a particular case of the motion

d
in a pipe. For consider a pipe for which = = 0, and which is divided symmetrically by the

F Pyar, dw
plane vz. From the symmetry of the motion, it is clear that we must have ay = 0 when = =0;

AND THE EQUILIBRIUM AND MOTION OF ELASTIC: SOLIDS. 305

but this is precisely the condition which would have to be satisfied if the fluid had a free surface
coinciding with the plane vx; hence we may suppose the upper half of the fluid removed, without
affecting the motion of the rest, and thus we pass to the case of a canal. Hence it is the same
thing to determine the motion in a canal, as to determine that in the pipe formed by completing the
canal symmetrically with respect to the surface of the fluid.

We have then, to determine the motion, the equation

& i
Ww ge ¢ gp sina cae
dx’ dy m
In, the case of a rectangular pipe, it would not be difficult to express the value of w at any point

in terms of its values at the several points of the perimeter of a section of the pipe. In the case

of a cylindrical pipe the solution is extremely easy: for if we take the axis of the pipe for that of
* w .

z, and take polar co-ordinates r, @ in a plane parallel to wy, and observe that 797 0, since the

motion is supposed to be symmetrical with respect to the axis, the above equation becomes

d’ 1 dw sin
w 24 gp sina _

dr rdr mm

Let a be the radius of the pipe, and U the velocity of the fluid close to the surface; then,
integrating the above equation, and determining the abitrary constants by the conditions that w
shall be finite when r = 0, and.w.= U when-r = a, we have

wo RAR (a? — 1°) + OU.
au

SECTION II.

Objections to Lagrange’s proof of the theorem that if udx +vdy +wdz is an exact
differential at any one instant it is always so, the pressure being supposed equal
in all directions. Principles of M. Cauchy's proof. A new proof of the theorem.
A physical interpretation of the circumstance of the above expression being an
exact differential.

10. THe proof of this theorem giyen by Lagrange depends on the legitimacy of supposing
u, v and w capable of expansion according to positive integral powers of ¢, for a sufficiently
small finite value of ¢, It is clear that the expansion cannot contain negative powers of ¢, since
u, v and w are supposed to be finite when ¢=0; but it may be objected to Lagrange’s proof
that there are functions of ¢ of which the expansion contains fractional powers of ¢, and that we do
not know but that w, v and w may be such functions. This objection has been considered by
Mr. Power*, who has shown that the theorem is true if we suppose w, v and w capable of
expansion according to any powers of ¢. Still the proof remains unsatisfactory, in fact inconclusive,

1

es

for these are functions of f, (for instance e~", ¢ log ¢t,) which do not admit of expansion according

* Cambridge Philosophical Transactions, Vol, vit. Part 3

Vor. VIII. Parr III. Rr

306 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

to powers of ¢, integral or fractional, and we do not know but that w, v and w may be functions of
this nature. I do not here mention the proof which Poisson has given of the theorem in his
Traité de Méchanique, because it appears to me liable to an objection to which I shall presently
have occasion to refer: in fact, Poisson himself did not think the theorem generally true.

It is remarkable that Mr. Power’s proof, if it were legitimate, would establish the theorem
even when account is taken of the variation of pressure in different directions, according to the

ey : y d ‘ F
theory explained in Section 1, if we suppose that = =0. To show this we have only got to treat

equations (12) as Mr. Power has treated the three equations of fluid motion formed on the ordinary
hypothesis. Yet.in this case the theorem is evidently untrue. Thus, conceive a mass of fluid which
is bounded by a solid plane coinciding with the plane yz, and which extends infinitely in every
direction on the positive side of the axis of 2, and suppose the fluid at first to be at rest. Suppose
now the solid plane to be moved in any manner parallel to the axis of y; then, unless the solid
plane exerts no tangential force on the fluid, (and we may suppose that it does exert some,) it
is clear that at a given time we shall have wu =0, v= f(a), w =0, and therefore wdx + vdy + wdz
will not be an exact differential. It will be interesting then to examine in this case the nature
of the function of ¢ which expresses the value of v.

Supposing X, Y, Z to be zero in equations (12), and observing that in the case considered

we have Zn, we get

Sapa eet ae Me ae pans Wbaise cleo tletas atone (24)

Differentiating this equation m — 1 times with respect to ¢, we easily get

d"v p\" dv 5
Cie (*) da®”
but when ¢=0, v =0 when w>0, and therefore for a given value of a all the differential
coefficients of v with respect to ¢ are zero. Hence for indefinitely small values of ¢ the value of
» at a given point increases more slowly than if it varied ultimately as any power of ¢, however
great; hence v cannot be expanded in a series according to powers of ¢. This result’ is independent
of the condition to be satisfied at the surface of the solid plane.

I think what has just been proved shows clearly that Lagrange’s proof of the theorem
considered, even with Mr. Power’s improvement of it, is inadmissible.

11. The theorem is however true, and a proof of it has been given by M. Cauchy*, which
appears to me perfectly free from objection, and which is very simple in principle, although it
depends on rather long equations. M. Cauchy first eliminates p from the three equations of
ip dp
dudy dyda
variables from w, y, %, ¢ to a, 6, c, ¢, where a, b, c are the initial co-ordinates of the particles.
The three transformed equations admit each of being once integrated with respect to ¢; and
determining the arbitrary functions of a, b, ¢ by the initial values of u, v and w, the three
integrals have the form

motion by means of the conditions that &e., he then changes the independent

on = Fw + Gow’ + How”, &e.,

* Mémoire sur la Théorie des Ondes, in the first volume of | equations (16), This equation may be obtained in the same
the Mémoires présentés & U'Institut, M. Cauchy has not had | manner in the more general case in which p is supposed to be a
occasion to enunciate the theorem, but it is contained in his | function of p.

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 307

w, w and w” denoting here the same as in Art. 2, and w, &c. denoting the initial values of
w, &c. for the same particle. Solving the above equations with respect to w’, w’ and w’”, the

resulting cquations are
a (= , ada , da 4 P
w =>|= o + —w," +— , &.
S\da“ * db“ ~ de :
where S is a function of the differential coefficients of 2, y and x with respect to a, b and e,

which by the condition of continuity is shown to be equal to Le p, being the initial density about
Po

: P : ; : ; da :
the particle whose density at the time considered is p. Since 7 &e. are finite, (for to suppose
a

them infinite would be equivalent to supposing a discontinuity to exist in the fluid,) it follows at
once from the preceding equations that if w, = 0, w,” = 0, w,” = 0, that is if w,da + v,db + wide
be an exact differential, either for the whole fluid or for any portion of it, then shall w’ =0,
w =0, w” =0, i.e. uda +vdy + wdz will be an exact differential, at any subsequent time,
either for the whole mass or for the above portion of it.

12. It is not from seeing the smallest flaw in M. Cauchy’s proof that I propose a new one,
but because it is well to view the subject in different lights, and because the proof which I am
about to give does not require such long equations. It will be necessary in the first place to prove

the following lemma.

Lemma. If w,, ,..-w, are m functions of ¢, which satisfy the m differential equations

dw

ie = Pio, + Qos... + Vien
dw,

=e = Pw, + Q,,Wo+-- + Viens

where P,, Q,...V,, may be functions of ¢, w,...w,, and if when w, = 0, w, = 0...w, = 0, none of the
quantities P,, ...V,, is infinite for any value of ¢ from 0 to 7, and if w,...w, are each zero when
# = 0, then shall each of these quantities remain zero for all values of ¢ from 0 to 7.

Demonstration. Let 7 be a finite value of ¢, then by hypothesis + may be taken so small
that the values of w,...w, are sufficiently small to exclude all values which might render any one of
the quantities P,...V,, infinite. Let Z be a superior limit to the numerical values of the

several quantities P,...V,, for all values of ¢ from 0 to 7; then it is evident that w,...w, cannot
increase faster than if they satisfied the equations

dw
Gp = Elwi + Wy 2 + wy);
tenets recess ean ees deepsea ditty a0)
dw, \
Fron L (ow + Wy... + @n)s

vanishing in this case also when ¢=0. But if w, + w,... + w, =Q, we have by adding together

the above equations

if now © be not equal to zero, dividing this equation by © and integrating, we have

Q = Cem;
RR2

308 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

but no value of C different from zero will allow Q to vanish when ¢ =0, whereas by hypothesis
it does vanish; hence Q=0; but Q is the sum of m quantities which evidently cannot be
negative, and therefore each of these must be zero. Since then w,...w, would have to be equal
to zero for all values of ¢ from 0 to 7 even if they satisfied equations (26), they must 4 fortiori
be equal to zero in the actual case, since they satisfy equations (25). Hence there is no value of
t from 0 to J at which any one of the quantities w,...w, can begin to differ from zero, and
therefore these quantities must remain equal to zero for all values of ¢ from 0 to 7.

This lemma might be extended to the case in which 2 = 9, with certain restrictions as to
the convergency of the series. We may also, instead of the integers 1, 2...m, have a continuous
variable a which varies from 0 to a, so that w isa function of the independent variables a and ¢,
satisfying the differential equation

d a
- = [¥@ w, thwda,
0

where y(a, 0, ¢) does not become infinite for any value of a from 0 to @ combined with any
value of ¢ from 0 to J. It may be shown, just as before, that if w = 0 when ¢ =0 for all values
of a from 0 to a, then must w = 0 for all values of ¢ from 0 to [. The proposition might be
further extended to the case in which a = ©, with a certain restriction as to the convergency of the
integral, but equations (25) are already more general than I shall have occasion to employ.

It appears to me to be sometimes assumed as a principle that two variables, functions of
another, ¢, are proved to be equal for all values of ¢ when it is shown that they are equal for a
certain value of ¢, and that whenever they are equal for the same value of ¢ their increments for
the same increment of ¢ are ultimately equal. But. according to this principle, if two curves
could be shown always to touch when they meet they must always coincide, a conclusion
manifestly false. I confess I cannot sée that Newton in his Principia, Lib. 1. Prop. 40 has
proved more than that if the velocities of the two bodies are equal at equal distances, the
increments of those velocities for equal increments of the distances are ultimately equal: at least
something additional seems required to put the proof quite out of the reach of objection. Again
it is usual to speak of the condition, that the motion of a particle of fluid in contact with the
surface of a solid at rest is tangential to the surface, as the same thing as the condition that the
particle shall always remain in contact with the surface. That it is the same thing might be
shown by means of the lemma in this article, supposing the motion continuous; but independently
of proof I do not see why a particle should not move in a curve not coinciding with the surface,
but touching it where it meets it. The same remark will apply to the condition that a particle
which at one instant lies in a free surface, or is in contact with a solid, shall ultimately lie in the
free surface, or be in contact with the solid, at the consecutive instant. I refer here to the more
general case in which the solid is at rest or in motion, For similar reasons Poisson’s proof of the
Hydrodynamical theorem which forms the principal subject of this section has always appeared
to me unsatisfactory, in fact far less satisfactory than Lagrange’s. I may add that Poisson’s

proof, as well as Lagrange’s, would apply to the case in which friction is taken into account, in
which case the theorem is not true.

13. Supposing p to be a function of at the ordinary equations of Hydrodynamics
af(p) Du = af(p) Dv df(p) Dw
= X-—, = Y- : Sate nteinseaeia asia rian
= dw Dt dy ee gta eee” “

The forces X, Y, Z will here be supposed to be such that Xda + Ydy+Zdzx is an exact
differential, this being the case for any forces emanating from centres, and varying as any functions

4

ee

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 309

of the distances. Differentiating the first of equations (27) with respect to y, and the second

: : : Du Dye
with respect to v, subtracting, putting for —— and —— their values, adding and subtracting

Dt Dt

— —, and employing the notation of Art. 2, we obtain
z

Dw” du , ty dv ,, du dv ,,

= 7) = |= 4 == cloisinatalyietaisfetstatatatersia|O
Dt dz dz” Comeap! a)
By treating the first and third, and then the second and third of equations (27) in the same
manner, we should obtain two more equations, which may be got at once from that which has
just been found by interchanging the requisite quantities. Now for points in the interior of

the mass the differential coefficients a &e. will not be infinite, on account of the continuity
% )

of the motion, and therefore the three equations just obtained are a particular case of equations (25).

If then wdx + vdy+wdzx is an exact differential for any portion of the fluid when ¢= 0,

that is, if w’, w” and w” are each zero when ¢=0, it follows from the lemma of the last

article that w’, w” and w” will be zero for any value of ¢, and therefore wdw + vdy + wdz

will always remain an exact differential. It will be observed that it is for the same portion

of fluid, not for the fluid occupying the same portion of space, that this is true, since equations
, tf

: : F ; D d
(28), &c. contain the differential coefficients Fi &c., and not ap &e.

14. The circumstance of wdxv + vdy + wdz being an exact differential admits of a physical
interpretation which may be noticed, as it is well to view a subject of this nature in different
lights.

Conceive an indefinitely small element of a fluid in motion to become suddenly solidified,
and the fluid about it to be suddenly destroyed; let the form of the element be so taken
that the resulting solid shall be that which is the simplest with respect to rotatory motion,
namely, that which has its three principal moments about axes passing through the centre
of gravity equal to each other, and therefore every axis passing through that point a principal
axis, and let us enquire what will be the linear and angular motion of this element just
after solidification.

By the instantaneous solidification, velocities will be suddenly generated or destroyed in the
different portions of the element, and a set of mutual impulsive forces will be called into
action. Let a, y, x be the co-ordinates of the centre of gravity G of the element at the
instant of solidification, w+’, y+y', x + x’ those of any other point in it. Let uw, v, w be
the velocities of G along the three axes just before solidification, u,v, w the relative velocities
of the point whose relative co-ordinates are a’, y’, 3’. Let u, 0, w be the velocities of G, uw, v,, w,
the relative velocities of the point above mentioned, and w,w’, w” the angular velocities just
after solidification. Since all the impulsive forces are internal, we have

W=U, D=V, D=W,
We have also, by the principle of the conservation of areas,
=m ty (w, — w') - x (v, — v')t =0, &e.,

m denoting an element of the mass of the element considered. But uw, = wx’ —w”y', w’ is

4 du ui dhe
ultimately equal to Fe eter et Magee a, and similar expressions hold good for the other
wv x

310 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

quantities. Substituting in the above equations, and observing that Smyx = 2m'z a
=Ima'y' =0, and [m2 = Smy? = mz, we have

We see then that an indefinitely small element of the fluid, of which the three principal moments
about the centre of gravity are equal, if suddenly solidified and detached from the rest of
the fluid will begin to move with a motion simply of translation, which may however vanish,
or a motion of translation combined with one of rotation, according as wdw + vdy + wdz is,
or is not an exact differential, and in the latter case the angular velocities will be the same
as in Art. 2.

The principle which forms the subject of this section might be proved, at least in the case
of a homogeneous incompressible fluid, by considering the change in the motion of a spherical
element of the fluid in the indefinitely small time dt. This method of proving the principle
would show distinctly its intimate connexion with the hypothesis of normal pressure, or the
equivalent hypothesis of the equality of pressure in all directions, since the proof depends on
the impossibility of an angular velocity being generated in the element in the indefinitely small
time d¢ by the pressure of the surrounding fluid, inasmuch as the direction of the pressure at
any point of the surface ultimately passes through the centre of the sphere. The proof I
speak of is however less simple than the one already given, and would lead me too far from
my subject.

SECTION III.

Application of a method analogous to that of Sect. 1. to the determination of the equations
of equilibrium and motion of elastic solids.

15. Att solid bodies are more or less elastic, as is shown by the capability they possess
of transmitting sound, and vibratory motions in general. The solids considered in this section
are supposed to be homogeneous and uncrystallized, so that when in their natural state the
average arrangement of their particles is the same at one point as at another, and the same
in one direction as in another. The natural state will be taken to be that in which no forces
act on them, from which it may be shown that the pressure in the interior is zero at all
points and in all directions, neglecting the small pressure depending on attractions of the
nature of capillary attraction.

Let «x, y, # be the co-ordinates of any point P in the solid considered when in its natural
state, a, (3, yy the increments of those co-ordinates at the time considered, whether the body
be in a state of constrained equilibrium or of motion. It will be supposed that a, and Y
are so small that their squares and products may be neglected. All the theorems proved in
Art. 2, with reference to linear and angular velocities will be true here with reference to linear
and angular displacements, since these two sets of quantities are resolved according to the same
laws, as long as the angular displacements are supposed to be very small. Thus, the most
general displacement of a very small element of the solid consists of a displacement of translation,
an angular displacement, and three displacements of extension in the direction of three rectangular
axes, which may be called in this case, with more propriety than in the former, awes of
extension. The three displacements of extension may be resolved into two displacements of
shifting, each in two dimensions, and a displacement of uniform dilatation, positive or negative.
The pressures about the element considered will depend on the displacements of extension only;

v

—

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 311

there may also, in the case of motion, be a small part depending on the relative velocities,
but this part may be neglected, unless we have occasion to consider the effect of the internal
friction in causing the vibrations of solid bodies to subside. It has been shown (Art. 7.) that
the effect of this cause is insensible in the case of sound propagated through air; and there
is no reason to suppose it greater in the case of solids than in the case of fluids, but rather
the contrary. ‘The capability which solids possess of being put into a state of isochronous
vibration shows that the pressures called into action by small displacements depend on homo-
geneous functions of those displacements of one dimension. I shall suppose moreover, according
to the general principle of the superposition of small quantities, that the pressures due to
different displacements are superimposed, and consequently that the pressures are linear functions
of the displacements. Since squares of a, 8 and vy are neglected, these pressures may be referred
to a unit of surface in the natural state or after displacement indifferently, and a pressure which
is normal to any surface after displacement may be regarded as normal to the original position
of that surface. Let — 4d be the pressure corresponding to a uniform linear dilatation 6 when
the solid is in equilibrium, and suppose that it becomes — m 40d, in consequence of the heat
developed, when the solid is in a state of rapid vibration. Suppose also that a displacement
of shifting parallel to the plane wy, for which a=ka, B=—ky, y=0, calls into action a
pressure — Bk on a plane perpendicular to the axis of 2, and a pressure Bk on a plane
perpendicular to that of y; the pressures on these planes being equal and of opposite signs,
that on a plane perpendicular to the axis of = being zero, and the tangential forces on those
planes being zero, for the same reasons as in Sect. 1. It may also be shown as before that
it is necessary to suppose B positive, in order that the equilibrium of the solid medium may
be stable, and it is easy to see that the same must be the case with 4 for the same reason.

It is clear that we shall obtain the expressions for the pressures from those already found
for the case of a fluid by merely putting a, 8, y, B for u,v, w, un and — 48 or —mAO for Ds
according as we are considering the case of equilibrium or of vibratory motion, the body being in
the latter case supposed to be constrained only in so far as depends on the motion.

For the case of equilibrium then we have from equations (8)

da dp dy
P,=-Ad-2B ($= -8), T.\=-B (= + Fk BW ce eects awe (29)
: da dB dy) : ee : :
é being here = 3 (= + ay + =a) ; and the equations of equilibrium will be obtained from (12) by
putting = 0, p = — Ao, making the same substitution as before for u,v, w and ». We have

therefore, for the equations of equilibrium,
djda dB dy Pa da da
1(4 — (= 2) ( , —— in) 7% He Cor 0
px+sC Sia dx dy dx da® dy? de st 7)

In the case of a vibratory motion, when the body is in its natural state except so far as depends
on the motion, we have from equations (8)

2 d dB d
any heer (<- 38), ee B (E+), we, wen Ehaeasded (31)

and the equations of motion will be derived from (12) as before, only a &e. must be replaced by

da ‘ ‘
—, &e., and X, Y, Z put equal to zero. The equations of motion, then, are

dt*
d‘a d jda dp dy da da da
ee * AV ed eee eee Oat
iar Ts aA a ( 3 ) a ae a dy’ "de

dw dy dz ), &C. serene (82)

a

312 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

16. The conditions to be satisfied at the surface of the solid may be easily deduced from
the analogous conditions in the case of a fluid with a free surface, only it will be necessary
to replace the normal pressure II by an oblique pressure, of which the components will be
denoted by X,, Yi, Z,. We have then, making the necessary changes in the quantities involved
in (14),

da da dp da dy
143 + B20 +m (4 = + Ze) =o &e.
Mrildbpat { a it as) "(ae * ae) Se

for the case of equilibrium, and for the case of motion such as that just considered it will only be
necessary to replace A by mA in these equations. If we measure the angles of which /, m, m are
the cosines from the external normal, the forees X,, Y,, Z, must be reckoned positive when /, m and
n being positive, the surface of the solid is urged in the negative directions of «, y, x, and in other
cases the signs must be taken conformably.

If the solid considered is in a state of constraint when at rest, and is moreover put into a state
of vibration, the pressures and displacements due to these two causes must be calculated separately
and added together. If m were equal to 1, they could be calculated together from the same
equations.

SECTION IV.

Principles of Poisson's theory of elastic solids, and of the oblique pressures existing in
fluids in motion. Objections to one of his hypotheses, Reflections on the constitution,
and equations of motion of the luminiferous ether in vacuum.

17. In the twentieth Cahier of the Journal de [Ecole Polytechnique may be found a memoir
by Poisson, entitled Mémoire sur les Equations générales de ( Equilibre et du Mouvement des
Corps solides élastiques et des Flwides, which contains the substance of two memoirs presented
by him to the Academy, brought together with some additions. In this memoir the author
treats principally of the equations of equilibrium and motion of elastic solids, of the equations
of equilibrium of fluids, with reference especially to capillary attraction, and of the equations
of motion of fluids supposing the pressure not to be equal in all directions.

It is supposed by Poisson that all bodies, whether solid or Huid, are composed of ultimate
molecules, separated from each other by vacant spaces. In the cases of an uncrystallized solid
in its natural state, and of a fluid in equilibrium, he supposes that the molecules are arranged
irregularly, and that the average arrangement is the same in all directions. These molecules
he supposes to act on each other with forces, of which the main part is a force in the direction of the
line joining the centres of gravity, and varying as some function of the distance of these points,
and the remainder a secondary force, or it may be two secondary forces, depending on the
molecules not being mathematical points, He supposes that it is on these secondary forces that the
solidity of solid bodies depends. -He supposes however that in calculating the pressures these
secondary forces may be neglected, partly because they become insensible at much smaller distances
than the main part of the forces, and partly because they act, on the average, alike in all
directions. He supposes that the molecular force decreases very rapidly as the distance increases,
yet not so rapidly but that the sphere in which the molecular action is sensible contains an immense
number of molecules. He supposes consequently that in estimating the resultant force of a
hemisphere of the medium on a molecule in the centre of its base the action of the neighbouring
molecules, which are situated irregularly, may be neglected compared with the action of those

on 4)

AND THE EQILIBRIUM AND MOTION OF ELASTIC SOLIDS. 315

more remote, of which the average may be taken. The consequence of this supposition of course is
that the total action is normal to the base of the hemisphere, and sensibly the same for one
_molecule as for an adjacent one.

The rest of the reasoning by which Poisson establishes the equations of motion and equilibrium
of elastic solids is purely mathematical, sufficient data having been already assumed. It might
appear that the reasoning in Art. 16 of his memoir, by which the expression for N is simplified,
required the fresh hypothesis of a symmetrical arrangement of the molecules; but it really does not,
being admissible according to the principle of averages. Taking for the natural state of the body
that in which the pressure is zero, the equations at which Poisson arrives contain only one
unknown constant /, whereas the equations of Sect. 111. of this paper contain two, A or mA and B.
This difference depends on the assumption made by Poisson that the irregular part of the force
exerted by a hemisphere of the medium on a molecule in the centre of its base may be neglected in
comparison with the whole force. As a result of this hypothesis, Poisson finds that the change
in direction, and the proportionate change in length, of a line joining two molecules are continuous
functions of the co-ordinates of one of the molecules and the angles which determine the direction of
the line; whereas in Sect. 111., if we adopt the hypothesis of ultimate molecules at all, it is
allowable to suppose that these quantities vary irregularly in passing from one pair of molecules to
an adjacent pair. Of course the equations of Sect. 111. ought to reduce themselves to Poisson’s
equations for a particular relation between 4 and B. Neglecting the heat developed by compression,
as Poisson has done, and therefore putting m = 1, this relation is 4 = 5B.

18. Poisson’s theory of fluid motion is as follows. The time ¢ is supposed to be divided
into a number n of equal parts, each equal to 7. In the first of these the fluid is supposed to
be displaced as an elastic solid would be, according to Poisson’s previous theory, and therefore
the pressures are given by the same equations. If the causes preducing the displacement were
now to cease, the fluid would re-arrange itself, so that the average arrangement about each point
should be the same in all directions after a very short time. During this time, the pressures
would have altered, in an unknown manner, from those corresponding to a displaced solid to a

Dp : é 5 é

normal pressure equal to p+ De” the pressures during the alteration involving an unknown
function of the time elapsed since the end of the interval +. Another displacement and another
re-arrangement may now be supposed to take place, and so on. But since these very small
relative motions will take place independently of each other, we may suppose each displacement to
begin at the expiration of the time during which the preceding one is supposed to remain, and we
may suppose each re-arrangement to be going on during the succeeding displacements. Supposing
now n to become infinite, we pass to the case in which the fluid is supposed to be continually
beginning to be displaced as a solid would, and continually re-arranging itself so as to make the
average arrangement about each point the same in all directions.

Poisson’s equations (9), page 152, which are applicable to the motion of a liquid, or of an
elastic fluid in which the change of density is small, agree with equations (12) of this paper. For
the quantity ¢ is the pressure p which would exist at any instant if the motion were then to

: dt D ; _ ae ? :
cease, and the increment, Ace or = 7, of this quantity in the very small time 7 will depend
: dyt D ; dwt
only on the increment, har or a 7, of the density x¢ or p. Consequently the value of ¥ ai

F ‘ , F ; ‘ dyt
will be the same as if the density of the particle considered passed from y¢ to yt + = 7r in the
time + by a uniform motion of dilatation. I suppose that according to Poisson’s views such a
motion would not require a re-arrangement of the molecules, since the pressure remains, equal

Vout. VIII. Parr III. Ss

314 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

dwt ’ ;
in all directions. On this supposition we shall get the value of at from that of R’— K in

d d d 1 dyxt
the equations of page 140 by putting = a = -- == axe = We have therefore
dvt a dyt
ee a aE
: ic 1 dxt. : 5
Putting now for 8B +P’ its value 2ak, and for matey ae value given by equation (2),
x

the expression for @, page 152, becomes
a du dv dw
w=p+-(K+k (H+ oe ~).
PY; ( ) dw dy dz
Observing that a(K +) = 3, this value of @ reduces Poisson’s equations (9) to the
equations (12) of this paper.

Poisson himself has not made this reduction of his equations, nor any equivalent one, so that
his equations, as he has left them, involve two arbitrary constants. The reduction of these two
to one depends on the assumption that a uniform expansion of any particle does not require a
re-arrangement of the molecules, as it leaves the pressure still equal in all directions. If we do
not make this assumption, but retain the two arbitrary constants, the equations will be the same
as those which would be obtained by the method of this paper, supposing the quantity « of
Art. 3 not to be zero.

19. There is one hypothesis made in the common theory of elastic solids, the truth of
which appears to me very questionable. That hypothesis is the one to which I have already
alluded in Art. 17, respecting the legitimacy of neglecting the irregular part of the action of the
molecules in the immediate neighbourhood of the one considered, in comparison with the total
action of those more remote, which is regular. It is from this hypothesis that it follows as a
result that the molecules are not displaced among one another in an irregular manner, in
consequence of the directive action of neighbouring molecules. Now it is obvious that the
molecules of a fluid admit of being displaced among one another with great readiness. The
molecules of solids, or of most solids at any rate, must admit of new arrangements, for most solids
admit of being bent, permanently, without being broken. Are we then to suppose that when a
solid is constrained it has no tendency to relieve itself from the state of constraint, in consequence
of its molecules tending towards new relative positions, provided the amount of constraint be very
small? It appears to me to be much more natural to suppose @ priori that there should be some
such tendency.

In the case of a uniform dilatation or contraction of a particle, a re-arrangement of its
molecules would be of little or no avail towards relieving it from constraint, and therefore it is
natural to suppose that in this case there is little or no tendency towards such a re-arrangement.
It is quite otherwise, however, in the case of what I have called a displacement of shifting.
Consequently B will be less than if there were no tendency to a re-arrangement. On the
hypothesis mentioned in this article, of which the absence of such tendency is a consequence,
I have said that a relation has been found between 4 and B, namely 4=58B. It is natural
then to expect to find the ratio of A to B greater than 5, approaching more nearly to 5 as the
solid considered is more hard and brittle, but differing materially from 5 for the softer solids,
especially such as Indian rubber, or, to take an extreme case, jelly. According to this view the
relation 4 = 5B belongs only to an ideal elastic solid, of which the solidity, or whatever we please
te call the property considered, is absolutely perfect.

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 315

To show how implicitly the common theory of elasticity seems to be received by some, I may
mention that MM. Lamé and Clapeyron mention Indian rubber among the substances to which
it would seem they consider their theory applicable*. I do not know whether the coefficient of
elasticity, according to that theory, has been determined experimentally for Indian rubber, but
one would fancy that the cubical compressibility thence deduced, by a method which will be
seen in the next article, would turn out comparable with that of a gas.

20. I am not going to enter into the solution of equations (30), but I wish to make a
few remarks on the results in some simple cases.
If & be the cubical contraction due to a uniform pressure P, then will
3P
7 Sr
If a wire or rod, of which the boundary is any cylindrical surface, be pulled in the
direction of its length by a force of which the value, referred to a unit of surface of a
section of the rod, in P, the rod will extend itself uniformly in the direction of its length,
and contract uniformly in the perpendicular direction; and if e be the extension in the
direction of the length, and ¢ the contraction in any perpendicular direction, both referred to
a-unit of length, we shall have

k

A+B A-—-2B
Soetoro ty te
P

also, the cubical dilatation = e—2e = vie
If a cylindrical wire of radius be twisted by a couple of which the moment is M, and
if @ be the angle of torsion for a length z of the wire, we shall have

2Msz
6 = —_..
a Br*

The expressions for &, c, e and @, and of course all expressions of the same nature, depend
on the reciprocals of A and B. Suppose now the value of.e, or @, or any similar quantity
not depending on A alone, be given as the result of observation. It will easily be conceived
that we might find very nearly the same value for B whether we supposed 4 =5B or 4=nB,
where 7 may be considerably greater than 5, or even infinite. Consequently the observation of two
such quantities, giving very nearly the same value of B, might be regarded as confirming the
common equations.

If we denote by E the coefficient of elasticity when A is supposed to be equal to 5B
we have, neglecting the atmospheric pressure},

$2 2P = 2Mzx
¥ ‘ ~ gEr'*
If now we denote by E, the value of EH deduced from observation of the value of e, and by

#, the value of E obtained by observing the value of @, or else, which comes to the same,
by observing the time of oscillation of a known body oscillating by torsion, we shall haye

2 1 ( 1 1 E B h 1 6 1
= aa 2= whelce: = = ———- =
5E, A ewer ig Minn Cena atte
* Mémoires présentés & l'Institut, Tom, rv. p. 469. + Lamé, Cours de Physique, Tom. 1.

316 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

If A be greater than 5B, E, ought to be a little greater than E,. This appears to agree
with observation. ‘Thus the following numbers are given by M. Lamé* £, = 8000, E, = 7500 for
iron; E,=2510, E, = 2250 for brass+. The difference between the values of E, and E, is
attributed by M. Lamé to the errors to which the observation of the small quantity e is liable.
If the above numbers may be trusted, we shall have

A = 60000, B= 7500, = 8 for iron;

B
a = 13:21 for brass.

The cubical contraction & is almost too small to be made the subject of direct observation},
it is therefore usually deduced from the value of e, or from the coefficient of elasticity E

: 2 A ; : k
found in some other way. On the supposition of a single coefficient E, we have — = 3, but
e

Q?
“te k Pe
retaining the two, 4 and B, we have = ed =9 (1 + 2 = which will differ greatly
e

aw |
from 3 if 3 be much greater than 5. The whole subject therefore requires, I think, a careful

examination, before we can set down the values of the coefficients of cubical contraction of
different substances in the list of well ascertained physical data. The result, which is generally
admitted, that the ratio of the velocity of propagation of normal, to that of tangential vibrations
in a solid is equal to \/3, is another which depends entirely on the supposition that 4A =5B.
The value of m, again, as deduced from observation, will depend upon the ratio of 4 to B;
and it would -be highly desirable to have an accurate list of the values of m for different
substances, in hopes of thereby discovering in what manner the action of heat on those substances
is related to the physical constants belonging to them, such as their densities, atomic weights, &c.

The observations usually made on elastic solids are made on slender pieces, such as wires,
rods, and thin plates, In such pieces, all the particles being at no great distance from the
surface, it is easy to see that when any small portion is squeezed in one direction it has consider-
able liberty of expanding itself in a direction perpendicular to this, and consequently the
results must depend mainly on the value of B, being not very different from what they
would be if 4d were infinite. This is not so much the case with thick, stout pieces. If
therefore such pieces could be put into a state of isochronous vibration, so that the musical
notes and nodal lines could be observed, they would probably be better adapted than slender
pieces for determining the value of md. The value of m might be determined by comparing
the value of mA, deduced from the observation of vibrations, with the value of A, deduced
from observations made in cases of equilibrium, or, perhaps, of very slow motion.

21. The equations (82) are the same as those which have been obtained by different
authors as the equations of motion of the luminiferous ether in vacuum. Assuming for the present
that the equations of motion of this medium ought to be determined on the same principles as
the equations of motion of an elastic solid, it will be necessary to consider whether the equations
(32) are altered by introducing the consideration of a uniform pressure I] existing in the medium

* Lamé, Cours de Physique, Tom. 1. pressibility of solids which would be obtained from Poisson’s

+ These numbers refer to the French units of length and weight. | theory is in some cases as much as 20 or 30 times too great. See

+ I find however that direct experiments have been made by | the Report of the British Association for 1833, p. 453, or Archives
Prof. Oersted. According to these experiments the cubical com- | des découvertes, §c. for 1834, p. 94.

+

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 317

when in equilibrium; for we have evidently no right to assume, either that no such pressure
exists, or, supposing it to exist, that the medium would expand itself but very slightly if it
were removed. It will now no longer be allowable to confound the pressure referred to a unit
of surface as it was, in the position of equilibrium of the medium, with the pressure referred to a
unit of surface as it actually is. The latter mode of referring the pressure is more natural, and
will be more convenient. Let the pressure, referred to a unit of surface as it is, be resolved
into a normal pressure I] + p, and a tangential pressure ¢,. All the reasoning of Sect. 111,
will apply to the small forces p, and ¢,; only it must be remembered that in estimating the
whole oblique pressure a normal pressure II must be compounded with the pressures given by
equations (31). In forming the equations of motion, the pressure [1 will not appear, because
the resultant force due to it acting on the element of the medium which is considered is zero.
The equations (32) will therefore be the equations of motion required.

If we had chosen to refer the pressure to a unit of surface in the original state of the
surface, and had resolved the whole pressure into a pressure [1+ p, normal to the original
position of the surface, and a pressure ¢, tangential to that position, the reasoning of Sect. 111.
would still have applied, and we should have obtained the same expressions as in (31) for the
pressures P,, 7, &c., but the numerical value of A would have been different. According to
this method, the pressure I] would have appeared in the equations of motion. It is when the
pressures are measured according to the method which I have adopted that it is true that
the equilibrium of the medium would be unstable if either 4 or B were negative. I must
here mention that from some oversight the right-hand sides of Poisson’s equations, at page 68
of the memoir to which I have referred, are wrong. The first of these equations ought to

- I (eu - du =) : Il du
p dat dy” de
other two equations.

It is sometimes brought as an objection to the equations of motion of the luminiferous
ether, that they are the same as those employed for the motion of solid bodies, and that it
seems unnatural to employ the same equations for substances which must be so differently
constituted. It was, perhaps, in consequence of this objection that Poisson proposes, at
page 147 of the memoir which I have cited, to apply to the calculation of the motion of the
luminiferous ether the same principles, with a certain modification, as those which he employed
in arriving at his equations (9) page 152, i.e. the equations (12) of this paper. That modi-
fication consists in supposing that a certain function of the time @(¢) does, not vary very
rapidly compared with the variation of the pressure. Now the law of the transmission of a
motion transversal to the direction of propagation depending on equations (12) of this paper
is expressed, in the simplest case, by the equation (24); and we see that this law is the
same as that of the transmission of heat, a law extremely different from that of the trans-
mission of vibratory motions. It seems therefore unlikely that these principles are applicable
to the calculation of the motion of light, unless the modification which I have mentioned be
so great as wholly to alter the character of the motion, that is, unless we suppose the pressure
to vary extremely fast compared with the function @(¢), whereas in ordinary cases of the
motion of fluids the function @(¢) is supposed to vary extremely fast compared with the pressure.

Another view of the subject may be taken which I think deserves notice. Before explaining
this view however it will be necessary to define what I mean in this paragraph by the word
elasticity. here are two distinct kinds of elasticity; one, that by which a body which is
uniformly compressed tends to regain its original volume, the other, that by which a body which is
constrained in a manner independent of compression tends to assume its original form. ‘The
constants d.and B of Sect. 111. may be taken as measures of these two kinds of elasticity. In
the present paragraph, the word will be used to denote the second kind. Now many highly

, instead of — , and similar changes must be made in the

contai -
p dv

318 Mr. STOKES, ON THE FRICTION OF FLUIDS IN MOTION,

elastic substances, as iron, copper, &c., are yet to a very sensible degree plastic. The plasticity of
lead is greater than that of iron or copper, and, as appears from experiment, its elasticity less. On
the whole it is probable that the greater the plasticity of a substance the less its elasticity, and
vice versa, although this rule is probably far from being without exception. When the plasticity
of the substance is still further increased, and its elasticity diminished, it passes into a viscous
fluid. There seems no line of demarcation between a solid and a viscous fluid. In fact, the
practical distinction between these two classes of bodies seems to depend on the intensity of
the extraneous force of gravity, compared with the intensity of the forces by which the parts
of the substance are held together. Thus, what on the Earth is a soft solid might, if carried
to the Sun, and retained at the same temperature, be a viscous fluid, the force of gravity at
the surface of the Sun being sufficient to make the substance spread out and become level at
the top: while what on the Earth is a viscous fluid might on the surface of Pallas be a soft solid.
The gradation of viscous, into what are called perfect fluids seems to present as little abruptness as
that of solids into viscous fluids; and some experiments which have been made on the sudden
conversion of water and ether into vapour, when enclosed in strong vessels and exposed to high
temperatures, go towards breaking down the distinction between liquids and gases.

According to the law of continuity, then, we should expect the property of elasticity to run
through the whole series, only, it may become insensible, or else may be masked by some other
more conspicuous property. It must be remembered that the elasticity here spoken of is that
which consists in the tangential force called into action by a displacement of continuous sliding:
the displacements also which will be spoken of in this paragraph must be understood of such
displacements as are independent of condensation or rarefaction. Now the distinguishing property
of fluids is the extreme mobility of their parts. According to the views explained in this article,
this mobility is merely an extremely great plasticity, so that a fluid admits of a finite, but
exceedingly small amount of constraint before it will be relieved from its state of tension by its
molecules assuming new positions of equilibrium. Consequently the same oblique pressures can be
called into action in a fluid as in a solid, provided the amount of relative displacement of the
parts be exceedingly small. All we know for certain is that the effect of elasticity in fluids,
(elasticity of the second kind be it remembered,) is quite insensible in cases of equilibrium, and
it is probably insensible in all ordinary cases of fluid motion. Should it be otherwise, equations (8)
and (12) will not be true, or only approximately true. But a little consideration will show that
the property of elasticity may be quite insensible in ordinary cases of fluid motion, and may yet
be that on which the phenomena of light entirely depend. When we find a vibrating string,
the small extent of vibration of which can be actually seen, filling a whole room with sound,
and remember how rapidly the intensity of the vibrations of the air must diminish as the distance
from the string increases, we may easily conceive how small in general must be the amount
of the relative motion of adjacent particles of air in the case of sound. Now the extent of
the vibration of the ether, in the case of light, may be as small compared with the length of a
wave of light as that of the air is compared with the length of a wave of sound: we have no
reason to suppose it otherwise. When we remember then that the length of a wave of sound in air
varies from some inches to several feet, while the greatest length of a wave of light is about .00003
of an inch, it is easy to imagine that the relative displacement of the particles of ether may be so
small as not to reach, nor even come near to the greatest relative displacement which could exist
without the molecules of the medium assuming new positions of equilibrium, or, to keep clear of the
idea of molecules, without the medium assuming a new arrangement which might be permanent.

It has been supposed by some that air, like the luminiferous ether, ought to admit of
transversal vibrations. According to the views of this article such would, mathematically speaking,
be the case; but the extent of such vibrations would be necessarily so very small as to render
them utterly insensible, unless we had organs with a delicacy equal to that of the retina adapted
to ‘receive them.

a

ty:

AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 319

It has been shown to be highly probable that the ratio of A to B increases rapidly according as
the medium considered is softer and more plastic. For fluids therefore, and among them for
the luminiferous ether, we should expect the ratio of 4 to B to be extremely great. Now if N be
the velocity of propagation of normal vibrations in the medium considered in Sect. 11., and 7’ that
of transyersal vibrations, it may be shown from equations (32) that

This is very easily shown in the simplest case of plane waves: for if B=y=0, a=f(#), the equations

2

hae whence a= (Nt-«)+W(Nt+a); and if a=y=9,

da?
: ; dB d’3 :
=f (x), the same equations give p ae B dat? whence 3 = (Tt —«)+&(Tt+.2). Conse-
oy ve

a
(82) give p = =1 (m4 +4B)

quently we should expect to find the ratio of N to 7’ extremely great. This agrees with a conclusion
of the late Mr. Green’s*. Since the equilibrium of any medium would be unstable if either
A or B were negative, the least possible value of the ratio of N* to TJ” is 4, a result at which
Mr. Green also arrived. As however it has been shown to be highly probable that 4 >5B even for
N
the hardest solids, while for the softer ones 7 is much greater than 5, it is probable that — is

ih
greater than ,/3 for the hardest solids, and much greater for the softer ones.

If we suppose that in the luminiferous ether R may be considered infinite, the equations

of motion admit of a simplification. For if we put mA (= + = + S
da y Fy

and suppose mA to become infinite while p remains finite, the equations become

) = —p in equations (32),

Pa _ ap B (ee Pai da)

pager aa? lay? apt aah | ns
da dB dy |
sa SE SY SO.
ri da dy 5s dz 4

When a vibratory motion is propagated in a medium of which (33) are the equations of
motion, it miay be shown that p = y(t) if the medium be indefinitely extended, or else if there be
no motion at its boundaries. In considering therefore the transmission of light in an uninterrupted
yacuum the terms involving p will disappear from equations (33); but these terms are, I believe,
important in explaining Diffraction, which is the principal phenomenon the laws of which depend
only on the equations of motion of the lumniferous ether in vacuum. It will be observed that putting
A= cc comes to the same thing as regarding the ether as incompressible with respect to those
motions which constitute Light.

G. G. STOKES.

* Cambridge Philosophical Transactions, Vol. v11. Part I. p. 2.

XXIII. Calculations of the Heights of the Aurore Boreales, of the 17th September
and 12th October, 1833; with Observations upon the Locality of the Meteor.
By Ricuarp Porter, M.A., late Fellow of Queens’ College, Cambridge,
and Professor of Natural Philosophy and Astronomy, University College,
London.

(Read December 8, 1845.]

Tue data I haye employed for the calculations of the heights of the arches of the Aurore
Boreales, which were seen on the nights of September 17th and October 12th, 1833, are chiefly
contained in the Conspectuses of the observations printed and distributed, together with various
recommendations, to members of the British Association for the advancement of Science.

In one instance, additional information is used from the Yorkshire Gazette; where Mr.
Phillips gave the breadth of an arch which he had omitted in the Conspectus.

In consequence of the attention of scientific men having been drawn to the subject, the
observations on these displays of the Aurora Borealis, were much more complete than had ever
been obtained before. The time in the various observations was reduced to Greenwich time,
by Mr. Phillips the Secretary of the Association, which thus facilitates the comparison of the
phenomena noted by different observers: nevertheless they have never before been carefully
discussed. The partial discussion communicated by Professor Airy to this Society in November
1838, and published in the Philosophical Magazine for December of that year, is the only
previous discussion of them, that I am aware of; and the height was investigated only by
a graphical method, which appears to have given results very inaccurate for many of the
observations.

Regular observations on the Aurora of September 17th were taken by Mr, J. Phillips at
York, by Mr. Clare, Mr. Hadfield and myself, at or near Manchester, by Professor Airy, at
Cambridge, and by the Hon. C. Harris, near Gosport.

On the 12th October, regular observations were obtained by Professor Sedgwick, at Dent,
near Sedbergh, by Mr. W. L. Wharton, near Guisborough, by Mr. J. Phillips, at York, by
Mr. Clare, Mr. Hadfield and myself, at or near Manchester, by Dr. Robinson, at Armagh, by
Professor Airy, at Cambridge, and by the Hon. Charles Harris, at Heron Court, near Christ-
church, Hants. ‘ :

The arches being perpendicular (or very nearly so) to the magnetic meridians of the places
of observations, a base for trigonometical calculation is more certainly obtained with respect to
them, than any other parts of the appearances. In the following calculations, I have accordingly
used observations on the arches only.

In the Conspectus for the 17th September, I find only two sets of contemporaneous obser-
vations, the one for Cambridge and Manchester, at 8".25™ Greenwich time; the other for York
and Gosport, at 11".0™. Manchester and York are too nearly of the same magnetic latitude
to furnish an adequate base. ‘To these I may add an observation of my own, of the altitude
of an arch and its extent on the horizon, for calculating the height from an observation at
one place only, by means of a subsidiary hypothesis that the arches are portions of small circles
round the magnetic axis.

PROFESSOR POTTER, ON THE HEIGHT OF THE AUROR# BOREALES, ETC. 321

The Conspectus for the 12th October, furnishes more sets of contemporaneous observations,
namely, Cambridge and York at 7".55" Greenwich time; Guisborough and Heron Court at
8". 20"; Dent and Manchester at 8",55™; Armagh and Manchester at 9".0™; and about 12 to
14 minutes later; Dent and Heron Court at 10".40™, Observations at Dent and Armagh, might
have been taken, but with a much diminished base line; and Armagh is situated on so distant
a magnetic meridian from that of Dent or Manchester, that the calculations have a greater
value with respect to the law of terrestrial magnetism, than as giving very accurately the height
of the Aurora.

The regular and perfect arches have their highest points so nearly in the magnetic meri-
dian, that if there be any determinable deviation from this, more accurate methods of observation
must be employed in order to measure it. If two places be situated on the
same magnetic meridian, the point in the arch which has the greatest altitude
above the horizon at the one place, will be the same as the point which has the
greatest altitude at the other. If the places are not situated on the same magnetic *%—\
meridian, this will not be the case; and in order to calculate the height of the
arch above the earth’s surface, from observations of the altitudes of the highest
points, we must obtain our base by projecting the places on an intermediate
magnetic meridian.

Let 4A and B be the two places, draw Aa, Bb perpendiculars on the magnetic
meridian, then ab will be the base to be used in the trigonometrical calculations ;
and putting v» = the magnetic variation, we have the formula in English miles,

ab= { difference of latitudes in degrees x cosv + difference of longitudes in
degrees x cos latitude x sinv} 69.

The lower sign to be used when the place having the greater latitude, has
the less West longitude. The are of the magnetic meridian thus found and
its chord, will not sensibly differ for any two of the places of observation; but
the observed altitudes will require correction for the curvature of the meridian, in order to reduce
the calculation to the case of a rectilineal triangle.

If C be the centre of the earth, A the point of the arch
supposed to be observed at a and 6, the projections as in the
last figure. ‘Then to solve the triangle Aab, we increase the
observed altitude at a by half the angle aCb, and diminish the
observed altitude at b by the same quantity, for the angles 4b’,
and Aab, Having found the distance 46, we find the distance
of A from the earth’s centre by solving the triangle 4bC; and
therefore know the height above the earth’s surface.

In this way I have calculated the following observations:

When the altitude of the arch was referred to a given star, I have calculated the altitude
of the star from the Right Ascension and Declination given in the Nautical Almanac, for 1833.

In such case there was no correction for refraction to be applied, as the star and arch
were equally affected.

In the observations on the 17th September, we have the following: the time in all cases
being Greenwich time.

From Professor Airy’s observations at Cambridge. ‘ 8",25™.—-The Aurora appeared in the
form of a large bright cloud, bounded on the lower side by the horizon, and on the upper
side by an arch of a small circle (not differing much from a great circle). The extremities
of the arch were in the N,E. and W.N.W. or nearly W. The upper boundary was lower
than Urse Majoris by 3 x distance from a Urse Majoris to (8 Urs Majoris,” &c.

Vor, VIII. Panr III, ‘Ten

322 PROFESSOR POTTER, ON THE HEIGHT

From Mr. Clare’s observations at Manchester. 8". 24™,—The arch 7° broad, includes
Dubhe, Arcturus, and Capella, so that Capella is on the extreme upper edge; Dubhe rather
above the middle of the breadth, and Arcturus rather below the middle, centre of the arch a
little E. of 3 Urse Majoris. Extent of the arch 130°.”

Now the altitude of (3 Urse Majoris at 8". 25™. was 24°. 17’ and 8 x dist. of a and B= 8°. 4,
therefore altitude of summit of arch = 16°. 13’ at Cambridge.

The altitude of a Ursa Majoris (Dubhe) at Manchester at 8". 24™, was 31°. 14’, and azimuth
22°, 34’ N. towards W. about 23°. from the magnetic meridian. Therefore the altitude of the
summit of the arch = 31°. 14’ + 3°.15’ = 34°. 29’ nearly.

The distance of Cambridge and Manchester projected on the magnetic meridian whose
variation is 24°. 30’ is 119.42 English miles.

These data give the distance of the arch from Manchester 123.27 miles, and the height
above the earth’s surface, of the upper edge, 71 miles. The breadth subtending 7° at Manchester,
we find it to be 15 miles. Therefore the height of the lower edge was 56 miles.

The above arch having disappeared, and the Streamers and Auroral light having diminished,
the appearances were subject to slight changes until 10°. 49}™.; when another arch was seen at
York by Mr. Phillips, and near Gosport by the Hon. C. Harris.

From Mr. J. Phillips’s observations at York.

“© 10", 49™. : : : ,
ri A low faint arch stationary, its upper edge nearly reaching to y and y Urse
115, 19™ Majoris; its vertex under Mizar (alt. about 18° in the middle).”

In the Yorkshire Gazette for 21st September, 1833, Mr. Phillips states its breadth to be 4°;
therefore the altitude of the under edge was 16°.
From the Hon, C. Harris’s observations, at 1 mile W.N.W. of Gosport.
“6 79h lm
afin: } Arch from N.W. to N.N.E. Its vertex under ¢ Urse Majoris, and the

t a s
% edge of its base half way between that star and the horizon.”

11%, 42™,

Now the altitude of ¢ Ursa Majoris at Gosport at 10".57™. was 21°. 32’, and therefore
the altitude of the lower edge was 10°. 46’.

The distance of York and Gosport projected on the magnetic meridian whose variation is
24°, 30’ is 197.66 miles.

These data give the distance from York 1011.53 miles, and the height above the earth’s
surface 389 miles.

In the Conspectus for the Aurora of October 12th, we have from Mr. Phillip’s observations
at York.

«74. 56™._The summit of the arch was now 3° below the stars 6 and y Urse Maj. &c.

«« 7h 57™.—Suddenly it appeared double, in consequence of the production of a very narrow
faint arch above that seen before, and separated from it by a dark band.

«7, 58™.—This upper arch rose, so as to include 8 and yy Urse Maj., in its midddle.

« gho™ Tt had vanished away, after rising still higher.”

From Professor Airy’s observations at Cambridge.

s¢ 7 54™.The upper boundary of the bright cloud was extremely sharp; it began to the
left of Arcturus, passed a very little above Arcturus, below yy Urse Maj. at exactly half the
elevation of yy Ursee Maj. (which was its highest point) and terminated E, of the N. at about
half the azimuth of 8 Aurige. &c.

s¢ 7h. 59™__A black line was discoverable very near the upper boundary and parallel to it.
The upper part rose and the lower fell a little, thus widening the black line. About Arcturus
the upper part rose most.

s¢gh_2™__The upper part after rising considerably had wholly disappeared, &c.”

OF THE AURORZ BOREALES, ETC. 323

We have here one of the rare cases which fix the identity of the phenomenon seen; the arch
appearing double at places so distant as Cambridge and York at the same time.

The distance of the projections of York and Cambridge on the magnetic meridian whose
variation is 24°. 30’ is 129.97 miles. he altitude of y Urse Majoris at York at 7h, 56™.
was 22°. 50’; therefore the altitude of the summit of the arch was 19°, 50’.

The altitude of -y Ursee Majoris at Cambridge at 7", 54™. was 21°. 3’; consequently the altitude
of the highest point of the arch was 10°.312’.

These data give the summit of the arch 199.93 miles distant from York, and its height
above the earth’s surface 72.2 miles.

From Mr. W. L. Wharton’s observations at Guisborough.
«« gh 20™.—Well defined arch, passing between a and (3 Urse Majoris its summit somewhat
above ¢ Urse Majoris, no radiations.”

From the Hon. Charles Harris’s observations at Heron Court, 4 miles N.W. of Christchurch,
Hants.

« gh go™ __Bright, irregular arch, like a luminous bank of fog, about 8° above the horizon.”

The distance of the projections of Guisborough and Heron Court on the magnetic meridian
whose variation is 24°, 30’ is 225.1 miles.

The altitude of ¢ Ursee Majoris at Guisborough at 8".20™. was 28°. 47’; therefore the summit
of the arch would have an altitude of about 29°. The breadth of the arch passing between
a and # Urs Majoris would be 5°; therefore the altitude of the lower edge would be 24°.

In the Hon. Mr, Harris’s observation we have the altitude of the lower edge 8° — refraction
= 7°. 53’.

From these data we find the distance from Guisborough to have been 167.34 miles, and the
height of the under edge to have been 70.9 miles. The breadth being 14.6 miles, the height
of the upper edge was 85.5 miles.

From the observations of Professor Sedgwick at Dent, near Sedbergh, Yorkshire.
‘¢ gh. 55",—The upper part of the arch, better defined than before, passed between a and 3
Urse Maj. and very near ¢ Ursee Maj. Its vertex in or near the magnetic meridian. &c.”

From my own observation near Manchester.

ce gh 534™.—The arch has its vertex under ¢ Urse Maj. and its upper edge touches Y
Urse Maj., altitude about 19°. 30’.”

At 8". 54™. Mr. Hadfield found, near Manchester, but on the opposite side, that the altitude
was 20°, and the extent on the horizon 120°.

The distance of the projections of Dent and Manchester on the magnetic meridian with
variation 25°. 30’ is 52.56 miles.

The altitude of ¢ Ursa Majoris at Dent at 8". 55". was 26°. 32’, therefore the altitude of the
arch passing near it we may call 26°.26’. Mr. Hadfield’s observation corrected for refraction
gives the altitude at 84. 54™, as 19°. 57’.

With these data we find the arch to have been 183.38 miles from Dent, and the height of
the upper edge to have been 84.97 miles.

The arch, or rather arches, appear to have been stationary from about 8".54™. to g?. 10",
for from Professor Sedgwick’s observations we have,

«gh 10™—Arch nearly as before.”

From Mr, Clare’s observations at Manchester, who has recorded the arch as double at 8", 54™.,
we have

«gh g™ The two arches remain in the same position.”

From Dr. Robinson’s observations at Armagh.

**9".1™.—Three parallel arches, the principal one has its upper edge on Polaris, and midway
between Capella and 8 Aurige ; its lower a little above 3 and y Urs Majoris.

vr 2

324 PROFESSOR POTTER, ON THE HEIGHT

«< gh 6™,Arch in the same place place, &c.”

The distance of the projections of Armagh and Manchester on the magnetic meridian with
variation 25°. 30’ is 128.76 miles.

By a careful examination of the course of the arch as seen at Armagh on a ccelestial globe,
the altitude of the summit must have been about 60°, and the altitude at Manchester as above
was 19°, 57’.

These data give the distance from Armagh 74.25 miles, and the height above the earth’s
surface 64.47 miles,

If we took the altitude at Armagh as 59°, and allowed
1°.30' for the point of the arch which appeared the highest at
Armagh, not corresponding with that which appeared the highest
at Manchester, on account of the elevation being so great at
Armagh, as shewn by the figure, we should have the height of the
arch above the earth’s surface 66.5 miles, and the distance from
Armagh 78.69 miles,

From Dr. Robinson’s observations at Armagh.
<< gh 11™.—Upper edge of arch has risen to Lyra and Capella, and a new arch has risen
beneath it, &c.”

From my own observations near Manchester.

©gh 741™._» Urse Majoris in the upper edge of the arch, the height of which by
measure = 21°. 10’.”

The altitude of » Urse Majoris at Manchester at 9". 144™. was 21°. 6’ confirming the altitude
I obtained by an instrument made purposely for observing the Aurora; as however there is a
discrepancy between the height above the earth’s surface deduced from these observations and
the previous ones, I will suppose the extreme upper edge had an altitude of 22°, that we may
be certain the discrepancy does not arise from an under valuing of the altitude at Manchester,

but must be sought in other causes,

From Dr. Robinson’s observation, the altitude of the upper edge must have been 714°,
from which we may deduct 14° for parallactic effect.

These data give the distance from Armagh 69.59 miles, and the height above the earth’s
surface 65.4 miles.

These results are remarkably in accordance with the others for the same places, but
considerably different from the calculations for other places for nearly the same time; so that
probably the method of projecting places of which the magnetic meridians are so distant as
Armagh and Manchester upon an intermediate magnetic meridian to obtain a base line, is
only approximately correct, from the course of the arch over the earth’s surface, rather than
for geometrical reasons.

Another arch was observed from 10", 34™. to 10",45™, at Dent, Guisborough, York, Man-
chester, and Heron Coutt.

From Professor Sedgwick’s observations we have,

«© 10",40™.—-The bright space arranges itself into an arch, commencing nearly N., passing
through » Urs Major.; about 25° high near the magnetic meridian (measured only by a
geological clinometer).”

From the Hon. Charles Harris’s observation.

« 10", 37™.—A low arch again formed, its base scarcely 5° above the horizon, extending to
about 7°, &c.” :

The distance of the projections of Dent and Heron Court on the magnetic meridian with
variation 25°, is 232.52 miles.

OF THE AURORZ BOREALES, ETC. 325

The altitudes of the summits of the arch being observed 25° at Dent and 7° at Heron Court,
correcting these for refraction, we find the distance of the arch from Dent to have been 136.33
miles, and its height above the earth’s surface 59.4 miles.

I some years ago shewed, in the Kdinburgh Journal of Science, that the locality of an
arch of an Aurora Borealis might be determined from observations at one place, by the help
of the hypothesis that the arch is a small circle round the magnetic axis, This hypothesis
cannot be accurate, from the change of the variation on the earth’s surface, and we must con-
clude that, strictly, the regular arches are only perpendicular to a series of magnetic meridians;
which for localities exterior to the earth’s atmosphere, may be found, when the meteor has
been more accurately observed, to differ from any assignable series on the earth’s surface.

As an approximation this method gives the height sufficiently in accordance with the
trigonometrical method, to induce us to attempt more accurate observations, when the theory
of terrestrial magnetism shall be sufficiently advanced to enable us to profit by them.

The required observations are the altitude of the summit of the arch, and its extent on
any given plane perpendicular to the magnetic meridian.

When the given plane is the horizon, the formula takes the following simple form:

weapin/ et. 4
F- (pil = m)? 35 4?

where 7 = earth’s radius,
R

e = trig. tang. of altitude of the summit,

distance of the arch from the earth’s centre,

m = (secant same angle)’,

l = (secant 3 extent on horizon)’,

p=1+eg, where g = trig. tang. of magnetic polar distance of the place of observation.

In the Aurora of the 17th September, I obtained the following observation with the view
to its being used with the above formula.

«gh 402™. Arch 38° or 39° high, and extending about 160° on the horizon.”
Taking the altitude 39°, and » = 3954 miles, the formula gives R = 4007.9 miles ; whence
R-—-r=height above the earth’s surface = 53.9 miles.

We saw that the height of the under edge was 56 miles, and of the upper edge 71 miles
at 8", 24™,

From the preceeding results, we must conclude that the meteor occurs immediately beyond
the ordinary limits assigned to the earth’s atmosphere, and from that to very great altitudes;
which is in accordance with the results of many previous calculations.

I shall conclude my paper with expressing my conviction that the Aurora Borealis will,
in some future time, from its connection with the earth’s magnetism, be subjected to much
more accurate methods of observation than have hitherto been attempted.

R. POTTER.

XXIV. The Mathematical Theory of the two great Solitary Waves of the First Order.
By S. Earnsuaw, M.A., of St. John’s College, Cambridge.

[Read December 8, 1845.]

TuovceH it is now about a hundred years since the general equations of fluid motion, expressed
in partial differential coefficients, were first given to the world, I am not aware that any
important case of fluid motion has hitherto been rigorously extracted from them. This however
has not arisen from want of effort, for the subject on account of its importance has successively
occupied the attention of the first mathematicians from the days of D’Alembert to the present
time; but rather from the peculiarly rebellious character of the equations themselves, which
resist every attack, except it have reference to some case of a very simple and uninteresting nature.

This want of success I am inclined to attribute chiefly to our experimental ignorance of the
peculiar and distinctive characters of different species of fluid motion. In this matter indeed
there was a tendency to ignorance produced by that little success which had attended mathematical
research; for as it was found that fluid motions of every sort, providing they are continuous,
are all expressible by the same partial differential equations, it was thought that those equations
ought to admit of being integrated in some general forms which should consequently include the
properties of every possible kind of continuous fluid motion. The natural consequence of this
idea has been that much effort has been unsuccessfully expended in attempts to obtain general
integrals. Two ways of approximation however are open to research ;—the one, in which the
approximations are made by neglecting certain terms on account of their supposed smallness in
comparison with the terms retained; and the other, in which ab initio hypotheses are made
as to the paths or velocities or some other character of the motions of the particles. With
regard to both these methods, it is evident that they must first be authorized by experiment,
before they are used in verifying or predicting results. The former however is peculiarly
liable to error, from our being uncertain in many cases, whether with the neglected terms, we
may not have discarded some of the peculiar and essential properties of the motion we are
investigating. And with respect to the latter method, recourse must be had to experiment
to ascertain what are the really distinctive characters of the various kinds of fluid motion.
Hence nothing seemed more likely to conduce to the advancement of the Theory of Hydro-
dynamics than the appointment of a Commission, by the British Association for the Advancement
of Science, the object of which was the discovery of the ‘‘ Varieties, Phenomena, and Laws of
Waves:” for if there be varieties of waves differing in their phenomena and laws, it was too
much to expect the mathematician (considering the exceedingly intractable nature of the
equations with which he has to deal) to discover what are the precise hypotheses which lead to
each variety. He must at least be allowed to know something of the peculiar phenomena of
each variety, before he proceeds to the integration of his equations; and there is no way in
which he could gain this knowledge except through the medium of experiments such as the
Commission, just alluded to, were directed to institute. The differential equations of motion
are too comprehensive to admit of general management. An hypothesis is in fact necessary to be

Mr. EARNSHAW, ON THE MATHEMATICAL THEORY, ETC. 327

made before we can advance a single step towards their integration ; and by the aid of it we may
only advance to a certain point, and no farther. If it be asked why we are thus stopped, the
answer seems to be this; the results obtained up to that point are still of too general a character,
embracing every variety of fluid motion which is compatible with the hypothesis on which we
_ started. Now among the large class of such motions, there may be some varieties which cannot

be analytically expressed by the same final formule; and consequently these require to be sifted
from the others and from one another, by additional hypotheses; each hypothesis pointing at the
variety or subdivision to which it belongs, and to no one else. Nothing in fact can be more
clear than this, that if there be varieties of fluid motion the laws of which do not admit of being
expressed in the same analytical forms, those varieties must be separately treated by the
mathematician; and to the oversight of this necessity I attribute the insignificance of the progress
which has hitherto been made in this subject.

I have thought it necessary to introduce these remarks, because some persons, especially
among such as have not made Hydrodynamics a special object of study, are apt to depreciate
investigations which set out upon a set of hypotheses which manifestly limit the range of the
results obtained. They prefer investigations which set out with fewer and broader hypotheses,
because they have the appearance of greater generality; and this character they continue to
ascribe to such investigations, though it is found that in carrying them out it may have been
found necessary to introduce a system of approximations by the neglect of certain terms. I am
persuaded that this view is utterly fallacious in the majority of cases of any importance in nature:
and that the wiser and better course when possible is, to consult experiment and thence obtain
authority for a set of hypotheses to start with, and to carry out these hypotheses to the end
without the introduction of analytical approximations. Our results will then be as comprehensive
as our hypotheses, and as far as they go may be relied upon with unlimited confidence. This
is the course which has been adopted in the following investigations. 'The experiments which
I have taken as a guide in framing my hypotheses are those of Mr. Scott Russell which are
printed in his “ Report on Waves” in the ‘‘ Report of the Fourteenth Meeting of the British
Association.” These experiments were conducted with well-contrived apparatus and great care,
and are as worthy of confidence as experiments on wave motion can be: and there seems to be
but one circumstance in them to be regretted, which is, that Mr. Russell having been led by his
results to adopt a certain empirical formula for the velocity of transmission of a wave, his
experiments seem in a great measure to have degenerated into an effort to establish the truth
of that formula, in which he appears to have overlooked or forgotten the probability that after
all it might only be an approximate result, and that the exact mathematical form might contain
elements not recorded in his tables, because not required in his formula. The consequence of
this oversight is that he has not recorded one element, very easy of observation, and of essential
importance; viz. the distance through which each particle was transferred in space by a wave in
passing it. Had this element been recorded, the experiments would have been much more
complete: and without it they are certainly defective as accurate tests of theory. It is true
Mr. Russell has given a rule for calculating this element; but he has not furnished us with the
requisite data. These are the volume of the fluid which is elevated above the general level, and
the breadth and depth of his canal. The last two are given, but the first is not given in any
one instance. He has indeed stated the volume of fluid originally put in motion, and seems to
have supposed that this would supply all that was wanted; entirely overlooking a fact, which
must have forced itself upon his attention in the very first stages of his experimental researches,
viz. that a single wave could never be generated alone, and that consequently all the fluid
originally displaced did not go to form the single wave of observation; which besides, as the
experiments themselves shewed, and as we shall prove theoretically, was continually wasting away,
and thereby rendering the data still more inaccurate as the experiment proceeded.

328 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF

And in fact Mr. Russell tells us he found it necessary to wait awhile after the completion
of the process of generating a wave till the main wave had separated itself from the residuary
waves, which always accompanied its genesis. To generate a single wave required, as we shall
see, the exertion of a peculiar law of pressure; and as no attempt was made to secure the
observance of this law in Mr. Russell’s experiments, the inevitable consequence was the genesis of
residuary waves. We shall also see from our theory, that the nature of the motions given to the
particles of the fluid in this kind of wave produces a natural tendency in the wave to generate
and cast off irregular disturbances from itself, working its own destruction as it proceeds. While
therefore I look upon these experiments as very valuable additions to our knowledge, I still regard
them as imperfect even to the extent to which they profess to have been carried. It is impossible
indeed to read the Synopsis which Mr, Russell has given in page 343 of his Report without
perceiving that he was too eager to adopt as results of experiments certain geometrical analogies,
of which there seemed to be some faint shadowings indicated in his observations.

In his Report Mr. Russell conceives that his observations authorized him to consider waves
as divisible into four distinct species; the first of which he has denominated “The great
solitary wave.” It is found to comprehend two varieties, the positive and the negative wave,
which though agreeing in some general characters differ in others. The object of the present
paper is to furnish the mathematical theory of this species. But how are we to sift this from
the other species? I haye examined the phenomena which Mr. Russell has recorded, and fixed
upon such as belonged to this species alone; and these I have made the basis of my calculations.
But it is obviously desirable that the phznomena thus selected should be of such a character
as admitted of easy and accurate observation. That the reader may judge in this matter I will
here propound them with Mr. Russell’s statement of the method by which he obtained the one on
which there might possibly be a doubt: merely premising that I suppose the wave to be
transmitted in a horizontal canal of uniform breadth and depth, and that the fluid is incompressible.

1st. The velocity of transmission of a wave is uniform.

2nd. The horizontal velocity of all particles, which are situated in a vertical plane,
intersecting the axis of the canal at right angles, is the same.

By a contrivance of peculiar ingenuity Mr. Russell was enabled to obtain the velocity of
transmission with great exactness; and the result at which he arrived, and which we shall assume
to be accurately true is, that abstracting from friction and the cohesion of particles, the velocity
of transmission is uniform and the wave is permanent. We shall in the end shew that this
hypothesis is not strictly accurate.

With respect to the verification of the other principle which I have assumed, Mr. Russell
thus writes: —*‘ The methods I had employed for such observations were the observation of the
motion of small particles visible in the water of the same, or nearly the same specific gravity with
water, or small globules of wax connected to very slender stems, so as to float at required depths,
The motions of these were observed, from above on a minutely divided surface on the bottom of
the channel; and from the side, through glass windows, themselves accurately graduated, the
side of the channel opposite the windows being covered with lines at distances precisely equal
to those on the window,- and similarly situated. These methods are the only methods of
observation I have found it useful to employ, but I have now increased the number and variety
of the observations sufficiently to enable me to adduce the conclusions hereinafter following, as
representing the phawnomena as far as their nature will admit of accurate observation.” “If the
floating spherules before mentioned be arranged in repose in one vertical plane at right angles to
the direction of transmission of a wave, and carefully observed during transmission, it will be
noticed that the particles remain in the same plane during the transmission, and repose in the
same plane after transmission, It is further found, as might be anticipated from the foregoing

THE TWO GREAT SOLITARY WAVES OF RUSSELL. 329

observations, that @ thin solid plane transverse to the direction of transmission, and so poised
as to float in that position does not sensibly interfere with the motion of translation or of
transmission.”

From this statement it would appear that we may safely assume, as an experimental fact,
the second principle which I haye proposed to assume as the basis of calculations. The
observations required to be made in establishing it are such as admitted of very accurate
verification; and seem also to have been made with care, and therefore the principle must be
either accurately true or very nearly so. By reference to the Report itself the reader will find
that this property of the solitary wave is not shared by any of the other three species of waves,
and is therefore yery proper to serye as a distinctive assumption to sift this species from the
general equations of fluid motion. The investigations which follow will therefore contain the
Mathematical Theory of Waves of the First Species, i.e. of the Positive and Negative Solitary Waves.

PROBLEM.

A quantity of incompressible fluid is in a state of repose in a straight horizontal canal, the
sides of which are vertical and parallel, and the bottom horizontal. A single wave is generated
by pushing in one end of the canal in a proper manner: to determine the subsequent motion of
the fluid, on the two hypotheses before mentioned, viz.

Ist. That the velocity of transmission of the wave is uniform.

And 2nd. That the horizontal velocity of every particle, in a transversal section of the
canal, is the same.

Let a horizontal line drawn along the bottom of the canal, parallel to the sides, be taken
for the axis of w; let the axis of y be vertical.

h = equilibrium depth of the fluid ;
k

ce = the velocity of transmission of the wave.

the depth from the top of a wave to the bottom of the fluid;

As the motion of each particle is manifestly in a vertical plane, it will not be necessary to
take account of the breadth of the canal, nor of the third co-ordinate of any particle; let
therefore wy be the co-ordinates and wv the velocities of any particle at the time ¢; and suppose
p the pressure of the fluid at the same point; the density of the fluid being taken as unity.

Then by our second hypothesis w is a function of w and not of y; consequently the equations
of motion are in this case,
Ag Die s—) GUN. UU Je cchttes Teta datcateotes voc(l)>
dp=—-g-—dv —ud,v—vd ,......:.,...--(2) 3
and the equation of continuity is,
O=du+d,v...... Polenta enadeer ais sloidse ocoae(B)s
and our first hypothesis gives,
OH Ay 4 CA,U soccereeeceee CSG OOCMEREEO vnereog (4)

From these four equations we are to obtain our results,

Yor, VIII. Parr III. Uu

330 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF

Integrating (3) with regard to y, remembering that w and therefore also d,w, is independent
of y, we find
We: TG ey ao dentaepocnasosuneadeck (2),

no arbitrary function of # being added to this integral, because manifestly v = 0 when y = 0,
whatever be the value of a; and no function of ¢ is added because from (4) ¢ enters with a only.

By means of this result eliminating v from (2) it becomes
dyp = — & + {dw + wdPu — (dyU)*} Yoreceecrreeereeees (6).

Now d,.d,p = d,. dyp 5 and as appears from (Q) dep being independent of y, d,.d,p = 0,
consequently d,p must be independent of #; from which it follows that the coefficient of y in (6)
though a function of w is not a function of x, and therefore not of ¢ by (4); and of course it is
not a function of y, consequently it is constant both with respect to a, y, and ¢;

‘, constant = d,d,w + udou — (d,U)*,.........24000000(7):

Before proceeding farther it is necessary to ascertain whether this constant have a positive
or negative sign. We may ascertain this as follows.

Let us use the letter § as the symbol of differentiation, taking w and y to belong to the same
particle through the time d¢; then it is well known that instead of the equation (2) we may use
the following which is exactly equivalent to it, viz.

dyp=-8-%Y,
which being compared with (6) gives,

ofy = — {d,d,uw +ud,u — (d,u)’ty,
y ( y
= — (constant) y.

Hence the force which urges the vertical motion of any particle varies as the distance of the
particle from the bottom of the canal, and has always the same sign. Consequently when the
original displacement of the fluid is such that any particle attains thereby a higher position than
it had when in equilibrium, the above force must act so as to bring it down to its original level ;
i.e. the force must then be negative. Hence for what Mr. Russell calls the positive wave the
above constant is positive. In a similar way it appears that for the negative wave the constant
has a negative sign. It is therefore now necessary to separate our investigation into two branches,
treating separately of these two varieties of the solitary wave.

OF THE POSITIVE SOLITARY WAVE.

In this case, representing the constant by n*, we have for discussion the equations

mn? = ddeu + UAL U — (AyU)? 5 ....ec creer eceees (8),

which belong only to the variety of wave we are now considering. The latter will furnish us
with the law of the vertical motion of each particle; and it shews that it is expressible in the
form of a sine or cosine of an angle the variable part of whose argument is nt.

THE TWO GREAT SOLITARY WAVES OF RUSSELL. 331

estes =pAlico’s' (oud a) S. oewhesed tase aiiette onidaie (10),
and v =6,y = — nA sin (nt — a) ...-...020+. (11).

Also d,u = — 2 =n tan (EG) ea iecnctewsees (12).
yy

If we knew the greatest and least values of y for any particle we should be able to deduce
results from these equations. Now for a particle in the surface, k and h are the greatest and
least values of y. If we call ¢,, ¢, the values of ¢ when the particle has these values for its y;
then v = 0, when ¢=¢, and y =k;

SG, SS Oona (CI) sag eenoeneooee (13),
and .. k= A from (10) ;
*. h =k cos (nt, — a) from (10)

=k cos (nt, — nt,) from (13);

ree 0
JH he= t, = — cos Bees (14).

Since v is positive or negative according as a particle is in its ascending or descending phase,
it appears from (11) that ¢ is less than a as long as the vertex of a wave is behind a particle; and
equal to a when the vertex is passing it; and greater than a when the vertex has passed it.
Hence the functions on the right-hand side of the equations (10) (11) (12) are to be treated
discontinuously, i.e. their variation is to be confined within certain limits; between these limits
however their variation is continuous. Since, from the nature of the case y cannot be zero
for a particle not originally at the bottom of the canal, it appears from (10) that m¢ ~ a must

7 : i :
always be less than = Equation (11) shews that the vertical velocity does not begin from

zero; but that it suddenly has a finite value, which gradually decreases till it is all lost; at
which moment the particle begins to descend, gradually regaining the lost velocity, which being
accomplished it is as suddenly lost, as it was suddenly generated. All this agrees exactly
with the recorded observations of Mr. Russell (see Report, p. 342). Equation (14) gives half

the time during which the vertical motion of any particle lasts. Consequently the time a wave
9

rt 2 h :
takes to pass a particle is = cos~* Roe (15). The quantity x is unknown at the present

stage of our investigation.

We must now proceed to integrate the equation (8). For this purpose we must remember
that (4) gives us

u= (ct —- «),
which being written in (8), using p for @(cf—«) for brevity, we have
n®?=-—cdp+ pd. p - (d.)’,

d.(c~) _ _dapde
c-b n+(d.p)

and .°.

from which by integration we find

c-pe=C Vn? + (d.p)’ ;

uu2

332 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF
C being an arbitrary constant, not containing ¢ because ¢ enters @ with # only in the form
ct —«.

The last equation being again integrated gives

0-4 C-G- Ont = De®.

D being another arbitrary constant, a function of ¢ such as makes the right-hand member
a function of ctf— 7;

si/aD, Sa Cpt
2 (e-g)=Cn (e+ ee).

But since @ is a function of ct — a, this equation by introducing ¢, and properly assuming
the origin of ¢, may be written

ct-x _tt-a
alain ae Cc +e cee 0 ee (16) ;
eee a
and .. 2d,u eG c -e c.) Sonar odocosace pete apenas steel (Uti):
The last equation enables us to connect w and a; for comparing it with (12) we have
et-2@ _ct—«
2tan(mt—a)=e © -e © ....... Eg Se 5 ee (18).

For a given particle a is constant, and consequently for that particle w so varies with ¢
as to preserve the truth of this equation.

Eliminating x between (16) and (18), we get
c-—u=Cn sec (nt —a).
Now for a particle in the surface w= 0 when ¢= ¢,;
. c=Cn sec (nt, — a)

= Cn sec (nt, — nt,) ;

C h
se Setk (nt, — nt,) = — from (14) ;
c k
ch
"Cn =—:
Parsee
tl ch
consequently ¢ -u = 7 ee (GUE ia) saint ea aelagatice ofthe - breseeee (19),

which gives the law of the horizontal velocity, as (11) gives the law of the vertical velocity of
a particle; and it is worthy of remark that neither of these is represented by a sine or a
cosine. An assumption therefore that they might be so represented would be improper: and
from this assumption we may date in some degree the erroneousness of the results which have
been obtained by some writers who have adopted methods of analytical approximation. We

have seen also that the argument m¢—a does not vary from — - to + . as some have supposed.

Equation (19) shews that the horizontal unlike the vertical motion of a particle is wholly in
one direction, and is a maximum when the particle has reached its greatest vertical displacement ;
after which it decreases to zero,

|

THE TWO GREAT SOLITARY WAVES OF RUSSELL. 333

Let now ,, x, be the values of # for a given particle at the times ¢,, ¢,.
Then (18) gives

eth — rn eth — Xr

2tan(mt,-a)=e © -e ©€ ,

Cth — te Cth — xh
and 0=2tan(mt,-a)=e © -e €
The last line shews that ef, = v,; and the preceding line gives, remembering that a = nf,»

cth— Xh \

erce =tan (7 ae bas
ag te fa)

7 h
and ,°. ct, — w, = C log, tan (G -3 cos-!7).
But ct,-—a,=0;

~ &, — @, — e(t, — t,) =

I
Q
a
08
-
i)
iI
Za
a1
|
=
}
is)
wn
1
|
ee

Now 2(v, — @,) is the distance through which a particle is horizontally transferred by the
transit of a wave; as this is an observable element we will denote it by 6

Bh B = 2(a, — @,).

Also the wave has travelled over the space c(¢,—¢,) in the time t, — t,.-

,

Now if X be the length of a wave, the wave in the time 2(¢, —¢,) has travelled over the
space \ + B,
“A+ B= 2clé, — 4),

and .. \ = 2C log, tan (= +3 cos"),

ch = 2ch 7 )
Consequently n= Gis Ae log, (= + g cos a} Bona aaocesel CO) 5
2c h
Also X} +6 = 2c(é,-4,) = = cos~! zs
beeasul
h k
% = +1=-——...... (21)

h
log, tan (= s 4 cos~"")

We may consider this equation as giving the value of the length of a wave; and then
(20) gives the value of m in terms of ec.

If we expand the terms of equation (21) we find,

B_k-h Kk (cos) k

h\*
= 26
a Sry (cos I + &e. ercivegeest tn 22) >

x h 6h

which shews that as & diminishes, (3 diminishes compared with ),

We may now proceed to determine the velocity of transmission; and the equation of a
wave surface.

334 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF

The equations for the pressure are,
d,p = — du — ud,u = (ce — u)d,u,
and dp=-—g+n’y;
p=-h(e-u)-gyt ny? + constant.

Now for a particle in the surface of the fluid p is constant; and if x be the value of y
for such a particle, then
constant = (¢ — uw)? + 29% — N°x?......000262 (23).

But the value of c—w is known in terms of w from (16), and consequently,

2

272 ct—a _et= ay 2
constant = 7 (e Oeene ee | #eee — Meee sense 2M)

is the equation which gives the form of a wave. ¢ is here to be considered constant.
Again, when = =h, u =0,

*, constant = c* + 2g¢h — n*h? from (23).

ch
Also when x =k, c-—w= 7 from 19, and consequently,

2 22

X :
constant = = + 2¢k —n°k*® from (23) ;

: o=e(1 a ) + 2n(h—&).- 0-8):

2
and .. @ +77 =
C+ ni A

And if in this equation we write the value of m from (20), we obtain the following final
equation for the velocity of transmission,

( =e )
is h+k
c= ee ei ato a irene (25).

em log. tan(= +1 =z)
| 8 (F+4e0 k

It is to be remarked, that if A be very nearly equal to &, the denominator of the fraction
on the right-hand side of this equation becomes equal to 1; and the numerator equal to 2k,
so that ¢ = /gk in that case; which is the empirical formula used by Mr. Russell. If A be

Qgk* ; k+k :
much less than &, then ea Gl Laer) is greater than gk; but in that case the

denominator is greater than 1, and consequently there is a tendency to compensation which causes
the value of ¢ to lean sensibly towards the value Vek; which accounts for the near agreement
of Mr. Russell’s formula with experiment ; and shews that he was mistaken in imagining the
velocity of transmission to be entirely independent of the length of a wave.

THE TWO GREAT SOLITARY WAVES OF RUSSELL. 335

Equation (21) shews that waves which give the ratio between # and k the same, have
their lengths exactly proportional to the spaces through which they respectively transfer a particle
by transit past it.

Equation (25) shews that in waves which have the same values of h and k, those will be
transmitted with the greatest velocity which are the longest; and those with the least velocity
which are the shortest.

We may conclude this portion of our investigations with the determination of the exact path
of each particle. The materials for this purpose are supplied by equations (10) and (19). In
both of them a is constant for our present purpose. The former gives,

y = A cos (nt — a)..........-.(26),

in which 4 is the maximum value of y for that particular particle.

Equation (19) gives

h
c- Se sec (nt — a);
k
ch
: a= et— J see (nt — a)

h
s (nt — a) — ~ log, Stan (nt — a) + sec (nt — a)} + constant;
n nk
t being eliminated between this and (26), we shall have the equation required; which is
manifestly not that of an ellipse as has been found by approximate methods; though as far as the
eye can judge in an experiment, it may not be distinguishable therefrom.

It is very easy to shew from (24) that the surface of a wave meets the level surface of the
quiescent fluid in a finite angle; and that under certain conditions it may have a point of contrary
flexure. The actual wave surface is only a symmetrical portion of the whole curve represented
by the equation (24). When a wave first reaches a particle dw = 0, and dy =a finite quantity ;
consequently the initial motion of each particle is vertically upwards with a finite velocity. When

it has described half its path d,v =e (1 -%). and dy = 0; consequently its motion is then

horizontal. At the termination of its motion d,aw = 0, and d,y = — (the initial velocity), sc that
the final velocity is vertically downwards, and is finite; which indicates that the motion ceases as
suddenly as it began. This seems to coincide either accurately, or very nearly so, with the
account Mr. Russell has given (Report, p. 342) of the observations he made on the motions of
individual particles in his experiments.

Before we proceed to compare the formula (20) with the results of experiment, it is necessary
to advert again to a circumstance which has been already alluded to. The formula of (20)
involves }. The value of this quantity not having been recorded in Mr, Russell’s tables, I have
been under the necessity, as the best substitute for exact measures, of having recourse to the
rule, which he has given in page 343 of his Report, for computing its approximate value. In
the notation of this paper, that rule may with sufficient accuracy be represented by the equation
AX =8h-—2k; for it is not necessary in computing the value of ¢ that ) should be known with
extreme accuracy, as the term in (25) into which it enters is very small, and has but little effect
upon the value of c. With these premises we give the following table, exhibiting a comparison
of theory and experiment.

336 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF

Velocity by Velocity by Proportionate
Theory. Experiment. Error,

A mere inspection of this table, the fifth column of which gives the proportion of the
error of theory to the whole velocity, will enable the reader to judge whether the theory
advanced in the preceding pages is borne out by experiment. I am not aware what degree
of accuracy Mr. Russell is disposed to ascribe to his observations, but I imagine he will hardly
maintain that the velocity of a wave could in any case be observed with greater accuracy
than the fortieth part of the whole. I aim therefore inclined to pronounce that the coincidence
of theory and experiment is exact.

The last column of the above table, though not necessary for the comparison of theory
with experiment, is added for the purpose of shewing that there was a considerable degree of
variation in the circumstances which characterized the several waves that are here selected as

tests of theory. It may also serve as a further test of theory, if ever the experiment should be
repeated.

al >

It is worthy of remark that the ratio 3 depends entirely on the value of the ratio —, and

not at all on the absolute value of either h or k,

It also appears that there is no means of determining the absolute values of X and 8, from
those of h and &: consequently we must consider either } or ( a necessary element in the
experimental determination of a wave of the kind we have been considering.

If 2 k and B are observed, then all the circumstances of the wave can be calculated; i.e., the
path, velocity, and position at a given moment, of each particle; and the place, form and velocity
of the wave.

We must now advert to a circumstance of considerable importance. The equation which
we have found for the pressure at any point within the moving portion of fluid is

p=-3(c-u)-gy+ n'y’ + constant.
Now for those particles of the wave which are immediately in contact with the quiescent
portion of the fluid, «=0; and consequently p= constant —gy+4m'y*; which varies partly
as the depth and partly as the square of the depth of any particle below the quiescent surface,

=-e

a

THE TWO GREAT SOLITARY WAVES OF RUSSELL. 337

But the pressure in the quiescent part varies with the depth only; and depends not at all on
the square of the depth; consequently there is a discontinuity of the law of pressure in passing
from the wave to the quiescent fluid. This is of course an impossibility; and therefore our
equations, though they may represent the properties of the wave with as much accuracy as the
experimental observations, cannot be regarded as the exact representatives of a possible wave
motion. But as they are rigidly deduced from the two hypotheses which Mr, Russell considered
to be experimentally justified, it follows as a necessary and indisputable consequence that it
is impossible for the particles of a permanent wave to move in the manner here assumed, viz.,
so that those which are in a vertical plane at right angles to the axis of the canal should always
continue in a vertical plane during the transit of the whole wave. This hypothesis, as we have
seen, leads us to an impossible result ; and it is of importance to notice that this impossibility could
not have been affirmed to be a necessary consequence of our hypotheses had methods of approxi-
mation been followed in our investigations, because it obviously depends on quantities which are
small.

It appears then that the pressure at the junction of the moving fluid with the quiescent
fluid cannot practically be such as our two hypotheses require it should be, yet as the hypothesis
respecting the continuance of particles in the same vertical plane is certainly known to be very
nearly true, as nearly true indeed as observation has been able to discriminate, we may expect
that it is the other hypothesis which deviates more sensibly from experiment. To the want
of permanency of the wave therefore we must look for the experimental confirmation of the
impossibility we have just discovered. We will therefore now turn to Mr. Russell’s experiments
for evidence upon this point.

At page 327 of The Report on Waves, we find what the author has designated the History of
a Solitary Wave of the First Order, from observation. A wave such as we have been
investigating was generated in a canal such as we have supposed. ‘The depth of the level fluid
was 5.1 inches; and k — h or the altitude of the crest of the wave above the general level was at
first 1.34 inches. An inspection of the table shews that the crest of the wave gradually fell, with
so rapid a degree of degradation, that in five minutes it was reduced to ‘08 inches, the wave
having in that time described 1160 feet. The velocity of the wave in the same time fell from 4.21
feet per second to 3.61 feet per second; the difference being ‘6 or one-seventh part of the whole
original velocity. It is evident from this statement that the degradation of the wave was a rapid
process, and that the consequent effect upon the velocity was considerable.

These effects, which are much greater than could have been caused by imperfect fluidity or
friction against the sides and bottom of the canal, I consider are fully accounted for by the
circumstance above-mentioned, viz. the impossibility there is that the pressure should be continuous
and the wave at the same time permanent if the motions of the particles are such as we supposed
them to be, and which experiment shews they very nearly are. We have certainly proved the
truth of these two alternatives ;—if particles continue in a vertical plane while a wave passes them,
then the wave cannot be permanent ;—and, if the wave be permanent then the motions of particles
once in a vertical plane cannot preserve them in a vertical plane while the wave passes them.

In proportion as one of our two hypotheses is more nearly true the other is farther from being
accurately true. Degradation of the wave is therefore the natural consequence of the law which
we have assumed for the motions of the fluid particles; and if that law be an experimental truth,
as we believe it is toa close degree of approximation, then the gradual destruction of the wave
is a necessary consequence, resulting not from friction alone, nor from imperfect fluidity, but
chiefly from the manner in which motion is initially communicated to the fluid particles.

Strictly speaking, our investigations have been conducted on two hypotheses which are
incompatible with each other; but experiment shews that, though they may not be accurately
true, they are approximately correct and compatible: and we claim for the results of our

Vor. VIII. Parr III. eX.

338 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF

investigation the same degree of accuracy as belongs to the hypotheses, because we have no where
infringed those hypotheses by analytical approximations. It is easy to shew that we cannot regard
our second hypothesis as being strictly correct. For if it were a possible hypothesis, then as the
first cannot be at the same time true, the quantity denoted by m* in equation (9) must be regarded
as a slowly varying function of ¢. The equation for p then assumes the form p = F(2,t) — gy
+4n’y’; which involves the same impossibility as before, because at any given moment, at the
junction of the wave with the quiescent fluid, the pressure depends on y* as well as on y, which
cannot be the case. Hence our second hypothesis is certainly not mathematically correct. «w must
therefore depend on y as well as on a.
We come now to the consideration

OF THE NEGATIVE SOLITARY WAVE.

In this case, we are to represent the constant of equation (7) by — n?; the equations therefore
which are peculiar to the wave we are now investigating are,

— n? = djd,u + UdZU — (d,U)?.0--ereeeees GOR
Oey SMe, Mee Ne Sane Soto aae olen (ese eaR(G')s

From the last we obtain
y = A(et-% 4 Oe at

and... dy = An(e"™~* = e~"*%).

Now by the nature of the case d,y=0 when the particle has gained its lowest position ;
but d,y can never become = 0, unless we use the upper sign; the upper sign must therefore be
used; and consequently we obtain

Wis let eeenM te) gas aves av sche naaktOOs

Doha e 8 Crremaiot: alaciae) er eke aS spel Ae 9)
v ett—* — exnt+a 4
Also d,u as ye = Veta y gotta wet e ee eee ree cecncs (42 VE

_ The form of (9’) shews that the force which regulates the vertical motion of each particle acts
upwards, and consequently if the particle oscillate (which it must do if it be part of a wave) its
motion at first must be downwards; it then comes to a minimum altitude above the bottom of the
canal and then rises again to its original level. Let h be as before, and & the altitude of the lowest
point of a wave above the bottom of the canal; then proceeding as in the corresponding part of
the investigation for the positive wave, we obtain

,

Mt, — A = Oveeeee onltnte nt Melaislo suite catefetis (LS: py
kk = 2A,

k nty — ntk ntk—nty ,
h=(e +e Veseeenees (14)

1 k
and .. 4,-t, = = 1B. tan (= + }eos-'=).

Since v is negative in the fore part of the wave, and positive in the hinder part, m¢ is less
than a as long the particle is situated in the fore part, greater than a when it is in the hinder part,
and equal to a when the vertex of the wave passes it.

i ee

THE TWO GREAT SOLITARY WAVES OF RUSSELL. 339

We must now integrate equation (8). Proceeding on the same plan as before, we obtain
n* — (dep)’ = (c — ) dig ;
“6 - p= Cy/ni — (d,)?.

And integrating this equation, and assuming a convenient epoch for the commencement of ¢,

we find
: av —ct
ce —u= Cncos Goons csc ner (ills) 2
—ct r
d,u =n sin te ceccesceeveeeee(17) 5
e n+ _ ptt-a : = (oy)
entre 4 eta sin 2,

diy. he"'se =a" = 25007 — 18)
and .. e +e = Gos (18s’).

Let a,, «, be the values of w for a given particle at the times ¢,, ¢,. Then (18’) gives
2, =ct,, and
@, — ct,

2 sec = ents — nl “4 ents — nls

2h fi hy
=> Te rom ( Ne
k
+ @, — Ct; = © coss* i’

Now while a particle is transferred by the wave through the space 2 (a, — xv,) (=) the wave
itself has travelled its own length (=A) in addition to this space; and the time occupied is
2 (4, = t,),

>

es Ota s +2,-—2,3

A
4 OS = 5 because et, = x,;

k
“.X=2C cos) _,
COs

But when # =a, and ¢=¢,, u=0, and consequently from (16’)
4 w, — ct k
c= Cn cos “= Cn. =;

h?
ch
s Cn meri
2ch ,
= a cos~! Fortier see see eee eee nee ees (20),

Again, d + 3 =2c(t, - ¢,)
2¢ T oY
FT log, tan (= + 40s i)

xx 2

340 Mr. EARNSHAW, ON THE MATHEMATICAL THEORY OF

7 k
log, tan ie + dos" ;)
eine, (21’).

~-1

h

ames)

— Cos

To find the velocity of transmission we must refer to the equations for the pressure. These
are

d,p = — du -— ud,u =(e —- u) d,u,
and dp= -g-n'y;
“ p=—-4(e-u)’-gy-—4n’*y’ + constant.
For a particle at the surface p is constant; and therefore
constant = (€ — u)? + 2g8 + 27S" ....ccceseencevees (23')

is the equation of the form of a wave: or restoring the value of w in terms of a, the
equation. of the curve of the wave is

Ch ,a—ct

[PE

+ 2gx +n’ x* =constant............ (24’),

in which ¢ is supposed constant.
When x=h, w=0;
. constant =¢? +2gh+4n7h* from (23').
ch
Also when x= hk, c-u= a
2 fp2

ch
*. constant "a 2gk + n*k* from (23);

2

O=e8 (F-3) — 2g(h — k) — n?(h? - );

Before submitting this formula to calculation a few words may be said respecting the
experiments of Mr. Russell on negative waves, which without questioning his experimental
accuracy in the least degree, I cannot but consider far less satisfactory than those which were
made on positive waves. For to generate a perfect solitary negative wave it was necessary that
a peculiar law of pressure should have been observed. Unless this law were observed it was a
necessary consequence that residuary and superfluous waves would be formed. Now the mode
of genesis which Mr. Russell employed seems to have been so little suitable to the nature of the
negative wave, that throughout its whole course it seems to have been continually casting off
superfluous (or, as Mr. R. calls them, companion) waves. This must haye produced a direct

THE TWO GREAT SOLITARY WAVES OF RUSSELL. 341

effect upon the wave itself, and an indirect effect in keeping the surface of the fluid in a state
of agitation till the return of the wave after reflection at the end of the canal; by which the
difficulty of accurately observing the exact time of transit would be greatly increased. Without
the aid of some supposition of this kind, I cannot account for the manifest irregularities
exhibited in Mr. Russell’s table of the observed velocities of negative waves. (Report, page 349).

Velocity by | Velocity by | Proportionate

k h Theory. Experiment. Error,
Inches, Inches, Feet, Feet.

96 1.00 1.59 1.38 iz
3.30 4.10 2,88 2.65 a
3.40 4.10 2.93 2.43 =
4.60 5.10 3.46 3.37 Fa

An inspection of the fifth column will at once acquaint the reader, that the errors here are
far larger than in case of the positive wave. Instances however might have been selected from
Mr. Russell’s table which would have exhibited a much closer agreement between theory and
observation. It is not necessary to repeat the remarks before made respecting the discontinuity
of the pressure, and the consequent destruction of the wave.

T will now conclude this paper with a few general remarks.

Mr. Russell states, “‘that the positive and negative waves do not stand to each other in the
relation of companion phenomena. They cannot be considered in any case as the positive and
negative portions of the same phznomena.” This is completely borne out by the foregoing
theory; which shews that the two waves are distinguished in our investigations by a circumstance
which prevents their coexistence; a certain constant being positive for one, and negative
for the other, thereby making it impossible for p to be the same for both at the junction of the
two parts, supposing them to be portions of one wave.

If it were possible for both waves to coexist at the same place, by meeting each other, or
by one overtaking the other, then we should have for the vertical motion of a particle in the
compound wave,

ofy = (vw? — n®)y.

This is obtained by uniting equations (9) and (9’). Hence if m’ be greater than m, the result
would be a negative wave; but if »’ be less than m, the result would be a postive wave; and if
n =n the result would be that y would be constant, or there would be no wave at all.

This explains the following phenomena observed by Mr. Russell.

“If a positive and negative wave of equal volume meet in opposite directions, they neutralize
each other and both cease to exist.”

“If a positive wave overtake a negative wave of equal volume, they also neutralize each other
and cease to exist.”

“Tf either be larger, the remainder is propagated as a wave of the larger class.” (Report, p. 351).

S. EARNSHAW.

XXV. On the Geometrical Representation of the Roots of Algebraic Equations.
By the Rev, H. Goovwin, late Fellow of Caius College, and Fellow of
the Cambridge Philosophical Society.

Read April 27, 1846.
I

1. Ir is usual to distinguish the roots of Algebraic Equations into three classes, viz, positive,
negative, and imaginary or impossible. Roots of all kinds may however be included under
one head, by considering them as composed of a modulus and a sign of affection, that sign
of affection being some power of — 1: thus if a be the modulus, positive roots will be expressed by

8

(- 1)°. a, negative by (—1)'.a, and imaginary by (- 1)*.a, and thus we may take (—1)*.a
as the general expression for the root of an algebraic equation, and if reasoning could be con-
ducted by means of such a symbol it would not be necessary to distinguish between real and
imaginary roots, but all would come under the same view; and speaking quite generally we
may say, that the root of an algebraic equation is a quantity with the negative affection developed
in any degree between zero and actual minus.

This mode of considering roots of course coincides with the ordinary mode of representing
the root of an equation by a (cos 044/71 sin @), which symbol will be real and positive
if @=0, real and negative if @ =7, and imaginary in other cases; but what has been said
appears to point out more clearly the true connexion between the different species of roots,
and to remove in some degree the artificial character which at first sight attaches to the
representation of real roots under an imaginary form.

2. We may also bring the roots of an equation under one view geometrically; for
considering the positive and negative roots only, we should represent them by setting off
distances in opposite directions from a given point along a given line: now instead of a line
passing through the point which we take as origin conceive a plane drawn through it, then

@

all the roots will be represented by lines in this plane; for the root (— 1)". @ or a (cos 0+\/— 1 sin @)
will correspond to a line of length a and which is inclined at an angle @ to the line along which
positive roots are measured; the conjugate root a (cos @ —./—1 sin @) will be a line similarly
situated on the opposite side of the positive line.

This is no new remark, but it has not, so far as I am aware, been followed into any of
its consequences; reflection upon it has led me to consider whether it might not be developed
into a theory which should throw some light on the nature of Algebraic Equations, that is, whether
it would not be possible so to represent geometrically the changes of value of a function of
wv, as to throw light upon the existence of the roots of the equation f(a) = 0.

With this view I have composed the following Memoir, and though I am not aware of
any practical step in the Theory of Equations which can result from my investigations, yet I
think they tend to throw considerable light upon existing knowledge, and to give us as it were
the rationale of some familiar theorems.

gs a aa

oats

Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION, ETC. 343

3. If we wish to represent the changes of value of f(«) taking into account only real values
of v, the mode adopted would be to construct the curve defined by the equation

Se— NAG nai tnouicfecode adectiss aco os stheateemercsaaessisecisen sells

but if we wish to give a general representation of the changes of the function, taking into
account both real and imaginary values of 2, we must construct the locus of the equation

SESW Hy/ Srvdviets. sek wah Not ets eatix(2);

where w y and # are to be considered as co-ordinates of a point in space as is usual. Now if
we restrict ourselves to values of = which are real, equation (2) will divide itself into two
equations, which will be the equations of a curve of double curvature, and the points in which

this curve meets the plane of wy will determine by their distances from the origin the roots
of the equation f(«) = 0.

I will observe here that f(v) will be considered throughout this paper (unless the contrary
is stated) as the representation of the quantity

COPD tac: Ve WS oh (eerceaedescaeersccee st Dys

where Pp, Po....s++++++- P, are real and either positive or negative.

4. The two cquations to which (2) corresponds may be expressed in several ways, which
I shall here put down together.

By direct expansion, equating real and imaginary parts, and dividing the second equation
by y, we have

x =f («) -f" (x) BT" w) 2 i — ke. |

o=f' (x) - rot +P) ~ Be.

If n be even and = ak these saIuAGiOns become

| RoaboHosAdSGLo 3000 see Nel ecto nssescaieacesseao)e

w= f(x) - - f(y Hs ‘ (w tiveness te G—inae
ip [2 if Mean +(-1)"y |]

o=f (~) - f(a) is +f’ (#) ce = osnodbosn Sa il) (Onin rip or |

and if m be odd and =2m-+1, they become >

x=f (wv) -f" (2) E mor = sececeehe + (— 1)" (2m 41.04 p,) y™

rf
y' ic wader Pent: 22a Vea lle (5).
O=f' (a) —f'" (a) = +f (a) = — weeeeeeee + (— 1)"
[Os
The equations also admit of a very neat symbolical expression, thus*:
* The method which I have given of representing the locus 0=f(e@tyV=1.s
of the equation z=f(a) taking into account values of a not 7
lying in the real plane, is applicable mutatis mutandis to curves Ls bg Wal aw ene
defined by an implicit relation between the co-ordinates. Thus, mE en A)
let the equation be ;
Alaa) ma ccs cash a¥spaueaih (A), 5 cosy 7 ett 1) siny a tf(a. +*),

then putting for « w+yV—1, this becomes which is equivalent to the two following,

-&

344

Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION

waf@tyV-1) ~
—d
ER =1 = F(a),

d Paha itl
(cos ia Vv-1 siny —) Sf (@);

which equation divides itself into these two

z= (cos y=) F(w) |]

S
I

If the symbols in these expressions be expanded it is evident that ioe (3) and (6) will

coincide.

There is another mode of expressing the equations in question hich will be found very
useful in the sequel, and that is by polar co-ordinates.

Put w=pcos@, y = psin @, then

# = f(p cos 0 +/-1psin 6),

which divides itself into two

F

f"(0)

sin 9 +

f')

(cos v ) F(a. =) =0

(siny A) rca. 2)=0

the differentiation indicated being partial with respect to 2.
Of course we might have treated in equation (4) in the same
manner as x, and this would have given the following

(cos y i) F(2-#)=0

(siny E)F(x.2)=0

The equations (B) and (C) may be considered as the complete
representation of the locus of (A).
For example, suppose

2 2
f(@.a)=a+h-1,

af (.%) _ 22 df (wz) _22
en ae ar a
B(x .%, =) _ 2 df(z.2)_ 2

da a? dz =F?

and equations (B) become

= f (0) + f’(0) p cos @ eae ® cos 204... + FO preosne

[2 p sin £04 tp "1 sin 20 \

fg ali))

fee eeeewene

A eae
a Spat 1 } Sater (B')
ay =0
and equations (C) become
CUT
a st+hR _ =1 i erawaeati (@),
ys =0

and it will be seen that the systems (B’)(C’) are equivalent to
these three,

2 =0 2 =o

Or the locus of the ordinary equation of the ellipse, thus
considered, comprehends an ellipse and two hyperbolas, the two
hyperbolas setting off in planes perpendicular to that of the ellipse
from the extremities of its axes.

I would refer here to two papers in the Cambridge Mathema-
tical Journal, by Mr. Walton, of Trinity College, (Vol. rr. p. 103°
and p. 155) in the first of which the complete representation of the
curve corresponding to a given equation between two variables is
considered, and in the second the real nature of a maximum or
minimum as being in fact a multiple point is noticed.

y =0

OF THE ROOTS OF ALGEBRAIC EQUATIONS. 345

or we may write these,

& = Pp, +P, p Cos O + p,_» p’ cos 20 + ...... + p" cos nO
0= p,-1 sin @ +p,» p sin 20 + ...... +p" sin woh

5. I now proceed to discuss these equations, and shall consider first the equation of the
projection of the curve on the plane of «vy.

This equation is in polar co-ordinates,

p’ ‘sin n@ +p, p"~* sin (wm —1)0 4+ ...... LD yeiSINO) = Ova seniseasecient(O)e
To find the asymptotes, I observe that p will be infinite when sin2@ = 0, except for @ = 0
and @ = 7; hence there will be infinite values of p for @ = ae =, geidiewe & ahs,
Again,
sin 20 +O sin (n—1)0 + ...... + a sin @ = 0;

“. mcosn@ — sin(n — 1). — = 0 when p=&
3 ,d8 p,sin(n —1)6.
cats dp — m cosnO

And if we put @ = ko, ke having any value from 1 up to m—1,
n

Hence there will be an asymptote corresponding to each infinite value of p, and these will
lie on the left of the corresponding infinite radius vectors looking from the pole. If however
we suppose the given equation deprived of its second term, that is, if p, = 0, then the polar
subtangent vanishes and the asymptotes pass through the origin and coincide with the radius
vectors; and since this condition may always be fulfilled, I shall generally suppose that p, = 0,
and then it may be stated that the projection of the imaginary branches of the curve on the
plane of wy has m — 1 asymptotes, which pass through the origin, are equidistant from each other,
and make the same angle with each other as the first of them makes with the axis of «.

The symmetry of these infinite branches with respect to the origin when p, = 0 seems to me to
point out a kind of geometrical explanation of the great simplicity introduced in the solution of
equations by first depriving them of their second terms.

6. To determine where p is a minimum, we have by differentiating (9) and putting ae =10;

np" cos nO + (m — 1)p,p"~* COs (1 ~1)O + veseee = Over serene ee (10),

which equation together with (9) will give the required values of p and @. Now if we make
6 =0, which satisfies (9), (10) becomes

mp" + (n— 1) pip" + ..000 = 0,

or, tT (p) =
which (if w be written for p) is the equation for eis the maxima and minima of the real
branch of the curve; hence p is a minimum for the projection of such points. Besides these
there may be other minimum values of p lying between the different pairs of asymptotes.

Vout, VIII. Paxr III, Yy

346 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION

7. Corresponding to the asymptotes of the curve in the plane of xy there will be infinite
branches in space, and it is easy to shew that these go off alternately to positive and negative

, 6 ka %
infinity. For from equations (8) we have, when @ = = and p is consequently very large,

= p” cos kr = (- 1)" p";

therefore for odd values of & the limiting form of the curve is given by

z=- 9",

which represents a parabolic branch going off to negative infinity for positive values of p, and
vice versa if m is odd, and going off to negative infinity on both sides of the origin if 7 is even.
And for even values of & the form is given by

id p”>
which represents a branch going off to positive infinity for positive values of p, and negative
infinity for negative values of p if m be odd, and to positive infinity in both cases if m is even.

This proposition it is easily seen includes the real branch of the curve, and hence if we
indicate by the mark + or — on an asymptote that the corresponding branch of the curve goes off
to positive or negative infinity respectively, the arrangement of the infinite branches will be

represented by the accompanying diagram.

n odd. m even.

8. I shall next prove the following theorem :

At points in the real branch of the curve for which the first p differential coefficients of f()
vanish, there are p imaginary branches going off on each side of the real plane or plane of wx,
and these are curved alternately in opposite senses, the one nearest the real branch being curved
in the opposite sense to that real branch.

Suppose the origin of co-ordinates such that the axis of passes through the point in question,
which may be done without in any way affecting the generality of the proof, then we shall have

ff) =, f"(0) =0...... f°) = 0,

and the equations (7) become

OF THE ROOTS OF ALGEBRAIC EQUATIONS. 347

p+l
z = f(0) ne a pcos (p +1) 0 -- cece.
0 pee (p+1)0 +......

The form of the curve very near the point in question will be given by taking only the terms
of the series above set down, and therefore we shall have

sin (p + 1)0=0,

(p+1)0=kzr;
pera 2a 31 pr.
p+’ p+’ ptr wocves p+i1°

omitting the values 0 and p +1 of & which correspond to the real branch. Hence there are Pp
imaginary branches going off from this point; and to determine the nature of the curvature
we have

Lo @
|p+1
- if (0)
= f(0) + (-1)* ——— pPth
FO EO se

The second term will be alternately positive and negative as & assumes successive value’,
and when it is further observed that the equation of the real branch is given by putting k =|
the whole of the theorem will be seen to be true.

In the case of simple maxima and minima, for which f’(#) = 0 and f’(«) does not vanisl
this proposition admits of more simple and obvious proof; for we have from equations (3),

x = f(0) +

cos ka. p?*?

2 = f(x) “F's Le

o=f'(a) - ro eae

and it is clear from these equations that when y =0 f'(w) = 0, that is, where there is a maximum
or minimum there is an imaginary branch; it is also evident that the imaginary branches can
never cross the real plane except at points for which fF (@) = 0, that is, either at maximum or
minimum points or points of inflexion: this last is an important consideration, because it shews
that, in tracing the general form of a curve, after having traced the real branch and those imaginary
branches which start from points at which f’(v) = 0 we may be quite sure that all the remainder
of the curve lies in isolated infinite branches situated symmetrically with respect to the real plane.

If we suppose as before the axis of x to pass through the maximum or minimum point,

we have,
y?
OE ROL

which shews that the form of the imaginary branch is that of a parabola curved in the opposite
sense to the real branch, or we may say that when there is a maximum in the real branch there
is a minimum in the imaginary, and vice versd.

YY 2

348 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION

It would not be difficult from this particular case in which only the first differential coefficient
vanishes, to derive the other more general proposition in which the first and any number of
subsequent differential coefficients vanish; at least we could conclude the existence of imaginary
branches curved in opposite senses though perhaps not their directions. For we may consider a
point for which f(x) f’(x)....-.f? (a) each = 0, as the case of p successive maxima and minima
degenerating into one point, and since these maxima and minima must necessarily occur alternately
there will be p imaginary branches curved alternately in opposite senses.

9. Let us now examine whether the ordinate x admits of any maximum or minimum values
besides those which it has in the real branch of the curve.

The general equation of the curve is

x = f(pelV>),
and the equation for finding the maxima and minima is
if, (pet) = 0,
which is equivalent to these two
2 n—-1
f'(0) +f" (0) pcos + Oa Ae) AB ceeds esi pes (n-1)0=0,

f’ (0) sin 0 + PO ae 20 + seveee + f"(0) — sin (xn — 1)0 = 0;

and we have also the condition of x being real, which is,

f (0) sin@ +f” (0) f sin ZONE Rae +rom sinn0 = 0:

Ee [n

or these may be written
Pn—1 + 2Pn-2p COSO + ...+.. + Mp’ cos (nm — 1)0=0
2p,» sin 0+ ...... + mp" sin (v7 —1)0=0 }...,..... (11).
Pn—1 Sin O + Py-z p SIN 20 +o. +p" sin n@ = 0

These three equations involving only two unknown quantities cannot be generally satisfied ; I
have not been able to shew directly that they never can be satisfied, though it seems possible that
such may be the case; I can however give a complete solution of the question so far as the
purpose of this memoir is concerned by proving that a maximum or minimum point is never
unaccompanied by a branch curved in the opposite sense, in fact, by extending to all branches of
the curve the proposition which has been proved above for the real branch.

10. The proof is as follows :
We have in general
z=f(wt+y J/ =n}
P+Q\/-1, suppose,

(cos y af),

=
=
®
be J
®
a=]
i]

Q= (sin y=) $0.

Prt a.

ee

OF THE ROOTS OF ALGEBRAIC EQUATIONS, 349

dP :
Now it will be easily seen, that if re represent the partial differential coefficients of P
v

with respect to v, then

in like manner,

lg 5 Soe rnsaicace cleus
dx dy

and similarly it may be shewn* that
aes Agee Sen):
dy dx

In order that 6x may vanish when a and y vary, we must have

dP dP
ane tea éy = 0,
dQ

dQ
aa eK dy a

Multiplying these equations by = and = and adding, we have, observing the relations
y

(12) (13),
ie 3 & z Oi
a ‘3 ay) Rie

dP
70) dQ =O)
dy dy

Hence also, GE =0 he = 0.
da dx

If the values of w and y which satisfy these equations also satisfy the equation Q = 0, this
will indicate a singular point in the curve, and we must determine the nature of this point; to do
this we have for the increment of x, supposing the terms of the first order to vanish,

d’P

Ie dP
2 = — dn’ 2 =———TOMtesieipssioccisesseciore(L&)is
Oz aa 8a + TET 2dudy + dy? oy (14)

(there is no term involving 6*y because its coefficient would be = which in this case vanishes) ;
y

* The roots of the equation f(a)=0 may be considered as ; dQ (n- ee
determined by the intersections of the curves P=0 and Q=0. (* wecar Til La dye?
These curves have the property of intersecting each other at p

- [ae dP dQ dt dQ
right angles; for the equations of the tangents to the two curves at | which in virtue of the relations FSET oo eT ee
& common point (, y) are ; y a ;

dP dP present two lines perpendicular to each other,
(1-7) dy t ("u-W77=%

350 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION

and we have also the relation,

Q,,, #Q. #Q
O0= Pa ow Ta Se gegen ee

SEs (cosy a \r@, F=-(sny F) £@,

£2 (usd) rer £2, --(iord) ron Be (oud) ro.
also Q= (siny 2) F(@):

12 innit) ron = (rg) 0

Hence, if we call the values assumed by (sin y =) f'' (2) and (cos y =) Sf’ (a) at the point
under consideration A and B respectively, (14) and (15) may be written thus:
2 Ss = Boat — 24 da dy — BSy? .......0.0000eeeeeeee (16),
0 = Ada’ +2 Bdady — Ady? o..202sc0000---se00e0 (17).

Let da = ds cos d, dy = ds sin @, then
os

ee ee
0=Acos2p+Bsin2g;
2
or 2 oe C cos (2 + Desi ssensrentesatse pen lS)y

O= sin (2 P+A) --..-- ee eee eee eee eee (19);
by putting 4 = B tan a,

Equation (19) determines two values for 2 + a, and one of these will make 6’ positive, the
other negative; hence at the point in question there will be two branches curved in opposite senses,
one will be a minimum, the other a maximum.

I have proved this proposition for simplicity’s sake in the case of a double point, but the same
mode of investigation may be applied to that in which the increments of x all vanish up to any
given order ; this I proceed to do.

It is not difficult to see that if the coefficients of the powers of d2 and dy in the increments
Sx, 2 ...... 6"—!x all vanish, then the value of 6"x may be written thus,

d d\”
1.2 seanwn et ee Co —— $02 bbe vesinwe testes
m3 (80 — + dy = P (20),

Le

OF THE ROOTS OF ALGEBRAIC EQUATIONS. 351

with the condition,
da a
( ¢ a y fee reeves soseonene ( ).

And it will easily appear that
P d" P d” P Nee
ae = (cosy a) Fo) ima 7 (me) S (), Pate pe (x)...
d"Q - d”Q d as d"Q F d iH
ce = (siny £) Fro), Faraidy = (8 ae) PO) Geenqa =~ (9 zs) TH @)-

d 1
Therefore calling the values assumed by sin (v =) f” (a) and cos (v =) f” (w) at the point

in question 4 and B respectively, equations (20) (21) may be written,

TED, fuk mo"z = Bdw" — mAdz" dy - Se BSa"-* dy? + &e.
0.= Ada" + mBSar-tby— "=D dqn-tgy
Let da =0s cos d, dy =os sin ~,
BD oves'e. meta B (coun gp - Bm = 2 cos"~* @ sin’? @ + «2... ) -— A(m cos"~'@ sin — ...)
0 =4 (cos" = aoe cos"~* @ sin” @ + ee + B (mcos”"'@ sin d — ...)
or 1.2...... m <= = Beosmp~ Asin mg,
8
0 =Acosmp + Bsin mg,
and lastly these expressions may be put under the form
§"
1.2 sesso m xa = C €08 (™ P+ a) veveescorees (22),
O = _— sin (MP +a) .es-eeeeeeee (23).

The last equation gives us
mopt+a=kr,

where k may have any one of the values 0, 1, 2...(m-—1); hence there will be m branches ;
also the sign of 6"x depends upon that of cos kz, or of (— 1)*, and will therefore be alternately
positive and negative; hence the m branches will be curved alternately in opposite senses.

Hence, therefore, if values of # and y can be found which will make dx =0 and Q = 0, there
will not be a maximum or minimum point properly speaking, but a multiple point in which
two or more branches of the curve meet, and these branches being, as has been proved, curved
in opposite senses, there cannot be an absolute maximum or minimum, that is, a maximum for every
branch or a minimum for every branch.

11. This proposition completes the theory of the roots of the equation f(#) =0; for it
has been shewn that the curve of double curvature corresponding to the equation x = f(w) admits
of po maxima or minima, and that it consists of m branches going off alternately to positive and
hegative infinity, hence the plane of wy or any plane parallel to it must necessarily cut the curve in
n points, and the distances of these m points from the origin will be the roots of the equation.

352 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION

It may be observed that the preceding investigation applies to multiple points in the
real plane by making 4 = 0.

12. A less general application of what has preceded presents itself in the case of an equation
of an even degree having its last term positive: in this case it is well known that there is some
difficulty in proving the existence of a root. But I observe that if x = f(«), where f(«) is of
even dimensions, x has necessarily a minimum value, and from the minimum point an imaginary
branch starts off on each side of the real plane, which will stretch out to negative infinity and
therefore cut the plane of wy in two points which will correspond to imaginary roots. Hence
we see as it were the rationale of such an equation having at least two roots, for f(#) must admit
of a minimum, and if this be negative the curve cuts the axis of # twice, if positive imaginary
branches go off from the minimum and these take us down to the plane of wy.

13. The roots which are thus determined by the intersection with the plane of wy of
imaginary branches starting from points of the real curve for which f’(«) = 0 are so related to the
real roots, that it has seemed to me to be desirable to denote them by a distinct name; I therefore,
for want of a better name, call such roots connected roots, and those which are determined by the
intersection with the plane of ay of other infinite branches which, as I have shewn, never cross the
real plane, I call isolated roots. Thus I should say of an equation of even dimensions, that it must
have two roots either real or connected.

14. But more generally we may distribute the m roots of an equation into real connected
and isolated roots. For suppose the real branch of the curve traced, and suppose that it has. p
points for which f’(w) = 0 and f’(#) does not vanish, then it is easy to see from what has been
said that there will be p +1 roots either real or connected; from the p maxima and minima there
go off 2p infinite branches which occupy 2p out of the 2” — 2 asymptotes*, leaving 2 — 2p — 2
asymptotes; between each pair of asymptotes there is an infinite branch which cutting the
plane of wy gives a root, therefore there are m — p —1 isolated roots; and thus we make up the
whole number of roots 2. I will just observe that » — p —1 is obviously even, because if m is
even p is necessarily odd, and vice versd. The same proposition may be extended to the case in
which other derived functions besides f’(7) vanish at any point, by the reasoning used in Art. (8);
for we may consider such a point to be the degeneration of a number of contiguous maxima and
minima, for each of which the proposition is true. It may therefore be stated generally, that if
there are p real values of w, whether all unequal or not, which make f’(w) vanish, then the
equation f(w) = 0 has p + 1 roots either real or connected.

15. It may be observed, that a pair of connected roots may be changed into a pair of real
ones by altering the position of the plane of ay, or speaking algebraically by changing the value
of the last term of the equation; and this fact points out the propriety of distinguishing between
connected and isolated roots, which latter are necessarily imaginary wherever the plane of wy cuts
the axis of x, since they are determined by the intersection of that plane with branches of the
curve, which, as we have seen, never cross the real plane.

16. The number of real and connected roots evidently depends upon the number of real
roots of the equation f’(«) = 0, and (as has been already in fact proved) if the number of real
roots of this derived equation be p, then the number of real and connected roots of the original
equation will be p + 1; consequently the number of isolated roots of the original equation is equal
to the number of imaginary roots of the derived,

" In Art. (5) [have spoken of n—1 asymptotes, here of 2n—2; | asymptote, here for convenience I have considered the same line as
the difference consists merely in this, that in the former case I have two stretching out to infinity on opposite sides of the origin.
considered the indefinite straight line through the origin as one

OF THE ROOTS OF ALGEBRAIC EQUATIONS, 353

17. Hence also we sce the truth of a theorem, of which I shall presently make use, namely,
that an equation has at least as many imaginary roots as any one of its derivatives; for the equation
f(z) = 0 has as many isolated roots as there are imaginary roots in f’(z) = 0, and therefore has at
least as many imaginary roots; f'(v) = 0 has in like manner at least as many imaginary roots as
f’(«) = 0, and so on: whence the truth of the proposition is clear.

18. If the plane of wy should happen to pass through a real maximum or minimum, which
is as we have seen properly speaking a multiple point, there will be several equal roots. The
condition of equal roots will be therefore that the plane of wy shall pass through a point for which
one or more of the differential coefficients of f(#) vanish, or which is the same thing, that f(«) = 0
and f’(#) = 0 shall have one or more roots in common; which as is well known is the test of equal
roots. Or we may shew directly that at a point for which there are m equal roots there are m
branches curved in opposite senses ; for let x = 0 for simplicity’s sake be the root which occurs m
times, then

F(@) = OS Pn + Par © + vooeee + Bt,
and the equations to the curve will be
x = p,p™ cos mO + ....,.5
0 =p, sin mO + ......3
therefore near the origin, sin m@ = 0,
«. m@=ka where k may = 0, 1...(m — 1),
and #=p,p”cos ka
k
=(- 1) Prp” 3
therefore there will be m branches curved in alternately opposite senses,

19. It will be seen, that a pair of equal real roots in the equation f’(0) implies a pair of
imaginary roots in the equation f(a) = 0, since f”(«) will also vanish for the same value of w
as that which makes f'(w)=0. And generally, if w be even, r equal roots of the equation
f (@) = 0 imply r imaginary roots in the equation f(~) =0; if 7 be odd, there will be r + 1 or
7” —1 imaginary roots according as f(#) and f"*'(#), which is the first derived function which
does not vanish, have the same or different signs.

This theorem requires no demonstration, as its truth will be seen at once on examination.

By means of it I am able to prove the ordinary proposition relative to the number of imaginary
roots belonging to an equation defective in any of its terms; the proof is as follows :

Suppose,
S(@) = Pn + Par B+ ooeeee + Mart + Nabt?t + oo... + ao’,

where » terms are wanting between the terms Ma“ and Na*+"t+!;
differentiating « times, we have

F(a) = (u-1) 00 2-1 M + (utvtl) (n+)... (v +2) Nao't's on...

+n(n-1)..(n-—pw+1)a"";
differentiating again,
fe (a) =(whtv41)-(Qutyv)... vt 1) Na" + ...... + 0.(m - 1)... (a — pw) are!
Hence the equation
f**'(@) = 0

has v roots equal to 0, and therefore the equation f*(#) = 0 has v imaginary roots if v be even,

and if v be odd, it has y +1 or y —1, according as f“(0) and f**’*'(0) have the same or opposite
signs, that is, according as M and N have the same or opposite signs. But, by a theorem cited

Vor. VIII. Parr III. Zz

354 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION

and proved in Art. (17), f(v) = 9 has at least as many imaginary roots as any one of its derived
equations; hence it will have at least y imaginary roots if v be even, and at least vy +1 or vy —1,
according as M and N have the same or opposite signs, if » be odd.

20. I will now illustrate what precedes by discussing some actual cases and tracing the
corresponding curves.

Let the equation be a quadratic, that is, let

S (a) = 0? = AV 4D HO cereeee eee s ene eee ses eeenes (24), N
of’ (@) =20-a,
ff" () =2;

and the equations of the curve of double curvature are
s=a-axv+b-y

0=2H%-a

if we eliminate w by means of the second of these equations, we have

a’
#2 6 ads

ée=

Hence the complete locus of the equation s = f («) will be in this case two parabolas in planes
perpendicular to each other, with their vertices coincident and their curvatures in opposite senses:
2

the height of the vertex above the plane of vy will be b — - , if this be positive the roots of the

given equation are imaginary, if negative they are real, because in the former case the plane of ay
cuts the imaginary branch, in the latter the real. We see in this simple instance what has already
been proved generally, namely, that x does not admit of a maximum or minimum yalue properly

speaking, because at the minimum point an imaginary branch goes off along which

= still
decreases.

The figure represents the curve corresponding to a quadratic equation,

OF THE ROOTS OF ALGEBRAIC EQUATIONS 355

a”

a bt ae ce
AB mae BC =b - 7? which in the figure is supposed positive: X, CX, is the imaginary
branch cutting the plane of wy in X, and X,, so that 4X), 4X, are the roots of the equation,

I may observe, that the mode of viewing the subject which is explained in this paper, though
rather complicated when considered generally, is of very easy application in the case of a quadratic ;
for the ordinary solution gives us the roots

2

ae VES if 6 be less than ee

iss)

4

2 2

and w=—+£\/=1 6- a if b be greater than = :

wl1a

a ane ;
Now v = a corresponds to the minimum value of «* — aw + b, and therefore the usual mode of

interpreting the symbol \/—1 would lead us to consider the preceding expressions as the distance

of the minimum point from the origin + a distance measured along the axis of # or perpendicular
2

to it, according as 6 is less or greater than ae

21. Let us take the case of a cubic, which I shall suppose to be deprived of its second term
for reasons heretofore assigned. We have then
(ay Uta I UE INO! cenoeunsacediseccereccetenereene (20).
f' (a) = 32° - 4,
f° @) = 62,
9 ied (x) = 6.
Hence the equations of the curve will be
z= @ —qu+r— sry’
0=8a -q-y

The curve will assume different forms according to the nature of the parameters q and r.
Let us consider the real branch of the curve ; then the condition dx = 0 gives us

3a°—q=0, or w= + Jt,

hence in order that there may be a maximum or minimum point q must be positive ; suppose this
to be the case, then there will be one maximum and one minimum, and for the value of x we

have
@
r= 2 ——
Y= i

re Us a * ve
T shall suppose r to be positive, and rR = Z, so that both values of x may be positive.

The curve in the plane of wy is evidently an hyperbola, the asymptotes to which are inclined
at an angle of 60° to the axis of «, and in the case here supposed of q being positive the real
principal axis will be the axis of a.

ZZ2

356 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION

These indications are sufficient to shew the whole course of the curve which is represented
in the annexed figure :

P,, P., are respectively the minimum and maximum point; the real branch of the curve
necessarily cuts the axis of # to the left of the origin; from the minimum point P, goes off an
imaginary branch which meets the plane of wy in X,, X,, thus giving two imaginary roots. It will

be remembered that the conventions which have been made are that q shall be positive, 7 positive,
2 3
and — ae it will be easily seen that the form of the curve will remain essentially the same

so long as the first condition is fulfilled, and the changes introduced by varying the latter conditions
may be represented by supposing the plane of wy shifted into different positions, Suppose for

instance the plane of wy to cut the real branch between P, and D (the point of intersection of the
< 3

- F a ore r :
curve with the axis of x); this will correspond to 7 positive, and Zz — = then there are three

real roots, two positive and one negative; if the plane of wy cuts the real branch between

By

9

: r ane
D and P., we have the case of 7 negative, and 4 <a = and there are one positive and two

negative roots; lastly, if the plane of wy cuts the real branch above P,, we have the case of 7

2 pe : OF . . .
negative, and ris — and we have one positive real root and two imaginary. I may just observe
that all the imaginary roots here spoken of are of the class which I have termed connected.

If we suppose q negative we have an entirely different form of curve, for in this case the real
branch has no maximum or minimum point, and therefore it is clear that one of the roots will be
real and the other two isolated and imaginary. Also the real principal axis of the hyperbola in the
plane of wy will be the axis of y, and not the axis of a as in the preceding instance. It is not
necessary to trace the curve, as its form is easy to imagine and it presents no varieties.

The preceding discussion includes every case of cubic equations.

22. We may discuss in like manner the general biquadratic equation. In this case,

f(a) = a8 + Qa? + 10 + 8 = Oveececgencnececees ces cos(28),
f (@) = 405 + 2qa +7,

On ee

OF THE ROOTS OF ALGEBRAIC EQUATIONS. 357

f(a) = 124° + 24,
f(a) = 24a,
ff" (a) = 24;
and the two equations to the curve are therefore,
saat qet+ra+s—y (6a +q)+y'
0 = 4a° + 2qu47r—- 4y'a }

Now the sign of s need not be considered, since (as has been obseryed before) a change in its
sign will only correspond to a change in position of the plane of wy, the figure of the curve
remaining the same; the combinations of sign of q and r will be as under,

+

and these different cases must be considered.

The equation for determining the maxima and minima of the real branch of the curve is,
3, 2 i
a +-v+—-=0
ae

which has one real root if q is positive, and if q is negative it has one or three, according as

iad
— is > or < than aN
8 27

First then, let q be positive and let also 7 be
positive, then it will be found that the curve will
be such as is represented in the annexed figure.
P is the minimum point of the real branch; the
dotted lines represent the imaginary branches, which
cut the plane of wy in the points X,, X., and in two
other similarly situated points on the other side of
the plane of wz which are not represented for fear
of complicating the figure.

If r be negative, the figure will be essentially
the same, but must be supposed to revolve through
two right angles about the axis of x.

; bys g

Secondly, let q be negative; then if "i be ieee
there will be no difference in the figure but this,
that the curve of projection on the plane of wy
a” lie nearer to the axis of w than the asymptotes, instead of lying further away, as in the

t case.

358 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION

But if 2 be < ci the form of the
8 27

curve will be essentially different, and
will be as in the annexed figure. If we
suppose the figure to correspond to the
case of 1 positive, then the figure for
negative will be found as before by sup-
posing everything turned through two right
angles about the axis of x.

23. The curve corresponding to the equation «” — 1 =0 is easily traced, and furnishes a good
illustration of what precedes. I shall trace this curve with polar co-ordinates.

We have, % = p" (cos 20 + / = 1sin VQ) = Wecs cocccacaceavrsces (30),

which divides itself into the two equations,

\. hi dt Set aia ce tt.2 (31);

# = p"cosn@ — 1

O= sinné
from the latter of these n0=kw where k =0, 1, 2...... (n - 1);
*.2=(- 1)‘p" — 1,
Hence the complete curve will consist of a series of parabolic curves defined by the equations
#=p"—1 and x= -—p"-1 alternately, and lying in planes 1z

passing through the axis of x and making with each other an

Tv
angle —.
n

The figure represents the curve; Oa,, Oa; ...... are the
branches stretching up to positive infinity, Oa, Oa, ......
those to negative infinity: the plane of wy intersects the
former set of branches but not the latter, and gives for the
roots) :A.N5; “AEX, 2225.0

If we suppose the plane of ay to intersect the branches
Oa, Oa ro-ns> we should have the case of the equation

a" +1=0;

and if the plane were to pass through O, we should have the
_ curye corresponding to «" = 0, in which case the roots would
be all equal to 0.

OF THE ROOTS OF ALGEBRAIC EQUATIONS. 359

24. The investigations of this paper have been restricted to ordinary algebraic equations,
nevertheless some of the results are of a more general character and need not be so restricted. The
proposition contained in Art. (8) is, I believe, perfectly general, as also is the proposition of
Art. (10) which is an extension of the former. In fact the theorems about maxima and
minima will be true for all such points as do not involve a failure of Taylor’s Theorem, which
never occurs in the case of a rational algebraical function. The propositions concerning the number
and position of infinite branches are of course applicable only to algebraic equations. I will just
notice one instance of an equation not algebraic: suppose

{MEY SSN a) =" acnbcsenauastedeen eee sacra (G2)

then zx

d ‘
(cos y aa sin wv

a

sin w f+? + + at ee

é ev +e"
= sin &

eupeuwebvscebsoscdseene (G5)

and 0= (sin y =) sin w
v

d

wee lye Fae cued

Is“ [5
=1COSE) C4 Opt ce assncecrleatsnne aos (O4):

In equation (34) the variables w and y are entirely separated ; the factor e” — e~” when equated
to zero gives, as will easily be seen, only one real value of y, namely y =0; this corresponds to the
real plane, and if we make y=0 in (33), that equation becomes

s=sin 2,
and we have the ordinary figure of sines in the real plane.

If we consider the ial cos w in (34), we have an infinite number of real roots for the equation,

3a
namely v = + = + ie ie =, &e., and substituting these in (33), that equation becomes

(ti

= oS
2

which shews that from the maximum and minimum points of the real branch of the curve imaginary

branches set off in planes at right angles to the real plane, which are in fact common catenaries, the

directrices of which are in the plane of wy, and which go off alternately to positive and negative

infinity.

25. In concluding this paper I will observe that I am not sufficiently well acquainted with the
literature of the subject to be certain as to how far the idea of it has been anticipated. I will
observe, however, that in the late Mr. Murphy’s Treatise on the Theory of Equations, (published
_ under the direction of the Society for the Diffusion of Useful Knowledge,) the existence of the roots
of Algebraic Equations is demonstrated upon principles similar to those which I have adopted ; it

i

360 Mr. GOODWIN, ON THE GEOMETRICAL REPRESENTATION, ETC.

is there proved, first, that after a rational function of 7 dimensions has attained a minimum value
corresponding to a real value of , it is possible to diminish the function still further by assigning
to w an increment of the form h + k/—1, and then it is shewn that by assigning to w a value of
like form, it is possible to give to a rational function of w of evem dimensions a series of
continually increasing or diminishing values, which propositions are akin to, but far less general
than, those which I have proved in Arts. (8) and (10). Nevertheless the mode of viewing the
subject is the same as that which I have adopted, and indeed suggested to me the possibility
of illustrating the theory of equations by reference to the curve of double curvature, which represents
the succession of real values of a function of # corresponding to values of the form x + y /=1:
apart from which geometrical illustration, the theory of the roots of equations which depends upon
the demonstrated impossibility of a maximum or minimum value of f(#), when the values of « are of

the form a + y/ =aF appears to me to be more luminous than any other which I have seen.

H. GOODWIN.

XXXVI. On a Change in the State of an Eye affected with a Mal-formation.
By G. B. Airy, Esq., Astronomer Royal.

[Read May 25, 1846.]

Twenty years ago, I had the honour of submitting to this Society a statement of the effects of
a mal-formation in my own left eye. The nature of the effect was this: that the rays of light
coming from a luminous point and falling upon the whole surface of the pupil do not converge
to a point at any position within the eye, but converge in such a manner as to pass through
two lines at right angles to each other, (a geometrical phenomenon, to which the term astigmatism
was very happily affixed by the present Master of Trinity College), and that these lines, in the
ordinary position of the head, are both inclined to the vertical in the manner described in my
paper (Cambridge Philosophical Transactions, Vol. 11.) The evidence of this astigmatism, and
the measure of it, are given by the simple observation of bringing the luminous point nearer
and nearer to the eye; the lines of focal convergence, according to the usual rules of focal
position, move in the same direction in the interior of the eye; and thus one line and the other
line are successively brought upon the retina; or the image of the point becomes successively
a line in one direction or in the other direction, these directions being at right angles to each
other. It was found in 1825 that the distances at which the luminous point must be placed to give
linear images were 3°5 and 6°0 inches; and the difference of the reciprocals of these numbers, or
0:119, is a proper measure of the astigmatism. The fault of the eye was corrected, as regards
the production of distinct vision, by the use of a lens of which one surface was spherically
concave, and the other surface cylindrically concave, and the radius of the cylindrical surface
was such as to give a power 0°119, or, in combination with a plane surface, to give a focal length
1 <a 7

_.— inch, or it was in inches

07119

Some years since, I found, from some unrecorded observations, that the general short-sighted-
ness of the eye had sensibly altered, but that the measure of astigmatism remained nearly the same
as at first.

Lately, having found that the spectacles constructed for me in 1825 do not very well suit the
present condition of the eye, I have made observations in precisely the same manner as in 1825, by
viewing a very fine hole pricked in a card, and causing that card to slide upon a scale whose end
rests upon the orbital bone of the eye, and measuring the distances at which the card is placed when
the point appears as a line. I have been careful to hold the body and head in the same general
position as before: the accuracy of the measures being sensibly affected by these circumstances.

As far as I can remember, the indication of the focal lines to the horizon, their length, and their
sharpness, are not in the smallest degree changed, But the distances of the luminous point which
produce them are sensibly changed. ‘They were formerly 3°5 and 6:0 inches: they are now 4°7 and
89 inches, The eye therefore has become generally less short-sighted than it was formerly.

But the measure of the astigmatism, which was formerly

1 1 d 1
— — — = 0119, 1s now — —- —
85 6:0 ? 47 8°9

Vou. VIII. Paur IIT. 3A

= 0°100.

362 Mr. AIRY, ON AN EYE AFFECTED WITH A MAL-FORMATION.

On examining the slightly, discordant observations, I am inclined to think that a distance somewhat
less than 4-7 is the true] one, and this would increase the measure of astigmatism above 0:100, and
would make it approach more nearly to the ancient value. It seems therefore that while}the short-
sightedness of the eye has materially diminished, the fault which produces the astigmatism has
undergone very little or no alteration.

Upon examining the right eye in the same manner, I find no perceptible fault. The image of
a fine hole is a luminous point very sharply defined. The distance of accurate definition is as nearly
as possible 4°7 inches, the same as the nearest distance at which the left eye forms a well defined line
for the image of a point. It would seem therefore that the normal formation of the two eyes is the
same, and that the abnormal alteration in the left eye is of the nature of a refraction through a dense

medium cylindrically concave, or through a rare medium cylindrically convex, superadded to the
normal refraction.

G. B. AIRY.

Royal Observatory, Greennich,
January 14, 1846.

XXVIII. A Theory of Luminous Rays on the Hypothesis of Undulations. By the
Rev. J. CHatuts, M.A., Plumian Professor of Astronomy and Experimental
Philosophy in the University of Cambridge.

[Read May 11, 1846.]

Ir a beam of Sun-light pass through a narrow aperture, about one-thirtieth of an inch in breadth
and be received on a glass prism the edges of which are parallel to the borders of the aperture,
a spectrum is formed by the transmitted light, which, when magnified and properly looked at,
exhibits, as is well known, a large number of dark lines parallel to the refracting edge of the
prism. If instead of passing through an aperture with parallel borders, the light passed through
a circular aperture, one-thirticth of an inch in diameter, a spectrum of diminished width would
be seen, but of the same length as before and crossed by the same dark lines. The trans-
mission through the prism has produced no change on the light: it has only brought into view
the parts of which the incident beam is composed. Taking, for instance, a portion of light
immediately contiguous to any one of the dark lines, the prism informs us that the incident
beam contains light of that particular refrangibility, abruptly terminated in a plane passing through
the axis of the beam perpendicular to the edge of the prism. The existence of this abrupt ter-
mination is owing to the cause, whatever it may be, which produces the dark line, and has nothing
whatever to do with the transmission of the light through a small aperture. Let now the prism
be turned about the axis of the beam to any other position. The spectrum will present exactly
the same appearance as before, and light of the same refrangibility (not necessarily the same light)
as in the former case, will still be bounded by a dark line. And so for every position the prism
be made to take by being turned about the axis of the incident beam. This experiment proves
that every beam of white light contains portions of light of a definite refrangibility, the sides
of which are turned in all directions from the axis of the beam. This fact is at once explained
by supposing light to consist of rays; and it does not appear possible to give any other explanation
of it. Although the experimental evidence applies immediately only to the portions of light con-
tiguous to dark lines, yet a very strong presumption is afforded by it that all light is in the
form of rays. The existence of the dark lines themselves is most simply accounted for by
supposing that certain rays of Sun-light are in some manner extinguished.

Admitting it to be a legitimate deduction from the facts of the dolar Spectrum, that light is
composed of rays, it is clear that no Theory of Light can be complete which does not take account
of this distinctive character. The facts are perfectly consistent with the Theory of Emission,
and the advocates of that theory might justly appeal to them as evidence in its favour. My
object in this communication will be to shew that rays of light are also to be accounted for
on the Undulatory Theory.

It must here be premised that it is not my intention to treat the Undulatory Theory, as
most optical writers of the present day have done, by a particular consideration of the molecular
constitution of the «ther. Not having been able to form the slightest conception how this view of
Undulations can be reconciled with the existence of rays of light, I propose to regard the «ther as a
continuous fluid substance, such that small increments of its pressure are proportional to small
increments of density, and to apply to it the usual hydrodynamical equations. The pressure

3A2

364 PROFESSOR CHALLIS, ON A THEORY OF LUMINOUS RAYS

being p and density p at the time ¢ at any point whose co-ordinates are a, y, x, it will be
assumed that p=a*p, a” being a certain constant.

In a former communication which I made to this Society, I gave the proof of a new fun-
damental equation in Hydrodynamics, by the combination of which with the ordinary equation
of continuity, an equation results which is indispensable in the present investigation. The process
for deducing this last equation is given in the Cambridge Philosophical Transactions, (Vol. v11.
Part 111. pp. 385 and 386): it is also obtained (p. 387) by independent elementary considerations.
Let V be the velocity and p the density at any time ¢, at a point where the principal radii
of curvature of the surface cutting the directions of motion at right angles are R and R’, and
let ds be the increment of a line ‘Gotieident with the directions at the time ¢ of the motions of
the particles through which it passes. Then the resulting equation I speak of is,

dp mest x)-
ae fs +pV(Z+ R Osreabiielaivicisteleivisiaiels sie (1);

the variation with respect to space being from point to point along the line s. Now the new
fundamental equation above mentioned, combined with the two other fundamental equations, gives
the means of obtaining a resulting equation, in which the variables are Wy, 7, y, x and ¢, the
principal variable y, being such a function of the others that = 0 is the equation of a surface
normal to the directions of motion, in whatever way the motion of the fluid may have originated.
It follows that the function yy, since it is given by a partial differential equation, contains arbitrary
functions of x, y, x and ¢, and that the normal surface is consequently arbitrary. The partial
differential equations applicable to the Undulatory Theory of Light are linear with constant
coefficients. For our present purpose, we have to enquire how far W is arbitrary when the
equations are of this nature: whether, for instance, the normal surface must necessarily be either
a plane or a spherical surface. The general equation which gives Wy by integration is too com-
plicated to be employed in this investigation, We may, foneyen dispense So the use of it
by combining equation (1) with the following general equation, which is obtained in p. 383 of
the communication already referred to:

2

dV V
* Nap. | == 108 — =e Bl(E) 5 wep vsycsacecigs sence sone 5
a’ Nap ogp + | ds + 3 F(é), (2)
the variation with respect to space being, as before, from point to point of the line of motion. By

differentiating this equation with respect to s and ¢ a we get,

atdp av av a!
+V—= 0,
pds tae ds fs

dV
rr ds pile ee Ost

Also equation (1) may be put under the form,
d Vi.d dV il) )
site eget +V (5 | =0.

pdt” pds? de RR
cane dp dp A : : aye F
Hence substituting for Sa and Aas from the preceding equations, and differentiating with respect
P pies

to s, the result is,

av @ leg Bars. syd dV dV /1 1) 1 1

Pe a ae re sola eng 2 RP

ae TM OY) Gat 0 vgaast * as ae *?? ae (i get (a * Re

d : ; :
for dR=dR'=ds. Now putting 04 for V, and integrating with respect to s after the substitu-
tion, it will be found that

ae

ON THE HYPOTHESIS OF UNDULATIONS. 365

L-(«- yo dq dq a (+ ara):

dt! dé) dé” ds dedt ”" de\R* FR’) ~
Lastly, for q put @ + y(t), the function x (¢) being such that F(t) — x” (#) = 0.

Then Ce ie and
ds ds
alae So) ai +e are oe aia at qe) TOs (3).

If the surface normal to the directions of motion be a plane, R and #’ are each infinitely great, and
the equation strictly applying to this case of motion, is

Ch _(p_@\€b , do TH _
dé - (a - 2) ores Gage Bee c ee eee reseesecsees (4).

It is well known that this equation is exactly satisfied by a particular integral applying to motion

dp

d
propagated in a single direction, namely, ce = F{ (a + <<) t- sf; and that at the same time a

8 ls
=4a.Nap.logp. From these two ale it follows that a given state of density and velocity
is carried through space by the propagation and by the motion of the particles, with the velocity

a+—. The rate of propagation is therefore strictly a, whatever be the velocity and density

of the particles. Unless this were the case the velocity and arrangement of density in a given wave
would change by propagation, however small the motion of the particles might be. Hence, in order
that equation (3), in which R and R’ are not supposed to be indefinitely great, may apply to motion
in which the type of the waves remains altogether unchanged by propagation, it must be of the same
form as equation (4). This will be the case if

o ib (2

Peak
at +) = SOD ese esreieerone ene grr oF (5);

the ae equation being the same as (4) with the difference of having a’ in the place of a. Also
~ = — - Nap. log p.

7 is now important to remark that the general partial differential equation having W for its
principal variable, to which I have already referred, is of the third order, and consequently its
integral, supposing it could be obtained, would inyolve three arbitrary functions of the co-ordinates
and the time. Hence the function \y may be made to satisfy three arbitrary conditions. The first I
shall suppose it to satisfy is, that the propagation of the motion be in a single direction; the next,
that the motion of the particles situated in a fixed straight line, which I shall call the axis of x, be
entirely in that line; the third condition I shall assume is, that for the motion along the axis of x
the equation (5) is satisfied. It will appear from the reasoning that follows, that a form of yy may
be found consistent with these conditions.

Let #, (#, t) be the condensation at the time ¢ at any point of the axis of x distant by x from the
origin, and let the condensation, for a reason that will appear afterwards, be assumed to be @ (x, ¢)
x f (@, y) at any point whose co-ordinates are a, y, x. For shortness sake I shall write @, and f for
these functions, treating , as a function of x and ¢ only, and f as a function of w and y only. Let
p be the density, and w, v, w, the components of the velocity in the directions of the axes of co-
ordinates, at the point wyz, and at the time ¢. It will be assumed that w, v, and w are always
small velocities, and their first powers only will be taken account of, ‘This being premised, we have

366 PROFESSOR CHALLIS, ON A THEORY OF LUMINOUS RAYS

a*dp ap, df =)
p= ee aameac as see Gel
d d
and to the first approximation, = =- ag Hence

af
u=— a dt+e
da Sd, ;
the arbitrary quantity ¢ being in’general a function of a, y, and . In the same manner,

d
v=- oF jy ats c.

. *d d d are f d
Also since — on - Be = (=) , we have to the same degree of approximation, =

d d . [ob dt ; Pa ,
= - at @, and we -a'f [edt +0" = Se +c’. But it is evident from the

assumed law of the condensation in any plane perpendicular to the axis of x, that the accelerative
force parallel to x at any point of this plane must to the first degree of approximation be equal to
fx the accelerative force at the point of intersection with the axis, and that the corresponding

velocities must be in the same proportion. Hence, ~ being the velocity at the point of the axis
s

d ;
of x, we shall have w =f. It follows that p=- a’ [p,dt, and that ce” = 0.
For reasons which will appear hereafter I shall also suppose that ¢ =0, and c'=0. Thus we

shall have,
ew es es dp
ESN OUR Were

It is to be remarked that these equations are the more exact, the smaller the ratios of w and v to w.
From the foregoing reasoning it follows that

d i
udu +ody +wde= pda +p ody + feds =4.Fp.

Hence wdw+vdy+wdzx is an exact differential; and it is well known that in a case of fluid
motion in which the first power of the velocity is alone retained, this condition must be fulfilled.
The assumed law of the distribution of the density consequently satisfies a necessary analytical
condition, and on this principle is justified. It follows also that dy,=d.f@, and by integration
that wv =foh +a function of ¢=0. Thus the equation of the surface normal to the directions
of motion is to a certain extent determined, and we may now proceed to obtain an expression

fo See
a ie
R R

1
The k ral ion for —
e known general expression for Rt 7?

de® dy + ae dz dy + Gs dx da dy’ dy’

dy dy ,ey dp dy j ty dy dp ty dy Re.

dz dzx* “dady‘ dw’ dy *dede ida ds “dydz' dy’ dz

dy? dy dyy-#
(a5 ay * 4

(Ge cy #y) fae ats dy dy dy ay |

Oe

——~

ON THE HYPOTHESIS OF UNDULATIONS. 307 —

dy, df dy df ay df
Iso, —— = @ — = a
Bt da e dx dx” arr dady Daag:
dy df Py ie df dy dp df
ay tay deine? aap rrr C ur ee
asa Ps yp ain of olan uit aos
dz dx dydz 2 dy”

Hence by substituting and reducing, it will be found that

1 4 aR aS? of pe apap @f af, af adfaf f Le
lat w) Gs dy” gt dx’) ~ dy?’ da da dy“ dady' dw dy g* dx \da* * dy
2 2 fe i
ie areas
po dx gd dz* dy
For our purpose we require an expression for at F for any point on the axis of x. Now

since by hypothesis w =0 and v = - : all times for all points on the axis of x, it follows from

d d
the equations u= ms and v = 9° — Z, that “f. 0 and Z =0 for these points. Hence the

foregoing equation gives,

G ‘ weg Se eS, a ARAB ee ee (6).

But equation (5) becomes for motion along the axis of s,

dp (1 1 F ip

[eee = —|j= C= yh) »

ae (ate) G ©) Te
Consequently putting a*(1 + k) for a’, and substituting from equation (6),

do 1 ae Cia @
Tat ipa 7g) P= Ola icawaneteaeenecete (7).

In this equation the coefficient of @, not containing w and y, is a constant, and we may assume it
equal to —m*. It hence follows that

&o Se Pree D) fe sae ceseargsccie=t gna beled «6ee00)(8):

dx?
&
I shall here take occasion to remark that since of = 0 when the velocity =
x

x

is a maximum, it

appears by equation (8) that @ =0 in the same case. Hence also ~= 0, and v= 0. Consequently
the assumption made heretofore that the arbitrary quantities ¢ and ec’ each =0, was equivalent to
assuming that the transverse velocity vanishes when the velocity is a maximum along the axis of

d :
It appears also from the expressions for uw, v, and w, that, when the velocity a =0, and @ is

consequently a maximum, w and v are each a maximum.
At the same time that the equation (8) is true, we have by equation (3) neglecting the small
terms, and by what has now been proved,

EDs ge MmaN a 0 2.02. speeluemnted (9).

368 PROFESSOR CHALLIS, ON A THEORY OF LUMINOUS RAYS

The equation (8) is satisfied by p=y(t)cos }mx+ (¢)} ; and the equation (9) by P=g(nz-na’2t).
Hence y, (#) = constant =, and y(#)=-—ma't. Consequently = = cos n(x —a’t), and the

velocity se =msin n(a't — %) =m sin ~ (a't — s) suppose.

It results from the foregoing reasoning that if the small vibrations of the zxther in the direction
of propagation follow the law expressed by the equation last obtained, the condensation in any plane
perpendicular to an axis of rectilinear propagation may vary at a given time from point to point,
and at the same time the propagation be uniform, A consequence of this result is that a very
slender cylindrical portion of the ther may continue in agitation while the contiguous portions are
at rest; and since the law above obtained is that which has been found by experience to apply to
the phenomena of light, the existence of rays of light, which was proved experimentally at the
commencement of this paper, is accounted for theoretically.

As far as we have hitherto proceeded, the function f has remained indeterminate. The con-
siderations I am now about to enter upon will serve to ascertain its form. Take a plane perpen-
dicular to the axis of x, in which the velocity parallel to the axis of is a maximum, and in which
consequently w = 0, and v =0. As the motion at any point of this plane is parallel to the direction
of propagation, and as the velocity of propagation is uniform, it follows that an equation like (5),
applicable to this point, is obtained by simply substituting f@ for @. Substituting also a* (1 + k)
for a'*, we have,

dp (1 1 op
tE (a+ pe) -HGS OS Re eae (10).
1 ee
At the same time the general expression for R + R gives,
(2 i) SOG Fh) tT 2 (ae)
aaa ae ae dy f \da® dy
1 1
d= dhs
a ih i
= nis + ay)

Hence by substitution in equation (10),

2 1 7 aed
v6. (* 03

rn

nd 1
ee, ee ee
Sar + ag —kn (U) ccupSouoneonsagar aaanne (12)

The function f must consequently be such as to satisfy this equation.

Again, as the phenomena of light shew that a ray of common light has similar relations to space
in all directions perpendicular to its axis, the function f, which is arbitrary, to apply to this kind of
light, must be assumed to be a function of the distance from the axis. That is, if r° = a + y',

ON THE HYPOTHESIS OF UNDULATIONS. 369

f is a function of r. And the equation (12) is quite consistent with this assumption. In fact, for
this case it becomes,

1 1
[Re

Jae f ie (i)
dry dr sf vig

which equation determines the particular form of f applicable to common light, This equation does
not appear to be exactly integrable. By putting it under the form,

af af at
dr® © fdr* * Fdr

nrJ/k _. z : Ae
v. satisfies it, when ris very small. By multiplying

it will be seen that the equation f= cos

d 2
equation (13) by f, and supposing of =, we shall have either f = 0, or = 4 kn?f=0. The

latter equation is satisfied at the axis of the ray: the other by a certain value / of r, which may be
called the radius of the ray. If S equal the condensation at the axis, and s the condensation at a
point distant by r from the axis, by what has been already shewn, s = Sif. Hence where r = /,

ds ‘ Ses 7 Ns c 98
both s =0, and a = 0; that is, at this distance there is neither condensation nor variation of con-
r

densation. Thus the parts of the fluid more distant from the axis than J may remain at rest, while
those at less distance continue in agitation. As a’*=a°(1+4), and as it is not probable that a’
differs much from a, k may be considered a very small numerical quantity. Hence the three first
terms of equation (13) will be small, since each would vanish if & vanished. Consequently / the

d
radius of the ray must be large compared to ) the breadth of an undulation. Because = is very
5 J

d
small for all values of r, and f and = vanish together where r = 7, it follows that the second term

of equation (13) is very small compared to the others at all distances from the axis. By neglect-
ing this term the equation becomes,
aig ae
dr? * rdr
which determines with sufficient approximation the function f.
df* df?
f and af
fda’ fdy’
quantities of the same order as the neglected term of equation (13), we obtain,
ef @f
du * dy i

which is a general equation, applying to a ray of light of any kind, and including as a particular
case equation (14). Since by hypothesis s = S'f, we immediately derive from (15),

By neglecting in the general equation (12) the terms containing > which are

SORE ES Osoceng0 cdo COBCCE HERO OCA CoOnE (15),

eee k 0 16
dat Tatas tS OMAN Re ar or oeseca sedi csinsesceouks (16),

a linear equation with constant coefficients, in which the principal variable is the condensation.

p ds

z F ye d : ?
The velocity (w) in the direction of w we find to be ay which, since s = S'f, becomes —.—.,
dz : S da

Vor. VEILI. Parr III. 3B

370 PROFESSOR CHALLIS, ON A THEORY OF LUMINOUS RAYS, ETC.

mr

ds Qn, do 2Q7

=+.— h that @ = —— cos — (af — x). Henc by ae ie

So v a iy But we have seen p SE (a ) ence oF ma’ sin = (a’'t-x)
ma’, 27 , a z A

=-—a'§. Therefore S = a sin = (at — x), and = ery cot = (wt —x). Call this quan-

ds ds
tity d. Then w= b— > andv= oa Let now s=o,+o0.. Then

4

@.(o,+0,) d’. (0, + 2)

+ kn*(o, + 2) = 03

dx” dy?
and as s in equation (16) is arbitrary, we may have separately,
do, d com 2
dx dy’ + kn o,=0,
da, Gas

and

and consider these equations to apply to two distinct rays. At the same time, since s = a, + o2,
pee epee ges Ng ey ae oe
da du da dy dy dy
rays resolved in the directions of the axes of co-ordinates are equal to the resolved parts of the
velocity of the original ray in the same direction. Similar reasoning would have applied if we had
assumed s = 6, + o2+0,+ &c. The general conclusion we may now draw is, that a ray may be
conceived to be composed of two or more rays in the same phase of vibration, and that if, after a
ray has been separated into distinct rays, the parts be put together in the same phase of vibration,
they will compose the original ray.
The foregoing Theory of Luminous Rays, conducts to a very simple and satisfactory explana-
tion of the phenomena of Polarized Light, which I propose to bring before the notice of the
Society at a future opportunity.

; that is, the sums of the velocities in the two

Cambridge Observatory,
May 11, 1846.

AAS Bed meas

XXVIII. 4 Theory of the Polarization of Light on the Hypothesis of Undulations.
By the Rev. J. Cuaruis, M.A., Plumian Professor of Astronomy and Experi-
mental Philosophy in the University of Cambridge.

[Read May 25, 1846.]

Tue Theory of Polarization contamed in this Paper is founded on the Theory of Luminous
Rays, given in my last communication, of which the present may be regarded as a continuation.
I shall, therefore, use the same symbols as in the former Paper, and suppose their signification to be
known, and for the sake of convenience, the reference numbers attached to the equations will be in
continuation of those of the other Paper.

Conceive a ray of common light to be submitted to some action which is not symmetrical with
respect to its axis, and which divides it into rays subsequently pursuing different paths. In general
the arrangement of the condensation in neither of these rays will be symmetrical about its axis: but
each may be supposed to consist of a symmetrical part having the properties of common light, and
a part which has a different arrangement. The unsymmetrical part is considered to be polarized.
A difference in the arrangement of the condensation in different directions transverse to the axis,
corresponds in this Theory to Polarization. By experiment it appears that a polarized ray has
a certain definite character, which is quite independent of the particular action producing the bifur-
cation of the original ray, being the same under modes of separation of very different kinds. The
explanation of the phznomena of polarization is therefore to be sought for in the modifications of
which the vibrations of a ray of common light are susceptible according to Hydrodynamical principles.
In the phenomena of common light there is nothing to decide whether the sensation of light is due
to the direct or the transverse vibrations. The phenomena of polarized light shew that it is to be
attributed to the transverse vibrations, and our attention must therefore be directed to the modifica-
tions which these may undergo. The direct vibrations very probably are productive of heat.

In the Theory of Luminous Rays it was shewn that a ray in which the condensation at any point
is s, may be supposed to be compounded of two rays in the same phase of vibration whose con-
densations are c, and o,, if s =o, +., independently of any relation between o, and o,. We are
therefore at liberty to assume another condition which these quantities shall satisfy. The assump-
tion I shall make is, that the bifurcation of a ray takes place so that the transverse velocity at each
point is converted into two velocities at right angles to each other, and that these are respectively
the velocities at the corresponding points of the two polarized rays. This law is most probably
deducible from purely Hydrodynamical principles; but in the present state of Hydrodynamics it
must be regarded as an hypothesis. By the reasoning and notation of the former Paper, the com-

2

a ee da d do do».
ponent velocities in one ray are ()——, and @ EO and in the other, @——, and ® —; and the
- dx dy dx dy

hypothesis we make is, that

doy do,
ay pated or dil ls eat Mpc eee ete (16).
do, do. dw dw dy dy

dw dy
3B

372 PROFESSOR CHALLIS, ON A THEORY OF THE POLARIZATION OF LIGHT

Let us now consider by itself the polarized ray in which the condensation is o,. Since
d d d do. d d
Ss =o6,+02, we have = = — - = s and = = = - ae Substituting these values in (16).

we obtain,

doy" ds do, doy ds do,

HO can wwsccvcnc ces 1
da de da dy dy" dy (7)
Also co, must satisfy equation (15). Consequently
do, do, 2
de tage eee Sogsocosnaese (18)

The equations (17) and (18) determine the function that o; is of w and y. For by eliminating

a

a from (18) by means of (17), an, equation results, which, as it contains only partial differential
Lv

coefficients of o, with respect to y, determines the form in which y enters into this function. The
form in which @ enters is similarly determined. The function expressing the value of o, is deter-
mined by equations exactly the same as (17) and (18), having only co, in the place of o,. In fact,

do,” da ds do, d
putting equation (17) under the form aaa -A a + B=0, we have A= aS aa + a ; and
d d do, do, da, doz ‘
= ae (a 3 aa == a a = a a The two roots of that equation are therefore

da dco. Dares Fi é 5
== and a and hence the process indicated above which determines c, determines a; also.
dx wv

Since the original ray is supposed to be one of common light, s is a function of r, and

p> a : :
fa f(r) s s = =f(r). = Substituting these values in equation (17), we have,
y a

dar do, av wees AV
dae * a -F0 (Fe mie dy r

If now the direction of the axes of co-ordinates be changed through 90° by putting — 2’ for y and
y' for x, neither this equation nor equation (18) will be altered in any respect, so that the same solu-
tion of the equations will result as before. Consequently by this transformation the function that
co, is of x and y is converted into the function that co. is of the same variables, and vice versa. It
follows that the original ray is divided into two equal polarized rays, such that if one be turned
about the axis of x through 90” it becomes identical with the other. Since also equations (18) and
(19) are not altered by changing the axes of co-ordinates through 180”, that is, by altering the signs
of x and y, it appears that o, and co, are symmetrical about planes passing through the axis of =.
These planes, from what has just been proved, must be at right angles to each other. Also it is
evident that a plane of symmetry of one ray must coincide with a plane of symmetry of the other.
Hence each ray will have two planes of symmetry at right angles to each other.

The above results would be more properly derived from the functions of w and y expressing the
values of c, and o,, if the integrations could be performed by which these functions would result
from equations (18) and (19). This it does not appear possible to do generally; but values of o,
and ¢, applicable to small distances from the axis of x may be obtained as follows. We have seen

k
that the solution of equation (13) for small values of r is f = cosn Mie v. Hence, as s = fiS, we

ON THE HYPOTHESIS OF UNDULATIONS. 373

bade /# ds Sn Ak ) Jt aw ds
have to the same approximation, s = S cos” —r; —=- .sinn =pPe=s al =
2 da /2 2 r dy

el. sinn J ore. Consequently substituting in (17), and putting the arc for the sine,
c

doe do, Snk ee ae)
antag 2 ( da ay
Such a value of o, is now to be found as will satisfy the equations (18) and (20) for small values

of w and y. The equation o,=mcos(gx + hy) will be found to answer. For substituting in
(18) we get the equation of condition,

VS Be DIE aL Olle gate cares niaasa dsials wicvw'aareisttsiots Soe (21),

and by substituting in (20) we have,

S'
m? (g° + h’) sin’ (gw + hy) - i

2

"km

(gv + hy) sin (ga + hy) = 0.

Hence, putting the arc for the sine,

Sn’ k
m (g° + h®) -— ; IO rs eerenotaetstatenees (22)
Comparing this equation with (21), it follows that m = = ; and consequently that

Ss tia }
a1 = 5 60s (gv + hy). Similarly we should find that o, = > cos (gv +h’y). But since s = o,+ 0%;

we must have,
Kk
S' cos n Jt r= cos (ga + hy) + 5 cos (ga + h’y).
Expanding to terms involving the squares of the small quantities,
n’kr = (ga + hy)’ + (g'x + h'y)?
= (g° + g*) a + (+h) y+ 2 (gh + gh’) vy.
This equation accords with (21) if g’ = and h’ =— g. Thus we have
S
n= C08 (gv + hy), =

It hence appears that o, becomes identical with co, by changing the directions of the axes of
co-ordinates through 90°. Since g? +h? = n*k, we may assume that g =” Vk cos @, and
h=n Vk sin@. Then,

o; =F 00s fn Vk («cos 0 + y sin 6),
S =e
and o, == cos Sn \/k (w sin @ — ycos6)}.

: , i : = S =
As @ is quite arbitrary, let it equal 90°. Then o, = conn ky, and o.= A cosm\/kv. The

axes of w and y are now evidently in the planes of symmetry, and these last values of o and

374 PROFESSOR CHALLIS, ON A THEORY OF THE POLARIZATION OF LIGHT

cs, shew that at small distances from the axis of x the motion in one polarized ray is parallel
to the plane zy, and in the other parallel to the plane xz. They may be said to be polarized in
these planes,

The foregoing reasoning proves that a ray of common light is divisible into two rays polar-
ized in planes at right angles to each other, and that these rays are necessarily equal. We have
next to shew that they are each of half the intensity of the original ray. Since

ds’ do, | dos ds do, do,
—= +—,; and — = — + :
dx dx dz dy dy _ dy
by squaring,
ds\ ds’ do, do ne do, do.
da? dy’ da® i dy’ dx dx
do," dasa gia do»
+—>+
dy’ dy” ane dy
= (So do, sa) (= dof

—; +— >] b ati 16).
q dy dae i?) 'y equation (16)
Let u,, v, be the velocities in v and y in one ray, %, v. in the other, and w, v in the original ray.
d d d
Then since 7 = as 5 v=) a ; u, = pet , &e.
dw dy da

w+ = (uj? + 0°) + (wu? + v.).

Hence the square of the velocity at any point of the undivided ray is equal to the sum of the
squares of the velocities at the corresponding points of the polarized rays. This is true of the
velocities in any tranverse section, and therefore true of the maximum transverse velocities.
Measuring, therefore, the intensity of a ray by the sum of the squares of the maximum trans-
verse velocities, it follows that the sum of the intensities of the polarized rays is equal to the
intensity of the undivided ray, and, their intensities being equal, that each is of half the intensity
of the undivided ray. This is conformable with experience.

Let us now proceed to estimate the intensity of a ray compounded of two rays polarized in
opposite planes, but not in the same phase. As above we shall consider the intensity to depend
entirely on the transverse velocity. In general for any ray not compounded,

df df Qn
OT Pe iste # nd d = we cos (alt = = +0).

. a F . : .
Now since the ratio — is a function of # and y independent of x and ¢, the direction of the
v

transverse velocity is independent of the phase of the direct velocity. Hence the transverse
velocities in two rays polarized in opposite planes, which by hypothesis are at right angles to
each other when the rays have the same phase, will be at right angles to each other whatever
be the difference of phase. Let therefore for one ray

af, df. df.
=~,—, a d for the other, w= @. —, t2= d.—.
uy = Pr, oe v= and for the other, w. Pp ie v p ag
Then since they are polarized in opposite planes, ial Brae » OF Uj Uy + %,%2 = 0. Consequently,
Aa ahs
! ay;
Ue SSE ee Phe Ne (28).

dw dw dy ‘dy

ON THE HYPOTHESIS OF UNDULATIONS. 375

BS mr Qr df,
Suppose now that w = >— cos — Tithe 6) =
PP Qa Xr ( da’
mr» 2a ¥ df,
v, = ae C08 Ge 2)
m' > 2 df.
PLES, I cage
Qa r d
mr 2@r_, df,
v, = — cos—(at—zx+c) —.
Qa r ( dq

r Cites df. ME as ; ee Obs
Then w, + wv. = — cos — (a’'t — x) (mE mi cos SF - sin — (a't - 2) m dive and if
Qa r du

=< d Qn r da’
fee ay»
an 228 LOE ira
rx Ties
ms m' cos eZ
du
nN df? js f
U or w+ & = fe SE + 2mm cose hf Lem ? GV os 7 (Wt 240’)
m’ sin c —
270" dy
Sovrtert =
o if tan if af
m —_ +m’ cos ¢ ——
y
2 d fe |S ee 2 ’
V or v4, + % - x (m4 me + 2mm cos 0 Sh dfs +m? TN eos 2 (ait - = +6").
Qa dy” dy dy” dy r

The two velocities U and V are not in this case in the same phase, and consequently the trans-
verse motion of a given particle, instead of being in a straight line, is in an ellipse or a circle.
The effects of the resolved parts of the velocities in the directions of the axes of w and y may be
assumed to be independent of each other, and the intensity of the compound ray will conse-
quently be as the sum of the squares of the maximum values of U and V; that is, as

a 9
gate — + 2mm’ cose = af + af, Be) +m’? (5 a

y dw dx dy dy! | dy?
which on account of the equation (23) is independent of cose. Hence the intensity is the same
whatever be the difference of phase, and therefore the same as when the two polarized rays have
the same phase. This agrees with experience.

Let us now proceed to consider the bifurcation of a polarized ray; for instance, the ray whose
condensation is ¢,;. Suppose it separated by any means into two rays whose condensations are 7,
and r,. We shall assume, as in the case of a ray of common light, that the sum of the conden-
sations at corresponding points of the divided rays is equal to the condensation at the corre-
sponding point of the original ray, and that the velocities at these points of the divided rays
are in directions at right angles to each other. We have then, by what has gone before,

p= Fr TET NCICB D6 ANAS DONDOOCHOnOCET (24),
dr, dr, dr, dr,

"— + —. =a oes eaaleeaasint (25),

da. dw dy dy

376 PROFESSOR CHALLIS, ON A THEORY OF THE POLARIZATION OF LIGHT

and +,, 7: must respectively satisfy the equations,

Gr, dr,
ae qe +nk7r,=0. eon=f26)s
Pr, dr.
or} d SE a hag =U cates fice. <a: (27)

From the system of equations (24), (25), (26), and (27), it is required to determine the forms of the
functions expressing the values of 7, and 7, that expressing the value of o, being supposed to be
known. It does not appear that this can be done generally; but as before, approximate solutions
may be obtained applicable to parts of the rays contiguous to the axis. The process for this pur-
pose will be analogous to that applied to the ray of common light.

Let 7,= mcos(gv+hy), and +r, = m'cos(g'a + h’y).
Then equations (26) and (27) are satisfied if
et+h?—-n?k=0, and g? +h? —n?k =0.

Also equation (25) is satisfied if gg’+ hh’ =0. And these three equations of condition are satisfied
if g=n\/kcos0, h = — nV/ksin@, gi =n/ksin@, h’'=n/kcos@. Hence since we have

c 8 5 2 S as
shewn that when the approximation is carried to the second powers of « and y, o,= = cosn/ky, we

shall have by equation (24),

=

= cos n Vky = mcos (nav Vk cos 0 — ny Vk sin 0) + m’cos (na Vksiné + ny Vk cos @) .... (28).

Hence, expanding to the second powers of # and y,

2 2y 2h rt 3
(1 7) =m+m' —— (wos 8 - y sin gy -™™*

(w sin 8 + y cos 8)’.

Ss ’
Therefore he m+m,

Ss ; Fc ‘ ; :
and Ack = (mcos*@ + m'sin*@) a — 2vy sin @ cos 0 (m — m’) + (m sin®*@ + m’ cos’@) y’,
ae ; S
or, substituting m +m’ for —,
2

(m cos? + m’sin*@) (y° — w*) + 2ay sin @ cos 0 (m — m’) = 0.......2.5- . (29).

It appears, therefore, that equation (28) is not satisfied to second powers of w and y for gene-
ral values of these variables, and the functions assumed for 7, and 7, are consequently true in
general only to first powers of x and y. It is however important to remark that the equation
(29), being put under the following form,

y° ; 2y sin@ cos @(m— m’)

®” @ ° mcos0+m sin’
shews that for two directions at right angles to each other, the assumed values of 7, and 7. are
true to the second powers of w and y. These two directions may be presumed to be the direc-
tions of the planes of polarization of the two rays. But because

ON THE HYPOTHESIS OF UNDULATIONS. 3h
7, = mecosn\/k (w cos@ — ysin@), and 7, = m' cos n Vk (wv sin@ + y cos 8),

— 6 with the plane of polarization of the original

~/ 3

these two planes evidently make angles @ and
ray. Hence putting cot @ for Yin equation (30), we find
a

m

2
— = tan’@,
m

¥

i
and we also have —= m + m’.

2

sin’@, and m’ = = cos’.

2)

Therefore m =

v9 |

The polarized ray is consequently divided in general into two unequal rays, the values of which
are assigned by these equations. If 9@=45", the two rays are equal ; which accords with
experience.

Suppose a polarized ray to be incident at the angle of complete polarization on a reflecting
surface, and let @ be the angle which the piane of incidence makes with the plane of polarization

-of the incident ray. Then A being the portion of the ray transmitted without bifurcation, which

bs

we will suppose to be independent of @, and J the portion bifurcated, the transmitted ray will be
A+TJsin°@, and the reflected ray Icos'@. If another equal ray, polarized in a plane at right
angles to the plane of polarization of the former be incident in the same direction, the transmitted
portion will be 4 + Jcos’@, and the reflected portion J sin*#. These two incident rays make up,
according to our Theory, a ray of common light, the transmitted portion of which is 2.4, and the
reflected portion J cos’@ + Isin°@, or I, which is independent of 0, as we know from experience
it should be. Respecting the law above found for the intensities of the two rays into which
a polarized ray is separated, Sir John Herschel remarks in his Treatise on Light in the Hneyelo-
peedia Metropolitana, (Art. 850), ‘* We must receive it as an empirical law at present, for which
any good theory of polarization ought to be capable of assigning a reason a priori.” Such a reason
is given by the Theory I am advocating.

Two polarized rays formed by the separation into two parts of a polarized ray derived
immediately from common light, possess in some respects the properties of polarized rays of the
latter kind, for instance, the two rays pursuing the same paths will not interfere whatever be the
difference of phase. This may be proved by the very same reasoning by which it has been already
proved that two rays of first polarization do not interfere, the reasoning being purposely adapted to
the case when m and m’ are unequal, At the same time the rays of second polarization differ in
this respect, that if they meet in the same phase they compose a plane polarized ray. When
6 = 45", we found that the two rays were equal. Yet their composition would form a polarized ray,
whilst two equal rays of the first polarization meeting in the same phase would compose a ray of
common light.

According to this Theory circularly and elliptically polarized light consists 0
polarized rays differing in phase, the two rays when in the same phase constituting a polarized ray of
the first kind. The reason Fresnel’s Rhomb does not produce elliptically polarized light, when com-
mon light is used, is that common light may be supposed to consist of two rays in opposite polariza-
tions, which produce exactly complementary effects. For the same reason common light produces no
coloured rings by transmission through a thin plate of a uniaxal or biaxal crystal cut nearly per-
pendicularly to its axis. Each of the polarized rays, of which common light may be supposed to

wow VIII. Parr III. ZC

f two oppositely

378 PROFESSOR CHALLIS, ON A THEORY OF THE POLARIZATION OF LIGHT, ETC.

be composed, does in fact produce coloured rings, but the two sets being exactly complementary,
the colours disappear.

Circularly and elliptically polarized light is capable of reflexion at the analyzing plate (in the
experiment above alluded to) because it consists of two rays polarized in opposite planes, which
cannot therefore both coincide at the same time with the plane of incidence. The analyzing plate
is necessary for the production of the colours, because the rays come out of the crystal in opposite
polarizations, and therefore not interfering. Those that fall on the plate in the same phase consti-
tute a single ray polarized in the plane of original polarization, and are therefore incapable of
reflexion when the plane of incidence on the plate is perpendicular to the plane of original
polarization. The rest of the rays fall on the plate in the form of circularly or elliptically
polarized light, and consequently from what we have already seen, are capable of reflexion.
This explanation does not require the supposition of the loss of half an undulation.

The Theory might be compared with experiment in many other instances; but perhaps those
I have adduced may suffice to gain for it the favourable consideration of mathematicians. TI will
only add, that having applied it in some degree to the phenomena of Double Refraction, I find
that it leaves the mathematical Theory of Fresnel unaltered, while it offers in several respects a
different physical explanation of the facts. Before I conclude it may also be proper to remark, that
I have argued on the supposition that the quantity & which enters into this Theory is a constant.

The reasoning would remain the same if k were a function of \, provided it did not vary with the
intensity of the ray.

Cambridge Observatory,
May 25, 1846.

XXIX. On the Structure of the Syllogism, and on the Application of the Theory of
Probabilities to Questions of Argument and Authority. By Aucustus Dr Morean.
Sec. R.A.S., of Trinity College, Cambridge, Professor of Mathematics in Uni-
versity College, London.

[Read Nov. 9, 1846.]

Since the time when the Aristotelian syllogism ceased to be regarded as an all-sufficient instru-
nent of inquiry, it has remained precisely in the state in which those who are called the schoolmen
left it. I have never heard* of any attempt to ascertain whether the forms which his followers
derived from the writings of the great master were the perfection of system and simplicity which
they were supposed to be. The uninquiring adherence of all writers on logic to the model
of the middle ages, proves one of two things: either that the model is human perfection, or that the
authority of the ancients has been followed as of course in the forms of logic long after it has been
abandoned in every other part. With such an alternative, it is not presumption to venture upon
the examination: and this is the more apparent when we consider that the general impression among
writers seems to be that there cannot exist any other theory of the syllogism except that derived
from Aristotle. If another can be produced, which is but self-consistent, true, and comprehensive,
the tacit assertion of all writers is overthrown, whether that system be or be not judged superior
to the one handed down.

I here venture to propose a derivation and classification of the forms of the syllogism, differing
very widely from that in use.

Section I. On the meaning of the simple term.

A TERM, or name, is merely the word which it is lawful to apply to any one of a collection of
objects of thought: and, in the language of Aristotle, that name may be predicated of each of those
objects. He uses this word predicate only as “that which can be said of.” When in later times
the negative proposition ‘* XY is not Y” was said to have Y for its predicate, the word ought to have
been non-predicate, or some equivalent. The proper predicate is not-Y, which I shall call the
contrary of Y.

When we use a term, such as “man,” we predicate, in Aristotle’s sense of the word, of every
individual which the notion can suggest, of John, Thomas, William, &c. If we extend the word,
and allow Y to be called the predicate of ‘‘ X is not Y,” we must then affirm that the word “ man”
predicates of every object of thought, either affirmatively or negatively: affirmatively of John or
Thomas, negatively of a certain tree, or quadruped, or book. Every name then, in this sense,
predicates of every thing: “ X is either Y or not-Y” is a proposition of universal identity.

The express introduction and consideration of contraries ought, I think, to have followed the
extension of the word predicate. Aristotle rejects and then admits: not-man, he says, is not
aname; and then he calls it an aorist name, which can be predicated both of existent and non-
existent things. I deny the justice of this distinction, for two reasons.

Names in logic are used subjectively ; they are the representations of the notion in the mind.
Now man and not-man are equally the names of things which, objectively speaking, are non-
existent. Nof-man, Aristotle would say, is a name which can be predicated of the speaking bird

* See the Addition at the end of this Paper.

3c2

380 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

and the singing tree in the eastern fable: but surely, with as much justice, man may be predicated
of the Shakspeare who wrote Paradise Lost, or the Caesar who conquered Darius.

In the next place, it is not true that the aorist or indefinite character of the mere contrary
actually exists in the use which we make of language. Writers on logic, it is true, do not find
elbow-room enough in anything less than the whole universe of possible conceptions: but the
universe of a particular assertion or argument may be limited in any matter expressed or under-
stood. And this without limitation or alteration of any one rule of logic. Let every one of the
possible points of space have one or more of the names X, Y, &c.: then if we can say, ** No X is Y,”
of course we can say ‘No Vis X.” But this is equally true if, by an understanding to that
effect, the universe of our proposition be one square described in a certain plane. Divide the points
of this square into Xs and not-Xs, and the not-X is no more an aorist term than the XX,

By not dwelling upon this power of making what we may properly (inventing a new technical
name) call the wniverse of a proposition, or of a name, matter of express definition, all rules
remaining the same, writers on logic deprive themselves of much useful illustration, And, more
than this, they give an indefinite negative character to the contrary, as Aristotle did when he said
that not-man was not the name of anything. Let the universe in question be “man:” then
Briton and alien are simple contraries; alien has no meaning of definition except not-Briton. But
we cannot say that either term is positive or negative, except correlatively. As to a claim of right
to be considered a prisoner of war, for instance, alien is the positive term, and Briton the negative
one. We separate formal logic from language, if we refuse to admit this. In many cases,
the language has the term which signifies the contrary, and wants the direct term: as in the
word parallels, for example. To this day the word intersectors has not found its way into the
idiom of geometry. In one case we give a name to the thing of course, and define the exception by
means of a contrary: in another we find it more convenient to reverse the process. I hold that the
system of formal logic is not well fitted to our mode of using language, until the rules of direct and
contrary terms are associated : the words direct and contrary being merely correlative. Those who
teach Algebra know how difficult it is to make the student fully aware that @ may be the negative
quantity, and — a the positive one. There is a want of the similar perception in regard to direct
and contrary terms.

Throughout this paper, I shall use the small letters 7, y, x, &c. for names contrary to those
represented by the capitals X, Y, Z, &c. Thus “every thing in the universe is either X or «,”
“* No X is w,” &e. are identical propositions.

Section II. On the simple proposition.

Tnere is no need to dwell on the usual matters given as to the distinction of universal and
particular, or of positive and negative. But, I think it cannot be denied, that the distinctions may,
for logical purposes, be considered as accidents of language. Any proposition which is either of the
four in one language, may be either of the others in another. Our language has, say the names X and
Y, and suppose that ‘Every X is Y” is true. Another language translates X by X’, but has no
term for Y, but only y’ for its contrary; the proposition is then “No Y’is y’.” In a third
language X’s have no specific name; they appear but as individuals of the name 1”: the proposition
is then **Some X's are Ys.” But if the last language had only possessed the name y”, it would —
have been ‘Some X's are not ys.”

Very often, having established such a proposition as ‘ Some Xs are Ys,” we, for that reason,
distinguish those Xs by a separate name, Z: and then we have “ Every Z is Y.” If language
were copious enough, particular propositions would seldom occur: and the idioms of every tongue
are probably influenced by its power of supplying universal terms, or of converting particulars into
the form of universals.

eS

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 381

I shall use, as is usual, the letters A and £ to signify positive and negative universals: and
T and O for the corresponding particulars: but with a modification presently noticed. I shall also
use the following notation, without which I should hardly have had patience for the many hundreds
of cases which this paper has required.

P)® signifies Every P is Q.

IP (Ale Ad cneane MINOW Zoey

PQ eee eenecan mole es are Os.

Pai Quasscbes een SOMew rs saresnot Qs:

I have taken for the convertible propositions, the symbols P. Q and PQ, which the algebraist is
accustomed to consider as identical with Q.P and QP: the same thing is true under these
meanings. But P) Q and P:Q, which are also used in arithmetic and algebra, convey no idea of
convertibility.

All expressions that have any meaning can of course be reduced to one of these forms.
Aristotle denies this, and divides all expression into significative and enunciative, meaning by the
latter that in which there is truth or falsehood. Thus prayer, he says, is speech, but neither true
nor false. This is surely not correct ;—Deliver us from evil may be either ‘“‘ To be delivered
from evil is our prayer,” or ‘* We are of those who pray to be delivered from evil,” or ‘ Evil is a
thing we pray to be delivered from.” Or, as the text, it would be “ Deliver us from evil, is the
passage on which I mean to comment :” and the sermon would probably give all the enunciations
above. In a request, command, inquiry, or announcement, the tone* of voice predicates.

In classifying all possible predications-by means of two names Y and X, their contraries must
be included. We must therefore consider all the relations that may exist between Y and Y,
X andy, y anda, wand Y. Between each of these there are six modes of enunciation: thus
between P and Q we have

P)Q, Q)P, P.q
1

2

Qe er LO QP) (PQs PONE.

4 5 6
But it will be best to arrange these by contradictories, or propositions one of which must be true
and the other false: as P)Q and P:Q, Q)P and Q: P, P.Q and PQ. These six modes
applied to each of the four variations of subjects, give twenty-four varieties, which are reducible
to eight, being identical three and three, as follows:

is) II

A)Y=X.y=4)a Pe, VES ONT iis
A OBNG = PGS ES PH) ACP Hs VCR
Vise ha — 0) OAS) VA STAG

Vee xe— Va ia -4) | ay = a; Y= 2X,
Though the use of the great and small letters may suit the eye, these lines should be read thus:
“Every X is Y” is identical with ‘no X is not-Y,” and “every not-Y is not-X,” and soon. hese
eight modes may all be derived from the four Aristotelian modes by changing both terms into
contraries ; which suggests the following notation ;

(A) eae) Yi vjy=Y) Xx (a).
(O) PS) TTR GOS (9).
(2) BEEN | wy “(e).
(J) ax Ve wy (i).

* To call a person by his name is a proposition, perhaps more. | John; therefore, you are the person I want to speak to.’’ he
There is certainly the full meaning of a syllogism init. Whena | least that can be said is, that he states the premises, and leaves
person calls—John! no one can say that any part of the following | John to draw the conclusion.
is not implied: ‘John is the person I want to speak to; you are |

382 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

This notation is established with reference to the order XY: the inversion of the order interchanges
small and large letters.

The propositions (4) and (Q) are the assertion and denial of complete inclusion of the first
name in the second. And (a) and (0), the assertion and denial of complete inclusion of contrary in
contrary, are, as appears, equivalent to the assertion and denial of complete inclusion of the second in
the first. Again, (2) and (J) are the assertion and denial of complete non-interference, or that each
name is wholly contained in the contrary of the other. But (e) and (7), the propositions which I
propose to add to those commonly received, may be explained as follows:

The proposition (7) or wy, affirming that there are individuals in the universe of the proposition
which are neither Xs or Ys, merely affirms that and Y are not contraries, and do not between
them contain the universe. The contradiction of this, (e), or a . y, affirming tiat it is false that there
are any individuals which are neither X or Y, might seem at first sight to declare that X and ¥ are
contraries. But it is not so, since the preceding is perfectly consistent with there being individuals
which are both Ys and Ys. In fact, to express that X and Y are contraries we must have both
v.yand X.Y.

The following tables show the relations of these propositions.

Leta, os clioomieeest en lla cmaroaoutey | Pe | nts” | Sra
A / OEe Li ao E IAa Oo | et
o| 4 Ee Taio I | E 4a | Ovio
a| oek li AO e| iad Oo | EF
o| a Ee i AIO Ainge Aa | oF IO

Or each universal proposition denies, besides its own contradictory, the two universals of a different
name; contains both particulars of the same namie ; and is independent of the other universal of the
same name and its contradictory. Each particular proposition denies only its own contradictory ; is
contained in both the universals of the same name; and is independent of either of the other three
particulars, as well as of the other universal (not its own contradictory) of a contrary name.

It is usual in modern works to say that a term which is universally spoken of is distributed.
But in truth every proposition distributes, wholly or partially, among the individuals of the
predicate, or of its contrary. It will be sufficient to call a term universal or particular, according
to the manner in which it is spoken of, It will then be found that every proposition speaks
in different ways of each term and its contrary; making one particular or universal, according as
the other is universal or particular, The manner in which the subject is spoken of is expressed; as
to the predicate, it is universal in negatives but particular in affirmatives. And of the two terms and
their contraries, each proposition speaks universally of two, and particularly of two.

Let § signify that the subject is changed into its contrary, P, the same of the predicate. Let
C signify that the copula is changed, from positive to negative, or vice versd. Let T' denote trans-
formation or interchange of subject and predicate: to avoid confusion, either this must be done last,
or the original subject and predicate are to retain those names after the transformation. Then we
have the following tables, Z standing for letting the proposition remain unaltered

1 EM RE Ld SP ioe L P || PT S SY Ml 2
PC || SCT | SC || PET | PC Cy SPCT | SPC | CT | C

Changes and combinations of changes that are written under one another, are in all cases of the same
effect : thus by writing PT’ and SPCT under one another, I mean that change of subject, predicate,
copula, and order, are always of the same effect as change of predicate and order only. Thus the
operation SPTC performed upon X ) Y gives y.v. But PT only gives y) X and y.v7=y) X,
as appears above. Further, when a single line separates two vertical pairs, the two pairs are
identical when performed upon imconvertible propositions : when a double line, the same with

“¥

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 383

respect to convertible propositions. Thus, as to convertible propositions, L, PC, T', SCT" are all of
the same effect: as to inconvertible propositions, 7', SCT’, SP and SC.

_ There is a point which developes itself very strongly when we come to consider the transforma-
tions upon instances; namely, the distinction between the assertion of a proposition subjectively and
objectively. The former mode is that which is always presumed: but in actual use of logie the
distinction must be drawn.

When we say, Every X is Y, as a proposition with meaning, and with or without truth as the
ease may be, we treat neither X nor Y as having any other existence except that which our minds
give them: but we imply that if X have any such other existence, so has ¥Y. But the syllogism
“X)Y and Y)Z therefore X) Z” is not valid merely by understanding X and Z to be taken in
the conclusion as in the premises. The middle term must eaist: not necessarily objectively, but it
must have a positive existence. It is no syllogism to say that X is Y, if there be such a thing, and
Y (if &c.) is Z; therefore XY is Z. And yet there is no offence against any of the ordinary rules of
logic: the middle term is strictly middle; it is ** Y, on the condition that Y exists” in both. Thus
‘¢ Homer was a perfect poet (if ever there were one); a perfect poet (if &c.) is faultless in morals ;
therefore Homer was &c.” The premises will sometimes be admitted; but they do not prove the
conclusion: the proper conclusion is a dilemma, ‘ Either Homer was faultless in morals, or there
never was a perfect poet.” The existence here spoken of is objective: but the same thing applies to
purely subjective cases. The terms of the conclusion may be conditional: but inference requires
that the middle term should be unconditional. Every X (if ever X existed) is Y; every Y is Z (if
ever Z existed): therefore every X (if ever X existed) is Z (if ever Z existed). This is a good
syllogism: but Y is here absolute.

When the syllogism can be converted into another, having for its middle term the contrary of
the first middle term, the same absolute existence must be claimed for the contrary. And here again
I remind the reader that the absolute existence spoken of is existence within the wniverse of the pro-
positions, Thus X ) Y and Y) Z give X) Z, or y) vw and x) y give x) a. A positive existence
is then required both for Y and y. There is an extreme case; y may not exist, that is, Y may
contain the universe; but then Y and Z are identical, and the conclusion A’) Z is identical with
X) Y and x) contains nothing.

Whatever sort of existence is spoken of is tacitly claimed for the terms of a proposition by the
proposition itself: the refusal of this claim, or the denial by assertion of non-existence, being a dis-
tinct thing from denial by contradiction. A certain meadow (the universe of the proposition) is
flooded during the hay-harvest: the proposition “*No part of the crop that was not flooded was
not saved” (of the form w.y) means logically that all which was not flooded || was saved, that all
which was not saved || was flooded, and that part may have been both flooded and saved. Some
reflexion (for want of habit of dealing with triple negatives makes the proposition rather complicated)
will shew that a person who is apt to think objectively of propositions, as all do who are not trained
in logical considerations, is much more likely to require the insertion of the words (if any) in two
places (|/) than he would be if the proposition were presented in the more simple form, ‘ All the dry
crop was saved.” Probably such a person would not require the conditional words here, merely
cause he would take it that the proposition asserts that some was dry: reserving the right to deny
by non-existence if there were none.

I suppose it is hardly necessary to remark that, in propositions, asserted as true, the same sort
f existence is claimed for both terms: for instance, that there is no objective first term with a sub-
ive second one. In such a proposition as “he is good” we may certainly say that “ good” by
itself is a purely subjective notion; a state of the mind in regard to an external object. But good
not the term of the proposition ; it is he (an external object) is one of these external objects to which
e mind attaches the idea of good. I can conceive opposition to this: what I say is that the oppo-
sition is not to me, but to the universal maxims of technical logic. For all writers admit that YY
necessarily follows from XY: which cannot be if ¥ be a name of the state of the mind and XY of an

384. PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

external object. Most of the Romans were brave; therefore some brave [men] were Romans,
No hint is ever given by writers on logic of the necessity, previously to conversion, of attaching the
subjective notion to an object.

Section III. On the quantity of propositions.

1)

Tne logical use of the word some, as merely ‘‘ more than none,” needs no further explanation.
Exact knowledge of the extent of a proposition would consist in knowing, for instance in ‘* some
Xs are not Ys”, both what proportion of the AX’s are spoken of, and what proportion exists between
the whole number of Xs and of Ys. The want of this information compels us to divide the expo-
nents of our proportions into 0, more than 0 not necessarily 1, and 1. An algebraist learns to
consider the distinction between 0 and quantity as identical, for many purposes, with that between
one quantity and another: the logician must (all writers imply) keep the distinction between 0 and
a, however small a may be, as sacred as that between 0 and 1 — a: there being but the same form
for the two cases. We shall now sce that this matter has not been fully examined.

Inference must arise from bringing each two things which are to be compared into comparison
with a third, Many comparisons may be made at once, but there must be this process in every
one, When the comparison is that of identity, of is or is nof, it can only be, in its ultimate or
individual case, one of the two following ;—‘‘ This X is a Y, this Z is the very same Y, therefore
this XY is this Z; or else “* This X is a Y, this Z is not the very same Y, therefore this X is not
this Z.” And collectively, it must be either ‘‘ Each of these Xs isa Y; each of these Ys is a Z;
therefore each of these Xs is a Z;” or else ‘* Each of these Xs is a Y, no one of these Ys is a Z,
therefore no one of these .Ys is a Z.”

All that is essential then to a syllogism is that its premises shall mention a number of Ys, of
each of which they shall affirm either that it is both X and Z, or that it is one and is not the other,
The premises may mention more: but it is enough that this much can be picked out; and it is in
this last process that inference consists.

Aristotle noticed but one way of being sure that the same Vs are spoken of in both premises:
namely, by speaking of all of them in one at least. But this is only a case of the rule: for all that
is necessary is that more Ys in number than there ewist separate Ys shall be spoken of in both pre-
mises together. Having to make m + 7 greater than unity, when neither m nor 7 is so, he admitted
only that case in which one of the two m or n, is unity and the other is anything except 0, Here
then are two syllogisms which ought to have appeared, but do not; and there are others ;—

Most of the Ys are Xs Most of the Ys are Xs
Most of the Ys are Zs Most of the Ys are not Zs
Some Xs are Zs .. Some of the Xs are not Zs.

And instead of most, or 4 +a, of the Ys, may be substituted any two fractions which have a sum
ereater than unity. If these fractions be m and n, then the veal middle term is at least the fraction
m+n-—1 ofthe Ys. It is not really even necessary that each Y should enter in one premiss or
the other: for more than the fraction m + — 1 of the whole may be found in each.

And in truth it is this mode of syllogizing that we are frequently obliged to have recourse to;
perhaps more often than not in our universal sylogisms. ‘*4// men are capable of some instruction ;
all who are capable of any instruction can learn to distinguish their right and left hands by name;
therefore all men can learn to do so.” Let the word all in these two cases mean only all but one,
and the books on logic tell us with one voice that the syllogism has particular premises, and mo con-
clusion can be drawn. But in fact, idiots are capable of no instruction, many are deaf and dumb,
some are without hands: and yet @ conclusion is admissible. Here m and are each very near to
unity, and m +” —1 is therefore near to unity. Some will say that this is a probable conclusion:
that in the case of any one person it means there is the chance m that he can receive instruction, and

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 385

n that one so gifted can be made to name his right and left hand: therefore m x m (very near unity)
is the chance that this man can learn so much.

But I cannot see how in this instance the probability is anything but another sort of inference
from the demonstrable conclusion of the syllogism, which must exist, under the premises given.
Besides which, even if we admit the syllogism as only probable with regard to any one man, it is
absolute and demonstrative in regard to the whole number of men with which it concludes.

This is not the only case in which the middle term need not enter universally: this however
is matter for the next Section: see also the Addition at the end. I now go on to another point.

Mathematicians, as such, are supposed to have a tendency to admit nothing but demonstration,
and to become insensible to ordinary evidence. Instances of this there may be, though whether the
temperament led them to mathematics, or mathematics brought on the temperament, has certainly
not been inquired into by those who make the charge. But to me it seems very clear, that if
ordinary logic do not produce this temperament in those who study it, there must be correctives else-
where. It is the only science I ever came in contact with, in which the want of demonstration is
formally made to amount to absolute rejection without further consideration. The mathematician,
having a given formula on hand, can and does satisfy himself not only that it is true, if it be true,
but that it is false, if it be false. But the young logician, when his premises do not yield their
inference legitimately, drops that inference as a fallacy: and few indeed are the books which speak
of the distinction between an invalid inference and a false conclusion in terms which shew that the
same distinction is a well recognized topic of the subject. It is, I think, for the mathematician to
try to correct the habit arising out of this omission, namely, the confusion between paralogism and
falsehood: and also to introduce his notions of probability, so as to establish some little power of
discriminating between the various degrees of fallacy which are all called by one name, whether that
name be falsehood or not.

If some Ys be Xs and some Ys be Zs we have no right to draw any inference: at least so says
many a one who thinks that mathematics would render him insensible to the evidence of high pro-
bability.

It will become of importance to reflect what the difference may be between the habit of not
looking for high probability when it exists, and that of not acknowledging it when it ought to be
seen—as soon as the following case is considered.

Let the whole number of Ys be s, the numbers which are Xs and Zs being severally m and n.
Nothing is known or suspected as to whether a Y being X is favourable or unfavourable to its being
also Z. It is required to ascertain what chance there is that there are Ys which are both Xs and
Zs, m+n not being so great as s. ‘I'hat is, when from ‘t some Ys are Xs and some Ys are Zs”
we decline to admit that some Xs are Zs, what is the chance that we reject a truth ?

Let p, signify the number of combinations of p out of g. If we pick out any m Ys to be Xs,
there are 7,_,, ways in which the Zs may be found among the rest. Consequently m, x 7,_, is the
whole number of ways in which “*Some Xs are Zs” is false. But the whole number of possible
cases is m, x m,; whence the chance of the falsehood is

Ns—m ber — m| = —n]
Pn °F Te] [s —m—n]

where [p] means 1.2.3... p. If s -m — 7 be not inconsiderable the substitution of
f/2mr.prtre-” for [p] gives
(s —m)(s—n) (s—m)'~"(s—n)*~" : pga). 2) —m) (1 - v) {¢ —p)'* (1 -»)! }
i 3) DE Se a

\ 8(s-m-—n) ‘s'(s—m-—n) GQ=-n=-vy

s—m—n ?

l—-u-y

if » and v be the fractions which m and m are of 5. For the calculation of this we have
Vor. VIII. Parr IIT. 3D

386 PROFESSOR DE MORGAN, ON THE STRUCIURE OF THE SYLLOGISM,

= l-p — l-v Se gt 8) es eee
comm. log G@ — = = — 4342945 (ents ae ty vee } ;
with which series we are to proceed until the term last obtained gives a sufficiently small product
after multiplication by s.

Now first observe, that since the base of the s‘” power is less than unity, s may, for any given
values of u and yp, be made great enough to make the probability that ‘* Some Xs are Zs” is false
as small as we please. Hence we have a right to assert the following :—

If, to our knowledge, a perceptible fraction of the Ys be Xs, and a perceptible fraction be Zs,
and if the number of Ys be great beyond perception; and if moreover we know nothing, except
what has just been stated, for or against a Y which is X being or not being Z,—we ought to treat it
as a moral certainty that some one or more of those Xs which are Ys are also Zs.

I do not say that the above case is a fair statement of the usual conditions under which
the syllogism with particular premises appears: nor does it matter to my argument whether it be
or not. What I say is, that it is a fair statement of the circumstances under which the rejection of
the conclusion ‘*some Xs are Zs” is ordered to be made in books of logic.

If « and » be small, the number of places of figures in w, @ to 1 being the odds in favour of one

5 r 43
or more Xs being Zs, may be stated as the integer next above = pvs at least. If s were 1000, and

1 . 5 5 r
wand v each ap this would be five; or the odds 10,000 to 1. Calculate more strictly, and it will

come out nearly 70,000 to 1. If a person then should distribute 100 sovereigns and 100 shillings
at hazard among a crowd of 1000 persons, not giving any one more than one coin of either sort, it is
about 70,000 to 1 that he gives one or more of them a guinea.

But to shew how wide the cases may be, which are equally rejected, let us take the following
supposition, which perhaps more nearly represents, in many cases, the rationale of the argument.
Representing all the Ys by aliquot parts of a certain line, it may be supposed that the A’s have
some connexion of contiguity in time, place, or other circumstance: let it then be a collection
of successively contiguous Ys which are As: and the same of the Zs. The state of the case
is now as follows.

There is a line of given length, which we shall take for our unit. Two given lines, each
less than the first line, are Jaid down in it at hazard, any one position of either being as likely
as any other. Let the lengths of the lines be « and py: it is required to find the probability that «
and yp’ shall not have a part exceeding y in common.

First, let 4 +p’ be less than 1, so that the lesser lines can be quite clear of one another. We
are to investigate the probability that they shall be so. Let « be on the left and «’ on the right;
and let a and w’ be the distances of their left and right extremities from the corresponding ends of
the unit. We must then have w+ # + + wu’ less than unity, in order that the lines may be clear
of one another. Now since # may be anything less than 1 — », and a anything less than 1 — ws
and all possible positions are equally likely, it will follow that the chances of the lines called a and
a lying between » and w + da, and 2 and 2’ + da’, will be dw+(1 — «) and da’ + (1 — wv’), and the
chance of the joint event is

dz. da’

G=p)G—«).

If we integrate this over all positive values of x and a’ in which @ + 2’ is less than 1 — a —ps
we shall have the probability in favour of the two lesser lines having no point in common when yp is
on the left, and «’ on the right. The result is easily shown to be the half of

(i-p-w)?

@—”) 0-2) eee cceccceocccrees (1).

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 387

Consequently, since there is the same probability that they shall have nothing in common when
u is on the right and »’ on the left, the expression just written is the probability that ~ and y’ shall
be quite clear of one another.

The condition that # and a’ shall not have so much as » in common, is expressed by saying
that v +’ less than 1 —w—,p' + v: v being less than ~ and than u’. Hence, by similar reasoning

(i —u —p' + v)?
(1 -u)Q- 4)
is the chance that « and «’ have not so much as vy in common.
Under these new circumstances, if « and p’ be each = - , the chance that they are clear of one
another is = or it is 64 to 17 in favour of it. That is, if a thousand persons were placed in a

row, and two being selected at hazard, 100 sovereigns were given successively, beginning with the
first, and 100 shillings successively, beginning with the second, it would now be about 64 to 17 that
no one received a guinea.

Section IV. On the Syllogism.

TueERe is much that is elegant and instructive about the theory of the four figures of the syllo-
gism, three of which belong to Aristotle. And the magic words Barbara, Camestres, &c. are models
of notation, almost every letter of the moods in the three latter figures being a rule of direction. The
following old epitaph on a schoolman selects, I think, one of the best parts of the system for ridicule:

Hic jacet magister noster
Qui disputavit bis aut ter
In Barbara et Celarent

Ita ut omnes admirarent

In Fapesmo et Frisesomorum
Orate pro animis eorum!

In proposing another system of classification, in connexion with the use of contraries, I
remark, first, that the ordinary method has two points of redundancy. ‘The distinct use of the
two forms of a convertible proposition, X.Y and Y.X, XY and VX, is made for the system
of figures, rather than the figures for it. It is desirable I think to confound them as much as pos-
sible; so that each may never fail to suggest the other. In the next place, if the use of contraries
be introduced, every one of the twenty-four modes of predicating would claim admission into a
system of figures, and their number would be increased to thirty-two.

Again, the first followers of Aristove, in adopting the rule that no syllogism should be admitted
in which the conclusion was not the strongest the premises would allow—in rejecting for instance
“X)Y and Y)Z therefore XZ,” because X)Z also follows—did not adopt the equally

obvious rule of admitting no syllogism in which a weaker premiss would lead to as strong a con-

clusion, They retained, for instance, “Y ) X and Y ) Z therefore XZ”, though Y) X andYZ would

produce the same conclusion. Now | think it desirable to adopt the rule of producing the strongest
conclusion with the weakest premises, not only because it will turn out that by so doing the number
of forms is diminished, even when contraries are considered, but also because a better and clearer
distinction is drawn between the necessary and the contingent.

I also drop the distinction of minor and major terms and premises. Aristotle meant them to
apply only to affirmative propositions, in which the predicate includes the subject. But the use of
them was extended, to the utter destruction of the meaning in negative propositions, or worse, to the

38D 2

388 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

danger of carrying a false meaning to the last named. The consequence of the distinction is that
these four syllogisms,

xX) Vha Lt a A X)YV

Vn & 2G) ih A)Y VBA

Vie.g X.Z xX.Z ARO. S
Figure 2 2 1 4
Name Camestres Cesare Celarent Camenes

which are identical in sense and effect, are made separate objects of study. In fact, as is known, all
the figures are in the first, the fourth being occasionally brought into it by a reduction which deserves
the name of Barbara in every case, and which the simplest use of contraries avoids.

For syllogisms I shall adopt such notation as

X)VY+Y)Z=X)Z.

Or if a weaker conclusion be taken,

X)Y+Y)Z>XZ.

As there are eight modes of predicating between XY and Y, and between Y and Z, it follows
that there are sixty-four combinations which may give conclusions. Of these, all but eight are
distributable two and two into counterparts, in which X and Z are interchanged, everything else
remaining the same; giving eight single, and twenty-eight pairs of counterparts. Of these, exactly
half, (four single, and fourteen pairs) are wholly inconclusive. Of the conclusive cases, two single
ones and two pairs are rejected, because as strong a conclusion can be obtained from a weaker pre-
miss. ‘There remain two single ones and twelve pairs, to which a systematic classification is to be
given. Instead of enumerating, I shall state a mode of deriving all the cases from a common
principle. :

Since every proposition is, but for accidents of language, a universal affirmative, as before
noticed, it will follow that there are really no forms of syllogism except those in which the
premises and conclusion are universal affirmatives, or can be made so by use of contraries and
invention of subgeneric terms. Now the only universal affirmative syllogism is

X)YVY+Y¥)Z=X)Z
considering the counterpart Z)¥+Y)X=Z)<X as identical in form. If we take universal
affirmative premises only, we have one which will have a particular conclusion, with respect to
the names X and Z,

Y)X+Y)Z=XZ,
which must be used in discovery of forms, (and will in fact give the two single syllogisms of
this system) though it will only ultimately enter as Y) X¥ + YZ= XZ. Now if we change one
or more of the terms X, Y, Z into their contraries, we have eight modes of transformation,
according as we use

XZY, XZy, wzy, «wsY, XxY, Xsy, #ZY, wZy.
First take the syllogisms
X)V+V)Z=X)Z and XY+Y)Z=XZ;

the latter of which is only the first, with the premiss X ) Y weakened, and would be reduced
to the first form by inventing a subgeneric name for the A’s there spoken of. Apply each of
these to the eight varieties just named, transforming premises and conclusion, when necessary, to
one of the eight standard forms of predication :

X)Z X:Z Z)X 2:X X.Z4 XZ wy ae.

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 389

To each result I attach a letter of notation, derived from the nature of the conclusion, with
letters subscript indicative of the premises. Thus 4,, signifies a syllogism with a conclusion 4
derived from premises of the forms 4 and a. The order of reference is XY, ZY, XZ, the middle
term being the predicate of both premises in the references.

Universal Syllogisms.

First Form. Transformation. Description.
DP UES QV ASE. OYA Ones VAR OWA Asa
A)y+ty)Z=X)Z Mee Fa sg = XG) ZF Ax,
vjyy+y)s = @)z Y)X+4+Z)VY=Z)X Cy
GDC ap VANES Fi) eg @.y +Z.VY=Z)X Qer
YY a Y)\s a X)x Fe AD ae Care Bigs
X)y +t y)x=X)z RLY! mye = XZ 1a
@)Y +Y)Z=2)Z v.y +VY)Z= aw. Caz
ve)y+y)Z=27)Z Y)X+s.y = @.% Cae

Of these forms four are distinct and the others are their counterparts.
counterpart, we have

Writing each with its

Aj, and a4, Ag, and a,,, Eyz and Egy, 4 and @,,.
Particular Syllogisms.

First Form. Transformation. Deseription.
AVY+Y)Z=XZ XY +VY)Z= XZ Tig
Ay+y)Z= XZ AG) ey) = XZ, Loe
vy + y)% = ws vy +Z)Y= asx tia
oY + Y)s = ex Y.X%+Z.Y= wz ior
XY +Y)s = Xz AY +Z.Y=X:Z On
Ay + y)x = Xz PLB EE YL CS AGS Ooa
@7Y¥ +Y¥)Z=a2Z Y:X+Y)Z=Z2:X oa
ay +y)Z=a2Z ay + 2.yY =Z24:X Oie

But though the eight universal syllogisms are counterparts, two and two, they admit of another
division into pairs, in each of which the terms are the contraries of those of the other. Thus 4,, is
connected with a,, in this manner, and A,, with a,,: and these are counterparts. And Ly,
and e,, and Ey, and e,,, which are not counterparts, have the same connexion, The eight par-
ticular syllogisms, which contain no counterparts, are divisible into four pairs with the same
connexion. Thus J;, is changed into i,,, Ig, into i,,, Oz, into 0,, and Op, into 0,4. It is
also worth notice that when the conclusion is negative, the premises are always of contradic-
tory forms, and when positive, of consistent ones: and that the substitution of the contradictory
forms in the premises is equivalent to that of contrary ferms in the conclusion. Thus from
AY+Y)Z=XZ; if we substitute contradictories in the premises, we have X.Y +Y: Z, the
conclusion of which is az.

Before proceeding further it may be worth while to endeavour to impress the notation upon the
reader. The letters 4, E, J, O, have the well-known meanings, thus restricted, that they belong
to a particular order of the terms, or a particular choice of subject and predicate. Thus, X and ¥
being the terms, in the order XY, or X being the subject and Y the predicate, the four capitals

390 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

stand for XY) ¥, X.Y (= Y.X), XY (=YX) and X:¥. And a, e, i, o stand for the same propo-
sitions when both subject and predicate are changed into contraries: that is, they stand for w)y,
w.y(=y-2x), vy (=yx), v:y. In these last, system is sacrificed to simplicity in using Y)X
and Y:.X for «)y and w:y. The following table shows what transformation takes place when the
terms and orders are successively XY, Xy, vy, vY, VX, yX, yx, Yo.

XY| Xy | zy | wY |VX| yX | yo | Ya
A E a e a E A e
O I 0 i 0 I 0) a

a e A E A e a E
0 a O I O i 0 I
E A e a E Wigs e A
If O i 0 iy 0 i O
e a E A e A E a
i 0 i 0 i O I 7)

Here we mean, for instance, that E of the order (X, Y) is the same thing as the A of (X, y,) or
the e of (a, y) &c,; or that ¥.¥, X)y are the same. As to the ¢ of (w, y) it is identically X.Y.
The eight operations by which the transformations of headings are made are those which I have
denoted by L, P, SP, S, T, PT, SPT, and ST, of which the simplest readings are

Forsinconyertibles £ 2 7 S FF S £ P,
For convertibles Te aS es eee. SP. OP.

Taking the terms XY, YZ, ZX, for the premises and conclusion, and the order of reference
XY, ZY, XZ, the notation given defines the syllogism. Thus, valid or not, the syllogism O,,
can be nothing but a.y+Z)Y = X:Z. In turning the fundamental syllogism into the form
X)Y+Y)Z=X)Z, I have altered Aristotle’s order of the premises, which would give Y) Z
+2X)Y=X)Z. Reasoning direct from his dictwm de omni et nullo, namely, that what is true
or false of all is true or false of every some, it would seem natural first to ascertain the fact
relating to the whole, and then to introduce the part which is to be considered. But in another point
of view it may be more natural to reverse this order. If there be three boxes P, Q, and R, of
which I want to ascertain by means of Q whether P will go into R, it seems to me more natural to
try first whether P will go into Q, and then whether Q will go into R. But if the question be
whether R will hold P, then perhaps it may be more natural to try first whether R will hold Q,
and then whether Q will hold P. It must be mere matter of opinion which should be taken; and
the idioms of the language which people speak produce the associations on which they will decide.

The syllogisms which we have got as yet, four universal and eight particular, contain all those
of Aristotle. Six of them indeed are enough for this purpose: that is to say, every syllogism of
Aristotle is either one of these six, or one of them with a premiss converted or strengthened, or both.
The six really distinct syllogisms of the old system, with the order XZ established in the conclu-
sion, are as follows, with the scholastic names of the forms which they have or can be made to have:

Ay, X)Y¥+Y)Z=X)Z | Barbara,

Tia | AY +Y)Z= XZ | Darii, Darapti, Disamis, Datisi, Bramantip, Dimaris.
Eur X)Y¥Y+Z.Y=X.Z | Celarent, Cesare, Camestres, Camenes. .
On | XY +Z.Y=X:Z | Ferio, Festino, Felapton, Feriso, Fesapo, Fresison.

Oo, | X:¥+Z)¥= XZ Baroko.

Ow | Y)X+¥:Z=X: Z | Bokardo,

2

4
(
fi

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 391

If we wish to have a notation which neglects the premises, we may call these 4, J, E, O,, O.,
O;, which may be separated into two connected sets, thus. The contradiction of a conclusion
coupled with either premiss must give the contradiction of the other premiss. It will be found that
if we call A, O., and O; opponents, and also E, I, and O,, each syllogism can be produced from
either of its two opponents, by coupling the denial of their conclusions with the affirmations of
their premises.

The six new syllogisms, reduced to the same order, will be

An, | KV wy =X)Z
tA ey +Z)Y= wz
X:¥+ s.y = XZ
te | Yasekes err are
Goya) a= a. x
0.4 DAY = cL,

The correlation of these two sets is by no means simple. Before examining it, observe that an
interchange of X and Z, though it alters A into a and O into 0, does not alter E and J, nor e and j,
The counterpart of a syllogism, made by this interchange, is represented by simply inverting the
letters of the premises, and interchanging 4 and a, O and 0, in the letters of the conclusion. Thus
the counterpart of i,, is iz,: that of Ag, is a,,- Now if we take the six Aristotelian syllo-
gisms, and make all the changes, and tabulate the results, we shall have as follows:

XYZ «YZ ayZ wys 2Y¥s XVx Xyzx XyZ

Aga Cea Cae aaa Ger Exp Epa Ax,
Ia Soa %e tia ign Or Ooa Toe
Exp Gen aaa Cae Cea Ass Ane Egy
On ion lia %e a Ij, To. Oos
Ooa tig top oa ie Loe ie On
Oro txo by O,0 ~ ORT Ty, To 0,;

The syllogisms written under each heading* are those which that written under the first
becomes, when the variation shown in the heading is made. Thus if XY and Z be changed into
a and x, O;, becomes o,,, or

AY+Z.Y=X:Z becomes e¥+2%.VY=:2, or V:X¥+Y)Z=Z: X.

The new syllogisms have their letters in Italics. Each form, old or new, or its counterpart,
occurs four times: but though the first column contains old syllogisms only, there is no column
which contains none but new ones. So that it cannot be said that the new syllogisms are, on any
one hypothesis, views of the old ones: though, in the column wy Z, five of them are so.

The following are the sets of opponents in the old and new system.

Old system Ay, O94 On, | Biya Ope 1G

New system dp, Ig, tog e

a

ca 0. Via

_ * The order of the headings follows a recurring law, the next the same process, and then forwards and so on until 2" have been
step of which would give YYZ again. If there be any odd num- made. Thus, if there were five, the changes would be made thus,
ber, n, of assertions, any one or more of which may be changed | 0 indicating no change, 0, 1, 2, 3, 4, 5, 4, 3,2, 1, 2, 3, 4, 6,4, 3, 2,
into its contrary, giving 2° varieties, all the varieties may be gained | 1,2, &c. In the case of three, it is

as follows : write them in order, change the first, in that the second, 0 1 2 8 2 1 2 8 2
in that the third, and so on to the end. ‘hen go backwards with XYZ w¥Z xyZ xvys 2¥x X¥x Xye NyZ| XYZ

392 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

But we have not yet closed our investigation; for we have to examine the remaining syllogism
of universal premises, or Y) X + Y) Z= XZ. If we go through the cases, in the last order of
headings, we shall find as follows.

First Form, Transformation. Deseription. Remarks.
Y)X+Y)Z=XZ Y)X+YV)Z= XZ df Derived from J,,
Y)v@+Y)Z=aZ Y.X4+Y)Z=Z2:X Opa AICP iMs a aia's ae met oe ih:
y)e +y)Z=2Z AX)V+ y.8 =Z2:X Og ) Smebssetetes Oi.
y)v + y)% = asx X)Y+Z)V= az tas Not yet obtained
Y)v+ Y)s = az Y.X+Y.Z= az ing Derived from i,
Y)X+Y)s=Xs Y)X+YV.Z=X:Z Qe sco ansone:scr On
y)X+y)e = Xz y.v +Z)VY=X:Z Ova Ssooceanoces fhe
y)X+y)Z=XZ ye + y.8 = XZ Te. Not yet obtained.

The derivation here mentioned is merely strengthening a premiss. We thus obtain the only two

remaining forms
44a KV 4B) Yim me lin hae y.a+y.2= XZ.

These cannot be derived from the twelve previously established by strengthening a premiss,
though their equivalents (the other six) can. These two last syllogisms differ from all the rest in
having no counterparts, and may therefore be called single syllogisms.

The old rules are of course true as to the old syllogisms: but most of them are inapplicable to
the new ones. Particular premises, indeed, never gave a conclusion, as yet: but premises both nega-
tive may, and in the case of i,,, the middle term is universal in neither premiss. Again, both premises
may be negative, and may give a positive form of conclusion. The following rules, however, will be
found to hold good.

From premises both particular, nothing follows. The middle term cannot be particular in both,
except in i4,: nor can its contrary be universal in both, except in J,,. One negative premiss
always yields a negative conclusion, and two negative premises an affirmative. When one
premiss is particular, the conclusion is particular. When e is in the premises the conclusion is
never in i.

I now take the two cases in which particular premises may give a conclusion: namely

Ty XY + ZY = XZ EXON, i= Ne Or

on the suppositions that the Ys mentioned in both premises are in number more than all the Ys.
If Y, and FY, stand for the fractions of the whole number of Ys mentioned or implied in the two
premises, and y, and y, for the fractions of the ys implied or mentioned, we shall by a repetition of
the process on VX + YZ = XZ (the other being obtained in the course of the process) arrive at the
following results or their counterparts: remembering that Y, + Y, is greater or less than 1, accord-
ing as y; + y» is less or greater. (See the Addition at the end of this paper.)

Designation. Syllogism. Condition of its validity.
In YX + YZ = XZ Y, + Y. greater than 1
0,, TERE SOD EOS GA Sp, En OOR CHOU SCO EAC IOOOC
Oa LES AEE Fie | nes I onthe concsnteccreen
04; DOB STN Ee, GA Y, + Y, less than 1
4 me ee Si TSN! NE Mee atestaiiee ects ociste stele
Oo: XG EEUU) S: = 2 Zig Wh ee epalbiterersiadataa cates setae

Too Aer Zi = XS Sonne SDSACEB ONCE Se

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 393

There are many remarks to be made on the demonstrative connexion of the parts of this system
with one another, and on the explanations in general language of the new varieties of syllogism.
The length of this paper, however, is a sufficient reason for stopping here with the formal part of
the subject, and proceeding to the consideration of the probabilities of argument and authority.

Section V. On the Application of the theory of Probabilities to questions of Argument
and Authority.

Writers on logic have made no effort to apply the mathematical theory of probabilities to the
balance of arguments; for we can hardly call by that name the simple statement that the pro-
bability of the conclusion of a syllogism is the product of the probabilities of the premises. How
far this is correct will appear in the course of the present section, which is intended to investigate the
manner in which the probability of a conclusion is to be inferred from opposing arguments and
authorities, of which the several probabilities are given.

Conclusions which are not absolutely demonstrated are established in our minds on two distinct
bases, argument and authority. Even if there be appeal to authority in establishing the
premises of an argument, the distinction is in no degree lost. ‘This we shall see as soon as the
terms are defined.

Argument is an offer of proof, and its failure is only a failure of proof: the conclusion may yet
be true. Authority is an offer of testimony, and its failure is a failure of truth: nothing can
furnish absolute reason for distrusting the authority on future occasions except the proof that the
conclusion asserted is false. A person who had made a hundred assertions, all supported by
inconclusive arguments, but all of which turned out to be frwe, would give a very high authority
to his hundred and first assertion.

We have an unfortunate use of language in the mathematical application of the word pro-
bability. We say that small probability and great improbability are identical terms ; which is not
true in their common meaning. In fact, a being what we call the probability of an event, a — 4 is
what we ought to call by that name: and if a— 3 be negative, we ought to call }-a the
improbability of the event. It would not be wise to introduce the same inaccuracy in the use of the
word authority: accordingly, « being the chance that an assertion of an individual, made on the
best of his knowledge and belief, is true, I shall call 4 the value of his testimony. When pu
exceeds 4, I shall say that he is authority for the conclusion. And, measuring absolute authority
by unity, I shall take 24 —1 as the measure of his authority, which is against the conclusion, if
2u4—1 be negative. . Again, if p be the number of times his testimony is given to a truth for once
which it is given to a falsehood (which we may call his relative testimony), and if a denote his
authority, we shall have the following equations, which will all be useful ;

-1

pt+l

wb l+a

Ta pe
l+a Pp

a= — = 5
2 fies

In forming our opinions upon argument, we are told to leave authority altogether out of sight,
and to consider only what is said, not who says it. It was Bacon, I believe, who first said that
assertion is like the shot from the long bow, the force of which depends upon the arm which draws
it; while argument is like the shot from a cross bow, which a child can discharge as well

Moye vill, Pant, II, 3E

394 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

asaman. But the simile is as inapt as the recommendation it contains is unwise ; for the endeavour
is to hit the mark, not merely to fire a shot: and the bow which most often succeeds in doing that
is the best. Closely examined, this direction to dispense with authority amounts to requiring
us to suppose that the proposer of an argument is as often right as wrong, and wrong as right, in
his conclusions. But what can be the wisdom of making believe that a person tells us ten truths to
ten falsehoods, if we know it for a fact that he tells us nineteen truths to one falsehood? If
absolute demonstration be given, no rule is necessary, for we cannot attend to authority. If some-
thing very near to demonstration be given, no rule is practically necessary, for we have what is called
moral certainty.

But, it may be said, why not throw away authority altogether? I answer that it is impossible :
and that any one who forms an undemonstrated conclusion independent of the authority of others,
can only do it by assuming some value for his own. All arguments, and all balance of arguments,
will leave three possible cases, Either one or more of the arguments for the conclusion will prove
it, or one or more of the arguments against will refute it, or all the arguments are inconclusive.
The conclusion is proved, disproved, or left neither proved nor disproved. But it is not one of the
three, true, false, or neither true nor false: it must be either true or false. And the mind must
come to some conclusion upon this point: it must, so to speak, distribute the inconclusiveness of
the arguments, in some way or other, between belief and disbelief. In whatever way this is done, it
amounts, as we shall see, to some assumption as to the authority either of the proposer or of
the receiver, or of some third person, or of all together.

There is but one way in which we can really deprive the proposer of an argument of any
authority; and that is, by depriving him of any peculiar authority. If Newton propose an
argument, to the conclusion of which Halley assents without knowledge of the argument, we have a
right to allow it to be reasonable that the argument should lend the same force to the conclusion as if
Halley had proposed it, and Newton had assented, also without knowledge. Admit this, so far
as the premises do not depend on the authority of the proposer, and we admit all the separation of
argument and authority which is practicable.

A conclusion is usually opposed, in argument, to what logicians call the contradictory, which
must be true if the conclusion be false, and vice versd. It is not often that it is opposed to
the contrary, which must be false when it is true, but not vice versd. I shall first consider the
proposition and its contradictory, as to authority, as to argument, and then as to the two in
combination.

Pros. 1. Required the joint value of authorities the separate values of which are given.

Let the first authority be one of the testimony p, or of m truths to m errors, « being

m+(m+n). Let p»’, m’, n’, take the place of u, m,n in the second authority: and soon. Now.

since the conclusion asserted cannot be true on one authority and false on another, our position with
respect to the conclusion is as follows: We have an urn of m white and m black balls, another of
m' white and n’ black, &c. from each of which we have to draw. The balls however are not free,
but are connected by such mechanism that no ball will leave its urn unless a simultaneous effort be
made upon one of the same colour in every urn. Now the number of ways of choosing one white ball
out of each urn is m m’ m”... ; and of choosing one black ball nn’ ’.... Hence the united testi-
mony for the conclusion is

, ” , ”
mmm’... . J nnn’...
and against it ——_—,———_,_,, ,
Mmmm ...+N2nNN...

ae Diese eee sa (aH Gs p)(Q — nu")... :
wee... +(1— mw) (1 -—p')-n’)... pep... +(1—uv) (0 -y) (1 -”)...
If a=2n-—1, &c. we have for the joint anthority expressed in terms of the separate
authorities,

TERT 7
mmm ... + nnn ...

ae

¥

_ ° tte tiwe a. ye

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 395

(i +a)(1 + a) (li+a’)...-(l-a)(i- a\iG@- ay) axe
(14+ a)(1+a@))(1+4’)... +(1-a)(1-@) (1 -a’)...

If p= u—(1— 4) &c., we find that the joint relative testimony is the product of the separate
relative testimonies; which is the easiest way of expressing the result. ‘Thus two authorities
of 3 and 4 truths to one error, amount to one authority of 12 truths to one error.

I need hardly say that the preceding conclusions are verified by their giving such results as the
following ;—that if one of the authorities be absolute, the joint authority is the same; that any
number of testimonies, each without authority either way, gives no authority either way ; that
inauthoritative testimonies do not affect the authority of the rest; and so on.

Problems of the preceding character are usually solved by the inverse method; or by the
determination of the probabilities of precedent states from an observed event. Others have noted,
I suppose, what has often struck me, namely, that the arrangement of conditions into an observed
event and its precedents, is sometimes made in a very indirect and unnatural manner. There
are however two classes of problems which give the same results: each inverse problem has a direct
problem of the other class connected with it. For instance, there are m and m’ white balls, and
n and 7’ black balls, in two urns. A white ball has been drawn; what is the probability that the
first urn was that which held it? The answer is well known to be

m(m' +n’) divided by m(m' +n’) + m'(m +n).

Now take the following problem. The black balls are absolutely fixed in the urns; and the white
balls are so connected that one will come out of neither, except when a white ball is touched in both,
which will only set free one, say the one which was touched first. With one hand in each urn, not
knowing one from the other, the chance of bringing out a white ball from the first urn (if we
try until a ball comes from one or the other) is the same as that above, namely, that a ball drawn
white was in the first urn. These two problems are really the same; the first says that a white ball
has been drawn, the second that a white ball must be drawn. And precisely the same sort and
amount of reflexion which must be employed to make this sameness apparent, must also be employed
before the problems above alluded to will lose that indirect and unnatural appearance to which
Thave referred. It should also be noticed, that any problem on an event to come may, by supposing
the event to have happened, not being yet known, be made a problem of inverse probabilities.

Pros. 2. Supposing the authorities to hias one another, required the method of allowing
for the bias.

When one authority expressly cites and defers to another, he does not thereby diminish his
own authority. For what we want to know of him is simply the value of his assent, which, unless
we have some specific reason, we have no more right to suppose less than his average when he judges
of another, than we have to suppose it greater. And, in fact, there are men who are better authori-
ties as to their judgment of others, than as to what they propose themselves. Neither, for a similar
reason, does it diminish the value of the second authority, that the conclusion asserted never would
have been known to him had it not been for the first. What we want to account for here is
undue bias, which I define to exist when there is a proportion of the conclusions of the second
authority which are no better for his testimony than they would have been if the first alone had
asserted them. The case of a number of authorities would lead to a complicated result. Suppose
three, the values of whose testimonies are p, pv’, u’; and let \’ and \” be the probabilities that the
second and third are unduly biassed by the first. Then the value of the joint testimony is

,

aes nee bp
Wr" a + (1 — NYA” ——_** __,
BYs ) we +(1—p~) (1-2)

pe”

eX (1A a a 77
( Yai +(1—-p)(l-y’)

t i we me
re io Cee wy ee ; Seth
( )( ae +(U-p)(i-v)Q-7-)
3B2

396 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

If there be only two authorities, the formula is reduced to

,

ap :
wu +(1—p) (= n')’

for the joint testimony, and the joint authority is

Aw+ (1 —-A)

ata —2dq' (1 —-a)

1+ aa’

Had it not been for the bias asserted, the authority would have been (a + a’) +(1 + aa’). When
a +a’ is positive and a’ negative, the joint authority is the greater for the correction of the bias.
This is as it should be; for the bias is then that of contradiction, and tends, until corrected,
to lessen the joint authority. I have only entered thus much into this part of the subject, merely
to show that the results of the preceding mode of treating the problem are confirmed by those
of common sense.

Pros. 3. To determine the joint effect of a number of arguments, the validities of which
are given, some for a conclusion, and some for its contradictory.

By the validity of an argument, I mean the probability that it proves its conclusion. The
argument being of a conclusion which is legitimately inferred from the premises, it is absolutely
valid, if all the premises be true: and what is here called its validity therefore means the product of
the probabilities of all the premises. Let a, a’, a", &e. be the validities of the several arguments
for the conclusion, and 6, b’, b”, &c. those of the arguments for the contradiction. If one argument
on either side be valid the conclusion of that argument is established. Hence the joint validity
of the arguments for is that of an argument whose validity is

1-(1-a)(1-d)(1- a’)... or Ya- Yaa + Yaa’ - ...

which is the probability that one or more of the arguments for proves its conclusion. Similarly the
arguments against amount to an argument the validity of which is

Pai byl 6 )e.. or Sb Seb + S bbe” = wes

And having thus shown how to reduce several arguments of the same kind to one, we may now
proceed as with one of each sort. If the process now coming be applied to several arguments
of each kind, the result obtained will, as we might predict, verify the correctness of the preceding
compositions.

Let there be then one argument of the validity a for, and one of the validity 6 for the contra-
diction, or against. Let the argument for, be as a drawing from an urn in which there are M valid
and N invalid cases: let that against, be as from another in which there are P valid and Q invalid
cases. Of course M: N::a@:1-—a and P:Q::b6:1-—6. If either argument be valid the
other must be invalid. Now it does not follow that if the argument for be valid, and be the case
marked, say 1, the invalid argument against may be any one of the cases 1, 2, 3... up to Q. For
it may happen that each particular mode of succeeding in one argument must be necessarily connected
with some particular mode or modes of failing in the other. To represent this, let us separate the
three cases, and assume as follows:

1. When the argument for is valid and that against invalid, let it be that M = m, + m, + ...5
Q= 4+ G+.--, and that when the first succeeds in one of the m, ways, the second must fail in one
of the g, ways; and the same of m, and q,, m, and q;, &c.

2. When the argument against is valid, and that for invalid, let NM =m, + m+...,
P=p,+ p.+... with the same connexion,

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 397

3. When both arguments are invalid, let N =n, +7, +..., Q=q) +q: +... with the same
connexion,

It is now clear that the number of compatible cases, in which the argument for is valid, must be
mq, + Mg.+... or Zmg. Similarly, Xp and Zn'q' are the numbers of cases in which the
argument against is valid, and in which both arguments are invalid. Hence we have for the
probabilities of the three cases, namely, that the conclusion is established, that the contradictory is
established, and that the arguments are inconclusive, the following expressions :

=mq Snp rnd
=mq + Unp+>n'q’ Smq + =Unp + Inq’ mq + np + Sn'_

To solve the question in the most general manner, would require that we should combine
the preceding results in all cases, that is, for all values, and all subdivisions, of M, N, P, Q.
Without attempting such generality, I may make the following observations. From what takes
place in other similar questions, it is highly probable we should find the result of this combination
either to agree with that in which any of the M cases may occur with any one of the Q cases, &c.
or to approximate to such an agreement as M, &c. are increased without limit. Next, that this
agreement actually takes place, when all the subdivisions are the same aliquot parts of their wholes.
With these presumptions, I content myself with their result, which amounts to supposing that any
one of the M cases may enter with any one of the Q cases, and so on. The probabilities then are,
for the three cases above-mentioned,

MQ > LD ee LC ey
1 MQ + NP + NQ’ MQ+NP+NQ’ MQ + NP +NQ’
a(1i — 5) b(i - a) (1 — a) (1-5)
or ° —=<g '
1— ab 1—ab 1-—ab

The third term is the chance of inconclusiveness, which necessarily renders this case indefinite: and
all we can say is, that the chance of the truth of the conclusion is
1-6
1-—ab
where the value of cannot be determined from argument (for all the arguments are used in

determining a and 5).
When the arguments are of equal force, or a = b, we have

ja+A(1 —a)},

a a l-a
(Oe iat Meera) 2 ices
Hence a~(1+ 4), which represents the probability that a verified conclusion was derived from
an argument of the validity a rather than from demonstration (when it must have been one
or the other), also represents the success of an argument of the validity a against an argument
of equal force on the other side.

So far as an argument is not demonstrative, it must rest on authority, including under that word
the authority of the recipient himself. Now a is in fact the testimony to the validity of the argu-
ment on one side, and 6 to that on the other. If these were testimonies to the truth or falsehood
of the conclusion, the joint testimonies to the truth and falsehood of the conclusion would then be

a(l — b) b(1 - a) ‘
a(1 — b) + b(1 - a)’ a(1— 6b) + b(1 -a)’
which, since a +b-—ab must be less than unity, are necessarily greater than the two first of

the three expressions. Or, if we attempt to consider argument entirely without reference to any
authority except that for the premises, the absolute testimony to the truth or falsehood of the

398 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM.

conclusion thus obtained is not so great as would be obtained from testimonies to the conclusion as
strong as those to the validities of the arguments.

Pros. 4. Given a number of arguments for a conclusion and for its contradictory, and
also a number of authorities, all of given probabilities ; required the resulting probabilities for the
conclusion and for its contradictory.

Let a and b have the meaning of the last problem, and let be the testimony which the joint
authorities give for the conclusion and against its contradictory. Let a and b be represented by
urns of m and p valid cases, and m and q invalid ones; and let » be represented by an urn of
» truths and w falsehoods. Then there are mqv cases in which the argument for is valid and the
conclusion true; mpw cases in which the argument against is valid and the conclusion false; nqv
cases in which both arguments are invalid and the conclusion true; ”qw in which both arguments
are invalid and the conclusion false. And these are all the possible combinations. Hence the
probability that the conclusion is true must be

(m + n)quv (1 — b)p

(m+n)qv+(p+qnw ” (1-be+(Q-ay(i-n)’
and the probability that the conclusion is false must be
(p+ qynw Ge ay'= 5)
(m+n)qv + (p+q)nw (1 —b)p+(1—a) (1 — 4p)”

To show the accordance of these formule with common notions, observe that they give the first
four of the following results :

1. In an impossible conclusion (or when » = 0) the first expression vanishes: or no argument,
however strong, can give any probability to an impossibility.

r 4 0
If » = 0 and a =1, we have incompatible hypotheses, and the expressions take the form — .
0
2. If a=1, the conclusion is certain; or absolute demonstration establishes its result, in
spite of any amount of authority against it.
3. If there be no authority, or if « = 4, then the probability of the conclusion is
1-6
1—b4+ 1—a’
and hence counter-arguments of equal strength, applied with no authority, give no authority to the
conclusion.

4. If a =5, the probability of the conclusion is 3; or counter-arguments of equal strength
leave previous authority unaffected.

5. If a+b =1, the effect of the arguments is simply that of one more authority: and that
independently of their inconclusiveness, which still remains.

6. If there be no argument against, or if 6=0, the probability of the conclusion is not @
(as stated by writers on logie*, who confound it with the conclusion made valid by the argument)

ye 1 fi .
—______———— ; or ~—., when there is no authority.
w+(i—a) (1-4) 2-4a

7. When there is no opposition, and no previous authority, any unopposed argument, however
weak, gives some authority to the conclusion; and every argument, however weak, increases the
probability derived from previous authority.

but

* Myself among the rest.

* irae

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 399

Authority apart, the odds for the conclusion are 1 — 6 to 1— a. When both arguments are of
great force, or 6 and a both near to unity, the ratio of the small quantities 1 — 6 and 1 — a, which
determines the probability for the conclusion, cannot be distinctly apprehended. When, then, there
is something as near to demonstration on both sides as can be found in a subject which does
not admit of absolute demonstration, the mind ought not to arrive at any conclusion more favour-
able to one side than the other. We constantly see the refusal of human nature to acquiesce in this
reasonable rule, and always with a determination to find out weakness in the argument on one side or
the other. It must be sometimes true that false conclusions shall be the exceptional cases, in which
arguments of the highest probability fail.

It also appears that moral demonstration on one side is not enough, if there be anything
resembling it on the other. All controversialists admit this in fact, by the stress which they lay on
answering the arguments of the opposite side. But they frequently do this as if it were a kind of
surplusage, a charitable (but not in any other sense necessary) allowance for the weakness of those
who do not see the force brought forward on their side of the question. Whereas it appears that it
may be perfectly necessary to answer an opponent who admits all they say to the full extent which
is demanded for it, supposing that to be anything short of absolute demonstration.

Pros. 5. To ascertain the manner in which the inconclusiveness of the arguments is divided
by the authorities between the probabilities of the truth and falsehood of the conclusion.

If we find \ from the equation,

a(1—b) (i -a)(i-6) (i — b)p E
1—ab 1— ab LG = Oyp3@ Seu pp)
we find )\ = Ono | Teta

(1 - b)n + (1 -a)Q—p)’
(1 +6)G —2#)- 6
(1 -b)p+ (l-a@) (1 ~ 4)”

* a :
From this it appears that ) is negative only when p is less than 7 , and 1 —) when xz is greater
a

1 A c ‘
than ee B In the former case we sce that unless the testimony of authority to the conclusion be
a 4

greater than the success of the argument for the conclusion against a counter-argument of equal
strength, the probability of the conclusion is less than that of the validity of the joint arguments.
If there be no authority, or if 4 =4, we have
l-a 1-6
ee I fey ee

1-b+1-a’ 1-b+1-a’

a result which demonstrates the unmeaning character of the result of Problem 3. For the incon-
_ clusiveness is divided between the truth and falsehood of the conclusion in the proportion of the final
probability of its falsehood to that of its truth. Or the more likely the conclusion is to be false,
the larger proportion of the inconclusiveness does its truth get.

But we find r
(l-b)p 1-6 (i —a)(i - 3b) 2n-1
Me iakiiaa)(l —“)) b-tedieae Lobel —o. (1-6) »40.-.4),0.—a),
which shews the addition made to the probability of the conclusion in passing from the case of argu-

ments without authority to that of arguments backed by the authority 24-1. In the case of

1-b l—p
arguments of equal strength, this is » —4, as it ought to be. When eit me.’ or when the

400 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

invalidities of the arguments for and against are in the proportion of the testimonies of authority
for and against, the same thing occurs; or the alteration of testimony in the above transition is

1'="Bi 2 i-
lies between 1 and ~
a it

more than the alteration of testimony is transferred ; in other cases less. The greatest transference

exactly transferred to the probability of the conclusion. When

l-p

is when x= Vv (

). in which case the amount of probability transferred is
“be

2
pes eS ee ee eee (Ve
1 Pei/Geey 2)
It appears from what precedes that in the formule, the invalidity of the argument against,

1—, enters for the conclusion, and the invalidity of the argument for, 1 — a, enters against the
conclusion, precisely in the same manner as the testimony for it, m, and that against it, 1—u. If
1-56 " l-a
then we cal] ———————— the ¢estimony of argument for the conclusion, and ——_———— that
1-b+1=a y of arg: f , Ses ea
against it, just as we call «4 and 1—«u the testimonies of authority for and against: and if also we call

1-6
the relative testimony of the arguments: then we may express the result of Problem 4 by

1-a
saying that the joint relative testimony of the combined arguments and authorities is the product of
all the separate relative testimonies, both of arguments and authorities.

It must be observed that the mode of entrance of the testimonies of argument makes it follow
that if, after obtaining a result from certain arguments and authorities, we use the probability
obtained as a new authority, in combination with additional data,—the final result will be the same
as if we had collected all the arguments separately and all the authorities, and then proceeded as in
Problem 4. This follows from the property of the functions p+(p + p’) and p’'+(p +p’), which
contain a mode of composition in which the order of the processes is indifferent, and their partial
collection allowable. If we denote the preceding functions by [ p] and [p’], we have

Clpl(a]}=(p9) Clpg]7] = [pr] &e.

When there are any number of arguments for, of validities a, a’, a’, &c., the chance that one or
more are valid is 1— (1 — a) (1— @) (1— a”) ..., and the testimony of argument against the conclu-
sion is (1 — a) (1 — a’) (1 —a’) divided by (1 —a@) (1-a@’)...+ (1-5) (1-8) +... Hence, the
arguments against having the validities 6, 6’, &c., and the authorities for and against being u,
mu, &e., and 1-4, 1—y', &c., and 4 being the probability for the conclusion derived from the
whole of the data, the principle of relative testimonies may be-expressed thus :

A i — 0 ele owl ie B we py”
= x

7 teeees

(Aveta Sas ta Cel ee

or as follows ;—let the probabilities of the conclusion, derived from the several arguments backed
by no authority, be considered as testimonies of authority to the conclusion, and used as in Pro-
blem 1. -

It may happen that, besides the validity a, obtained directly from the premises, there is sepa-
rate testimony of authority to the validity of an argument. Let it be &: then instead of a must be

ag

ak+@-a)(@-8"
I now return to the question of the dismissal of authority, which was partially entered on at the
beginning of this paper. I assume that the mind will form an opinion upon any proposition which
is laid before it. Even if the assertion were in a sealed packet, with no reason whateyer to suppose
it one rather than another of all that could possibly be made, an opinion would be formed as to its
truth namely, that it is an even chance whether it be true or false. And this opinion is a just one;

used

|

:

>
a j
4
,
4

-

4

a

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 401

for, since every assertion has its contradictory, and one of these two must be true and one false,
it follows that the numbers of possible truths and falsehoods must be equal. When the packet is
opened, this opinion will probably change: duly, in a manner depending upon previous associations
of knowledge, or unduly, from what are then properly called prejudices. That every mind must
form some opinion, may almost be concluded from the notorious fact that most minds, indeed nearly
all uneducated ones, have little power except of absolute belief, or absolute unbelief. Their reason-
ing power is a spirit-level in awkward hands; the bulb is always at one end of the instrument or
at the other. Now when it is recommended to dismiss authority, or to allow no authority, I appre-
hend that the advisers are not aware that they are promoting the specific plan of assuming that the
proposer of the argument is a person of ten truths to ten errors. They rather wish to dismiss ¢esti-
mony, which it is clear, if it be that a conclusion must be formed, cannot be done.

Nor is it by any means true that the proper way of doing without authority is to assume the
measure of authority = 0. If we wish to find the value of an argument, be the authority what it
may, or as if the authority be unknown, we must allow for the effect of any possible authority, put-
ting every value on equal terms with the rest. Let du be the chance that the testimony of authority
lies between « and » +d, then the chance of the conclusion being true concomitantly with the
authority lying between » and » + du is

a - b) udp
(1-6) m+ (1—a) (1 w)’
which, integrated from « = 0 to « =1, gives for the probable truth of the conclusion
r { log r 1-6

where r = :
r-1 r-1 l-a

If we assume that the chance of the testimony lying between » and « + du is M oud,

where M is the reciprocal of Ji gpudn, we have for the probable truth of the conclusion
M f rupyudu
om+t1l—-w

aud some other supposition except @u =1, is absolutely necessary: it is absurd to suppose equal
chances for all values of the authority; to take the unknown proposer for instance, to be just as
likely to be infallible as to be of no authority at all. What form should be assumed for pe must
be matter of opinion. If it be desired to try it on the supposition that » is most likely near to some
specific value X, then, m and m being two integers in the proportion of X and 1 —X, the assumption
pu =p" (1 -— »)" will represent the hypothesis, if m and m be considerable. And the greater m
and m are taken, the smaller the chance that the testimony differs from by so much as a given
quantity.

To give a case somewhat more like the proper notion of human authority than that in which all
values of the testimony are equally probable, let us take pu =au(1-—4), M=6. The above
integral then becomes (after multiplication by J),

r :
Gay {6rlogr+2+ 3r—Gr° + 7°}.

es
If r =1 this becomes 4, as we might expect.

In the above conclusions, r is the relative testimony of the argument, on the supposition of no
authority. If p be that of the authority, the joint relative testimony to the conclusion is rp: let
us now see how far this is affected in the case of a moral certainty by the supposition that the chance
of the testimony of authority lying between p and w + dp is (1 — p)"dp, where m+ (m + m) is the
previous fixed value of 4. Now we have
fig (| _ ue) n SAG aa jr) 1 rp = pynts

Te A
pi — (el a = py" = ; :
rua+r+l-yp r r ite SATO ia WOT

Vou. VIII. Parr III. 3F

402 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

And M is [m+n +1]+[m].[m], where [m] means 1.2.3...m. If the probability required

be denoted by P,,,,,, we have, multiplying by Mdp and integrating from 4 = 0 to uw = 1,
m+11 (m+2)(~+1) 1 (n+3)(m+2)(m +1) 1
- = a == — ae
m m(m—1) 7° m (m —1)(m—-2) 7

Pa,n=!

M—3, n+3°

Now P,,_35 »+3 is less than unity, so that if r be considerable, any degree of approximation may
be obtained by this method, carried to more terms if necessary, and if the value of m will permit.
Take the first three terms: then if the testimony of authority were given = m—+(m +), instead of
being most likely to have something near that value, the approximation to P,, , would then be

n 1 n® 1

Subtract the second from the first, and we have
1 3mn +2m+n* 1
mr mtn 1).
Write (m + 7)» and (m +) (1 — «) for m and m, and we have, supposing m and m+ con—-
siderable numbers,

1 1 2n+1)Q-
i= Le Z eed nearly.
m+n \ur per {
Except then when p» is very small, the principle of relative testimonies is sufficiently accurate,
in the case above supposed, taking for the testimony of authority the most probable value of that

testimony.

Pxrozs. 6. Given arguments and authorities for a proposition and for its contrary, required
the probability for the truth of each proposition, and for the falsehood of both.

The contrary is thus distinguished from the contradictory: both the proposition and the con-
trary may be false, though both cannot be true: while either the proposition or its contradictory
must be true. As far as the arguments alone are concerned, the problem is that of Problem 3: for
either one of the arguments is valid and the other invalid, or else both are invalid. But there is a
difference in the meaning of authorities; for, » being the testimony to a proposition, 1 — is not
necessarily the testimony to its contradictory. Let , and v be the testimonies of authority to the
conclusion and its contradictory, and a and 6b the probable validities of the arguments. There are
then five cases, two favourable to the truth of the proposition, two to that of the contrary, and one
to that of the falsehood of both; 1. The argument for may be valid, in which case the proposition is
true, the contrary false, and the argument against invalid. 2. The argument against may be valid,
in which case the contrary is true, the proposition false, and the argument for invalid. 3. Both
arguments may be invalid, and the proposition true. 4. Both arguments may be invalid and the
contrary true. 5. Both arguments may be invalid and the proposition and contrary both false.
Treating these in the manner in which the preceding problems have been solved, and which it is now
unnecessary to repeat, we have the following expressions for the probability of the proposition, of
its contrary, and of both being false,

(1-b)(1-v) 1 : (1-a) (1-1) v — G-a4)d-yd)
(1-b)(1-v)4+(-a)-4)u+(1-a)(1-b)(1-4)(1-v) (1-b)(1-v)a+(1-a)-4)u+(1-a)(1-b)(1-n)-») (1-6)(1-v)+(1-a) (1-p.v+(1-a)(1-6) (1-2 )(-¥) ©
If there be no authorities, or if » = v= 4, these become
1-5 l-a (1 — a) (1-5)
1-b+1-a+(1-a)(1-6) 1-64+1-a+(1i-a)(i— |) 1—6+1—-@+ (1— a)
If the arguments be of equal strength these become
1 1 l-—a

ee

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 403

ne aL A : 2 0 ;
except when @=1 (an absurdity in this case), in which they take the form O° But we get this

result, that if (1 — b)+(1 - a) =p, then the more nearly the arguments become demonstration, the
more nearly is it certain that either the proposition or its contrary must be true, the probabilities
for one and the other being as p(1—v)p and (1—,)». This is a singular result: for, since of
two exceedingly strong arguments, one on each side, one must be invalid, it is not easy to explain
from @ priori notions why there is so great a probability that one or the other must be valid.
That it is so appears from the probabilities of the validities* of the two arguments, and of the
invalidity of both, namely,

a(i — b) b(1 - a) (i - a) (1 -— 8)
a(i—6)+b(1—a) +(1—4) (1-6) a(i—b) +b(1-a) + (1—a) (1-6) a(i—b) +b(1—-a) + (1-a) (1-8) |
in which p : 1 is the limiting ratio for the probabilities of the two validities. The same remark
may be made with reference to the authorities: when two very high authorities affirm contraries, the
higher the authorities the more likely is it that one or the other is right.

When there is no argument for the contrary, or b = 0, the three expressions become
(1) (ea) (1-#)» (1a) (1-1) (1-v)
(1-v)4+(1-a)(1-#)¥+(1-a)(-)(1-¥) (1-v) #+(1-a)(1-#)v+(1-a)(1-e)(1-v) (1-4 +(1-a)(1-) + (1-a)(1-4)(1-v)?
when there are no authorities these become

1 1-@ I cs (a7
3-2a’ 3-2a° 3— 2a’
or when an argument is proposed, simply, the chance thereby given to the contrary is the same as
that of neither being true.

It will seem strange at first, that the probability for the conclusion is not ———: for it will be

said, an argument and none for the contrary, is precisely the same position as an argument and none
for the contradictory. But the suppositions as to authority are different. Looking to authorities
only the chances of the three cases are

#(U— ») v(1 — p) (l-p») QU — ») f

w(l—v) +v(1—p”) + (I=) (1-v) pv) + (1—p) + Ap) -v)? w(l=v) +v(-4) + (=p) 1-9)’

and in the case of no authorities, there is the chance 4 for each of the cases. Now in treating the

contradictory the testimony of no authority is 3.

Let us now suppose that there is authority for each of the three cases, and also argument, or
generally, let us take the following problem :

Pros. 7. Let there be a dilemma of any number of horns, one or other of which, but
only one, must be true; required the probabilities of the several horns, arguments and autho-
rities being given for each.

Let a, b, c, &c. be the probable validities of the several arguments, p, v, €, &c. the testimonies of
authority. This problem, treated as before, gives the following result. Let
z=(1 —b)(i-ec)...n(1 -v)(1 — &) +(1 —a)(1 —c)... (1 —yu)v(1-&)...
+(1-a)(1-5)...(i-pw A -v)é...
then the probabilities of the several horns containing the truth are

* I should have made this remark before, in regard to the contradictories, but for having written the denominator in the transforme
shape 1—ab, I have always found the best rule to be, never perform operations in denominators.

3F2

404 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

(1-0) Q-e)... a (I= v) G8)... (=a) G6)... =H) v(1-&)... (1-a)(1-8)... (=m) (1 W) Ee
= = >

This problem contains all that have gone before, except the second. But this may not be appa-
rent at first. In fact, if I had commenced this paper with the general case now in hand, and had
then descended to the particular cases, the method of descending might have appeared exceptionable,
requiring the authority of an independent consideration of the particular results arrived at. Suppose
a dilemma of two horns, such as a proposition and its contradiction. If the testimony of authority
for the proposition be ,, there is in this case the testimony 1 — tmplied for the contradiction. But
this does not enter the formula: it is only the form belonging to the case of what is virtually repre-
sented in the general formula above, namely, that there is the testimony 1 — implied in favour of
one or other of the horns following the first, because there is the testimony » given for the first.
No express testimony is given to the contradiction: so that it enters with the testimony 3. And if
there be only two horns, and the testimonies be « and 4, it will be found that the preceding expres-
sions agree with the answer to Problem 4. There was no need in that case to suppose testimonies
and yp, because, as the testimony to each horn is a definite testimony to the other, they would but
have amounted to a joint testimony for the proposition.

If we want the case of the last problem, we have to take three horns, making ¢= 0 and = 4.
Or we may if we like suppose argument and testimony offered for the third case, namely, that both
the proposition and its contrary are false.

If we wish to construct the general case upon the supposition that no one need be true, all we
have to do is to add one more horn with an argument 0 and a testimony 4.

The easiest way of representing the result of the general case is as follows. Let 4,, represent
the probability of the m horn from argument only, and M,, the same from authority only. We
have then (using a, a, &c. and p, p, &c.),

1 Pon
A, = i — ‘ M,, = WS Mn
> Sere.
1-4, ih Mn
A, M.
and the probabillty of the m™ horn is SU,

The term 4,,M,, or may be called the exponent of probability of the m' case :

Dr i 3

and the probability of that case is its exponent divided by the sum of all the exponents. This
exponent is proportional to the number of balls in the urn the exits of which are favourable to the
case. It is the product of two relative testimonies, that of the authority, and that of the argument
alone, to establish the conclusion against its contradictory, that is against everything opposed to it.

Now suppose a complex dilemma of this kind, namely, that m of the horns, neither more nor less,
must be true, and the rest false. An examination of this problem leads to the following result.
The product of the exponents of any m cases, divided by the sum of the products of all the
exponents, m and m together, is the probability that the m cases chosen are the true ones. Hence
can be readily found the probability that any one case is among the true ones. If there be four
cases, for instance, of which two must be true, and if e,, e2, e,, e,, be the exponents, the probability
that the first case is true is

e€,(e, + es + e,)
€\€s + Cs + C,€, + Cy€s + Cre, + C56,

If it should be that m cases or fewer, but not more, may be true, then the probability that any
m — p cases and no others, shall be true, is the product of the exponents of those m — p cases, divided

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 405

by 1 + De, + De .e, + Dee, + ... + Dee, ...€,, the 1 being omitted if all cannot be false. The
various restrictions which might be imposed, as that only an even number can be true, that no two,
three, or any number of contiguous cases can be true together, &c. &c. may be easily contained under
this one rule. In every set of cases that can be true together, multiply together the exponents of
those cases; the product is the numerator of the probability that those cases only are true, and
the sum of all the products is the denominator.

This rule applies to one case which we have not yet considered. When several arguments were
proposed together, all for, or all against, a conclusion, it was supposed that they were perfectly
independent. But it may happen that two or more arguments are so connected that some must be
valid together and invalid together, or that some are valid when others are invalid, and vice versa,
or that the validity of one makes another valid, but the invalidity of the first has no influence on the
validity of the second. All these cases, and a great number of others, including in fact, under one
view or another, any question that may be proposed, may all be solved by the following Rute.

There is any number of events, each of which may happen in any number of ways, the separate
probabilities of which are given, but so connected that there are specific necessary coincidences,
or failures of coincidence. Take all the combinations which can happen, and compute the
probability of each combination, as if its events were entirely unconnected. The resulting products
are proportional to the probabilities of the several cases arising.

Thus, if there were three urns, the first giving white, black, or red (with chances w, 6, 7) ;
the second white or black (with chances w’,b’) ; the third white or black (with chances w”, b”), but so
connected that black cannot be drawn from the first, nor white from all three, nor red from the first
except when different colours come from the second and third, and it be required to find the chance
of having a red ball, we proceed thus. Enumerate the possible cases, which are WWB, WBW,
WBB, RBW, RWB, and the probability of a red ball is

7 (bw + w'b”)
w (w'b" + bw" + Ub") + 7 (b'w" + wb")

I have taken such an example, because it seems as if the condition that a black ball cannot be
drawn from the first is equivalent to taking away those black balls, in which case the chances
of the others cannot be w andr. But if the black balls be previously removed, then for w and r

cop and — » Which will not affect the formula. In the same way any addition

of other coloured balls, with the condition that they cannot be drawn, though it will affect the
probabilities of the independent events made use of in the solution of the problem, will not affect
the ratio which expresses the final result.

I have given so many proofs of particular cases of this principle that it is not necessary
to say any thing on the general proof. But I shall observe that the circumstance noticed in
combining argument and testimony, namely, that instead of the validity of an argument entering for
the conclusion, the invalidity enters against,—is an immediate application of the preceding rule.
For it is not the validity of an argument which is necessary to the truth of a conclusion, but the
invalidity of it which is necessary to its falsehood. Thus, in Problem 4, the necessary cases are, either
1. Argument against invalid, and testimony for true, giving (1 — b)n; or 2. Argument for invalid,
and testimony against true, giving (1 — a) (1 —,).

The application of the principles on which the preceding rule is established, would, I suspect,
give much clearer views of many problems than the ordinary method of employing inverse con-
siderations.

we must write

A. DE MORGAN.
University College, London,
October 3, 1846.

406 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM,

ADDITION.

Sixcx this paper was written, I found that the whole theory of the syllogism might be deduced
from the consideration of propositions in a form in which definite quantity of assertion is given
both to the subject and the predicate of a proposition. I had committed this view to paper, when I
learned from Sir William Hamilton of Edinburgh, that he had for some time past publickly taught
a theory of the syllogism differing in detail and extent from that of Aristotle. From the prospectus
of an intended work on logic, which Sir William Hamilton has recently issued, at the end of his
edition of Reid, as well as from information conveyed to me by himself in general terms, I should
suppose it will be found that I have been more or less anticipated in the view just alluded to. To
what extent this has been the case, I cannot now ascertain: but the book of which the prospectus just
named is an announcement, will settle that question. From the extraordinary extent of its author’s
learning in the history of philosophy, and the acuteness of his written articles on the subject, all who
are interested in logic will look for its appearance with more than common interest.

The footing upon which we should be glad to put propositions, if our knowledge were minute
enough, is the following. We should state how many individuals there are under the names which
are the subject, and predicate, and of how many of each we mean to speak. Thus, instead of **Some
Xs are Ys,” it would be, ‘* Every one of a specified Xs is one or other of 6 specified Ys.” And the
negative form would be as in “No one of a specified Xs is any one of b specified Ys.” If propo-
sitions be stated in this way, the conditions of inference are as follows. Let the effective number
of a proposition be the number of mentioned cases of the subject, if it be an affirmative proposition,
or of the middie term, if it be a negative proposition. Thus, in “ Each one of 50 Xs is one or other
of 70 Ys,” is a proposition, the effective number of which is always 50. But ‘No one of 50 Xs is
any one of 70 Ys” is a proposition, the effective number of which is 50 or 70, according as X or F is
the middle term of the syllogism in which it is to be used. Then two propositions, each of two
terms, and having one term in common, admit an inference when 1. They are not both negative.
2. The sum of the effective numbers of the two premises is greater than the whole number of exist-
ing cases of the middle term. And the excess of that sum above the number of cases of the middle
term is the number of the cases in the affirmative premiss which are the subjects of inference. Thus,
if there be 100 Ys, and we can say that each of 50 Xs is one or other of 80 Ys, and that no one of
20 Zs is any one of 60 Ys;—the effective numbers are 50 and 60. And 50 + 60 exceeding 100 by
10, there are 10 A’s of which we may affirm that no one of them is any one of the 20 Zs mentioned.

The following brief summary will enable the reader to observe the complete deduction of all the
Aristotelian forms, and the various modes of inference from specific particulars, of which a short
account has already been given.

Let a be the whole number of 4s; and ¢ the number specified in the premiss. Let ¢ be the
whole number of Zs; and w the number specified in the premiss. Let 6 be the whole number of
Ys; and w and » the numbers specified in the premises of # and x. Let X,Y,, denote that each of
+ Xs is affirmed to be one out of w Ys: and X,: Y,, that each of ¢ Xs is denied to be any one out
of u Ys. Let X,,, signify m Xs taken out of a larger specified number m: and so on. Then the
five possible syllogisms, on the condition that no contraries are to enter either premises or conclusion,
are as follows :—

1. DG) =f ZX, ae Ga Z,, = RTS elite
2 XY + Vol, =Xisv ie Zo= Zreo-b wre
oy eYOAG FZ, = eat Z, = Lain eek wt

4. ~ X,Y, + Z,:¥,= x,

{

|

AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 407

The condition of inference expresses itself; in the X,,, of the conclusion, m must neither be
0 nor negative. The first case gives no Aristotelian syllogism; the middle term never entering
universally (of necessity) into any of its forms, under any degree of specification which the usual
modes of speaking allow. The other cases divide the old syllogisms among themselves in the fol-
lowing manner: they are written so as to show that there is sometimes a little difference of amount
of specification between the results of different figures, which amount may change in the reduction
from one figure to another. The Roman numerals mark the figures.

2 Sth OS ty Y)Z,+ X)Y,=X)Z,, | Barbara I.
Sih Dali XG) Ya) Ze = Zane Bramantip IV.
t<a, v=b WZ AGS = AG 7, Darii 1.
tei nO A,Y, + V)Z,= Ze x, Dimaris 1V.

3. T1058 =D Y)X,+ Y) Z, = Z,, Xu, Darapti ITI.

u<b, v=6b LAC BD GA Aa Ge, Disamis III.
w=b,v<b Wy) xp YZ, = ZX: Datisi IIL.

4. t=a, v=b, w=c VRsZ, eA) a AG Zi Celarent I.
Si Dei WS TL EVAAS PQ) Cn PCG A, Cesare II.
2ST, DSlin DSO 2Q) DC RORA RD Ce CA AG Camestres II.
pS YSly, Dae XO) VEeE OY ZS) LO Camenes IV.

v=b, w=ec VOSA A DG) G Ay Z Ferio I.
DSslipn DS G Z.Y + XY, A: Z Festino II.
t=a, v=b, DENA SEL) AES ALE OG Baroko M1.

5. u=b, v=6b, w=c Y.Z24+Y)X,= X,,:2 Felapton III.
uw=b, ¥=b, w=Ce Zin Va Va) XG = 2X Zi Fesapo IV.

sete 0 Y.Z+ V,X, = X,,:Z Feriso III.
v=b, w=c Z.Y + ¥,X& =X, 22 Fresison IV.
u=b, w=c Y,:24+Y)X,= X,,:Z Bokardo I11.

This system is complete in itself, if contraries be excluded. That in the body of this paper is
also complete, if all specification be excluded, except which is contained in the usual words some and
all. An attempt to combine the two systems would be useless, because its forms of expression
would not be those of common language. For instance, the following must be one form of an
affirmative proposition in the combined system ‘Of ¢ Xs and ¢ ws every one is one or other
of u Ys and w' ys.” It would be useless to investigate the conditions of inference as to forms
which are not those of speech in any language.

But at the same time there is a certain approach to the preceding forms, if we take in not
merely the logical force of our common propositions, but also what is usually implied. He who
says, “Some Xs are Ys,” is generally held to mean that the other X’s are not Ys. ‘The complex
syllogisms, in which the alternatives left by the common forms are supposed to be definitely settled,
are worthy of attention: and their theory is as follows.

With respect to the name Y, the name X may be of seven different kinds, distinguishable with-
out numerical specification. These are as follows: neither term containing the whole universe.

408 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM, ETC.

(1.) The two terms may be identical, or Y) X and X)Y. Let this be denoted by D.
Taking the order XY, we have, to constitute D, the proposition A, a. And denoting coexistence by
+, as before, we may write D = 4 + a.

(2.) X may be entirely contained in, but not repletive of, Y, or we may have X ) Y and Y: X.
Let X be now called a subidentical of Y, and let D, denote this form. We have then D = 4 +o.

(3.) X may entirely contain Y, and more; or Y) X and X:Y. Let X be now called a
superidentical of Y, and let D’ denote this form. We have then D' =a + O.

(4.) & may be the contrary of Y, both together filling up the universe of the proposition with-
out anything in common; or X.Y and w.y, Let this form be called C: we have then C= E +e.

(5.) X and Y may have nothing in-common, but may not together fill up the universe of the
proposition; or X.Y and wy. Let them be called subcontraries, and let C, denote this form.
We have then C,= E +i.

(6.) Xand Y may have something in common, and may together fill up the universe; or XY
and w.y. Let these be called swpercontraries, and let C’ denote the form. We have then C’ = e + J.

(7.) Each of the two may have something in common with the other and something not in
common, both together not filling up the universe; or XY, wy, X:¥, Y: X. I cannot propose
any name for this case with which I am in any degree satisfied: but as all the particular forms
are here concerned, I will for the present call X and Y in this case complete particulars each of the
other. Let P represent this form; we have then P=J+O+i+0.

In arranging for a syllogism, let the order be XY, ZY, XZ, the conclusion being described by
what X is as related to Z, X coming from the first premiss ; and both terms of the conclusion being
described with respect to the middle term, Y. On examining the cases in which complete premises
give a complete conclusion, I find as follows.

1. If one of the concluding terms be a complete particular of the middle term, there is no
complete conclusion except when the other concluding term is either identical with or contrary to
the middle term. And then each concluding term is a complete particular of the other.

2. The following table shows the result of all the other cases.

Z

po

D. D D' C, Cc Cc

(DN (CCL. D: D, GC G6) DD’)
D| D D D Cc Cc C

oy | D; D D' (C,C) | (D,D) C Cc’

ac, c_ | @p)| (cc) | D D,
GA ie Cc C D' D D,
¢\(DD)| Cc Cc’ D’ Do. ACO)

This is a table of double entry, in which from the description of X and Z with respect to Y, we
see set down that of X with respect to Z, when one can be affirmed: and, when nothing can be affirm-
ed, all that can be denied, in parentheses. Thus, if X be a supercontrary of Y, and Za subcontrary,
X must be a superidentical of Z. But if X and Z be both subidenticals of Y, it may be denied that
& is either the contrary or a supercontrary of Z.

I will not lengthen this addition by putting down in words all the rules which are expressed in
the preceding table.

A. DE MORGAN.
University College, London,
February 27, 1847.

XXX. Supplement to a Memoir On some Cases of Fluid Motion. By
Grorce G. Sroxes, M.A., Fellow of Pembroke College.

[Read Nov. 3, 1846.]

In a memoir which the Society did me the honour to publish in their Transactions*, I showed that
when a box whose interior is of the form of a rectangular parallelepiped is filled with fluid and made
to perform small oscillations the motion of the box will be the same as if the fluid were replaced by a
solid having the same mass, centre of gravity, and principal axes as the solidified fluid, but different
moments of inertia about those axes. The box is supposed to be closed on all sides, and it is also
supposed that the box itself and the fluid within it were both at rest at the beginning of the motion.
The investigation was founded upon the ordinary equations of Hydrodynamics, which depend upon
the hypothesis of the absence of any tangential force exerted between two adjacent portions of a fluid
in motion, an hypothesis which entails as a necessary consequence the equality of pressure in all
directions. The particular case of motion under consideration appears to be of some importance,
because it affords an accurate means of comparing with experiment the common theory of fluid
motion, which depends upon the hypothesis just mentioned. In my former paper, I gave a series
by means of which the numerical values of the principal moments of the solid which may be substi-
tuted for the fluid might be calculated with facility. The present supplement contains a different
series for the same purpose, which is more easy of numerical calculation than the former. The com-
parison of the two series may also be of some interest in an analytical point of view, since they appear
under very different forms. I have taken the present opportunity of mentioning the results of some
experiments which I have performed on the oscillations of a box, such as that under consideration.
The experiments were not performed with sufficient accuracy to entitle them to be described in
detail.

The calculation of the motion of fluid in a rectangular box is given in the 13th article of my
former paper. I shall not however in the first instance restrict myself to a rectangular parallelepiped,
since the simplification which I am about to give applies more generally. Suppose then the problem
to be solved to be the following. A vessel whose interior surface is composed of any cylindrical
surface and of two planes perpendicular to the generating lines of the cylinder is filled with a homo-
geneous, incompressible fluid; the vessel and the fluid within it having been at first at rest, the
former is then moved in any manner; required to determine the motion of the fluid at any instant,

‘supposing that at that instant the vessel has no motion of rotation about an axis parallel to the gene-
pposing P

rating lines of the cylinder.

I shall adopt the notation of my former paper. w, v, w are the resolved parts of the velocity at

“any point along the rectangular axes of , y, x. Since the motion begins from rest we shall have

udw + vdy + wdzx an exact differential dp. Let the rectangular axes to which the fluid is referred

* Vol. VIII, Part I., p. 105.
Vou. VIII. Part III. 3G

410 Mr. STOKESS SUPPLEMENT TO A MEMOIR

be fixed relatively to the vessel, and let the axis of x be parallel to the generating lines of the cylin-
drical surface. The instantaneous motion of the vessel may be decomposed into a motion of transla-
tion, and two motions of rotation about the axes of y and x respectively; for by hypothesis there is
no motion of rotation about the axis of w. According to the principles of my former paper, the
instantaneous motion of the fluid will be the same as if it had been produced directly by impact, the
impact being such as to give the vessel the velocity which it has at the instant considered. We may
also consider separately the motion of translation of the vessel, and each of the motions of rotation ;
the actual motion of the fluid will be compounded of those which correspond to each of the separate
motions of the vessel. For my present purpose it will be sufficient to consider one of the motions of
rotation, that which takes place round the axis of = for instance. Let w be the angular velocity
about the axis of x, w being considered positive when the vessel turns from the axis of w to that of
y. It is easy to see that the instantaneous motion of the cylindrical surface is such as not to alter
the volume of the interior of the vessel, supposing the plane ends fixed, and that the same is true of
the instantaneous motion of the ends. Consequently we may consider separately the motion of the
fluid due to the motion of the cylindrical surface, and to that of the ends. Let @, be the part of

due to the motion of the cylindrical surface, @, the part due to the motion of the ends. Then we
shall have

P= p, + de CO mee wrwebedivnesseccccesecceses (1). a

Consider now the motion corresponding to a value of @, way. It will be observed that way
satisfies the equation, ((36) of my former paper,) which @ is to satisfy. Corresponding to this
value of @ we have

U=oY, VEoOr, WH=

Hence the velocity, corresponding to this motion, of a particle of fluid in contact with the cylindrical
surface of the vessel, resolved in a direction perpendicular to the surface, is the same as the velocity
of the surface itself resolved in the same direction, and therefore the fluid does not penetrate into,
nor separate from the cylindrical surface. The velocity of a particle in contact with either of the
plane ends, resolved in a direction perpendicular to the surface, is equal and opposite to the velocity
of the surface itself resolved in the same direction. Hence we shall get the complete value of @ by
adding the part already found, namely way, to twice the part due to the motion of the plane
ends. We have therefore,

p= way + 2H, =2p, —way, bY (1)---ceeeersenveeeeeees (2),
ADO Oe it OU ne niniebie mana wns telcesonSa= 6s<56 aoe (3).

Hence whenever either @, or @, can be found, the complete solution of the problem will be
given by (2). And even when both these functions can be obtained independently, (2) will enable
us to dispense with the use of one of them, and (3) will give a relation between them. In this case
(8) will express a theorem in pure analysis, a theorem which will sometimes be very curious, since —
the analytical expressions for @, and @, will generally be totally different in form. The problem —
admits of solution in the case of a circular cylinder terminated by planes perpendicular to its axis, —
and in the case of a rectangular parallelepiped. In the former case, the numerical calculation of the
moments of inertia of the solid by which the fluid may be replaced would probably be troublesome, — mG
in the latter it is extremely easy. I proceed to consider this case in particular. 7

Let the rectangular axes to which the fluid is referred coincide with three adjacent edges of the
parallelepiped, and let a, 6, c be the lengths of the edges. The motion which it is proposed to cal-

ite

ON SOME CASES OF FLUID MOTION. 411

culate is that which arises from a motion of rotation of the box about an axis parallel to that of z
and passing through the centre of the parallelepiped. Consequently in applying (2) we must for a
moment conceive the axis of x to pass through the centre of the parallelepiped, and then transfer the

a : a b
origin to the corner, and we must therefore write (« ~ 5) (y ~ 5) for way. In the present case
2 2
the cylindrical surface consists of the four faces which are parallel to the axis of w, and the remain-
ing faces form the plane ends. The motion of the face wy and the opposite face has evidently no
effect on the fluid, so that @, will be the part of @ due to the motion of the face wx and the opposite
face. The value of this quantity is given near the top of page 133 in my former paper. We have
then by the second of the formule (2)

nob ny nob nary

nee Ca De Ge aes Naa a b
=o ra nab nab cos a a (« = 5) (y ia ;) DODO (4),

p= = yee

the sign =, denoting the sum corresponding to all odd integral values of m from 1 to 0. This
value of @ expresses completely the motion of the fluid due to a motion of rotation of the box about
an axis parallel to that of x, and passing through the centre of its interior.

Suppose now the motion to be very small, so that the square of the velocity may be neglected.

Then, p denoting the part of the pressure due to the motion, we shall have p = — wee Also
ae

dt

a SUE rCe the axes to be fixed in space, since by taking account of their
motion we should only introduce terms depending on the square of the velocity. In fact, if for
the sake of distinction we denote the co-ordinates of a fluid particle referred to the moveable axes by

a’, y', while a, y denote its co-ordinates referred to axes fixed in space, which after differentiation with
respect to ¢ we may suppose to coincide with the moveable axes at the instant considered, and if we

in finding

t d
denote the differential coefficient of @ with respect to f¢ by (<) when w, y, ¢ are the independent

variables, and by = when w’, y’, ¢ are the independent variables, we shall have

dp\ dp dpdx dpdy d te
() -2 I LACLEDE CNL

Cit de at Sug ae We ae as
d d d d
for : s mean absolutely the same as a a and are therefore equal to w, v respectively.
da’ dy 4 : Sy ‘
Now me aa? depending on the motion of the axes, are small quantities of the order w; their

values are in fact wy, — wa; so that, omitting small quantities of the order w*, we have

d
We shall therefore find the value of p from that of @ by merely writing — p > for w. In order

* It may be very easily proved by means of this equation, | effect on the motion of the box as the solid of which the moment
combined with the general equation which determines p, that | of inertia is determined in this paper on the supposition that the
whether the velocity be great or small the fluid will have the same | motion is small,

8G2

412 Mr. STOKES’S SUPPLEMENT TO A MEMOIR

to determine the motion of the box it will be necessary to find the resultant of the fluid pressures on
its several faces. As shown in my former paper, these pressures will have no resultant force, but
only a resultant couple, of which the axis will evidently be parallel to that of x. In calculating
this couple, it is immaterial whether we take the moments about the axis of x, or about a line
parallel to it passing through the centre of the parallelepiped: suppose that we adopt the latter
plan. If we reckon the couple positive when it tends to turn the box from the axis of « to that

of y we shall evidently have — de i Py=0 (2 - 5) dadzx for the part arising from the pressure on

e b Xe
the face vx, and he pe, (y — 5) dydzx for the part arising from the pressure on the face yz.

It is easily seen from (4) that p,_, = —p,-,, and p,_, = — P,.s So that the couples due to the pres-
sures on the faces vx, yx are equal to the couples due to the pressures on the opposite faces
respectively. In order, therefore, to find the whole couple we have only got to double the part
already found. As the integrations do not present the slightest difficulty, it will be sufficient to

write down the result. It will be found that the whole couple is equal to ——s where

64pa'e_ 11l—e *
2 Gab rtetereesreees (5).

b
c= P30) +

5

This expression has been simplified after integration by putting for =, — its value =
n 9

It appears then that the effect of the inertia of the fluid is to increase the moment of inertia
of the box about an axis passing through its centre and parallel to the edge c by the quantity C.
In equation (40) of my former paper, there is given an expression for C which is apparently very
different from that given by (5), but the numerical values of the two expressions are necessarily
the same. If we denote the moment of inertia of the fluid supposed to be solidified by C, we

shall have C, = ie +6*); and if we put

Cc
i =?1, G. =f),
and treat (5) as equation (40) of my former paper was treated, we shall find
1
f(r) = (1 + 7°) 7? $1 — 87° + 27° (1.260497 — 1.254821 E, — versin 20,)} ......+6. (6),
ae
n
where, tab. log tan 8, = 10 — .6821882 — .
i:

The equation (6) is true, (except as regards the decimals omitted,) whatever be the value of 7;
but for convenience of calculation it will be proper to take 7 less than 1, that is, to choose for a
the smaller of the two a, b. The value of f(r) given by (6) is apparently very different from
that given at the bottom of page 134 of my former paper, but any one may easily satisfy himself
as to the equivalence of the two expressions by assigning to r a value at random, and calculating
the value of f(r) from the two expressions separately. The expression (6) is however preferable to
the other, especially when we have to calculate the value of f(r) for small values of r. The
infinite series contained in (6) converges with such rapidity that in the most unfavourable case, that
is, when r=1 nearly, the omission of all terms after the first would only introduce an error
of about .000003 in the value of f(r).

ON SOME CASES OF FLUID MOTION. 413

For the sake of showing the manner in which f(r) alters with 7, I have calculated the following
values of the function. The expression (6) shows that f’(r) =0, when r=0; and f’(r) is

also =0 when r =1, since f (-) = jG)

The experiments to which I have alluded were made with a wooden box measuring inside §
inches by 4 square. The box weighed not quite 11b., and contained about 4}|bs. of water, so
that the inertia of the water which had to be overcome was by no means small compared with that
of the box. The box was suspended by two parallel threads 3 inches apart and between 4 and
5 feet long: it was twisted a little, and then left to itself, so that it oscillated about a vertical
axis midway between the threads. The points of attachment of the threads were in a line drawn
through the centre of the upper face parallel to one of its sides, and were equidistant from the
centre. The weight of the box when empty, the length and distance of the threads, the time
of oscillation, and the known length of the seconds’ pendulum are data sufficient for determining
the moment of inertia of the box about a vertical axis passing through its centre. When the
box is filled with water the same quantities determine the moment of inertia of the box and the
water it contains, whence the moment of inertia of the water alone is obtained by subtraction. It
is suposed here that the centre of gravity of the box coincides with the centre of gravity of its
interior volume. In the following experiments a different face of the box was uppermost each
time. In Nos. 1 and 2 the long edges of the box were vertical, in Nos. 3 and 4 they were hori-
zontal. In all cases the inertia determined by experiment was a little greater than that resulting
from theory: the difference will be given in fractional parts of the latter. The difference was

ye. Do: i. eee f
a in No. 1, = in No. 2, fe in No. 3, and aa in No. 4. On referring to the table at the end

of the last paragraph, it will be seen that the ratio of the moment of inertia of the fluid to what it
would be if the fluid were solid is about three times as great in the last two experiments as in the
first two.

I had expected beforehand to find the inertia determined by experiment a little greater than
that given by theory, for this reason. In the theory, it is supposed that both the fluid itself and
the surface of the box are perfectly smooth. This however is not strictly true. The box by its
q roughness exerts a tangential force on the fluid immediately in contact with it, and this force
_ produces an effect on the fluid at a small distance from the surface of the box, in consequence of
_ the internal friction of the fluid itself. We may conceive the effect of this force on the time of
_ oscillation in a general way by supposing a thin film of fluid close to the surface of the box to be
_ dragged along with it. Consequently, the moment of inertia determined by experiment will be a

little greater than it would have been had the fluid and the surface of the box been perfectly
smooth.

A414 Mr. STOKES’S SUPPLEMENT TO A MEMOIR, ETC.

These experiments are sufficient to show that in the case of a vessel of about the size and shape
of the one I used, filled with water, and performing small oscillations of the duration of about
one second, (as was the case in my experiments,) the time of oscillation is not much increased by
friction; at least, if we suppose, as there is reason for supposing, that the effect of friction does
not depend on the nature of the surface of the box. They are not however sufficiently exact to
allow us to place any reliance on the accuracy of the small differences between the results of
experiment, and of the common theory of fluid motion, and consequently they are useless as tests
of any theory of friction,

7

G. G. STOKES.

TRANSACTIONS

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

VotumeE VIII. Part IV.

CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS;
AND SOLD BY
JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. & J. J. DEIGHTON ; AND MACMILLAN & CO., CAMBRIDGE.

M.DCCC.XLVII,

XXXI. Ox a New Notation for expressing various Conditions and Equations
in Geometry, Mechanics, and Astronomy. By the Rev. M. O'Brien, late
Fellow of Caius College, Professor of Natural Philosophy and Astronomy
in King’s College, London.

[Read November 23, 1846.]

Tue notation Dw’. uw, the meaning and use of which is explained in the following pages, denotes
a line of a certain length perpendicular to the lines denoted by the symbols wu and w’. It is derived
from the consideration of the rotation of a rigid body, in which the line w is fixed, about the line u,
being, in fact, the differential coefficient of « with respect to the directions of the axes of co-ordi-
nates, the line w’ being constant, as will be explained.

It will be found, that this notation and a corresponding notation, Aw’.w, have several
important properties, that they express with great simplicity several conditions and equations in
various parts of Mathematics, and especially in Mechanics, and that they simplify in a remarkable
manner several complicated investigations.

The present paper contains an explanation of the meaning of the notation, and its application to
Statics, and to the determination of the Rotation of a rigid body about its centre of gravity.

Of the Notation Du’. u.

1. Let us assume the symbols a, 3, -y to denote the lines OA, OB, OC, each a unit of
length, drawn from an origin O at right angles to each other, so forming a
system of three rectangular axes. Let a, y, x denote any three abstract
numbers; then wa, yB, xy will denote three lines, drawn along (or parallel
to) the three axes, and numerically equal to w, y, x respectively.

Let OP be any line drawn from O, and let us assume the symbol w to
denote OP in magnitude and direction; then, if wa, yB, xy be the co-ordinates
of P, we have, according to well-known principles,

U=BA+ YB + By .-..oece0e0s (1).

(Fig. 1.)

Cc

2
ve Zz
VA
xn /
/»
We shall now suppose that the axes OA, OB, OC are capable of motion
about the point O, always however remaining at right angles to each other.
We shall also suppose that a, y, x are not affected by this motion, or, in other words, that the
position of P relatively to OA, OB, OC, does not alter. In fact, we assume that the point P and
the axes OA, OB, OC are fixed in a rigid body which is capable of motion about the point O.
Let 6 denote any indefinitely small displacement arising from a motion of this kind; then from
(1) we have

B

OU = WO + YOR + BOY vosscccocscccrecoseecsonees (2).

Now, since a is invariable in length, da denotes a displacement of the point A at right angles to
OA: for, let OA’ be the line denoted by a+0a; then, since

OA = OA + AA’, we have a + 6a=a+ Ad’, and therefore (iia Se ues

ba= AA’. But, since OA’ = OA (a being invariable in length), 0 ——————____ |

and since the angle O is indefinitely small, 44’ is perpendicular to = -->______|

OA. Hence da denotes a displacement of A at right angles to OA. #
Vor. VIII. Parr IV. 3H

416 Mr. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS

In like manner 58 and dy denote displacements of B and C at right angles to OB and OC
respectively.

2. Let the displacement da be resolved into two others, (Fig. 3.)

Ac and Ab’, of which Ac is parallel to OB, and Ab’ to CO. ‘
In like manner let 3 be resolved into Ba parallel to OC, and
Bc’ to AO; and let dy be resolved into Cb parallel to OA, and
Ca to BO.

Also let us denote the numerical magnitudes of these re-
solved displacements by e, b’, a, c’, b, a’, respectively.

Then, since 0.4, OB, OC always remain at right angles to
each other, it is evident that a =a’, b=0', andc=c'. Hence,
giving these displacements their proper signs of direction,
namely 8, —y; ; — 4 4 — 3, respectively, we have,

da =Ac+ Ab'= Be - yb
SB = Bat Bel = ya—ac [rrr ers res tees ee eee cee ees (3).
dy = Cb + Ca’ =ab - Ba

The quantities a, 6, c here denote any arbitrary numerical differentials.

Making these substitutions in equation (2), we find,
du = (2b -ye)at (we — xa) B + (ya — wh) y wc. eee ee (4).

3. Now it is evident from the nature of the motion which 6 denotes, that dw represents an
indefinitely small line at right angles to w; therefore, if X be any numerical arbitrary quantity, du
will represent any line (not necessarily small) at right angles to w. The sign \0 therefore, written
before w, changes w into the symbol of a line at right angles to w, and therefore has somewhat the
same effect as the sign Wz 1, or (—)3. Since however there may be an infinite number of
different perpendiculars to w, it remains to put the sign Xd in such a form as shall indicate
what particular perpendicular Xdw represents. We shall do this in the following manner.

4. Multiplying (4) by X, and putting Aa = a’, Ab= y’, Ac = &’, we find
Adu = (xy — 8'y) a + (ax' — a's) B + (ya - ya) y...... (5).

Now it is evident from this expression, that \Ow vanishes when w =a’, y=y', * = %'; in other
words, if we assume
, , , ,
u=Wat+y B + xy;
it follows, that Adu = 0, when w = wu’. Therefore \dw denotes a differential* of w taken on the sup-
position that zw’ is invariable.

On this account we shall replace 6 by the sign D,, defining D,, to denote a differential taken
on the supposition that uw is invariable. We have then,

Du = (xy'— s'y) a+ (v2 - a's) B+ (ya'-y'2) y.
If we interchange a, y, x, and a’, y’, x’ respectively, this equation becomes

Dw = (xy — xy) a + (a's — wz’) B + (ye - yo')y.
Hence we find, that
Du = — Du.
From this equation we may shew that the operation D,, is distributive with respect to uw; that
is to say, that

* Meaning by the word differential here any quantity proportional to an indefinitely small difference.

IN GEOMETRY, MECHANICS, AND ASTRONOMY. 417

Dy ye @W) = Dyuw + Du;
for we have

, ”
Dy, ¢(u) = — D, (v'+ w”),
, ”

- Dw—- Du
Du + Dy.

The operation D,, is therefore distributive with respect to w’.

To indicate that D,, is distributive with respect to w’, we shall elevate the subscript index w’,
and write it in the same line as D, putting a dot between w’ and the symbol on which the operation
is performed ; that is to say, we shall write

Du'.u instead of Dw.

5. Having thus settled the form of the notation, we shall now interpret the meaning of the
expression for Dw’. u, namely,

Du.u = (xy'— #y) a + (w2'—- ax) B + (ya'— ya) y ... 2-202 (6)5
from which, as we have seen, immediately follow the two equations
Dit = — Dit. Bir cassscescepltteess saaceu Cl)

D(w'+u").w= Dw.u + Du’.u......... (8).
ist. To determine the direction of the line Dw’. w, let
Du.usvat+y B+2,y
and therefore, by (6),
w= xy 'y
Y= UB — UR) occeseoanenceeee---s (9):
s ,
z= Yu —Yyu
From these equations we have immediately
@e+yyt+s2,s=0,
vey y+ #8 = 0.
Whence it appears that the line drawn to the point (w,y x) from O, is at right angles to the line
drawn to (wy) and the line drawn to ('y'x’); in other words, Dw’. w is at right angles both to w
and w. This determines the direction of the line Dw’. wu.

andly. To determine the magnitude of Du’.w, let r,, 7, and r’ denote the magnitudes of Dw’. w,
u, and w’ respectively, and let be the angle made by w and w’: then, by the equations (9),

2 2 et RES 2 2 2 2 f2 SEN) = ary’ art ye o’\2
Or ty? +x? = (a? + yt 2°) (vw? t+ y? + 2) -— (wat yy + ee),
or 7? = 74"? — (rr’ cos @)*,

and therefore r,= 77" sin @ .........+0. Bare ez (10):

Hence the numerical value of the line Dw’. uv is the product of the numerical values of the lines
u and 7 multiplied by the sine of the angle they make with each other.

6. Since r7’sin@ is the area of the parallelogram formed upon the lines w and wu’ as sides, it
follows, that Dw’. w is a line numerically equal to the area of the parallelogram formed upon w and
w, and perpendicular to its plane.

It follows from (7) that Dw.w’ denotes a line equal in magnitude to Dw’.u, but opposite in
direction.

8H2

418 Mr. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS

7. If w be any numerical quantity, we have

Daw.u=-Du.pw = -—pDu.w=pDu.u.
Hence we have
Date UA ap accasccoesesaiees<s-i(Ll):

From which it appears that a numerical coefficient of w’ may always be brought outside the
sign D.
Hence Du'.w= D(w'a+ y'B+3'y).u
= Da'a.u+ Dy B.u+ D2'y.u, by (8);
and therefore by (11) Du’.w=a@Da.u+ YDB. wt 2 Dy. t recceeeereee (12).
8. In the equation (6) putting all the co-ordinates, except # and y, equal to zero, we find

Da'a. yf = wyy, and .. Da. =-y: and in the same way we may shew that DB.y =a, and
Dy.a=[3. We have therefore

Da.B=y, DB.y=a, Dy.a= Bowes (13).
From these equations we find by (7),
DBra——1,) Dy G—— a, Dany = —B.-c..,e0 (14).

Also we evidently have,

DARE 10 Woaereratae stains eieicivte's aie's)s ose (15).
And therefore

Daisae=205 DB. B= 0, IDiryliry ='0\snscencas (16)s*

9. Du'.u is a line proportional to, and drawn in the same direction as the small displacement
ou, which displacement takes place on the supposition that w’ is invariable: in other words, the dis-
placement dw results from giving a small angular motion, round the axis w’, to the rigid body in
which 0.4, OB, OC and P are fixed. From this consideration we may easily see that Dw’.w is at
right angles to w’ and w, and is proportional to 7 sin 6*.

It is plain from figure (3), that the rotation by which the displacement dw is generated is right-
handed, supposing that we look along the axis of rotation (w’) towards the origin. We may say,
therefore, that Dw’. w is generated by right-handed rotation round the axis w’.

10. Since Da.B=vy, and Da.y = - Bs it follows that (Da)*. 8 = — 3: and in ite same
way we may shew that (Da)*. y = — y; but, since Da.a=0, we have (Da)’. a = 0, instead of — a.
Hence (Da)’ written before 3 or y is equivalent to the sign —, and therefore Da. is equivalent
to the sign (—)4, or \/—1; but this is not true of Da. written before a. Similar remarks may
be made respecting D3, and D+.

In general, we may see from what has been said above, that (Dw’)*. w= — w when the numeri-
cal value of w’ is unity, and w’ is perpendicular to w: in this case, therefore, Dw’. is equivalent to
(=i, or 4/1,

In this case, therefore, a line numerically equal to ~, drawn at an angle @ to w, and at right
angles to w’, is expressed by the formula

ucos@ + (Dw’. uw) sin@, or 6° ?"u.

11. When two ore more of the symbols Da., D., Dry. come together, the order in

which they are written must not be changed: thus DB.Da.B=a, but Da.DB.B =0.

* The ratio of Du'.u to rsin @ is arbitrary; we may therefore assume it to be 7, and then we have Du’.u=rr'sin§, This
is equivalent to the assumption that, Aa=a", Ab=y', Ae=2', in Article 4.

IN GEOMETRY, MECHANICS, AND ASTRONOMY. 419

Of the Notation Auv'.u.

12. In obtaining the notation Dw’. u we supposed the axes a, B, y to be varied in position,
but not in length, always remaining at right angles to each other; we shall now obtain another
notation by supposing the axes to undergo a different kind of variation.

Let 6 denote any variation (whether in length or position) of the axes a, B, y, vyz being sup-
posed invariable: then

éu=xdat yop ate voy.
Let us assume that
da=adh, SB=y'dh, Sy = 20h,
where 6h is a small displacement in the direction of the line w’, or va +y/B + z'y.

Thus we have
Ou = (wa'+ yy'+ x2’) oh.

wu’ + yy + xx’ is therefore the differential coefficient of w, when the axes a, , ry suffer the vari-
ations adh, y'dh, x'dh respectively, i.e. when the points 4, B, C (fig 1.) receive displacements
proportional to a’, y’, x’ respectively in the direction of the line u’, We may therefore represent
this differential coefficient by the notation A,,w, since the magnitude and direction of the variation
of w depends upon w’, or is, so to speak, a function of uw’. We have therefore

Ay = aa + yy + 82.

It is evident from this expression that we may interchange w and w’. Also the operation
A, is clearly distributive, and we shall therefore, as before, write Aw'.w instead of A,,w. Hence
we have,

AU. h = BG + YY + BR rar reccrycececagecers (17),
or Aw.u=rr cos@’...... ag enets saree Saiecisissc (18).
(Nad te: NWSW oo oeee sess ds vsnetiveadsvee (19)5

and A (w+ wu") .w= Aw. + Aw’. U .cccccescvccerers (20):

13. The following formule are also evident, namely,
INR CV encnanonononatnocbono i Ca).
If w be at right angles to w, then
NU Oreasent traced Micntecteen (22).
Hence it follows that, whatever w’ be,
Ns (Oe) Olga ies sesso oes (23).
14, We may express wyx and w by the following formule,
Aa.u=a, AB.u=y, Ay.w=8 10, (24),
u=aha.u+ BAB.Uw+yAy, bere (25).
(25) may be expressed by saying that
ahat+BAB+yAy=1.

420 _ Mr. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS

15. Hence we may easily shew that
Aa.(DB.u)=Ay.u, AB.(Dy.u) =Aa.u, &e. &e.,
or, omitting 2,
Nae) Sr Nevins ANG Dy — Nees Ay.Da.= AB.
AB.Da.=-Ay., Ay.DB.=-Aa., Aa. Dy.=- AB.
Also [or from (23)] it follows, -that
Aa.Da.=0, AB.DB.=0, Ay. Dy.=0... (27).

} +0).

16. It is easy to see that the displacement which gives rise to the differential coefficient
Aw'.u, is caused by a uniform expansion of the rigid body (in which the axes and the point P are
fixed) in the direction of the line w’, the modulus of expansion being proportional to the numerical
magnitude of w’. That plane containing the origin which is perpendicular to w’ is unaffected by
this expansion.

Instances of the application of the Notation Dw.u and Au.u to Statics.
17. The expression w, or

wat yp + xy,
determines completely the position of the point P; on this account we
shall call « the symbol of the point P.

In like manner, if X, Y, Z be the three components of any force,
and if

(Fig. 4.)

U=Xa+YVB+ Zy,

U is the symbolical expression for the force, representing it com-
pletely in magnitude and direction. We shal] therefore call U the
symbol of the force whose components are X, Y, and Z.

For brevity we shall generally say, ‘the force U” instead of, “the
force whose symbol is U ;”and, in like manner, “the point u,” instead
of, “ the point whose symbol is u.”

(I).

18. If the forces U, U’, U", &c. keep a rigid body at rest, the six equations of equilibrium
are contained in the following equations, viz.

22 LOLS: ORCA CPOE PECTIC EOL DADC e COPED CTO EOOEDOOCER C3) 5
PED PUNO. cronies aslewnisissioeiuth iri lasiunnieiaitian eer C9)s
For TU=alX+ PB rY + y=Z,
and therefore (28) is equivalent to the three equations
2X50, S¥=0, SZ =0.
Again, by equation (6) we have,

Du.U=(Zy - Ys)a+ (Xx - Za) B+ (Va- Xy) y:

and therefore (29) is equivalent to the three equations
=(Zy-— Ys) =0, S(X¥s-Ze)=0, T(Ya—- Xy)=0.

- —————

IN GEOMETRY, MECHANICS, AND ASTRONOMY. 42)

(II).

19. ‘To deduce the equations (28) and (29) immediately from the parallelogram of forces.
We must premise the following Lemmas.

20. Lemmai. If w and w’ be any two points situated on the line of direction of the force
U, then Du. U= Dw’. U.

For the line («’— w)* coincides in direction with the line U (forces being supposed to be
represented by lines) ; and it is therefore evident from Art. 5, that D(u’-— u).U=0, i.e. Du’.U
= DIB SOR

21. Lemma 2. If three forces P, Q, R, applied to a rigid body at the points p, g, 7 respec-
tively, balance each other, then the conditions of equilibrium are

Pt Qe 0) ssvcecizcs sess consists (30).
DPT PIA ED 1Q DT ie =O) seesecese =e (31).

For P, Q, and R must meet in the same point; let w be that point: also — R must be the
resultant of P and Q, and therefore, expressing the parallelogram of forces symbolically, we have,
—-R=P+Q, or,

P+Q+R=0.
Now performing the operation Dw. on this equation, we have

Du.P+Du.Q+ Du. k=0,
and therefore, by Lemma 1,

Dp.P+Dq.Q+Dr.R=0.

Hence the conditions (30) and (31) must hold if P, Q, and R balance each other.

And, conversely, if (30) and (31) be true, the forces P, Q, and R will balance each other. For
let « be the point of intersection of P and Q; then, by Lemma 1, we have Du. P= Dp. P, and
Du.Q=Dq.Q; and therefore by (31), we have

Dr. Rk =—- Du.(P + Q)=Du.R, by (380).

Hence D (7 — u).R =0, and therefore the line r — w coincides with R in direction, i.e. w is a
point in the line of direction of #. Hence P, Q, and R meet in the same point wv. Also by (30),
-R=P+Q, ie. — R is the resultant of P and Q. Hence P, Q, and R balance each other if
the conditions (30) and (31) be satisfied. These conditions therefore are necessary and sufficient for
equilibrium.

22. From these Lemmas we may now prove that the equations (28) and (29) are the neces-
sary and sufficient conditions of equilibrium of a rigid body, acted upon by the forces U, U’, U",
&e. at the points wu, w’, w’, &c.

Choose any three points}, p, q, 7, in the rigid body; resolve U into three forces acting along
the lines wu -— p, w-—q, u-—7, (i.e. the lines drawn from p, gq, and 7 to w); let P, Q, R denote
these forces respectively ; in like manner resolve U’ into P’, Q’, R', acting respectively along the
lines w'— p, u'—q, w—r: treat U" similarly, and so on.

Then the forces U, U', U", &c. are reduced to the three sets of forces,

P, P’, P", &e. acting at the point p,
nO cee
Tee Te AR Cae Cann oi cons ca'csaees r,

* (w'—u) expresses in magnitude and direction the line drawn from the point u to the point u’.
+ These points are supposed not to lie in the same right line.

4292 Mr. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS

And, by the parallelogram (or rather, the polygon) of forces, these are equivalent to the three
forces,
=P at p, SQ atq, TH at r.
Hence the conditions of equilibrium of these three forces are the conditions of equilibrium of the
forces U, U', U", &e. Therefore, by Lemma , the conditions of equilibrium of the forces U, U’;
U", &e. are

2P+2Q+2R=0............. (32).
Dp.=P+Dq.2Q+Dr.=R=0...... (33).
Now, since U is the resultant of P, Q, and R, we have
P+Q+R=U,

and therefore (32) becomes, =U = 0.

Also we have

Du.P+ Du.Q+Du.R=Du.0vU,

and therefore, by Lemma 1,
Dp.P+Dq.Q+Dr.R= Du.v.

Hence (33) becomes >Du.U=0,

It appears therefore that the necessary and sufficient conditions of equilibrium of the forces
U, U’, U"", &e. acting at the points w, w’, wu,” &c. of a rigid body, are

(III.) g

23 The equation (29) includes the whole theory of couples.

For, suppose the forces U, U’, U", &c. to constitute a set of couples, in other words, suppose,
that

Sb

U=-U, U"=-U", &c. &e.
Then the equation (29) evidently becomes
D(w-u).U + D(w” =u"). U" + &e. = 0...... (34).

Now, by Art. (5), if » and R be the numerical magnitudes of wu’ - w and U, and @ the angle
contained by w’ — uw and U, then the numerical magnitude of D (u’- w).U is Rr sin@; which is the
moment of the couple consisting of U and U’; for r sin @ is evidently the perpendicular distance
between U and U’. Also D(u’—w).U is a line perpendicular to w’— uw and U, and therefore to
the plane of the couple (U, U’), Hence D(u'— w).U is the awis of the couple (U, U’).

The equation (34) therefore indicates, that the symbolical sum of the axes of a set of couples which
balance each other must be zero. Which includes all the propositions of the theory of couples.

(dvV.)
24. When the forces U, U’, U”, &c. do not balance each other, to find the condition of

their having a single resultant, !
Suppose that # is the resultant, and 7 its point of application ; then since — R, U, U’, &e.

balance each other, we have, by (28) and (29),
=U-R=0, TDu.U-Dr.R=0,

“ Respecting this equation, we should have remarked, that Du.U is the symbol of the wais of the couple which transfers the
force U from the point u to the origin. See Article 24, page 423.

IN GEOMETRY, MECHANICS, AND ASTRONOMY. 423

or, putting TU=V, and >Du.U=W, for brevity,
Jip VV IDR SRE IIe
and therefore, Dr.V=W ........... (35).
Which equation indicates that V and W are at right angles.

V is evidently the resultant of all the forces, supposing them transferred to the origin in their
proper directions ; and W is the axis of the resultant of the couples introduced by transferring the
forces; for Dw. U is evidently the axis of the couple consisting of U acting at wv, and — U acting
at the origin; and therefore 2Dw.U is the sum of the axes of all such couples, and therefore
the axis of the resultant couple. Hence the condition of the forces having a single resultant is,
that the resultant force (V) shall be at right angles to the axis (W) of the resultant couple.

This condition is simply expressed by the equation,

O=AV. WwW;
which is got immediately by performing the operation AV on (35). See Article (13).

25. If we transfer the forces U, U’,-U", &c. to any point v, instead of the origin, the
resultant couple will be 2D(w—v).U instead of 2Du.U. Now TD(u -—v).U = >Du.U
—-Dv.>U=W-Dv.V. Hence, if we assume W to denote the resultant couple when the forces
are transferred to v, we have

W,=W-Dv.V.

We may determine the minimum numerical value of W_ as follows :

Let AV be the projection of the line W on the line V; then W—)V is perpendicular to V, and
is therefore expressed by a symbol of the form Dv’. V, where v’ denotes a line which we do not
require to know.

Hence, we have W=)V+ Dv’. V, and therefore

W,=rV + D(v' -v). V.

Since v is arbitrary, D(v' — v).V denotes any line whatever at right angles to V: hence the
numerical value of W, is least when D(v'—v).V=0; and therefore W,=V. To determine },
since W —}V is at right angles to V, we have

AV.(W=XV) =0, and .. X ==——_..

Hence the axis of the couple of minimum moment is
AV Ww
AV.V-

We may observe that the equation W/=)V indicates that the axis of the couple of minimum
moment (W_) is parallel to the resultant force (V).

These instances suffice to shew the application of the notation Dw’.u, and Aw’. wu to Statics.

Application of the Notation Du .u and Au'.u to the Calculation of the Motion of
a Rigid Body about its Centre of Gravity.
26. Let w’ be the symbol of the position of any particle (dm) of a rigid body at any time (¢),
2 ,
and U the accelerating force which acts upon dm: then, since om _ is evidently the symbol of

; dul : Ae
the effective force on dm, the forces Udm, and — qe dm applied to dm, and similar forces to the

Woo, VIII. Parr IV, 3I

424 Mr. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS

other particles, must satisfy the conditions of equilibrium. We have therefore by equations (28)
and (29),

Pu’ Pu’
=(U-— = ya -— = 0.
=(U ar) m=, EDu (u- SE) dm =o

Let ~ be the symbol of the centre of gravity of the body, and assume wu’ = u + w; then these
equations become (observing that udm = 0),
du d'u
m—~ = 2Udm =Du.—~ m= Du. Udm.
dt ; dt
Which equations are equivalent to the six equations of motion of a rigid body.
Since wu is the symbol of 5m with respect to the centre of gravity as origin, the second of these
equations determines the motion of rotation of the rigid body about its centre of gravity, and, as
far as this equation is concerned, the centre of gravity may be regarded as a fixed point.

d du du du du du
i I a a a — Da
Also, since ai u di os ant Du de Te
this equation may be written in the following form,
d
(BD Fim = [Du.Udm....... (36).

27. ‘To effect the integration denoted by = in the first member of equation (36).
Take the principal axes through the centre of gravity as the co-ordinate axes, and let 2, y, x,
be the co-ordinates of dm: then we have

u=vat+yB+ zy,

and therefore, since #, y, * are independent of ¢,

du da dB dy
ap hee ae! df dé (37).

Now, referring to Art. 2, we may see immediately, that, if w, denote the velocity of the
point B parallel to OC, w, the velocity of C parallel to OA, and w, the velocity of A parallel to
OB (in other words, ,, w2, ws, are the angular velocities about the axes OA, OB, OC, of the

planes BOC, COA, AOB respectively), then we have
a=w,dt, b= wodt, c = a,dt,

and therefore the equations (3) become

da

Te oo w;3 = Woy *

d

- SOT TY Ng ew cles oes ocissee =e (38)
dy

We may here observe in passing, that, if we assume

@=w,a + w.[3 FWY csercccrceccees (39),

_ * Ifwe put these values in (37) the coefficients of a, B, 7 ate wz —wyy, w3.%— 2, w,y—gx; which are the well known expres-
sions of the velocities of any point of a rigid body moving abouta fixed point.

IN GEOMETRY, MECHANICS, AND ASTRONOMY. 425

the equations (38) become

da dp ¥Y
DM EO aE Te Pwnage eee:
and therefore (37) becomes
du
di =_Dioe hr ecnieteses See reeees (41)

Now, referring to Art. 5, Dw.w is a line drawn perpendicular to w and w, whose numerical
magnitude is mr sin 8, where m and r are the numerical magnitudes of w and wu respectively, and 6

the angle made by w and w. Hence, the equation (41) indicates that the velocity ois due to

the rotation of the rigid body about the axis w with the angular velocity m. In other words, the
symbol w represents completely the motion of the rigid body ; for w represents, in direction, the
axis of instantaneous rotation, and, in numerical magnitude, the angular velocity of the body about
that axis. _

Returning to equation (36), we find by (37), observing the properties of principal axes,

d
EDu. dm = SimD(wa + y+ x7y).(x Bee )

da dp 2 dy
= Da.7, Sdma* + DB. Edmy + Dy. oT Zdme*.

Now, by Art. 8, and by equations (38), we have

nse 23 + wa.

da dp d
Da. 7, = way + o/s DB. 7, = wa + ony, Dy. oF

Hence we find,
d
TDu. bm = wad (y’ + 2°) dm + w, 3S (2* + 0°) dm + wsyd (a + y”) dm
= Aw,a+ Bu. + Cosy.
Hence the equation (36), cleared of the sign =, becomes

d
di {Awa se Bw.(3 + Cwszry} = =Du . Udm* eee (42).

28. We shall now apply this equation to the problem of Precession and Nutation.

To effect the integration = in the second member of (42) when the force U arises from the
attraction of a very distant body, which may be supposed to be collected into its centre of gravity.

Let w’ be the symbol of the centre of gravity of the distant body, and let m’ denote its absolute
_ force; then since w’—w denotes the line drawn from dm to m’, the attraction of m’ on
m is

a m’ (u'— w) :
~ fA(w-u). (w= uh?’

* If we perform the operation in the first member of this, by Whence it follows that the first three of Euler's six equations
_) means of equations (38), it becomes follow immediately from (42).

dw, ims The last three of Euler’s equations follow immediately from
45 ot Cla Tad 5 at i a —(C—A)eg oy te the equations (38), in the same manner as I have shown in my

Mathematical Tracts.

a iis 6%
312

» 426 Mr. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS

for this represents a line drawn in the same direction as u'—u, and having its numerical magnitude
equal to

R®’
where R is the numerical magnitude of w'—w [observing that A (w'’— x). (u'— u) = R* (Art. 13, |
j

equation 21) ].
Now A (w= u).(w =u) = Aw.u'-2Au. w+ Au.u

=r?—2Au.u' 47°, (Arts. 12 and 13).
uae
Therefore, since = is very small, we have
r

fe ‘ , U7 A ;
SA (w-u).(w-u)tt= rl +3 “| very nearly.

, ,
m Au.u 4
Hence u-T(1+s = ) ww;
” r
and therefore, since Du. w= 0, and Du’.u = — Du.w,

jf 3
TDu.Udm= - i Du’. (Sudm +— Tudu.u' dm).
r r
Now =wom = 0, since the origin is centre of gravity: also, by Art. 12, equation (17), we have,
observing the properties of principal axes,
Sudu.udm = >(wa+yPB + xy) (wa't yy'+ 22’) dm
=aaSda°dm + y BSy%m + zy Ex°%dm
=u >rdm - (Aa‘'a+ By B+ Czy),
since Sa'dm = Tr’dm — U(y?+x*)dm, &e. &e.

rac
Hence, since Dw’. u'=0, we have Du. Udm = ae Du. (Ava + By B+ C2'y)*.

Thus (42), cleared of the sign =, becomes,

,

3m

Du’. (Aw'a + By'B + Cz'y) ... (44).

,
yr

d
ag (4e# + Bo.f + Cws'y) =

29. To find the Solar Precession and Nutation by means of this equation.
Let + be the north polar axis of the Earth; then B = A, and C exceeds A by a small quantity,
\A suppose, and therefore C = 4(1+). Hence, observing that wa + o.8 + ay = 0, va+yB

+2’y =u’, and Dw.w =0, Dw.u'= 0, (44) becomes
dw d(@,y) “8m,

ae * dt 75

revolves about its polar axis with a uniform angular velocity, and that the Earth moves round he

* Performing the operation Dw’ (i.e. a’ Da.+y' DB.+2'Dy.) The coefficients of a, 8, y here are the well-known expressions fo
the second member of (43) becomes the moments of the attraction of Sun or Moon about the princip

3m’ be wy f axes of the Earth.
sa (B-C)y'2 a+(C—A)2'a'B+(A-—B)ea'y'y}-

IN GEOMETRY, MECHANICS, AND ASTRONOMY. 4.27

Sun in a circle uniformly ; in other words, we shall suppose that w,*, ry and 7” are constant. Hence.
observing that 2’= Avy. w’, (45) becomes

dw 3m

ae 5 A(Aw’. y) (Du. y).eeceeeee eee (46).

Now let a’ and 3’ be two unit axes at right angles to + and to each other, one of which (a’)
points to the first point of Aries: also let 8” be a unit axis pointing to the north solstice: then,
if we assume w@ to denote the obliquity of the ecliptic, and n't the Sun’s longitude, we evidently
have

p’= Posm + ysing, w= 1 (a'cosn't + 3’ sin n't),
and «. w= 71" ja cosn’'t + B'cos w sinn’t + y sin w sinn’t?.

Hence we have

Au.y =1'sing sin n't,
Duy = - Dy.w=r (a'cos gsin n't — (3'cos n't),

‘ fall aie td
and therefore, observing that ’* = ve? (46) becomes

dw i Ae i eis ° ee
Gp = 3m *h sin w fa cos w sin n't — Bcosn'tsinn’tt......... (47).

By integrating this equation we find w, i.€., wa + w.3 + w,y 3 and therefore, by equating the
coefficients of a, 3, yy, we find w,, w,, w;; from which it appears that w, is constant (as has been
shewn before), and @,, w. are small quantities.

Now, if m denote the numerical magnitude of w, we have

n= Joe + of + we.
Also the sine of the angle which the axis ry makes with the axis w is

Vw Sr wo.”
—————
V w2 + we? + wg

But, since n't varies but little in one revolution of the Earth, it follows from (47), that we may
regard w,, w2, w;, a8 invariable for one day in quantities multiplied by X.

Hence it follows, that in a day the axis yy describes a conical surface round the axis @ (i.e. the
instantaneous axis) with a uniform angular velocity m; and therefore the mean daily motion of the
axis ry must be the same as the motion of the axis w; or, in other words, observing that the numerical
magnitudes of yy and w are 1 and m respectively, we have, as far as the mean daily motion of vy is
concerned,

Hence by (47) we find
dy 8n?_. :
on =—- Asin gw ja'cos@ sin n't — B'sin n't cos nth... (48).
iL

Which equation completely determines the motion of yy the Earth’s north polar axis.

1 wy,

: 7s 5 . . a . :
* It is so easy to see that the coefficient of y in the first member of (45) is (1 +A) i? and that in the second member it is zero
a

therefore w, is constant, whether \ be small or not.

428 Mr. O'BRIEN ON A NEW NOTATION, ETC.

It appears from this equation that the north pole has two velocities, namely

72
X sin @ cos w sin’n’t, parallel to a’, i.e. perpendicular to the solstitial colure ;

72

and ~——sinw sin n't cosn’t, parallel to 3, i.e. in the plane of the solstitial colure, and paral-
Ls lel to the equatorial plane.

Hence the length of the path described parallel to a’ in any time ¢ is

,

n a , ,
an » sin w cos mw (nt — 4cos2n’'t),

and the path parallel to ’ is

,

3n ; ,
—Asing@w cos2nt.
4n

Which are the well-known values of the solar precession and nutation of the pole.

The calculation of Lunar Nutation may be effected very simply by the above method; in fact
the equation

dy 3m’ “ee
= aie prs k (Au -y) (Dw.-y);
still holds, and we have only to make the proper substitution for w’ to suit the Moon’s motions, and

then integrate as above.
‘ M. O'BRIEN.

Upper Nornood, Surrey,
Nov. 1846.

(Nors.) In a series of papers on Symbolical Geometry by Sir W. Hamilton, which are at present being
published in the Cambridge and Dublin Mathematical Journal, a very remarkable interpretation is given
to the product of two symbols. According to this interpretation }(uw'+ uu) means the same thing as Aw. w
in the present paper, and 3 (wz'— ww) means the same thing as Du. w’.

ee

XXXII. On the Principle of Continuity, in reference to certain Results of Analysis.
By J. R. Younc, Professor of Mathematics in Belfast College.

[Read December 7, 1846. ]

THE mathematical axiom that ‘‘ what is true wp fo the limit is true a# the limit,” is necessarily
implied in the general principle of Continuity. The recognition of this truth is essential to the very
conception of continuity; of which indeed a sufficiently clear idea may be conveyed by the simple
enunciation of the axiom itself. In Geometry the continuity here mentioned refers to magnitude
only, irrespective of shape: in Analysis it refers simply to value. And in both, the limit spoken of
is that, whatever it may be, at which the continuous series of individual cases terminates; or, if
the expression be preferred, at which it commences.

It is plain that different continuous series may start from, or terminate in a common boundary :
or the terminal limit of one series may be the commencement of another ; each series being governed
throughout by its own independent law. But there is a liability to suppose the limit unique when
it is in reality multiple, or ambiguous; and indeed to confound the true limits with some unique
isolated form, having no connexion whatever with either series.

Thus:—the tangent of 2, when wv commences in the first quadrant and continuously increases,
arrives at its limit when wv reaches 90°. In like manner, the tangent of w, when # commences in the
second quadrant and continuously diminishes, arrives at its limit when reaches 90’. But the two
limits (which are very liable to be confounded) are perfectly distinct. In the former case the limit is.
tan 90° = + ©: in the latter case, tang0°= —o. And, viewing the tangent independently,—that
is, as altogether unconnected with a continuous series, and therefore as uncontrolled by any law of
continuity,—the tangent of 90° is ambiguously + 2: and we cannot select one of these values, to
the exclusion of the other, without destroying the independence here supposed, and subjecting the
tangent to the operation of a law binding it in connexion with a continuous series of tangents.

Again : the limit or extreme case of the continuous series of values of the progression

1—a+a°— c+ wv — a + &e. ad inf............. (1),
furnished by the continuous variation of « from some inferior value up to wv = 1, or from some
superior value down to # = 1, has been supposed in each case to be properly represented by

1-1+1-—-1+1-1+4 &e. ad inf...................(2).

But it has already been shown by the writer of these remarks*, that so far from this being the com-
mon limit, the two limits are totally distinct :—the one having for value 4, and the other infinity :
whilst the series (2) is not comprehended at all among the continuous cases of (1), but is entirely
unconnected with, and independent of, those cases: its value is ambiguously 1 or 0.

In order that the influence of the law of continuity, which connects together all the individual
cases of (1), may not be overlooked or evaded in the extreme one of those cases, it will be desirable

y ls re 5 A
to change the notation: writing 1 — — for 2, when the limit 1 is to be reached through continu-
z

1 ar f
ous ascending values of «, and 1+-— when it is to be reached through continuous descending
z

values of 2.

* Philosophical Magazine for November and December 145,

430 PROFESSOR YOUNG ON THE PRINCIPLE OF CONTINUITY.

It will then only be necessary to suppose x to approach infinity as its limit; the connexion of
which limit, with the continuous set of values that it terminates, being preserved by the actual
exhibition of x in its final state, or under the form ©; a symbol which, it will be observed, thus
spontaneously presents itself; and is not arbitrarily introduced to effect a purpose.

Hence, when the limit 1 is reached through continuous ascending values of x, the extreme case
of the series (1) is

1- (1 -+) + (1 = ~)- (: - a) + &e. ad inf.... (3);

and when it is reached through descending values, the extreme case of the series is

1 tN 1 \* .
1- (1+ ~) + (1+) - (1+2) + &e. ad inf....... (4):

And the values of these, as shown in the publication referred to, are respectively } and infinite.
For any finite number of terms, these series do not differ sensibly from one another, nor from

; ; iL Set UNS
the neutral or independent series (2). But since we know that (1 - =) =-, and (1 + =) =e,
e

it follows that, after a finite number of terms, the three series are totally distinct: and we thus see
that in such extreme cases as those we are now considering, it is not allowable,—as generally sup-
posed,—to neglect the terms infinitely remote from the commencement of the series: for it is only
in the infinitely remote region that the distinguishing peculiarities of the series become fully
developed. And it is because of this, that in contemplating these extreme or limiting cases, differ-
ent orders of infinity become unavoidably forced upon our attention. Thus, in the infinitely remote
region of the series (3), it is obvious that there are places for the terms

fae N pdfs shA, fi LYM asec ofe 2)

of which the numerical values are

And all these terms, as far as the zero-term, being significant, necessarily affect the numerical
expression for the sum of the whole; and cannot be neglected with impunity in a correct estimate
of the value of the altogether boundless series (3).

The theorems proposed by Cauchy, for testing the convergency of infinite series, do not apply
to the limiting cases, such as those here noticed. These theorems have in fact been the occasion of
error in the treatment of those cases; and it is one object of the present communication to invite
attention to this circumstance.

In discussing the series

2 3 4

Nt ae I, a rk
aie ge A /Sopdcorceaeacine (5).
Cauchy observes* that it will be convergent, or divergent, according as the numerical value of w is
inferior, or superior to unity; but that when the limits #=1, # = —1 are actually reached, the
series will be divergent in the first case, and convergent in the second}+. This is not a correct
account of what happens at the limits: if # ascend from an inferior numerical value (that is from
a fractional value, either positive or negative) up to w =1, or w = —1, the limiting cases will be
convergent, like all the preceding cases: but if the same limits be reached through descending |

* Cours d’ Analyse, p. 153. + [bid., p. 155.

IN REFERENCE. TO CERTAIN RESULTS OF ANALYSIS. 431

values of w, the extreme cases will then, on the contrary, be divergent. The truth of this will
appear by writing these extreme cases with the proper symbol or indication of continuity, intro-
duced, or rather preserved, as in the instances above. For we thus get the converging series

+ (1-5) +4 (1- 5) 43 (1-2) + be ad inf.

eo) 2 © 3 © : ”
and the diverging series

+ (142) +h (1+5) #4 (14 2)'4 be. ad ing

° . © 2 © : e

That the former of these is convergent is obvious: and that the latter becomes divergent, in its

infinitely remote terms, will be seen from the following considerations :—
a

As noticed above, (1 + =) =e; so that, in the infinitely remote region, there occur the

terms

oy,

e ; e
., and, in fact, after —.
co ¢o

which evidently diverge after the term

Similar reasoning applied to
14+”+42a°+2.3a°+4+2.38.4a4+ &c. ad inf. ...... (6),

another of the series considered by Cauchy, and which he affirms to be equal to 1 when 2 becomes
zero, will show, that instead of 1, the value is infinite, For, writing the zero in the allowable

1 . . . .
form —, we find among the terms infinitely remote, the following: viz.
ne )

2h Bid 0s 5 DiS Gaels NCO) seis CO
= ==: ————,,———-, &e.
o co
in which, as «&’ may exceed © in any ratio, the numerator may exceed the denominator in any
ratio; so that the terms at length become infinitely great; that is to say, the extreme case,
corresponding to w = 0, is like all the other cases, divergent.

‘The preceding reasonings, in which terms infinitely remote, and infinites of different orders, are
considered, may perhaps be regarded as too vague and subtil to justify an unhesitating recep-
tion of the conclusions to which they lead: and although they do not appear to me to be fairly
chargeable with this objection, yet I wish them to be regarded—less as demonstrations of the truth
of these conclusions, than as confirmations, supplied by the laws of analysis—when these are allowed
to have their full and unrestricted scope—of the general axiom which stands at the head of this
paper; and in virtue of which, if it be demonstrated, that an assigned analytical formula correctly
expresses the sum of an infinite series for all cases short of a certain extreme case—however closely
to this case we approach,—then we may safely infer that it equally, and as correctly, expresses
the sum in the extreme case also: a fact which is as necessarily true as any of the axioms of
Euclid; and which I think can be questioned only by those who overlook the controlling influence
of the law of continuity over these terminal cases. It would be very wrong, in utter neglect of this
law, to confound the series

17 — 2° + 3° — 47 + &e.,
for instance, with what
1* — 2a + 3*a* — 4a + &e.
becomes in the extreme case of w = 1; and thence to assert, as indeed has been done, that its sum is

Vou, VIIT. Parr IV. 8K

432 PROFESSOR YOUNG, ON THE PRINCIPLE OF CONTINUITY,

zero, when in reality the sum is +. The erroneous sums assigned to divergent series, will be
found in many other instances besides this, to belong, not to the independent series themselves,
but to the extreme cases of certain general forms. Yet the errors adverted to, and which formed
the subject of a communication submitted to the British Association in 1844*, are not always of this
character : the value *596347..., for instance, assigned by Euler, and many succeeding writers, to the
series

1-14+2-—-2.34+2.38.4- &e,

neither belongs to this series nor yet to the extreme case of the general series (6), in which
1 : ] P é

xv = 1-—-— since we have seen that when # becomes — even, the infinitely remote terms must
ee) ee)

still diverge.

In the Mémoires on Series and Definite Integrals, which Poisson has published in different
Cahiers of the Jowrnal de [ Ecole Polytechnique, a fault analogous to that above noticed is very
frequently committed+. It is the common practice of this distinguished analyst arbitrarily to
introduce the ascending powers of a foreign variable, in connexion with the terms of an isolated and
independent series, and then to employ the extreme case of the general form thus obtained, when 1,

or rather 1 — = is put for the new variable, instead of the original series. In this way he con-

: : : 1
verts the neutral series 1 —- 1 + 1 — 1 + &c. into a convergent series, and thus gets A for the sum ;

which is of course erroneous. He applies the same process to periodic series in general; thus, in
fact, destroying their periodicity—at least in the infinitely remote terms—and thence obtains sum-
mations that are palpably wrong. Thus, in referring to a particular series of this kind, in his last
great work, he says, “* Elle est de l’espece des séries périodiques, qui ne sont ni convergents ni
divergents, mais qu’on peut néanmoins employer en les considérant comme les limites de séries
convergentes, c’est-a-dire en multipliant leurs termes par les puissances ascendantes d’une quantité
infiniment peu différent de Vunité”{: the inaccuracy of which principle I have, I think, suf-
ficiently discussed elsewhere ||.

It is of importance to observe, however, that there is one class of series in reference to which
the adoption of this principle is allowable, as its application will be unattended with error :—I mean

: , : tees 1 :
convergent series. For since, as already shown, the foreign multiplier 1 — —_ becomes effective
©

only in the terms infinitely remote, and as all these in converging series are themselves zero, these
multipliers produce no modification of the character of the series, nor any change in its sum. In
periodic series however error must of necessity arise from replacing them by the limits of converging
series; inasmuch as these latter always tend to some determinate value—either finite or infinite:
whereas an infinite periodic series, from its very nature, tends to indeterminateness. To attribute a
unique value to such a series is therefore absurd.

I have here spoken of the sums of converging series as sometimes tending to infinity, which
tendency some may suppose to be opposed to convergency : a simple reference however to the series
1+a +4 o* + &e. will I think correct this supposition, since it will be admitted that this continues

1 1 : r
convergent for all values of w from # = = up to #=1- Tasks for which extreme value the sum is

infinite 9. I have also ventured to call the infinites, to which the extreme cases of certain convergent

* See also Proceedings of the Royal Irish Academy, 1845, | || Philosophical Magazine, Dec. 1845.

No. 49, where the communication referred to is printed at length. | ere
+ Journal de lV’ Ecole Polytechnique, Cahiers 17, 18, and 19. § The series 1 + Tacs tie.3t & also, is convergent
+ Théorie de la Chaleur, p. 199. for all real values of 2, and tends to infinity as w does.

Fa

7 .

IN REFERENCE TO CERTAIN RESULTS OF ANALYSIS. 433

series thus tend, determinate: because if we reflect upon the peculiar character of a strictly diverg-

ent infinite series, we shall perceive, that however remotely into the region of infinity its terms be
considered to extend, yet we can never, even in imagination, reach a stage beyond which the series
ceases to be accumulative, and may be rejected as zero: the portion so rejected would, on the
contrary, still be infinite; and this is a peculiarity which sufficiently distinguishes a divergent
series from a convergent series with an infinite sum. It has place even in those slowly diverging
series of which the individual terms continually tend to zero, as, for example, in the series

Ter ileee dt yal
1+-+-+-+-+ &.
DS Ae

1 2 3 4
and +

+ + + &e.
Ue ont) Duce! ther {8}

for however remote the m term may be, 2 terms more of the first of these series will be

1 1 1 1
OOnO0 ser}

+ + +.
n+ n+2 n+3 2n

and » terms more of the second,
m+ nm+2 2n
os ; Soon = =
4(n+1jf?—1 4(n+4+2)?—-1 4(2n)?-1°

and these additional 2 terms will, in the first case, exceed

1 1
mx— or -,
2n 2
and in the second case,
1 1
1 8

82 — —
2

A diverging infinite series therefore tends to no limit, either finite or infinite; and this
consideration is perhaps sufficient to justify the language of the continental analysts, who say that
such series have no sum.

It would seem desirable however to divide series into other classes besides convergent, divergent,
and periodic; in order to distinguish those which come under the influence of continuity, from
those which, like the series just considered, are entirely isolated and independent. The latter class
might be called independent or neutral series; and the former dependent series. Hutton* appears
to have called the series 1-1+1-—1+41-— &c. a neutral series, simply because it is neither
convergent nor divergent. In the sense in which it is here proposed to use the term, no reference
is made either to convergency or divergency: but merely to the fact of the series not being united
to a set of others by the bond of continuity. A neutral series may therefore be either convergent,
divergent, or periodic: the series

1+ —-+——+ + &e
OSS RUF LD. 18
ana
te Se Se = pb Be

9 83

12 — 2? 4 3? — 4? + &e.

l= Ppt — t+ &e.

* Mathematical Tracts, Vol. 1. p. 178.

434 PROFESSOR YOUNG, ON THE PRINCIPLE OF CONTINUITY,

are all neutral. But, as already remarked, since the first of these is convergent, its sum does not
differ from that of the corresponding dependent series*.

It is not in reference to series only that this distinction between neutrality and dependence has
been overlooked. It has been improperly neglected in the treatment of an extensive class of definite
integrals; all those, namely, that are analogous to periodic series, in respect to the indeterminate-
ness which they involve. It has already been shown what contrivance Poisson resorts to, in order
to get rid of this indeterminateness in the series: he destroys the indeterminateness of the integrals
by a similar artifice. The series were rendered determinate by multiplying their terms by the
ascending powers of a foreign factor; thus bringing them under a law of continuity from which
originally they were wholly free. The integrals are rendered, in like manner, determinate by
introducing, under the sign of integration, a new variable:—an exponential multiplier, in virtue of
the variation of which, a bond of continuity is, as before, imposed upon the expression, and its
indeterminateness thus overruled. The following definite integrals quoted from Poisson, and those
who have espoused his principles, are all essentially indeterminate :—

0 0

ao a ao @
pi da sin ra, i da cos ra, | daz" sin ra, up dx x"~' cos ra,
0 0

-@ C)
| dx x" sin ra, fae w"-*cos rv, &c., &c.;
0 0

the exponents of w in the last four being positive. A very little consideration will suffice to
convince us of this: we need only revert to the ordinary ideas involved in the method of quadratures :
for if in any of these forms the expression under the integral sign—omitting the dr—represent the
ordinate of a curve, we at once see that for a = co—one of the proposed limits—that ordinate, and
therefore the area, or the entire integral, must be indeterminate. By introducing the factor e~™,
for which there is of course not the slightest warranty, these forms become changed into the follow-

ing :—

wo 2 a) :
if dxve- sin ra, tf dwe- cos ra, ih dx ex" sin ra,
0 0 0

an) a) ao
yf dae 2" cos ra, i) dre“ a" sin ra, di daze“ x"~* cos rv,
0 0 0

in reference to which the ordinates, at the limit « =, all vanish, irrespective of the value of a.
If the integrations be now executed, each result will be a general expression involving a; and if we
seek what this expression becomes when a, by continuous variation, arrives at zero, we shall truly
obtain the limit of the integral; that is to say, we shall obtain the last of the continuous series of
values which the integral passes through as @ diminishes continuously, from some superior value,
down to zero. These results therefore are all valid, as limits of the changed integrals; but have,
in strictness, nothing to do with the integrals originally proposed; these latter being neutral, or
independent ; and therefore not included in the continuous series of values adverted to.

The impossibility of reconciling some of the erroneous, but prevalent conclusions that have been
arrived at respecting the foregoing integrals, with certain known elementary truths, has led one or
two recent writers to pass too sweeping a condenmation on integrals of this kind ; and to reject, as
false, integrations that may easi!y be proved to be true. I shall advert to some of these presently.
But it may not be altogether out of place previously to remark, that much needless ingenuity seems
of late to have been expended in proving that sin ¢ and cos cannot be zero; although such
is unhesitatingly affirmed to be the case by the late Mr. Gregory+, and—with misgivings how-

* See Note (B), at the end of this Paper.
+ “ Both the sine and the cosine of an infinite angle are equal to zero.” Gregory’s Examples, p. 477-

IN REFERENCE TO CERTAIN RESULTS OF ANALYSIS. 435

ever—suspected by Mr. De Morgan. But it is proper to state that Poisson nowhere countenances
this notion; nor is it implied in his principles, though it has been thought to follow from them.
It is true that Poisson makes

i <) ; @
yp da sin = 1, and [ du cos v= 0.
0 0

It is also true that these integrals are respectively 1 —cos ©, and sin ©: but it does not follow
that we have any right to equate Poisson’s results with these. Poisson virtually selects a particular
value out of the innumerable values of cos o> ; and a particular value out of the innumerable values

‘of sin © ; these selected values are each zero. He does not deny the existence of the other values,
nor say that sinc and cos@ are zero only, as others have said: he expressly declares that he
takes that particular value of cos ¢ which unites in continuity with the values of

oo ah
ii dxe- sin x@;
0
and that particular value of sin 2 which unites in continuity with the values of
wo
if dxe-™ cos a,
i)

it being understood that @ varies from some superior value down to zero ; and his doctrine is that,
by taking the extreme limit thus reached, he gets, in each case, ‘‘ une valeur unique qu’on peut
employer dans V’analyse.” The fault of Poisson consists solely in his bringing indeterminate
expressions under the control of arbitrary conditions, in virtue of which that indeterminateness
is destroyed, and unique values deduced ; and in consequence of which these unique values—as in
the instance of the series 1 ~ 1+ 1 —1 + &c.—are frequently not even among the indeterminate
set: but this great man must not be charged with the palpable error of making the sine and cosine
of an infinite are zero*. It should also, in justice to the same illustrious analyst, be observed
further, that some English authors, under the impression that they have been carrying out Poisson’s
views, have also, on other points, employed reasonings, and arrived at conclusions, which those views
do not justify. The results which Poisson assigns to the integrations noticed in this paper are all
true as far as they go. He chooses one out of an infinite variety of equally admissible values, and
disregards all the others:—a fault which appears to me to be analogous to that which would
be committed by arbitrarily selecting one of the ” roots of an equation of the n'" degree, to be
employed in physical applications, and rejecting all the others. But, from a pretty careful exami-
nation of Poisson’s different Mémoires on Series and Definite Integrals, I can find no foundation for
the statement recently made, that ‘‘ Poisson would admit 1° — 2° + 3? -4° 4+......=0.” He rejects
diverging series: and in applying his principles to cases where divergency might be suspected, he
takes care, in order to justify his mode of proceeding, to remove the suspicion, by showing that the
series must be convergent. (See Théorie de la Chaleur, p. 188.)

Resuming now the consideration of the definite integrals, I have to remark, that among those
that have been rejected are

2 sin aw 2 COSax
Hi dw and f dx;
0

av 0 1 +a"
the grounds of this rejection being that these integrals have not the values hitherto assigned

* “Les sinus et cosinus d’un are infini sont évidemment des | périodique, que s’etendent a ]'infini : ces intéyrals n'ont aussi des
quantités indeterminées.”’ Poisson: Journal de l' Ecole Polytech. | valeurs déterminées, que quand on les regarde comme les limites
Cah. xix. p. 407. d'autres intégrales, dont les élémens convergent vers zero, et sont

“La maniére dont nous avons considéré les séries périodique _ nuls a J’infini.” Thid., p. 431,
infinies, s’applique également aux intégrals définies de quantités

436 PROFESSOR YOUNG, ON THE PRINCIPLE OF CONTINUITY,

to them, but are, on the contrary, indeterminate—like those already noticed*. That they are not
indeterminate however will be obvious from again adverting to the notion of quadratures: the
ordinates of the curves are evidently determinate throughout the whole extent of the integration,—
that at the superior limit o> being zero. The first of these integrals has been proved—by what
appears to me to be perfectly valid reasoning, though it has recently been objected to—to be

‘ T wT ‘
altogether independent of the valwe of the constant a, and to be equal to = ey according as
the sign of this constant is positive or negative. Poisson indeed, following Euler and others, says

< T
that the values of the integral are = 0, or —

wo 139

, according as the constant is positive, zero, or ~

negative}. But it should be remembered, in obedience to the law of continuity, that if a become
zero, by passing through neighbouring values, and vanish positively, the value of the integral is still

. . 2 : s Tv * . .
=. and if it vanish negatively, the value of the integral is still — >> as in all the continuous series
[9] .

of cases which these terminate.

The integration of the second of the preceding forms has however been effected by methods
which are really objectionable, notwithstanding the accuracy of the results obtained by them: and it
may not be uninstructive briefly to direct attention to this circumstance.

Legendre commences his process by at once destroying the generality of the proposed integral—

2Qhar

taking for the limits, not @ = 0, and v=o, but «=0, and v@ = , k being a whole number; and

then, at a convenient stage of the investigation, making & infinite. By means of this artifice, the
indeterminateness, which the method employed would otherwise have introduced at the limit
wv = 9, is overruled by an arbitrary condition}. The true result however necessarily comes out ;
because that result is independent of all condition as to how the limit ¢o is reached.

In the other method of integration, the indeterminateness adverted to is not evaded, but is
allowed to enter into the process: it is however wholly disregarded; and thus, by a sort of com-
pensation of errors, the true result is again obtained. This, I presume, is the method to which
Sir. W. R. Hamilton alludes, at page 16 of his profound and remarkable paper on Fluctuating
Functions§, where an accurate investigation of this integral is given ||.

It may be proper to add, that when by applying differentiation to a determinate form, whether
an infinite series or a definite integral, we are led to indeterminateness, the step must be regarded
as inadmissible, and unless corrected, as leading to a false result. It is not difficult to see the reason
of this. In each case a certain constant is considered to be infinite; for which extreme value
a particular function of the variable, that for all other values of the constant would have entered the
original expression, disappears ; but which function if preserved, instead of being obliterated as zero,
would reappear in an indeterminate form, after differentiation. The suppression however of the
evanescent function in the original, precludes this reappearance; and thus leads to a defective
result. This, I think, is rather an interesting fact: it shows that the differentials of certain forms
of analysis require indeterminate corrections, in a manner somewhat analogous to that by which the
ordinary determinate corrections are introduced into integrals ; and the omission of which indeter-
minate corrections has led to so many erroneous summations of certain trigonometrical series. From

“ Transactions of the Society, Vol. vi11. Part 111. Earn- | applies whenever the subject of integration, in integrals of this
shaw’s Paper on sinc and coscs. It may be remarked here, in | kind, becomes zero for v=co.

reference to the two integrals in the text, that the function under + Chaleur, p. 288.

the sign of integration becomes in each case zero at the superior + Legendre: Exercises de Calcul Intégrai, ‘Tome 1. p. 357.
limit ¢:: and that therefore, as was before observed of periodic § Transactions of the Royal Irish Academy, Vol. x1x. Pt, 11,
series, the foreign factor, e~“*, which Poisson introduces merely to || For the faulty process, see Gregory’s Examples, p. 481.
destroy indeterminateness at this limit, is inoperative, and may s See a Paper by the author in the Phil. Mag. Vol. xxviii.

therefore be admitted without incurring error: and thesame remark | p. 213.

IN REFERENCE TO CERTAIN RESULTS OF ANALYSIS. 437

this omission, too, we further see how it happens that, in enquiries of this kind, we may be led from
premisses absolutely wrong, and that by a train of correct reasoning, to conclusions absolutely
right. We have only to take the results of differentiation here noticed, each with the indeterminate
constant suppressed, and which are thus erroneous, and to apply the reverse process of integration, in
order to arrive at correct forms. Thus Poisson, starting from the false equation

+ = cos 8 — cos20 + cos 34 — cos 40 + &c.*,

2
in which he supposes 9 < 7, multiplies by d@ and integrates ; thus obtaining the true equation
(ele tee sin2@ sin3@ sin 40
— =sin@ — + —— + &e. ... (4),

2 2 3
and from a second integration, the other true equation

Te want cos2@ cos3@~ cos 460
—-—— =cos@ — + =

12. 4 4 on iG ne

Again: proceeding from the false equation,

0 = sin 0 — sin 30 + sin 50 — sin70 + &ce.,
he arrives, in a similar manner, at the true results
cos3@ cos5@ cos70@

= cos @ — + - + &c.... (B),
3 5 7

ca]

sin3@ sin 50
+
95

— &e.;

and ay = sin @ —
4

in reference to which however, from neglecting the principle of continuity, he commits the error of
supposing (4) to fail when @ = 7, and (B) to fail when @ = aa although, in virtue of that principle,

beth must necessarily hold +.

As supplementary to the foregoing observations on the principle of continuity, I would wish to
add a remark or two in reference to what has been called discontinuity :—a term which, I think, is
sometimes injudiciously employed in analysis. Many. expressions called discontinuous, should rather
be considered as composed of different continuous groups united together under one general form.
Distinct continuities, so to speak, may be comprehended in one and the same function ; and it is
obvious that these may be separately discussed, and the aggregate of the entire group estimated,
without at all introducing the idea of discontinuity. For instance, certain functions, submitted to
integration, become infinite between assigned limits of #:—-would it not be better, and indeed morc
accurate, to say, of such functions, that each consists of two continuous series of values, within the
proposed limits, both series terminating at the same absolute value of w, than to say that the
function becomes discontinuous for that value? To obtain the definite integral in such a case, we
should only have first to integrate over one of the continuous series of values, then to integrate over
the other continuous series, and to unite the results, taking special care that the terminal or initial
value of w, which unites the two series, obeys the Jaw of continuity impressed upon each, And in
this way may the integration be correctly executed, however often infinity may occur between

the proposed limits. The definite integral if aw~'da may serve for illustration. The function

I suppose Poisson considers the powers of his arbitrary multiplier,
‘‘infiniment peu différente de lunité,”” to be virtually present in
these series, to destroy their periodic character. But this does not

* That this equation is false, has already been shown by the
author inthe Phil. Mag, for December, 1845, But it is sufficient
fo observe, both with respect to this equation and that next
quoted, that it is impossible, from the character of sin < and | interfere with the principle in the text.
tos 2. that the series-side of either can be a determinate quantity. + See Journalde i’ Ecole Polytech.,Cahier Xvi11, pp. 313

a,

438 PROFESSOR YOUNG, ON THE PRINCIPLE OF CONTINUITY,

x7! becomes infinite for a = 0; so that we have two continuous series of values, each terminating,
or each commencing at # = 0; which value of w however is united to one series by the sign plus, and
to the other by the sign minus. Hence, integrating over the former series, we have log m — log 0;
and integrating over the latter, we have log (— m) — log (— 0). Consequently

ath

| «de

—m

Slog n — log 0} — Slog (— m) — log (- 0)}

SL oak AM a
bo —los —— ion —-
FO = om

There is of course nothing new in thus dividing a definite integral into portions: but the
treating of these portions, when their boundaries are infinite, as distinct continuities, allowing the
influence of each continuous law to operate throughout the entire range, the limits included :—this
mode, I say, of treating what are called discontinuous functions, is not that generally adopted ;
though the neglect of it has occasioned a difficulty that has appeared to interfere with the clearness of
the idea of a definite integral when considered as the limit of a summation. Moreover, from this

same neglect, Poisson and others have been led to very erroneous values for the definite integrals
+n

included in the form a a? da. Thus Poisson affirms that
“—m
+1 Fon
J w-' da = — (2n + 1) r\/— 1%,
= |
af
an imaginary quantity, instead of zero as above: and the value of vi a? da he states to be -2,
-1
instead of infinite, as it is found to be by the method here proposed, which gives

+1
i a"*da=(+o0-1)-(-® +1) =20@ -2,
-1

and many other such errors might, if necessary, be adduced from his writings.

But the examples already given of the influence of the principle of continuity in extreme
or limiting cases of general forms, and of the mistakes committed by analysts from disregarding this
influence, will, I think, be considered as sufficient to invite more general attention to this matter:
and I shall rejoice if the brief and imperfect sketch I have here attempted to give of the views and
principles, by conforming to which such mistakes may be avoided, meet with acceptance from the
Cambridge Philosophical Society. Ihave been induced to submit it to the indulgent consideration of
that distinguished body, chiefly because the topics embraced in it have already furnished matter for
two Papers printed in the Cambridge Transactions :—one by Professor De Morgan, and the other
by the Rev. Mr. Earnshaw. I have ventured to entertain the opinion that the views and investi-
gations of these excellent analysts do not preclude the necessity for a further consideration of the
interesting and somewhat delicate points of analysis which they have discussed: an opinion which is
strengthened by the fact, that the Papers referred to are in a considerable degree opposed to each
other, both in principle and in result. It is scarcely necessary to add, that in the present communi-
cation I have contemplated the subject under an aspect more or less different from that in which it —
has been considered either by Mr. De Morgan, or by Mr. Earnshaw; and I think it probable,
from the study of the three Papers, that the truth may be elicited; and something like consistency
and stability be at length given to a portion of analytical science which has long been affected with
much uncertainty, vagueness, and perplexity.

* Journal del’ Ecole Polytechnique, Cah. xvuit. p. 318.

IN REFERENCE TO CERTAIN RESULTS OF ANALYSIS. 439

Of the two Nores which follow, the second has already been referred to, by anticipation, in the
text: the first is intended further to confirm and establish the accuracy of the general theory which
pervades this Paper.

Note (A.)

Ir appears from the preceding observations that in certain infinite series involving a quantity subject to con-
tinuous variation we are presented in the extreme or limiting cases, with instances of what may be called
insensible convergency and insensible divergency. The peculiarity of such cases consists in this:—that, within
a finite extent of a certain infinite range of terms, the convergency or divergency of the series is insensible ; so
that for such a finite extent the series does not sensibly differ from what I have proposed to call a neutral or
independent series. When however we pass beyond this finite range, and in imagination contemplate the terms
infinitely remote, we at once recognize the accumulated effect of these insensible variations; and the con-
vergency or divergency of the series becomes abundantly apparent. The infinitely remote term at which this
fact discovers itself, is alike the termination of one infinite range of terms and the commencement of another :
the completion of which, if the expression may be allowed, shows the effect of the insensible variations through
a second infinite range, and so on.

We are thus unavoidably led to the contemplation of different orders of infinites and different orderS
of zeros—things altogether beyond the reach of actual ocular examination. But those who take that com-
prehensive view of the scope and powers of analysis, which its own well-established results, and the practice of
those most deeply imbued with its spirit so fully justify, will not, I think, found any objection to the reason-
ings in the foregoing Paper on this circumstance. In fact, in the common doctrine of vanishing fractions, the

very same principles are virtually recognized: the symbols 5 and =, which ought perhaps rather to be

A fe) ' iy he
written , and zw may each represent any ratio whatever :—even infinity: so that the reasonings adverted

to involve in them nothing repugnant to generally received conclusions. The symbols here noticed, when
really determinate, are so solely in consequence of their being governed by the principle of continuity.
This is pretty generally admitted: but there are certain other results of analysis, which the same principle
equally controls, but over which its influence is little suspected. Every one admits the truth of the equation
@=1, whatever be the value of a, without any reference to the law of continuity: yet if we reason from this
equation—still keeping the conditions of continuity out of sight—we shall speedily be led to conclusions of a

very startling character, as follows :—
1

@=1; ..a=1° =1%,

that is to say, unity raised to the power infinity is equal to any quantity whatever !

Nore (B.)

Iv was observed at page 432 that every neutral converging series might, without error, be replaced by the
corresponding dependent series. This observation might have been rendered more comprehensive ; for
diverging series, whose terms continually tend to zero, might also have been included, since the dependent
series, corresponding to these, have infinite sums, as well as the independent diverging series themselves.
These infinites however are not strictly identical in the two cases: and by saying that the one series may
replace the other, nothing more is meant than that the sum in either case will be infinite. Poisson, Abel, and
others, have shown that

1 1
9 log (1 +20 cos p +a?) =acos p— 5a" cos 2p + 5 a’ cos 3p — Kc.

Mor. VLll. Parr LY, 3L

440 PROFESSOR YOUNG, ON THE PRINCIPLE OF CONTINUITY, ETC.

whenever the second member is a converging series. Abel says, “‘ Pour avoir les sommes de ces séries
lorsque a=+1 ou —1, il faut seulement faire a converger vers cette limite *:” and he then writes

| log (2 +2 c0s g) =cos @—3 cos 29 +5 C083. — Ke.

[4]

1 log (2 -2 cos) =~ cos p — 5 c0s 29 — 5 c08 3 — &e.

which he says, “a lieu pour toute valeur de excepté pour »=(2+1)7 dans la premicre expression, et pour
=2p7 dans la seconde.” Now the second members of these equations are not the limits of the proposed
general form, any more than 1—1+1—&ce. is the limit of 1-a+a*—&c. A limit always implies continuity,
and is never exempt from the control of that principle: putting therefore the condition of continuity in
evidence, the preceding expressions should be written

1 1 1 1\? 1 Le
Hog (2 +208) =(1- ) e089 -5 1-2) cos2p +5 1-2) cos 3 — &e.

1 1 1 INE 1 UNG
3 os (2-20059)=-(1 - 5 Joo 9-5 (1 -=) cos 29 — 5 (1 -3) cos 3p — &e.

which are true, whatever be the value of @, for the series are always convergent. As ¢ tends to the values
excepted to by Abel, the series tend to infinity ; which they actually attain when these excepted values
are reached, as the first members sufficiently show. We thus see that ;

1 1 1 TENG a( TNS
i =(1—-— —(1-— &e.
1+2(1 =)+5(1 =) +qQ =)* c

DL oot Dey gl
14+4-4+=4+-+&c.
ta tm t ge oe

1+n
l-—n

1 1 1 i \ oe a\e
1+2(1-2)+i(1- 2) +7(1-Z) +e.

is infinite as well as

In like manner, from the development of log , we should infer that

is infinite as well as

Dbatadeyas]
I+gtgta+&e.,
and thence that
1 2 1 3
7 3 yaar s(1-a)+5 s(t + &c

is infinite as well as

1

el eae +&c

so that any of these diverging infinite series may be replaced by the corresponding dependent converging series,
and vice versd, without numerical error. And @ priori considerations, in reference to this class of diverging
series, would lead us to the same conclusion. The equations [4] are thus universally true without any
exception whatever.

* Guvres Completes. Tome 1. p. 89.

Belfast, September, 1846,

XXXIII. On the Theory of Oscillatory Waves. By G. G. Stoxes, M.A.,
Fellow of Pembroke College.

[Read March 1, 1847.]

Iw the Report of the Fourteenth Meeting of the British Association for the Advancement of
Science it is stated by Mr. Russell, as a result of his experiments, that the velocity of propagation
of a series of oscillatory waves does not depend on the height of the waves*. A series of oscillatory
waves, such as that observed by Mr. Russell, does not exactly agree with what it is most convenient,
as regards theory, to take as the type of oscillatory waves. The extreme waves of such a series
partake in some measure of the character of solitary waves, and their height decreases as they
proceed. In fact it will presently appear that it is only an indefinite series of waves which
possesses the property of being propagated with a uniform velocity, and without change of form:
at least this is the case when the waves are such as can be propagated along the surface of a fluid
which was previously at rest. The middle waves, however, of a series such as that observed by
Mr. Russell agree very nearly with oscillatory waves of the standard form. Consequently, the
velocity of propagation determined by the observation of a number of, waves, according to Mr.
Russell’s method, must be very nearly the same as the velocity of propagation of a series of
oscillatory waves of the standard form, and whose length is equal to the mean length of the waves
observed, which are supposed to differ from each other but slightly in length.

On this account I was induced to investigate the motion of oscillatory waves of the above form
to a second approximation, that is, supposing the height of the waves finite, though small. I find
that the expression for the velocity of propagation is independent of the height of the waves to a
second approximation. With respect to the form of the waves, the elevations are no longer similar
to the depressions, as is the case to a first approximation, but the elevations are narrower than the
hollows, and the height of the former exceeds the depth of the latter. This is in accordance with
Mr. Russell’s remarks at page 448 of his first Report+. I have proceeded to a third approximation
in the particular case in which the depth of the fluid is very great, so as to find in this case the
most important term, depending on the height of the waves, in the expression for the velocity of
propagation. This term gives an increase in the velocity of propagation depending on the square
of the ratio of the height of the waves to their length.

There is one result of a second approximation which may possibly be of practical importance.
It appears that the forward motion of the particles is not altogether compensated by their backward
motion; so that, in addition to their motion of oscillation, the particles have a progressive motion in
the direction of propagation of the waves. In the case in which the depth of the fluid is very great,
this progressive motion decreases rapidly as the depth of the particle considered increases. Now
when a ship at sea is overtaken by a storm, and the sky remains overcast, so as to prevent astro-
nontical observations, there is nothing to trust to for finding the ship’s place but the dead reckoning.
But the estimated velocity and direction of motion of the ship are her velocity and direction of
motion relatively to the water. If then the whole of the water near the surface be moving in the
direction of the waves, it is evident that the ship’s estimated place will be erroneous. If, however,
the velocity of the water can be expressed in terms of the length and height of the waves, both
which can be observed approximately from the ship, the motion of the water can be allowed for in
the dead reckoning.

* Page 369 (note), and page 370. + Reports of the British Association, Vol, vi.

8L2

442 Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES.

As connected with this subject, I have also considered the motion of oscillatory waves propagated
along the common surface of two liquids, of which one rests on the other, or along the upper
surface of the upper liquid, In this investigation there is no object in going beyond a first
approximation. When the specific gravities of the two fluids are nearly equal, the waves at their
common surface are propagated so slowly that there is time to observe the motions of the individual
particles. The second case affords a means of comparing with theory the velocity of propagation of
oscillatory waves in extremely shallow water. For by pouring a little water on the top of the mercury
in a trough we can easily procure a sheet of water of a small, and strictly uniform depth, a depth,
too, which can be measured with great accuracy by means of the area of the surface and the quantity
of water poured in. Of course, the common formula for the velocity of propagation will not apply
to this case, since the motion of the mercury must be taken into account.

1. Iw the investigations which immediately follow, the fluid is supposed to be homogeneous
and incompressible, and its depth uniform. The inertia of the air, and the pressure due to
a column of air whose height is comparable with that of the waves are also neglected, so that
the pressure at the upper surface of the fluid may be supposed to be zero, provided we afterwards
add the atmospheric pressure to the pressure so determined. The waves which it is proposed to
investigate are those for which the motion is in two dimensions, and which are propagated with
a constant velocity, and without change of form. It will also be supposed that the waves are
such as admit of being excited, independently of friction, in a fluid which was previously at rest.
It is by these characters of the waves that the problem will be rendered determinate, and not by
the initial disturbance of the fluid, supposed to be given. The common theory of fluid motion,
in which the pressure is supposed equal in all directions, will also be employed.

Let the fluid be referred to the rectangular axes of 2, y, x, the plane xx being horizontal,
and coinciding with the surface of the fluid when in equilibrium, the axis of y being directed
downwards, and that of # taken in the direction of propagation of the waves, so that the ex-
pressions for the pressure, &c. do not contains. Let p be the pressure, p the density, ¢ the
time, u, v the resolved parts of the velocity in the directions of the axes of w, y; g the force of
gravity, h the depth of the fluid when in equilibrium. From the character of the waves which
was mentioned last, it follows by a known theorem that wdv+vdy is an exact differential dd.
The equations by which the motion is to be determined are well known. They are

dp? dp?
p= apy ph —2{(S2)'s (2) I, Jo ete ety,
ae Ten ‘ - @);

dp = 0, when y= hy ceseeccercccseceencencesventeereseessenenes (3);
dy

dp dpdp dd dp
— 4+ —£ — + —= —_=0, when p=, ...,..sereevesesseres 4) ;

dt 'dad«' dydy ° fie oe @)

where (3) expresses the condition that the particles in contact with the rigid plane on which the
fluid rests remain in contact with it, and (4) expresses the condition that the same surface of
particles continues to be the free surface throughout the motion, or, in other words, that there is
no generation or destruction of fluid at the free surface.

Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES. 443

If ¢ be the velocity of propagation, w, v and p will be by hypothesis functions of # — ct and y.
: d d
It follows then from the equations wu = o, v= aa and (1), that the differential coefficients
y

of @ with respect to vw, y and ¢ will be functions of #—ct and y; and therefore @ itself must
be of the form f(#-cet, y)+Ct. The last term will introduce a constant into (1); and if
this constant be expressed, we may suppose @ to be a function of w—ct andy. Denoting « — ct
by wv’, we have

dp _ dp dp _ dp

da da’ dt da’
and similar equations hold good for @. On making these substitutions in (1) and (4), omitting
the accent of z, and writing —- gk for C, we have

d 2 d 2
p=ep(y+hk) +ecp are e{(3) + a \, Be ate dele 52 (5),
gD dp dp dp _ z
Eso ees when p = 0. .,......0.- . (6).

Substituting in (6) the value of p given by (5), we have

Bee 108 20, de we
Fay ° da da dx dy dady

: e2\28 pg OED ot Ds (5°) se-o (7);

dx) dat da dy dedy \dy) ay °°"
dp dp)?
Fis -3 J ia AH VU. serves cesrvecsses S .
when g(y+k) + oe 21 (3f) + ioe \ Omrnene (8)

The equations (7) and (8) are exact; but if we suppose the motion small, and proceed to the
second order only of approximation, we may neglect the last three terms in (7), and we may

d
easily eliminate y between (7) and (8). For putting ¢’, @,, &c. for the values of ap ; 2,

da” dy
when y=0, the number of accents above marking the order of the differential coefficient with
respect to 2, and the number below its order with respect to y, and observing that / is a small

quantity of the first order at least, we have from (8)

Ey+k)+e(p + oy) -3(b* + 6) =0,
phone | y= : g'+ =o: («+ ; ¢)+ a Ca (9).

Substituting the first approximate value of y in the first two terms of (7), putting y = 0 in the
next two, and reducing, we have

Eb, OP" — GP, — FP, (Hh + P) + 20 (P'P" + P,h/) = 0. ++» (10).

@ will now have to be determined from the general equation (2) with the particular conditions (3)
and (10). When @ is known, y, the ordinate of the surface, will be got from (9), and * will
then be determined by the condition that the mean value of y shall be zero, The value of p, if
required, may then be obtained from (5).

* The reader will observe that the y in this equation is the ordinate of the surface, whereas the y in (1) and (2) is the ordinate of
any point in the fluid. The context will always show in which sense y is employed.

444 Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES.

2. In proceeding to a first approximation we have the equations (2), (3) and the equation
obtained by omitting the small terms in (10), namely,

ae ee Ppa when y = 0.
dy dx”

The general integral of (2) is

oO) = SAcwt ray,

the sign = extending to all values of A, m and m, real or imaginary, for which m*+m*=0:
the particular values of @, Ca + C', Dy + D’, corresponding respectively to m =0, m=0, must
also be included, but the constants C’, D' may be omitted. In the present case, the expression
for @ must not contain real exponentials in w, since a term containing such an exponential would
become infinite either for = — ©, or for e= + ¢, as well as its differential coefficients which
would appear in the expressions for w and v; so that m must be wholly imaginary. Replacing
then the exponentials in « by circular functions, we shall have for the part of @ corresponding
to any one value of m,

(Ae™’ + A’e-"”) sin ma + (Be"” + B’e-"”) cos ma,

and the complete value of @ will be found by taking the sum of all possible particular values of
the above form and of the particular value Cz + Dy. When the value so formed is substituted
in (3), which has to hold good for all values of x, the coefficients of the several sines and cosines,
and the constant term must be separately equated to zero. We have therefore

= 0n eA le A, Bee BR:
so that if we change the constants we shall have
p= Cat Ee" + eo ™") (A sin ma + Bos ma), ... (12),
the sign = extending to all real values of m, A and B, of which m may be supposed positive.

3. To the term Cw in (12) corresponds a uniform velocity parallel to 2, which may be supposed
to be impressed on the fluid in addition to its other motions. If the velocity of propagation be
defined merely as the velocity with which the wave form is propagated, it is evident that the
velocity of propagation is perfectly arbitrary. For, for a given state of relative motion of the
parts of the fluid, the velocity of propagation, as so defined, can be altered by altering the value
of C, And in proceeding to the higher orders of approximation it becomes a question what
we shall define the velocity of propagation to be. Thus, we might define it to be the velocity
with which the wave form is propagated when the mean horizontal velocity of a particle in the
upper surface is zero, or the velocity of propagation of the wave form when the mean horizontal
velocity of a particle at the bottom is zero, or in various other ways. The following two definitions
appear chiefly to deserve attention.

First, we may define the velocity of propagation to be the velocity with which the wave form
is propagated in space, when the mean horizontal velocity at each point of space occupied by the
fiuid is zero. The term mean here refers to the variation of the time. This is the definition
which it will be most convenient to employ in the investigation. I shall accordingly suppose
C =0 in (12), and e will represent the velocity of propagation according to the above definition.

Secondly, we may define the velocity of propagation to be the velocity of propagation of the
wave form in space, when the mean horizontal velocity of the mass of fluid comprised between
two very distant planes perpendicular to the axis of # is zero. The mean horizontal velocity of
the mass means here the same thing as the horizontal velocity of its centre of gravity. This
appears to be the most natural definition of the velocity of propagation, since in the case considered
there is no current in the mass of fluid, taken as a whole. I shall denote the velocity of propaga-
tion according to this definition by ce’. In the most important case to consider, namely, that in

2

lh dallas

eee

-
he

peat oS

——o

_

n!

:

Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES. 445

which the depth is infinite, it is easy to see that c’ = c, whatever be the order of approximation.
For when the depth becomes infinite, the velocity of the centre of gravity of the mass comprised
between any two planes parallel to the plane yx vanishes, provided the expression for w contain
no constant term.

4. We must now substitute in (11) the value of @.
p= (e272 4 e-™ UM) (4 sin ma + Bos m2) ......2..+.- (18) ;
but since (11) has to hold good for all values of «, the coefficients of the several sines and cosines

must be separately equal to zero: at least this must be true, provided the series contained in (11)
are convergent. The coefficients will vanish for any one value of m, provided

h

™

256 —
C=

—mh
ms em gam See ee er rrr (14).

Putting for shortness 2mh =p, we have

du ap e&—e

which is positive or negative, « being supposed positive, according as

3
eur<e—er><2(u4 i sorte
Ne ed
and is therefore necessarily negative. Hence the value of c¢ given by (14) decreases as mw or m
increases, and therefore (11) cannot be satisfied, for a given value of c, by more than one positive
value of m. Hence the expression for @ must contain only one value of m. Either of the terms
Acos mx, B sin ma may be got rid of by altering the origin of 7 We may therefore take, for
the most general value of @,

Pa Pt ice PEI GF Soot. are ehe vee vaacescae (19):

Substituting in (8), we have for the ordinate of the surface
m Ac ; p
y= - an Coa Ficmatn COB Uk tan saateace sep eiod sae essec) (10)

k being = 0, since the mean value of y must be zero. Thus everything is known in the result
except A and m, which are arbitrary.

5. It appears from the above, that of all waves for which the motion is in two dimensions,
which are propagated in a fluid of uniform depth, and which are such as could be propagated into
fluid previously at rest, so that wda + vdy is an exact differential, there is only one particular kind,
namely, that just considered, which possesses the property of being propagated with a constant
velocity, and without change of form; so that a solitary wave cannot be propagated in this manner.
Thus the degradation in the height of such waves, which Mr. Russell observed, is not to be
attributed wholly, (nor I believe chiefly,) to the imperfect fluidity of the fluid, and its adhesion to
the sides and bottom of the canal, but it is an essential characteristic of a solitary wave. It is true
that this conclusion depends on an investigation which applies strictly to indefinitely small motions
only: but if it were true in general that a solitary wave could be propagated uniformly, without
degradation, it would be true in the limiting case of indefinitely small motions; and to disprove
a general proposition it is sufficient to disprove a particular case.

6. In proceeding to a second approximation we must substitute the first approximate value of
p, given by (15), in the small terms of (10). Observing that k= 0 to a first approximation, and
eliminating g from the small terms by means of (14), we find

Sh, - oD" - GA mic sin 2mMa = 0 vcseseeeeee (17)

446 Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES.

The general value of @ given by (13), which is derived from (2) and (3), must now be restricted to
satisfy (17). It is evident that no new terms in @ involving sin ma or cos ma need be introduced,
since such terms may be included in the first approximate value, and the only other term which can
enter is one of the form B(e""-” +¢""-”) sin 2ma. Substituting this term in (17), and
simplifying by means of (14), we find
3m A’
B = e (e™* = e- m4) .

Moreover since the term in @ containing sin ma must disappear from (17), the equation (1+) will
give c to a second approximation.
If we denote the coefficient of cos ma in the first approximate value of y, the ordinate of the
surface, by a, we shall have
ga ea

Bae Me a

and substituting this value of 4 in that of @, we have
en k-y em (h-y) erm (h-y) + en 2m (a-y)
= pow > Se ; 2 end PE ee Se
p=-ac gmk — gamh sin mv+3marc (em — ey
‘Lhe ordinate of the surface is given to a second approximation by (9). It will be found that
g PP y
(are e7™") (em ae e72mh re 4)
2 (e" a ent )®

sin 2m ... (18).

y = a cos ma — ma COS 297 Dh ee coaiane'sae cans

ma*
em pa e72mh ° 5

7. The equation to the surface is of the form

Y = 4 CoS MH — Ka? COS QMA vassccrccsccaceccesscecers (20);

where K is necessarily positive, and a may be supposed to be positive, since the case in which it is
negative may be reduced to that in which it is positive by altering the origin of # by the quantity
T Xr Ps 7
— or —, d being the length of the waves. On referring to (20) we see that the waves are sym-
m 4
metrical with respect to vertical planes drawn through their ridges, and also with respect to vertical
planes drawn through their lowest lines. The greatest depression of the fluid occurs when a =0

5 A r 3
or = +, &c., and is equal to a — a’ K: the greatest elevation occurs when a = + Fs or = + a &c.,

and is equal to a+a°K. ‘Thus the greatest elevation exceeds the greatest depression by 2a°X.
When the surface cuts the plane of mean level, cos mv —aK cos2ma=0. Putting in the small

i ; ’ ; 7 7
term in this equation the approximate value ma = a” we have cos mw = —aK = cos 5 + ak

—+

x oe)
5 4 Q9

whence a@ = + (> +

“ (2 aKkyr
4 2Q9

i &c. We see then that the breadth of each hollow,

measured at the height of the plane of mean level, is 3 + Laas , while the breadth of each elevated

T

: oi) tN y CIN

portion of the fluid is — — sa hi
2 Tv

It is easy to prove from the expression for K, which is given in (19), that for a given value
of X or of m, K increases as h decreases. Hence the difference in form of the elevated and
depressed portions of the fluid is more conspicuous in the case in which the fluid is moderately
shallow than in the case in which its depth is very great compared with the length of the waves.

Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES. 447

8. When the depth of the fluid is very great compared with the length of a wave, we may
without sensible error suppose / to be infinite. This supposition greatly simplifies the expressions
already obtained. We have in this case

Di =F Ce ne ESI 4 a ee sassiacaseeegeins cepaitiesonceensecieeel (20) 5

Y =A COS ML — LMA” COS AMA versscecrcosesescscessee (22)s

m
k=0, ats

the y in (22) being the ordinate of the surface.
It is hardly necessary to remark that the state of the fluid at any time will be expressed by
merely writing # — ct in place of w in all the preceding expressions,

9. To find the nature of the motion of the individual particles, let # + & be written for x, y +
for y, and suppose w and y to be independent of ¢, so that they alter only in passing from one
particle to another, while £ and y are small quantities depending on the motion, Then taking the
case in which the depth is infinite, we have

d

== u = — mace ™"*" cosm (w + — — ct) =— mace~™ cosm (w — ct) + mace-™ sinm (w— ct) .&
+m’ace~™ cos m (w — ct) .n, nearly,

d : ;

a =v =mace—"*" sin m (a + & — ct) = mace ™ sin m (w — ct) + m?ace-™ cos m (w — ct) .E&

— m’ace—™ sin m (# — ct) .y, nearly.
To a first approximation
E=ae-™ sinm(a—ct), n=ae—™ cos m (w- ct),
the arbitrary constants being omitted. Substituting these values in the small terms of the preceding
equations, and integrating again, we have
& =ae-™ sin m (@ — ct) + m*a°cte-?™,
yn = ae~-™ cos m (a — ct).

Hence the motion of the particles is the same as to a first approximation, with one important
difference, which is that in addition to the motion of oscillation the particles are transferred forwards,
that is, in the direction of propagation, with a constant velocity depending on the depth, and
decreasing rapidly as the depth increases. If U be this velocity for a particle whose depth below
the surface in equilibrium is y, we have

Qa\% 3 an
U= mace *™ = a? (=) Bes se bop ccc ACABCE EBERRON (ed)

The motion of the individual particles may be determined in a similar manner when the depth
is finite from (18). In this case the values of & and y contain terms of the second order, involving
respectively sin 2m (w — ct) and cos 2m (w — ct), besides the term in & which is multiplied by ¢.

The most important thing to consider is the value of U, which is
e2MY-2) 4 2m(y—A)

(e™
Since U is a small quantity of the order a’, and in proceeding to a second approximation the

velocity of propagation is given to the order a@ only, it is immaterial which of the definitions of
velocity of propagation mentioned in Art. 3, we please to adopt.

Vor, VIII. Parr IV. 3M

U=miae — EP SATs aiden tebe cet ckevaccevoeseee (24)

e7 mye 5

448 Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES.

10. The waves produced by the action of the wind on the surface of the sea do not probably
differ very widely from those which have just been considered, and which may be regarded as
the typical form of oscillatory waves. On this supposition the particles, in addition to their
motion of oscillation, will have a progressive motion in the direction of propagation of the waves,
and consequently in the direction of the wind, supposing it not to have recently shifted, and this
progressive motion will decrease rapidly as the depth of the particle considered increases. If the
pressure of the air on the posterior parts of the waves is greater than on the anterior parts,
in consequence of the wind, as unquestionably it must be, it is easy to see that some such pro-
gressive motion must be produced. If then the waves are not breaking, it is probable that equation
(23), which is applicable to deep water, may give approximately the mean horizontal velocity
of the particles; but it is difficult to say how far the result may be modified by friction. If
then we regard a ship as a mere particle, in the first instance, for the sake of simplicity, and put
U, for the value of U when y=0, it is easy to see that after sailing for a time ¢, the ship
must be a distance U,t to the lee of her estimated place. It will not however be sufficient to
regard the ship as a mere particle, on account of the variation of the factor e~*"’, as y varies from
0 to the greatest depth of the ship below the surface of the water. Let 6 be this depth, or rather
a depth something less, in order to allow for the narrowing of the ship towards the keel, and suppose
the effect of the progressive motion of the water on the motion of the ship to be the same as
if the water were moving with a velocity the same at all depths, and equal to the mean value
of the velocity U from y=0 to y=0. If U, be this mean velocity,

1 La? $
y= sf Udy = — (1 - “eal

On this supposition, if a ship be steered so as to sail in a direction making an angle @ with the
direction of the wind, supposing the water to have no current, and if V be the velocity with which
the ship moves through the water, her actual velocity will be the resultant of a velocity V in
the direction just mentioned, which, for shortness, I shall call the direction of steering, and of
a velocity U, in the direction of the wind. But the ship’s velocity as estimated by the log-line
is her velocity relatively to the water at the surface, and is therefore the resultant of a velocity V in
the direction of steering, and a velocity U, — U, in a direction opposite to that in which the wind
is blowing. If then E be the estimated velocity, and if we neglect U’*,

E=V -(U,- U,) cos8.

But the ship’s velocity is really the resultant of a velocity V+ U,cos@ in the direction of steering,
and a velocity U,sin@ in the perpendicular direction, while her estimated velocity is E in the
direction of steering. Hence, after a time ¢, the ship will be a distance U,¢ cos@ ahead of
her estimated place, and a distance U,¢sin@ aside of it, the latter distance being measured in a
direction perpendicular to the direction of steering, and on the side towards which the wind is
blowing.

I do not suppose that the preceding formula can be employed in practice; but I think it
may not be altogether useless to call attention to the importance of having regard to the magnitude
and direction of propagation of the waves, as well as to the wind, in making the allowance for
lee-way.

11. The formule of Art. 6 are perfectly general as regards the ratio of the length of the waves
to the depth of the fluid, the only restriction being that the height of the waves must be sufficiently
small to allow the series to be rapidly convergent. Consequently, they must apply to the limiting
case, in which the waves are supposed to be extremely long. Hence long waves, of the kind
considered, are propagated without change of form, and the velocity of propagation is independent
of the height of the waves to a second approximation. These conclusions might seem, at first sight,

Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES. 449

at variance with the results obtained by Mr. Airy for the case of long waves*, On proceeding
to a second approximation, Mr. Airy finds that the form of long wayes alters as they proceed,
and that the expression for the velocity of propagation contains a term depending on the height
of the waves. But a little attention will remove this apparent discrepancy. If we suppose
mh very small in (19), and expand, retaining only the most important terms, we shall find for

the equation to the surface
2

y= acosmaxe — 3 cos 2m a.

mh
Now, in order that the method of approximation adopted may be legitimate, it is necessary that

the coefficient of cos 2ma in this equation be small compared with a. Hence

a
7 and therefore

2

- a (bY 7c ee
= must be small, and therefore ; must be small compared with (=) . But the investigation

of Mr. Airy is applicable to the case in which z is very large; so that in that investigation
u

h\? ‘ ; e
is large compared with (3) . Thus the difference in the results obtained corresponds to a
h

difference in the physical circumstances of the motion.

12. There is no difficulty in proceeding to the higher orders of approximation, except what
arises from the length of the formule. In the particular case in which the depth is considered
infinite, the formule are very much simpler than in the general case. I shall proceed to the third
order in the case of an infinite depth, so as to find in that case the most important term, depending
on the height of the waves, in the expression for the velocity of propagation.

For this purpose it will be necessary to retain the terms of the third order in the expansion
of (7). Expanding this equation according to powers of y, and neglecting terms of the fourth, &c.
orders, we have

Eb - EP + CH EG/)U + (Chu Eh, £ +209" + 6.9,)
+ 20(f/ "+ fb)! + b,0/ + 0,0,)¥ - P°P - 2b, — 97>, = 0 verve (25)-

In the small terms of this equation we must put for @ and y their values given by (21) and (22)
respectively. Now since the value of @ to a second approximation is the same as its value to

a first approximation, the equation g@@,— c’p” =0 is satisfied to terms of the second order. But
2

the coefficients of y and :, in the first line of (25), are derived from the left-hand member of

the preceding equation by inserting the factor e~”’, differentiating either once or twice with
respect to y, and then putting y= 0. Consequently these coefficients contain no terms of the
second order, and therefore the terms involving y in the first line of (25) are to be neglected.

d
The next two terms are together equal to ¢ aa (p? +7). But

p+ op; = ma? ec,
which does not contain #, so that these two terms disappear. The coefficient of y in the
second line of (25) may be derived from the two terms last considered in the manner already
indicated, and therefore the terms containing y will disappear from (25). The only small terms

* Eneyclopadia Metropolitana, Tides and Waves, Articles 198, &e.
3M2

450 Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES.

remaining are the last three, and it will easily be found that their sum is equal to m‘a%c* sin ma, so
that (25) becomes
EP, — 8h + marc sin Ma =0,......00000sec006 (26).
The value of @ will evidently be of the form Ae~™ sin ma. Substituting this value in (26),
we have
(m?c* — mg) 4 + m‘a’c’ = oO.
Dividing by mA, and putting for A and c* their approximate values — ac, = respectively in

the small term, we have
me = g + ma*g,

3 3 2 42
whence Qe (=) (: im 4 ma) a (=) (1 * ere ).
m Lor r2

The equation to the surface may be found without difficulty. It is

y =acosma — 4 ma’cos2mw + 3 ma’ cos 8Ma*, 0.00. 406006+24(27):
we have also k=0, p= —ae (1 — 2 m’a’) e~™ sinma.

The following figure represents a vertical section of the waves propagated along the surface
of deep water. The figure is drawn for the case in which a = m . The term of the third order

in (27) is retained, but it is almost insensible. The straight line represents a section of the plane
of mean level.

13, If we consider the manner in which the terms introduced by each successive approximation
enter into equations (7) and (8), we shall see that, whatever be the order of approximation, the
series expressing the ordinate of the surface will contain only cosines of ma and its multiples,
while the expression for @ will contain only sines. The manner in which y enters into the
coefficient of cos rma in the expression for @ is determined in the case of a finite depth by
equations (2) and (3). Moreover, the principal part of the coefficient of cos 7 mw or sin rmw will
be of the order a” at least. We may therefore assume

p= dy" aA, (MOY 4. g-™4-) sin rma,

y = acosma +>, a’ B,cosrma,
and determine the arbitrary coefficients by means of equations (7) and (8), having previously
expanded these equations according to ascending powers of y. The value of c° will be deter
by equating to zero the coefficient of sin mw in (7).

Since changing the sign of a comes to the same thing as altering the origin of # by 4), it is
plain that the expressions for 4,, B, and c* will contain only even Pee of a. Thus the values
of each of these quantities will be of the form C+ Ca’ + C,a'+.

It appears also that, whatever be the order of approximation, the waves will be symmetrical with
respect to vertical planed passing through their ridges, as also with respect to vertical planes
passing through their lowest lines.

* It is remarkable that this equation coincides with that of the | sell to waves of the kind here considered, Reports of the British
prolate cycloid, if the latter equation be expanded according to | Association, Vol. v1. p. 448, When the depth of the fluid is not
ascending powers of the distance of the tracing point from the | great compared with the length of a wave, the form of the surface
centre of the rolling circle, and the terms of the fourth order be | does not agree with the prolate cycloid even to a second approx~-
omitted, The prolate cycloid is the form assigned by Mr. Rus- | imation.

Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES. 451

14. Let us consider now the case of waves propagated at the common surface of two liquids,
of which one rests on the other. Suppose as before that the motion is in two dimensions, that the
fluids extend indefinitely in all horizontal directions, or else that they are bounded by two vertical
planes parallel to the direction of propagation of the waves, that the waves are propagated with
a constant velocity, and without change of form, and that they are such as can be propagated into,
or excited in the fluids supposed to have been previously at rest. Suppose first that the fluids
are bounded by two horizontal rigid planes. Then taking the common surface of the fluids when
at rest for the plane wz, and employing the same notation as before, we have for the under fluid

rp &h
0,

—— + —

Os. paneot SLO Ea Dey Poe PEO:
da? dy* i)

dp
dy: = OL WIENS = bs) caseaeives poasteiieb een c's 6 (29),
dp

p= Hepy t+ eps

neglecting the squares of small quantities. Let h, be the depth of the upper fluid when in equi-
librium, and let p,, p,, p,, C, be the quantities referring to the upper fluid which correspond to
Pp; p; , C referring to the under: then we have for the upper fluid

eh, FP
z SEED NO) o ciclaichctctal givierele'y'e‘e‘e'e’s volae bas DOCOC DoscDOne coomnBadan >
dx dy’ oo
a =r ONWHEN MY) =P Stietdcodncaccnseacssteesetsesabsitecsescsess (OL)y
do,

D,='C;+ apy + ep, 7

We have also, for the condition that the two fluids shall not penetrate into, nor separate from each
other,

Lastly, the condition answering to (11) is

dp dp r rp io, ‘t
2 (p dy = py dy) = (05% = 0; =) = 0 cavccccvvresvescey (33),
dp do,
when C~C,+e(p-p)ure (pe -p,2) =o see eeeeeeeee (34).

Since C — C' is evidently a small quantity of the first order at least, the condition is that (33)
shall be satisfied when y=0, Equation (34) will then give the ordinate of the common surface of
the two liquids when y is put = 0 in the last two terms.

The general value of p suitable to the present case, which is derived from (28) subject to the
condition (29), is given by (13) if we suppose that the fluid is free from a uniform horizontal motion
compounded with the oscillatory motion expressed by (13). Since the equations of the present
investigation are linear, in consequence of the omission of the squares of small quantities, it will be
sufficient to consider one of the terms in (13). Let then

Pm A (GMO 5 g-™9) alan 0 ony onda eoabe sods ovo osundaanens (86)

452 Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES.

The general value of , will be derived from (13) by merely writing — h, for h. But in order
that (32) may be satisfied, the value of @, must reduce itself to a single term of the same form as
the second side of (35). We may take then for the value of @,

$,=A, (CAL RE A ils LLNS ay77, 1 Nt A et (36).
Putting for shortness
et ot em = S, em a e7 mh = D,
and taking S|, D, to denote the quantities derived from S, D by writing h, for h, we have from (32)
DAD) A= Oseapasienitsos <eqein cee ree Mee tte eee. cca see sees (OT):

and from (33)
p (gD —meS) A +p, (gD, + MEOS)A, = 0 vicsecceeseeceeeesees (38)

Eliminating A and 4, from (37) and (38), we have

The equation to the common surface of the liquids will be obtained from (34). Since the mean
value of y is zero, we have in the first place

We have then, for the value of y,

A =A COEING DS weniate \olcure cote oe calacaneanteria ce ccsdenne oc dpaivss tents on CAL);

where
mep,d S'-pAS DD p,A 8S, - pAS
aa = 7B jet See ety " (42).
& p-p, - ¢ pSD,+pSD
Substituting in (35) and (36) the values of 4 and A, derived from (37) and (42), we have
p=- 5 (Cree At) sinniat Raanenceenvee ane mean aeaee (43),
ac h h :
%=p (em tp Mt) Sin Ma ...005.0005 Brose sce ane's (44).

Equations (39), (40), (41), (43) and (44) contain the solution of the problem. It is evident that
C remains arbitrary. The values of p and p, may be easily found if required.

If we differentiate the logarithm of c? with respect to m, and multiply the result by the product
of the denominators, which are necessarily positive, we shall find a quantity of the form Pp+Pp,
where P and P, do not contain p or p,._ It may be proved in nearly the same manner as in Art. 4,
that each of the quantities P, P is necessarily negative. Consequently c will decrease as m increases,
or will increase with }. It follows from this that the value of @ cannot contain more than two
terms, one of the form (35), and the other derived from (35) by replacing sin ma by cos ma, and
changing the constant A: but the latter term may be got rid of by altering the origin of w.

The simplest case to consider is that in which both h and h’ are regarded as infinite compared
with A. In this case we have

p= -—ace-™ sin ma, p,=ace™ sin ma, ef = PoP & 5 ¥ =a cos M2,
PrP

the latter being the equation to the surface.

Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES. 453

15. The preceding investigation applies to two incompressible fluids, but the results are
applicable to the case of the waves propagated along the surface of a liquid exposed to the air,
provided that in considering the effect of the air we neglect terms which, in comparison with those
retained, are of the order of the ratio of the length of the waves considered to the length of a wave
of sound of the same period in air. Taking then p for the density of the liquid, p, for that of the
air at the time, and supposing h, = «© , we have

ge Nie ee - ( ae,
Cm pS+pD mS\" ae ueely:

If we had considered the buoyancy only of the air, we should have had to replace g in the

formula (14) by Pe; g. We should have obtained in this manner

Pe AC nt IEE - 85 (1 - 4)
m ps mS Pp :

Hence, in order to allow for the inertia of the air, the correction for buoyancy must be increased
i : D :
in the ratio of 1 to 1 + x The whole correction therefore increases as the ratio of the length of a

wave to the depth of the fluid decreases. For very long waves the correction is that due to
buoyancy alone, while in the case of very short waves the correction for buoyancy is doubled.

Even in this case the velocity of propagation is altered by only the fractional part P of the whole ;

and as this quantity is much less than the unavoidable errors of observation, the effect of the air in
altering the velocity of propagation may be neglected.

16. There is a discontinuity in the density of the fluid mass considered in Art. 14, in passing
from one fluid into the other; and it is easy to show that there is a corresponding discontinuity in
the velocity. If we consider two fluid particles in contact with each other, and situated on opposite
sides of the surface of junction of the two fluids, we see that the velocities of these particles resolved
in a direction normal to that surface are the same; but their velocities resolved in a direction tan-
gential to the surface are different. These velocities are, to the order of approximation employed

p

A ; nia d do
in the investigation, the values of as and ae when y=0. We have then from (43) and (44), for

the velocity with which the upper fluid slides along the under,
(S, 5)
mac \— + —] Cos mwa.

D

17. When the upper surface of the upper fluid is free, the equations by which the problem
is to be solved are the same as those of Art. 14, except that the condition (31) is replaced by

d d*
aa) -¢ ev: =a Os mV E N/a auetae tate ateleisie.s ejeie aa c <a 40)) is
dy dx d

and to determine the ordinate of the upper surface, we have

dg, = 0,

C,+&p,9 +p,

where y is to be replaced by —h, in the last term. Let us consider the motion corresponding to
the value of d given by (35). We must evidently have

p, = (4 "4 + Be-”Y) sin ma,

454 Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES.

where A, and B, have to be determined. The conditions (32), (33) and (45) give
DA+44,-B =0,
p(gD-—mec’S) A+ p,(g+ me’) 4 —- p,(g- mc’) B,=0,
(g + me’) eA, — (g - me’) eB, =0.
Eliminating 4, 4, and B, from these equations, and putting

aoe

a

we find
(p S'S, + p,DD,) C - p(SD, + SD) C+ (p — p,) DD, = 0. ... (46).

The equilibrium of the fluid being supposed to be stable, we must have p,<p. This being
the case, it is easy to prove that the two roots of (46) are real and positive. These two roots
correspond to two systems of waves of the same length, which are propagated with the same
velocity.

In the limiting case in which P =O, (46) becomes

SSC - (SD, + S,D)€+ DD, =0,

é D D : is ales A :
the roots of which are = and —, as they evidently ought to be, since in this case the motion of

S Ss
the under fluid will not be affected by that of the upper, and the upper fluid can be in motion
by itself.
SD +S.D ert) _ _-mieny
The former of these roots corresponds to the waves propagated at the common surface of the fluids,
while the latter gives the velocity of propagation belonging to a single fluid having a depth equal
to the sum of the depths of the two considered.

When p, = p one root of (46) vanishes, and the other becomes

; D F
When the depth of the upper fluid is considered infinite, we must put = =1 in (46). The

(p — p,) D
+p,
propagated at the upper surface of the upper fluid, and the latter agreeing with Art. 15.
When the depth of the under fluid is considered infinite, and that of the upper finite, we

‘

two roots of the equation so transformed are 1 and , the former corresponding to waves

must put sad =1 in (46). The two roots will then become 1 and sleet BEL . The value of the
p pS, + p,D

former root shows that whatever be the depth of the upper fluid, one of the two systems of
waves will always be propagated with the same velocity as waves of the same length at the sur-
face of a single fluid of infinite depth. This result is true even when the motion is in three
dimensions, and the form of the waves changes with the time, the waves being still supposed to
be such as could be excited in the fluids, supposed to have been previously at rest, by means of
forces applied at the upper surface, For the most general small motion of the fluids in this case
may be regarded as the resultant of an infinite number of systems of waves of the kind con-
sidered in this paper. It is remarkable that when the depth of the upper fluid is very great, the
root (= 1 is that which corresponds to the waves for which the upper fluid is disturbed, while
the under is sensibly at rest ; whereas, when the depth of the upper fluid is very small, it is the
other root which corresponds to those waves which are analogous to the waves which would
be propagated in the upper fluid if it rested on a rigid plane,

‘

Mr. STOKES, ON THE THEORY OF OSCILLATORY WAVES. 455

When the depth of the upper fluid is very small compared with the length of a wave, one
of the roots of (46) will be very small; and if we neglect square and products of mh, and (, the
equation becomes 2 pD¢— 2 (p — p,) mh, D = 0, whence

— eh, oc cae ro ae Ae saan Benak db) lees (47).

‘These formule will not hold good if mh be very small as well as mh,, and comparable with it,
since in that case all the terms of (46) will be small quantities of the second order, mh, being
-regarded as a small quantity of the first order. In this case, if we neglect small quantities of the
third order in (46), it becomes

4pG—4mp(h+h)C+4(p—p,) mhh, = 0,

whence a au th, + A/ (h-hh) + ale i,t madosooUnd s6AgnECeC (48).
y P

Of these values of c*, that in which the radical has the negative sign belongs to that system of
waves to which the formule (47) apply when h, is very small compared with h.

If the two fluids are water and mercury, P is equal to about 13.57. If the depth of the

_ water be very small compared both with the length of the waves and with the depth of the
mercury, it appears from (47) that the velocity of propagation will be less than it would have
been, if the water had rested on a rigid plane, in the ratio of .9624 to 1, or 26 to 27 nearly.

G. G. STOKES.

Vor. VIII. Panr IV. SN

XXXIV. On the Internal Pressure to which Rock Masses may be subjected, and
its possible Influence in the Production of the Laminated Structure.
By W. Hopkins, Ese., M.A., F.R.S., &c.

[Read May 3, 1847.]

Onr of the most curious phenomena in the constitution of rock masses, consists in the laminated
structure which pervades so large a portion of the older sedimentary formations, producing what
is called their slaty cleavage. In some cases, this lamination is comparatively coarse and ill-defined,
but in others (as in the roofing slates) it is so fine and regular as to leave no doubt of its being
the result of some kind of molecular action of the constituent particles on each other, analogous to
that of crystallization, and not the direct and immediate mechanical effect of external forces
acting on the mass. But still it would seem very possible, that these external forces may
maintain the mass in a state of internal constraint which may possibly be a condition favourable
to the production of the laminated structure, and observations have lately been made which seem
to afford some confirmation of this notion. Professor Phillips, some years ago, and Mr. Sharpe,
more recently, have recognized some curious and interesting facts respecting the frequent distortion
of fossil shells, and other organic remains, from their original well-known forms; and these distortions
appear to indicate certain relations between the positions of the cleavage planes and the directions
of the internal pressures which must have produced the distortions in question. These distortions
of determinate organic forms indicate, in fact, corresponding distortions in those elements of the
mass in which they are respectively comprized. To explain the nature of the tensions or pressures
acting on any such element and the distortion produced by them, let us denote by s a small plane
surface passing through any point P. Generally, there will be an action between the particles
(M) on one side of this small plane, and 1’, those in contact with them on the opposite side.
If s be sufficiently small, we may represent the whole action of M on M’ by ps, a force having
a determinate direction, which we may suppose to make an angle 6 with the normal to s. Then will

pscosdé, and pssind,
be the normal and tangential parts of the whole action of M on M’, and
—pscosd, and —pssind,

will manifestly be the same parts of the reaction of M’' on M. If the normal force be a pressure,
it will only tend to preserve the contact of the particles immediately on opposite sides of s; but if
that force be a tension, then will ps cos 6 tend to separate these particles by motions normal to s,
and in opposite directions. In all cases there will be likewise forces equal to ps sin 6, and
— ps sin 6, tending to separate any one particle immediately on one side of s, from the particle
originally in contact with it on the other side of s, by communicating to these particles, motions in
opposite directions parallel to the plane of s. If this plane assume different positions by moving
about P as a fixed point, the normal and tangential forces acting on it will have different values,
assuming maxima or minima values for certain determinate positions of s, and it is on these

particular positions of s that the distortion of a small portion of the mass about P, and that of any —

organic form contained in it, will depend. Generally, The linear dimensions of the element will be

Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 457

altered by extension or compression, and it will also be twisted, so that if it were originally a
rectangular parallelopiped it will become an oblique-angled one, and these changes of form will be
indicated by the corresponding distortions of the organic remains. Now, if the directions of the
cleavage planes were originally determined by the state of internal tension and pressure of the mass,
it would seem probable that they would be perpendicular to the directions of greatest, or to those
of least normal pressure, or that they would coincide with the planes of greatest tangential action.
‘These hypotheses must be tested by the evidence derived from the organic forms, as will be
explained in the sequel, but for that purpose it will be necessary in the first place, to investigate
the relative positions of the lines and planes just mentioned. This investigation will form the first
Section of this memoir.

SECTION I.

Relative positions of the lines of maximum and minimum tension, and planes of
maximum tangential force in the interior of a continuous mass.

1. Taxrne any point P of the mass, let it be made the origin of co-ordinates wyx. Let the
small plane s be conceived as before to pass through P, and let the forces upon it in the positions
specified be denoted as follows, all being referred to a unit of surface.

(1.) When a perpendicular to the plane coincides with the axis of w, let

The normal force =A; The tangential force = le parallel to y,

(2.) When a perpendicular to the plane coincides with the axis of y, let

(Gu parallel to x,

The normal force = B; The tangential force = {%
ser eee rotons a.

(3.) When a perpendicular to the plane coincides with the axis of x, let

A” parallel to w,
BGs Pelee stoletss''o y:

Between the six accented quantities there are three essential relations, which are easily found.
On the three co-ordinate axes at P, construct an indefinitely small parallelopiped whose edges
are 6x, dy, and dx. The six equations of equilibrium of this element will express the conditions
that the sums of all the resolved parts of the forces parallel to the co-ordinate axes shall respectively
be equal to zero; and that the moments of the forces with reference to three axes, shall also
severally be equal to zero. Let us take the three latter conditions, lines through the center of
gravity of the element and parallel to the co-ordinate axes being taken for the axes of the com-
ponent couples. The tangential force parallel to the axis of w on the side dv.dx being 4’,

The normal force = C; The tangential force = |

that on the opposite side will be — (4’ + — dy); and the couple resulting from these forces
y

about the axis parallel to x, will be
dA

oy.
ia oy) dwdx c-

A'dadz. Sy + (A' +

oF, omitting small terms of the fourth order,

A’ da dy dx.

458 Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES.

,

Similarly, the couple arising from the forces B' and B' + ane 6a about the same axis parallel to 2,

will be
— B sadysz.

Also the moment of the normal forces A, B, C, with reference to the above-mentioned axes,
will be zero, always omitting small quantities of the fourth order. Consequently the whole moment
of the forces on the parallelopiped with reference to the axis parallel to that of x, will be

(A' — B') dadydz;

which must = zero by the conditions of equilibrium; and therefore

A'= B’.

In exactly the same way we find, by taking the moments with reference to the axes parallel

respectively to those of y and 2,

A" =C’,

j ile = Ge
By means of these three relations the six accented quantities are reduced to three independent
quantities. ;

2. Let us now conceive a plane to meet the three co-ordinate planes so as to form with them
a tetrahedron, whose vertex is at the origin P. Suppose the exterior normals to the three faces
formed by the co-ordinate planes to point respectively towards the positive directions of « y and
x; and let a 8 and y be the angles which the normal to the base of the tretrahedron makes
with the co-ordinate axes of w y and x. Also let s denote the area of the base, and s’ s” and s’”
the areas of the sides of the tetrahedron perpendicular respectively to the axes of w y and x, all
these quantities being indefinitely small.

Again, let ps denote the whole resultant force acting on s, and let X and v be the angles
which its direction makes with lines parallel to the co-ordinate axes of # y and x, this direction being
exterior to the tetrahedron. Then, in order that the tetrahedron may be in equilibrium, we must —
have

ps.cos\ = As’ + A's” + A”s'”,

ps.cosp = Bs” + B's’ + Bs’,

ps.cos v = Cs’ + C's’ + C's";
but

, u" m

s s B s
—=cosa, —=cosp, —=cosy;
8 8 s v"

making these substitutions, and also putting

fire Cl Dy
Vig = ox = i.
A = 8B =F,

we shall have
p.cos\ = Acosa + F'cosB + E cosy, |

- p.cosp= Bcos3 + Fcosa + Dos yy feeseereseeeeeeeee (@);
p.cosy = Ccosy +E cosa + Deos 8;

formule in which the notation agrees with that of M. Cauchy (Ewercises de Mathématique, —
Vol. 11. p. 48).

Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 459

If 6 denote the angle between the direction of p and the normal to s, we shall have p cos dé
for the whole normal force acting on the area s in a direction exterior to the tetrahedron, and
p sin 6 the whole tangential force acting on the same area, Our first object will be to determine
a B and y, or the position of the base (s) of the tetrahedron, so that the normal action upon it,
p cos 6, shall be a maximum. We shall afterwards have a similar investigation with reference to
the tangential force p sin é.

We have cos 6 = cos) cos a + cos p cos 3 + cosy cosy,
whence we immed ately obtain
pcos d = Acos’a + B cos* 8 + C cos’ y + 2D cos 3 cosy + 2E cosa cosy + 2F cosa cos 3, ... (1);
and since COST ICOS Sets COSdnyE— al al sents stun te ceria oe sistsilesistacislnn os scisiei esis oleae (2);
we have (LZ being an arbitrary multiplier),
(A + L) cos’ a + (B + L) cos* B + (C + L) cos’ y + 2Dcos cosy + 2E cosa cos vy

+ 2F cos a cos 3 = maa.
Hence,

{(4 + L) cosa + E cosy + F cos 8} sina = 0
{(B + L) cos B + Dcosy + Fos at sin B = 0 ) ver--ees (0).
ia eh meatus Em Sess) eo

/
To satisfy these equations together with

cos’ a + cos’ 3 + cos’ y = 1,

we must equate the first brackets to zero. We thus have four equations from which LZ may be
eliminated, and a 3 and vy determined.

If we multiply the first factors on the left-hand sides of equations (b) by cos a, cos B and cos
respectively, and add them together, we have by virtue of equations (a),

L = — Pcos 0,
and substituting for Z in equations (6b), we have
pcosdcosa = A cosa + F'cos 3 + Ecosy,

= pcos);
cos 6 cos a = cos X.
Similarly, cos 6 cos 3 = cos p,
cos 0 cos y = cos v5
whence cos? 6 = 1,
6=0,

which shews that when the resultant force p is a maximum or minimum, its direction coincides
with that of the normal to the plane s. Consequently, also, the tangential force p sin d then
becomes = zero.

This value of 6 gives, L=-p,

and substituting for LZ in equations (b) we have,
(A - p) cosa + F' cos 8 + E cosy = 0 |
F cosa + (B - p) cos B + Deosry = 0 (rrtttttt (c),
E cosa + D cos 8 + (C — p) cosy = 0 \

460 Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES.

and eliminating cos a, cos 3, and cos y by cross multiplication, we obtain
(A - p) (B= p) (C —p) - D’ (A-p) - EX (B-p)- F°(C-p) + 2DEF=0.

If we take the three values of p deducible from this equation and substitute them successively
in equations (c), those equations combined with (2) will give three distinct systems of values for
cosa, cos 3, and cosy, belonging (as is well-known) to three lines perpendicular to each other.

Hence, it follows, that there is at every point (P) of a continuous solid mass under extension
or compression, a system of three rectangular axes, such, that if the small plane (s) at P be so
placed that its normal shall coincide with one of those axes the whole resultant action on s shall be
normal to it, the tangential action upon it being then equal to zero. These three axes are called
the awes of principal pressure or tension with reference to the point P.

3. M. Cauchy, in the Memoir above referred to, converts equation (1) into the equation
to a surface of the second order, by putting

1
pcosd= + 3» rosa = a, rcos 3 =y, TCOSy = x.

The inverse of the square of any radius vector will manifestly be a measure of the normal
action on a small plane through P perpendicular to the radius vector, the axes of principal pressure
or tension coinciding with the axes of this surface of the second order. We may remark, that of
the three principal pressures or tensions above determined, one will be a maximum and another a
minimum, while that of an intermediate value will be neither, though it satisfies the conditions
d.pcoso d.pcos6 ene ke
ft = 0, and ae =0. Itis, in fact, that value of p cos 6 which is represented by the —

a
inverse of the square of the mean axis of the surface, and that mean axis, considered as a parti-
cular radius vector, is a maximum with reference to one principal section, and a minimum with

d 7/1 d 1
reference to the other to which it belongs, so that though a (=) =0, and — (=) = 0 when
med

. dB\r

7
y = mean axis, all the conditions of a maximum or minimum are not satisfied.

I make these remarks here because a similar mode of geometrical representation may be found
useful in explaining the results obtained in the succeeding part of the investigation. a

4. I shall now proceed to investigate the positions of the plane s passing through P, when the”
tangential action upon it is greatest, é.e. when p sin 6 = maa.

To simplify our formule, we may here take the axes of principal pressure or tension as the
co-ordinate axes. In this case there will be no tangential force on the plane s when it is perpendi-
cular to any of these axes, and consequently, we must have

DS — 07) ais —=10, ai 0;
and, therefore, equations (a) give
p’ = 4’ cos’ a + B’ cos* B + C* cos’ y,
and equation (1) gives,
pcosd = Acos'a + B cos’ B + C cos’.
Hence we have
p’ sin’ } = A? cos’ a + B® cos’ B + C* cos’ y — (A cos*a + B cos” B + C cosy)’,
the quantity which is to be made a maximum subject to the condition
cos’ a + cos’ 3 + cos’ = 1.
By virtue of the last equation, we have
p® sin? 6 = (cos a-+cos? B + cos? ry) (4? cos® a + B? cos® 8+C? cos* y) — (A cos* a + B cos 8 + C cos" y)s

—

Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 461

which by reduction gives
p’ sin? d = (A — B)’ cos’ a cos’ 8 + (A — C)? cos? a cosy + (B — C)’ cos’ B cos* y = max »+2((3).
Hence, if Z be an arbitrary multiplier, we obtain
{(4 — B)’ cos’ B + (A — C)’ cos’ + Li cosasina = 0
(4 - B) cosa + (B- C)*cos'y +L} cosBsinB=04 ay,
}(4 — C)’ cos’ a + (B- C)’ cos’ B + Lt cosy siny =0
Let us first suppose these equations satisfied by equating, in each case, their first factors to

zero; and for brevity put
P=A-B, Q=A-C, R=B-C;

« P=Q-R.

Now, substituting 1— cos’ a — cos’ 3 for cos’ ry, and eliminating Z between the first and third,
and the second and third equations, we obtain

(P? — Q? — R?) cos? B — 2Q* cos? a + Q? = 0,
(P? - Q — R?) cos? a — 2R* cos? 3 + R?=0;
or since P*=(Q-—- R)’,
P*— Q — R’ = - 2QR;
. 2R cos’ 3 + 2Qcos*a — Q=0,
2Qcos*a + 2R cos’ 3 —R = 0,
which cannot hold simultaneously unless @= R, and .. P=0; or A=B. This mode, therefore.

_of satisfying equations (d) is not admissible.

Again, we may satisfy those equations by
sina=0, cosB=0, cosy =0,
a system of equations which also satisfy (2). In this case the normal to the small plane s will
coincide with the axis of w, i.e. with an axis of principal pressure, and therefore, these values
ought to give the tangential force = zero, as they do. Zero is in fact a minimum value of that force.
Similar conclusions hold with reference to the axes of y and x for the following systems of values.

cosa=0, sinfB=0, cosy =0;
cosa=0, cosf=0, siny =0,
Finally, we may satisfy equations (d) by
cosa = 0,
P* cos’ a + R’ cos’ y + L =0,
Q’ cos’ a + R? cos’ B + L = 0.
Eliminating L, we have
cos’ 3 = cos* y ;
2ICOR, Graal;
“, B= = + 45°.
Two other systems of values may evidently be obtained in a similar manner, and thus equations
d) and (2) are satisfied by the three following systems of values:
a= 90", Bay = = 45°,
B= 90°, ST A I | ee eee (e).
7 = 90°, ee |

462 Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES.

5. If 7, T:, and 7, be the corresponding values of the tangential force, we have
TP =4(B-O%, TP=1(4-0C), TR=1(4A-B)*

If A, B, C be taken as they always may be, so that A shall be tke greatest and C the least,
T. will be the greatest of these values, and I shall shew that it in fact is the only one which
satisfies the conditions of being a maximum. ‘To do this, and to explain the relations of these
particular values of the tangential force to its general values, it will be convenient to have recourse
to a geometrical representation, analogous to that before spoken of with reference to the normal
forces. For this purpose assume

5
: e
psind = T,—,

Pe

a@=rcosa, y=reosP, x=7rcosy;
where 7’, denotes a constant force, -and ¢ a constant line. Then equation (3) becomes
Tic = Paty + Qa a Rig? 5. ....cese- 00 (4)5

the equation to a surface such that the inverse of the square of the radius vector from the point P,
will be proportional to the tangential force on the plane s when perpendicular to that radius vector.

To find the intersections of the surface and the co-ordinate planes, put w =0, y = 0, and x =0,
consecutively ; we thus have

a
-= & 262)
yx R

ez= + qe
th
= + —2
vy Pp’?

as the equations to the intersections, each of which consists of two equal hyperbolas referred to the
asymptotes as axes of co-ordinates, as repre- (Fig. 1.)
sented in the annexed diagram. PA, making
equal angles with the two co-ordinate axes in
the plane of the paper, is the semi-axis major,
and minimum radius vector in the hyperbola
whose vertex is 4, Its position and that of
PA’ correspond to the first, second, or third of
the systems of values (e) of a 3 and y, ac-
cording as the plane in which the hyperbolas
lie is that of yx, wz, or ay. Also the values

f ates th r ctivel
of —— in these cases respectively are
AP P y

Roi Q 1 Pil
20, ie Di mere nie Tye

which are proportional to
Rk, Q P;
orto B-C, A-C, A-B,
or to the three tangential forces previously

designated by
T; ? T., Ts.

Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 463

In each case PA is a minimum value of the radius vector with reference to the hyperbola
of which it is the semi-axis major, and consequently PA is the position of the normal to the plane s,
when the tangential force upon it is, in the same relative sense, a maximum. It still remains
to be determined whether Pd is also a minimum value of the radius vector in a section of the
surface made by a plane through PA and the co-ordinate axis perpendicular to the plane of
the paper. For this purpose, let this last-mentioned axis be first taken as that of wv, and let

w=rcos#, ye=rsin@sing, 2#=rsin@.cos¢,

r @ and @ being the ordinary polar co-ordinates. Substituting these values in the equation (4)
to the surface, and putting @ = 45°, we obtain
ica) Ba Qe: PE Qt UREN
Th 7 = 2S aint -(—E* -—) Si hogondsosodeaod( @))s
oi 9 (2) 4

< ~

the polar equation to the section through the axis of w and the axis of either hyperbola in the
plane of yzx.
Similarly, putting
y =7rcos0, z=rsin @sin d, wv = rsin @.cos d,

we obtain

Ge 22 Tee

P? 2 2:
eee ee Ae

: =F) Bit Bos sessee ee. (6),

the polar equation to the section of the surface by a plane through the axis of y and the axis of
either hyperbola in the plane of vz.

Again, putting

x=rcos@, y=rsin@sin d, w= rsin @ cos,
we have
e ere Fete 2 R i. ae
i = ee arene (= zs — =) Gite: @bponensconsaase( (Th)
Tr 2 2 4

the equation to the section through the axis of x, and the axis of either hyperbola in the plane of vy.
d
Differentiating (5), (6), and (7), and putting a = 0, we obtain in the several cases,
{(P? + Q’) -— {2(P’ + Q’) — R*} sin’ 6} sin @ cos @ = 0,
{(P? + R*) — {2 (P? + R®) — Q’| sin’ O} sin @ cos @ = 0, peeerereeeeee (A).
$(Q’ + R*) — §2(Q® + R*) — P*} sin’ 0} sin @ cos @ = 0,
Each of these equations may be satisfied by
sin@=0, cos@=0.

The first corresponds to r= ¢@, the axis from which @ is measured being an asymptote to the
curve. The second gives @ = 90°, and therefore r = AP, which is consequently either a maximum
or a minimum value of r with respect to the curve in which r and .@ are the variable co-
ordinates. Now since r = © when @=0, or 180°, and x = AP when 6 = 90°, it is manifest that

; dr .
AP must be a minimum value of r, provided 76 ® not rendered zero by any value of @

: dr
between 0 and 90°; but if, on the contrary, do become zero for some value of @ between those

limits, the corresponding value of r must be a minimum, in which case PA will be a mawimum
Vou. VIIT. Parr IV. 30

464 Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES.

value, since maxima and minima occur alternately. To ascertain whether any value of @ between
dr , ‘
0 and 90° does render TT) = 0, we must see whether such value can be derived in any of the

three cases above, by equating to zero the expressions within the brackets in equations (f).
Taking the first of those equations, we have

P+ Q@
PF Qk
which will give a value of @ between 0 and 90°, provided the fraction be positive and less than
unity. Now the difference between the numerator and denominator

=P?+Q-R,
and A, B, and C being taken in order of magnitude, and 4 the greatest, Q (= 4A—C) is greater than
R(=B-C). Consequently P* + Q’ — R° is positive, and the denominator of the above fraction
is positive and greater than the numerator, and sin @ is possible. The value of 7 corresponding to
the value of @ thus obtained, will be a minimwm, and therefore PA will in this case be a maximum.
Hence it appears that PA is a maximum value of the radius ‘vector with reference to the section of
the surface by a plane through the axis of 2, while it is a minimum with reference to the section,
perpendicular to the former, made by the plane of yx. In this case then PA is neither a maximum
nor minimum value of the radius: vector of the surface.

Exactly the same conclusion may be drawn from the third of equations (f), in which case the
two sections to which PA is common, and one through the axis of z, and that made by the
plane of vy. }

The annexed figure (2) represents the curve in each of the above cases referred to r and @,
CPC’ being in the first case the axis of w, and in the
second the axis of x. PB and PB’ represent the two
minima radii vectores in these sections.

It remains to consider the second of equations (f),
which gives

sin? @ =

(Fig. 2.

sin’@ = ae é
2(P*? + R’) - Q
Here, the denominator — the numerator = P®? + R? — Q’.
Now P=Q-R, (Art. 4);
- P+ R’- Q@=2R’-—2RkQ
=-2R(Q-R),
which, since Q is greater than R, shews that the deno-

minator is less than the numerator. Consequently there
is no value of 9 between 0 and 90°, in this case, which

dr
renders — = 0, and PA is here a minimum value of

dé
the radius vector, i.e. in the section made by a plane through the axis of y. PA is also a minimum
for the section made by the plane of wz. Consequently if figure (1) represent the plane of az, each
of the four equal lines PA is an absolute minimum value of the radius vector of the surface,

1 : : aK
and PH represents the absolute maximum value of the tangential force. The positions of these

lines correspond to the following system of values of a 8 and vy,
G=90))” y=a = 45°,

4

Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 465

the second of the systems (c), (Art. 4), and the corresponding value of the tangential force is
=3(4-0),

the absolute maximum value of the Eis: force acting on a plane of indefinitely small and
constant area (s), passing through any assigned point P of the solid body, and capable of assuming
any angular position about that point. For this maximum value, the normal to the small plane,
maylie in two positions, both in the plane of ws, and bisecting the angleb etween the axes of
a and x, the one above and the other below the axis of wv, those axes being so taken as to coincide,
the former with the direction of the greatest principal tension at P, and the latter with that of the
least. If one of the principal tensions be changed into a presswre, it must be regarded as a
negative tension, and therefore as the Jeast principal tension, and its direction taken as the
axis of x. In this case we shall have T,=4(4+C). If there be two pressures, the greatest
will =-—C. If all the principal forces be pressqwres, the least pressure will be — A, and the
greatest pressure — C, and therefore 7,=4(C- A). Thus 7, will in all cases be the algebraical

2
difference of the greatest and least principal tensions, considering pressures as negative tensions. *

6. As an elucidation of the subject, I shall consider a few particular cases.

(1) Suppose B= C; the normal tensions will be the same for all positions of the plane s, in
_ which its normal lies in the plane yx, and there will be an infinite number of positions ‘of the
plane s corresponding to the maximum tangential action, such that the locus to the normals of s
will be a conical surface whose axis is that of w, the semi-vertical angle of the cone being 45°.

(2) If the mass at any proposed point (P) be acted upon only by two tensions acting as principal
tensions, these must be considered as the axes of a and y, the axis of x, that of least principal
_ tension (supposed here = zero) being perpendicular to the plane of the two tensions.

(3) If there be only two principal tensions, as in the last case, but one of them become a
pressure, the direction of this latter must be taken as the axis of x, that of least tension.

(4) If both these principal tensions become pressures, the line perpendicular to the plane in
which they act, must be taken for the axis of , (the axis of greatest principal tension), and the
direction of the greatest pressure for the axis of z.

The axes of # and x, those of greatest and least principal tensions being known, the two positions
of the plane of maximum tangential action are immediately known.

gh eee rs 4

(5) Let PQRS represent a plane section of an elementary parallelopiped of the body parallel
to two opposite sides, and suppose PQRS a square. Let the forces on the
element be entirely tangential and parallel to the plane of the paper, there
being no force perpendicular to that plane. Then (Art. 1) the tangential
force on each side of the element will be the same; let it =f and act on each
side in the directions indicated by the arrows. Also let Pqrs be the section
of the same element, supposing the forces f not to act; then it is manifest that
_ these forces produce an extension = SQ — Sq in the direction SQ, and a com-
t pression = Pr — PR in the direction RP perpendicular to SQ. In fact the
i forces f may be resolved into f. cos 45° parallel to §Q, extending each particle in that direction,
and an equal force compressing the particles perpendicular to SQ. The former will act as a

_ principal tension, the latter as a principal pressure. If A and —C be their values referred to a
unit of surface, we must have

(Fig. 3.)

ee ra

A. QR sin 45° =f. QR cos 45°,
and C. QR cos 45° =f. QR sin 45°;

A=f, and C=f;

* Since 7? =(A—CY, Ty=4(A—C). All notice of the negative sign is omitted in the text, as altogether unessential,

BO?

466 Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES.

and therefore the greatest tangential force
=}(4+0)=f,

and it acts in the planes PQ and PS;; i.e. it is the impressed tangential force, as it is sufficiently
manifest it ought to be.

(6) It is of some importance with reference to the particular application of these investigations
which is here contemplated, to remark that when the general mass is so acted on by external forces
that its different elementary portions are subjected in very different degrees to the kind of distortion
represented in fig. 3., there may be a great extension or compression at particular points without
a correspondent increase or decrease on the same scale in the general dimensions of the mass.
Indications of such local extension and compression seem to be frequently indicated by the distortion
of organic remains.

SECTION II.

7. Orcanic remains, more especially shells, are usually found in greatest abundance along
those surfaces within a fossiliferous mass, which we recognize as planes or surfaces of separation
between contiguous beds. These shells, especially the flatter ones, will generally be found with
their flatter surfaces parallel to the surface of the bed on which they lie, and such may also be

expected to be the case as a general rule, with respect to shells contained within a bed instead of —

being between two contiguous beds. The first pressure to which these shells was subjected must
have been that due to the weight of the superincumbent beds deposited upon them, while the whole
remained undisturbed. If the shell yielded to this pressure it would become flattened, and fre-
quently also extended in length or breadth, or in all directions according to the nature of the shell.
It would seem probable that the proportions of the linear horizontal dimensions would not be
much altered by this vertical compression, but the possibility of its being otherwise should not be

forgotten by the observer. It may also be remarked, that should any horizontal elongation take

place from this cause in one direction more than another, that direction can only have reference to
the shell itself, and not to any fixed lines in space, unless it can be shown that the position in which
the shell was originally imbedded bore some relation to such lines, as for instance, that the median
lines from the beak to the margin in different shells should have been parallel to some common direction.
Any such law, however, would seem to carry with it the highest degree of a priori improbability.
When the mass became elevated and dislocated, especially in the degree in which such has been
the case in most of the ancient fossiliferous rocks, it would generally be subjected to great
pressures and tensions; but it is of the first importance to remark, that none but comparatively
small pressures or tensions could be called into action in the direction of the strike of the beds,
by their elevation into straight, or approximately straight anticlinal ridges; and that, consequently,
two of the directions of principal tension or pressure must lie in a vertical plane perpendicular to
the direction of the anticlinal line and strike of the beds, with which the third axis of principal
tension must coincide. Now in this elevation, it is highly probable that the mass will generally

be extended in some directions, and I consider it almost certain that it must, in most cases, be

compressed in other directions, these compressions and extensions taking place in the above
mentioned vertical plane perpendicular to the strike of the beds. Hence, we may conclude that

generally the minimum tension will be a pressure, as in (3) of last Article. The axes of greatest.

and least tension through any point will lie in a vertical plane perpendicular to the strike of the
beds, and consequently the intersections of the planes of greatest tangential action with the planes
of the beds will be horizontal lines. Through every point there will be two planes of maximum
tangential action perpendicular to each other, and therefore, dipping one of them in the same
direction as the beds, and the other in exactly the opposite direction, the strike of all these planes
being the same.

Mr. HOPKINS. ON THE INTERNAL PRESSURE OF ROCK

MASSES.

407

8. Let us now consider how the distorted forms of organic remains may indicate the

directions which must have been
those of maximum and minimum (Fig. 4.)
tension or pressure, and the po-
sition of the planes of maximum
tangential action at some former
epoch, posterior to the elevation
which raised the general mass
into anticlinal ridges. In the
first place, suppose the planes of
maximum tangential action to
coincide, at least approximately,
with those of stratification. Let
MN represent one of these planes
on which, between two beds, the
fossil shell AB is found, the un-
distorted form of the shell being
known. ALN is supposed to co-
incide with the dip of the beds,

and the median line of the shell to lie in the direction of their strike, the plane of the paper
being vertical. Also, let C7’ and CP be the directions of maximum and mimimum tension

respectively, each inclined at an angle of 45° to MN.
Then the continuous line will represent the distorted
form of the shell, of which the original form is indicated
by the dotted line. Fig. 5, in which the plane of the
paper represents the plane of a bed, will represent the
distorted form of the upper valve of the same shell. It
is important to remark, that this angular distortion will
take place in the direction of the dip (MN) of the beds,
and perpendicular to their strike (SCS),

Again, let the planes of stratification be perpendicular
to one of the directions of principal tension, then will
MN the direction of the dip, be a direction of maximum
tension or of maximum pressure In the former case an
imbedded fossil will be elongated, and in the latter case
compressed, in the direction MN, but without any of
that angular distortion represented in the previous case
(Fig. 4), unless it should be accidentally produced by
direct compression, in which case, however, it .will have
no such necessary reference to the directions of dip and
strike as above mentioned.

Conversely, if it be observed that the organic forms
lying between two contiguous beds, have undergone
great angular distortion, we may conclude that the
planes of stratification must have coincided more or less,

(Fig. 5.)

approximately, with those of maximum tangential action at the time when the distortions were
produced; but if the observed distortions indicate only direct compression or extension, unac-
companied by angular distortion, we may conclude, that the planes of stratification, at the time
just mentioned, must have coincided at least approximately, with the directions of maximum or

of minimum pressure.

468 Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES.

9. The application of these conclusions to the leading object of this communication, the possible
influence of internal pressure in producing cleavage structure, may be made very briefly. If we
recognize the probability of this influence, if not as a primary cause, yet as effective in determining
the positions of the planes of cleavage, we must, I think, almost necessarily suppose, as I have
before remarked, that those planes must coincide more or less accurately either with planes
perpendicular to the directions of maximum pressure, or with those perpendicular to the direc-
tions of minimum pressure, or with the planes of greatest tangential action. Now, let us sup-
pose the organic forms lying on the surface of a bed to have suffered great angular distortion,
and therefore the planes of stratification and of greatest tangential action to have been at least
approximately coincident ; then, if the planes of cleavage nearly coincide with those of stratification,
we may conclude that the tangential action and not the direct pressure or tension has been the effective
agency in determining the position of the cleavage planes; and the conclusion will be strengthened
if we find that, as a general rule, the angular distortion is greater the more nearly the planes of
stratification and of cleavage are coincident. Again, suppose the observed distortions to consist in
direct compression or extension, without considerable angular distortion, and therefore the planes of
stratification to have been perpendicular either to the directions of greatest pressure or to those of
greatest tension, and consequently inclined at an angle of 45° to the planes of greatest tangential
action; then, if the cleavage planes be also inclined at an angle of nearly 45° to the planes of stratifi-
cation, we shall be again led to the same conclusion as above. If, on the contrary, it should be found
that when the cleavage planes and the planes of stratification are nearly coincident, the distortion con-
sists only in direct compression ; or if, with great angular distortion, the cleavage planes should be
inclined at about 45° to those of stratification (cases exactly opposite to those previously supposed,)
we must conclude that direct pressure has been the influential cause in determining the position of
the planes of cleavage.

FF —2? P

In the memoir already referred to, Mr, Sharpe has collected, I believe, nearly all the evidence
which has hitherto been obtained on this subject, consisting principally of observations made by
himself and Professor Phillips, and has given drawings of several characteristic distortions, principally
of spirifer disjunctus, a frequent and well-known shell. in some of the older formations in which the
cleavage structure is very distinctly developed. In the most remarkable specimens of Mr.
Sharpe’s collection (for the inspection of which I am indebted to him) the distortions are very striking,
and, for the most part, of that kind which I have termed angular distortion. Now all the most
remarkable instances of this kind, as Mr. Sharpe has stated in his memoir, are those in which the
planes of stratification and those of cleavage are approximately coincident, the angles between
them varying from one or two to ten or fifteen degrees; whence I should conclude that the
cleavage planes must have approximately coincided with the planes of greatest tangential action, —
and consequently that it is to this kind of mechanical action, and not to direct pressure, that
the influence in the production of the cleavage structure must be attributed. Mr, Sharpe has
also described and figured other specimens taken from beds in which the planes of stratification
are inclined to those of cleavage at angles varying from forty to sixty degrees, and in these cases
the distortions (as described in his memoir) consist in a shortening of the axes of the shells in
directions perpendicular to the intersections of the planes of stratification with those of cleavage,
such as would result from direct presswre in that direction. So far this evidence is perfectly in
accordance with that previously cited, for it indicates that the direction of maximum pressure must
have approximately coincided with the planes of stratification, and therefore that these planes must
have been inclined approximately at an angle of forty-five degrees to those of mavimum tangential
action. Consequently these latter planes must have approximately coincided with the cleavage
planes in this case as well as in the former one. This latter evidence, however, furnished by Mr.
Sharpe’s specimens is not, probably, nearly so complete with respect either to the number of dis-
torted shells or the distinctness of their distortions, as that furnished by the shells first men-
tioned as so curiously and distinctly characterized by great angular distortion. Still, the

|

Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 469

evidence hitherto adduced appears to be, on the whole, consistent with itself and strongly in favour
of the conclusion that whatever may have been the influence of pressure in producing a laminated
structure, that influence must have been due to the tangential action parallel to those planes, and
not to direct pressure perpendicular to them. In fact, I regard the specimens above mentioned,
in which there is great angular distortion combined with an approximate coincidence of the planes
of cleavage and of stratification, as almost decisive against the latter conclusion.

In the search of further evidence, the observer should direct his attention especially to those cases
in which the inclination of the cleavage planes to the bedding is either small or nearly 45°. In
the former case, according to the above inferences, he may expect to find great angular distortion
of the fossils lying (as they will very generally be found to lie,) with the plane of separation
of the two valves parallel to the surfaces of the beds; and in the latter case he may expect to
find the shells characterized more especially by direct compression or extension (more probably the
former,) in the plane of the bed, and in directions perpendicular to the intersections of that plane
with the planes of cleavage. At the same time it should be remarked that-the angular distortion
may be accompanied by a lengthening or shortening of the shell, (more probably the former.) in the
direction of the dip, and also that a considerable direct compression is not likely to be produced
without some degree of angular distortion; but still, if the above conclusions be true, angular dis-
tortion in the one case, and direct compression or extension in the other, ought especially to
characterize the actual forms of the organic remains.

It might be objected against the theory to which the preceding conclusions tend to lead us, that
if tangential action has been an effective cause in the production of the laminated structure, there
ought to be two systems of cleavage planes at right angles to each other, since there are two such
systems of parallel planes in which the tangential action is a maximum; and this might, I think, be
regarded as a valid objection to a theory which should assign the mechanical action here considered
as the primary cause of the laminated structure ; but the objection may probably be obviated in a
great degree, if we regard this kind of mechanical action only as a secondary cause, for it is very
conceivable that it might have greater effect in aiding the development of the structure in question
along one of the systems of planes of greatest tangential action than along the other. Whatever
may be the apparent force of this objection, however, the discussion of it may be regarded, perhaps,
somewhat premature till further observation shall have ascertained more distinctly what indications
may be found of the existence of a second set of cleavage planes less developed than those which
more immediately attract our notice. The point is deserving of the attention of the geologist.

The adoption of the opinion, that the mechanical agency above described has been one efficient
cause of the laminated structure, necessarily involves the conclusion of that structure having origi-
nated at some epoch posterior to the great movements which have determined the general configura-
tion of the external surface, and the geological structure of large portions of the earth’s crust, which
are observed to possess this laminated character. It is also a necessary inference that the line
of strike of the planes of lamination must coincide with that of the planes of stratification (Art. 7).
The results of observation undoubtedly corroborate this latter inference, for it would appear that
we may state as a general fact, that the strike of the planes of cleavage is parallel to the directions
of the anticlinal lines of the district. The amount and character of the local deviations from this
law are not yet in any case, I believe, accurately determined. Observed facts appear, also, to
corroborate the above conclusion respecting the epoch at which the laminated structure was super-
induced ; for the persistency with which the strike and dip of the cleavage planes are frequently
maintained through disrupted and contorted strata, distinctly implies that the lamination must have
been produced after the elevation and disruption of the general mass. These general facts are in

harmony with the theoretical view of the subject which has been here presented ; how far the more

detailed results of observation will be found so remains to be determined.
Geologists are well aware that currents of electricity have been assigned as a probable cause of
the laminated structure, and that this hypothesis has received great support from the results of

470 Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES.

experiments made in the first instance by Mr. Fox, and afterwards repeated by Mr. Hunt, as
described by the latter gentleman in an interesting memoir, contained in the Memoirs of the Geo-
logical Survey of Great Britain, Vol. 1., on the influence of magnetism on crystallization. It
would be foreign to my object to enter into any discussion of a theory founded on these experi-
mental results; and indeed detailed discussion of any theory on the subject would be, I conceive, ,
at present entirely premature; but I would remark that views of the subject founded on these
results, and those founded on the observed facts respecting the distortion of organic forms, are
by no means to be considered as opposed to each other. On the contrary, it is very possible
tion we may hereafter be better able to account for the phenomena of lamination by the joint opera-
that of the causes to which they would be referred respectively according to these two views of
the subject, than by the independent operation of one of those causes only.

In concluding this communication, I would especially remark that the advocacy of any parti-
cular theory on the subject of cleavage structure has formed no part of my object. Our ignorance
of the physical causes of crystallization, or the manner in which such supposed causes may ope-
rate, is too great to admit of our forming at present any theory on the subject which might not be
deemed altogether premature. All I would here insist upon is this—that the facts observed by
Professor Phillips and Mr. Sharpe indicate certain determinate relations between the distortions of
organic forms and the positions of the planes of lamination of the beds in which those forms are
discovered, relations which seem to imply that the forces which produced the distortion had also
their influence in determining the planes of lamination. My object has been to point out the
accurate mechanical conditions of the problem, and thus to indicate the points to which the
attention of future observers should be especially directed in order that their observations may
afford conclusive tests of the truth of any theory which may hereafter recognize the efficiency of
the mechanical agency explained in this paper, as one of the causes of the laminated structure.

W. HOPKINS,
Campriner, May 3, 1847.

XXXV. On the Partition of Numbers, and on Combinations and Permutations.
By Henry Warsurton, M.A. M.P., F.R.S., F.G. S., formerly of
Trinity College.

[Read March 1, 1847.]

Introduction,

Tue researches of which an account is here presented, had their origin in the following
manner. In the Autumn of 1846, having communicated a Theorem (which will be found in the
sequel) on the Partition of Numbers to Professor A. De Morgan, I received from him an obliging
reply, wherein he intimated a wish that I would turn my attention to Combinations, as a depart-
ment in Mathematics, which, he thought, much needed cultivation. I acted upon this suggestion,
and shortly afterwards sent to Mr. De M. results, and subsequently from time to time further
results, which he wished me to render public. These I placed at his disposal; and, with my
concurrence, he drew up an account of my Researches, in a Paper which was read before the
Society on the 1st of March, 1847.

After the reading of this Paper, further suggestions presented themselves to me, of which I
drew up an account, and this was laid before the Society by way of Supplement to the former Paper
of Professor De Morgan. Still further improvements again occurred to me; and it then seemed to
me desirable that both Papers should be withdrawn, to give me an opportunity of revising my own
researches, and of incorporating the revision in one Paper to be communicated to the Society.

Many important original observations on the same heads of inquiry, proceeding from Professor
De Morgan himself, were contained in the Paper which he drew up; and I should much regret
_ if, in consequence of the course which I have suggested of withdrawing that communication, those
observations were to be lost to the Society and the public.

It was as impossible for me, as for any other person, to hold communication with that gentle-
man on Mathematical questions, and avoid deriving great advantage from his sagacity and erudition
in Mathematics. I have not, I trust, abused those advantages by appropriating to myself anything
which belongs to him; but I have endeavoured, while possessing those advantages, to carry on my
researches with originality and independence.

SECTION I.
On the Partitions of Numbers.

1. From a recollection of the important application made by Waring of the Partitions of
Numbers to the developement of the power of a Polynome, I was led to investigate their properties,
in the hope of discovering some ready method of determining in how many different ways a given
Number can be resolved into a given number of parts.

Assuming the Unit to be the lower limit of the magnitude of the parts, I found that if the
Number to be partitioned, N, were expresscd in terms of a certain Modulus, m, so that N was

Vou, Vii, Parr IV. s.r

472 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

= mt-+v7, the number of the ways of resolving N into p parts could be expressed in the form of a
rational and integral function of the factor, 4 Thus, in the case of Bi-partition, 2 being the
Modulus, and the Number being 2¢, or 2¢+ 1, ¢ is the number of the partitions. In the case of
Tri-partition, 6 being the Modulus,

For N = 6¢, the number of the partitions is 37°,

Be = SO SEL Scns riva spppedooonancosesones, Ul ere Sal

Saas SORES) aSsee% davetsebaeeadoa Beagsaar | bee seal A
= OLEES magnus abassesoaremencass Gl 4 Obra Ls

and in the case of Quadri-partition, when the Modulus is 12, and ¢ becomes of 3 dimensions, I also
ascertained the formule. But perceiving that, since the modulus and the dimensions would increase
with the number of the parts, the functions obtained would be so many, and of such complexity as
to be of little or no practical utility, I abandoned that method, and sought for some other. Having
at last discovered the method here proposed, (Arts. 7 and 8, Sect. 1,) I communicated the same to
Professor A. De Morgan, trusting to his known Mathematical erudition for obtaining the information
I required—whether the method was novel. By his reply; I was made aware that the Partitions
of Numbers had received a share of his attention, and that he had written a paper on the subject,
which was published anonymously in the 4th Volume of the Cambridge Mathematical Journal. He
further stated that, after the date of that publication, he had also discovered the Theorem which I
communicated to him; though he had not announced it; and since I have no doubt of the entire
accuracy of that statement, he must participate fully in any credit that may attach to the discovery
of the formula in question.

In this Section of my present Paper, I have limited myself, as regards these partitions, to
what I considered necessary for the proof and illustration of the Theorem in question. Other
matters bearing on the question of Partitions occur in the Section on Combinations.

2. The number of the different ways in which a Number, N, can be resolved into p parts,
when no number is admitted as a part, but such as is either equal to, or greater than, the arbitrary
number, 7, may be denoted by [N, p,]. We may term » the lower limit of the parts, or parti-

tion, or, for brevity, the lower limit. By a p — partition of MN, I mean any set of p numbers, —

having M for their sum.
A partition included among those, the number of which is denoted by [N, p> may consist of

parts exclusively equal to, or exclusively greater than »; or it may contain some parts equal to,

and some parts greater than y.
[N, p,] includes the whole of [N, p41];
[N, p,+:] includes the whole of [N, p,+2];

and, generally, the partitions which have » for their lower limit, include all those partitions in
which the lower limit is greater than ».

3. If to or from each part in every partition of N whose lower limit is y, a given number 8
be added or subtracted, N will be increased or diminished by the amount’ p@; but the number of
the partitions, and the number of the parts in every partition, will remain unchanged: i.e.

[ENO A[ENic= i O5y plreilc.cecsianasvadonces stacey (L)»

This involves the conclusion, that we recognize 0, and negative numbers also, among the
admissible parts; unless we expressly assume that they are to be excluded. It also involves the
recognition of negative numbers, as the subjects of partition, unless their exclusion be expressly
stipulated.

_ ail ee

AND ON COMBINATIONS AND PERMUTATIONS. 473

4, Since N=y+(N-—~y), therefore take all such (p —1) — partitions of N — y, as have y
for their lower limit; and with every such set of parts let 4 be conjoined as an additional, or p™
part. We shall thus obtain all such p — partitions of N, as, besides having for their lower limit,
agree also in containing at least one part equal to 7. From these partitions, therefore, are excluded
all those p — partitions of N which have for their lower limit y +1. Hence,

[ewer NG eel = (gp pel). cancers asa ss ccace-uenescees: (11);
= also) [Vi a, p+ 15 — [N+ pl, -a||= LN; pa ).-2-20---csencene (11*).

5. In [N, p,], let N<py. If negatives be admitted, then let N be the greater negative ;
and by the addition to y of a positive quantity greater than y, and to WN of a positive quantity
greater than py, let the lower limit and the number to be partitioned be rendered positive. Since
no positive integer less than py can be resolved into p parts having a positive integer, 4, for their
lower limit, no partition of the kind indicated by the notation can be effected. Therefore,

WHET NV 975 uN Pell =nORs tote ast arcs cabiisenasmsares es. (111).
Then if 7 be positive, and p is also a positive integer, [0, P, | = 0.
The two following extreme cases, [N, 0,], and [0, 0,] require explanation.
By (.) [N +”, iL] -[Ni, 141] =i Ne O21
But, when N and » are positive integers, [N +, 1,] =1,
and) (PV 75 Issn — ws
S| Paly Qh ]) SeU Sal Oh as apgnepc ch accoa6 ose soe Odor aBOEe (1v).
INS, Nyy (Ce) NG PG tbl = os WP
But (NV, 1y]=1; and, by (im) [N, ty.) =0;
Ree Og Owl nl OL == Mncstameromctiseckaeceteocetcetetomeecenes sc: (v).
Hence also, if N=py, [N, p,]=[p» p,] =[0 p] =[p, pi] =1.
6. Professor De Morgan (as he informs me) has, in the Cambridge Mathematical Journal,

traced Equation (11.) to its consequences, in the case where the number of parts is preserved
constant, and the variation is thrown on the number to be partitioned.

Ma this case, [N; p,| — [Ns 2,41) = onccececercceeeneecersaeees =[N —y, p—1,],
[N, Prin] -(Ms poa2]=[N-1-1, p- Lail =([N-7-p, p —1,]-
LN, Py +0) = LAN PD, +642] =[N-7-6, p- i) =[N-7-pé, p -1,];
ee LN, Pr, | = LN, fo-eoral = St [LN - =) 5) p — il
Here is the lower limit of the parts admitted, and 7 + @ +1 is the lower limit of the parts

excluded ; that is to say, all those partitions are excluded which have every part greater than 9 +0.
Write Y’ for 7 +0. Then let N=pY +r, r being a remainder less than p, and Y the integer

y

N
nearest to, but not exceeding —. When Y’ becomes Y, [N, py4,] = 0;
8 p Py'+

“ [N, Py) = 0s pr] + 9°" (N - 1 - pe p-1,,]
= a [N -- px, p -1,]
=([N, py4,] + ie) [N-(1+p[n-1])- px, p 5 i hoppanes: (v1.)
= sy [N-(1+p[y-1]) -— ps, p 7 Ul etieeneas (vu)

3P2

474 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

Thus, [31, 5,] = S°[20-5%, 4],

= [20, 4] +[15, 4] + [10, 4] + [5, 4]
ie: 2 NOD =e OAR Ee ee eee eects el.

7. By a different transformation of the equation of differences, (11), we arrive at a different
summation ; in which the number remains constant, while the parts vary. In that equation, if we
write q for p, we have,

[V+ 9+1]-[N, 4] =[V +n q+i1]=[N-a q+1];
J nt
[N +2n, Bag ap Bs be Ba Slbed Reis Hed: Get lege oat 1 q+2]-
1

eee ter eeereenee eee eee ee eee ee eee eee eee ee

[N + pn, hig Deg Na tpi nm g+ (p-I)]=([N+pn, q+p)=(N-a» q+p]s
1] yt 1
[N+pm 9+P]-[N, a] = St[N-an 9 +8].
n z

Now [N, q,] vanishes, either when N < qy, or when q = 0; the exception to the latter case
being when N=qy. If N <qy; then, since

Ba eS) 2s gle qn+q+P, 9+P] =
and LN, &) =F - n+ 4 uv] =0,

it follows that S?[N - qn, q +2] =0.
z 1
And we have only 0+0=0,
But if g =0,
[NV +p, Pil = [N; 0, | = SP [N, %]3
. (NV +p, pi] = S? LN, &]...... Betas anealeipenienceds say a CRTLE)S
or [N; pi] = S?(N =p, 2] «--0.c-00e- Dente ection seco sscan| (WELT Sis

aptlao (GN, ype Se [oN — 90995 | coeMiig ed nana gaways anects Suan dent (IX):

Thus [31, 5,] =[16, 0] + [16, 1] + [16, 2] + [16, 3] + [16, 4] + [16, 5].
Or, 101 = 0 + 1 + 8 ch 21 + 34 + 37.
The following very elementary proof of this proposition has also suggested itself to me,
We shall exhaust all the ways of resolving N into p parts, having 1 for their lower limit,
if we take
Ist, p—1 units, and the remainder N — (p — 1) entire, not less than 2.
2d, p-2 bins and the a eae of the remainder N — p + 2, not less than 2.

se ewenee eee e ee rer eee eee eee eer eee eee eee eee Pee eee mee wee ree ewe nee

mnie p —m units, and the m — eartitogs of N-—p+m, not eee than 2.
Lastly, p—p=0 units, and the p— partitions of N-p+p=WN, not less than 2*.

* When p> x the greatest value of m is N—p; and the | of the partitions, the two cases of pnot>>, and p>, are b

partition of N, corresponding to that value, is [(2—) units, | comprehended in formula (v111*).
and (N —p) repetitions of the number 2]. As regards the nwmber

AND ON COMBINATIONS AND PERMUTATIONS. AT 5

From each of the parts, in every one of these partitions, deduct 1.
Then we shall have

[N, aJ=[N-p, 1] +[N-p, 2] +......... [MN —p, m] +... (NM - p]s
Also, [V +p, p,] =[N, 11] +[N, 2:] +... aco. |G IIE
=[N, po].
N= Only ae 20) alee | ANG | 7 | 8 | 9 | 10 | 11 | 12
p= 0 Ty 08 80 | ©] OG CO} Ol ON Oy Ow
a Of aL la] ] aha ai ae Sat) Se ate
=p | ©] @) fipa =| Neal call alone all etre
= 2 |) OO} olf Wi Tis 3 | 4| 5] 7| 8] 10] 12
= AT a) OPO waar ae \ 3; 5LS| 9 11 | 15
= & O)) Ol OG) Of, ie) Thee | oman mar, | 10 | 13
= 6-) O) Ol Ole-O| Ol ©} Deh i) eel Sinz
= Fh oh oleo) ol) ol oleae sel ell eh blew
=6 | of} Glo) oO} oO} oO] OG] wi a al) ey sel se
=@® | @] oO} Ol @) O] ©] @] oO} wil 1) rm] ai &
=i | oO Ol] oO] OF Ol al Gl ww) Mla ale
=a oO! Ol of OF GI Ol Glo} @| @) ao} a} a
=12 O} lO) OO] Ol CO) -Oll wl) wi) wih w)) at
Sum of } 1/ 1} 2) 3] &| 7/11 115] 22 | 30} 42 | 56 | 77
Columns.

8. I shall proceed to shew the application of these latter formule to the construction of a
Table of the Partitions of Numbers, and point out the leading properties of such a table: and
since all partitions, whatever may be their lower limit, are reducible to partitions whose lower limit
is 1, I shall confine my observations to a table whose lower limit is the Unit.

To the Equation of Differences, 11, we may give the following forms:

[M, ml t =[N=-4, ad = Nee Tema sb ser aassracd= caek (X)>
or [N+p, p,]-[N+p-1, p = it |PANG fail cosaboaorGecnodeosaroule se

and these will best serve for the construction of the table.

The annexed table is one of double entry, N being the index of the columns, and p of the
lines. [N, p,] is the term in column N, line p. In formula x, the change to N—1, and p—1,
marks that we are to recede simultaneously one column and one line, that is, diagonally. The
diagonal will cut line 0 at the head of column N — p, and [N — p, p,] is the term on the p" line of
that column. Thus the term on the p line in the vertical column is the difference between the
terms on the p'® and (p — 1)" lines on the diagonal. Suppose that all the terms are known in the
vertical column N, and that we have determined all the terms on the diagonal, proceeding from the
head of that column to the (p—1)" line inclusive. Then the term in the diagonal on the p' line,
that is, [V +p, p,], is equal to[N+p-1, p—1]+[N, p,]; and in the same way the term

1

476 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

on the (p +1)" line of the same diagonal may be found; and so in succession, to any required
extent, until they become constant: (vide § 9).

The consequence of the preceding equation is, that any term in the table, say that on the p™
line, in column N, is equal to the sum of all the terms from line 0 to line p inclusive in column
N-p; which is the column at the same distance backwards from column N, that the line 0 is
from the line p.

y. If in the table we draw a zig-zag line from [0, 0] to [12, 6], it will be seen that all the terms
below that line are of constant recurrence, and are identical with the numbers 1, 1, 2, 3, 5, 7, 11,
15, &c., which arise from the summation respectively of all the terms in the columns...... 0, 1, 2,
3, 4, 5, 6, 7, &c. 1 proceed to explain this. Let any diagonal line proceed from the head, say
of column A, advancing simultaneously one column and one line. When that diagonal cuts the line
p, N will be equal to 4 + p.

Now [4+ p, p] = S?[4, =z],

and when p ="4, or > A, its value becomes constant, and is [24, 4] =[4, 0] + [4,1] +... [4, A],
that is, it becomes equal to the sum of all the terms in column 4. Thus one-half of the whole
table is occupied by terms =0; and an additional fourth of it by these constants; and were it
thought requisite to compute a table of the partitions of numbers, it is only the terms that occupy
the remaining fourth of the whole space of the table, that would actually require computation by
the method of differences: and of this fourth the three first lines are so obvious, as merely to
require being transcribed.

SECTION II.

On Combinations.

1. Tuer well-known Theorem in Combinations enables us to determine in how many different
ways w elements can be taken at a time out of s elements, all dissimilar. It is the coefficient
of wv in the developed power of the binome, [1 + #]’, which, in this case, affords the solution of
the problem.

2. In the first case of combinations which I now propose to investigate, the combining elements
are also of s different kinds ; but there may be more than one element of the same kind : for instance,
a of the elements 4, 3 of the elements B, and so on; and the question proposed is,—In how many
different ways uw of the said [a + 8 + &c.] =o elements can be taken at a time, on condition that
those which are plural in their respective kinds, may be repeated in the same combination ?

2

5. Combining elements of the form proposed are found in the s geometrically progressing
polynomes,
[1+ do + A’a*+....., A*w*] x [1 x Bo+...... Boa?) x &e.,

and all the possible combinations of these elements, taken 0, 1, 2,......5 U,-+----o, at a time,
are respectively found aggregated, each with a positive sign, in the coefficients we obtain of

MnO soe wads ences) Woe

when the product of the said polynomes is developed according to the powers of «. That develope-
ment, supposing all the coefficients to be complete, is of the form

1+ S[A]o+ S[4°4+ AB] a*+ S[44+ 4B + ABC]a*+......
+ S [4% + 4° B+ A? (B+ BC) + &e.] a + 2.0.

AND ON COMBINATIONS AND PERMUTATIONS. 477

If 1 be now substituted for each of the elements 4, B, C, &c., the polynomes will respectively,
become [1+ @7+a°+......a°], H+v+2'?+...... x], &c., and the coefficient of wv“ in the de-
veloped product, now that 1 has become the value also of each term of the form A?B’C’ in that
product, will represent, not only the sum, but also the number of all such terms: that is to say,
of the different combinations which can be formed with the o elements, taken w at a time.

4. That coefficient is an explicit function of w, which I now proceed to determine.
The product of these geometrical polynomes, is

Pete a = ght!

: sOcCers
l-g 1-@
that is, (0 —@)7* [1 — a+] [1 —2***). &e. .......-. (x1.)
s 1 s" 1*
But [1-@]S=1+se+ Tappa soba ap v" +
1 5-1 Qs- LAS s—}]1 ,u
=p [ae eta oe. fet 1h a 2]:

For the sake of brevity,
Let w+1 be represented by w,;
7 Ed eR ERC oOoperOSEe by a;

[E}4o il Socponceccssencce eh /6),.8 Kae
ist. When each of the s kinds of elements, 4, B, C, &c. admits of unlimited repetition, the
required coefficient of w", will be

Beale yp

wy? or ry’ ©0008 obs cre u lace ces nesses voesoacesseesenecescs (x1*));

and in this case, of plural elements, all kinds admitting of unlimited repetition, a solution of the
combination problem, to the same effect as the preceding, has, as Mr. De Morgan informs me,
been given by Hirsch.

2dly. When the elements of one kind, 4, are limited in number to a, but the elements of
the other (s—1) kinds may be repeated without limit, the required coefficient, (which is that of
(1 — w)~*[1 -«]), will manifestly be

1
rye far [ee = a, |? |"), cnevencascerwacwanscsconseeceen ee (XII);

from which expression however, the second term is to be excluded, in case [w,—a,] should be
negative.

3dly. When the elements of two ral A and B, are limited in number to a and # re-
spectively, but the elements of the other s —2 kinds may be repeated without limit, the required
coefficient, (which is that of w" in the developers of (1 — ~)-* [1 -— #] [1 - #*]), will be obtained
by performing on formula (x11) with 8, the same operation that was before performed on formula
(x1*) with a, The result will manifestly be

e etl li —a,]'""|'+ [u,-a a.
718 -[u,-B)- *t

atty date op REDD

* I use the factorial notation, in which
s*|! represents 8 (8 4-1) (8 +2) ..... o.. [s+(u—1)],
and 68 [-! ......600 8 (8—1) (8 —2) scessreeeeee [8 —(U—1)]s

478 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

from which expression, however, every factorial expression is to be excluded which has a negative
quantity for its first factor.

4thly. By operating on formula (x111), with y, in a similar manner, and on the result of that .
operation with 65 and so on in succession, until there remain no factors undisposed of, we shall
obtain for the coefficient of the developement of [1 — @#]~* x [1-—a*] [1-.*], &c. the following
expression, subject to the same rule as before, of omitting every factorial which has a negative
quantity for its first factor :

ws)! — [u,— a4] + [u,— 9, - B,}"|) — &e.
vel ra [ice Sivan [u,-a,-—y,]*-’|’ = xe. |
ie? |} | = [u, -y,}]} Hl [u, = B,- y 3) ral 0 wevses

— &e. + &e.

5. Now if a, B, y, &c., are all equal, that is to say, if the required coefficient is that of x”

,

— xv FY
= h formula (xtv) will become

1
in the developement of ( 4

2} -1
1 fet ate arp s Le fy 26)" ]'= so
a pon dea. eae ties Audie (xv) 3

Ets (= i)? tn ; Le, 7) Ga; ||! + &e.

where, for any determinate value of uw, the maximum of @ is the integer nearest to, and not

Pee ac wt 5 tn c : * . .
exceeding rr but if w attain its maximum (which is sa), then the maximum of @ is the
a
2 . Sat+1
integer nearest to, and not exceeding 7
=

Example of formula (x1v).

How many different combinations can be formed by taking 2, or 8, at a time, of the 10 elements,
of 4 different kinds,

A, BB, CCC, DDDD?
1
Answer, JE LSE AD err EAE ae os hae Ea

Answer for u = 8.

9:10.11 —7.8.9+ 4.5.6
| O10 3.4.5]
-— 5.6. 2.3.4'=9.
1.2.3 C3 *
—4.5.64+2.3.4
+1,2.3

Example of formula (xv).

How many different combinations can be formed by taking 2, or 8, at a time, of the 10 elements,
belonging to 5 different kinds,

AA, BB, CC, DD, EE?

AND ON COMBINATIONS AND PERMUTATIONS. 479

1
Answer for 2; ————_ .3.4.5.6=15.
WEL EEISC!
1
nis cdeaeivee 8; ———— [9.10.11.12 —5.6.7.8.9+10.8.4.5.6] = 15.
1.2.3.4

6. Incases of Combination, such as those to which formulas (xtv.) and (xv.) apply, when it is
required to determine the number of Combinations corresponding, not merely to one or two powers
of x, but to the entire range of the values of w, from 0 to [a+ 8 ++, &c.] =o in the former case,
and from 0 to sa=o in the latter, the expression (xt.) for the product of the s Polynomes suggests
the following method for determining arithmetically the entire series of the Coefficients. The

method will be best explained by an example.

How many Combinations can be formed from the Six Elements 4, BB, CCC, taking 0, 1, 2>
3, 4, 5, 6 of them at a time.

Different Values of u

Coefficients of (1 — a) ~%
Subtract

Coefficients of (1 — w) * x [1- 2°]
Subtract

Coefficients of (1- w) ~* [1 — a] [1 -#°]
Subtract

Coefficients of

(1 -@)* [1-4] [1 - 2] [1 -2']

The law of the terms in the last line, which contains the answer, deserves notice: viz. that the
terms corresponding to the indices w and 6 — wu, are equal.

How many different Combinations can be formed from the Four Elements 44, BB, taking
0, 1, 2, 3, 4 at a time?

Coefficients of (1 — a) ~* eee 2a eS. | |
Subtract

Coefficients of (1 — w)~* [1 — a]
Subtract

Coefficients of [1 — w]~* [1 - a]?

7. From the given numbers of the Combinations formed by + elements of ¢ different kinds
which combine v at a time, and by « — 7 elements, of s —¢ different kinds, which combine wv — v
at a time, it is required to determine the numbers of the Combinations formed by those elements

Vox. VIII. Parr IV. 3Q

480 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

conjoined; that is, by o elements, of s different kinds, which combine w at atime; assuming the ¢
kinds to be different from the s — ¢ kinds.

Let the Combinations which can be formed by
taking w* at a time of the o elements, be {w, oc}.
Ben GWU) eee teereenrayOl ute meceetates wil Ny. Oma te
Imagine some determinate values given to the variables ~ and v. Every Combination }w —v,
« — 7} may be paired with every Combination fv, +}; and thence will arise }w—v, o-—7}

x fv, rt different pairs, each containing w elements of the s kinds. If w remains constant, while
v» varies, there will be a pair for every value of v from 0 to wu, if w< 7; and from 0 to 7, if w>7.

Thus we have, for w<7,
fu, of = {u, o—7} f{0, re + fu-1, o-7} x iT pl es es ae
seoeenecsdeewens eave dectussenal tt 9 gost fu- 1, Th + 40, a7} fu, T}......(XVI).
For u>7T,
Su, of = {u, o—7} {0, re + fu—1, o— Th {1, TE + o.eeeee eee eeee(XVMI).
Ziuyectwaedebagestiogs + lr ch ls -7} fr — is zt + fu- 7 ¢—Th \7, rt.
Now in each of these expressions write (¢ —w) for w; and we shall have in the former

o —-u>oa-—T; therefore some of the terms at the commencement of this formula fail; in the second
expression we shall have o —u<o — T.

For c-u>o-T;
{ ou, ot =fo-7T, ¢0- rt Sr — U, 7} a teases siniee fo- T— Uy; oT} Sr, rt coseceses, (XVID
For ¢ -—u<o-T,
fo =U, t= fo —U, o- rt fo, rt bh epeowesivetees So -T-u,o-7} \r, rt weoeeeeews ss | GRO
I shall apply this method to the two examples before given, where we have for the Combinations —
of A, BB, CCC, taken 0, 1, 2, 3, 4, 5, 6 at a time,
1, 3, 5, 6, 5, 3, 1 Combinations;
and for the Combinations of
DD, EE, taken 0, 1, 2, 3, 4 at a time,
1, 2, 8, 2, 1 Combinations,

From these numbers, we have to determine the number of the Combinations of all the Elements
A, BB, CCC, DD, EE, taken together 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 at a time.

* I write {z, o} to denote the number of Combinations formed tuo}= Sil {ur}xf{u-v,0—7} J;
by o elements, plural or singular, of any kind or numbers of kinds,
when those elements are taken w at a time: { } is used instead of |
{ J, in order not to confound Combinations with Partitions.

and the terms of the table itself are given by the equati
[v, v] =[v, v] [w-v, 0}; which means that the term on line
column v, is equal to the product of the term on line v, col n
», by the term on line (w—v), column 0,

+ The formula is perhaps better given in the condensed form,

AND ON COMBINATIONS AND PERMUTATIONS. 481

ed |
Jue {= 1 2| BI By a | 9 the total.
u Ju-v,5o-—7 | | | | Ju, o |
In i L
0 1 1={ O,cf
1 3 2 5={ 1c]
2 5 6 14={ 2,6 | ;
s 6 10°) 9) || 2 Xl Gh cet
4 5 A20SHI GME te\| 80a matoat
5 8 10 |18| 10! 3] 44={ 5,¢ |
6 1 Gaon ete 5a SOL Onion
7 2 On elOn me Gsle2i— sl bviercall
8 A 3| 6| 5|14={ 8c]
9 3 BEM WGyecrt
10 1| 1=]10,c}

8. From the foregoing investigation, I deduce the following important Corollary; that ¢
Elements of various kinds, and singular or plural in their several kinds, will form the same
number of different Combinations, whether they combine wu, or o — u, at a time.

For if we compare the ist, 2nd, 3rd, &c. terms in formula (xvr.) with the last, penultimate,
and ante-penultimate, &c. terms respectively in formula (xviir.) ; and if we make a like comparison,
term by term, of the (xvir.) with the (x1x.) formula, we shall find that the first term and the last,
the 2nd term and the penultimate, the 3rd and the ante-penultimate, and so on, are identical, in the
series $0, ot, 41, lath coaodadace $o-—1, ot, {o, o{, provided it can be shewn that the same law
applies to the terms of each of the two component series,

NOS chs sess ats caupsaccpton air =>, alig aie sere
and {0, o-7}, 41, (ae teh Heobe fo-7-1, o—Th, fo-—7, o-T}.

But when the 7, and the (¢ —7) elements each consist of only one kind, the number of the
Combinations that can be formed by taking 0, 1, 2, 3, &c. of these elements at a time, is invariably
1, 1, 1, 1, &c. and this series is identical, whether it be taken in direct, or in reverse order.
Therefore the law will apply to the series formed by the elements of two single kinds conjoined ; and
therefore to the elements of three kinds conjoined; and therefore universally, of whatever number
of different kinds the elements may consist.

Hence it appears that, to diminish the labour of computation in the application of formulas

(x1v.) and (xv.) to particular cases, we ought always to make a selection of the least of the two
numbers uw and o — u, before substituting one of them for the variable in either of these formulas.
The theorem just established may also be enunciated in the following terms :
If the product of any number of geometrically progressing Polynomes, each of which has a
limited number of terms, and # for the common ratio of the terms, be developed according to the
powers of w; then, assuming o to be the sum of the dimensions of all the Polynome factors, the
Coefficient of w“, in the product, will be equal to the Coefficient of a7“.

9. Hitherto, the Combinations I have been considering, have been subject only to the condition,
that they all contained w of the given Elements. But we may impose the further one, that the

3Q2

482 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

number of the kinds from which the « Elements are taken, shall be x; or this additional limitation,
that m of the x kinds shall each contain v elements ;

" ”
IN ncvvecvescvccersccvervosrcsses AR. SOME ARO SpOOEeS

and so on; and the Elements from which such Combinations are to be formed, may admit either of
limited, or unlimited repetition.

10. If the given Elements are of s kinds, and may be repeated in each kind without limit, the
Coefficient of a", in the product of the geometrically progressing Polynomes, will consist of terms
in which there are w elements of one kind, ... of 2 kinds, ... of = kinds, ... » never exceeding u,
and finally, when « becomes equal to, or greater than s, becoming equal to s. Consequently, the
models or types, after which these several terms in the Coefficient of a* are formed, will depend
altogether on the partitions of the number w into 1, 2, 3, ... 2 parts. If « <s, the number of
these terms will depend on the number of the partitions of w enumerated in the expression,

[u, 11] + [et 2] + ---0-- [Ms | = [2 4, uw, ].

When x becomes equal to s, the number of these partitions will be [2 s,s]. When w >, the

number of the partitions will be
[u, 1] +[v+2)+...... [wv 3] = [wt+s, 5].

See Article 7, Section I., of the present Paper.

Thus, if the Elements are of 6 kinds, and they are to be combined together 7 at a time, there
will be in all [13, 6] =14 types, in accordance with which all the Combinations, containing
7 Elements each, will have to be constructed; and these types are the following partitions of the
number 7.

Number of

nda Partitions of 7. | Corresponding Type.

1 Tlie?
2 6,1 | A8B
5,2 A> B2
4,3 | A‘B3
5, As A°BC
4, 2,1 AAB2C
3,3,1 | A3B3C
8,,2,'2 ASB2C2
4,1,1,1, | A4&BCD
3,2,1,1, | A3B2CD
202 OF 1, | AaBtoD
8,1,1,1,1,| ABCDE
2,2,1,1,1, | A2B?CDE
2,1,1,1,1,1| A?BCDEF

10.* Let one of the x— partitions of w be
Os, Os ete CIN) GUG Bis «save. (IM) 0, Us oe Urstsceaas (IM )y BcCoy
So that 97. +- m2 4-910 + .ssccuccsece = >
and mv +m’ v + mv! + 1.2... = Ue

AND ON COMBINATIONS AND PERMUTATIONS. 483

z}—1
Since the s kinds of Elements can be combined x at a time in |

aE different ways, and since

the different = parts of the above partition admit of being permuted, and in that way differently

17 1
Sp ae iret ways, the number of the
different Combinations of the proposed form, in case of unlimited repetition, will be
Sal ge er lies
ite a let 12 1 & = Pe a ae ae a: E s\s\a) 0) ¢:s\v\s/e's'sia (
and if corresponding to every different x — partition of w, we construct a similar expression, the sum
of these will give the total number of the Combinations which can be formed from the s kinds of
Elements, when in each Combination there are w Elements of = kinds.

distributed among the = kinds of Elements, in

1 7” Xx).

11. In the case of unlimited repetition, the aggregate of all the terms, containing w Elements of
# kinds, admits of Summation. For, if in each of the x — partitions of the number w, the parts be
permuted one with another, the number of all these permutations will be

[w—1]*-*|-" és eesti tla
’ aes = = eee ce etme eet ees cee (xx1),

_ equivalent terms in the developement of the Binomial [1 +1]“~'. This will appear from the

* : é u 2.8
following consideration. In the case of i | partition the parts can be permuted in one way.
or 1 — -
(w-1) -
or Bi-
factorial successively up to 2*~* [1], or =*~* [vw —1]; and the formula (xx1.) will be the Integral.

In the case of | partition the parts can be permuted in (w—1) ways. Integrate the

Consequently, the number of the different Combinations, containing « Elements of z kinds,
will be
eo laa (Ge
=a 6 SS
7 elle i= al
Exampie, How many Combinations, containing eight Elements of three kinds each, can be
(None iy +3 7.6

_ formed from four kinds of Elements, unlimited in number. Answer {as eae 84.

ndioooSdbacecancacicomor . (xxi),

Now the sum of all the terms of the form (xxu1.), from x = 1 to x = w, ought to be equal to the
ull
_ Coefficient of wv", or to Pre and accordingly, if we give to the product (xx11.) the form }z, s}
.
_ x }x-1,u-11}, it will appear to be a particular case of the general theorem, Article 7 of the present
ull

s
rT? the first

_ Section, last demonstrated; so that $+ [fzx, st x {s-1,w-1}]= {ustu-1} =
=z

. term in the developement being 0,

a 12. Suppose the Elements in the s given kinds to be limited in point of number. Let it be
_ required to form, from these elements, Combinations, each containing (w) elements of kinds, with
_ this further limitation, that

m of the x kinds shall contain v elements, each :

i ! . 1 and so on.

™m wee reeves resrersesreseeseny UV cosccsverveece

Ist. If none of the given kinds contain as many elements as are denoted by any one of the
numbers v, v', v’,,.., no such Combinations as are required, can be formed from the given
elements.

484 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

2ndly. If any of the given kinds contain fewer elements than are denoted by the least of the
numbers v, v’, v”... , such kinds, it is manifest, may be wholly omitted from consideration.

srdly. If any of the given kinds contain more elements than are denoted by the greatest of the
numbers v, v', v” ..., the excess above such greatest number may be wholly omitted from con-
sideration; and, in the same manner, if any of the given kinds contain a number of elements
intermediate between two of the numbers v, v’, v”’, ... , the excess above the least of these two
numbers may, in the course of the operation hereinafter directed, be wholly omitted from con-
sideration.

Thus the given set of elements admits of reduction to ¢ kinds, containing at least v elements
each :

+ 7’ kinds, none of which contain v elements, but each of which contains at least v’ elements :
+ T" kinds, none of which contain v’ elements, but each of which contain at least v” elements, &e.
Thus there will be ¢ kinds to supply m kinds in each Combination with v elements each :
t+=m+T =t' kinds, to supply m’ kinds in each Combination with v’ elements, each: ¢ — m'
+ T” =t" kinds, to supply m” kinds in each combination with ” elements each : and so on,
Therefore since m kinds have been chosen out of ¢ kinds;
m' Borer eee CROUL OL a cess.’
Gt, eve aaree saan sisic EOI OT ATae rcv is aOCCs
the number of the Combinations of kinds that will be formed, in which the several kinds will
contain the requisite number of elements, will be
t™ -1 pel -1 vm -1
ry x rp x | 7 x &e. Reese aaenan Sees ae les siennse OXLUD)
Examptre.—F rom the elements 7’, E°, D', C*, B’, 4, how many Combinations of the form
or type,

can be constructed ?
Since 3 is the highest number in the type, reduce the given elements from
G 3G5h Bewray cay iy f

to $ So 68, | 85408), ils '
then to Be Qi. Oy 8, 0s
then to they ne ley fia utils

Then since m=1; m'=1; m”=2;

t=4; f=4; ft =4.

Blak of

ey ae

-1 vm | -1

4 4 4.3
Re) =-x-
J 1

a

13. When, however, after previous reduction, if requisite, the limited number of elements is
the same in each of the given kinds, and it is required to determine how many Combinations can
be formed from those elements in accordance with a given type, either all or none of the kinds
will contain the number of elements requisite to form any required Combination: and the formula
applicable to the case of unlimited repetition, viz. (xx.) 4

s* | -1
ri ira lay oo eeeeeed
is to be applied.

Exampite.—Giyen the elements TATE Oa 10

and the type Aegan 2

AND ON COMBINATIONS AND PERMUTATIONS. 485

How many Combinations can be formed in accordance with that type?
Here S=4; 2=3; m=1; '=2,
Bhai ol
Given the same elements.—How many Combinations can be formed in accordance with the type
bp, 1, 1?

Since 6 is greater than the given limit 5, the Answer is 0.

And the number of Combinations = = 12,

14. Before closing this Section on Combinations, I shall beg to notice that all of the theorems
it contains, admit of an important application, and that is, to the properties of Composite Numbers.

It is known, for instance, that, if the elements 4, B, C, &c. represent primes, a composite
number, of the form 4*. B’. CY. &c., will have the total number of its divisors represented by (a@ + 1)
(8 +1) (y +1) &c.; but if the question be, how many divisors such a number has that are of x
dimensions, the answer to that question will be obtained by- means of formula (xiv). But it will
suffice to have hinted at these analogies.

SEC LON Tine
On Permutations.

1. Wuewn there are s different kinds, each containing only a single element, these elements,
taken w at a time, will form s“|~’ different Permutations; where s"|-* = 1"|’ x the coefficient of a"
in [1+ ]' developed. But when any of the s kinds contain more than one element, and the
plurality of the elements is short of infinity, it is only in the particular case where w is equal to the
united number of all the elements belonging to the s kinds, that the number of the permutations
has hitherto been determined. In this case, if there be a elements of one kind, of a second kind,
-y of a third kind, &., and a+ +-y + &c. =o, the number of the permutations formed by the ¢

5 : r ate
hings taken all at a time, is See os according to the well-known theorem.
; : . &e.

2. The latter formula denotes the number of permutations which the a elements of the kind 4,
the 3. elements of the kind B, the ry elements of the kind C, &c. are capable of forming, when,
instead of being permuted indiscriminately, the A’s, the B’s, the C’s, &c. change their order of
Sequence in respect of one another, but in respect of the elements of their own several kinds, preserve
an immutable order of sequence. If the a elements A, not to the full extent of 1* | 1, but to some
limited extent, undergo the permutations P(a); and in like manner the 3 elements B to the limited
extent P(/3); and the y clements C to the limited extent P(r), &c. the number of the permutations
which the o things will then together form, will be

] 1° 1

Sota SE G)s PUB) Pye ccitecs .cvaves vee (XXIV);

_ 8, To determine generally the number of permutations which can be formed from any given
set of elements, taken w at a time.

Let any partition of uw be p+ q+ 7 + &e. =u.

It has been shewn, in Articles 12 and 13 of the preceding Section, how to determine the number of
all the Combinations which can be formed from a given set of elements, when each Combination is to
consist of ~ elements of x kinds, and is to accord with any particular partition of w, or type. If

486 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

that partition be p, g, 7; then, in the number of combinations so determined, are included, not

only those of the form A’ BIC’, D? EF", &c. (in which the kinds of elements are changed), but

B? C1, D' E' F®, &e. (in which the kinds remaining the same, the order of

Let the total number of the combinations corresponding

every such combination will give rise to
ul

different permutations, if we denote Sasa by P;

also those of the form A’
sequence in the numbers p, q; 7 is altered).
to one such partition of w be denoted by Q. Then since

petatr|? [2
| ee a | Pak a ad pr toh Be
every different partition of w, or type, will give rise to Q x P permutations. We must therefore
determine by Articles 13 and 14, the number of the combinations corresponding to all the different
partitions of w, and also the corresponding permutation factors, and take the product; and the sum
of all these particular products, or S'[Q x P], will give the total of the permutations which can be
formed from the elements taken w at a time.

ist Exampte. Given the set of Elements 4‘, B’, C®. Required the number of all the Com-
binations and Permutations of those Elements, when 7 are taken at a time.

Here u = 7; 8 = 33 and, since all parts are to be excluded which exceed 4, = in this case

varies only from 2 to 3.

< tm? -1 Ugmi\—t 3 i- = ® |

Given t | em | Coots Permutation al e

Vlements. mpl mit anon factor. pe
1 lose factor. tions.

Parti- \
tions of 7 \

and Exampxe, Given the set of Elements 4°, B’, 0°; required the number of all the Com-
binations and Permutations of these Elements, when 8 are taken at a time.
Here wu = 8; 8 = 3; and, since all parts are to be excluded which exceed 5, = in this case varies
only from 2 to 3. The combinations are here obtained by the formula
Re | Re
i:

—* oe ee eer ag ——_—

Combina- Permutation No. of Per-

Given
tions. Factor. mutations.

Elements.

=6 oe = 560 336

=3 |) tear = 70 210

Parti- =6 i+ = 168 1008
tions of 8 6 aa = 280 1680

4. The method I have just described, of determining in succession the permutations corre-
sponding to the different partitions of the number w, must, in cases of limited repetition, have
been adopted to determine also the number of the combinations, when wu elements are taken at a

.

a

AND ON COMBINATIONS AND PERMUTATIONS. 487

time, had not the formula xrv and xv, or the method described in Article (6) of the former Section,
afforded a readier method of attaining the same object. In the case of Permutations, I have not
succeeded, except in certain cases, in readily determining by means of an explicit function of w,
the number of the permutations formed by w elements of s kinds.

5. It isa known formula, that when the elements in all the s kinds admit of unlimited repe-
tition, the number of the permutations which can be formed by taking w elements at a time, is
expressed by s”.

If we take the form I have assumed, in Articles (10) and (12) of the former Section, for the
resolution of « into x parts, where these parts are represented by

5 U5 V>«-- (mM) v', v', v', ... (m') v', v", v0”, ... (m’),
and) 7) om mM” 4. vane wcccs sme = 2,

now

and mv + m'v' + mv" 4+...... =U;
we shall, in the case of unlimited repetition, have the combination factor,
1*| 1
12 |e 1” |1, 2”

Q, =

13

and the permutation factor,
uf1

al
= PPro pepe en ene (xxv).
the partitions being those in which the lower limit of the parts is 1; and x extending from 1 to x,
when wu < s; and from 1 tos; whenw=or>s.

But, by Art. 7, Sect. I, p. 5, [w, 1] +[w, 2] +......[u, 2] =[u, x];
and the product Q x P therefore coincides with the expression given by Lagrange, in his demon-
stration of the Polynome theorem, for the terminus generalis of the expansion of

P,

pe eR ON Cy Oe! x €° a. spsicvntes'svei (8))
*, The terminus generalis, so multiplied, is

be IES NG Aap
ir] a all =) ,
where p, g, 7, &c., are all the different parts obtained by s — partitioning w, the lower limit of the
parts being 0; and for every determinate set of values assigned to p, q, r, &c., these letters
receiving every different order of sequence possible.

when multiplied by the factor 1°

P+iti+...@y=rps (

In the case, therefore, of unlimited repetition, the number of permutations which can be formed
by taking w elements of s kinds at a time, is the coefficient of wv in the product of the s infinite

series,
2

at
“s + &e.] [1+ 7 += + Ge] os... (6)5 multiplied by 1

ar

[it+a+
iF
6. It is manifest, therefore, that if with respect to the elements of any one kind, 4, we restrict
the number of elements to a; and in another kind, B, to (3; and so on, we must make a correspond-
ing restriction in the terms of 1, 2, or more, of the above Polynomes. And this leads to the following
theorem: viz. that the number of the permutations which can be formed by the elements of s kinds,
whose respective limits are a, 3, yy, &c., when those elements are taken w at a time, is the
coefficient of w" in the product of the s Polynomes,

Messte ss seeeos(XXVI)s

) PRS
Vou. VIII. Part IV. 3R

a a w a? alone u
(1 + rar "hc =) (1 +U+—+... mm) &c., multiplied by 1

488 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

This theorem, I fear, is not likely to facilitate much the practical computation of the permutations of
plural elements, though perhaps it may lead to curious Algebraic results, Professor De Morgan, since
I made known to him this Theorem, has done much to remove the difficulties which beset the com-
puting of permutations by means of it. But I doubt whether the method will be rendered more simple
than that derived from a direct consideration of the problem of permutations, given in Article 2 of
the present Section. Thoroughly examined, the two methods must in the end prove identical.

I had some expectation that by giving to the Polynomes the form

(e* — a) (ce? — b) ( - 0), &e. = &” — ae"! + abe** — Xe.
—b + ac
—c + be,
some facilities might be afforded to the computing of permutations in certain cases; but I do not

at present believe that any such results are to be anticipated.

7. The theorem, xxv1, has led me to the determination, in one particular case, of an
explicit function of w, for expressing the number of the permutations formed by s kinds of elements
taken w at atime: the case is that where, in all the kinds, the elements are dual. If we develope

[40+ Aa + A,w* + &e.]}, :
by Arbogast’s method, we obtain for the coefficient of x",

Pld Sif

igs lee

Arne D(4,)"-") -

ray
where D* (4,)"~* is the coefficient of #* in
[41 + doa + Aza® + &e. J";

and a second developement leads to a double series, in which, if 4, and 4, are made equal to 1,

and 4, to 4, and all the other terms 4,, 4,, &c., are made equal to 0, we obtain terms
2:

expressing the coefficient of a” in (1 +a+ =) ; and that series multiplied by 1"

' gives the
number of the permutations in the case stated *.

* I here transcribe the coefficient of a, in [14+ a+ 4,02 + A,a%+x-]' obtained by Arbogast’s method, slightly modified.

1
0. |-———s*|-
rp! s

_

fe ea
re2yr* |-* 43.

i)

1
ray s*-?|-1 Ag.

s

] Ag)?
eg Raa Ay + s"-?|—1 a +

1
A peep Leb Ae+ oopA, 4G).

1 A,? As
& felt trey | ana [emi sr]:

6. iat [&c.] (the law of the terms is obvious).

AND ON COMBINATIONS AND PERMUTATIONS. 489

The series may be expressed by

S

u**| sis re] -1
(ar

2]? ) > eoavecavmveseea(XXVII.)

a : , U
the limits of @, when w is even, being 0 and a?

u—1
nodacacenies Wile naneodOeirGl

This gives the number of permutations formed by s dual sets of elements, when taken w at
a time.

Thus when uw = 6, we have

ier ee aid ee pi Gale
2 2.4 2.4.6
And when wu = 7,
Pie eae fie eros | (il maa

2 2.4 2.4.6
When wu = 6, let s = 3.

HebipesoO Al eBeoail
RC ee
ENED BS Hey ag 2

= 90.

The number of the permutations

When uw = 7, let s = 4.
Te Ord pAvi8 Qi Mr Baa 2iw

The number of permutations is then = a a 7.6.5.4.3 = 2520.

It is not improbable that a further developement of the series obtained by Arbogast’s method,
and a subsequent equating of particular terms to 0, might lead to other symmetrical and curious
results.

8. In Art. 7 of Section II., I have given a method, from the known combinations of two
independent and separate sets of elements, differing from one another in kind, to determine the
combinations of the two sets, when united. I proceed to apply a precisely similar method to Per-
mutations.

If » of 7 elements form {v, 7} combinations, and (w—v) of (¢ — 7) elements form {x — v,
o — T{ combinations, they will together form

Sv, rt x {u—v, o — 7} combinations.

Take some one of the {v, +} and some one of the {w—v, o— 7} combinations. If it be a
condition that the elements belonging to these two sets, separately considered, shall preserve their
original order of sequence, but that those of one set of elements, as compared with those of the other,
may change their places, the permutations formed by the two sets of this particular combination

united, will be
1

1°
But, if the v elements, and the w — v elements, considered separately, may change their places,
and the former may undergo P(v) changes, and the latter P(w — v) changes, and if the same may
be predicated of the elements contained in each different pair of combinations, fv, rv} and {u —»,

o—7h
8R2

490 Mr. WARBURTON, ON THE PARTITION OF NUMBERS,

then the number of the permutations, corresponding to every pair of determinate values given to v
and uw —v, will be

*P Su, of = {v, rt {u—v, o— 7} P(v) P(w - »),
hel A
And for the line w of the table, w remaining constant, and v only varying, we have
i 1

Plu, at = 87] P fo, tT} Pfu—», o-hril:

And the equation of the terms of the table of double entry, in which w is the index of the line,
and v of the column, is

u+ti—vi’|!
[u, v]=[v, v] x [w—v, 0] ea eceentee aera,

I give an example of such a table, applied to the case of the two sets of elements,
A, BB, CCC; and DDDD, EEEEE.

Permutations of 4, B’, C*; whose Combinations are 1, 3, 5, 6, 5, 3, 1
and of Dt, E° Sie sven nesvacignaoesesoess Te Oh Omnis tees ary cyan

8
48 19 111
192 152 38 494
240 | 640 760 380 60 2111
558 | 1920 3040 2280 720 60 8634
1176 | 5208 | 10,640 10,440 5040 840 33,435
2184 |12,544 | 32,984 35,560 26,880 6720 116,998
3402 |26.208 | 89,376 | 148,428 107,520 40,320 415,380
3780 | 45,360 |207,480 | 446,880 468,720 201,600 | 1,378,820
55,440 | 395,010 | 1,141,140 | 1,552,320 859,320 | 4,003,280
526,680 | 2,370,060 | 4,824,820 | 3,104,640 | 10,325,700
3,423,420 | 9,729,720] 9,369,360 | 22,522,500
15,135,120 | 22,702,680 | 37,837.8004
37,837,800 | 37,837,8001)

0
1
2
3
4
5
6
7
8
9

a
ime SO

_
i)

_
>

~
oO

* P {u, o} means the permutations of the elements contained in the combination {u, o}.
+ The truth of Article 9, Section 3, is here exemplified.

AND ON COMBINATIONS AND PERMUTATIONS, 491

9. The following corollary results from the preceding Article.

1° i
The formula FF) Py ee expresses the number of the permutations, not only of the

a+ P+ + &c. elements, when they are all permuted together, but also of [a+ B+y+&e]-1
elements, when permuted together: of which the following is the demonstration.
g-1}1
Since, by xxviu1, Pi{o-1, of =P§r-1, tT} Phao-—t, 6- zt TE
ic ai 19
yo-} 1
+ P[r, 7] P§o-7-1, oa
assume, for a moment, that
Pir, rt} = P{r—1, +}; and let each =J,

and that P{ao—7, o-r}=Pjo-—7-1, o—7}; and let each= XK.

Then, P{o-1, o} Sri oad era raster
a
aie ae ae
a;l
But, P$o, of = Pfr, tt Pha —-7, Sapa
stagiigr wcll

Hence the law enunciated will be true of the two sets of elements conjoined, if it be ever true
of each of the two sets separately. But it is true of two separate sets, when each consists of elements
of only one kind; for then, whatever may be the number of the elements permuted at a time, the
number of the permutations is constantly one. Consequently, the law holds true when there are two
kinds of elements conjoined ; consequently, when there are three kinds ; and therefore universally.

Hence, in the product of the Polynomes,

E % a at j a :
ie A A ace Tae
1.2 1/2 Se 1

the penultimate coefficient = o x ultimate coefficient.

HENRY WARBURTON.
May, 1847.

ADDENDUM.

Since this Paper was corrected for publication, a member of the Society, distinguished for his mathematical
erudition, has caused the Author’s attention to be drawn to the work of Bézout* on Elimination, as containing
a formula similar in structure to the Author’s formula xy.

In the Author’s researches in Combinations, his concern has been exclusively with such of the terms of a
polynome function of the s quantities, 4, B, C, &c., as were of some one, say, the u'", dimension, By such

modes of investigation as occurred to him, he obtained an expression representing the number of such terms,

* Théorie Générale des Equations Algébriques, par M. Bézout, 4to, 471 pp. Paris, 1779.

492 Mr. WARBURTON, ON THE PARTITION OF NUMBERS, ETC.

with the special object of applying it to denote the number of the combinations which can be formed with
plural elements.

Bézout’s object, at least the sole use to which he applies his formule throughout his work, is elimination.
His concern is with all the terms, in the aggregate, of how many dimensions soever, belonging to such a
polynome function as is above described. By a mode of investigation, entirely different from that of the
Author, he obtains a formula expressing the number of the terms which, in a polynome, complete in all its
terms of every dimension from 0 to w, are not divisible by any of the factors, A+, B®, C*, &c.* He finds that,
[w+1]‘|*

Ty ; and the number of

iu the complete polynome, the number of all the terms is represented by
the terms not divisible by any of the said factors, by
[w+1}'|'- [e+ 1-e]}'|'+[u+1-a—fA]|'- &e.

= [ee FA (G54 [peer ee, MUU naceaie(e(eclclelstetsisietsisiee tare the (1.)

— &c,
and this is the formula to which the Author’s attention has been directed. It agrees in its general structure
with the Author’s formula xiv: the points in which it differs will presently come under notice.

In his 4th problem, Bézout considers a particular case of an incomplete polynome, meaning thereby a
polynome in which the highest dimension of one of the s quantities, A, is a, of another B, is 3, and so on;
a, 3, &c., belng less than u, the highest dimension of the polynome itself: and he here makes the observation,
that there are as many terms in such a polynome as there would be of terms not divisible by any of the factors
A“%*!, B8*1, &c., in the polynome, supposing it to be complete; but he gives no formula coextensive with
the generality of that observation. By following out that observation, we may, by two steps, deduce the
Author’s formula x1v. from that of Bézout.

cs

The first step is the following. The terms which in the polynome, if complete, would be non-divisible
by any of the factors d*+', B®*1, &c., amount in point of number to
, (w+1]'|'—[wt+1—(¢4+1)}'|'+[u+1-(@+1)—(8+1)]'|'- &e.
Ty —[w+1-—(8+1)]'|' &e. efavetele fe chfe. sem (eae)
— &e.
and such, therefore, is the number of the terms in the incomplete polynome function of s quantities, where
a, 8, 7, &c., are the limits of the dimensions of A, B, C, &c., respectively ; the highest dimension of the
polynome itself being w.

The second step is the following. If from a polynome whose highest dimension is w, all the terms of the
dimensions not exceeding («— 1) be deducted, the remainder will be the terms which the polynome contains
of the wu dimension. Hence the number of the terms of the w'* dimension in the incomplete polynome
will be obtained, if in (2) we substitute « for (w+1), and deduct the result from (2). That is to say, the
required number of terms will be A(2), meaning, by A (2), [1— E“] (2); i-e.,

[w+ 1] |?—[e4+1—(@+1)]7 |) + [w+ 1-(a4+ 1) -(8+1)]"|!- &e.
mp | Zan NE 96 Fes pee [BET I Jiorioini.3 (3.)
— &e.
which agrees with the Author’s formula xtv.

Considering that Bézout’s work has now been published nearly seventy years, it will no doubt excite the
surprise of many members of the Society, that a deduction from Bézout’s formula so easy as the foregoing,
should not have been made long ago, and applied to the solution of the problem of the combinations of
plural elements.

* The complexity of Bézout’s notation rendered it inexpedient to retain it in its original form. To facilitate comparison, the letters
have been assimilated to those used by the Author.

—— ol

XXXVI. On a Peculiar Defect of Vision. By HENry Goong, M.B.,
of Pembroke College.

[Read November 9, 1846, and May 17, 1847.]

Tue following details of a case of defective vision may not be uninteresting.

About ten years ago I first perceived a defect of vision in the right eye, the extent of which,
before that period, I believe to have been inconsiderable: the defect being that small objects,
when viewed at the distance of greatest distinctness, appear as two. My attention having been
called to Professor Airy’s Paper on his own eye, I find that my eye, tested in the manner he
proposes, exhibits a similar defect. This method is to view with the defective eye a pinhole in
a card, which slides along a graduated scale, one extremity of the scale being applied to the cheek-
bone, and the other directed towards an illuminated sheet of paper.

The following are the appearances observed :
1. When the card is quite close to the eye, the image of the pinhole is perfectly circular.

2. As the card is removed to a greater distance, the image becomes gradually elongated
in the form of an ellipse, with a sharp dark line in the long diameter, most distinct at the distance
of 4.5 inches, and best visible in a minute hole.

3. At 6.13 inches the image has become extended into a bright well-defined line, of the breadth
of the pinhole as estimated by the sound eye, and crossed in the centre by a dark line perpendicular
to the former dark line which has disappeared: if several pinholes be pricked near one another, the
dark band holds the same relative position in all of them.

4, As the card is removed to a further distance, the bright line becomes gradually shortened,
and at the distance of more than a foot appears as two bright spots only, situated one on each side
of the dark band ; but, at the same time, in the direction of, and as it were overlying the dark band,
a bright line gradually appears, short at first, and becoming elongated with the removal of the card,
so that at about 10 inches or more the appearance is that of a cross, most strongly illuminated in
the position of the two bright spots before described. At 12.2 inches this cross appears as a regular
quadrangular figure with concave sides, the two spots being most strongly illuminated. If a dark
spot on a sheet of white paper be viewed in the same manner, the appearance is necessarily the
same; but owing to the greater distinctness of the two spots, the remainder of the figure is easily
overlooked, and the appearance is that of a double spot; consequently, if a page of small type be
viewed at this distance, the print appears double.

5. When the card is at 25 inches, and all greater distances, the image is a bright line perpen-
dicular to that seen at 6.13 inches, the two spots representing that line having almost coalesced into
one, causing the bright line to be brightest in the centre.

Distant luminous objects with clear defined outlines, such as the Moon, appear as a succession of
well-defined images overlying one another with their centres in this line.

6. The more distant line is inclined to the mesial plane of the body at an angle of 21°, and the
upper part falls inwards towards this plane.

It appears in the above, that a short distance within the nearer focus a dark line occupies the
position of the bright line seen at that focus, while beyond the focus at all distances the line continues
illuminated. ‘The same holds with regard to the second focus.

494 Mr. GOODE, ON A PECULIAR DEFECT OF VISION.

7. On viewing two dark or two bright lines drawn in the form of a cross, and held in the
position of the lines above indicated, the vertical line appears broad and very faint at the distance
of 6.13 inches, the horizontal line appearing clear and well defined ; while the reverse is the case
beyond 25 inches.

There is no apparent defect in the left eye. If several holes be pricked near one another, and
viewed by this eye at the distance of 5 inches, which is somewhat within the range of distinct vision,
a dark central spot is observable in the centre of each: also a narrow luminous slit appears traversed
in the direction of its length by two central parallel dark lines.

It is probable that the defect of the eye is inherited, as my mother has a defect of a similar
nature in both eyes. A circular pinhole viewed with either eye at the distance of 7.5 inches appears
as an ellipse with the major axis parallel to the mesial plane of the body, while at the distance
of 5 inches the image is an ellipse with the major axis perpendicular to the former.

Stnce the period when the above measurements were taken, I have made frequent use of the
eye; owing, most probably, to this circumstance, a very considerable amelioration has taken place in
it; the first focus, which in the month of June last was at 6.13 inches, in the month of December
was at about 10: the second focus was readily ascertained in the month of June to be at between
24 and 25 inches; but in December it was impossible to determine the exact position of it by the
simple observation of a pinhole; because, instead of appearing as a sharp distinct line, as before, the
image was always confused by the presence of the luminous square above described. The image
was, in fact, a rhomb, with the longer diagonal, distinguished by its brightness, in the direction
of the further line, while the line seen at the nearer focus never disappeared, but became shortened,
remaining always the brightest part of the image, and forming the shorter axis of the rhomb. How-
ever, by means of the instrument described below, the second focus was ascertained to be at a
distance of between 27 and 28 inches. Since December, up to the month of March of this year,
no change whatever has taken place in the eye, notwithstanding the constant use of it.

The length of the line, as observed at either focus, is, of course, dependent on the aperture of
the pupil, and the distance from the retina, before or behind it, of the line of convergence of the rays
refracted from the other focus.

The differences in the eye observed in June and December, are exactly such as occur, when
similar observations are made on a sound eye, to which is applied, in one case, a cylindrical convex
lens of short focal length, and in the other a lens of weaker power.

The instrument above alluded to as serving to determine the distance of the foci is simply that
of Scheiner. Let a tube which slides within another in the manner of a telescope be closed at
its extremity by a card pierced by a single pinhole, while the other extremity of the apparatus
is closed by a card pierced by two holes, the distance of which from one another is less than the
diameter of the pupil of the eye of the observer. When the extremity pierced by the single hole is
presented towards a luminous surface, and the other is applied close to a sound eye, if the distance
of the single hole is equal to the most convenient distance of distinct vision, free from any exertion, |
the hole will appear single; but if the distance be greater or less than this, the hole will appear
as two; as is well known. This instrument may be applied to the determination of the two foci of
a defective eye, by observing that, in order to ascertain the distance of either focus, the line passing
through the two pinholes must be perpendicular to the direction of the line, which forms the image
of the point at that focus.

There are, however, two inconveniences attending the use of this instrument; namely, firstly,
that if the eye, on which the observation is made, be at all long-sighted, so that the pinhole requires
to be placed at a considerable distance, the two pencils of light falling on the two pinholes are nearly
parallel, and therefore the pinhole may be moved through a considerable space backwards and for-

2. ————e———eEe—eEeEeE
eed

Mr. GOODE, ON A PECULIAR DEFECT OF VISION. 495

wards, without much affecting the position of the resulting images. The other inconvenience is, that
the eye is naturally endowed with a power of adapting itself to different distances, and that this
power is very little under command in an eye which is not habitually used, a circumstance, perhaps,
frequent with those who have one eye defective: such an eye, when tested by any instrument, will
at one instant appear to have one focal length, and at another instant another. When a cross drawn
on paper is held at a distance between the two foci, I find that I can at will discern either the
horizontal line, or the perpendicular, without altering the position of the paper.

There are, therefore, no means of attaining the requisite measurements beyond an approximation,
and the rest must be ascertained by direct experiment with a series of cylindrical lenses.

Having calculated an approximation of the glass I required, I applied to M. Chamblant,
a working optician at Paris, who occupies himself solely with the construction of lenses and
spectacle glasses with cylindrical surfaces, and after several trials I succeeded in obtaining a glass,
which gives me distinct vision of objects both far and near alike, thus shewing that the error of
malformation ‘is independent of the state of adaptation of the eye. The glass I use is plano-
cylindrical, the cylindrical surface concave, with a radius of curvature of nine French inches. The
axis of the cylinder when presented to the eye, coincides of course with the direction of the line at
the nearer focus.

A plano-convex glass also, with the axis perpendicular to the direction of the line at the first
focus, and the curvature of which is the same, gives distinct vision, provided that the object
is placed sufficiently near to the eye; or even a glass much stronger, when the object is very
close to the eye.

Considering that the inclination of the lines at the foci might have a physiological importance,
I devised the following method of determining it accurately. If a number of pinholes be pricked
in a card, in a straight line, and the card be fixed in such a manner that it may be made to revolve,
and have an illuminated surface behind it, when a defective eye is placed at the proper distance, it
readily recognizes the position of the card in which all the lines representing the images of the
pinholes lie in one straight line, being the line in which the holes are pricked: care must be
taken that the body is held perpendicularly. It is easy now to determine the inclination of this
line to a hair stretched vertically by a weight.

Within the last few months I have met with three or four cases of defective vision similar
to my own; only two of which are of sufficient magnitude to be worthy of mention,

One is that of Mr. Parry, who has served many years as a medical officer in the army. This
gentleman’s left eye is perfect, except in being somewhat presbyopic, but from the time of his
earliest recollection he has never had distinct vision with his right eye; he has never been able
to read with it, though he has an indistinct vision of objects at all distances.

His eye, tested by a pinhole in a card, perceived the hole as a horizontal line at the distance
of 37 centimetres (about 144 inches); the line is inclined at an angle of 87 degrees to the
mesial plane of the body, and meets this plane produced inwards and upwards. At some distance
beyond this the hole appeared enlarged, and of a rhomboidal figure, but never as a line.

When he viewed two lines drawn in the form of a cross, he saw well enough the horizontal
line at 144 inches, and for some distance beyond, but at no distance could he discern the vertical
line. he error therefore seemed to consist in an exceedingly feeble refractive power in horizontal
planes: I therefore tested his eye with plano-cylindrical convex glasses, in order to obtain data
for calculating the forms of glasses to be used for viewing objects at different distances; and
we found that with a glass of 24 French inches radius, the two lines of the cross, at 12 or 14 inches
distance, appeared of nearly equal brightness. This glass was rather too strong, while 3 inches
gave a glass rather too weak. To view distant objects, therefore, I caused to be made a glass

_ cylindrical concave on one side, with a radius of 74 French inches, cylindrical convex on the

other, with a radius of 44; the axes of the cylinders of course crossing at right angles, and
Nor, VIIT, Parr LY. 38

496 Mr. GOODE, ON A PECULIAR DEFECT OF VISION.

the axis of the convex surface being in the vertical direction. This glass appears to fulfil the
required conditions: it enables Mr, Parry to read inscriptions at a few yards distance, and also
to have a distinct perception of very distant and minute objects, such as are presented in an exten-
sive landscape.

In order to ascertain if it were possible to detect any error of curvature on the surface of the
cornea, I observed the appearance of the reflection of a small luminous square held a few inches
from the eye; but in the central part of this structure the reflected image was perfectly square,
while the distortions produced at the circumference were equally produced in the sound eye; and
there was no reason to conclude that the defect of vision arises from any defect in the cornea.

Mr. Parry finds his sight considerably improved by looking through a small hole in a card, so
as to admit pencils only to fall on the central parts of the cornea; or, still better, by looking
through a narrow vertical slit, provided that the illumination of the object be sufficient to
compensate for the smallness of the pencils admitted. He finds that a very slight pressure
on the eyeball, applied at the outer angle of the eye, improves the vision. I also find the same
when gentle pressure is made at the upper and outer part of the ball. It is to be observed, that
the application of a narrow slit to a sound eye produces an effect nearly analogous to that produced
by a plano-cylindrical lens.

The second case is that of a student, who stated, that in observing small objects at 20 or 30 yards
distance, he saw a second image of the objects, one image, however, being much fainter than the
other. He considered that his sight had become impaired by too intense application to books, having
only observed that his eyes were defective after several years close study.

On testing his eyes by a pinhole in a card, he saw the hole as a horizontal line most distinct
at about 35 centimetres distance; beyond this the hole appeared indistinct. Also when he viewed
two lines in the form of a cross, when they were held at 35 centimetres distance, he perceived most
distinctly the horizontal line, and at some distance beyond this the vertical line. The line seen
at the nearer focus was exactly perpendicular to the mesial plane of the body. I ascertained that
the distinctness of his vision was considerably improved by applying to the eye a plano-cylindrical
concave glass, of about 16 French inches radius.

Since the above paper was read, I have met with three gentlemen in the University, all of whom
have one of their eyes affected with a malformation similar to my own; or with “ astigmatism,” as it
has been called. The amount of the “astigmatism” in all of them appears to be corrected
by a plano-cylindrical glass, the curvature of which is 12 inches radius.

In one of these gentlemen it is the more perfect eye that is thus affected. This eye, as observed
in some other cases, gives diplopic vision of objects at a certain distance. Another stated that
the vision of his eye was perfect until a few years since.

HENRY GOODE.

iA

XXXVII. Contributions towards a System of Symbolical Geometry and Mechanics.
By the Rev. M. O'Brien, Professor of Natural Philosophy and Astronomy
in King’s College, London, and late Fellow of Caius College, Cambridge.

[Read March 15, 1847.]

1. Tue important distinction which has been made by an eminent Authority in Mathematics
between Arithmetical and Symbolical Algebra, may be extended to most of the Sciences which
call in the aid of Algebra. Thus we may distinguish between Symbolical Geometry and Arith-
metical Geometry, Symbolical Mechanics and Arithmetical Mechanics. This distinction does not
imply, that in one division numbers only are used, and in the other symbols, for symbols are
equally used in both, but it relates to the degree of generality of the symbolization. In the
Arithmetical Science the symbols have a purely numerical signification, but in the Symbolical they
represent, not only abstract quantity, but all the circumstances which, as it is usually expressed,
affect quantity. The Arithmetical Science is, in fact, the first step of generalization, and the
Symbolical the complete generalization.

In this view of the case, I have ventured to entitle the following Paper ‘‘ Contributions towards
a System of Symbolical Geometry and Mechanics.” The Geometrical System about to be proposed
consists, first, in representing curves and surfaces by symbolical formule, and secondly, in using
the Differential Notation to denote Perpendicularity, according to the principles explained in a Paper
read a few months since at a Meeting of the Society. The proposed Mechanical System is analogous in
many respects to the Geometrical: examples of it have already been given in the Paper just quoted,

2. The following well-known principles are those upon which the B

c
Geometrical System is based. Un me

ist. If ABCD be any polygon, then 4D = AB + BC + CD. A D
This may be regarded as the definition of +.

2ndly. Giving the usual definition of — it follows, that, in the triangle S
ABC, ‘ge
AC — AB = BC.

3rdly. Where it follows that, if « denote any right line, — a denotes an equal right line
measured in an opposite direction.

4thly. If m denote any number, ma denotes a line m times the length of a drawn in the same
direction as a. This follows immediately from the first principle.

These principles, with some others which we need not specify here, form the basis of the
Geometrical System about to be proposed.

3. It will be convenient to consider that every line is traced by the motion of a point, and
this will Jead us to distinguish between the beginning and end of a line, the beginning being the
extremity from which the tracing point starts, and the end the other extremity.

38s 2

498 Mr. O’BRIEN’S CONTRIBUTIONS TOWARDS A SYSTEM

4, When we say that a symbol, @ for instance, represents a straight line, we mean that @
defines the magnitude and direction of the line, but not its beginning ; in other words, the line is
supposed to be drawn of a given length and in a given direction, but not from a given point.

However, if the contrary be not specified or implied, we shall always suppose the line to begin
at the origin, i. e. at a certain point chosen for the purpose of reference.

5. We shall use the term Direction Unit to denote a straight line of a unity of length drawn
in any particular direction, We shall always use the letters a, 3, ry to denote direction units, and,
unless the contrary be stated, we shall also suppose these three directions to be at right angles to
each other: in other words, we shall assume a, /3, ry to represent three straight lines drawn at right
angles to each other, and each a unity of length.

6. We shall divide symbols into two classes, Nwmber Symbols and Line Symbols, the former
representing numerical quantities positive or negative, the latter straight lines in magnitude and
direction.

7. We shall define the position of a point in space by the Line Symbol representing its
distance from the origin: thus, whenever we speak of the point a, we mean the point whose
distance from the origin is represented in magnitude and direction by the symbol a.

In our idea of distance here we suppose direction, as well as magnitude, to be included.

8. If a, b, ¢ be any line symbols, it follows, from the first principle above stated, that
a+6-+c represents the distance of the end of the line e from the beginning of the line a; the end
of a being supposed to coincide with the beginning of b, and the end of 6 with the beginning of ec.

In like manner a — b denotes the distance of the end of a from the end of b, a and b being sup-
posed to have the same beginning.

Hence, if a and 6 be the symbols of any two points 4 and B, a—b is the symbol of the
right line drawn from B to A, and b — a the symbol of the line drawn from A to B.

9. If « be any number symbol, and a any direction unit, wa represents a straight line of the
length w drawn in the direction a.

Hence, if r be the length of a right line drawn from the origin, w y = the lengths of the
co-ordinates of the end of that line, and a 3 + the direction units of the three co-ordinate axes, the
three co-ordinates will be represented by the symbols wa, y/3, xry, and the line by the symbol

wat yp + BY.

This symbol also defines the position of the point whose co-ordinates are w y %.

If a 6 be the direction cosines of the line, its symbol becomes

r(aa+bB + cy).

The coefficient of 7 is evidently the direction unit of the line.

10. Let » and * be the lengths of any two lines AP and AP’ drawn P
from a point A, and let e and ¢’ represent their direction units; then the symbols _—
of these lines will be re and 7’¢’, and therefore the symbol of the line PP’ willbe 4 P

re! — re.

If » =7 and ¢’ —e¢ is indefinitely small, this expression becomes
y ’ P
rde.

Now in this case PP’ is at right angles to AP, and therefore it follows that rde is the symbol
of an indefinitely small line perpendicular to the line re.

“¢ Ka terelips

7

OF SYMBOLICAL GEOMETRY AND MECHANICS. 499

The length of this small line is rd@, assuming d@ = angle PAP’; but the direction unit of a
line is expressed by dividing the symbol of the line by the symbol of its length; hence the direction
unit of the small line is

rde de
7d0 or da

AL BAe : ‘ 5 . a
Hence the direction unit of a perpendicular to a line re is mak

In the Paper already referred to, which was read before the Society some months since, the
reader will find this method of representing perpendicularity by the differential notation fully
developed, and the notation Dw.w’', thence derived, explained, together with an auxiliary notation,
Awu.u'; both of which we shall have occasion to make use of hereafter.

11. The following is the method we shall adopt of representing curves and surfaces sym-
bolically.
| To represent a curve or line we shall suppose a variable parameter to be involved in the symbol
of a point, in which gase it is clear, that the point will be indeterminate in position, but restricted
so far, that it will always be found upon some curve or line. The symbol of a point therefore,
when it involves a variable parameter and is thereby made indeterminate, becomes a symbolical
formula defining some line or curve, and may be called the formula of that line or curve.

In like manner the symbol of a point, when it involves two variable parameters, becomes a
symbolical formula defining some surface, and may be called the formula of that surface. This
virtually amounts to defining lines and surfaces by symbolical polar equations.

It is important, however, to observe that we suppose the variable parameters here spoken of to
be nwmber symbols. If the variable parameter be a direction wnit, it must be regarded as equi-
valent to two number symbols.

12. The following are examples of this method of representing curves and surfaces.
The general symbolical formula of a straight line in space is
ute,
where w is the symbol of a given point, 7 a mwmerical variable parameter, and ¢ a given direction
unit.

For take O4 =u (O being the origin) OB =.c, draw a line through 4
parallel to OB, taking upon it AP equal in length to r. Then AP is repre- B
sented by the symbol re, and therefore w+ re is the symbol of the point P,
which, since 7 is indeterminate, may be any point of the line drawn through 4
parallel to OB. £

It appears, therefore, that w + 7e is the formula a straight line drawn through the point whose
symbol is w, in the direction represented by e.

rgo™

13. In like manner the general symbolical formula of a plane is
uwtretre
r and r being numerical variable parameters.

For take OA =u, OB =c, OC =e’, draw AP parallel to OB and equal in
length to r, PQ parallel to OC and equal in length to r’. Then, it is evident,
hat w+re+7'e’ is the symbol of the point Q; and that, since and r are
determinate, Q is any point of the plane which contains the point 4 and is
parallel to OF and OC.

Hence wu +re+7'e is the formula of a plane which is parallel to the directions represented by
¢ and ¢’, and contains the point whose symbol is wu.

}

’

500 Mr. O’BRIEN’S CONTRIBUTIONS TOWARDS A SYSTEM

14. The following is an example of the case where the variable parameter is a direction unit.
The formula of a sphere is
u+ Te,
where ¢ is the variable parameter, and 7 determinate.

For w + 7e represents a point whose distance from the point w is indeterminate in direction but
determinate in length, being always equal tor. ‘Therefore the formula uw + 1e defines a sphere
whose centre is w and radius 7.

We shall now illustrate this method of Symbolical Geometry by the following propositions,
without attempting any systematic arrangement, as our only object is to shew the nature and use
of the method.

15. To deduce the equation of the plane from the formula of the plane, namely, wv +7e+7'e.
Let w y x be the co-ordinates of the point represented by w+re+e, wy, x, of the point
represented by w, let a 3 ry be the direction units of the three co-ordinate axes, and let
e=dat+bB+cey, ¢€ =dat+l'B+c'y.
Then we have
watyB+eysutretre

watyB+zy+r(aatbBrey) +r (dat B+e'y);

tl

and .*., equating coefficients of a, B, y, w=4,+rat+r a,
y,+rb +78,
=e, ¢retrc,

Sti Kae
i}

whence eliminating 7 and 7’ we find an equation of the form
40+ By+Cs=D.

16. To express the formula of the plane by means of the symbol D.

If v be an indeterminate line symbol, and ¢ a determinate direction unit, Dv.e denotes a line
of any length drawn at right angles to ¢ in any direction. Hence it is evident that the formula

re+ Dv.c, or (7+ Dv.)e,
represents the plane whose perpendicular distance from the origin is 7e.

17. The formula of the right line drawn through the two points represented by w and wu is
evidently

u + m(u' — w)
where m is a numerical variable parameter.

18. Hence if « be the formula of any curve the formula of the tangent at the point w is

u+mdu.

19. To shew that the formula of the osculating plane of the same curve, at the point w, is

u+mdu+ndu.

Let PP’ and P’P” be two consecutive chords of the curve; produce PP’ to any point Q, and
draw QQ’ of any length parallel to P’P”: then Q’ is any point of the
plane containing the two chords, which plane, when the chords are
indefinitely small, becomes the osculating plane.

Let w w’ w” be respectively the symbols of the points P P’ P”; then
the symbols of PP’ and P’P” are respectively wu’ — uw and w" — wu’, and
therefore the symbols of PQ and QQ! are m(w' — u) and n(u" — w’),
m and » being arbitrary numbers. Hence the symbol of the point Q’,
and therefore the formula of the plane containing the two chords, is

u+m(u —u)+n(u"’ —w).

i
b

OF SYMBOLICAL GEOMETRY AND MECHANICS. 501

Now in the limit we may put
u—-u=du, w —-w=du+du.
Hence the formula of the osculating plane is of the form
u+mdu+nd*u,
putting m in place of m + n.
20. From this formula to deduce the ordinary equation of the osculating plane.
Let w y x be the co-ordinates of P, then
u=ratyP+ zy,
and therefore the formula of the osculating plane is
(2+ mda + nd’x)a + (y + mdy + nd*y) B +(x + mdz + n&2)y,
whence, if # y, x, be the co-ordinates of any point of the osculating plane, we find
2=a2+mde+ndia, y=yt+mdy+ndy, %,=% + mdz + nd’ x,
from these equations, eliminating the variable parameters m and m, we find the common equation
of the osculating plane.

21. Respecting the geometrical meaning of the symbols dw and d?u it is worth observing,

he
J

that dw represents in magnitude and direction the element (ds) of the are of
the curve defined by the formula wu, and d’wu represents what is called a
double sagitta, as we may prove very easily; for, let P P’ P” be three conse-
cutive points of the curve indefinitely near each other; complete the parallel-
ogram PP” and draw the diagonal P’Q. Let wu’ wv’ be the symbols of the
points P P’ P”, then w’ — w’ represents the line P’ P”, and therefore the line
PQ, and uw’ — wu represents the line PP’; hence, it follows, that the line P’Q
is represented in magnitude and direction by (w” — w’) — (w’ — w), or, passing
to differentials, by d°w. P'Q is a double sagitta of the are PP” *.

22. From this we may derive the following remarkable theorem.
If uw be the formula of any curve in space, s the numerical length of the are of the curve

measured from any fixed point to the point w, » the numerical magnitude of d’u, and ¢ the
f ; ds* ;

direction unit of d’w, so that d’w=ve; then 2— expresses the numerical length of the chord of

v

curvature drawn in the direction e.

The direction ¢ is perfectly arbitrary, depending on what the independent variable in the
differential d*w is supposed to be. If we consider s to be the independent variable, it is evident
that PP’ and P’P” are equal in magnitude, and therefore the chord of curvature becomes the
diameter of curvature.

23. Another remarkable theorem is the following:
d P
The symbol d (=) represents a line drawn from the point of contact towards the centre of
8
curvature, and numerically equal to the angle of contingence.

This may be proved as follows. dw represents a line whose length is ds drawnin the direction
: du
of the tangent at the point w, therefore rs represents the direction unit of the tangent. Hence,
; :

if we draw two direction units from the same point parallel to two consecutive tangents, the symbol

* If we take s as the independent variable, in which case PP’= P’P”, P’Q will be perpendicular to PP", and d?u will represent
the double sagitta pointing towards the centre of curvature.

502 Mr. O’BRIEN’S CONTRIBUTIONS TOWARDS A SYSTEM

of the elementary line joining the ends of these two direction units will be a(=) . Now this
s

elementary line is evidently parallel to the normal drawn to the centre of curvature, and numerically
equal to the angle of contingence (as the angle made by two consecutive tangents is commonly
called).

‘ 5 du ‘ “esa :
24. Let v be the numerical magnitude of a($*) ; € its direction unit, and p the radius of
curvature; then, according to a well-known theorem,
ds
Pit

Hence, we may immediately deduce the well-known expressions for p.

We haveu=va+yB + xy;

du dx dy dz du
1 a(t) = ot 8) + of) v0) - ff) 0
' fale Nag) oe edie as) > aa)

Hence
ds

p= J fae) + fa(@)hfa(Sy

which is the well-known expression for p, the independent variable being arbitrary.

; d ; , Ose
25. Ifm be any number, it is clear that md () represents in magnitude and direction
s

any line drawn from the point of contact through the centre of curvature. Hence, the formula of q
that normal which lies in the osculating plane is

du
lee
ee (=) ‘
m being the variable parameter.
26. The symbol of the centre of curvature is evidently,

uU+—e,
v

d
e being the direction unit, and v the numerical value of d (=) :
s

. : : d
27. If 8 denote any arbitrary variation (as in the Calculus of Variations), then d (=) denotes
8

; : ere yee
any small line at right angles to the direction unit meee to the tangent. Hence, the formula
8

wma (5).

It is obvious that this is also the formula of the normal plane, for it is the symbol of any point:
in the normal plane.

of any normal at the point w is

OF SYMBOLICAL GEOMETRY AND MECHANICS. 503

To deduce from this expression the common equation of the normal plane.
Let wv y,%, be any point in the normal plane; then

vaty B+z,y ae

|

i

+

3

oo
aa
=>
o|s
———

i
———
R
ete
3
oo
te
=|
———
a
a
+
(as
<
+
3
o2
oe
Se
Soh) || Ss
ees
S—
ces)
+
——
ze
+
3
oo
Qa
|
——
———
S)

Hence we have

di
a,-a=md—, y,-y=mo—, 8, k= mo,

hee

Bee. dx dy dz m, {(ds)?
and .°. (= 0). $s) + Ga a)a. astern
which is the common equation.

28. It is however much more convenient to use the symbol D in expressing perpendicularity.
Dv.du denotes a line of any length perpendicular to du, supposing v to be any arbitrary line
symbol. Hence the formula of the normal plane is

u+ Dv.du.
29. The formula of the normal perpendicular to the osculating plane is
u+mDdu.du,

because dw and d’w both respresent lines lying in the osculating plane.

30. We shall now give a few examples of the application of this method to surfaces and to
some common geometrical problems.

If w be the formula of a surface, the formula of the tangent plane at the point w, is
.u+mdu,
m being a numerical variable parameter.

For du represents the elementary line joining any two contiguous points of the surface, and
therefore mdw represents a line of any length touching the surface at the point w.

31. The formula of any normal plane (i. e. any plane containing the normal at the point w) is

evidently
u+ Dv.du,

v being any arbitrary line symbol.

u, being the formula of a surface, must involve two variable parameters: let them be m and
(both numerical), and let d,,« and d,w represent the respective partial differential coefficients of %
with respect to m and m: then the formula of the normal at the point w, is

u+pDd,,u.d,U,
p being a numerical variable parameter.
32. The formula of a plane containing the three points uw, w’, w”, is
u+m(u —u)+n(u" — ux),

or what is the same thing,

wow

mu + mu +m’ uv".

Where m, m’, m” are numerical parameters subject to the condition m +m’ +m” = 1.
Vou. VIII. Parr IV. Cay

504 Mr. O’BRIEN’S CONTRIBUTIONS TOWARDS A SYSTEM

33, If wu be the formula of a right line (involving of course one variable parameter), the

formula of a plane containing that line is evidently
u + Mv,

m being a numerical variable parameter, and v any determinate line symbol.

If the plane be also restricted to contain a given point w’, its formula is

utm(u—u), or mutm uw,

where m +m’ = 1.

34. Let the symbols of the angular points of a triangle be w, u’, u'’; then the symbol of the

oint mid-way between w and w” is 4(u' +w”’), and the formula of the line drawn through
Pp . . y . 2 ? 8
this point and w is

—— )
u+m —u),

3m m i om
or Per bar ESE ae Be)»

Now if we put m= 2, this formula becomes symmetrical with respect to w, w’, wu”; which shews
that the point whose symbol is 4(w + «’ +”) is common to the three bisectors of the sides of a
triangle drawn from the opposite angles.

35. We shall now give a few examples of this method applied to Mechanics. We have
already (in the Paper read a few months since) shewn how the fundamental principles of Statics
may be proved and expressed with great simplicity by means of the symbol D. We have also
shewn how the motion of a rigid body about its centre of gravity may be investigated by means
of this notation, and exemplified its use in the problem of Precession and Nutation.

36. We may investigate the equations for finding the motion of a planet in the plane of its
orbit, and the motion of that plane, as follows.
Let w be the symbol of the position of the planet at any time ¢, then the symbol of the force
acting on the planet will be
Hu
aa
Let 7 be the radius vector of the planet, a, 8, y, three direction units at right angles to each
other, a being the direction unit of wv (and .. « =ra), and vy being perpendicular to the plane of
the orbit: let w, denote the angular velocity of (3 and y about a, w, that of y and a about £,
w, that of a and about y; then w, is the angular velocity of the planet in its orbit, w, is the
angular velocity of the plane of the orbit about the radius vector, and w, is evidently zero. Hence,
(see Equations 38, former Paper,) we have

da d d
ae 7 oF = wy — oe he — wf.
P sot ath 3 d 1
Now u=ra; wherefore differentiating and substituting for a 5 Ls and = we have,
du dr da
—_—_— = 7 —
ws at
dr
= de” + Tw [3 3
fu dr dr da_ d(rws) dB

-gphagae Caicd: tap 2 Plas

OF SYMBOLICAL GEOMETRY AND MECHANICS. 505

dr dr d (ws)
= (G- rar’) a+ {Fes oF dé ke + 130) +:

This is the general symbolical expression for the force acting on the planet, and it consists
of three parts whose direction units are a, B, yy, that is, which act, along the radius vector,
perpendicular to it, and perpendicular to the plane of the orbit. Hence, if P, Q, S' be the
forces which act on the planet in these three directions respectively, we have,

d’r P
= Tip TWs">
_ dr d(rws) _ 1 d(r*as)
Bidiee ae os Ardh,?

= T W315
which are the general equations for determining the motion of the planet, and of the plane of the
orbit.
37. To determine the motion of a particle acted on by a central force varying inversely as the
square of the distance.
Using the same notation as in the preceding Article, it is clear that the symbol of the force is

pa
oe,
and therefore we have
Pu pa
de pe Tee teteeteseseeeees (1).

Performing the operation Dw on each member of this equation, and observing that

Du.a=rDa.a=0, we have

Pu
Du ° =0 F
dt’
du
and therefore Dw aie CON STIENN moaconnod asoneacuonSHoad (2);
sbadi: 5 ; 5 : du du ;
for the former equation is evidently the differential of the latter, observing that D Wn aaa *,
du dr ath : ;
Now uw=ra, ay ae rw{3 (writing w instead of w;), and therefore, since Da.a=0, and

Da.fs=ry, the equation (2) becomes
7° wry = constant.
Hence, vy is an invariable direction (i. e., the motion is in one plane) and 7*w is constant, equal

to h suppose.
d

Now — =-—wa, and therefore (1) becomes

dup dB ywdp

dé rw dt h dt’

* It is obvious that d(Du.v)=Ddu.v+Du.dv.
3712

506 Mr. O’BRIEN’S CONTRIBUTIONS TOWARDS A SYSTEM

and therefore
du ph
=== CONSEANE yo ceseeseeeeeerreseree 3
Fr wees @);
; du, 3 h i om ahe
and therefore putting for aE its value, observing that rw = ae and assuming e to be the direction

unit of the constant, and e its numerical magnitude, we have

d h

a+ BaF Bree... “80a meapotnss (4),
performing on this the operation AB, observing that AB.a=0, AB.B=1, we find

hep.

Piet Gad Socboanuee6er (5),

which is the polar equation of a conic section, the origin being focus, e being the eccentricity,
and e perpendicular to the axis major ; for AB.« is the cosine of the angle which 8 makes with e,
i. e. the cosine of the angle which the radius vector makes. with a perpendicular to e.

If we perform the operation Aa upon (4), we obtain

Aa.e denoting the cosine of the angle which the radius vector makes with a perpendicular to the
axis major.

38. To determine the motion of the particle when it is acted upon by any disturbing force
U in addition to the central force.

In this case instead of the equation (1), we have

Cu pa
de raga Cie eiiicissa css neeseasectnesie (6):
Treating this equation as we did (1), we find
Pu
Du.— = 5 (Oe
Use Du.U;
d(h
and .*. SO) _ Du. I a Rhee hs as canna TD

d
for Du. a =Prwy =hry, using h to denote 7°w.

By integrating equation (7), we find A and -y, and thus by integrating one equation we de-
termine three elements of the orbit, for -y, being perpendicular to the plane of the orbit, determines
both the inclination and the position of the node.

If we integrate (7), after having performed the operation Avy on each side, we find
h = [Ay.(Du.U) dt.
Now Ay.(Da.U) = AB. U%*,
hence, since w = 1a, we have

ie (r is 8). Udit sc taven aes cacestns (8)s

“ For U=a (Aa. U)+B(AB.U)+y (Ay. U),
and therefore performing successively the operations Da and Ay, we find
Ay (Da.U)= 4p. U.

OF SYMBOLICAL GEOMETRY AND MECHANICS. 507

and from (7), we have
iM
y= z [Du. (Oho ChpkScecepnBao nos dae copooCuDUOBp scedoconodsenseasene ())s

(8) and (9) give h and vy separately.

We may observe respecting these formule for h and ry, that Af.U expresses the resolved
part of the disturbing force U in the direction (6, i.e. perpendicular to the radius vector and in
the plane of the orbit; and Dw.U is the symbol representing in magnitude and direction the
moment of the Couple which transfers the force U from the point ~ to the origin. (See former
Paper.)

39. To integrate the equation (6) directly as we did the equation (1), we have only to take
the same steps, (observing that / is now variable,) as follows, (6) becomes

du» dp
d@ h dt *

+ U,
d
hence, (integrating 5 s by parts), we have

aes =" p+ eee Bdt + [Udt,
and therefore by (8),

- r

dup
ae — (ABU) Bate fUdt v2..:..0...0 (10).

This is the symbolical expression for * velocity of the disturbed body. To find the parallax,
put in (10),

and then, performing the operation Af3 on both sides, we find

1 4
~= 53+ AB. ie (AB. U)B + uf ae,
which determines the parallax.
40. To determine the eccentricity and position of the axis major.
We have seen, that when there is no disturbance,
p
at B + ee,

assuming this equation to be still true, e and ¢ being now variables, and comparing it with (10),
we find

ee= fli (AB. vp +0 at OT ete «3 (11) ;

A) (ABU) B+U.. EPR sass cathe cuseniee steel (12).

Now e¢ is the eccentricity, and « is the direction unit of a line at right angles to the axis
major in the plane of the orbit: hence, (12) or (11) determines at the same time “the eccentricity
and the position of the axis major.

The Dynamical investigations just given are good instances of the nature of the Symbolical
method here proposed.

or

M. O'BRIEN.
Upper Norwoop, Surrey.
January, 1847.

XXXVIII. On the Symbolical Equation of Vibratory Motion of an Elastic Medium,
whether Crystallized or Uncrystallized. By the Rev. M. O’Brien, late
Fellow of Caius College, Professor of Natural Philosophy and Astronomy
in King’s College, London.

[Read March 15, 1847.]

Preliminary Observations.

Tue object of the following Paper is twofold; first, to shew that the equations of vibratory
motion of a crystallized or uncrystallized medium may be: obtained in their most general form,
and very simply, without making any assumption as to the nature of the molecular forces; and,
secondly, to exemplify the use of the symbolical method and notation explained in two Papers read
before the Society during the present academical year.

First, with regard to the Method of obtaining the Equations of Vibratory Motion.

This method consists, first, in representing the disarrangement (or state of relative displacement)

of the medium in the vicinity of any point wyz by the equation

dv dv dv dy dv

éu= a” + rd + To + 7a ot + een dake

(where v=Ea+B+Cy, & 7 @ denoting, as usual, the displacements at the point wyx, and
a B ¥y the direction wnits* of the three co-ordinate axes), and, secondly, in finding the whole force
brought into play at the point ayx (in consequence of this disarrangement) by the symbolical
addition of the different forces brought into play by the several terms of dv, each considered
separately. It is easy to see that these different forces may be found with great facility, without
assuming anything respecting the constitution of the medium more than this, that it possesses direct
and lateral elasticity. By direct elasticity we mean that elasticity in virtue of which direct or
normal vibrations take place, and by Jateral that in virtue of which lateral or transverse vibrations
take place.

The forces due to the several terms of dv are obtained by means of the following simple
considerations :—

Let AB be any line in a perfectly uniform medium, and conceive the medium to be divided
into elementary slices by planes perpendicular to AB;
let OM (=~) be the distance of any slice PP’ from
any particular point O of AB, and suppose this slice to
suffer a displacement equal to }e* (e being a constant)
in the direction AB, and the other slices to be similarly
displaced. Then it is evident that the medium suffers
by these displacements a uniformly increasing ewpansion
in the direction OB, and a uniformly increasing con-
densation in the direction OA, the rate of increase both of

the expansion and condensation being c. Now in all known substances, whether solid, fluid, or

* i.e. Three lines, each a unit of length, drawn parallel to the three axes.

Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION, ETC. 509

gaseous, a disarrangement of this kind would bring into play on the slice O a force along the
line AB proportional to the rate of increase c, i.e. a force Ac, A being a constant depending upon
what we may call the direct eiasticity of the substance.

Again, suppose that the slice PP’ receives a displacement $ca* in the direction OC perpen-
dicular to AB, and the other slices similar displacements. Then the line AB will become curved
into a parabola A’OB’, and all the lines of the medium parallel to 4B will be similarly curved, the

: 1 : :
radius of curvature being equal to—, and perpendicular to 4B. Now in all known substances* a
e

disarrangement of this kind would bring into play upon the slice O a force in the direction OC
proportional to the curvature c, i. e. a force Bc, B being a constant depending upon what we may
call the lateral elasticity of the substance.

Lastly, suppose that MP =y, and that the point P of the medium receives a displacement cay
parallel to 4B, and the other points similar displacements. Then the slice PP’ will, in consequence
of this kind of displacement, turn through an angle tan7' (ew) into the dotted position, and the
other slices will suffer similar rotations, those on the other side of O, such as QQ’, turning the
opposite way. Now it is easy to see that a disarrangement of this kind produces a uniformly
ncreasing expansion in the direction OC, and a uniformly increasing condensation in the direction
OC’, the rate of increase both of the expansion and condensation being c. But the expansion and
condensation here described are quite different from those previously noticed, since they are pro-
duced, not by displacements parallel to C'C, but by /ateral displacements, i.e. perpendicular to C'C.
On this account all that we can assert without further investigation is, that the force brought into play
upon an element at O by this disarrangement acts along the line C’C, and is proportional to ¢, i.e.
equal to Cc, where C is a constant evidently depending in some way both upon the direct and
lateral elasticity of the medium.

There is, however, a very simple way of finding the precise value of the force brought into
play by a disarrangement of this kind; for, if we turn the axes of w and y in the plane of the paper
through an angle of 45°, it will be found, that this disarrangement is nothing but a combination of
the two kinds of disarrangement previously noticed, and from this it immediately follows, in the
case of an uncrystallized medium, that the force brought into play at O is (4 — B)e; in other
words, the coefficient C, which must be multiplied into ¢ in order to give the force brought into
play by the disarrangement cwy, is equal to the coefficient of direct elasticity (4) minus the
coefficient of lateral elasticity (B).

In the case of a crystallized medium it may be shewn that siwv relations, corresponding to
the relation C= A — B, are most probably true, and are essential to Fresnel’s Theory of Transverse
Vibrations; that is to say, the medium is capable of propagating waves of transverse vibrations, if
these six conditions hold, but otherwise it is not.

In employing the above considerations to determine the equations of vibratory motion, the
directions AB and C’C are always taken so as to coincide with some two of the three co-ordinate
axes, and it is this circumstance that makes the method peculiarly applicable to crystallized
media. Indeed, if it were necessary to take the lines 4B and CC’ in any directions but those of
the axes of symmetry, the above considerations would not apply without considerable modification.

The equations of vibratory motion obtained by this method for an uncrystallized medium are
the well-known equations involving the two constants 4 and B. The equations obtained for a
crystallized medium are perfectly free from any restriction of any kind, are applicable to all kinds
of substance, whether we suppose its structure to be analogous to that of a solid, fluid, or gas, and
hold for all kinds of disarrangement, whether consisting of normal, or transverse displacements,

or both.

* Fluids and gases possess lateral elasticity as well as solids, only in a comparatively feeble degree,

510 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF

When we introduce the six relations between the constants above alluded to, and moreover
assume that the vibrations constituting a polarized ray are in the plane of polarization, we arrive at
Professor Mac Cullagh’s equations *. If, on the contrary, we suppose the vibrations to be perpen-
dicular to the plane of polarization, we arrive at equations which agree exactly with Fresnel’s
Theory in every particular +.

If we introduce these six relations into the equations for crystallized media deduced from
M. Cauchy’s hypothesis, that the molecular forces act along the lines joining the different particles
of the medium, it will be found that these equations are immediately reduced to the equations for
an uncrystallized medium. From this it follows that M. Cauchy’s hypothesis cannot be applied to
any but uncrystallized media. In fact, it may be easily proved, that, if the equations derived from
M. Cauchy’s hypothesis be true, a crystallized medium is incapable of propagating transverse

vibrations.

Secondly, respecting the use of the Symbolical Method and Notation above alluded to.

Tur application of the Symbolical Method and Notation to the subject of vibratory motion
is very remarkable, and leads to equations of great simplicity. In the case of an uncrystallized
medium, the three ordinary equations of motion are included in the single symbolical equation.

ae Bae cits shot -B (ah po+y2) (SF oe

dt da * dy? * dal “da dz) \dw * dy* dz
If we employ the notation A w’.u, and assume the symbol @ to represent the operation
d d d
da * Pay + ae
the equation of motion becomes
d?
= BADD) v+ (4 -B) DAD;
or, by using the notation Dw’. w also, it may be put in the form
d*
Gf 7 [ABAD - BBD) 40.

The symbol ¥ written before any quantity U, which is a function of wys, has a very
remarkable signification; the direction wnit of the symbol 3U is that direction perpendicular to
which there is no variation of U at the point wyx, and the numerical magnitude of WU is the
rate of variation of U when we pass from point to point in that direction.

The symbols A¥.v and D¥W.v have also remarkable significations. AW.v is a numerical
quantity, representing the degree of expansion, or, what is called the rarefaction of the medium
at the point wyx. DZ.v represents, in magnitude, the degree of lateral disarrangement of the
medium at the point ayx, and, in direction, the awis about which that displacement takes place.

These two symbols may be found separately by the integration of an equation of the form

d?U @fU @U &U
= =C (Ge oaoe hoe)
dt? da day dz

* Given in a Paper read to the Royal Irish Academy, December 9, 1839, p. 14.
+ On this subject see a Paper by the late Mr, Greene in the seventh Volume of the Cambridge Transactions, p. 121.

i ES SB ee

(mA

ry

VIBRATORY MOTION OF AN ELASTIC MEDIUM. 511

When the six conditions above alluded to are introduced, the equation of motion for a crys-
tallized medium becomes
d d
da ody

.
}

d*v
dt

+bD.| (B22 - Bi, a+ (3% - 2,4) a+ (2.35 - B, zr}:

Where 4, 4, A; are the three coefficients of direct elasticity with reference to the three axes
of symmetry, and B, B, B, B, B, B,' the six coefficients of lateral elasticity with reference to
the same axes.

If the vibrations be transverse, this equation is reducible to the form

dv

ag 7 7 (PRY (PEa+b’nB+eCry)

= (Aa

2

d
or - = —(DD.) @ahat+ VU BAB + Cy Ay), eececeeeee (A)s

assuming the vibrations of a polarized ray to be perpendicular to the plane of polarization.

The well-known condition that a plane polarized ray may be transmissible without subdivision,
and the expression for the velocity of propagation, may be immediately deduced from this equation.

If we assume the vibrations of a polarized ray to be im the plane of polarization, the equation
becomes :
dv
eae DD.(Paha+ BAB + CyAy) DB. cresccccesceeee (B).
The equation (4) agrees in all respects with Fresnel’s Theory, and the equation (B) includes
Professor Mac Cullagh’s three equations. It is curious that (4) and (B) should differ from each
other only in the order of the operations performed on v in the second member.

Investigation of the Symbolical Equation of Vibratory Motion of an Uncerystallized
Medium.

1. Ler w(=aa + By + yx)* be the symbol of any particle (P) of an elastic medium in a
state of equilibrium, v( =a + Bn ++) the symbol of the displacement of the particle at any
time t, w+ du (du = ade + Bdy + ydx) the symbol of the equilibrium position of a contiguous
particle (P’), and v + dv (dv = ad& + Bdn +8) the symbol of the displacement of P’ at the
time ¢; then we have
dv dv dv dv da
“hale ow + rie att 4 7a ba? + aaa Sady + &....0.06(1).

This equation expresses the disarrangement, or state of displacement, of the medium in the
immediate vicinity of P, for dv is the relative displacement of P’ with reference to P, and by
giving different values to dv dy dx in (1), corresponding to the different particles near P, we find
the displacements of those particles relatively to P.

2. Inconsequence of the disarrangement of the medium in the vicinity of P, represented by (1),
a force will be brought into play upon P; owr object is to find this force.

Vou. VIII. Parr IV. 3U

512 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF

Now, by a well-known principle, the force on P resulting from the disarrangement

dv dv
dv =— oa + 7 &e.,
is the resultant (or symbolical sum) of the forces due to the separate disarrangements
dv dv dv
bv = 7” ou ag > dv = —_ os, &e.

Hence, if we find the forces due to the several terms of the expression (1), and add them
together, the resulting sum will express, in magnitude and direction, the whole force brought into
play upon P by the disarrangement (1). This we now proceed to do.

3. To find the force brought into play on P by the disarrangement,

dv dé as
do = danas dat Ba

om + ry — Oa.
d Ce as Z

d ; 5
a Z da represents a small line, proportional to da, drawn in the direction a; therefore the
@

disarrangement indicated by
dé

dv=a— da
dx
is a uniform expansion of the medium in the direction a. This brings no force into play upon P.

B +80 represents a small line, proportional to da, drawn in the direction 8; therefore the
@

disarrangement indicated by

a N

d SS
bv = Bde

takes place as follows: Suppose the medium when at cae

rest to be divided into physical lines parallel to the :
direction a, let MN be any one of these lines, M being the point when it meets the plane per- —
pendicular to a containing P, and let MN’ be a line parallel to the plane of wy, making an

d ’ : A
angle tan7! (Z) with MN. Then the disarrangement consists in the displacement of the line
v

MN into the position MN’, and a similar displacement of all the other physical lines. This
disarrangement evidently brings no force into play upon P.

3 d
The same reasoning applies to the remaining term y an ba.

4. Reasoning therefore in this way it is clear, that the disarrangement represented by the
first three terms of the expression (1) brings no force into play upon P. J

5. To find the force brought into play on P by the disarrangement represented by

bo = 5 S2 dat = b TS (ak + Bn + 7h) 80% :
4a a 5a* represents a small line, proportional to 3a?, drawn in the direction a; therefore
the disarrangements indicated by
dv=ta ld éa*

VIBRATORY MOTION OF AN ELASTIC MEDIUM. 513

is a uniformly increasing expansion of the medium in the direction a, the rate of increase of the
Py

expansion being ewe Hence, according to well-known principles, this disarrangement brings

ae. “Ue
into play on P, a force proportional to = in the direction a, that is, a force whose symbol is

da? P
Aa ans , A being some constant.
aa?

yj 9 . . ° . . .
i oa? represents a small line, proportional to dw*, drawn in the direction (ee
aD

therefore the disarrangement indicated by

2
‘

d
Again, $6 F;

dn. .
dv=48 EE ba"

is a curvature of the physical line MN (see Art. 3.), and a similar curvature of all the other

physical lines, the symbol of the indew of curvature “A =
(i.e. a line equal to the reciprocal of the radius of curva-
ture drawn towards the centre of curvature) being ae
d’n
du

Hence, according to well-known principles, this disarrangement brings into play upon P a
; ay. Swa ;
force proportional to = in the direction 3, that is, a force whose symbol is

d’y ;
BB an? B being some constant.

In the same manner we may shew that the force brought into play by the disarrangement

is represented by the symbol

Hence the whole force brought into play by the disarrangement

Cor
éu = 4 dv Oa",

is represented in magnitude and direction by the symbol

A8

(F) {4ak + BBn+ Oh,

d 2
or (=) §(4 - B) af + Bo}.
6. To find the force brought into play by the disarrangement represented by

dv a
Ov = ade Owdy= Dap (aE + Bn + yQ dowdy.

3u2

514 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF

Let a’y’ be co-ordinates referred to two new axes (a’f') in the plane of wy, making respectively
angles 45° and 90° + 45° with the axis of 2; then

bw= (da — dy’), dy = Fz a’ + By)
} fh i = —(a
a7, ame B-F@ + B).

Making these substitutions, we find
d? 1 1
= ee ' >=(n-£&) + da” — dy”).
Meal aay gag ee et Ce 8")
Hence, by what has been already proved, the force brought into play will be

& eae ie
aay -B)a Fp Ne Jan D+ B- B) 7h

ad
= (4 - DY agar int

7. We may now write down the symbol of the whole force brought into play by the dis-
arrangement represented by the expression (1), neglecting terms beyond iefioge of the second order,
It eal be as follows,

FB AGe) tla) * (Ge) f°

dy? Y da
+ (4 - B) & a (<
+ qagy (BE + an) + (yn BO + ak + 70)

The coefficients of a, B, y in this expression are the well-known differential formule for
_the three forces (parallel to the three axes) brought into play by the displacements &, y, ¢,

The part of F which is multiplied by 4 —B, may be put in the form
d d d—é dn dé
Fey tore a) Geta ak

Hence, the equation of motion of the medium (which includes the three ordinary equations)
assumes the following form,

ae ~? {(ga) + (Gp) + Ge) e+ 4-® (a8 ay* 7a) (Gat ay? ae)

8. This equation may be put in a remarkably simple form by the use of the notation Aw’. u.
Let us assume the symbol ¥ to represent the operation
d d d
a ae ae B dy shiry ae ’
then, since v =a + Bn +, we have
d d d
AD.v= ea + sg ;

VIBRATORY MOTION OF AN ELASTIC MEDIUM, 515
d\? d\* ay?
? aera elles es
- re las # dy * ds

Hence the equation of motion becomes

oe (AR Bye A By DAD Co ay

d

We may also put it in somewhat a different form by using the notation Du'.w; for
(AD.D)v — DAD.» = - (DD). v*,

therefore (3) becomes
;
4 d*
: _ SAW AR (DBI © ose. Cabs ccanrseeeesrene (A):
9. The symbol #3 has a very remarkable meaning which we shall now proceed to explain.
denotes the rate of variation when & alone is varied, that is, the rate of variation im the

a

direction a. ‘To indicate this, we shall employ the notation d, instead of —; i.e., if U be any

x
quantity which is a function of # y x, and which therefore varies when we pass from one point
to another of the medium, then d,U denotes the rate of variation of U, when we pass from point
to point in the direction a.

Now this rate of variation may be affected, like an ordinary velocity, with a sign of direction ;
and it may be resolved or compounded in the same manner, and by the same rules, as an ordinary
velocity.

Hence, we may see immediately the meaning of the expression
BU, or ad,U+ Bd,U + yd,U;

for ad,U is the rate of variation of U in the direction a, affected with its proper sign of direction a,
{d,U is the rate of variation in the direction 8, and yd,U in the direction -y, each affected
with its proper sign of direction, Hence, compounding these rates of variation as if they were
ordinary velocities, it follows, that the symbolical sum

ad,U + Bd,U + yd,U
expresses, in magnitude and direction, the complete rate of variation of the quantity U.
10. We may shew this differently as follows.

Let a, 8, y, be any three direction units at right angles to each other; then it is easy to
prove, that

ad, + B,dg + 7,d, = ad, + Bdg + yd, eoscesconcescnsveeres (5)
Let us now choose a, (3, y,, so that a, shall be in the direction of the normal to the surface
dU =0,
at the point w yz; in other words, supposing U to denote some disturbance or displacement of

the medium, a, is chosen so as to be perpendicular to the surface called the front of the wave, for
U =0 is evidently the differential equation of that surface.

* For let uw and wu’ be any two lines, and let @ represent the Now Auw.w =r, and udu’. u=r'%aa; therefore
direction unit of uv’; then, if u’=r'a, and usaa+bp+ecy, we have Du'.(Du'.u) or (Du')* wesw Au u-(dw.u')u.
— Du'.u=r' (by—cf), and... Du’. (Du'.u)=r'* (—bB—cy).

516 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF

a, being thus assumed, we have dj, U = 0, d, U = 0, and therefore by (5),

ad, U =ad,U + Bd,U + yd,U = DU.

Now a, is the direction of propagation of the disturbance U, therefore d, U is the rate of
variation of U in the direction of propagation, and ad, U is that rate affected with its proper
sign of direction.

Hence, the symbol 39 U expresses, in magnitude and direction, the complete rate of variation
of the quantity U, that is to say, the direction of JU is that direction perpendicular to which there
is no variation of U at the point vyx, and the magnitude of BU is the rate of variation of U in
that direction.

It is manifest, therefore, that the symbol 39 has a very important signification, especially in
investigations relating to the propagation of waves.

11. Returning now to the equation (4) we shall, in the first place, interpret the meaning of
the symbols AW.v, and DD.».

Let & », ¢ be the resolved parts of the displacement vin the directions a, B, vy, respectively ;
then, choosing (as we may do)* the direction yy, so that G = 0, we have

= Ea, a7 1,2,>

and therefore, since 3 =ad,, we find
AD pile d,£,,

DD .v = yd, 0,-

Let OX and OY be the directions a, and 8,, yy, being perpendicular to the plane of the
paper; let O be the point (wyx) of the medium, POQ the
line of particles which, in a state of equilibrium, lie in the
direction a,, and OT the tangent to POQ at O. Then since
OX is the direction of propagation, and since the disturbance
(v) consists of two parts, namely & in the direction OX, and
n, in the direction OY, it is evident, that dé, is the ewpansion
(i.e. the degree of ewpansion, or, what is called the rarefaction)
of the medium at the point O; also d,y, is the tangent of the
angle 7OX, and therefore measures the degree of lateral displacement of the medium at the
point O.

This lateral displacement consists in the rotation of the line OT' about the axis y,> and a
corresponding rotation of all the other lines of particles which constitute the medium in the im-
mediate vicinity of the point O, these lines being supposed to be parallel to OX in a state of
equilibrium. Hence it follows that the symbol D¥.v represents, in direction, the aavis round
which the lateral displacement takes place, and in magnitude, the degree of lateral displacement.

Thus it-appears that the symbols A¥.v and aD¥.v have a very important signification in
investigations relating to the propagation of waves, the former expressing the degree of expansion —
of the medium at the point wyx, and the latter representing, in magnitude, the degree of lateral —
displacement at the point wyx, and, in direction, the awis about which that displacement takes —
place.

12. Hence it is evident that the symbol AW. v defines the kind of disturbance which constitutes _
normal waves, and DY.v that which constitutes transverse waves. ~

* a, is the direction of propagation, as in the preceding article, and y, is chosen at right angles, not only to a,, but also to the
direction of vibration at the point vyz. In this case ¢, is clearly zero.

VIBRATORY MOTION OF AN ELASTIC MEDIUM. 517

13. AW.v, and D#.v may be found separately by the integration of a differential equation
of the form

2 a da
CU (ee U =) EEO):

dé? da®* dy * dx*
For performing the operation A. on both sides of (3), and putting AB.v = U, we find
&
= (AB. ») = B(AD. DB) AD.» + (4- B)(AD.D)AD.v;

or, wea (AD. D) U;

and performing the operation D®., and putting D®.v = U, we find (observing that DB. B = 0),

oe B(AD.D) U:
non ameme (s (24 (8

hence, AW.v and D®.v may be found separately by the integration of an equation of the
form (6), C being equal to A in one case, and to B in the other.

Investigation of the Equation of Vibratory Motion of a Crystallized Medium.

14. WueEN the constitution of the vibrating medium is crystalline, we may obtain a differential
equation similar to that we have found for an uncrystallized medium, and by exactly the same
method ; the only difference will be in the constants introduced, in regard to which we must bear
in mind, that the elasticity of the medium is no longer the same in all directions, and therefore the
constants 4 and B, which we may call the coefficients of direct and lateral elasticity respectively,
will be different with respect to different directions. ‘The method we have employed to find the
force brought into play by a disarrangement of the medium requires us to consider this difference
of elasticity only with respect to the three directions a, 3, y, assuming that the medium is still
symmetrical with respect to these three axes, #.e. supposing them to be the awes of elasticity.

15. Hence, reasoning as in article (5), the force brought into play by the disarrangement,

aE dn ae
=l _— a —_
dv=4 ta 7, ow + Bape ty + > dz*}
d’ dy #C
will be Ava da? + A,B dy? + Ayy dz? eer erewe ses verecccesceseses (UV),

A,, A,, A;, being the coefficients of direct elasticity for the three directions a, (3, +.
Again, reasoning as in the same article, the force brought into play by the disarrangement,

oma fa (FE aye + Fo ax’) + p (rast + 2 308) ey (FS Pra i

will be

a (mE sn SE) 20 (mts ni f8) oy (0 BH).

B,, Bi, B., B, &c., denoting the coefficients of lateral elasticity for the three directions a, , ry.

We make a difference here between B, and B,’, because the disarrangements,

518 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF

are of a different nature, though they consist of displacements in the same direction a; for the
former disarrangement consists in the rotation (i.e. the curvature) of physical lines parallel to 6
about the axis ry, and the latter of physical lines parallel to y about the axis B. The same re-
marks apply to B, and B,, B,and B,. Fresnel virtually assumed that B, = B!, B. = B., B; = By.

16. By reasoning as in article (6), we might easily shew, that the force brought into play by
the disarrangement,

dv d
Sv= daca Sady = dnd (a& + Bn + yQ) ow dy,

BE pest

dady 5 Ne dudy

But we shall shew this somewhat differently, in order to find out what relation subsists (if any)
between C C’ and the constants already introduced.

BC

a
The disarrangement a ve dady, is of the following nature.
dady

Let OX and OY represent the directions a and 8, and O the point (wyx); take OM = 6a,
draw S/S" parallel to YY’, and PMF’ making the tangent of the

aE ae :
angle PMS equal to tay dw. Then it is clear that the dis-

d°
arrangement Hee dwdy causes the physical line SMS’ to
CREE)

assume the position PMP’. In like manner, if ON = — da, and
TNT ' is parallel to YY’, the physical line J'NT" will, in con-
sequence of the disarrangement, assume the position QNQ’, the
angles QNT and PMS being equal. The physical lines (taken
parallel to YY’) between §\S’ and TT’ will suffer similar deviations, the tangent of the angle of
deviation being proportional to da.

17. The effect of a disarrangement of this kind is obvious; for it produces a uniformly
increasing expansion of the medium as we go along the line OY, and a uniformly increasing con-
densation as we go along the line OY, the rate of increase both of the expansion and condensation
being, as it is easy to see,

d=
dady

The effect of this will be to bring into play upon the particle O a force in the direction OY
proportional to this rate of increase, #.e. a force whose symbol is,

ae

BC eau: C being some constant.
: : a : :
In like manner we may shew that the disarrangement B ieaytY brings into play upon O
a force whose symbol is,
d’y
UJ v .
al aay » C" being some constant,

VIBRATORY MOTION OF AN ELASTIC MEDIUM. 519

2

d ; :
18. We may easily shew that the disarrangement yy ——*- dady brings no force into play

dudy
upon O; for it is perpendicular to the plane of the paper, and its nature ae 4
is as follows. Draw two physical lines QQ’ and PP’ through O equally PNG | is
inclined to XX’; then in consequence of the disarrangement the lines OP \
and OP’ will become bent wpwards (i.e. upwards considering the plane x S Vis ze

of the paper to be horizontal), and the lines OQ and OQ’ will be bent
downwards; also the curvature of POP’ will be exactly the same as
that of QOQ’, only opposite in directions. Hence the two forces
brought into play on O by the curvature of the two physical lines PP’ Q

and QQ’ will be equal and opposite ; and the same may be said of every Ne
other pair of physical lines drawn through O equally inclined to XX’. It is therefore manifest
that no force will be brought into play on O by this disarrangement.

19. Thus it appears that the force brought into play by the disarrangement,

dv a
aaa ovdy, or jada (a€ + Bn + yQ)dwdy,
. PE , an
will be pe ines +aC anda
Hence the force brought into play by the disarrangement,
do... @ d
ov = gag + tga Oyox + Fe éxdw
will be expressed by a symbol of the form
e 4) ain 9 a ;
Bap EOCbOnaS rer (ay:
20. Hence, collecting these three results, the general equation of vibratory motion will be
dv
Za Rs yr ,
dE (Of ABA E aa OF PORNO acs GOONOG COPEL IOAN OOODOF (8)

21. We have seen that, in the case of an uncrystallized medium, the constant C (i.e. the
constant to which the different C’s in U" become equal when the medium becomes uncrystallized)
is equal to d — B; in other words, C is the difference between the coefficients of direct and lateral
elasticity ; and it is easy to explain how this is on simple mechanical principles, which appear to
apply to a crystallized medium as well as to an uncrystallized, and which therefore will furnish us
with certain probable relations between the coefficients involved in equation (8). These relations,
as we shall presently shew, have a very important physical signification.

22. Referring to the figure in p. 11, we may explain the physical meaning of the relation,
C=A - B as follows :—

The disarrangement represented in this figure consists of an increasing expansion of the
medium as we go along the line YY’, caused, not by direct displacements (i. e. displacements
parallel to Y'Y), but by dateral displacements (i.e. displacements perpendicular to Y’Y). Conse-
quently the force brought into play upon O by this increase of expansion will be modified by the
lateral elasticity of the medium, which tends to restore the physical lines PP’, QQ’, &c. to their
equilibrium positions S'S", 7'7", &c. In fact the unequal expansion caused by the disarrangement

You. VIII. Pazr IV. 3X

520 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF

is resisted, and, to a certain extent, balanced (so to speak) by the lateral elasticity, and therefore
the unequal expansion has not its full effect in producing force upon O, but a certain part is spent
upon the lateral elasticity.

If there was no lateral elasticity the force on O would be the same as if the displacements were
direct (i.e. parallel to Y’Y), for then the unequal expansion would produce its full effect ; in other
words the force brought into play on O would be

GE B,
dx dady
we

dudy

observing that the rate of increase of the expansion of the medium as we go along Y’Y is

To find the force actually brought into play upon O, allowing for the lateral elasticity, we
must diminish this force by a certain quantity depending upon the lateral elasticity, which quantity
a

ady

It is clear therefore that the force actually brought into

must of course be proportional to

play upon Q is expressed by a symbol of the form
(4-P)— Ls ge?
P being a certain constant depending upon the me elasticity. Art. 6 shows that P = B.

This evidently explains the physical meaning of the relation, C = A — B, for this relation
rie

dowdy is, not the force

indicates that the force brought into play by the disarrangement a dad
; wey,

2

es 3, which is the force due to the full effect of the unequal expansion, but the force
AEA)

(A ee) a — ae which is equal to the former force diminished by a quantity depending on the
lateral Te and proportional to the rate of increase of the expansion.

23. From this explanation of the meaning of the relation C = A — B, it is very probable,
I think, that a similar relation holds when the medium is crystallized ; for it does not seem essential
to this explanation that the medium shall be perfectly uniform in all directions; all that seems
really necessary is, that the medium shall be symmetrically arranged with reference to the two axes
XX’ and YY’. We must take care, however, in applying this explanation to a crystallized medium,
to give A and B their proper values, namely A, and B,; for by A is to be understood the coefficient
of direct elasticity in the direction OY, that is A,, and by B the coefficient of Jateral elasticity
brought into action by the unequal rotation of physical lines parallel to OY about the axis of x,
that is B, (for B, is the coefficient of lateral elasticity for the ewrvature of such lines about the axis
of x). Hence the relation, C = A — B, transferred to a crystallized medium, is C,; = 4, — B,, and
therefore, writing down this relation for the six C’s in the expression 7”, we have the following six
relations, viz. :—

C2 4, SB Rey’ laeag 4 i 2A B,
Ops ay= By CO; 4,= B! Cha BR’

24. We shall now shew, that, if these six relations hold, the forces brought into play by any
system of transverse vibrations constituting a single wave, are always perpendicular to the direction
of propagation of the wave ; but if these relations do not hold, the forces will not be perpendicular
to that direction. In other words, we shall shew that these sia relations are essential to Fresnel’s
Theory of Transverse Vibrations.

ows

VIBRATORY MOTION OF AN ELASTIC MEDIUM. 521
If we substitute in (8) the values of the C’s given in (9), we find the following value of U”, viz.,

* da (Zt +H)+ ABT, dy x (s+ =) + ile 2)

2

d @&
yas B07 + BEB) ~ Gq Baba + BlEy) - 5p (BEB + BL na),

and therefore (8) becomes

Poy a d ee +)
ee A by =
dt? (Aq dx + 4B e a +) (E+
d d : “le
ie ae pa )e te 7d) dz
--- (10).
nlod-ri)etom (ok ot)
one V7 dx “dy da
d d¢
wae peeing eet can
vi (vz dite sd aie Bi (y= dy Ze dy
B h d ob h “ DD. g Bs
y using the notation in Art. 8, &c., and o serving t at a = Boat > oe = LE
= —- DD.P, &e. &e. the equation (10) becomes
iv d d d
Te (40 Gy +48 dy + Ary 7z)AB-”
(11).
dy d¢é a 2) ( dé =
DB.; |B, — — B,; — B, — - B, —
+DD.| ( dx Bs) at (Bo BEY eg) OE Bins or Dai

For transverse vibrations we have A ¥.v = 0, and therefore,
dv dyn de d¢ = dé o)
= DB .4( B'S Loy ee ste Ae Nae),
aed {(2. aaa = ae (2, Team 9 das ez Eda) Vo

Now Du'.w is the symbol of a line perpendicular to w’ and w; hence (12) indicates that the

Pov, ; See se ; ;
force df is perpendicular to the direction of 9, and that direction, as we have seen, is the direc-

tion of propagation. It follows, therefore, that if the relations (9) hold, the forces brought into
play by transverse vibrations are always perpendicular to the direction of propagation.

25. We shall now shew that this cannot be the case except the conditions (9) hold.
If the conditions (9) do not hold we must add to the second member of (12) an expression of
the form

d* ;
fads ONY + EB) + —_ —_—_(E,(a + E,/Ery) + ca (B,£B + Ena) = V, suppose

E, Ey E, E/ E, Ej being the unknown corrections to be made in the second members of the
relations (9).

Performing on V the operation AW., we find,
a dy & ag a? ( dé | 2,2"),
-V=—— = — E, + ——|E,
past ees MEF (#32 + eer rors. ec de 7) dady\ dy") de
Now, if the second member of (12) + V is perpendicular to the direction of @, the same must
3x2

522 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF

be true of V, and therefore AW. V must be always zero, Consequently AW. V must be zero in
the particular case where { = 0, and & and » are functions of w and y only, in which case we have

& eee
Bis | es cee
AD.V al ian tS)

PPPs) Pas) hae | :

/ d& dn d¢
observing that Pas = + ak = 0) = 0.
That this expression should be always zero evidently requires E, and E,' to be each zero; and
in the same way we may shew that the other #’s are each zero.
Hence, it appears, that the forces brought into play by transverse vibrations are not perpen-
dicular to the direction of propagation, except the conditions (9) hold.
These conditions are therefore essential to the truth of Fresnel’s Theory of Transverse Vibrations.

26. Hence it follows that (12) is the most general form of the equation of vibratory motion,
when the transmission of a wave of transverse vibrations through the medium in every direction is
possible.

27. Experiment shews that the six constants involved in (12) are reducible to three in the case
of ordinary Biaxal Crystals; for it appears, that, when the plane of polarization of a ray coincides
with the plane af, the velocity of transmission in the direction a is the same as that in the direction
B. Now, first, let us assume with Fresnel, that the vibrations are perpendicular to the plane of”
polarization (a); then, for the directions a and $ the equation (12) becomes, in each case
respectively,

dt * da*?
and eno = DB xt By 0), or of By a
Hence, if ¢ denote the common velocity of transmission in the two directions, we have
Bia =107s
In exactly the same way we may shew, that
B, = B/ =a’,
Be BY ale,

where a is the velocity of transmission in the directions 8 and -y of a ray polarized in the plane
Bry, and 6 the velocity in the directions ry and a of a ray polarized in the plane ya.
Hence the equation (12) becomes

ov
r —
dt
dv

ee ri —-(DD.)* (Wada +P BAB + yAry)v,

= —(DBD.)(@Ea + Un + CC),

eal,

VIBRATORY MOTION OF AN ELASTIC MEDIUM. 523

If, however, we suppose that the vibrations of a polarized ray are im the plane of polarization,
we may shew as above, that

fi be Sa
B, = B/ = 8,
B, = BJ =e,

and therefore (12) becomes,
dy »(dn d¢ ~(4¢ dé 9 (dé |
qe 7 DB. ie eee (S-S) e+e ad 2 yt;

2

d
or =o — DB. (Paha +B BAB + cy Ay) DB. v.e.ceeseccesserees (14).

@v
28. Taking the equation (13), we shall now find under what circumstances the force ae
de

is in the direction of vibration.
Let us choose a, 3, yy, as in Art. 10, a, being the direction of propagation, and #, that of
vibration ; and let » = 7,3. Then, as in the article just referred to, we have M = a,d,.

Rees faeeantiitinn What tie oles’ way Belin the direction Bis Av ko (eos
ow, e condition ne rorce ——- ma e in € dalrection — aS = Os
é ae") VB OV ae dé

a

already perpendicular to a, and this condition makes it perpendicular to vy, likewise), or by (13),
Ay,.(Da)’. (@adat+¥ BAB + Cy Ay), = 0.

But, by the general proportions of the notation D and A, we have Ay,.Da,.= Af,., and
therefore Ary,.(Da,.)*= AB,.Da,.= — Ay,.. Hence this condition becomes

’ a’(Aa.y) (Aa. B) +8 (AB.y) (AB-B) + (Ay. (Ay-B) = 0.
This is the well-known condition of Fresnel that the force brought into play by a transverse
vibration may be in the direction of that vibration; for

Aa.ry, = cos (ay,) Aa.B = cos (af) &e. &c.

To find the velocity of propagation in this case, we have, performing the operation A, on
both sides of (13),

OM = fai(Aa. A)’ + 6(AB-BY + & (Ay-BY dain,
and therefore the square of the velocity of propagation is
a’ (Aa. B)’? + (AB. B)’ + (A.B),
which is Fresnel’s expression.
29. We may treat the equation (14) in exactly the same way.
M. O'BRIEN.
Upren Norwoop, April, 1847.

XXXIX. A Theory of the Transmission of Light through Transparent Media,
and of Double Refraction, on the Hypothesis of Undulations. By the
Rev. J. Caauuis, M.A., Plumian Professor of Astronomy and Experimental
Philosophy in the University of Cambridge.

[Read May 17, 1847.]

I~ a former communication to this Society, I ventured to advance a new Theory of the
Polarization of Light, founded on a Mathematical Theory of Luminous Rays. (Cambridge
Philosophical Transactions, Vol. vi11. Part 111. pp. 361, and 371.) As the Theory was not then
applied to the phenomena of Double Refraction, I propose in this Paper to attempt to give it
that extension. ‘The course of the reasoning will require a general consideration of the transmission
of light through transparent media. I shall therefore commence with this part of the subject.

1. It will be assumed that the ether is of the same uniform density and elasticity within
any transparent medium as it is without; and that the diminished rate of propagation in the
medium is owing to the obstacle which its atoms oppose to the free motion of the etherial particles,
Considering the proximity of the atoms to each other, and that the retarding effect of each atom
at a given instant, extends through many multiples of its linear dimensions, it is presumed that
the mean retardation, though resulting from the presence of discrete atoms, may be regarded as
continuous. It will also be supposed that the mean effect of the presence of the atoms is to
produce an apparent diminution of the elasticity of the wther, the motion in all other respects
being the same as in free space. Let a@=the velocity of propagation without the medium,

and ~ = that within. Then, p being the density in a line of rectilinear propagation, at a point
mn

Bi 5 : aG@ da
distant by v from the origin, the effective accelerative force = — a . If there were no
be pda
; : a’dp :
retarding effect of the atoms, the accelerative force would be — ah Hence, the accelerative
pda
; 2 ‘ 1\ dp F j
force of the retardation (2) is equal to a* (1 - =) aan For this force another expression may
wl) pdi

be obtained by the following considerations. Ifv be the velocity of the ether at the time ¢ at

the point whose co-ordinate is #, we have by known equations,

a a
v = — Nap. log, p = (<¢-2).
eee: toe p= z

Now the accelerative force of the retardation at a given point must vary conjointly as the
number of atoms in a given space, that is, as the density of the medium, and as the effective
accelerative force of the «ther at that point. Hence, X being a certain constant, and 6 the density
of the medium,

dv dv
R=-K56 (5) =~ Ko very nearly.

PROFESSOR CHALLIS, ON THE TRANSMISSION OF LIGHT, ETC. 525

Consequently by the foregoing equations,

Dep we wr eee

uw da pe pdx
Comparing this expression for # with the former, we have

(1 -*) eiaie eget bee, or pwe—-1=Ko.

we) pda we pda

2. Hitherto we have supposed the atoms of the medium to be absolutely fixed. If, as it is
reasonable to suppose, they are moveable by the mechanical action of the ztherial vibrations, the
retardation produced by them will differ from that obtained above. Assuming the mean effect of
the presence of the atoms in this case also to be an apparent diminution of the elasticity of the
zther, the accelerative force of the retardation will vary as the density of the medium and the
difference of the effective accelerative forces of the ther and the atoms at a given position. That
is, if v’ be the velocity of an atom, where the velocity of the vibrating «ther is v, we shall have

d dv’ .
R=-K6 ae ae) » very nearly. And, as before, R= a’? (1 --) —— = - (2 - 1) a

d : do. sido. &
Hence, putting g for the ratio of re to di? it follows that p?-1= Kd(1 — q).

3. Since the retardation will be less and the velocity of propagation greater when the atoms
are moved than when they are fixed, « will be less in the former case than in the latter, and
consequently q is a positive quantity. As it is known from experience that the rate of propagation
of light in a given direction in a medium, is uniform and independent ef the intensity of the light,

: dv’ dv 5 ;

the ratio of to er must be the same at different points of the same wave, and the same also
for vibrations of different magnitudes, if the breadths of the waves be given. But to account for
the phenomenon of dispersion, g must be a function of X the breadth of the wave. For our
present enquiry it is not necessary to ascertain the form of this function. It is only necessary to
assume that in crystallized media q is different for different directions. The theoretical reason for
this probably is, that the retardation depends on the elasticity of the medium, and that the elasticity
of crystallized media, and consequently the mobility of their particles, depends on the direction.

4, What has been said above respecting the transmission of light through transparent media,
will suffice for the consideration of the theory of Double Refraction, on which I am now about to
enter. It will be assumed that in any medium which does not retard the progression of the
luminous rays equally in all directions, there are at least three directions at right angles
to each other, in which the retardation will take place in the manner hitherto supposed. Let
a7, b?, ¢* be the constants of elasticity for plane waves in these three directions, and let a be the
velocity of the waves in free space. Then, q,, 925 qs, being the values of qg for the same directions,
the time of vibration being given, we have,

2 2 2
-. =1+ Kd(1-4q,), — =1+Kd(1-q@), s =1+ Kd(1-q).
‘ ‘ ‘

5, When an atom of the medium is displaced in one of the three rectangular directions above
mentioned, the direction of displacement coincides by hypothesis with the line of propagation of the
waves. Although in general this will not be the case, waves may still be propagated in all directions
in the medium. Tor supposing plane waves of given breadth to be propagated simultaneously in
the three rectangular directions, (which may be called the axes of elasticity,) the resulting effect on a

526 PROFESSOR CHALLIS, ON THE TRANSMISSION OF LIGHT

given particle of the ther, according to the principle of the coexistence of small vibrations, may be
a vibration in a certain resulting direction, of the same period as that of the component vibrations.
Consequently waves which would produce the same vibration of the extherial particle may be pro-
pagated in that direction. But the displacement of the atoms of the medium does not necessarily
take place in the same direction. If this displacement be resolved in two directions, one coinciding
with the direction of vibration of the etherial particles, and the other perpendicular to this, the
resolved part of the displacement in the latter direction, will give rise to xetherial vibrations which
will be propagated laterally and produce no sensation of light. With reference to phanomena
of light, the other part alone requires to be taken account of. The above considerations will
enable us to determine the effective elasticity in any direction in the medium, in terms of the
elasticities in the directions of the axes.

6. Let v be the velocity of a particle of the zther, the vibrations of which are due to waves
propagated in a direction making angles a, 8, ty, with the axes of elasticity; and let v’ be the
resolved part in that direction of the velocity of an atom of the medium situated where the
velocity of the ather is v. Then by Art. 2, the accelerative force of the retardation is equal to

-x3(2-F) , or -K80-g%.
If now the velocity v be resolved in the directions of the axes, the accelerative forces of retardation
corresponding to the resolved parts of the velocity will be,

dv
dt’

And by the considerations in Art. 5, the accelerative force of the retardation in the given direction
of propagation, is equal to the resultant of these forces. Hence

d
— KS (1 —q,) cosa - Kd(1-q) cos B=, — K3 (1 ~ q) cosy

d
= K3 (19) Oe = — KBE. {0 — 9) costa + (1 ~ 4) 008" B + (1) c08" 7}.

Let now 72 be the constant of elasticity in the direction of propagation. Then by the equations in
Art. 4, we have,
2 tha ae a
<-1=K3(1-4); meine — H); p71 = 80 ORs eet = Kd(-4q),

‘ ‘

Hence, by substitution in the foregoing equation,

2

w -l= ( - 1) cos’ a + [ -- 1) cos’ 8 + (= - 1) cos’ ry.
2 a; b? c?
Consequently,

1
rr a? b? C2

‘

cos*a cos (3 cos"
ae ‘

The surface of which this is the equation in polar co-ordinates, may be called the surface of
elasticity. It is evidently that of an ellipsoid. The radius vector 7, represents the velocity of
propagation of plane waves in any direction coinciding with that of r

7. We have now to find the velocity of propagation in a filament of the xther, corresponding
to a ray of light. In considering the motion in a filament of a medium the elasticity of which
varies with the direction, I shall proceed in a method analogous to that employed in my Paper
on Luminous Rays. (Camb. Phil. Trans. Vol. vit. Part 111. p. 365). It will be supposed that
in the filament there is an axis of no transverse velocity. This is taken for the axis of s. The

THROUGH TRANSPARENT MEDIA, AND ON DOUBLE REFRACTION. 527

condensation at any point of the filament is assumed to be @, (x, ¢) x f(«, y), which for shortness
sake, will be written ff, @, being treated as a function of x and ¢ only, and f as a function of
wv and y only. Let p be the density, and w, v, w the components of the velocity in the directions
of the axes of co-ordinates, at the point wyz, and at the time ¢. Also let @%, b%, c* be the co-
efficients of elasticity in the directions of the axes of w, y, * respectively. First powers only of the
velocities w, v, w, and of the condensation p —1 will be taken account of. This being premised,
we have,

+ jan

a _ ade _ ua 2g, df |
pdx P ie

=

d to the fi sh
and to the first approximation, imaa OD ae Hence,

u=-—a” D jp ats,

the arbitrary quantity c being in general a function of w, y, and x. So also
1, Of '
= —b? =~ dt +C’.
ve WT fyats

ok i dw e*dp Poe
Again, since ($) =-—-——~ , we have to the same degree of approximation,

dw

Fran acdeke , SU Oi Sy) sayy eg ong IMG ”

But from the supposed law of condensation in any plane perpendicular to the axis of x, it follows,
that the accelerative force parallel to this axis at any point of the plane, must to the first degree
of approximation, be equal to f x the accelerative force at the point of intersection with the axis,

: ae . : d :
and the corresponding velocities must be in the same proportion. Hence, ‘e being the velocity
z

at the point of intersection with the axis, we shall have
sis Consequentl fb dt OS®
= —. —c” = an =
de ie quently Ppat=9,

Assuming now that C and C’ are each equal to zero when @ =0, we obtain,

WE 12 hf
uaa ?a » andv=-— Pigg? Hence,
Cae bee Saf dp
uda+vdy + wdz= ge PG, + aein ag oY pes dz.

4 ; a” b”?
In this case wdw + vdy + wdz is not an exact differential. Let — =A, and — = /, and suppose
c? c

1
that f= PIB, the function /’, containing w only, and the function #, containing y only. By
this supposition, a factor which will render the above quantity an exact differential may be found,
which, though not the most general, will suffice for our present purpose. By differentiating,
df 1 dF, df 1 dF,
fdv” hP, dw’ fdy~ UF," dy
Vou. VIII. Parr IV. Bie

528 PROFESSOR CHALLIS, ON THE TRANSMISSION OF LIGHT

dF. dF
Hence, udv + vdy+ wdz = wf Bia du + a dy +fbas

siden dF, dF, d
= Fe fang iF 7 dat F, dy dy) p + FF. ® ax}

1

ee ee ee
=F! FY .d.F,F.d.

. = a 5 :
Consequently the required factor is F, ".F, ‘; and the differential equation of the surface

cutting at right angles the directions of the motion, isd. /,F.p=0. If =0, be the equation
of this surface, we have ~=/,F,¢ + a function of ¢ We may now proceed to find a value

1 1 qi ae op
of R + R° the sum of the reciprocals of the principal radii of curvature of the surface at any

1

a 1 :
point, by substituting in the general expression for Rt RP? viz.,

si eZ sy ee oy 2a dy dy dy dy |

dy? dy? dy dx* dy’ d3z* da* dy’ dz? da®* da* dy . dy’
lar eae ial

dx ds* ~dwdy dw dy dadz dw’ dz ~‘dydz dy dz

eh de 20 po d
Now dy = F, B FD ‘ u, and a = Fy gh F, ‘.v; and therefore a =0 if w=0, and
wv y é
1 1 c
co sh 0 ifv=0. As we shall require the value of R’R only for points where «=0 and
dy
d d
v =0, we shall suppose in the general expression that = 0 and = =0. Hence
yy daw dF, dF,
Jomwl) utdare, tale dain rs uc dh dy’
R*R 7 ns rp
= 14%2
dz dz

Taking now the equation (3) obtained in page 365 of the Paper on Luminous Rays, and sup-
posing it to apply to any point of the plane perpendicular to the axis of x in which w= 0 and v = 0,
we shall have, neglecting small terms,

BLP _ on ESP LAP le =e =) =0.
dt® dz* dz R R

a. fp malgllPe Pale, =0, which, for the reasons given
dt? dz"

in the Paper just cited, it is required to be, we must have
d.fo 1 1 re ig _ fs a’. fp
dz pate "as
@.fp
42
mee dx

ie,

That this equation may be of the form

c?.

if ec? =c*(1 +h).

ae Fe ee tt, te a

ante

THROUGH TRANSPARENT MEDIA, AND ON DOUBLE REFRACTION. 529

dp /1 1 d’ Meer? 1 1
Hence, Fas GF + =) =i ae and by substituting the value of Rt? obtained above,
k. zo = = _ —, '
dz Fda’ F.dy’
C0)
dz”

x

For a point on the axis of z this equation becomes +n'p=0, the constant n” being

5 2 2
such that if X be the breadth of the waves, 2 = wi Hence, substituting — np for Ze in
the foregoing equation, the result is

d F, dF, nen
Fda Tag +kn*® = 0.

1

1
By taking account of the equality f= F,".F,', we obtain by substitution in the above
equation,

af Gf B.i(h=1) df 1. =1)0 df?
: “ de “ E a * 2 =
da® + dy? + 7 aa ae ae sae ; dy? + kn f =0.

If now for the same reasons as those given in p- 369 of the Paper on Luminous Rays, the
2

terms involving Fda? and fay be neglected, we have, finally,
df oh
hA.at+l ey
da? + dy? +khnf

The general result from this course of reasoning is, that a ray of which the condensation in the
transverse direction is detined by a function of w and y, which satisfies this equation, may be
propagated in a medium whose elasticity varies with the direction of propagation. The reasoning,
however, only applies to a function of w and y, which is the product of a function of # and
a function of y. It is evident that f cannot be a function of w + y*, and, consequently, that the
ray cannot be one of common light.

8. It is found by experience that a polarized ray may be transmitted in certain transparent
erystallized media. I shall assume that in these media the retardation of the propagation produced
by the presence and inertia of the atoms, is such as corresponds to an apparent diminution of the
elasticity of the «ther, different in degree in different directions. I shall assume also that there
are three rectangular axes of elasticity, and that, in accordance with the result contained in
Art. 8. of this Paper, the surface of elasticity is an ellipsoid. On these suppositions the ray
cannot be one of common light, because the effective elasticity is different in different directions
transverse to the axis of the ray. But the suppositions are consistent with the transmission
of a polarized ray. For according to the Theory of Polarization contained in my Paper in the
Camb. Phil. Transactions, (Vol. vi1t. Part 111. p. 372), the condensations for a polarized ray must
be disposed symmetrically with reference to two planes at right angles to each other passing
through the axis of the ray. Consequently the force of retardation and the effective elasticity
must act symmetrically with reference to two such planes. And this will evidently be the case:
for any section through the centre of the surface of elasticity is an ellipse, the radii of which
drawn from its centre, are symmetrically disposed with reference to its axes. It is possible that
the function f for a polarized ray may be such as that supposed in the preceding Article,
namely, the product of a function of » and a function of y. All, however, that can be affirmed
respecting this function from the reasoning in the Paper above referred to is, that for small
distances from the axis of the ray, it is a function of one co-ordinate only, the axis of w and y

8Y2

530 PROFESSOR CHALLIS, ON THE TRANSMISSION OF LIGHT

being supposed to be in the planes of symmetry. Let, therefore, in the last obtained equation,
f be a function of # only. Then,
af kn
Bee ewe
Now in the Paper on the Polarization of Light, (p. 373), the particular value of f for a polarized

ray was found to be cos n\/ka. By substituting this value in the equation above, we obtain

= ¢ It would appear therefore that a polarized ray

the equation of condition h=1, or a
cannot be transmitted in the medium, the transverse elasticity being different from that in the
direction of propagation, if the velocity of propagation really be c’, or c’./1+k. For the
transmission of the polarized ray it is necessary to suppose an alteration of the rate of propagation.
This may be conceived to take place as follows: First, suppose h = 1, and a polarized ray in which

Qa =a
the breadth of the waves is )\, or es to be transmitted with the velocity V1 +k. 'Then

suppose the elasticity in the direction of the plane of polarization to be altered from ec” to a”, and a
polarized ray to be still propagated. By hypothesis the nature of the medium is such as to allow
of this taking place. Now as f, and consequently the transverse section of the ray, do not alter
by the supposed change of elasticity, the only way in which the condensation can be altered is
by a change of . ‘The time of vibration of a given xtherial particle remaining constant, the rate
of propagation will be altered in the same ratio. Let therefore f =cos n'\/ka, and let X’ be
the new value of X. By substitution in the foregoing equation, we obtain the equation of
condition

and the velocity of propagation
, Wok, v , ae Ta a ,
=OV1l thx =e V/1tkh x5 =a'V/1 +k.
ce
The foregoing reasoning involves the inference that the rate of propagation of a ray in a medium
is not solely due to the effective elasticity in the direction of its axis, but is affected also by the

circumstance that the medium is incapable of transmitting any but a polarized ray, and that
for such a ray k is a constant.

9. We are now prepared to find the equation of a surface, the radius-vector of which
drawn in any direction from a fixed point, shall represent the velocity of propagation of a ray
in that direction. As we found the velocity of propagation to be that due to the elasticity in the
direction of a line drawn perpendicular to the axis of the ray in the plane of polarization, the
process will evidently be the following. Cut the surface of the ellipsoid of elasticity by a plane
perpendicular to the direction of propagation. The semi-axes of the section will be the radius-
vectors in that direction of the surface required. Let a, b, ¢ be the semi-axes of the ellipsoid.
Its equation in rectangular co-ordinates referred to the axes and the centre will be

ao <P) <=?
a ete :

Let the directions of the rectangular axes be changed by substituting for w, y, and x the
following values:

z=auv +By +y%,
yaaa + By +x,
v=a’ a + B’y ap ry",

a

THROUGH TRANSPARENT MEDIA, AND ON DOUBLE REFRACTION. 531

and in the result make x = 0, in order to obtain the equation of the section. This equation
will thus become

(aa + Byy (a a’ us Byy all a’ fs B’y'y
—————— + =
a? b? ce

= 1.

Supposing in this equation a and y' to be referred to the axes of the section, and r, 1”, to
be the two semi-axes, we shall have

1 Ge Ste RE
Bo eo?
1 ee
Fema san te
, “au
Re

The equation of the surface, the radius-vectors of which in a given direction are r and 2”, is
consequently the following :

2 2 yr? a &b Ce
Ee ug : 1 (= a B 2 a” aE 2 z ak ae sk F ap ap” a’? 3"? M a’ 3” aL a”? Be
i ae a B Ce at ht mA Ae
2Q'2 a” 2 a 2 2 yr
aB+a"B a’ +a" A" _ |
ae bc?

By combining with this the equations

a +B + =1,
a® Be B” + yy”?
a’ +B +a" =

as a’ 3 a’ p" 2
ea pees aie

ll
et
4

we obtain,
S 12 a 172. ae hed 2 Pe ” 2 ha lis W AINe
1 apiny t=? 1-7") , ap - ey", @B"-a"By , BY — a" BY _
ro P\ a@ iy é aoe ac Be

Again, from the equations
2 2 2
a +a +a =1,

Baa ere ieee,
pry? +y”
aB+a' Pp’ + aoe = (0)

tl
=
.

we have,
a’ +a"? + 2aBa B =a'*B” = (l= a’ - a’) (1 - B’ - B”)
= 1 a = 6 = a? — B? + a3? + a%B* 4 a’ 3? + a B*.
Hence, 0=7'-a*- B’ + (af - Ba)’;
or, ens. Sie + BP -y=1-7y ane ry”;
so (aB” - a" By =", and (a'B” - a” BY’ =

532 PROFESSOR CHALLIS, ON THE TRANSMISSION OF LIGHT, ETC.

The equation consequently becomes

12 fh Mn 2 12 2
ae PD ie Fue Tin
a* b? c CaDe hp Gave Oe

1
r

\ (ee

2
ae

Transforming it into rectangular co-ordinates by putting «* for r°-y*, y° for ry", & for ry”,
and a +y°+2° for 7°, there results

2

Yts e+e WV+yH wv

< y
2 2 Sin
1 e b2 = Ce +@tty+#) (Sat e+ 7a) = 03

or, (a? + y+ 2) (aa? + OY? + x”) — 7b (a? ty?) — @e (a? +) — BE (y’ + 2”) + PD? = 0.

This is the equation of the wave surface in Fresnel’s Theory of Double Refraction. It is very
remarkable that principles and reasoning so widely different from those of that Theory should have
led to the same result. It is needless to go farther in the investigation, as all subsequent deductions
may be made in the same manner as in the received Theory.

10. In conclusion, I beg leave to refer to an objection which may be raised against the
Theory of Polarization which I have brought forward. It may be urged, that as a wave is
conceived in this Theory to be composed of a vast number of rays in the same phase of vibration,
the transverse vibrations of the different rays will mutually destroy each other, leaving only the
direct vibrations, which by hypothesis do not produce the sensation of light. To this it may be
replied, that it is only the awis of a ray which can be considered as subject to the law of
refraction ; for the motion of the xtherial particles along the axis is rectilinear, and coincident in
direction with the line of propagation, while at every other part of the ray the direction of the
motion of a given particle is continually varying, and is generally not coincident with the line of
propagation. Admitting the independent motion of each ray, it is possible that by refraction
through the eye, the directions of the axes of different rays may be brought to pass nearly through
the same point of the retina, in obedience to the common law of refraction, while the separate rays,
not being subject in other parts to this law, may not be altered as to the diameters of their trans-
verse sections. ‘The constancy of the transverse section is, in fact, a necessary consequence of
a supposition already made in the course of this Theory, namely, that the quantity & is a fixed
numerical quantity, the same for rays propagated in media as for rays propagated in free space.
As, however, I am not at present provided with the means of ascertaining the nature and value
of that quantity, this part of the subject must be considered as open to further inquiry.

J. CHALLIS.

CAMBRIDGE OBSERVATORY,
May 17, 1847.

TPR AEN SAG PPorns

OF THE ©

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

VouumE VIII. Part V.

CAMBRIDGE:
PRINTED AT THE UNIVERSITY PRESS;
AND SOLD BY
JOHN WILLIAM PARKER, WEST STRAND, LONDON;
J. DEIGHTON ; AND MACMILLAN & CO., CAMBRIDGE.

M.DCCC.XLIX.

XL. On the Critical Values of the Sums of Periodic Series. By G. G. Svoxes, M.A.,
Fellow of Pembroke College, Cambridge.

[Read December 6, 1847.]

THERE are a great many problems in Heat, Electricity, Fluid Motion, &c., the solution of
which is effected by developing an arbitrary function, either in a series or in an integral, by
means of functions of known form. The first example of the systematic employment of this
method is to be found in Fourier’s Theory of Heat. The theory of such developements has
since become an important branch of pure mathematics.

Among the various series by which an arbitrary function f(#@) can be expressed within
certain limits, as 0 and a, of the variable x, may particularly be mentioned the series which

: * Te ; a : .
proceeds according to sines of — and its multiples, and that which proceeds according to
a

cosines of the same angles. It has been rigorously demonstrated that an arbitrary, but finite
function of v, f(#), may be expanded in either of these series. The function is not restricted
to be continuous in the interval, that is to say, it may pass abruptly from one finite value to
another; nor is either the function or its derivative restricted to vanish at the limits 0 and a.
Although however the possibility of the expansion of an arbitrary function in a series of sines,
for instance, when the function does not vanish at the limits 0 and a, cannot but have been
contemplated, the wtility of this form of expansion has hitherto, so far as I am aware, been
considered to depend on the actual evanescence of the function at those limits. In fact, if the
conditions of the problem require that f(0) and f(a) be equal to zero, it has been considered
that these conditions were satisfied by selecting the form of expansion referred to. The chief
object of the following paper is to develope the principles according to which the expansion of
an arbitrary function is to be treated when the conditions at the limits which determine the
particular form of the expansion are apparently violated; and to shew, by examples, the advantage
that frequently results from the employment of the series in such cases.

In Section I. I have begun by proving the possibility of the expansion of an arbitrary
function in a series of sines. Two methods have been principally employed, at least in the simpler
cases, in demonstrating the possibility of such expansions. One, which is that employed by
Poisson, consists in considering the series as the limit of another formed from it by multiplying
its terms by the ascending powers of a quantity infinitely little less than 1; the other consists in
summing the series to 2 terms, that is, expressing the sum by a definite integral, and then con-
sidering the limit to which the sum tends when m becomes infinite. The latter method certainly
appears the more direct, whenever the summation to ” terms can be effected, which however is
not always the case; but the former has this in its favour, that it is thus that the series present
themselves in physical problems. The former is the method which I have followed, as being that
which I employed when I first began the following investigations, and accordingly that which best
harmonizes with the rest of the paper. I should hardly have ventured to bring a somewhat
modified proof of a well-known theorem before the notice of this Society, were it not for the
doubts which some mathematicians seem to feel on this subject, and because there are some points
which Poisson does not seem to have treated with sufficient detail.

Vout. VIII. Parr V. 8Z

534 Mr. STOKES, ON THE CRITICAL VALUES OF

I have next shewn how the existence and nature of the discontinuity of f(#) and its derivatives
may be ascertained merely from the series, whether of sines or cosines, in which f(w) is developed,
even though the summation of the series cannot be effected. I have also given formule for
obtaining the developements of the derivatives of f(#) from that of f(x) itself. These develope-
ments cannot in general be obtained by the immediate differentiation of the several terms of the
developement of f(v), or in other words by differentiating under the sign of summation.

It is usual to restrict the expanded function to be finite. ‘This restriction however is not

necesssary, as is shewn towards the end of the section. It is sufficient that the integral of the
function be finite.

Section II. contains formule applicable to the integrals which replace the series considered

in Section I. when the extent @ of the variable throughout which the function is considered is
supposed to become infinite.

Section III. contains some general considerations respecting series and integrals, with reference
especially to the discontinuity of the functions which they express. Some of the results obtained
in this section are referred to by anticipation in Sections I. and If. They could not well be
introduced in their place without too much interrupting the continuity of the subject.

Section IV. consists of examples of the application of the preceding results. These examples
are all taken from physical problems, which in fact afford the best illustrations of the application
of periodic series and integrals. Some of the problems considered are interesting on their own
account, others, only as applications of mathematical processes. It would be unnecessary here to
enumerate these problems, which will be found in their proper place. It will be sufficient to
make one or two remarks.

The problem considered in Art. 52., which is that of determining the potential due to an
electrical point in the interior of a hollow conducting rectangular parallelepiped, and to the elec-
tricity induced on the surface, is given more for the sake of the artifice by which it is solved than
as illustrating the methods of this paper. The more obvious mode of solving this problem would
lead to a very complicated result. :

The problem solved in Art. 54. affords perhaps the best example of the utility of the
methods given in this paper. The problem consists in determining the motion of a fluid within
the sector of a cylinder, which is made to oscillate about its axis, or a line parallel to its axis.
The expression for the moment of inertia of the fluid which would be obtained by the methods
generally employed in the solution of such problems is a definite integral, the numerical calculation
of which would be very laborious; whereas the expression obtained by the method of this paper
is an infinite series, which may be summed, to a sufficient degree of approximation, without much
trouble.

The series for the developement of an arbitrary function considered in this paper are two, a
series of sines and a series of cosines, together with the corresponding integrals; but similar
methods may be applied in other cases. I believe that the following statement will be found to
embrace the cases to which the method will apply.

Let w be a continuous function of any number of independent variables, which is considered
for values of the variables lying within certain limits. For facility of explanation, suppose w a
function of the rectangular co-ordinates 7, y, x, or of w, y, x and #, where ¢ is the time, and
suppose that w is considered for values of w, y, x, ¢ lying between 0 and a, 0 and b, 0 and e,
0 and T’, respectively. For such values suppose that w satisfies a linear partial differential equation,
and suppose it to satisfy certain linear equations of condition for the limiting values of the
variables. Let U=0, U’=0 be two of the equations of condition, corresponding to the two
limiting values of one of the variables, as vz, Then the expansion of w to which these equations
lead may be applied to the more general problem which leads to the corresponding equations of

condition U= F, U' = F’, where ¥ and F’ are any functions of all the variables except «, or of
any number of them.

or

THE SUMS OF PERIODIC SERIES. 5

©

SECTION I.

Mode of ascertaining the nature of the discontinuity of a function which is expanded
in a series of sines or cosines, and of obiaining the developements of the
derived junctions.

1. By the term function I understand in this paper a quantity whose value depends in any
manner on the value of the variable, or on the values of the several variables of which it is com-
posed. Thus the functions considered need not be such as admit of being expressed by any
combination of algebraical symbols, even between limits of the variables ever so close. I shall
assume the ordinary rules of the differential and integral calculus as applicable to such functions,
supposing those rules to have been established by the method of limits, which does not in the least
require the possibility of the algebraical expression of the functions considered.

The term discontinuous, as applied to a function of a single variable, has been used in two
totally different senses. Sometimes a function is called discontinuous when its algebraical expression
for values of the variable lying between certain limits is different from its algebraical expression for
values of the variable lying between other limits. Sometimes a function of w, f(«), is called con-
tinuous when, for all values of w, the difference between f(wv) and f(# +h) can be made smaller
than any assignable quantity by sufficiently diminishing h, and in the contrary case discontinuous.
If f(x) can become infinite for a finite value of x, it will be convenient to consider it as dis-
continuous according to the second definition. It is easy to see that a function may be discon-
tinuous in the first sense and continuous in the second, and vice versd. The second is the sense in
which the term discontinuous is I believe generally employed in treatises on the differential calculus
which proceed according to the method of limits, and is the sense in which I shall use the term in
this paper. The terms continuous and discontinuous might be applied in either of the above senses
to functions of two or more independent variables, If I have occasion to employ them as applied
to such a function, I shall employ them in the second sense; but for the present I shall consider
only functions of one independent variable.

In the case of the functions considered in this paper, the value of the variable is usually sup-
posed to be restricted to lie within certain limits, as will presently appear. I exclude from
consideration all functions which either become infinite themselves, or have any of their differential
coefficients of the orders considered becoming infinite, within the limits of the variable within which
the function is considered, or at the limits themselves, except when the contrary is expressly stated.
Thus in an investigation into which f(«) and its first m differential coefficients enter, and in which
f(@) is considered between the limits w = 0 and w = a, those functions are excluded, at least at first,
which are such that any one of the quantities f(#), f’(w)...f"(v) is infinite for a value of «
lying between 0 and a, or for a=0 or w= a; but the differential coefficients of the higher orders
may become infinite. The quantities f(w), f’(#) ...f"(w) may however alter discontinuously
between the limits v = 0 and «=a, but I exclude from consideration all functions which are such
that any one of the above quantities alters discontinuously an infinite number of times between the
limits within which « is supposed to lie.

The terms convergent and divergent, as applied to infinite series, will be used in this paper in
their usual sense; that is to say, a series will be called convergent when the sum to » terms
approaches a finite and unique limit as ” increases beyond all limit, and divergent in the contrary case.
Series such as 1-1+1-—..., sin w+sin2v+sin 3a+4..., (where w is supposed not to be 0 or a
multiple of 7,) will come under the class divergent; for, although the sum to m terms does not
increase beyond all limit, it does not approach a unique limit as increases beyond all limit. Of

322

536 Mr. STOKES, ON THE CRITICAL VALUES OF

course the first m terms of a divergent series may be the limits of those of a convergent series: nor
does it appear possible to invent a series so rapidly divergent that it shall not be possible to find a
convergent series which shall have for the limits of its first m terms the first m terms respectively of
the divergent series. Of course we may employ a divergent series merely as an abbreviated mode
of expressing the limit of the sum of a convergent series. Whenever a divergent series is employed
in this way in the present paper, it will be expressly stated that the series is so regarded. .

Convergent series may be divided into two classes, according as the series resulting from taking
all the terms of the given series positively is convergent or divergent. It will be convenient for
the purposes of the present paper to have names for these two classes. I shall accordingly call
series belonging to the first class essentially convergent, and series belonging to the second
accidentally convergent, while the term convergent, simply, will be used to include both classes.
Thus, according to the definitions which will be employed in this paper, the series

17? 41,3
wt+tart+tas+...
is essentially convergent so long as a°<13 it is divergent when #>1, and when # = 1; and it is

accidentally convergent when w= — 1.

The same definitions may be applied to integrals, when one at least of the limits of integration

2 sin v .
dw is

is co. Thus, ifa>0, f,°w-*dw is essentially convergent at the limit «, while di

a
only accidentally convergent, and f° sinadw«, not being convergent, comes under the class of
divergent integrals. These definitions may be applied also to integrals taken between finite limits,

when the quantity under the integral sign becomes infinite within the limits of integration, or at

oe ; Kj
one of the limits. Thus flog da is convergent, but Hf — divergent at the limit 0.
ett

2. Let f(w) be a function of w which is only considered between the limits = 0 and w = a,
and which can be expanded between those limits in a convergent series of sines of ld and its

multiples, so that

Q2Q72 ~ arr
woe + Ay SIN —— + oecceeecceeecee (1):
a

‘ a 3
f(v) = A, sin a + 4A, sin
a

Nv

To determine 4,, multiply both sides of (1) by sin dw, and integrate from 2 =0 tow=a.

Since the series in (1) is convergent, and sin ee does not become infinite for any real value of «,

; . nre :
we may first multiply each term by sin —— da and integrate, and then sum, instead of first
a

summing and then integrating*. But each term of the series in (1) except the m will produce

in the new series a term equal to zero, and the n‘ will produce aA. Hence
2 a . NV
A, =— | w) sin —— dav.
= [Fe sin da,

3 2 id ee . are
and therefore F(a) = - ss hi f(#) sin eat dw .sin ie See ee (2)-

0

" Moigno, Lecons de Calcul Différentiel, &c. Tom. 11. p.70-

THE SUMS OF PERIODIC SERIES. 537

3. Hence, whenever f(a) can be expanded in the convergent series which forms the right-hand
side of (1), the value of 4, can be very readily found, and the expansion performed. But this
leaves us quite in the dark as to the degree of generality that a function which can be so expanded
admits of. In considering this question it will be convenient, instead of endeavouring to develope
f(z), to seek the value of the infinite series

NTL

U
(e ~ Te
=f fe’) sin — — da’, sin
F a

agQy;w

provided the series be convergent ; for it is only in that case that we can, without further definition,
speak of the sum of the series at all. Now if we had only a finite number » of terms in the series
(3) we might of course replace the series by E

/
, Qrae . Qare re , nee

2 See! Nae He . 5
- [°F () {sin sin — + sin sin ... + sin sin aa da’.... (4).
a, a a a a

a

As it is however this transformation cannot be made, because, the series within brackets in the
expression which would replace (4) not being convergent, the expression would be a mere symbol
without any meaning. If however the series (3) is essentially convergent, its sum is equal to the
limit of the sum of the following essentially convergent series

NTL

i,
Nx A
dn [sin wi « teen teneeaes erascteee cae (5) 5
a

2 a eee
72s" f f(a!) sin

when g from having been less than 1 becomes in the limit 1. It will be observed that if (3) were
only accidentally convergent, we could not with certainty affirm the sum of (3) to be the limit of
the sum of (5). For it is conceivable, or at least not at present proved to be impossible, that
the mode of the mutual destruction of the terms of (3) in the infinitely remote part of the series
should be altered by the introduction of the factor g", however little g might differ from 1. Let us
now, instead of seeking the sum of (3) in those cases in which the series is convergent, seek the limit
to which the sum of (5) approaches as @ approaches to 1 as its limit.

a

4. The transformation already referred to, which could not be effected on the series (3), may
be effected on (5), that is to say, instead of first integrating the several terms and then summing,
we may first sum and then integrate. We have thus, for the value of the series,

Bay, . nea’ , F
= fF) {Be sin st sin "Sh aa sAneroossnaccocesancscconns (Os

a

The convergent series within brackets can easily be summed. The expression (6) thus becomes

1 a ; 1-g 1-2 :
sal, f@) a (a — a) Fz, 7 (a +) ; da. erin (UA):
1 —2g cos ——_—~ 4g? 1 — 2g cos ——__—__ +g”

a a

/
Now since the quantity under the integral sign vanishes when g = 1, provided cos =— be
not =1, the limit of (7) when g=1 will not be altered if we replace the limits 0 and a of a by
any other limits or groups of limits as close as we please, provided they contain the values of w’
which render w' £ w equal to zero or any multiple of 2a, Let us first suppose that we are con-
sidering a value of w lying between 0 and a, and in the neighbourhood of which f() alters
continuously. Then, since w + never becomes equal to zero or any multiple of 2a@ within the
limits of integration, we may omit the second term within brackets in (7); and since a’ — wv never
becomes equal to any multiple of 2a, and vanishes only when «’ = 7, we may take for the limits

538 Mr. STOKES, ON THE CRITICAL VALUES OF

of x’ two quantities lying as close as we please to 2, and therefore so close as to exclude all values
] ; , T

of a for which f(a’) alters discontinuously, Letg=1—h, « =«%+&, expand cos we by the
a

ordinary formula, and put f(a’) =f(v) + R. Then the limit of (7) will be the same as that of

2-h hd&
rae [if(@) + Rt 7 SE ae
kh +e( -...)

a?

the limits of & being as small as we please, the first negative and the second positive. Let now

g (= - | = £°,

a*

so that a is ultimately equal to = , that is to say when g is first made equal to 1, and then the
Tv
limits of &, and therefore those of &’, are made to coalesce. Let now G, L be respectively the
1d Se ae : ‘ ,
greatest and least values of (1 — 3h) - {f(w) + R} within the limits of integration. Then if
a
b that was = tale e + C, where tan-! denotes an angle lying between —~ and =
we observe at (apo A 5 gle lying 3 and —,
putting — &,, & for the limits of &’, we shall see that the value of the integral (8) lies
between

G (tan + tan7} =) and L (ton + tan! _ $

3!
h
but in the the limit, that is to say, when we first suppose f to vanish and then &, and &, G and
L become equal to each other and to b f(e), and tan-* : + tan7? = becomes equal to 7. Hence,
f(e) is the limit of (7). i

Next, suppose that the value of « which we are considering lies between 0 and a, and that
as w passes through it f(«’) alters suddenly from M to N. Then the reasoning will be exactly as
before, except that we must integrate separately for positive and negative values of é’, replacing
f(v) +R by M+ # in the latter case, and by N+ R' in the former. Hence, the limit of (7) will
be 4 (17 + N).

Lastly, if we are considering the extreme values c = 0 and x =a, it follows at once from the
form of (7) that its limiting value is zero.

Hence the limit to which the sum of the convergent series (5) tends as g tends to 1 as its limit
is f(w) for values of # lying between 0 and a, for which f(x) alters continuously, it is }(M + N)

for values of w for which f(x) alters suddenly from M to N, and it is zero for the extreme values
0 and a.

5. Of course the limiting value of the series (5) is f(0) and not zero, if we suppose that ¢
first becomes 1 and then w passes from a positive value to zero. In the same way, if f(z) alters
abruptly from J/ to N as @ increases through w,, the limiting value of (5) will be M if we suppose
that g first becomes 1 and then » increases to x, and it will be N if we suppose that g first
becomes 1 and then wz decreases to #,. It would be futile to argue that the limiting value of (5)
for # =0 is zero rather than f(0), or f(0) rather than zero, since that entirely depends on the
sense in which we employ the expression limiting value. Whichever sense we please to adopt, no
error can possibly result, provided we are only consistent, and do not in the course of the same —
investigation change the meaning of our words.

THE SUMS OF PERIODIC SERIES. 539

It is a principle of great importance in these investigations, that a function of two independent
variables which becomes indeterminate for particular values of the variables may have different
limiting values according to the order in which we suppose the variables to assume their particular
values, or according to the nature of the arbitrary relation which we conceive imposed on them as
they approach those values together.

I would here make one remark on the subject of consistency. We may speak of the sum of an
infinite series which is not convergent, if we define it to mean the limit of the sum of a convergent
series of which the first 2 terms become in the limit the same as those of the divergent series.
According to this definition, it appears quite conceivable that the same divergent series should have
a different sum according as it is regarded as the limit of one convergent series or of another. If
however we are careful in the same investigation always to regard the same divergent series, and
the series derived from it, as the limits of the same convergent series and the series derived from it,
it does not appear possible to fall into error, assuming of course that we always reason correctly.
For example, we may employ the series (3), and the series derived from it by differentiation, &c.,
without fear, provided we always regard these series when divergent, or only accidentally convergent,
as the limits of the particular convergent series formed by multiplying their m" terms by g”.

6. We may now consider the convergency of the series (3), in order to find whether we may
employ it directly, or whether we must regard it as the limit of (5).
By eae os parts in the m" term of (3), we have

- [fw sin “7 da’ =~ — = fw) eo

ame

1 ome

af" (a) si

24a sane ee ta
—— a) sin —— da’... f
7 ne It (>) a (9)

Suppose that f(#) does not nee vanish at the limits g=0 and w =a, and that it alters
discontinuously any finite number of times between those limits, passing abruptly from M, to N,
when # increases through a,, from M, to N, when w# increases through a, and so on. Then, if
we put § for the sign of summation referring to the discontinuous values of f(a’), on taking the
integrals in (9) from w=0 to w= a, we shall get for the part of the integral corresponding to the
first term at the right-hand side of the equation

2 NTA
= {FO - (-)"f(@ +S (N - 1) cos a pee sccneonaa (HO),
It is easily seen that the last two terms in (9) will give a part of the integral taken from 0 to a,

pee L :
which is numerically inferior to—, where LZ is a constant properly chosen. As far as regards
4 n

these terms therefore the series (3) will be essentially convergent, and its sum will therefore be
the limit of the sum of (5).

Hence, in examining the convergency or divergency of the series (3), we have only got to

° - Are ° : P
consider the part of the coefficient of sin - of which (10) is the expression. The terms f(0),
a

f(a) in this expression may be included under the sign S if we put for the first a =0, M = 0,
N= f(0), and for the second a=a, M=f(a), N=0. We have thus got a set of series to con-
sider of which the type is

2 1 @
SN M) = — cos ee ae, 2 Juvaivilhacodanesicsaaans (11).
T n a

a

540 Mr. STOKES, ON THE CRITICAL VALUES OF

If we replace the product of the sine and cosine in this expression by the sum of two sines,
by means of the ordinary formula, and omit unnecessary constants, we shall have for the series
to consider

“ : es
Let now U% = sins +/FSiN VF... + — SIM, .oscpeervosescoesecsereseseee (13),
n
du sin (2 +1)
then sae Ep eager y pape aes aa
dz 2sin ds

sin (x +4) %
indz
between — 27 and + 27, so that the quantity under the integral sign does not become infinite

and since w vanishes with x, in which case is finite, we shall have, supposing = to lie

within the limits of integration,

wat

*sin(n +1) 2
pea 8,
fy sin 32 2
and we have to find whether the integral contained in this equation approaches a finite limit as

increases beyond all limit, and if so what that limit is. Since w changes sign with , we need not
consider the negative values of x.

First suppose the superior limit » to lie between 0 and 27; and to simplify the integral write
22 for s, n for 2 +1, so that the superior limit of the new integral lies between 0 and 7; then

A * sin 22 *sinns 2 sin 2x
the integral = li : dz = ih —dz= ie (1 + Ra) dz,
) sing peliee esinus f

s—sing : : Betake ete vate : :
where R = ——— , a quantity which does not become infinite within the limits of integration.
z sing
Hence, as is known, the limit of {* sinm*.Rdz when m increases beyond all limit is zero, Hence,

if J be the limit of the integral,
Rvs = sin nz a. "sin ¢
I = limit of [ : dz = limit of f oe dg.

ti)

Now, * being given, the limit of mz is ©, and therefore
sin
I= - “pa ;

Secondly, suppose x in (14) to be equal to 0. Then it follows directly from this equation, or
in fact at once from (18), that «= 0, and consequently the limit of w = 0.

The value of w in all other cases, if required, may be at once obtained from the consideration
that the values of w recur when x is increased or diminished by 27.

Hence, the series (12) is in all cases convergent, and has for its sum 0 when x = 0, and 4 (7 — =)
when g lies between 0 and 27.

Now, if in the theorem of Article 4. we write x for 2, and put a =a, f(z) = 4 (a — 8); we find,
for values of x lying between 0 and 7, and for x = z,

Pak 1 c
limit of B= g*sinns =4(r-2);

THE SUMS OF PERIODIC SERIES. 541
and evidently

as 1 ;
limit of = — g" sinnx = 0, when zx =0,
n

that is of course supposing = first to vanish and then g to become 1. Also the limit of
> = sin mz changes sign with x, and recurs when x is increased or diminished by 27. Hence,
the series (12), which has been proved to be convergent, is in all cases the limit to which the sum of
the convergent series Sg" sin nz tends as g tends to 1 as its limit. Now the series (11) may be

decomposed into two series of the form just discussed, whence it follows that the series (3) is
always convergent, and its sum for all values of a, critical as well as general, is the limit of the
sum of the series (5), when g becomes equal to 1.

The examination of the convergency of the series (3) in the only doubtful case, that is to say,
the case in which f(a) is discontinuous, or does not vanish for x = 0 and for z =a, is more curious
than important. For in the analytical applications of the series (3) it would be sufficient to regard
it as the limit of the series (5); and in the case in which (3) is only accidentally convergent, we
should hardly think of employing it in the numerical computation of f(«) if we could possibly
help it, and it will immediately appear that in all the cases which are most important to consider
we can get rid of the troublesome terms without knowing the sum of the series.

The proof of the convergency of the series (3) which has just been given, though in some
respects I believe new, is certainly rather circuitous, and it has the disadvantage of not applying
to the case in which f’(x) is infinite*, an objection which does not apply to the proof given by
M. Dirichlet+. It has been supposed moreover that f”(«) is not infinite. The latter restriction
however may easily be removed, as in the end of the next article.

7. Let f(«) be a function of « which is expanded between the limits 2 =0 and w =a in the
series (3). Let the series be

. Te . 2are
A, sin — + A, sin
a a

- nae
.-. + A, sin

a Pad phe sdeusdease ccd i) 5

and suppose that we have given the coefficients 4,;, 4,..., but do not know the sum of the series
f(«). We may for all that find the values of f(0) and f(a), and likewise the values of w for

which f(a) is discontinuous, and the quantity by which f(w) is increased as w increases through
each of these critical values.

For from (9) and (10)

An (i nara
nA, = = {F00) —(-1)"f(a) + S(N = M) cos eae
a
R being a quantity which does not become infinite with m. If then we use the term dimi¢ in an
extended sense, so as to include quantities of the form C cos my, (of course C (- 1)" is a particular
case,) or the sum of any finite number of such quantities, we shall have for x = © ,

limit of 2A, = {70 -(-1)"f(a) + S(N - M) cos

9 naa\
7 L

joo (16).

0

* This restriction may however be dispensed with by what is proved in Art. 20, + Crelle’s Journal, Tom. tv. p. 157.

Vor. VIII. Parr V. 4A

542, Mr. STOKES, ON THE CRITICAL VALUES OF

.

Let then the limit of 2.4, be found. It will appear under the form

Ci + C,(-1)'+ SCcosny ..... dif adits: atlas Ba fen LT)
Comparing this expression with (16), we shall have
T Tv
f0) ==, fla) =-2¢,;

and for each term of the series denoted by S' we shall have
a= sell »5 N-M= - G:

In particular, if f() is continuous, and if the limit of m 4, is ZL, or L, according as n is odd or
even, we shall have

Ly = ~{£(0) +f(a)}, Le= = {£(0) - f(a};

whence
f(0) = = (L,+L,), f(a) = * (Lo SS Baya Rebealdaen. dead

If f(w) were discontinuous for an infinite number of values of w lying between 0 and a, it is
conceivable that the infinite series coming under the sign S§ might be divergent, or if convergent
might have a sum from which x might wholly or partially disappear, in which case the limit of
nA,, might not come out under the form (17). It was for this reason among others, that in Art. 1,
T excluded such functions from consideration.

If f(w) be expressible algebraically between the limits «= 0 and «=a, or if it admit of
different algebraical expressions within different portions into which that interval may be divided,
A, will be an algebraical function of », and the limit of m4, may be found by the ordinary methods.
Under the term algebraical function, 1 here include transcendental functions, using the term alge-
braical function in opposition to what has been sometimes called an empirical function, or a general
function, that is, a function in the sense in which the ordinate of a curve traced libera manu is a
function of the abscissa. Of course, in applying the theorem in this article to general functions, it
must be taken as a postulate that the limit of m4, can be found, and put under the form (17).

The theorem in question has been proved by means of equation (9), in which it is supposed
that f”() does not become infinite within the limits of integration. The theorem is however true
independently of this restriction. To prove it we have only got to integrate by parts once instead

: , : R ;
of twice, and we thus get for the quantity which replaces — the integral
n

2 a nora
ee if f(a’) cos and da’,
To

which by the principle of fluctuation* vanishes when m becomes infinite. There is however this
difference between the two cases. When the series (15) has been cleared of the part for which the

* I borrow this term from a paper by Sir William R. Hamilton
On Fluctuating Functions. Transactions of the Royal Irish
Academy, Vol. xrx. p. 264. Had I been earlier acquainted with
this paper, and that of M. Dirichlet already referred to, I would
probably have adopted the second of the methods mentioned in the
introduction for establishing equation (2) for any function, or

rather, would have begun with Art. 7, taking that equation as
established, I have retained Arts. (2)—(6), first, because I
thought the reader would enter more readily into the spirit of the
paper if these articles were retained, and secondly, because I
thought that Section r11, which is adapted to this mode of viewing
the subject, might be found useful.

THE SUMS OF PERIODIC SERIES. 5 543
limit of 2 A, is finite, by the method which will be explained in the next article, the part which

: 3 é mune 1 1
remains will be at least as convergent in the former case as the series Sear en aeriesc) whereas
2 ne

we cannot affirm this to be true, and in fact it may be proved that it is not true, in the case in
which f’(#) becomes infinite. Observing that the same remark will apply when we come to
consider the critical values of the differential coefficients of f(#), I shall suppose the functions and
derived functions employed in each investigation not to become infinite, according to what has
been already stated in Art. 1.

8. After having found the several values of a, and the corresponding values of N — M, we
may subtract the expression (10) from 4,, provided we subtract from the sum of the series (15)
the sums of the several series such as (11). Now if X be the sum of the series (11),

na («+ a) 1 . mar(wv-a)
+ = — sin :
a n a

x =i ats sin \. --- (19).

i aa "
But it has been already shown that = — sinnz = 4(7-—=) when z lies between 0 and 27, =0
n

when x=0, and = — 5(r + #) when # lies between 0 and —27. Now when @ lies between 0 and a,
a(@+a) ,. 7m (@ — a) ,. ;
pale +2) lies between 0 and 27, and Cae) lies between — 27 and 0; and when @ lies between
a
7T(e+a ; A 7T(v-—a P aks
a and a, mae) still lies between 0 and 27, and lt) now lies between the same limits.

a
Hence
v& .
=-(N-M) -—, when lies between 0 and “f
a

weseee (20),
a-—-x

=, (N-M)

Fa when wv lies between a and al
We need not trouble ourselves with the singular values of the sum of the series (15), since we
have seen that a singular value is always the arithmetic mean of the values of the sum for values
of z immediately above and below the critical value. This rule will apply to the extreme cases in
which w = 0 and a =a, if we consider the sum of the series for values of w lying beyond those
limits. The rule applies to the series in (19), which is only a particular case of (15), and con-
sequently will apply to any combination of series having this property, formed by way of addition
or subtraction; since, when we increase or diminish any two quantities M,, N, by any other two
M, N respectively, we increase or diminish the arithmetic mean of the two former by the arithmetic
mean of the two latter.

It has been already stated that we may, with a certain convention, include quantities referring
to the limits = 0 and w=a under the sign of summation §. If we do so, and put = for the
sum of the series (15), and B, for the remainder arising from subtracting the expression (10)
from A,, we shall have

SGX we SB din —:
a

and the sum of the series forming the right-hand side of this equation will be a continuous function
of w. As to SX, the value of each series contained in it is given by equation (20),

To illustrate this, suppose = the ordinate of a curve of which is the abscissa. Let OG be
the axis of a; OA, MB, ND, Gb right lines perpendicular to it, and let OG = a. Let the curve

4A 2

544 Mr. STOKES, ON THE CRITICAL VALUES OF

of which © is the ordinate be the discontinuous curve 4B, CD, EFG. Take Gb equal to BC,

|

‘
and on the positive or negative side of the axis of # according as the ordinate decreases or
increases as @ increases through OM, and from O measure an equal length Oc on the opposite
side of the axis. Take Gd, Oe, each equal to DE, and draw the right lines AG, Ob'b, ce'G,
Od'd, ee'G. Then it will be easily seen that if X,, X,, Xz be the values of X corresponding to
the critical values of v, w =0, 7 =OM, «= ON, respectively, X, will be represented by the right
line AG; X, by the discontinuous right line O6', c’G; and X, by the discontinuous right line
Od’, e'G. Take MP equal to the sum of the ordinates of the points in which the right lines lying
between OA and cB cut the latter line; MQ equal to the sum of the ordinates of the points in
which the right lines lying between ec’ B and d’E cut the former, and so on, the ordinates being
taken with their proper signs. Let P, Q, R, S be the points thus found, and join AP, QR, SG.
Then SX will be represented by the discontinuous right line AP, QR, SG. Let the ordinates of
the discontinuous curve be diminished by those of the discontinuous right line last mentioned, and
let the dotted curve be the result. Then = — SX will be represented by the continuous, dotted
curve. It will be observed that the two portions of the dotted curve which meet in each of the
ordinates MB, NE may meet at a finite angle. If there should be a point in one of the con-
tinuous portions, such as AB, of the discontinuous curve where two tangents meet at a finite angle,
there will of course be a corresponding point in the dotted curve.

If we choose to take account of the conjugate points of the curve of which SX is the ordinate,
it will be observed that they are situated at O, and midway between P and Q, and between # and S.

9. There are a great many series, similar to (3), in which f(w) may be expanded within
certain limits of 2. I shall consider one other, which as well as (3) is of great use, observing that
almost exactly the same methods and the same reasoning will apply in other cases.

The limit of the sum of the series

Lage 3 ive BR a ;
af feyda’ + 7 Sa" f fle’) cos

n

rae Norv
da’. cos
a

THE SUMS OF PERIODIC SERIES. 545

when g from having been less than 1 becomes 1, is f(«), v being supposed not to lie beyond the
limits 0 and a. For values, however, of v for which f(«) alters discontinuously, the limit of the
sum is the arithmetic mean of the values of f(w) for values of « immediately above and below the
critical value. I assume this as being well known, observing that it may be demonstrated just
as a similar theorem has been demonstrated in Art. 4.

10. Let us now consider the series

~ [F@) die’ + - = [fe cos

n

Ct a D
da’ .cos ——.......... (22).
a a

We have by integration by parts

,
Te

, , ,
a 2 Ne 2a n nr xv
de =— f(a’) sin = + —_ Ff (2')\cos'— da’
pach ) a eat (7) a

3

- [F(2’) cos - _ Jf" (2) cos

and now, taking the limits properly, and employing the letters M, N, a and S’ in the same sense
as before, we have

a

R

met (23),

DCO nre , Q . 20
=f Ff (v’) cos —— da’ = —- — S(N - M) sin —— +
avy a nT a
R being a quantity which does not become infinite with m. It follows from (23), that the series
(22) is in all cases convergent, and its sum for all values of «, critical as well as general, is the
limit of the sum of (21).

It will be observed that if f(v) is a continuous function the series (22) is at least as convergent

as the series = = . This is not the case with the series (3), unless f(0) = f(a) = 0.
n

NTP

If the constant term and the coefficient of cos in the general term of (22) are given, f(x)

itself not being known, except by its developement, we may as before find the values of w for
which f(7) is discontinuous, and the quantity by which f(w) is suddenly increased as @ increases
through each critical value. We may also, if we please, clear the series (22) of the slowly con-
vergent part corresponding to the discontinuous values of f().

11. Since the series (3) is convergent, if we have occasion to integrate f(#) we may, instead
of first summing the series and then integrating, first integrate the general term and then sum.
More generally, if p(w) be any function of « which does not become infinite between the limits
w = 0 and « =a, we shall have

f f(a) $ (a) da = : = [Fe sin

the superior limit w of the integrals being supposed not to lie beyond the limits 0 and a; and the
series at the second side of the above equation will be convergent. In fact, even in the case in

,
Nee

a

es . Tre
da’. f go (x) sin —— da,
f a

: 1 :
which f(a) is discontinuous the series will be as convergent as the series = a: A second inte-

gration would give a series still more rapidly convergent, and so on. Hence, the resulting series
may be employed directly, and not merely when regarded as limits of converging series. ‘The
same remarks apply in all respects to the series (22) as to the series (3).

12. But the series resulting from differentiating (3) or (22) once, twice, or any number of
times would not in general be convergent, and could not be employed directly, but only as limits
of the convergent series which would be formed by inserting the factor g’ in the general term.

546 Mr. STOKES, ON THE CRITICAL VALUES OF

This mode of treating the subject however appears very inconvenient, except in the case in which
the series are only temporarily divergent, being rendered convergent again by new integrations ;
and even then it requires great caution. The series in question may however be rendered con-
vergent by means of transformations to which I now proceed, and which, with their applications,
form the principal object of this paper.

The most important case to consider is that in which f(#) and its derivatives are continuous,
so that the divergency arises from what takes place at the limits 0 and a. I shall suppose then, for
the present, that f(w) and its derivatives of the orders considered are continuous, except the last,
which will only appear under the sign of integration, and which may be discontinuous,

Consider first the series of sines. Suppose that f(v) is not given in finite terms, but only by
its developement

UT v

f (2) = =A, sin siaeoleceetetins atta ceacaa scents athe veenetes (28)s

where 4, is supposed to be given, and where the developement of f() is supposed to be that which
would result from the formula (3). I shall call the expansions of f(#) which are obtained, or which
are to be looked on as obtained from the formulz (3) and (22) direct expansions, as distinguished
from other expansions which may be obtained by differentiation, and which, being divergent, cannot
be directly employed. Let us consider first the even differential coefficients of f(~), and let 4”,

‘ - NZ, t , :
A; ... be the coefficients of sin 7“ in the direct expansions of f’ (x), f'(«) ... The coefficient of
a

sin —— in the series which would be obtained by differentiating twice the several terms in the
a

2
series in (24) would be — = A,. Now

BAN a MY Apher G1 OL
= 7 d 3
A, = ") f(#) sin 2/5 Oe
and we have by integrating by parts
nr

2 n* r* 2

,
av
d a’ =

Anes a a Re Bk
wv’) sin ahs
a If ( ) = Z (2) cos = oa (a) sin .

/
NTL

du.

Nae oe asi
+ off" (#) sin

Taking now the limits, remembering the expression for 4”, and transposing, we get

a

y, bee _ §f(0) -(- )"f(@t - a PAD: ane trent EPs (25).
Any even differential coefficient may be treated in the same way. We thus get, » being even,
BB a\* 1 Bl 2 H-=3 Ap 4
(-yiag=(")4,-= (=) yo-cuver+= (7) ro@-Core@s--.
+(- 1)? = “ms {f*-?(0) — (- 1)"f*?(a@)}. --- (26).

13. In the applications of these equations which I have principally in view, (0), f(a), f” (0)...
are given, and 4,, 4,, A,... are indeterminate coefficients. If however 4,, 4,.-. 4, .-. are given,
and f(0), f(a) ... unknown, we must first find (0), f(a) ..., and then we shall be able to sub-
stitute in (25) and (26). This may be effected in the following manner.

pena,

THE SUMS OF PERIODIC SERIES. 547
We get by integrating by parts
Sf(@) sin

, 3 ,
a narxr a@\* Nrxe ne
da’ = — —f(a’) cos + (=) aw’) sin
amt « ) nT F( )

CEN yet
seals (+) ff (@) cos
a no

NTP

a a

fo}
Multiplying now both sides by = and taking the limits of the integrals, we get

ie = £#(0) - (- 1)"f(a)} - =. (“) § 7G) = (17 @} beeees CP

Hence, if m be always odd or always even, 4, can be expanded, at least to a certain number of
terms, in a series according to descending powers of m, the powers being odd, and the first of
them —1. The number of terms to which the expansion in this form is possible will depend on the
number of differential coefficients of f(a) which remain finite and continuous between the limits w = 0
and 2 =a. Let the expansion be performed, and let the result be

1 1 1
Lhe (Why = ap 10k as (ON ae ace ies Se
n n n

1 1 1
ee
nm nN

7 +... when 7 is even|
Comparing (27) and (28), we shall have

f0)= = (+E). f(a)= = (O-%),|

3

f'(0) =- = (Os PE fa) a (Ob Ba nse o hea (20),

4a*
5

FO= T(A+k), f@= 7.0,- Ey,

and soon. The first two of these equtions agree with (18).

If we conceive the value of 4, given by (27) substituted in (26), we shall arrive at a very

simple rule for finding the direct expansion of f*(xv). It will only be necessary to expand A, as

far as

;21> admitting (— 1)" into the expansion as if it were a constant coefficient, and then, sub-
n

tracting from 4, the sum of the terms thus found, employ the series which would be obtained by
differentiating the equation (24) » times. It will be necessary to assure ourselves that the term

in — vanishes in the expansion of A,, since otherwise f“(#) might be infinite, or f*~'(«) discon-
n

tinuous without our being aware of it. It will be seen however presently (Art. 20) that the
former circumstance would not vitiate the result, nor introduce a term involving n~".

1 1 cz 3 : :
Should A, already appear under such a form as — +"; (—1)"— + mc", &c., where c* <1, it
n n'

will be sufficient to differentiate equation (24) times, and leave out the part of the series which
becomes divergent. For it will be observed that the terms ec", m*c", in the examples chosen,

ane ,
decrease with — faster than any inverse power of 7,
n

14. Let us now consider the odd differential coefficients of f(w), supposing f(v) to be
expanded in a series of cosines, so that

548 Mr. STOKES, ON THE CRITICAL VALUES OF

NTe

f(x) = By, + =B, cos GCM testtstesteseesseesencesrenses cesses (30).

NT Ue

Let 4’,, A”... be the coefficients of sin in the direct expansions of f’(w), f’’(«) ... in

nv

series of sines. If we were to differentiate (30) once we should have — B,, for the coefficient

of sin Now
a
nr 2 5, Nr2e., 2 _ nre 2 eT ee
—-— = a) cos dx = — — f(x) sin = av) sin dz';
— IF (a!) cos = (fle) sin —— + = ff (2’) sin —
and taking the limits of the integrals, and introducing B, and 4’,, we get
a3
A ae een ne a iain ole cae eae wecint cnsaltouos seas vecinecadese (GU)e

a

Hence, the series arising from differentiating (30) once gives the direct expansion of f(a) in a
series of sines.

2 . Te
The coefficient of sin

in the series which would be obtained by differentiating (30) « times,

Berl be
» being odd, would be (— 1) 2 (=) B,. By proceeding just as in the last article we obtain

0? a= (Sp Bee (™)" ro-core}-=(") y’o-cor'o}+-..
“HO mor = hues
+(-1)?2 wile tk (0) — (—1)"ft-®(a)}. ... ee (32).

When f’ (0), f’(@), &e., are known, this series enables us to develope f“(a) in a direct series of
sines, the direct developement of f(a) in a series of cosines being given.

15. If we treat the expression for B, by integration by parts, just as the expression for A, was
treated, going on till we arrive at the integral which gives 4“, and observe that the very same
process is used in deducing the value of A* from that of B, as in expanding the latter according to
inverse powers of m, and that the index of m in the coefficient of A* is — u, and that A, vanishes when
m becomes infinite, we shall see that in order to obtain the direct expansion of f*(w) we have only

1 A Le evs pA
got to expand B, as far as ne? (the coefficient of =r will vanish,) and subtract from B, these
n
terms of the expansion, and then differentiate (30) « times.
The expansion of B,, at least to a certain number of terms, will proceed according to even

1 ; _ 1
powers of — , beginning with 5° If we suppose that
n

1 1 1 :
act Os = +... when 7 is oad, ]

1 1 *
B, =E,=+ E;— + E,— +... when m is even,
n n* n

THE SUMS OF PERIODIC SERIES. 549

and compare these expansions with that given by integration by parts, we shall have

f' (0) =- 7 (+8) f' (@=- = (0,- *)|
; ; Seeieiets we» (34),
f'(0) = = (O,+ E;), f(a) = = (0; — z»,|

: : : ; De ’
and so on, the signs of the coefficients being alternately + and —, and the index of — increasing
a
by 2 each time.

16. The values of f“(0) and f*(a) when f(x) is expanded in a series of sines and , is odd,
or when f(z) is expanded in a series of cosines and y is even, will be expressed by infinite series.
To find these values we should first have to obtain the direct expansion of f“(#), which would be
got by differentiating the equation (24) or (30) » times, expanding A, or B, according to powers

ofr and rejecting the terms which would render the series contained in the np" derived equation

divergent. The reason of this is the same as before.

17. The direct expansions of the derivatives of f(w) may be obtained in a similar manner in
the cases in which f(a) itself, or any one of its derivatives is discontinuous. In what follows,
a will be taken to denote a value of w for which f(w) or any one of its derivatives of the
orders considered is discontinuous; Q, Q;, .-. Q, will denote the quantities by which f(x), f’ (2), ...
f“(a) are suddenly increased as w increases through a; S' will be used for the sign of summation
relative to the different values of a, and will be supposed to include the extreme values 0 and a,
under the convention already mentioned in Art. 6. Of course f(a) may be discontinuous for a
particular value of w while f*(#) is continuous, and vice versa. In this case one of the two Q, Q,
will be zero while the other is finite.

The method of proceeding is precisely the same as before, except that each term such as

f(x) cos —— in the indefinite integral arising from the integration by parts will give rise to a series
a

nra . : is ;
such as — SQ cos —7* in the integral taken between limits. We thus get in the case of the even
a

derivatives of f(~), when f(a) is expanded in a series of sines,

“ “ 9 Ml m2
(-1)? At = (=) yes =. (7) aga (=) SQ, sin
; a a \a a a a

a
2 nar\"-3 ee Pee ty ait
at ) SQ, cos —— — ... + (- 1)2° .—. SQ,-1 sin efat eset. (85)
a a a a A

In the case of the odd derivatives of f(v), when f(«) is expanded in a series of cosines, we get

wt na\" 2 (n7w\""! _ nra 2 (nr? . nT a
71) 3" Ae = (“) B,+- ( } SQ sin —— +- (’ — ) SQ, cos Sion
a a\a a a\a a
pl

wal) nT a ~
-1)? - §¢ sin —— . ...... (36).
+ (=1)7 — SQ,-1 ; (36)

When the several values of a, Q, Q,... are given, these equations enable us to find the direct
expansion of f“(). he series corresponding to the odd derivatives in the first case and the even
in the second might easily be found.

Vor. VIII. Parr V. 4B

550 Mr. STOKES, ON THE CRITICAL VALUES OF

If we wish to find the direct expansion of f*(w) in the case in which A, or B, is given, we
have only to expand A, or B, in a series according to descending powers of n, regarding cos my
or sin my, as well as (— 1)”, as constant coefficients, and then reject from the series obtained by the
immediate differentiation of (24) or (30) those terms which would render it divergent. This readily
follows as in Art. 15, from the consideration of the mode in which 4* is obtained from A, or B,.

The equations (35) and (36) contain as particular cases (26) and (32) respectively. It was con-
venient however to have the latter equations, on account of their utility, expressed in a form which
requires no transformation.

18. If we transform A, and B, by integration by parts, we get

2 nira 2a _ nara 20 Ne "
BSA Ros —,, §Q, sin — = SQ, cos — — + 022, reveee (37);
hd a n* a a nm a a

nara 2a a 2a

nr
SQ, cos ——— +
a

22

ga
B,=-— SQ sin
nmr nN 1

. nra
Sa Qe Sin esa Bee leas (38),
W 7 a

where the law of the series is evident, if we only observe that two signs of the same kind are always
followed by two of the opposite kind. The equations (37), (38) may be at once obtained from (35),

‘ - ; : 1
(36). The former equations give the true expansions of 4, and B, according to powers of — 5

because when we stop after any number of integrations by parts the last integral with its proper
coefficient always vanishes compared with the coefficient of the preceding term.

- : ‘ Ty ads :
Hence 4, and B,, admit of expansion according to powers of =) if we regard cos mry or sin ny
as a constant coefficient in the expansion. Moreover quantities such as cos mry, sin my will occur
3 é Fi . 5 1
alternately in each expansion, the one kind going along with odd powers of — and the other along
n

with even. If we suppose the value of 4, or B,, as the case may be, given, and the expansion
performed, so that

1 5‘ 1 1
A, = SF cosny = + SF, sin ny. + SF, cos ry. + woe) ve0- (39);
”
i 1 1 ; 1
B, = SG sin ny. Ss SG, cos ny . ne + SG, sin my. = He sas Tiyee shes (40),
and compare these expansions with (37) or (38), we shall get the several values of a, and the
corresponding values of Q, Q,, Q,.... We may thus, without being able to sum the series in

equation (24) or (30), find the values of 2 for which f(a) itself or any one of its derivatives is
discontinuous, and likewise the quantity by which the function or derivative is suddenly increased.
This remark will apply to the extreme values 0 and a of a if we continue to denote the sum of the
series by f(#) when @ is outside of the limits 0 and a.

19. Having found the values of a, Q, Q,..., we may if we please clear the series in (24)
or (30) of the terms which render f(«) itself, or any one of it dervatives, discontinuous. If we
wish the function which remains expressed by an infinite series and its first » derivatives to be
continuous, we have only to subtract from 4, or B, the terms at the commencement of its expansion,

ending with the term containing ——, and from f(w) itself the sums of the series corresponding
n

to the terms subtracted from A, or B,. These sums will be obtained by transforming products of
sines and cosines into sums or differences, and then employing known formulz such as

THE SUMS OF PERIODIC SERIES. 551

cos cos3z wT re
+

2 3 BS: Si eek 9) ARON JO Fh ary So00ne (41),

which are obtained by integrating several times the equation

sins +3 sin2e+4+4sin3%+...=1(7-2), froms=0toz=27,

or the equation deduced from it by writing 7 — for x, and taking the semi-sum of the results.
It will be observed that in the several series to be summed we shall always have sines coming
with odd powers of and cosines with even. Of course, by clearing the series in (24) or (30) in
the way just mentioned we shall increase the convergency of the infinite series in which a part
of f(a) still remains developed.

When 4, or B, decreases faster than any inverse power of m as increases, (as is the case for
instance when it is the ‘" term of a geometric series with a ratio less than 1,) all the terms of

its expansion in a series according to inverse powers of m vanish. In this case, then, f(«) and
its derivatives of all orders are continuous.

20. In establishing the several theorems contained in this Section, it has been supposed that
none of the ere nicer of f(#) which enter into the investigation are infinite. It should be
observed, however, that if f*(«) is the last derivative employed, which only appears under the sign
of integration, it is allowable to suppose that f“(«v) becomes infinite any finite number of times
within the limits of integration. To show this, we have only got to prove that

[ro sin yeda or [F) cos vada
0 0

approaches zero as its limit as y increases beyond all limit. Let us consider the former of these
integrals, and suppose that f*(x) becomes infinite only once, namely, when v =a, within the limits
of integration. Let the interval from 0 to a be divided into these four intervals 0 toa -(, a-¢€
to a, a to a+, a+’ to a, where ¢ and C are supposed to be taken sufficiently small to
exclude all values of w lying between the limits a—¢ and a+ ¢’ for which f*~'(«) alters discon-
tinuously, or for which f*(«) changes sign, unless it be the value a. For the first and fourth
intervals f“(«) is not infinite, and therefore, as it is known, the corresponding parts of the integral
vanish for y= co. Since sin yw cannot lie beyond the limits +1 and —1, and is only equal to
either limit for particular values of w, it is evident that the second and third portions of the
integral are together numerically inferior to 7, where

Dm Afi hae a) oC) te Ne a fe (a.t a),
the symbol 4 ~ B denoting the arithmetical difference of A and B, and ¢ being an infinitely small
quantity, so that f(a —e), f(a +e) denote the limits to which f(#) tends as w tends to the limit a
by increasing and decreasing respectively. Hence the limit of the integral first considered, for

y=, must be less than 7. But J may be made as small as we please by diminishing ¢ and
(G and therefore the limit required is zero.

The same proof applies to the integral containing cos ya, and there is no difficulty in extend-
ing it to the case in which f*(«) is infinite more than once within the limits of integration, or at
one of the limits.

21. It has hitherto been supposed that the function expanded in the series (3) or (22) does
not become infinite; but the expansions will still be correct even if f(w) becomes infinite any
finite number of times, provided that /f(#) dw be essentially convergent. Suppose that f(#) be-
comes infinite only when a=a. ‘Then it is evident that we may find a function of w, /’(w), which
shall be equal to f(w) except when w lies between the limits a — ( and a+ (’, which shall remain

4B2
y

552 Mr. STOKES, ON THE CRITICAL VALUES OF

¢ a+g"
finite from w=a-—( to e=a+(', and which shall be such that f F(w)da = [ ‘ Ff (2) da.
a-¢ a-¢

in the

: * ° A . NTe
Suppose that we are considering the series (3). Then, if C,, be the coefficient of sin
expansion of F'() in a series of the form (3), it is evident that C, will approach the finite limit 4,

lees )
when ¢ and ¢’ vanish, where 4, = Sh Ff(@) sin “x dz. But so long as ¢ and ¢’ differ from
aH

zero the series =C, sin

nT, 5
is convergent, and has F(w) for its sum, and F(x) becomes equal
a

to f(~) when ¢ and @’ vanish, for any value of w excepta. We might therefore be disposed to
conclude at once that the series (3) is convergent, and has f(«) for its sum, unless it be for the
particular value «=a; but this point will require examination, since we might conceive that the
series (3) became divergent, or if it remained convergent that it had a sum different from f(x), when
¢ and ¢’ were made to vanish before the summation was performed. If we agree not to consider
the series (3) directly, but only the limit of the series (5) when g becomes 1, it follows at once
from (7) that for values of a different from a that limit is the same as in Art. 4. For w =a the
limit required is that of $$ f(a—e) + f(a+e)} when e vanishes. If f(w) does not change sign
as wv passes through a the limit required is therefore positive or negative infinity, according as f(x)
is positive or negative ; but if f(w) changes sign in passing through © the limit required may be
zero, a finite quantity, or infinity. The expression just given for the limit may be proved without
difficulty. In fact, according to the method of Art. 4, we are led to examine an integral of
the form

=e ife-D+fe+b} pan =

where ¢ is a constant quantity which may be taken as small as we please, and supposed to vanish
after h. Now by a known property of integrals the above integral is equal to

¢ hdé
\f(a-&) +f(a+ &)} 5 eee?

Bu t [aes ws pe which is equal to tan S, becomes equal to . when h vanishes, and the limit of

1 c
- where &, lies between 0 and ¢.
Tv

& ae h mia must be zero, since it cannot be greater than ¢, and ¢ may be made to vanish
after h.

22. The same thing may be proved by the method which consists in summing the series

. ne NnTre 5 . .
> sin —— sin — — tom terms. If we adopt this method, then so long as we are considering a
a a

value of w different from a it will be found that the only peculiarity in the investigation is, that the
quantity under the integral sign in the integrals we have to consider becomes infinite for one value
of the variable; and it may be proved just as in Art. 20, that this circumstance has no effect on
the result. If we are considering the value w =a, it will be found that the integral we shall have
to consider will be

a en os Cree ee (Chea) Cece

To

where pv is first to be made infinite, and then ¢ may be supposed to vanish. If f(a +e) + f(a —€)
approaches a finite limit, or zero, when ¢ vanishes, as may be the case if f(#) changes sign in
passing through ©, it may be proved, just as in the case in which f(«) does not become infinite,
that the above integral approaches the same limit as 4 }f(a+e)+f(a~e){. In all cases

THE SUMS OF PERIODIC SERIES. 553

however in which f(#) does not change sign in passing through «, and in some cases in which
it does change sign, f(a + «) + f(a—e) becomes infinite when ¢ vanishes.

In such cases put for shortness
24 2a
f(a + — §) +f(a- —) = F@),
Tv vie
sin v&
E

or which is the same those of (NE az taken from 0 to x, from 7 to 27... be denoted by

7 og 29
dé taken from 0 to—, from — to —.,..
v v v

and let the numerical values of the integral i

I,, I,... Then evidently 1, > 1,>J;... Also, if ¢ be sufficiently small, F(&) will decrease from
~=0 to £=% if we suppose, as we may, F(£) to be positive. Hence the integral (42), which
is equal to

{, F(&) —-LF(E,) + Ig F(E) — --.}, .2000e Beppcncoree cere Cay

3 |-

aac : Tv Q7
where £,, & ... are quantities lying between 0 and — , — and — ... is greater than
v v v

1
i {T, F(é,) a I, F (&)t,

if we neglect the incomplete pair of terms which may occur at the end of the series (43), and
which need not be considered, since they vanish when y= ce. Hence, the integral (42) is

1
a fortiori > — (I, - I,) F(é,). But &, vanishes and F(&,) becomes infinite when v becomes infinite ;
Tv

and therefore for the particular value # =a the sum of the first m terms of the series (3) increases
indefinitely with 7.

If a coincides with one of the extreme values 0 and a of x, the sum of the series (3) vanishes
fora =a. This comes under the formula given above if we consider the sum of the series for
values of w lying beyond the limits 0 and a. The same proof as that given in the present and last
article will evidently apply if f(#) become infinite for several values of a, or if the series considered
be (22) instead of (3). In this case, the sum of the series becomes infinite for w= a when
a=0 or =4.

23. Hence it appears that f(v) may be expanded in a series of the form (3) or (22), provided

only [f(«) da be continuous. It should be observed however that functions like (sin “| > Which

become infinite or discontinuous an infinite number of times within the limits of the variable within
which they are considered, have been excluded from the previous reasoning.

Hence, we may employ the formule such as (26), (35), &c., to obtain the direct developement
of f’(r), without enquiring whether it becomes infinite or not within the limits of the variable for
which it is considered. All that is necessary is that f() and its derivatives up to the (x — ri)
inclusive should not be infinite within those limits, although they may be discontinuous.

24. In obtaining the formule of Arts. 7 and 13, and generally the formula which apply to
the case in which A, or B, is given, and f(#) is unknown, it has hitherto been supposed that we
knew a priori that f(#) was a function of the class proposed in Art. 1 for consideration, or at
least of that class with the extension mentioned in the preceding article. Suppose now that we
have simply presented to us the series (3) or (22), namely

* . ar@r % nav
>A, sin —— or B, + SB, cos F
a a

554 Mr. STOKES, ON THE CRITICAL VALUES OF

where 4, or B, is supposed given, and want to know, first, whether the series is convergent,
secondly, whether if it be convergent it is the direct developement of its sum f(), and thirdly,
whether we may directly employ the formule already obtained, trusting to the formule themselves
to give notice of the cases to which they do not apply by leading to processes which cannot be
effected.

25. If the series SA, or SB, is essentially convergent, it is evident a fortiori that the series
(3) or (22) is convergent.
c ‘ Cree ‘ ;
If A, = S —cosny +C,, or if B, =S nn MY +C,, where =C, is essentially convergent, the
n
given series will be convergent, as is proved in Art. 6,
In either of these cases let f(x) be the sum of the given series. Suppose that it is the series
NTL

of sines which we are considering. Let E, be the coefficient of sin

of f(v). Then we have

in the direct developement

Te

f(w) = 24, sin "7" = Sx, sin arte

and since both series are convergent, if we multiply by any finite function of 2, p(x), and integrate,
we may first integrate each term, and then sum, instead of first summing and then integrating.

Taking @(@) = sin exe

als and integrating from #=0 to =a, we get E, = A,, so that the given

series is the direct developement of its sum f(#). The proof is the same for the series of
cosines.

; ; 5 1 5 7
26. Consider now the more general case in which the series = — 4, is essentially convergent.
n

The reasoning which is about to be offered can hardly be regarded as absolutely rigorous ;
nevertheless the proposition which it is endeavoured to establish seems worthy of attention.
Let w, be the sum of the first terms of the given series, and F(m, w) the sum of the first 2 terms
Nav ;

: a
of the series > — —— 4, cos
nr

[nem — Un) de = F(n +m, x) — F(n, x) =\(m, x), suppose. ...... (44).

Then we have

é 4 1 : dee
Now by hypothesis the series = — A, is essentially convergent, and therefore a fortiori the
n

; a Nae,
series > — are A, cos —— is convergent, and therefore (co, #) = 0, whatever be the value of m.
7 a

Let the limits of xv in (44) be # and a + Aw, and divide by Aa, and we get

1 r+Ar Ay (m,
=f (Un 4m — Uy) de = Ay (n, 2) |
hal, Aw

and as we have seen the limit of the second side of this equation when we suppose 7 first to
become infinite and then Aw to vanish is zero. But for general values of « the limit will remain

the same if we first suppose Aw to vanish and then m to become infinite; and on this supposition
we have

limit of (t#,4n—U,) =0, for n = 2;
so that for general values of w the series considered is convergent.

To illustrate the assumption here made that for general values of w the order in which m and
Aw assume their limiting values is immaterial, let (y, 2) be a continuous function of « which

THE SUMS OF PERIODIC SERIES. 555

becomes equal to \/(m; #) when y is a positive integer; and consider the surface whose equation
is =\W(y, ~). Since (co, #) = 0 for integral values of y, the surface approaches indefinitely to
the plane wy when y becomes infinite; or rather, among the infinite number of admissible forms of
W(y, @) we may evidently choose an infinite number for which that is the case. Now the assertion
made comes to this; that if we cut the surface by a plane parallel to the plane ws, and at a
distance m from it, the tangent at the point of the section corresponding to any given value of «
will ultimately lie in the plane ay when 7 becomes infinite, except in the case of singular, isolated
values of vw, whose number is finite between 2 = 0 and w =a. For such values the sum f(#) of the
infinite series may become infinite, while /f(w)da# remains finite. The assumption just made
appears evident unless A, be a function of m whose complexity increases indefinitely with its
rank, i.e. with the value of n.

Since the integral of f(«) is continuous, f(w) may be expanded by the formula in a series

; “ no TO 2 z
of sines. Let E, be the coefficient of sin —— in its direct expansion; so that,
a

. Are
f(«) = 24, sin ——,

a

. are
SE, sin ——,
a

f(@)

where both series are convergent, except it be for isolated values of w. Consequently, we have,
in a series which is convergent, at least for general values of a,

Nee se pepstieh-eterieieri-ereteD))»
|

NTL

0 = (4, - E,) sin alec bees Sanese fala claieis'a (46).
a

The series (45) may become divergent for isolated values of 2, and are in fact divergent for
values of # which render f(a) infinite. But the first side of (46) being constantly zero, and the
series at the second side being convergent for general values of w, it does not seem that it can
become divergent for isolated values. Hence according to the preceding article the second side
of the equation is the direct developement of the first side, i.e. of zero; and therefore EK, = 4,,
or the given series is the direct developement of its sum, which is what it was required to prove,
The same reasoning applies to the series of cosines.

It may be observed that the well known series,

4 + COS @ + COS 2H + COS BM ....e0ssceeereserns deletes (AT)

forms no exception to the preceding observation. This series is in fact divergent for general
values of wv, that is to say not convergent, and in that respect it totally differs from the series in
(46). When it is asserted that the sum of the series (47) is zero except for # = 0 or any multiple
of 27, when it is infinite, all that is meant is that the limit to which the sum of the convergent
series 4 + 2g" cos mw approaches when g becomes 1 is zero, except for # = 0 or any multiple of 27,
in which case it is infinity.

27. It follows from the preceding article that even without knowing a priori the nature of the
function f(v) we may employ the formule such as (35), provided that if m~ be the highest power

1 : 5 4 . : Las
of — required by the formula, and n~"C, the remainder in the expansion of A,, the series > . G.
n

be essentially convergent. For let G, be the sum of the terms as far as that containing ~“ in the
expansion of A,, those terms having the form assigned by (35), that is to say cosines like cos mry

1 é , vi -y 2
coming along with odd powers of —, and sines along with even powers. Then 4, = G, + n-"C,,
n

556 Mr. STOKES, ON THE CRITICAL VALUES OF

Let =G, sin de

we
wail F(a);

then f(#) — F(w) = 3n-"C, sin ane Wife wer ycidy eOeopigy
Now if @(w) = =wu,, where the series Ew,» ee are both convergent, we may find @’(2)
a
by differentiating under the sign of summation. This is evident, since by the theorem referred to
d
in Art. 2 (note), we may find {= de by integrating under the sign of summation. Conse-

quently we have from (48)

#1 =
fe-1(@) — FY-"(a) = +(7) SS cage ie Cae ee ee a ey
a n

: : 1 : ; : :
and since the series =-— C, is essentially convergent, the convergency of the series forming the
n

right-hand side of (49) cannot become infinitely slow (see Sect. III.), and therefore, the n™ term
being a continuous function of w, the sum is also a continuous function of a, and therefore
f"(«) — F*(@) is a function which by Art. 23 can be expanded in a series of sines or cosines.
But F“(«) is also such a function, being in fact a constant, and therefore f*(wv) is a function
of the kind considered in Art. 23, which is what is assumed in obtaining the formula (35).

It may be observed that these results do not require the assumptions of Art. 26 in the case in
which the series 2C,, is essentially convergent, or composed of an essentially convergent series

: ec. c : ° *
and of a series of the form SS’ — sin my or 2S — cos ny, according as C, is the coefficient of a
n n

cosine or of a sine.

SECTION II.

Mode of ascertaining the nature of the discontinuity of the integrals which are analogous
to the series considered in Section I, and of obtaining the developements of the
derivatives of the expanded functions.

28. Let us consider the following integral, which is analogous to the series in (1),

x) ain aed BJO neato AL BOY;

where (8) = = fre’) sin Ba' da’... .25.05000-.(51):

Although the integral (50) may be written as a double integral,

ff fe) sin Ba sin Ba’ dBda'..........0000 (52),

THE SUMS OF PERIODIC SERIES. 557

the integration with respect to # must be performed first, because, the integral of sin Ba sin 3a’ ds
a

not being convergent at the limit ©, He sin 3w sin Bx'dB would have no meaning. Suppose,
0

however, that instead of (52) we consider the integral,

: [fi faye™ sin Baw sin Ba'dBda' .........+-. (53),

where h is a positive constant, and ¢ is the base of the Napierian logarithms. It is easy to see that
at least in the case in which the integral (50) is essentially convergent its value is also the limit to
which the integral (53) tends when h tends to zero as its limit. It is well known that the limit of (53)
when h ies is in general f(wv); but when »=0 the limit is zero; when #=a the limit is } f(a) ;
and when f(2) is ee darian it is the arithmetic mean of the values of f(«) for two values of wv
infinitely little greater and less respectively than the critical value. When w>a it is zero, and in
all cases it is the same, except as to sign, for negative as for positive values of wv.

We may always speak of the limit of (53), but we cannot speak of the integral (50) till we
assure ourselves that it is convergent. Now we get by integration by parts,

. , , 1 , , 1 - , . , 1 , : ,
Sf@) sin Ba da’ = — alo) cos Ba’ + Bt (2’) sin Ba’ — B If'@ ) sin Ba'dz ...... (54).
When this integral is taken between limits, the first term will furnish a set of terms of the form

Cc aN ;
B cos a, where a may be zero, and the last two terms will give a result numerically less than —,

< ; dp .
where ZL is a constant properly chosen. Now whether a be zero or not, [cos Basin Bx “ is

convergent at the limit ©, and moreover its value taken from any finite value of 3 to B = © is
the limit to which the integral deduced from it by inserting the factor ¢~"® tends when h vanishes.
The remaining part of the integral (50) is essentially convergent at the limit oo. Hence the
integral (50) is convergent, and its value for all values of #, both critical and general, is the limit to
which the value of the integral (53) tends when / vanishes.

29. Suppose that we want to find f”(v), knowing nothing about f(x), at least for general
values of w, except that it is the value of the integral (50), and that it is not a function of the
class excluded from consideration in Art. 1. We cannot differentiate under the integral sign,
because the resulting integral would, usually at least, be divergent at the limit co. We may
however find f(a) provided we know the values of # for which f(w) and f'(2) are discontinuous,
and the quantities by which f(#) and f(a) are suddenly increased as w increases through each
critical value, supposing the extreme values included among those for which f(#) or f (0) is
discontinuous, under the same convention as in Art. 6. Let a be any one of the critical values
of w; Q, Q, the quantities by which f(a), f’(w) are suddenly increased as a increases through a;
S the sign of summation referring to the critical values of #; ,(() the coefficient of sin Bw in
the direct developement of f’(«) in a definite integral of the form (50). Then taking the integrals
in (54) between limits, and applying the formula (51) to f’(w), we get

$:(B) = — P’p(B) + - BSQ cos Ba - = 8 sin Ba.

We may find @,(() in a similar manner. We get thus when p» is even

- 1)? ,(B) = Bp(B) -= pe SQ cos Ba + - BY-? SQ, sin Ba + ...

M419 ;
+ (-1) =~ §Q,-1 sin Ba Nekiies vey isis Nich +0e(55),
Vor, VIII. Parr V. 4C

558 Mr. STOKES, ON THE CRITICAL VALUES OF

where sines and cosines occur alternately, and two signs of the same kind are always followed by
two of the opposite. The expression for p* (3) when p» is odd might be found in a’similar manner.
These formule enable us to express f“(#) when @(f) is an arbitrary function which has to be
determined, and f(0), &c. are given.

30. If however (3) should be given, and (0), &c. be unknown, #((3) will admit of expansion
according to powers of 3-1, beginning with the first, provided we treat sin (3a or cos Ba as if it
were a constant coefficient ; and sin (3a, cos Ba will occur with even and odd powers of (3 respec-
tively. The possibility of the expansion of @((3) in this form depends of course on the circum-
stance that @(«) is a function of the class which it is proposed in Art. 1, to consider, or at least
with the extension mentioned in Art. 23. It appears from (55) that in order to express f“(#) as a
definite integral of the form (50) we have only got to expand @((), to differentiate (50) « times
with respect to «, differentiating under the integral sign, and to reject those terms which appear
under the integral sign with positive powers of or with the power 0. The same rule applies
whether « be odd or even.

31. If we have given (a), but are not able to evaluate the integral (50), we may notwith-
standing that find the values of w which render f(@) or any of its derivatives discontinuous, and
the quantities by which the function considered is suddenly increased. For this purpose it is only
necessary to compare the expansion of @(/3) with the expansion

2

(8) Sea SQ cos Ba - ae SQ, sin Bae peeces seers (56),

Tv

given by (55), just as in the case of series.

We may easily if we please clear the function @((3) of the part for which f() or any one of
its derivatives is discontinuous, or does not vanish for 7 =0 and «=a. For this purpose it will be
sufficient to take any function /(«) at pleasure, which as well as its derivatives of the orders
considered has got the same discontinuity as f(a) and its derivatives, to develope F(x) in a definite

integral of the form [°%®) sin Bx df by the formula (51), and to subtract Fw) from f(@) and
0

(8) from (@). It will be convenient to choose such simple functions as 7 + ma + na*;
lsinz +mcosa; le~*+me-**, &c, for the algebraical expressions of F'(#) for the seyeral
intervals throughout which it is continuous, the functions chosen being such as admit of easy
integration when multiplied by sin Bada, and which furnish a sufficient number of indeterminate
coefficients to allow of the requisite conditions as to discontinuity being satisfied. These conditions
are that the several values of Q, Q,, &c. shall be the same for F'() as for f(«).

@
32. Whenever il f(#) da is essentially convergent, we may at once put a= © in the
0

preceding formule. For, first, it may be easily proved that in this case, (though not in this case
only,) the limit of (53) when h vanishes is f(#); secondly, the limit of (53) is also the value of
(52); and, lastly, all the derivatives of f(w) have their integrals, (which are the preceding
derivatives,) essentially convergent, and therefore co may be put for a in the developements of the
derivatives in definite integrals.

When f(a) tends to zero as its limit as # becomes infinite, and moreover after a finite value
of w does not change from decreasing to increasing nor from increasing to decreasing,

fre f@) sin Bada’

Pee

THE SUMS OF PERIODIC SERIES. 559

will be more convergent than [ f(#’) sin Bx'da’, and the latter integral will be convergent, and
0

its convergency will remain finite* when 6 vanishes. In this case also we may put a = ©.

Thus if f(x) = sin Jx(6° + «*)-", we may put a = o because f(w) has its integral essentially
convergent: if f(w) = (b+ )~4, we may put a= © because f(x) is always decreasing to zero
as its limit. But if f(@) = sin /w (b + w)~4, the preceding rules will not apply, because f(«),
though it has zero for its limit, is sometimes increasing and sometimes decreasing. And in fact in
this case the integral in equation (51) will be divergent when $=/, and p(B) will become infinite
for that value of 6. It is true that f(z) is still the limit to which the integral (53) tends when
h vanishes; but I do not intend to enter into the consideration of such cases in this paper.

33. When oo may be put for a, and f(w) is continuous, we get from (55)

(= 1) u(B) = BY p(B) ~ = A F(0) + = BF" O) ~ 02 + (—1)= BPO). sovnee (57).

In this case @((3) will admit of expansion, at least to a certain number of terms, according to
odd negative powers of 3. If we suppose @(() known, and the expansion performed, so that

(8) = H,B- + HB" + HB + ...

and compare the result with (49), we shall get
f(0) = aa Oy es = = Hs: f'(0) = = Hy: ere, aan (58).
34. The integral
i Wh((6)) @OR(EMICTE) coodecoccianaccecon concen toe Sedgsansnnbogaons- (59),

where W(B) = = (Fe) (aay BWC HA” Vac eegeedesa seancobneceomeosocece ((i8)\,

which is analogous to the series (22), is another in which it is sometimes useful to develope a function
or conceive it developed. For positive values of w the value of (59) is the same as that of (50).
When w =0 the value is f(0); and for negative values of w it is the same as for positive. It is
supposed here that the integral (59) is convergent, which it may be proved to be in the same
manner as the integral (50) was proved to be convergent.

Suppose that we wish to find, in terms of (3), the developement of f*(w) in a definite
integral of the form (50) or (59), according as « is odd or even. We cannot differentiate under
the integral sign, because the resulting integral would be divergent. We may however obtain the
required developement by transforming the expression w (8) by integration by parts, just as before.
We thus get for the case in which u is odd

wth
(-1)* 4(B) = BY (B) + = Bt SQ sin Ba + = B** SQ, 00s Ba - «.
1

+(-1)* : SQ, zinta seo (61),

where @, (3) is the value of @() in the direct developement of f*() in the integral (50). In the
same way we may get the value of yy,(3) when u is even, wy, (/3) being the value of y/() in the
direct developement of f*(w) by the formule (59), (60).

* See next Section,

4C2

560 Mr. STOKES, ON THE CRITICAL VALUES OF

The equation (61) is applicable to the case in which y/(f) is an arbitrary function, and a,
Q, &e., are given. If however \/((3) should be given, we may find @, (8) or y,(B) by the same
rule as before.

In the case in which y/(8) is given, we may find the values of a, Q, &c., without being able to
evaluate the integral (59). For this purpose it is sufficient to expand (3) according to negative
powers of (3, and compare the expansion with that furnished by equation (61).

35. The same remarks as to the cases in which we are at liberty to put cc for @ apply to (60)
as to (51), with one exception. In the case in which f(«) approaches zero as its limit, and is at
last always decreasing numerically, or at least never increasing, as # increases, while [f(«) dw is
divergent at the limit co, it has been observed that @((3) remains finite when # vanishes. This
however is not the case with (3), at least in general. I say in general, because, although

c
[ f(a) da increases indefinitely with its superior limit, we are not entitled at once to conclude from
=0
oa
thence that [ cos (3x f(x) dx becomes infinite when 3 vanishes, as will appear in Section III. It may
0

be shown from the known value of [a eos Bada, where 1 >” > 0, that if f(a) = F(a) + Ca,
0

where F'(w) is such that [F(«) da is convergent at the limit co, y/((3) becomes infinite when 6
vanishes; and the same would be true if there were any finite number of terms of the form Ca~".
There is no occasion however to enquire whether \, (3) always becomes infinite: the point to consider
is whether the integral (59) is always convergent at the limit zero.

In considering this question, we may evidently begin the integration relative to a’ at any
value w, that we please. Suppose first that we integrate from a’ = 2, to a’ = X, and let w(8) be
the result, so that

aw (3) = = (fe) cos Bx’ da’.

Let w,() be the indefinite integral of w(8)d: then, ¢ being a positive quantity, we get from
the above equation

,

@,(B) - (ec) = = [* Fe) Ssin Bw’ — sin cat =

sin Ba’
a
remains finite (Art. 39.) when ( vanishes, as may be proved without much difficulty, its value

cannot become infinite, and therefore a ,(/3) does not become infinite when (3 vanishes. Now

fa(B) cos Ba dB = w,(R) cos Bx +x fa,(B) sinBedp, ......... (62),

when « is positive; and when w = 0,

Ja (6) d(B) = w, (8):
hence in either case fw(Q) cos Bwdf is convergent at the limit zero. Now the quantity by
which a (3) differs from y,((3) evidently cannot render (59) divergent, and therefore in the case
considered the integral (59) is convergent at the limit zero.

da’ is a convergent integral, and its convergency

Now put X=c. Then since f f@)

By treating a) e~"® cos Bad in the manner in which fa(B) cos Pwd is treated in

0
(62), it may be shown that the convergency of the former integral remains finite when / vanishes.
Hence, not only is the integral (59) convergent, but its value is the limit to which the integral
similar to (53) tends when h/ vanishes.

THE SUMS OF PERIODIC SERIES. 561

When f(z) is continuous, and ¢« may be put for a, we have from (61)
Bel B+ 2

(1) QB) =P) + = BPO) = = BPO) +e t(D

us

Bf*-?(0). .-. (63).

If y() be given we can find the values of f’(0), f’"(0) ... just as before.
36. ‘The integral

= [> [cos B (a! ~ 2) f(a!) dB ae’, et eee Gaye

cumichy © Ce

Tv

in which the integration with respect to 2’ is supposed to be performed before that with respect
to 8, so that the integral has the form

[x®) cos Ba dB + fo) SN /GRGYE — ssacesuoncossesosy (i)

may be treated just as the integral (59); and it may be shown that in the same circumstances we
may replace the limits — a, and a by — c, + 0 respectively. If we suppose x() and o(()
known, we may find as before the values of w for which f(a), f’(#)... are discontinuous, and the
quantities by which those functions are suddenly increased. We may also find the direct develope-
ment of f’(v), f”(x)... in two integrals of the form (65); and we may if we please clear the
integrals (65) of the part which renders f(«), f’(#) ... discontinuous.

37. In the developement of f(w) in an integral of the form (50) or (59), or in two integrals
of the form (65), it has hitherto been supposed that f(#) is not infinite. It may be observed
however that it is allowable to suppose f(#) to become infinite any finite number of times, provided
[f(a) dx be essentially convergent about the values of # which render f(«) infinite. This may be
shown just as in the case of series. Hence, the formule such as (55) which give the develope-
ment of f“(a) are true even when f“(@) is infinite, f*~'(#) being finite.

SECTION III.

On the discontinuity of the sums of infinite series, and of the values of integrals
taken between infinite limits.

38. Ler TERS, Sata Cale TLL, SER REYE AT ee eed Sescjeceees (OO);
be a convergent infinite series having U for its sum. Let

Dat DMs tek (ete eater hs ieee a nncls chcineciessan'aensicnel(Al)s

be another infinite series of which the general term v, is a function of the positive variable h, and
becomes equal to uw, when h vanishes. Suppose that for a sufficiently small value of 4 and all
inferior values the series (67) is convergent, and has V for its sum. It might at first sight be
supposed that the limit of V for h = 0 was necessarily equal to U. This however is not true. For
let the sum to m terms of the series (67) be denoted by f(m, h): then the limit of V is the limit
of f(n, h) when n first becomes infinite and then h vanishes, whereas U is the limit of f(m, h) when
h first vanishes and then m becomes infinite, and these limits may be different. Whenever a dis-
continuous function is developed in a periodic series like (15) or (30) we have an instance of this ;
but it is easy to form two series, having nothing to do with periodic series, in which the same

562 Mr. STOKES, ON THE CRITICAL VALUES OF

happens. For this purpose it is only requisite to take for f(m, h) — U,, (U, being the sum of the
first n terms of (66),) a quantity which has different limiting values according to the order in which
n and h are supposed to assume their limiting values, and which has for its finite difference a
quantity which vanishes when becomes infinite, whether h be a positive quantity sufficiently small
or be actually zero.

For example, let

2nh
= = BSE en Tidinelic heed cewise thee 68),
Fash) — 0, = (68)
which vanishes when 2 = 0. ‘Then
2h

Af f(ash) — Un} = Par — M1 = Ga ay Gh dB aT)"

1 1
Assume U,=1- » so that wu, = AU,_, = ———_,,
nm+1 m (n+ 1)
and we get the series
1 1 a
toe Eterm eUertic's cinedinceiccites ser swetelde 69),
1.22.3 ard er ern ae (69)
1+5h h(h+2)n°+h(4—h 1-h
a ( ) ( a A. (70).

2(1+h)" n(n+1)f{(m-Dh4+1t (nh+1) tenes

which are both convergent, and of which the general terms become the same when fh vanishes.
Yet the sum of the first is 1, whereas the sum of the second is 3.

If the numerator of the fraction on the right-hand side of (68) had been prh instead of
2nh, the sum of the series (70) would have been p +1, and therefore the limit to which the sum
approaches when f vanishes would have been p+1. Hence we can form as many series as we
please like (67) having different quantities for the limits of their sums when hf vanishes, and yet
all having their m™ terms becoming equal to w, when h vanishes. This is equally true whether the
series (66) be convergent or divergent, the series like (67) of course being always supposed to be
convergent for all positive values of h however small.

39. It is important for the purposes of the present paper to have a ready mode of ascertaining
in what cases we may replace the limit of (67) by (66). Now it follows from the following theorem
that this substitution may at once be made in an extensive class of cases.

Turorem. The limit of V can never differ from U unless the convergency of the series (67)
become infinitely slow when h vanishes.

The convergency of the series is here said to become infinitely slow when, if x be the number
of terms which must be taken in order to render the sum of the neglected terms numerically less

than a given quantity e which may be as small as we please, ” increases beyond all limit as A
decreases beyond all limit.

Demonstration. If the convergency do not become infinitely slow, it will be possible to find
a number 7, so great that for the value of & we begin with and for all inferior values greater than
zero the sum of the neglected terms shall be numerically less than e. Now the limit of the sum of
the first 2, terms of (67) when h/ vanishes is the sum of the first 2, terms of (66). Hence if e’ be the
numerical value of the sum of the terms after the 7," of the series (66), U and the limit of V cannot
differ by a quantity so great ase+e’. But e and e’ may be made smaller than any assignable
quantities, and therefore U is equal to the limit of V.

Cor. 1. If the series (66) is essentially convergent, and if, either from the very beginning, or

after a certain term whose rank does not depend upon h, the terms of (67) are numerically less than
the corresponding terms of (66), the limit of V is equal to U.

THE SUMS OF PERIODIC SERIES. 563

For in this case the series (67) is more rapidly convergent than (66), and therefore its
convergency remains finite.

Cor. 2. If the series (66) is essentially convergent, and if the terms of (67) are derived from
those of (66) by multiplying them by the ascending powers of a quantity g which is less than 1,
and which becomes 1 in the limit, the limit of V is equal to U.

It may be observed that when the convergency of (67) does not become infinitely slow when h
vanishes there is no occasion to prove the conyergency of (66), since it follows from that of (67).
In fact, let V,, be the sum of the first 2 terms of (67), U,, the same for (66), V, the value of V for
h=0. Then by hypothesis we may find a finite value of m such that V — V,, shall be numerically
less than e, however small h may be; so that

V=V,, + a quantity always numerically less than e.

Now let A vanish: then V becomes V, and V,, becomes U,. Also e may be made as small as we
please by taking m sufficiently great. Hence U, approaches a finite limit when 2 becomes infinite,
and that limit is V,.

Conversely, if (66) is convergent, and if U=V,, the convergency of the series (67) cannot
become infinitely slow when / vanishes.

For if U,/, V,/ represent the sums of the terms after the 2" in the series (66), (67) respectively,
we have
V2 VeVi, O= 0,4 U7;

whence V, =V-U-(V,-—U,) + U;.

Now V—U, V,-—U, vanish with h, and U,/ vanishes when m becomes infinite. Hence for a
sufficiently small value of f and all inferior values, together with a value of m sufficiently large, and
independent of h, the value of V,’ may be made numerically less than any given quantity e however
small; and therefore, by definition, the convergency of the series (67) does not become infinitely
slow when h vanishes.

On the whole, then, when the convergency of the series (67) does not become infinitely slow
when h vanishes, the series (66) is necessarily convergent, and has V, for its sum: but in the
contrary case there must necessarily be a discontinuity of some kind, Either VY must become infinite
when h vanishes, or the series (66) must be divergent, or, if (66) is convergent as well as (67),
U must be different from Vy.

When a finite function of 2, f(#), which passes suddenly from M to WN as « increases through a,
where a > a > 0, is expanded in the series (15) or (30), we have seen that the series is always
convergent, and its sum for all values of w except critical values is f(#), and for w =a its sum is
4(M+N). Hence the convergency of the series necessarily becomes infinitely slow when a — @
vanishes.’ In applying the preceding reasoning to this case it will be observed that h is a —a,
V,is M, and Uis 4(M+N), if we are considering values of @ a little less than a; but h is a —a
and V, is N, if we are considering values of # a little greater than a.

When the series (66) is convergent, as well as (67), it may be easily proved that in all cases
U=V,-L,
where L is the limit of V,’ when h is first made to vanish and then m to become infinite.

40. Reasoning exactly similar to that contained in the preceding article may be applied to
ao
integrals, and the same definitions may be used. ‘Thus if | F'(w, h) dw is a convergent integral,
a

we may say that the convergency becomes infinitely slow when h vanishes, when, if X be the
superior limit to which we must integrate in order that the neglected part of the integral, or

564 Mr. STOKES, ON THE CRITICAL VALUES OF

[°F@, h) da, may be numerically less than a given constant e which may be as small as we
x

please, X increases beyond all limit when h vanishes.

The reasoning of the preceding article leads to the following theorems.

If V= ify F(w,h) da, if V, be the limit of V when kh = 0, and if F(x, 0) = f(x); then, if

a
the convergency of the integral V do not become infinitely slow when h vanishes, Hi f(x) de must
a
be convergent, and its value must be V,. But in the contrary case either V must become infinite

c ]
when h vanishes, or the integral [ f() da must be divergent, or if it be convergent its value
a

must differ from V,.

When the integral [°F@) dw is convergent, if we denote its value by U, we shall have in all

cases

a, ds

where ZL is the limit to which if F(«, h) da approaches when h is first made to vanish and then
x
X to become infinite.

The same remarks which have been made with reference to the convergency of series such as
(15) or (30) for values of 2 near critical values will apply to the convergency of integrals such
as (50), (59) or (65).

The question of the convergency or divergency of an integral might arise, not from one of the
limits of integration being oo , but from the circumstance that the quantity under the integral sign
becomes infinite within the limits of integration. The reasoning of the preceding article will
apply, with no material alteration, to this case also.

41. It may not be uninteresting to consider the bearing of the reasoning contained in this
Section and a method frequently given of determining the values of two definite integrals, more
especially as the values assigned to the integrals have recently been called into question, on account
of their discontinuity.

Consider first the integral

®sinax
u =f Dy, Jae seesesencenecencesiens ses (71),

0 av

where a is supposed positive. Consider also the integral

© sinaa
v= if ee dx.
CY)

av

It is easy to prove that the integral v is convergent, and that its convergency does not become
infinitely slow when h vanishes. Consequently the integral w is also convergent, (as might also be
proved directly in the same way as in the case of v,) and its value is the limit of w for h = 0.
But we have

dv “ A &
—=-f e~**sinavde = — ——;
dh ; a’ + h*

h
whence » =C—tan'!-;
a

THE SUMS OF PERIODIC SERIES. 565

: : Z 7
and since v evidently vanishes when h = ©, we have C = =. whence

T = T
v = — — tan > u=-,
2 2
7 : : : : F
Also u = 0 when a = 0, and uw = — = when @ is negative, since « changes sign with a. By the

x sin an
da when

value of « for a = 0, which is asserted to be 0, is of course meant the limit ie

a is first made to vanish and then X made infinite.

It is easily proved that the convergency of the integral « becomes infinitely slow when a
vanishes. In fact if

2 sinaax
u = i da,

x av

we get by changing the independent variable

» sin v
w= [ dau:
bee Gi

but for any given value of X, however great, the value of uw’ becomes when a vanishes if

© sine

dx,

an integral which might have been very easily proved to be greater than zero even had we been
unable to find its value. It readily follows from the above that if «’ has to be less than e the value
of X increases indefinitely as a approaches to zero.

42. Consider next the integrals
® Cos anda hy £08 auvdax
w= f : Jaeiec —————— , seccecccccesccssee (72):
Sita) + a 0 1+?

It is easily proved that the convergency of the integral v does not become infinitely slow when
h vanishes, whatever be the value of a. Consequently w is in all cases the limit of v for h = 0.
Now v satisfies the equation

dv

da*
It is not however necessary to find the general value of »; for if we put h=0 we see that w
satisfies the equation

a h
SiMe =f e-"* cosaada = — Pad’ apa cposoaasaceccde 4 Ue) E

ad?
ca w= 05 PRRGAN: cst eeeey DOREE Oa ae (74),

so long as a is kept always positive or always negative: but we cannot pass from the value of wu
found for positive values of a to the value which belongs to negative values of a by merely writing
-a for a in the algebraical expression obtained. For although w is a continuous function of a, it

Gn. é : r
readily follows from (73) that aa discontinuous. In fact, we have from this equation

r
(3°) = (<") -f vda — 2tan~! —
dG} cn \A@) ga-y A h

A e .
Now let / first vanish and then A. Then v becomes uw, and ih vda yanishes, since v does not

ot. VIII. Pant V. 4D

566 Mr. STOKES, ON THE CRITICAL VALUES OF

become infinite for @ = 0, whether h be finite or be zero. Threfore = is suddenly decreased by
a :

7 as @ increases through zero, as might have been easily proved from the expression for w by means
of the known integral (71), even had we been unable to find the value of w in (72). The equation
(74) gives, a being supposed positive,

u = Ce7 + Ce",

vw Tv e . .
whence C’=0, C= so) oa e~*. Also, since the numerical value of w is unaltered when the

; : 7 P :
sign of a is changed, we have w= ; e* when a is negative.

It may be observed that if the form of the integral «w had been such that we could not have
inferred its value for a negative from its value for a positive, nor even known that zw is not infinite
for a=—, we might yet have found its value for @ negative by means of the known continuity

‘ du F é
of w and discontinuity of os when a@ vanishes. For it follows from (74) that w = C,e7+C,e~* for a
a

F : 7 30
negative; and knowing already that w = 5 e~* for a positive, we have

7 T
g Oi t Cos a re tae

7 T 3
whence C, = Fu C,=0, w= 3 e’, for a negative.

Of course the easiest way of verifying the result wv = 3 e~" for a@ positive is to develope e~* for

x positive in a definite integral of the form (59), by means of the formula (60).

SECTION IV.
Examples of the application of the formule proved in the preceding Sections.

43. Brrore proceeding with the consideration of particular examples, it will be convenient
to write down the formule which will have to be employed. Some of these formule have been
proved, and others only alluded to, in the preceding Sections.

In the following formula, when series are considered, f(a) is supposed to be a function of «
which, as well as each of its derivatives up to the (1 — 1)™ order inclusive, is continuous between
the limits # = 0 and w = a, and which is expanded between those limits in a series either of sines

Te nT v

or of cosines of and its multiples. 4, denotes the coefficient of sin when the series is one

Nee

of sines, B,, the coefficient of cos when the series is one of cosines, 4," or B," the coefficient of

av

_ are nre . : ane : P -
sin —— or cos in the expansion of the »'" derivative. When integrals are considered, f(«)
a

THE SUMS OF PERIODIC SERIES. 567

and its first « — 1 derivatives are supposed to be functions of the same nature as before, which
are considered between the limits 2 =0 and # = <3; and it is moreover supposed that f(x)

decreases as # increases to ©, sufficiently fast to allow Ov to be essentially convergent at the.
limit ©, or else that f(v) vanishes when 2 = &, and after a finite value of x never changes from
increasing to decreasing nor from decreasing to increasing. (8) or y/(B) denotes the coefficient

of sin Bx or cos Bw in the developement of f(x) in a definite integral of the form is PB sin Brdz
0

or i (RB) cos Bada, P,() or y,,(B) denotes the coefficient of sin Ba or cos Ba in the deve-
0

lopement of the »™ derivative of f(w). The formulz are
(-1? B= (7) 4,-= (=) fO- Cove}
ed ed TO ek OO) ee CR)

a\a

(aa = (M2) 2 (22) LO = (FY + oo even esceesnnnee(B)

I

Z yt 7 ("2)"2. 3 g ee $f(0) -(-'f' @} - .. odd) pecreceeeeen (C)s

2 nam\* 2 (nm\"* 5
(-1)7 BY = (=| ae = (*) {f' (0) — (= 1)"f'(a)} — «2. (ue even)..eeesseneee (D),
except when = 0, in which case we have always
Bs =~ {FO =f},
a
B, being the constant term in the expansion of f*(w) in a series of cosines.
In the formule (4), (B), (C), (D) we must stop when we have written the term containing

the power 1 or 0, (as the case may be,) of “*. The formule for integrals are
(= 1)? (B) = BYP B) ~ = BF (0) + = BAF") ~ 2a (He ODW)eveveeserreene()
(1? GB) = BP (B) — = BF) += BEF) = oes (H ev0D) voeveeseene ,
(= 1) 7 (8) = BB) += BPO) — = PF" 0) + oo OFM) srreeserenl

(= 1)24,(B) = BY (B) + = BE2F (0) ~= Pr" (0) + 22. (u CED) vereeseeseeal)s

where we must stop with the last term involving a positive power of 3 or the power zero.

44. As a first example of the application of the principles contained in Sections I. and IT,
suppose that we have to determine the value of @ for values of « lying between 0 and a, 0 and
respectively, from the equation

d’ d’
ci + - ? o\ Olsens teosh bowessspaicosve= evanekt (0)s
daw’ dy

4D2

568 Mr. STOKES, ON THE CRITICAL VALUES OF
with the particular conditions

—" =w(v—-4a), when y=0 or =b............(76),

—* = —w(y-4b), when 2 =0 or =@............ (77).

This is the problem in pure analysis to which we are led in seeking to determine the motion
of a liquid within a closed rectangular box which is made to oscillate,

For a given value of y, the value of @ can be expanded in a convergent series of cosines of

7" and its multiples; for another value of y, @ can be expanded in a similar series with different
a

coefficients, and so on. Hence, in general, @ can be expanded in a convergent series of the form

nTe
= Y,, cos

a

where Y,, is a certain function of y, which has to be determined.
In the first place the value of @ given by (78) must satisfy (75). Now the direct developement
d F : : : Hs :
of ae in a series of cosines will be obtained from (78) by differentiating under the sign of sum-
y

ad
mation ; the direct developement of a will be given by the formula (D). We thus get

PY, nr 2 nar v
> _ Yo —— 31 (= 3)" —1b)} cos =0;
{Grn Yet 2 - (=F - gO} cos ™

and the left-hand member of this equation being the result of directly developing the right-hand

member in a series of cosines, we have
@Y, nx
dy" a’

according as m is odd or even. This equation is easily integrated, and the integral contains two

arbitrary constants, C,, D,, suppose. It only remains to satisfy (76). Now the direct developement

4
w= -—(y- $b) or =0

>

Y,
of rH will be obtained by differentiating under the sign of summation, and the direct develope-

4 ,
ment of w(# — 4a) is easily found to be - 3, _ cos , the sign =, denoting that odd values
7 a
only of m are to be taken. We have then, both for y = 0 and for y = b,
ay, bs 40a Re
dy ~ an? aa

according as- is odd or even, which determines C’, and D,,.

It is unnecessary to write down the result, because I have already given it in a former paper ™*,
where it is obtained by considerations applicable to this particular problem. The result is con-
tained in equation (4) of that paper. The only step of the process which I have just indicated
which requires notice is, that the term containing (« — 4a) (y— 4b) at first appears as an infinite

* Supplement to a Memoir ‘ On-some Casesof Fluid Motion,’ p. 409 of the present Volume.

THE SUMS OF PERIODIC SERIES. 569

series, which may be summed by the formula (41). The present example is a good one for
showing the utility of the methods contained in the present paper, inasmuch as in the Supplement
referred to I have pointed out the advantage of the formula contained in equation (6), with respect
to facility of numerical calculation, over one which I had previously arrived at by using develope-
ments, in series of cosines, of functions whose derivatives vanish for the limiting values of the
variable.

45. Let it be required to determine the permanent state of temperature in a rectangle which
has two of its opposite edges kept up to given temperatures, varying from point to point, while
the other edges radiate into a space at a temperature zero. The rectangle is understood to be a
section of a rectangular bar of infinite length, which has all the points situated in the same line
parallel to the axis at the same temperature, so that the propagation of heat takes place in two
dimensions.

Let the rectangle be referred to the rectangular axes of #, y, the axis of y coinciding with one
of the edges whose temperature is given, and the origin being in the middle point of the edge.
Let the unit of length be so chosen that the length of either edge parallel to the axis of w shall be
x, and let 2(3 be the length of each of the other edges. Let « be the temperature at the point
(x, y), h the ratio of the exterior, to the interior conductivity. Then we have

cu au

da + dy? SOM std oa\nininnss siecle, ee eoraeie uelave (79);

- —hu = 0, when y = — [B.....-....00006 (80);

du

dy eh r= 05, whenjy = B..eem; seme cee -as (81),
ts f(4)) se WHEN ti 350) eaepemeteseris as Seer GES
um = Fy), when @ = @)...-..-..-.- Sonor ant (83),

f(y), Fy) being the given temperatures of two of the edges.

According to the method by which Fourier has solved a similar problem, we should first
take a particular function Ye”, where Y is a function of y, and restrict it to satisfy (79). This
gives Y = Acos\y + Bsin)y, A and B being arbitrary constants. We may of course take, still
satisfying (79), the sum of any number of such functions. It will be convenient to take together
the functions belonging to two values of X which differ only in sign. We may therefore take, by
altering the arbitrary constants,

u = ZhJA(AO-? — e-M7-*)) 4 Ble — c-”)} cosry,
+ SPO (ARH Ma Meo) 4 DE — es) SIN AY... .geeceees (84),

in which expression it will be sufficient to take only one of two values of X which differ only by
sign, so that \, if real, may be taken positive. Substituting now in (80) and (81) the value of
given by (84), we get either C = 0, D = 0, and

NBR EADIN Sims Br aioecedeecccceesisos ose 000805
or else A = 0, B=0, and

AB.cotrAB = —h.......00. Dnesiseanaaiaes (86).

It is easy to prove that the equation (85), in which \/3 is regarded as the unknown quantity,

: wey :
has an infinite number of real positive roots lying between each even multiple of = , including zero,

570 Mr. STOKES, ON THE CRITICAL VALUES OF

and the next odd multiple. The equation (86) has also an infinite number of real positive roots
lying between each odd multiple of ; and the next even multiple. The negative roots of (85) and

(86) need not be censidered, since the several negative roots have their numerical values equal to
those of the positive roots; and it may be proved that the equations do not admit of imaginary
roots. The values of \ in (84) must now be restricted to be those given by (85) for the first line,
and those given by (86) for the second. It remains to satisfy (82) and (83). Now let

fy) + f(-y =2f.%, FY) -f(-Y¥) = 2f2(y);

F(y) + F(-y) =2FiQ), FY) - #(-y) =2F.):

then we must have for all values of y from 0 to 8, and therefore for all values from — £ to 0,

TAL cosry=fi(y), ZBL cosry = F,(y).----- eee ee (87),
SCM sin py =f2(y), TZDMsin wy = F,(y) «.--..-. ee. (88),
where Z, = Could _ e773 M = et" — eres

» denoting one of the roots of the equation

uB.cot up =- LiBinsesivache ster ess sax(89))

and the two signs = extending to all the positive roots of the equations (85), (89), respectively.
To determine A and B, multiply both sides of each of the equations (87) by cos \’ydy, ’ being
any root of (85), and integrate from y=0toy= (3. The integral at the first side will vanish, by

1
virtue of (85), except when X’ =A, in which case it will become ri (2A 6 + sin 2A), whence A

and B will be known. C and D may be determined in a similar manner by multiplying both sides
of each of the equations (88) by sin u’ydy, m’ being any root of (89), integrating from y = 0 to
y = 3, and employing (89). We shall thus have finally

u = 4>A(2AB + sin2rB)—*(e*" — e~ 2")? § (Ae) — e-Me-9) MO) cosdydy
AY —Ar B
+(e” — e7”) f F,(y) cos ry dy} cos dy,
0

+ 43 (2up — sin 2nB)-(e" — e-#)-? { (ett — eM)" f(y) sin uy dy

B : :
+(e" —e) f F,(y) sinpydy} sin py .....-..- (90).

46. Such is the solution obtained by a method similar to that employed by Fourier. A
solution very different in appearance may be obtained by expanding w in a series ZY sin mx, and
employing the formula (B). We thus get from the equation (79)

ay Qn

— -v Y+— -—(-1)'F(y)} =0

ie = {f(y) - (- WD" F(y)} = 9,
which gives

tae . : np hes
Y=Ae'l + Be7"™! — eal ff¥y) -(- 1)"F(y')t (9-7) — e"-9)) dy’;
0

THE SUMS OF PERIODIC SERIES. 571
d
whence, ou = SY’ sin naw, where

Gj
Y'=nAe¥ —nBe-"d — =) {f(y’) -(- 9 FYy)} (6-4 9) dy’.
0

The values of A and B are to be determined by (80) and (81), which require that

dY
—— = =
dy hY =0 when y = +3.

We thus get
(n+ hye A-(n— hye B-~ [FY — (I FY)} [4 A) COM 4 (n= hy eP-N} dy 0,

and the equation derived from this by changing the signs of h and 8; whence the values of
A and B may be found. We get finally

b= SY sm na, 222-020. BCAA an cacti aSeE CECE (91),
where

F=—finshyer—(n— meron en) [fm thyeO 4 (w= hy en
thy’) — (- "Fy (y’)t dy’

fs : ie (e"o-v) my) e-"(9-9)) Sh (7) - (- 1)’F, (y')t dy

+ a §(m + h) e" + (nm — h) e- "Pt! (e"Y — e-"Y”) Me {(m +h) e"®-Y) + (mn —h) eg Pony
thy’) — (- 1)" F.(y')} dy’

- feo = eY-P) $F.(y) = (= 1) FY} dy, scceccesceecesceccescee (92).

47. The two expressions for w given, one by (90), and the other by (91) and (92), are
necessarily equal for values of w and y lying between the limits 0 and 7, — and 2 respectively.
They are also equal for the limiting values y= — and y=, but not for the limiting values
wv =0 and w=7, since for these values (91) fails; that is to say, in order to find from this series
the value of wu for ~= 0 or x =7, we should have first to sum the series, and then put v=0
or ®=T.

The comparison of these expressions leads to two remarkable formule. In the first place it
will be observed that the first and second lines in the right-hand side of (92) are unchanged when
y changes sign, while the third and fourth lines change sign with y. This is obvious with respect
to the first and third lines, and may be easily proved with respect to the second and fourth by
taking — 4’ instead of y' for the variable with respect to which the integration is performed, and
remembering that f,(y), /’\(y) are unchanged, and f,(y), F'.(y) change sign, when y changes sign.
Consequently the part of w corresponding to the first two lines of (92) is equal to the part expressed
by the first two in (90), and the part corresponding to the last two lines of (92) equal to the part
expressed by the last two in (90). Hence the equation obtained by equating the two ex-
pressions for w splits into two; and each of the new equations will again split into two in con-
sequence of the independence of the functions f, F, which are arbitrary from y=0 to y = p.
As far however as anything peculiar in the transformations is concerned, it is evident that we may
suppress one of the functions f, F', suppose J’, and consider only an element of the integral

572 Mr. STOKES, ON THE CRITICAL VALUES OF

by which f is developed, or, which is the same, suppose f,(y’) or f.(y’) to be zero except for values
of the variable infinitely close to a particular value y’, and divide both sides of the equation by

SAY) dy’ or fiy)ay’.
We get thus from the first two lines of (90) and the first two of (92), supposing y and y’ pesitive,
and y/ the greater of the two,
4 ei @-*) a e~hr-)

20\B8+sn201B &™=e%*

1 $ (e”! os e~"!) (nm +h) et B-y) + (n a h) e "F-¥)t

igs (n +h) e"® — (n—h)e~*

where the first & refers to the positive roots of (85), and the second to positive integral values
of n from 1 to @.

cos Ay cos \2/

sinna, ... (93),

Of course, if y become greater than y’, y and y’ will have to change places in the second side
of (93). This is in accordance with the formula (92), since now the second line does not vanish ;
and it will easily be found that the first and second lines together give the same result as if we had
at once made y and y’ change places. Although y has been supposed positive in (93), it is easily
seen that it may be supposed negative, provided it be numerically less than y/.

The other formula above alluded to is obtained in a manner exactly similar by comparing the
last two lines in (92) with the last two in (90). It is
Aw elz—#) s eThr—-2)

= , sin py sin wy’
2u.8 —sin2pn6 Cat eet uy oa

1 s (ec! — 7") J(m + h)c"8-Y + (m — h)e2P-¥
Big (n + he" + (nm — h)e'?

where the first > refers to the positive roots of (89), the second to positive integral values of 7,
and where x is supposed to lie between 0 and 7, y’ between 0 and (, y between 0 and y’, or, it
may be, between —y' and y’. Although # has been supposed less than 7, it may be observed
that the formulz (93), (94) hold good so long as w, being positive, is less than 27,

SIMD pees se (94),

48. Let it be required to determine the permanent state of temperature in a homogeneous
rectangular parallelepiped, supposing the surface kept up to a given temperature, which varies
from point to point.

Let the origin be in one corner of the parallelepiped, and let the adjacent edges be taken for
the axes of w, y, x. Let a, b, ¢ be the lengths of the edges; f\(y, x), Fi(y, #), the given tem-
peratures of the faces for which w = 0 and « = a respectively; f,(%, v), F(z, v) the same for the
faces perpendicular to the axis of y; f;(x, y), F3(a, y) the same for those perpendicular to the
axis of x. Then if we put for shortness v to denote the operation otherwise denoted by

a ha ad
—++35
dx‘ dy dz’
as will be done in the rest of this paper, and write only the characteristics of the functions, we
shall have, to determine the temperature w, the general equation yw«=0 with the particular
conditions
u=fy, wheng=0; w= Fy, when & = G......,0c050008+0+(95) 3
u =f, when y=0; wu =F), when y = B.......0ceee+e--0-(96) 5

% = fxs when ¥=05 °° = Fog when & = C..2ecpes abibeamaa (OTs

THE SUMS OF PERIODIC SERIES. 573

It is evident that w is the sum of three temperatures w,, w,, «;, where uw, satisfies the conditions
(95), and vanishes at the four remaining faces, and w,, u, are related to the axes of y, x as u, is
related to that of z, each of the quantities w,, w., u; representing a possible permanent temperature.

- nT 4 .
sin ane where Z,,, is a function

Now u; may be expanded in a double series 2=Z,,, sin aod
a

of x which has to be determined. Let for shortness

mar nr pr
iE —=), — =
a b
then the substitution of the above value of w, in the equation Vw, = 0 leads to the equation
BZaw
dz a GF Zrn = 0,

where g? =p» + v*, which gives Z,,, = A,,,¢% + B,,,¢~%3; and the constants 4,,, B,,, are easily
determined by the condition (97). We may find w, and w, in a similar manner, and the sum of
the results gives w. It is thus that such problems are usually solved.

We may, however, expand wu in a series of the form =~ Z,,, sin zw sin vy, even though it does
not vanish for 2 = 0 and # =a, and for y=0 and y=6; and the formule proved in Section I.
enable us to make use of this expansion.

Let then u = ZZ sin pe sin vy,
the suffixes of Z being omitted for the sake of simplicity. We have by the formula (B)
Pu
dx

Let f,(y, =) —-(-1)"F\(y, x) be expanded in the series SQsinvy by the formula (3), so that
Q will be a known function of x, m, and m. Then

2 0
= Sf -wW>Z sin vy + Ch —(-1)"F,}} singe.

2p. ; F
sini Srf{-wWZ+ 783 sin we sin vy.

Pu s ats du. :
The value of Tail be expressed in a similar manner, and that of dat 8 found by direct
y

differentiation. We have thus, for the direct developement of yu, the double series
LZ bysite 2Qv 2

Sa = P es 7 j p

23 {oo (w+ v)Z 4 Fi tah @} sina sin vy

where P is for « what Q is for y. The above series being the direct developement of Yu, and Yu
being equal to zero, each coefficient must be equal to zero, which gives

LZ
dz q

where g means the same as before. The integral of the equation (98) is

2

2
Z + = Pie ZQ6 ye, W609)

a

1 z 1 Zz
Z = Ac™ + Be% —-e" | e PT dz+-e°"% | &Tdz,
ree oe
2T' denoting the sum of the last two terms of (98). It only remains to satisfy (97). If the
known functions f,(a, y), F(a, y) be developed in the double series 22G sin ww sin vy,
SEA sin pe sin vy, we shall have from (97)
A+B<=G,
1 bd 1 ~
ae “a _ _ gfe -@Tdz +-e~% °Tds = H.
Ac” + Be ass Tdx + -6 ifr

0 0

Vox. VIII. Panr V. 4 E

574 Mr. STOKES, ON THE CRITICAL VALUES OF
A and B may be easily found from these equations, and we shall have finally

1 z ; "|
(ef — e-) Z =G (et) — ce -2°-) 4 A (c® — 6 @) +. gfitiaa = ea i (c¥ — e-¥) T'dz'!
0

1 c ' ,
+ Fs a e#) “fh (e’-*? =} equa T dz i
z

T’ being the value of 7’ when s = =’. It will be observed that the letters Z, P, Q, T, A, B, G, H
ought properly to be affected with the double suffix mm. It would be useless to write down the
expression for w in terms of the known quantities f\(y, #), &c.

It will be observed that « might equally have been expressed by means of the double series
=z X,, sin vy sin@s, or 2 Vey sin zw sing, where p is any integer. We should thus have
three different expressions for the same quantity w within the limits w = 0 and a =a, y=0 and
y=b, x=0 and x=c. The comparison of these three expressions when particular values are
assigned to the known functions f,(y, *) &c. would lead to remarkable transformations. The
expressions differ however in one respect which deserves notice. Their numerical values are the
same for values of the variables lying within the limits 0 and a, 0 and b, 0 and c. The first
expression holds good for the extreme values of z, but fails for those of w and y: in other words,
in order to find from the series the value of w for the face considered, instead of first giving # or y
its extreme value and then summing, which would lead to a result zero, we should first have to
sum with respect to m or m, or conceive the summation performed, and then give # or y its extreme
value. The same remarks apply, mutatis mutandis, to the second and third expressions; so that
the three expressions are not equivalent if we take in the extreme values of the variables.

49. Many other remarkable transformations might be obtained from those already referred
to by differentiation and integration. We might for instance compare the three expressions which

a 6 Cc
would be obtained for ch ah i! udzdydzx, and we should thus have three different expressions
0 0 0

for the same function of the three independent variables a, b, c, which are supposed to be positive,
but may be of any magnitudes. Some examples of the results of transformations of this kind may
be seen by comparing the formule obtained in the Supplement alluded to in Art. 44 with the
corresponding formule contained in the Memoir itself to which the Supplement has been added.
Such transformations, however, when separated from physical problems, are more curious than
useful. Nevertheless, it may be worth while to exhibit in its simplest shape the formula from
which they all flow, so long as we restrict ourselves to a function w satisfying the equation Vu = 0,
and expanded between the limits w = 0 and # =a, &c. ina double series of sines.

The functions f,(y, ¥) &c., which are supposed known, are arbitrary, and enter into the
expression for w under the sign of double integration. Consequently we shall not lose generality,
so far as anything peculiar in the transformations is concerned, by considering only one element of
the integrals by which one of the functions is developed. Let then all the functions be zero
except f,; and since in the process f, has to be developed in the double series

4 me b , U7 . , ’ s
a >>>) J J, Fe(v’, y’) sin pa’ sin vy'da'dy’ . sin wa sin vy,

consider only the element f,(2’, y) sin x2 sin vy'da'dy’ of the double integral, omit the da’ dy’,
and put f,(«’, y’) = 1 for the sake of simplicity. If we adopt the first expansion of w, and put
q° for n* + v*, we shall have :

ae a
Z = A(et?-*) — e-1-*), (6 — e-*) A = = sin wae sin vy’;
a

4 elle) a e7uc-2)

pie USE ee pis ree x
whence ue >>>» Ge ge Sin ma’ sin vy sin wa sin UYsdnved>o-te ---(99).

THE SUMS OF PERIODIC SERIES. 575

By expanding w in the double series }2Y sin yp» sin wz we should get

2 w (ee! — e eb) dod e—8b-y¥) , ’
u = =z— d ; sin wv’ sin wa sin we .........(100),
ac s e’—e *

where s* =? + @?, and y/ is the greater of the two y, y’, The third expansion would be derived
from the second by interchanging the requisite quantities. In these formule x may have any
positive value less than 2c.

.

We should get in a similar manner in the case of two variables w,

2 e’ (4-7) e7Y (2-2) 1 (HY s e*Y) (et (o-¥) _ EHO)
Sr

eg Pe ga sinvy’ sin vy = —= Sin w, -.~ | (101);

a —-va —
ES a re)

where w is supposed to lie between 0 and a, y’ between 0 and b, and y between 0 and y’. ‘This
formula is however true so long as & lies between 0 and 2a, and y between — y/' and y’.

b pb pa
If we compare the two expressions for f if dh udydy'dax obtained from (101), taking =, for
(oy 0

the sign of summation corresponding to odd values of m from 1 to ©, putting a = rb, and
2

1
replacing >, af by its value = 5 we shall get the formula

ut
1 hee ei pope. ew a
2 =, Piers et nD = Tae = gr cerceeereteeeehese nee (102),
Il+e 7”

which is true for all positive values of 7, and likewise for all negative values, since the left-hand
side of (102) is not changed when — 7 is put for r. In integrating the second side of (101), sup-
posing that we integrate for y before integrating for y’, we must integrate separately from y = 0
to y=y’, and from y=y' to y=5, since the algebraical expression of the quantity to be integrated
changes when y passes the value /’.

It would be useless to go on with these transformations, which may be multiplied to any
extent, and which cease to be useful when they are separated from physical problems to which they
relate, and of which we wish to obtain solutions.

It may be observed that instead of supposing, in the case of the parallelepiped, the value of
u known for all points of the surface, we might have supposed the value of the flux known, subject
of course to the condition that the total flux shall be zero. This would correspond to the follow-
ing problem in fluid motion, w taking the place of the quantity usually denoted by @, “To
determine the initial motion at any point of a homogeneous incompressible fluid contained in a
closed vessel of the form of a rectangular parallelepiped, which it completely fills, supposing the
several ‘points of the surface of the vessel suddenly moved in any manner consistent with the
condition that the volume be not changed.” In this case we should expand w in a series of cosines
instead of sines, and employ the fomula (D) instead of (B). We might, again, suppose the value
of w known for the faces perpendicular to one or two of the axes, and the value of the flux known
for the remaining faces. In this case we should employ sines involving the co-ordinates perpendi-
cular to the first set of faces, and cosines involving the others.

The formule would also be modified by supposing some one or more of the faces to move off
to an infinite distance. In this case some of the series would be replaced by integrals. Thus, in
the case in which the value of w at the surface is known, if we supposed @ to become infinite we
should employ the integral (50) instead of the series (3), as far as relates to the variable w, and the:
formula (b) instead of (B). If we were considering a rectangular bar infinitely extended both
ways we should employ the integral (65). Of course, if we had already obtained the result for the

4E2

576 Mr. STOKES, ON THE CRITICAL VALUES OF

case of the parallelepiped, the shortest way would be thence to deduce the result for the case of the
bar infinite in one or in both directions, but if we began with considering the bar it would be best
to start with the integrals (50) or (65).

50. To give one example of transformations of this kind, let us suppose 6 to become infinite

: now 7 ; me
in (101). Observing that y= —-, Av =-—, we get on passing to the limit
g b b 8 P g
2 w ¢’ (2-2) ep 1 J P
fi = z— sin vy’ sin vydy = —Z(e*¥ — 7") e-"Y sinpw. ... (108).
TJ, e — en a

Multiply both sides of this equation by dvdy, and integrate from # = 0 to # =a, and from y =0
toy= ©. With respect to the integration of the second side, it is only necessary to remark that
when y becomes greater than y’, y and y’ must be made to change places in the expression written
down in (103). As to the integration of the first side, if we first integrate from y =0 to y = F,
we get, putting f(v, w) for the fraction involving 2,

a pe ; ; d
—f f(y, 2) sin vy’ (1 — cos of).
wy v

Now let ¥ become infinite; then the term involving cos y ¥ may be omitted, not because cos vy ¥
vanishes when Y becomes infinite, which is not true, but because, as may be rigorously proved,
the integral in which it occurs vanishes when Y becomes infinite. If we write 1 for a, as we may
without loss of generality, we get finally

v

SR wen GCI 1 ;
SIG ey (1) — 05,7 ) pp ons anipicis-innp erie nen (LOK):
i l+e” y vy 7 07

51. Hitherto in satisfying the general equation Vw =0, together with the particular conditions
at the surface, the value of w has been expanded in a double series involving two of the variables,
and the functions of the third variable which enter as coefficients into the double series have been
determined by an ordinary differential equation such as (98). We might however expand w in
a triple series, and thus satisfy at the same time the equation Vw =0 and the conditions at the
surface, without using an ordinary differential equation at all, but simply by means of the terms
introduced into the series by differentiation, which are given by the formule at the beginning of
this Section; and then by summing the triple series once, which may be done in any one of three
ways, we should arrive at the same results as if we had employed in succession three double
series, involving circular functions of 2 and y, y and x, x and « respectively, and the corresponding
ordinary differential equations. I am indebted for this method to my friend Prof. William
Thomson, to whom I showed the method given in Art. 48.

Let us take the case of the permanent state of temperature in a rectangular parallelepiped,
supposing the temperature at the several points of the surface known. For more simplicity
suppose the temperature zero at the surface, except infinitely close to the point (a’, y’) in the face
for which x = 0, so that all the functions f, &c. are zero, except f,(w, y), and f,(«, y) itself zero
except for values of a, y infinitely close to a’, y’ respectively ; and let {/f;(~, y) dwdy =1, provided
the limits of integration include the values v = «’, y= y'. Let w be expanded in the triple series

TTA pnp SIN wa SIN vY SIN WR, --+-0..02--- AAP (OES).
where wu, v, @ mean the same as in Art. 48. Then
du 2

he Zp} — Za 2nW’ Anny Sin pa sin vy + Fee f(a, y)} sinws. ...... (106).

: eral ga he es ake - : :
Now the expansion of f,(#, y) in a double series is ab == sin pa’ sin vy’ sin ww sin vy, that is to
a

THE SUMS OF PERIODIC SERIES. 577

say with this understanding, that the result is to be substituted in (106); for it would be absurd to
speak, except by way of abbreviation, of a quantity which is zero except for particular values of
a and y, for which it is infinite. The values of —— and ee

da’ dy’
entiation. We have therefore for the direct developement of yw in a triple series

will be obtained by direct differ-

; OT Ts : é : :
Vu = SEZ} (w+ 0? + Dw’) Mn + abe it uo’ sinvy’t sinww sin vy sin we,

.
But yz being equal to zero, each coefficient will have to zero, from whence we get A,,,,, and then
8 Dw i aiite mee : f
— == >> = sina’ sinvy’ singe sin vy sin We. ...... (107).
abe w+ +o

One of the three summations, whichever we please, may be performed by means of the known
formule
_o sin Ds c eh (o—*) = ea kc—)
w+ ie 78 ke ¢—ke
Ae 7 k cos vy, A 6 ee +e Few)
2h kh? + 2 Ge ee
which may be obtained by developing the second members between the limits s = 0 and z =e,
y,=0 and y = 6 by the formule (2), (22), and observing that the expansions hold good within
the limits written after the formule, since e*'°~*) — ¢~ “~* has the same magnitude and opposite
signs for values of x equidistant from ec, and e*?~”) 4 ¢~*-”) has the same magnitude and sign for
values of y equidistant from 6. If in equation (107) we perform the summation with respect to p,
by means of the formula (108), we get the equation (99): if we perform the summation with respect
to n, by means of the formula (109), we get the equation (100).

SWORDS S50 sh5cc anos sen (OS)

5 EN) Sy SS Aopen (CU)

52. ‘The following problem will illustrate some of the ideas contained in this paper, although,
in the mode of solution which will be adopted, the formule given at the beginning of this Section
will not be required.

A hollow conducting rectangular parallelepiped is in communication with the ground: required
to express the potential, at any point in the interior, due to a given interior electrical point and to
the electricity induced on the surface,

Let the axes be taken as in Art. 48. Let a’, y’, x’ be the co-ordinates of the electrical point, m
the electrical mass, v the required potential. Then v is determined first by satisfying the equation

MBA Bisade
vv = 0, secondly by being equal to zero at the surface, thirdly by being equal to z infinitely

close to the electrical point, r being the distance of the points (a, y, %), (2, y', *), and by being
finite and continuous at all other points within the parallelepiped.

Let v =— + ,, so that v, is the potential due to the electricity induced on the surface.
r P :
Then v, is finite and continuous within the parallelepiped, and is determined by satisfying the
. . m ’ .
general equation yv, = 0, and by being equal to —— at the surface. Consequently v, can be
r

determined precisely as w in Arts. 48 or 51. This separation however of v into two parts seems
to introduce a degree of complexity not inherent in the problem ; for v itself vanishes at the
surface; and it is when the function expanded vanishes at the limits that the application of the
series (2) involves least complexity. On the other hand we cannot immediately expand v in a
triple series of the form (105), on account of its becoming infinite at the point (a’, y', %’)-

'
578 Mr. STOKES, ON THE CRITICAL VALUES OF

Suppose, therefore, for the present that the electricity is diffused over a finite space: then
it is evident that we may suppose the electrical density, p; to change so gradually, and pass so
gradually into zero, that the derivatives of v, of as many orders as we please, shall be continuous
functions. We may now suppose v expanded in a triple series, so that

v= ZEZA ha sin u@ sin vy sin we ;
and we shall have
Vv = — TEDW + FP + @) Ann, Sin wa sin vy sin ws.
But we have also, by a well-known theorem, Vv = — 4p; and

p= 2ZZR,,, sin wae sin vy sin we,

aad sthond p sin wa’ sin vy’ sin wx da! dy’ ds’,

p being the same function of a’, y’, 3’ that p is of w, y, x. We get therefore by comparing the two

where Rin, =

abe
expansions of Vv

Aninp = 40 (ue +P + @)* Ranys
whence the value of v is known. We may now, if we like, suppose the electricity condensed
into a point, which gives

8m . ine Pas i
mnp = ~~ SIN pa’ sin vy! sin we’,
abe

32am
Tee VTE (we + 2 + @*)—'sin pa’ sin py’ sin ws’ sinwsin vysinws.........(110).
a

One of the summations may be performed just as before. We thus get, by summing with
respect to p,
8am it (e* — e~#) (et"- a ed Wins

ab q ef — «2

sin wa’ sin vy’ sin wae sin vy......(111),

where q° =," + v*, and s is supposed to be the smaller of the two x, x’. If = be greater than x’,
we have only to make x and x’ change places in (111).

53. The equation (110) shows that the potential at the point (v, y, x) due to a unit of electri-
city at the point (a’, y’, =) and to the electricity induced on the surface of the parallelepiped is equal
to the potential at the point (a’, 7’, x’) due to a unit of electricity at the point (a, y, x) and to the
electricity induced on the surface. This however is only a particular case of a general theorem
proved by Green *.

Of course the parallelepiped includes as particular cases two parallel infinite planes, two paca
infinite planes cut at right angles by a third infinite plane, &c. The value of v being known, the
density of the induced electricity at any point of the surface is at once obtained, by means of a
known theorem.

If we suppose a ball-pendulum to oscillate within a rectangular case, the value of fp belonging
to the motion of the fluid which is due to the direct motion of the ball and to the motion reflected
from the case can be found in nearly the same manner. The expression reflected motion is here
used in the sense explained in Art. 6 of my paper, ‘On some Cases of Fluid Motion+.” In the
present instance we should expand @ in a triple series of cosines.

54, Let a hollow cylinder, containing one or more plane partitions reaching from the axis to
the curved surface, be filled with homogeneous incompressible fluid, and made to oscillate about its

* Essay on Electricity, p. 19. + See p. LL of the present Volume.

THE SUMS OF PERIODIC SERIES. 579

axis, both ends being closed: required to determine the effect of the inertia of the fluid on the
motion of the cylinder.

If there be more than one partition, it will evidently be sufficient to consider one of the sectors
into which the cylinder is divided, since the solution obtained may be applied to the others. In
the present case the motion is such that wda# + vdy + wdz, (according to the usual notation,)
is an exact differential dp. The motion considered is in two dimensions, taking place in planes
perpendicular to the axis of the cylinder. Let the fluid be referred to polar co-ordinates 7, 9 in a
plane perpendicular to the axis, 7 being measured from the axis, and @ from one of the bounding
partitions of the sector considered, being reckoned positive when measured inwards. Let the radius
of the cylinder be taken for the unit of length, and let a be the angle of the sector, and w the
angular velocity of the cylinder at the instant considered. It will be observed that a = 27
corresponds to the case of a single partition. Then. to determine fp we have the general equation

2 2
ee
dr ridr rd

= (1) Bec Eee eee eee (ENF

with the conditions

1d
ee wr, when 0 =0 or = GiSaamanosuegureascs can (EN

rd0
dp
ae OMe Lert —Wllirver\teste rete stam tisanionelaaee eetesen (14)
and, that @ shall not become infinite when 7 vanishes.

Let r = e~*, and take 0, d for the independent variables; then (112), (113), (114) become

LS eel vanes < aiun siaphaweinioniunisisna sicieip si eiciteiie “ee
ant de? a1):
“t hens when OG —1OWOn ="aeceences weclececee ee (116),
d

sf =o, WHET A =0)9. wicinaistetctase sisioaaeceaese Marine ac os (117).

Let be expanded between the limits @ = 0 and @ = a in a series of cosines, so that

A,, A, being functions of >. Then we have by the formula (D) and the condition (116) applied
to the general equation (115)
PA,

= 0
dr* ;
aA na * 20
—~ — (— } A,- — {1 -(- 1)"}e" = 0;
d»* (**) a t ( dhe
whence Ay = AA + By,
ae nny 1-(-1)"
N= Ane aan a af pe)" ( JF om,

n* a — 4a”

Since @ is not to be infinite when 7 vanishes, that is when X becomes infinite, we have in the

first place 4, =0, A,=0. We have then by the condition (117)

2
8wa

"eae? — ba)”

580 Mr. STOKES, ON THE CRITICAL VALUES OF

when n is odd, and B, = 0 when is even. If then we omit B,, which is useless, and put for \
its value, we get

The series multiplied by r? may be summed. For if we expand sin2 (0 —}a) between the
limits 9 = 0, @ = a in a series of cosines, we get

8acosa no O
i z ast 3 ssi Slee u ‘
sin (20 —a) = ca a ie a
= nawO
7 * COS
whence = 8wa’d r° sin (20 — a)......,..(120).

ee
°na(n?n —4a°) 2cosa

In determining the motion of the cylinder, the only quantity we care to know is the moment
of the fluid pressures about the axis. Now if the motion be so small that we may omit the square
of the velocity we shall have, putting @ = — wf(r, 8),

iG w(é) i fr, @),

where p is the pressure, y(#) a function of the time ¢, whose value is not required, and where
the density is supposed to be 1, and the pressure due to gravity is omitted, since it may be taken
account of separately. The moment of the pressure on the curved surface is zero, since the
direction of the pressure at any point passes through the axis. The expression (119) or (120) shows
that the moments on the plane faces of the sector are equal, and act in the same direction; so that
it will be sufficient to find the moment on one of these faces and double the result. If we consider
a portion of the face for which @=0 whose length in the direction of the axis is unity, we shall

d
have for the pressure on an element dr of the surface aft 0)dr; and if we denote the whole

moment of the pressures by — o
in the direction of @ positive, we shall have

reckoned positive when it tends to make the cylinder move

, c=2f fer, 0)rdr.

Taking now the value of f(r, 0) from (120), and performing the integration, we shall have

1

TE Se roa (ESI)
(nm — 2a)nr(nz + 2a)® G2)

1
C = — tana —- 16a°S,
4

The mass of the portion of fluid considered is da; and if we put

C= dak”,

. 89
and write — for a, so that s may have any value from 0 to 4, we shall have

~

x? 1 ‘ sr 88" 1
= — tan — =
St

SL END ECTS ES hse roe dneusot Leis
2 mw °m=s)n(n+s)P (2)

THE SUMS OF PERIODIC SERIES. 581
When s is an odd integer, the expression for k’? takes the form oo — ©, and we shall easily find

4 88° 1
w « "(n—s)n(n+s)

Apooceontiadcancdce.( (EE).

where all odd values of 7 except s are to be taken.

The quantity &’ may be called the radius of gyration of the fluid about the axis. It would
be easy to prove from general dynamical principles, without calculation, that if & be the corre-
sponding quantity for a parallel axis passing through the centre of gravity of the fluid, h the
distance of the axes

in fact, in considering the motion of the cylinder, which is supposed to take place in two dimen-
sions, the fluid may be replaced by a solid having the same mass and centre of gravity as the fluid,
but a moment of inertia about an axis passing through the centre of gravity and parallel to the
axis of the cylinder different from the moment of inertia of the fluid supposed to be solidified.
If K’, K be the radii of gyration of the solidified fluid about the axis of the cylinder and a parallel
axis passing through the centre of gravity respectively, we shall have

If we had restricted the application of the series and the integrals involving cosines to those
cases in which the derivative of the expanded function vanishes at the limits, we should have

expanded @ in the definite integral { (0, B) cos BXdB, and the equation (115) would have
Pp § q
0

given
(8; B) = E(B)e” + x(B)e™,

& x denoting arbitrary functions, which must be determined by the conditions (116). We should
have obtained in this manner

¢Pl0—4a) _ @~PO-}a)

4m 7? 1
p = ait Bt) te) cos ( log~) dGirentacce ceo. (126),

32 — dp

k? = eS ee
1 + e~** 3(B* + 4)? eee ewe weee eee enee

ma
It will be seen at once that k* is expressed in a much better form for numerical computation by
the series in (122) than by the integral in (127). Although the nature of the problem restricts
a to be at most equal to 27, it will be observed that there is no such restriction in the analytical
proof of the equivalence of the two expressions for @, or for k”.

In the following table the first column gives the angle of the cylindrical sector, the second
the square of the radius of gyration of the fluid about the axis of the cylinder, the radius of the
cylinder being taken for the unit of length, the third the square of the radius of gyration of the
fluid about a parallel axis passing through the centre of gravity, the fourth and fifth the ratios of
the quantities in the second and third to the corresponding quantities for the solidified fluid.

Vor. VIII. Panr V. 4F

582 Mr. STOKES, ON THE CRITICAL VALUES OF

ee K

*50000 "05556
"45385 ‘03179
*39518 "03492
"84775 "07442
*31057 “13044
*28101 "18261
"25703 *21700
*23720 “22858

*22051 "22051

55. When a is greater than 7, itewill be observed that the expression for the velocity which
is obtained from (119) becomes infinite when 7 vanishes. Of course the velocity cannot really
become infinite, but the expression (119) fails for points very near the axis. In fact, in obtaining
this expression it has been assumed that the motion of the fluid is continuous, and that a fluid
particle at the axis may be considered to belong to either of the plane faces indifferently, so that
its velocity in a direction normal to either of the faces is zero. The velocity obtained from (119)
satisfies this latter condition so long as a is not greater than w. For when a <7 the velocity
vanishes with r, and when a = z the velocity is finite when r vanishes, and is directed along the
single plane face which is made up of the two plane faces before considered.

But when a is greater than a the motion which takes place appears to be as follows. Let
OABC be a section of the cylindrical sector made by a plane perpendicular to the axis, and
cutting itin O. Suppose the cylinder to be turning round O in the direction indicated by the
arrow at B. Then the fluid in contact with OA and near O will be flowing, relatively to OA,

a

towards O, as indicated by the arrow a. When it gets to O it will shoot past the face OC; so
that there will be formed a surface of discontinuity Oe extending some way into the fluid, the fluid

THE SUMS OF PERIODIC SERIES. 583

to the left of Oe and near O flowing in the direction 40, while the fluid to the right is nearly at
rest. Of course, in the case of fluids such as they exist in nature, friction would prevent the
velocity in a direction tangential to Oe from altering abruptly as we pass from a particle on one side
of Qe toa particle on the other; but I have all along been going on the supposition that the fluid
is perfectly smooth, as is usually supposed in Hydrodynamics. The extent of the surface of
discontinuity Oe will be the less the smaller be the motion of the cylinder; and although the
expression (119) fails for points very near OQ, that does not prevent it from being sensibly correct
for the remainder of the fluid, so that we may calculate kh” from (122) without committing a
sensible error. In fact, if yy be the angle through which the cylinder oscillates, since the extent
of the surface of discontinuity depends upon the first power of ry, the error we should commit
would depend upon y*. I expect, therefore, that the moment of inertia of the fluid which would
be determined by experiment would agree with theory nearly, if not quite, as well when a> as
when a < 7, care being taken that the oscillations of the cylinder be very small.

As an instance of the employment of analytical expressions which give infinite values for
physical quantities, I may allude to the distribution of electricity on the surfaces of conducting
bodies which have sharp edges.

56. The preceding examples will be sufficient to show the utility of the methods contained in
this paper. It may be observed that in all cases in which an arbitrary function is expanded
between certain limits in a series of quantities whose form is determined by certain conditions to be
satisfied at the limits, the expansion can be performed whether the conditions at the limits be
satisfied or not, since the expanded function is supposed perfectly arbitrary. Analogy would lead
us to conclude that the derivatives of the expanded functions could not be found by direct differ-
entiation, but would have to be obtained from formule answering to those at the beginning of this
Section. If such expansions should be found useful, the requisite formulae would probably be
obtained without difficulty by integration by parts. This is in fact the case with the only
expansion of the kind which I have tried, which is that employed in Art. 45. By means of this
expansion and the corresponding formule we might determine in a double series the permanent
temperature in a homogeneous rectangular parallelepiped which radiates into a medium whose
temperature varies in any given manner from point to point; or we might determine in a triple
series the variable temperature in such a solid, supposing the temperature of the medium to vary in
a given manner with the time as well as with the co-ordinates, and supposing the initial temperature
of the parallelepiped given as a function of the co-ordinates. This problem, made a little more
general by supposing the exterior conductivity different for the six faces, has been solved in
another manner by M. Duhamel in the Fourteenth Volume of the Jowrnal de Ecole Polytechnique.

Of course such a problem is interesting only as an exercise of analysis.
G. G. STOKES.

ADDITIONAL NOTE.

Ir the series by which r* is multiplied in (119) had been left without summation, the series
which would have been obtained for k would have been rather simpler in form than the series
in (122), although more slowly convergent. One of these series may of course be obtained from
the other by means of the developement of tan w in a harmonic series. When s is an integer,
k* can be expressed in finite terms. The result is

ki? =8 8-'-* log.24+8 87! m~* f2-' 447... 4 (8-1) ~7} + 4ar-® fo-8 447%... 4+(8—1)-*}— 4, (8 odd),
Ki? w 8a-*7-* §1-1 + 87! + (8 —1)7'$ +4 {1-2 +3-°... + (8 - 1)7*} -4. (seven).

A : ; as reo ake. :
Moreover when 28 is an odd integer, or when a = 45°, or = 135°, &c., ke” can be expressed in
finite terms if the sum of the series 1-2 + 5~* + 97* + ... be calculated, and then be regarded as
a known transcendental quantity.

4F2

XLI. A Mathematical Theory of Luminous Vibrations. By the Rev. J. CHALuIs,
M.A., F.R.A.S., Plumian Professor of Astronomy and Experimental Philo-
sophy in the University of Cambridge.

[Read March 6, 1848.]

In three preceding communications to this Society I endeavoured to explain some of
the principal phenomena of Light on the Hypothesis of Undulations, regarding the «ther as
a continuous and elastic fluid, and applying to it the usual Hydrodynamical Equations. I
propose now, on the same principles, to investigate the particular nature of the ztherial vibrations
which produce light, and the laws of their propagation under given circumstances. As this com-
munication is intended to be supplementary to the three former, I shall take occasion to advert
to any reasoning they contain, to which I may be able to add elucidation or confirmation.

1. Let a?(1+s) be the pressure at any point wy# of the ether at any time ¢, s being
a small numerical quantity, the powers of which above the first are neglected; and let w, v, w,
be the resolved parts of the velocity at the same point and at the same time, in the directions
of the axes of co-ordinates. Then, retaining only the first powers of w, v, w, we have, as is known,

ds du ds dv ds dw

?,— + —=0, (1 2, —+4+—=0, (2 7.— + — =0, (3

Ce ga gp NO a, Biigy PaO On rnhie gat Geer @
ds du dv dw

d —+— + —+— =0. 2... seentet sooscnceoe 4

sa dt ? da HF dy dz (4)
The last of these equations gives by means of the other threc,

ee eS d’s “*) a i

dé J da * dy? * ds 10 Sean yo seeats coe sacs sel ).

Suppose, for the moment, that s has been obtained from this equation by integration. Then
for the velocities we have,

ds , d. fsdt
=c— a? |— =c-a@. mat fe nals onveeaas vase stves 6
uw=Cc-a we dt=c-a ae (6),
ds dfsdt
v=c —a’ |— dt =¢ -— @. = aeis.eeee sco gsocnOEre (7),
fag dy )
wo 9 es » Gfedt
w=c —-@ fat =e -a@ ae ae Maule ooistioivectevaceeeen (8) 3

wnere ce, c, and ec” are functions of co-ordinates only. It is to be observed that these values
of u, v, w are perfectly general, being obtained prior to any consideration of the way in which
the fluid was put in motion, and consequently apply to all points of the fluid in every instance of
motion in which powers of the velocity and condensation above the first may be neglected. Now
the motions of the ztherial medium are vibratory, or, at least, not permanent. There is no known
cause to produce motions in the ather, which either wholly or in part remain permanently the
same at the same points of space for any length of time. And even if, from causes with which

Proressorn CHALLIS, ON A MATHEMATICAL THEORY, ETC. 585

we are unacquainted, such motion should exist, provided it be small compared with the velocity «a,
we may abstract from it in considering vibratory motion. This will appear as follows. Since
equation (5) is linear, we may suppose s to be composed of parts due to separate causes, among
which may be included the cause that produces the permanent part of the motion. But the
condensation due to this cause being represented by o, we shall have,

a da is de Be @ do dc’ . , do dc’
aie dey CET Raa une ra ar ergs
That is, o is either constant throughout the fluid, or is a quantity of an order already neglected.
Hence the values of , 9%, @ in equations (6 8 in th h b
ne value —, —, — in :
ence the values o dae ag ds in equations (6), (7), (8), remain the same whatever be

the permanent motion. Hence also w—c, v—c’, w—c’, or the parts of the motion which
are not permanent, are the same whatever be c, ec’, ec’. We may, therefore, either put c = 0,
e =0, c’=0; or, suppose wu, v, w, to stand respectively for w—c, v-—c, w—e’.

This being premised, let y= —a°fsdt. Then

wet pee bl yo gull
TGipe made 7 FE?

and uwdax + vdy + wdz = (dy), an exact differential. Also by means of equation (4), we derive,

Gyr (dy Oy ay
oy iii (Z + Ad + 7)=° ponnsndosococde nce (9).

2. 'The motions of the xther which correspond to the phanomena of Light are vibratory,
Hence in treating the Undulatory Theory of Light hydrodynamically, the quantity wda +vdy
+ wdzx must be an exact differential, by what is shewn above, without reference to the manner
in which the fluid was put in motion, the reasoning being prior to, and entirely independent of,
any such considerations. The condition of integrability is to be satisfied generally. One obvious
method of doing this, is to suppose the motion to consist of separate motions which tend to
or from fixed centres, and are functions of the distances from the centres. But the phenomena of
Light do not accord with this supposition, since, instead of spreading equally in all directions
from a centre, it is generally propagated in the form of rays. Another way of satisfying the
condition of integrability in a general manner, is to suppose yy to be the product of two functions
g and f, such that @ does not contain « or y, and f does not contain x. For on these
suppositions,

d d d
udaw + vdy+ was = (SE do + iL dy) + F58 ae,

which is an exact differential of fd with respect to co-ordinates. The consequences of thus
satisfying the condition of integrability, which are of a very remarkable kind, I now procced
to develope.
3. The above values of u,v, w give,
du df dv ef dw _ ,&o
da ‘da’ dy a

Let us now, for a reason that will presently appear, suppose that f does not contain ¢. ‘Then

586 Proressor CHALLIS, ON A MATHEMATICAL THEORY

: 5 do ds f &op :
since W = of = - a’Jsdt, it follows that I ae = -a’s, and aus ai Ee Hence substi
tuting in the equation (4), we obtain,

Cig) tp @& af af

Te = a as + ve (Sa+ oa) p Wen cecuctecesses (10).

Now the nature of the question under consideration requires that this equation should be linear.
Let therefore the coefficient of @ be equal to a constant — 6°. According to this supposition
gd may be a function of x and ¢ only, and f a function of a and y only; as, in fact, they ought
to be, in consequence of suppositions already made on these quantities. Thus equation (10) resolves
itself into the two following,

ap Ep
de ro ie A My eee near ee see ret ccscrsccaces (11),
af af b?
ete ere Md CE eee eee 12),
da * dy’ +e (2)
The equation (11) is transformable into the following:
io e
aude =- 4a — Chee taiteeatn im aiotae ietaiounis sip iste inc Sane ae (3),

in which wu =x + at and v=z—at, (See Peacock’s Examples, p. 466). For convenience sake
he F
put e for re Then, regarding e as a small quantity, the integral of (13) may be obtained in a

series as follows.

fp dp _
Let sida vl 0; then ia a F'(u), and p= F(u) + G(v).

a ‘
Hence sale =e §F(u) + G(v)} approximately.

e = G'(v) +e {F,(u) +u G(v)}

p = F(u) + G(v) +e f{vF\(u) + uw G,(v)} 5 and so on.

Thus =F (wu) + G(v) +e fuF\(u) +uG(v)} + Ss fy? F.(u) + uw? G (v) } + &e.

where F,(u) = [F(u)du, F:(u)=fF, (udu, G,(v) =f G@(v) dv, &e. Each of the functions
F and G separately satisfies the given equation. Let us, therefore, for the purpose of drawing
some inferences from this integral, suppose that F=0. Then,

2 ig 2

$ = G(r) + ewG, (”) +. G, (v) +

us
12). 3

Gy (v) + We. «22.220. (14).

4. It appears by this result that @ does not admit of being expressed exactly so long as the
form of the function G is entirely arbitrary. No inference, therefore, can be drawn from the
integral (14) in its general form. The nature of the series, however, suggests at once a
particular form of G, which gives to @ an exact expression, and which, as we shall see, applies
to our present enquiry; viz. the form Ae”. As we have already introduced the condition that
the velocity and condensation be small, and consequently that @ be small, whatever be x and ¢,

A AE

OF LUMINOUS VIBRATIONS. 587

it is clear that G must be a circular function. Let therefore G(v) = Ac”V-1 + Be“ V-4; or
what is equivalent, let G(v) = mcos(nv +c). Then,

m d.
G, (0) = fmeos (nv +0) do = ™ sin (nos c) = —% Teens.
m n* dv
ie m d’.cos(nv + c)
G, (v = f—sin nvo+e dv = ~~ cos nv +e) = —,——__,—
( ) n ( ) n* ( ) ni dv? >
= me m. m d’.cos(nv + c)
G; (v) = { A cos (nv +c) dv = — im (nv +c) = — 54a ardeal a
ae. — &e
Consequently,
m d.cos(nv+ oe m d’.cos(nv+c) eu*
m cos (nv +c) — — .——-_———- . eu + — .——____—_—~. —_
v= ( ) n* dv + dv NS 2
m d@.cos(uv+c) eu*
ausar 3 + &c.
2 dv Ts okey

m cos }n (»-<) +ct
= mcos jn (x — at) -< (2 +al) +ch
mos {(n — <) « - (» +<) at +c}.

€. 27 e 4c”
Let, now, »-—= —. Then valence ay tt =
n r n r

+4e. We have, therefore, finally,

2 ”
f= mf cose at 1 + +c¢)=y.

>

The velocity in the direction of z is fae. Hence, if m, = — ‘ia

w =m f sin — (s ~at Via +e) Neweneseae Sr tenrad (15).

a since (Art. 3) f. 2D at a’s=0, it follows that
pel Mord at J A 1+ sin 2 (wat 14 1+ +e) orn (16).

he Pc oe meas
It hence appears that the velocity of propagation of the wave whose breadth is A, is a / ‘ 5°

The value of e depends on equation (12). If the velocity of propagation be independent of ),
A 2

er r
we shall have ey =k, a numerical constant, and consequently e =

wv
—_
5. Since equation (9) is linear with constant coefficients, it will be satisfied by the sum

of any number of such values of \) as that just obtained, jf; e, m, A, and c’, being different
for each. Hence we have generally,

588 Proressor CHALLIS, ON A MATHEMATICAL THEORY

yak {fm cos 2% wat 1+ +0}
Tv
. 2a en?
AY 8 w <3 ffm, sin (eat / 14% 40}

1 dy / ern? . Qa / er”
ae a Ae Ne ee pool (ra Seer

It follows, since in each of the terms under the sign = the quantities which are independent
of = and ¢ are at our disposal, that we may satisfy by this integral any state of the fluid in
the direction of x, subject to the limitation that the condensation and velocity are at all times
small. The course of the reasoning shews that the particular form of the function G which
has conducted to the above results, has not been adopted as an analytical artifice, but is really
the only form which determines the velocity of propagation, and gives a definite solution of
the Problem. The particular kind of motion it represents, and the component character of the
whole motion as consisting of an indefinite number of such motions, are accordingly to be regarded
as physically true. These results explain the fact of the composition of light.

6. Before proceeding farther, it will be worth while to compare the foregoing investigation
with that which I have given in my Paper on Luminous Rays. (Cambridge Philosophical Trans-
actions, Vol. vi11. Part 111. p. 363). It may be remarked, that the two investigations agree in
their results, but differ in the course of the reasoning. Inthe Paper referred to, the velocity of
propagation is assumed to be uniform (p. 365), and the form of the function expressing the nature
of the vibrations is deduced from this assumption (p. 368). In the present communication the
form of that function is first obtained by d@ priori reasoning from the Hydrodynamical Equations,
and the uniformity of the rate of propagation is then strictly deduced. The inferences in the
former Paper (p. 365), drawn from the supposition that the velocity of propagation is uniform
when the motion is not small, still hold good. It may also here be remarked, that the consider-
ations in p. 366 on which the arbitrary quantities c, c’, c’ were made to vanish, are superseded
by the more general reasoning in Art. 1. of this Paper.

. I proceed now to the consideration of equation (12), viz.
P q

af af
da dy?

+ 4ef= 0.

As this equation does not contain ¢, there is no propagation of motion in any direction
parallel to the plane of zy; or, the propagation in the direction of x takes place without
lateral spreading. A value of f expressed in finite terms is not therefore required, as in the
case of the integration of equation (11), for deducing velocity of propagation. It may however
be argued, that as a particular form of @ was found, by which the vibrations in the direction
of x were defined, prior to any consideration of the manner in which the fluid was put in motion,
so a particular form of f exists by which the condensation and velocity in directions transverse to
the axis of x are defined, and which is equally independent of the arbitrary disturbance. As
this form may, or may not, be capable of expression in exact terms, I shall first apply to
equation (12) the process already applied to equation (11), for the purpose of ascertaining
whether any exact value of the integral satisfies the conditions of the Problem.

8. The equation (11) coincides in form with (12) by putting —1 for a*, and 4e for 6.
That is, since e = aa? We shall have —e in the place of e in the integral of (12). Hence
a

OF LUMINOUS VIBRATIONS. 589

ifu=nm+yV/ —1 and v=a-y Veal by integrating as before,

f =F (u) + G(v) —e fv F, (uw) +uG, (v)} + — Sv? F. (wv) + wu °G,(v)} — &e. ....... (17).

The impossible quantities are got rid of by making G the same function as F. If then, to
obtain an exact value of f, we suppose that F'(w) = de and G (v) = Ac, we shall have,

°

iy OH! at
f=A(e 4+”) - 73 (ve + wel”) + (ve + wet) — &e.

etl:
Ae™(1 ev eu" evs Bey Ack eu ew eu
SA es eee tC) Ae” (l=. — -- Z
I Waksofee Sia eniay hn a PEC MEET ey.

ev eu
Ame ak Ake

= AcketyN=)- = w-yVA) 4 4, gk (w= YVAN) ~ & (ty Va)

= Age), +R yVA A Ae Oo w=, Ht ava

(-D: e
=2A k —|y.
€ cos (x + z)Y

Since, from the form of equation (12) w and y are interchangeable, we shall also have

(«Ey

ine Age . COS (x + 7) v.

Therefore generally,

k—‘)x e , (K-5)y me)
v= Bn con (e+ ) y+2A Tos (x + a wv.
As the quantities #4 and k’ may be any whatever, this solution is so far indeterminate. But it
is clear that the value of f must not, from the nature of the question, increase indefinitely
with w and y, and that consequently the exponentials must be made to disappear. Hence we

shall have k = k’ = \/e, and
f= 24 cos2/ey+ 2A cos2 few cecescereees Fer Bocicen (ues)

This then is the general form of f expressed in finite terms, and subject to the limitation of
being free from exponentials. Other forms may be adduced, apparently, but not really, different
from this, which equally satisfy the equation (12). For instance f= 4 cosqwcos q'y, provided
q+q*=4e. But this is reducible to the form of the terms of equation (18), by a change in the
direction of the axes of w and y. (See Theory of the Polarization of Light, p. 373.)

I shall have occasion hereafter to advert to equation (18). At present I have only to remark
that the above form of f does not correctly define the motion transverse to the axis of x, at least
for all values of w and y, for this reason. At the boundary beyond which the motion does
not extend in directions transverse to x there must be neither condensation nor variation of

df C
condensation, otherwise there will be transverse propagation. Hence f, aa and oF must
vanish together. But plainly this is not the case with the value of f obtained above,

9. From the above reasoning we may conclude that the form of f we are seeking for, is
. . fe q . . Pay MER Beet: Mho
not expressible in finite terms, and must consequently be obtained in an infinite series. ‘The

Vor. VIII. Parr V. 4G

590 Proressor CHALLIS, ON A MATHEMATICAL THEORY

only way in which a particular value of f is deducible from the general integral (17) without
assigning arbitrary forms to the functions F’ and G, is to suppose F'(w) and G(v) to be
arbitrary constants. Let, therefore, F (wu) =e, and G(v) =c. Then

F, (uw) = cu, G,(v) =¢'2,
Fw =, G,( ==,
&e. = &e. &e. = &e.
Hence f = (c +c’) (1 — euv + a= uv — “== .wv® + &c.),
=(e+e)Q-er+ oe - — Sh Sifts) j ocsconcondsoceadde (19),

by putting 7° for a +y’. Determining the arbitrary quantities so that f= 1 when r=0, we
bs

d
have e+e¢’=1. Also = =0 whenr=0, and =-—2e. Hence f has a maximum value at
r

dr
the axis of x, and is a function of the distance from that axis.

10. It appears, therefore, that the required form of fis derived from equation (12), by

supposing f to be a function of a +y’. That equation accordingly becomes,

df df

SH + HOS HO. cee cer ec ccc cee ccccer enc vecees .

dr rdr f 20)
Equation (19) is the integral of this equation in a series, the only mode in which it appears
to be expressible. By putting f= 0, we have for determining the corresponding values of r the
equation,

3 sy tien’ er"
We Canee Tec e

from which it appears that there are an unlimited number of possible values of r for which
f vanishes. Since there is no lateral propagation, the motion does not extend beyond a certain

limiting distance from the axis, at which f and a both vanish. It is not, however, apparent
ie
from equation (20) that these quantities may vanish together, that being an approximate equation
i A d
which does not give the exact value of = when f= 0. To ascertain whether this will be the
.

case, recourse must be had to equation (12) in the Paper on Luminous Rays, (p. 368), which
was obtained without neglecting small terms. On putting 4e for kn* that equation becomes,

1 1

@.— @#.-

ae pny
da HRT Saker sei demis setae (21)

whence it is clear that if f= 0, os also vanishes. Since a = 0 both when f=1 and f=0, for
* Y ~

OF LUMINOUS VIBRATIONS. 591

some intermediate value =. must be a maximum. Hence if a curve were described having for
its equation y=f(v), it would have a point of contrary flexure between the values of & corre-
sponding to y=1 and y =0, and would resemble the trochoid. It is also to be remarked that
the second term of equation (22) must be regarded as very small compared tothe other terms,
in order that that equation may be equivalent to a linear equation in # and y, excepting where f is
very small. By the omission of the second term, equation (22) becomes identical with equation (20).
Hence, with the exception just mentioned, the curves which represent the integrals of (20) and (22)
coincide; and as we found that the curve corresponding to (20) cuts the axis of abscisse in an
unlimited number of points, the same must be the case with the curve corresponding to equa-

d
tion (22). But for the latter curve we have shewn that f=0 and = = 0 at a point of intersection.
r
Hence the motion does not extend beyond the least value of r corresponding to f = 0.
11. The integral of equation (21) is derived from that of (12) by putting for f and —e for e.

Hence the integral of (21) in a series is,

1 2 pt er?
BSG 4 Se ee SE
of IPRER TE AOR Ge

3e%rt 1968 7°
Bo Tg gy FSG rerese none var (28):

Whence f =l-er+

This series diverges from the approximate series (19) after the second term. Let 7 be the least
value of r corresponding to f=0. Then,

\ o7]6
0=1-eP + piste - 9 eBh + Xe.
a 36
er
Hence e/’ is a numerical quantity. Let el’ = q. Then, as we have also —— = k, it follows that
mo
qn’ é f Df ; nN
ers Hence & is a constant quantity for all vibrations, if the ratio 7 be a constant. Now
it may be thus argued that \ and / have a constant ratio to each other. These quantities must
be related in some way, otherwise the motion is not defined. Let F(A, J, S) = 0 express this
relation, § being the maximum condensation corresponding to f=1. As there are no other
quantities concerned in this relation, and as ) and / are the only linear quantities, this equation

i ke
is equivalent to * = F,(S8). And we have above, * = Jt. Hence + Jt = F,(S). But

it has already been shewn (Art. 4) that & is independent of §. Hence F, (8) is a constant, the
same for all vibrations. Hence also & is the same for all vibrations.

12. We have now found for f a particular value which satisfies the hydrodynamical conditions
of the question, but does not admit of being definitely expressed. It can only be expressed in an
infinite series, the terms of which do not necessarily converge. If, therefore, the phenomena of
light be expounded by a definite form of f, this can agree with equation (23) only under
certain limitations. Now, by equation (18), we have a definite form of f obtained in a general
manner, without reference to the mode of disturbance. If in this equation 24 = 24’ = 4, we
obtain, ;

20° a 9¢
fe 3 (1 - 2ea* + a - &e.) + 3 (1 - 2ey’ + “a — &c.)

462

592 Proressorn CHALLIS, ON A MATHEMATICAL THEORY

sine ty) +l +y) — ke.

This value of f agrees with that given by equation (23) only to two terms. Consequently the
exact integral (18) may be employed only for small values of w and y. With this limitation, it
gives a value of f definitely expressed, and at the same time satisfying the hydrodynamical
conditions. These results point to the inference that the phenomena of light depend exclusively
on the motions contiguous to the axis of =; for it may be presumed that so far as the motions
correspond to the phenomena of light, they admit of being defined by exact expressions. The

ratios = and. : as applied to the Jwminous ray, will each be very small.

13. It may here be remarked, that in my Paper on the Polarization of Light, the equation
f = cos V/ 2er corresponds to common light, and the equations f= cos 2 V/ ex, f= cos2 Vey, to
light polarized in the planes of az and yz, subject to the limitation of taking r, w, and y, very
small. The first equation was obtained by assuming f to be a function of r, because common
light is observed to have the same relations to space in all directions perpendicular to the direction
of its propagation, and the other two were deduced from the first, by asswming the bifurcation of
a ray of common light to take place, so that the sum of the condensations at corresponding points’ of
the two parts, is equal to the condensation at the corresponding point of the original ray, and the
velocities are the parts of the original velocity resolved in directions at right angles to each other.
Since in the present Paper the same values of f have been arrived at by @ priori considerations,
that particular property of common light, and its resolution in that particular manner, may be said
to be accounted for on hydrodynamical principles.

14. The foregoing theoretical conclusions serve to explain some general phenomena of light.
In Article 7. it was argued that the motion transverse to the axis of the fluid filament, must be
defined by a particular form of f independent of the arbitrary disturbance of the fluid, and in
Art. 9, a form of this function was found without assigning particular forms to the arbitrary
functions, which in Art. 10. was proved to be consistent with the hydrodynamical conditions. As
this form indicates that the condensation is arranged alike in all directions about an axis of pro-
pagation, it follows that light which comes directly to the eye from its origin, of whatever kind
the disturbance may be, is common light, the distinctive property of which is, that it is alike
affected in all directions perpendicular to the direction of propagation. This inference is confirmed
by the fact that Light from the Sun, from Stars, from a lamp, from the electric spark, from
lightning, &c. is common light. The dispersed light by which objects are rendered visible, which
originates in the disturbances passively caused by the presence of the individual atoms of the
medium on which any ray impinges, should according to the theory be common light: and such
it is found to be. Moon-light and light from the Planets come under the same description.

Again, the form which the ray assumes at its origin determines it to have direction, for it is
clear that the direction of its propagation must from the first be coincident with the axis about
which the condensation is symmetrical. Hence as direction is determined without reference to the
mode of disturbance, there may be an unlimited number of directions of propagation, as there may
be an unlimited number of rays, (see Art. 5), due to the same disturbance. In fact, the state of
the fluid at the first instant, whatever it may be, can be satisfied by having at disposal in the

9
; . | 2% ue ey
equations V=a,§$ =msin ag (at —x+c), the quantities m, A, c, and by an unlimited number

of rays unlimited as to direction, notwithstanding that the functions @ and f are defined for each
ray. This agrees with the fact, that light coming immediately from its origin, is seen in all
directions.

OF LUMINOUS VIBRATIONS. 593

15. Hitherto we have reasoned on the supposition that no extraneous force acted on the ether.
It is quite possible that a ray, after taking its original form and direction, may be modified
subsequently in both these respects, by the action of forces, and retain the new form and direction
after the action of the forces has ceased. For instance, in the case of the ordinary reflexion of a
ray, forces act upon it for a short time and through a short space at the surface of the reflecting
medium, which, as they do not act symmetrically with reference to the axis of the ray, alter the
form of f. The analytical fact that this function is given generally by the integration of a partial
differential equation, and therefore not necessarily always of the same form, is quite consistent with
such an alteration, But on the principle that the transverse motion in the modified ray, so far
as it corresponds to phanomena of light, is still defined by an exact expression, the new form.
of f will be consistent with equation (18). Consequently, as A and 4d’ in that equation are
arbitrary, the new ray will either be completely polarized, or will consist partly of a common ray
and partly of a polarized ray. We cannot however suppose any alteration of the function @,
unless the forces be such as to destroy the luminous character of the ray; for on the particular
form of @ which we found in Art. 4, depends the uniformity of propagation, a property which a
ray of light is supposed to retain under the modifications here contemplated. It is unnecessary
to point out the accordance of the above theoretical inferences with observed facts.

16. A ray may also be modified by forces which act upon it continuously, as is the case on its
intromittence into a transparent medium, the modifying forces being the retardations which the
vibrations suffer by encountering the atoms of the medium. This kind of modification I have
considered in my Paper on the “ Transmission of Light through Transparent Media, and on
Double Refraction.” (Cambridge Philosophical Transactions, Vol. vit. Part 1v. p. 524.) I
have seen no reason to correct the Theory therein contained, and have only to remark, that the
approximate equation in p. 529, which determines f, may be arrived at by reasoning similar to that
in Arts. 2 and 3 of this Paper, as follows. We have, as in Art. 7 of the Paper cited,

» ads du 4 a den ige 3 as aes
_—+—e= 2,— +— =90, Cc? 10,
dav aan 2 dy dé dz dé
. [edt , d.fedt deka
Hence, yee Sedeee v=b?. fp 4 w=ec?, fe us
da dy dz

no arbitrary function of co-ordinates being added for the reasons heretofore given in Art. 1.

Uu v w . . . . £ .
Consequently a de + or, dy +-—,d must be an exact differential. This will be the case if
a 2 ec

s=fq', f being a function of # and y, and q@’ a function of x and ¢. For then,

; u df v df
fsdt= [fpdt=fo; so that, a das ae he

f f

w do u v w d d dp
— r a — di — = 2 : — dz =(d. 5
Lies , and aa da + oe dy + AD dz=q (= da + "i dy) +f Fe (d. fo)

dy
ds_ 7p du_, ng tt 4 _ np Fh
Also ear =f aa? da 21d Poe z dy = P ay? 4 dz ae ;

594 Proressor CHALLIS, ON A THEORY OF LUMINOUS VIBRATIONS.

PO pec coor Se (4. Fi 32 of) ¢

de” * ds fdeey dy
which equation resolves itself into the two following. (See Art. 3),
af af
—+4 C60 ead nis ccs cleancusreeees
h. dat? +1, dy? + 4ef=0, (24),
FP » th r2
a ens aan MOC) = 0. cnsrovccenvenssorecsves (25)

The former of these equations is the one it was required to obtain. By reasoning like that by which
equation (18) was derived, the analogous integral of equation (24) is,

f= Acos2 Joos A’ cos2 Vey.

Hence it appears that a ray of common light cannot be transmitted in the medium so long
as h and J are different quantities. Hence also two rays of opposite polarizations cannot in

general be transmitted in the same direction with the same velocity, for in that case they would, if
they were equal, be equivalent to a ray of common light. But equation (25), integrated in the

. ce ne
same manner as equation (11), gives for the velocity of propagation, ¢ r/ ee Le which, if —>

be equal to the constant &, is the same for rays of opposite polarizations. In \cpdoastadh of this
2 2

Sarl Sec A wp OX nk
apparent contradiction, it is to be said that if —-~=4, and consequently e = oe the value of f

™
2
for a ray polarized in the plane of wz is cos Vee, which is not independent of h, and

therefore not independent of the nature of the medium; whereas experience shews that a polarized
ray remains the same under all circumstances, and is in no way affected by the medium through
which it passes. That the value of f may be that which belongs to a polarized ray, we must
OS ri

have \./h = ’ the breadth of the wave; or, = But the velocity of propagation corre-

ec
sponding to ) is c',/1 +k. Hence the time of vibration of a given particle, or the colour
of the light, remaining the same, the velocity of propagation must be altered in the ratio of
»’ to A, and consequently becomes a J/ 1+. This result was obtained by somewhat different
considerations in Art. 8. of my Paper on Double Refraction.

J. CHALLIS.

Camprince OBsERVATORY,
March 2, 1848.

XLI* Supplement to a Paper “ On the Intensity of Light in the neighbourhood of a
Caustic.” By Grorce BIDDELL Airy, Esq., Astronomer Royal.

[Read May 8, 1848.)

In a Paper “On the Intensity of Light in the neighbourhood of a Caustic” communicated to
the Cambridge Philosophical Society about ten years ago, and printed in the 6th Volume of their
Transactions, I shewed that the expression for the intensity of light near a caustic would depend
on the infinite integral

Tw 1
J. c08 = (w* — m.w)* | from w=0 to wnat

where m is a quantity proportional to the distance of a point from the geometrical caustic, measured
in a direction perpendicular to the caustic, and estimated positive towards the bright side of the
caustic : and I gave a detailed account of the method of quadratures by which I had computed the
numerical value of this infinite integral for the values of m —4°0, — 3:8, &c. as far as + 4°0; and
I exhibited in a table the computed values of the integral.

The computation by quadratures was exceedingly laborious, and I did not resort to it without
trying other methods of a more refined nature. But in every attempt at expansion of the formula
I was met by the integral of a sine or cosine with infinite limits. The reasonings upon which
several mathematicians have attempted to establish the value of such an integral appeared to me so
little conclusive, that I preferred at once to abandon the expansions which introduced them, and
to rely only on the infallible but laborious method of quadratures.

On my stating to Professor De Morgan, after terminating the calculations, the scruples which
had led me to reject the expansions, he expressed himself so strongly confident of the correctness
of the conclusions upon the point which I had considered doubtful, that I was induced to undertake
the numerical computation of the series given by expansion of the formula. I proceeded at once as
far as it was possible to go with 7-figure logarithms, when I was interrupted, and the computations
were laid aside for some years, I have lately taken them up again, and have completed them as
far as they can be carried with 10-figure logarithms. It is the result of this calculation, and the
comparison of this result with that formerly obtained from quadratures, that I now beg leave to

present to this Society.

Before entering upon the numerical investigations, I will transcribe a letter which Professor
De Morgan at my request has written to me, and which he has permitted me to publish. It
contains an explanation of his views upon the evidence for the numerical certainty of the results
obtained by such integrals as those to which I have alluded.

* [ retain this notation in preference to that which is commonly | any notation which requires the expression of a differential at the

employed, partly because it is familiar to me, and because I have | end is for that reason objectionable,
used it in the paper to which I refer, partly because I think that

596 Mr. AIRY’S SUPPLEMENT TO A PAPER ON THE INTENSITY OF LIGHT

“ University College, London, March 11, 1848.

“In reply to your request that I would send you a sketch of the method which I
communicated to you some years ago, for finding the numerical value of es cos (w* —- mw) dw, I

send you the following. I am not aware that there is anything about it peculiarly my own, or
other than what would suggest itself as a matter of course to any one familiar with the current
methods in definite integrals.

‘‘ The series which I furnished depend ultimately upon the following formula :—

a s bs cos n@
iff e789. cos (r sin 8.w).w" 'dw,=T, = gist

@ Z F ‘< sin 20
He e709. sin (7 sin @.w).w' 'dw =T,

0 r" ?

in which r and @ are independent of w, rcos@ is positive, m is positive, and I’, stands for

pede, as usual. Under these conditions the theorems do not or need not rely upon any

notion of algebraical as distinguished from numerical equality. Calling either of them /pw.dw,
common arithmetical calculation would establish any degree of approximation between the conver-
gent series P0.a + pa.at+ p2a.a+ and the asserted value of the definite integral, if a were

taken small enough. And this for any value of 6, from 0=0 to 0= 5 -— B, B being of any

Tv . . .
degree of smallness. But when 6 = re the numerical character of the equivalence is lost, and the
equations assume the same character as 1 —1+1—1 + ...... =, and are subject to the same
discussion. ‘
“The above equations were first obtained by substituting a + bx/—1 for a in

por n=) 23. r,
J ew dw = a’
which is an equivalence of numerical character even after the substitution, if a be positive, and 6
(be it positive or negative) numerically less than a. For the use of the expansions of ¢~ 4y\/ —1 and
(a+b = 1)~" in powers of 6 would produce an equivalence such as

A, + Ak + 4k +...... = By, + Bk + Bak’ + ......

where k = alt A, = B, is a numerical equivalence, and 24, is a convergent series. But,
when 6 is numerically greater than a, a convergent series would be rendered divergent im inte-
gration: and, when this happens, I do not see any way to place the divergent series so obtained
upon the same footing as those of ordinary algebra.

“It is not however necessary to depend upon this introduction of divergency. If we call the
two integrals C, and S,, and differentiate both with respect to 0, we have
dC,

qe rsin@.C,,, —7cos@. S41,

—=rsin@.§,., + 7cos@. Cyiis
d@ i+ +1

IN THE NEIGHBOURHOOD OF A CAUSTIC. 597

aC... dS,

whence TC. = Ty sin 9 + qe cos 0,
ds, . dC,

Shen rT) sin @ — 7) cos 0,

from which it easily appears that for what value soever of m the equations first given are true, they
remain true when that value is increased by a unit. And that they are true when m = 1 is proved
by common integration by parts.

‘Tf instead of w we write w*, k being positive, and then for kn write m, we have

i 6.0 A ed 1 nO a8
J, 677289 cos (7 sin O. w*).w"- dw = eh ae r *,
? _rcosé.wk * . k = 1 - no on
los -sin (r sin@.w*).w"-'dw = -T, .sin—.r *.
kG) &k
“If r=1, and we call these integrals C, and §,, let us take
Zaye
cos (sin 9. w* — mw) = cos (sin @.w*). {1 — pti bates ot
, ; mw
+ sin (sin 6. w°). {mw — eh aeeaeceite
2.3
Multiplying by e~°°°:.dw, and integrating, we have
2 m>
fe OR cos (sin 8. * — mw)dw = C, + S..m — G - §, 5g name

Tv . . . .
“Tf we now make 0 = 3? and observe that in this case C, vanishes whenever m is an odd

multiple of 3, and .S,, whenever m is an even multiple, we obtain

3 6
7) m m m
cos (w? — mw) dw = C, — S, — -— C, —————. + 8, = 600009
I ( ) Lid, ff 2 Bites sh SUSE Oh NABER. iG
4 m m*°

go Ry TE yg 8s
50.3.4 AE Say A aT aera (0)

1 7 1 m 1 7 0 m°®
= -—J', cos (5-3) — —TJ,sin (5-2) -55 —-—T, cos (-3) eae
33 3°2 35 3°2/°2.8 3 % 3°2)2.3...6

1 . {2 1 5 m* 1 af: =) m
- -.- = cos |-. ° -_-- St ac =F ~ cccvce
ta wag 7) m4=T, ( ) Ts (5 2) 2.8...7

; 3°2)"S.a.4 8 °% 2
1 . m* 41 m® 741 °
=-T,.cos—. 1--. -— a ae gg aeenne ee eee let ig ene g = @| | ane + oe.
BF 6 81 S8 6. Sasa 6) 68) 8 8: “ANSTO j
r 7 2 im bine m 8 5 2 m"°
+ - -COS —.3M—-—. +-.-. —— meme or
; 6 8226 BV be B BBB vet 8 88 2eBrce 10

«I may observe that the precautions which I have taken, to shew that the algebraical cases
are limits of arithmetical ones, are not absolutely necessary in this instance. For if we resolve

Vor. VIII. Parr V. 4H

598 Mr. AIRY’S SUPPLEMENT TO A PAPER ON THE INTENSITY OF LIGHT

fy cos w'dw into its successive positive and negative portions, we have A, — 4,+-4;— A,+... in
which by 4:,,; is meant the portion of the integral (taken positively) which occurs from
w? = (2n +1) 2 to w* = (2n + 3) = The greater m is made, the smaller is the interval of this
partial integration; and these successive portions diminish, and diminish without limit, so that the
series is convergent, and the error always Jess than the first term rejected. And [sin wsdw may

be treated in the same way.
«A. DE MORGAN.”

The following numerical values occurring in the application of Professor De Morgan’s final
series may be conveniently placed here :—
Log I, = 0-4279627493.

Log TV, = 0°1316564916.

3
1
With these series I have computed the values of [cos = (w> —m.w) (m =0tom= “| ; for

w=-—5.6, —5.4, &c. as far as + 5.6: and I now exhibit a table of the results, compared
with those deduced from quadratures as far as the latter were carried, Each term of the series
was computed to 6 decimals, and one figure was struck off in the sum.

Values of Values of Values of Values of Values of Values of
Integral by Integral by Integral by Integral by Integral by Integral by
Quadratures. Series. Quadratures. Series. Quadratures. Series.

+ 0°00011 + 0:10377 | + 0°10377 + 0°56490 | + 0°56490
+ 0:00018 + 0713461 | + 0713462 + 0°35366 | + 0°35366
+ 0°00028 + 0°17254 | + 0°17254 + 0°11722 | + 0°11722
+ 0°00041 + 0°21839 | + 0:21839 — 0712815 | — 0°12815
+ 0:00063 + 0°27283 | + 0°27283 — 0°36237 | — 0°36237
+ 0°00093 ; + 0°33621 | + 0°33622 — 0°56322 | — 0°56323
+ 0'00138 : + 0'40839 | + 0°40839 — 0°70874 | —0°70876
+ 0°00204 + 048856 | + 0°48856 — 0°78018 | — 0°78021
+ 0°00298 | + 0:00297 + 0°57507 | + 0°57507 — 0°76516 | — 0°76516
+000431 | +0:00429 + 0°66527 | + 0°66527 — 0°66054 | — 0°66044
+ 0°00618 | + 0'00621 + 0°75537 | + 0°75537 — 0°47446 | — 0°47419
+ 0°00879 | + 0:00878 + 0°84040 | + 0°84040 — 0:22645
+ 0°01239 | + 0:01239 + 091431 | + 0°91431 + 0°05193
+ 0°01730 | + 0°01730 + 097012 | + 0°97012 + 0°32258
+ 0°02393 | + 002393 + 1°00041 | + 1:00041 + 0°54475
+ 0°03277 | + 0:03277 4 + 0°99786 | + 0°99786 + 0°68182
+ 0'04442 | + 0°04442 + 0°95606 | + 0°95607 + 0°70818
+ 0°05959 | + 0:05959 + 0'87048 | + 0°87048 + 0°61515
+ 0°07908 | + 0°07908 + 0'73939 | + 0°73939 + 0°41460

It is impossible to make the calculation for larger values of m, positive or negative, even with
10-figure logarithms, on account of the divergence of the first terms of the series. For the values

IN THE NEIGHBOURHOOD OF A CAUSTIC. 599

+ 5.6, the largest term in the series is 169:044826: and it is necessary to proceed as far as the
45" power of m, The result + 0°000114 for m = — 5:6 is obtained by combining the sum of
positive terms + 614°149962 with the sum of negative terms — 614°149848: and the result
+ 0°414595 for m = + 5°6 is obtained by combining the sum of positive terms + 614°357203 with
the sum of negative terms — 613°942608. For values of m ereater than + 5:6, the calculation must
be made in natural numbers.

The agreement of the values of the integral, computed by methods so totally different, is not
a little remarkable. On the one hand, it may be received by some persons as a proof of the
correctness of that part of the theory of the series which asserts the evanescence of the integral of

4 Ae? 1 P - F
a cosine when the limits are 0 and 5 : on the other hand it may be considered to afford evidence

of the great care with which the quadrature computations had been made.

‘For the last two or three sets of numbers compared, there is a trifling discordance. It will
be remarked that in my account of the computation by quadratures I have shewn that difficulties
begin to arise in the accurate computation for the values of m approaching to 4:0, (unless the
actual summation were carried to higher values of w than I carried it in those computations).
That the source of the discordances is in these difficulties and the consequent inaccuracy of the
quadratures, and not in the inaccuracy of the series, is evident from the following consideration.
The numbers computed by the two methods agree well for the values of m — 4-0, — 3°8, — 36: and
as the quadratures there present no difficulty, it is reasonable to suppose that both sets of numbers
are accurate (within such limits as are possible for the sums of numerous figures). Now the terms
of the series combined to form the value of the integral for m = + 4:0, + 3:8, +3°6, are exactly
the same as those by which the value of the integral for m = — 4°0, — 3°8, — 3°6, is formed: the
only difference being that they are combined in a different manner, and therefore, from the evident
accuracy of the series for m = — 4:0, — 3:8, —3°6, we are entitled to infer the accuracy of the
series for m = + 40, + 3°8, + 3°6.

G. B. AIRY.

Roya OxsservaTory, GREENWICH,
March 24, 1848.

4H 2

XLII. Some Remarks on the Theory of Matter. By Rosert L. E..is, M. A.,
Fellow of Trinity College, Cambridge.

[Read May 22, 1848.]

In the present state of Science, there are few subjects of greater interest than the enquiry
whether all the phenomena of the universe 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 circumstance that the application of mathematics to natural philosophy 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, have 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 phenomena 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 principles
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 causation, ordinary mechanical force is the first term.

3. With respect to the first point, namely, the impossibility of explaining all phenomena mecha-
nically, 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 conceivable mechanical forces, we
are not in a position to decide @ priori that any given phenomenon might not be thus produced.
Non constat, 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 conclusive answer
can be given, but the following considerations will I think naturally suggest themselves.

Mr. ELLIS, ON THE THEORY OF MATTER. 601

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 undulatory
theory of light has, as we all know, given consistent and satisfactory explanations of a great
variety of phenomena. Thus it appears, and the same remark might be educed from other
though similar considerations, that we are by no means absolutely estopped by the imperfection
of our mechanical philosophy, 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 distinct nature.
And to this view we find that most of those incline 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 insufficiency not only of a mechanical but even of a chemical physiology
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 rejecting 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 apyc:, 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
aad contains, so to speak, less that is discontinuous, than any which should recognise 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 altogether 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 arguments 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 difference

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

602 Mr. ELLIS, ON THE THEORY OF MATTER.

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 cognoscible. 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 deter-
mined,) 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 mind: and a distinction, analogous or identical
with that between primary and secondary qualities, has accordingly been expressed by the anti-
thesis between that which is a parte hominis and that which is a parte wniversi. That the
distinction 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 conception 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 tendency to motion. A body changes its form in virtue of the 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 soon. We thus arrive at a point of view at which all
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 essentially to the
domain of mechanics: and if chemical affinity be a cause of local motion, that is, if in virtue of its
action || a particle of matter finds itself at a given time in a position different 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

* Or that which in its formation it was to be, td ci xv | with more or less success attempted in the seventeenth century,
elvat. the restoration of science. Vid. Leibnitz, Epist. ad Thomas, 1.

| The doctrine of the cognoscibility of substantial forms, ¢ That is, all that are commonly enumerated as primary
which is intimately connected with this distinction, is as Leibnitz | qualities.

in effect remarks, as it were the common character of those who || As, for instance, in the phenomenon of crystallization.

Mr. ELLIS, ON THE THEORY OF MATTER. 603

liquid when the acid is introduced into it begins to act on the luminiferous vibrations which exist
near it in a different manner from that in which it had previously acted. The whole change, whe-
ther 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 con-
clusion 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.

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 apprehended—the natural result of this state of things is the formation 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 any thing
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 thecry. 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 Democritus;—a system which as we know found no 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 presented 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 manifest, and on the other hand mecha-
nical 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
notion. 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

* Specific differences of motion seem for more than one reason | with in the writings of modern historians of philosophy, Zeller's
not to have been used in giving an account of the differences of | Philosophie Der Griechen, 1. § 10.

bodies. + The physical theories of Des Cartes, though not properly
t See for a more favourable, and J think, a juster view of the | atomistic, since he proceeded on the hypothesis of a plenum, yet
philosophy of Democritus than that which we commonly meet | in many respects are akin to those of which we are speaking.

604 Mr. ELLIS, ON THE THEORY OF MATTER.

of Boscovich’s theory*. To Boscovich appears to belong the credit of having perceived that
if the atoms were conceived of simply as unextended 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 uncon-
sciously—almost all modern investigations 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 namely in
which the centres of force are discrete and at insensible 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 harmony, 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 illustration. 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 @ 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 C should have the power of enabling 4 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 C 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 no existence in rerum natura, it is clear that the inference
from the conclusion that no phenomenon can be imagined not resoluble en derniére analyse, into
local motion to the assertion that mechanical force is the only agency to be recognised in the

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

Mr. ELLIS, ON THE THEORY OF MATTER. 605

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 indirectly, manifest itself
in modifications of local motion. Furthermore, if for an instant we call this kind of action
(force)*, 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)*, which consists in the power of
modifying (force)*. 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)*, 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). 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 speculation which here present themselves.
The expansion of bodies by heat may however be particularly mentioned, because notwithstanding
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 ordinary conduction, being propagated from one part of the body to that which is next it.—
It is easy to conceive that by an alteration 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
established, we should have a case of the action of (force)?.

11. In conclusion, it may be well to remark that mathematical 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 equa-
tions of higher orders, I venture to predict with a degree of confidence, which doubtless I shall
not communicate to many, that if we ever succeed in establishing a mathematical theory of chemistry,
it will be as much conversant with equations of the third or of a higher order, as physical astro-
nomy is with equations of the second.

R. L. ELLIS.

May 1, 1848.

* Lam, of course, not to be understood as suggesting a materialistic explanation of phenomena of thought or volition.

Vor. VIII. Parr V. 41

XLII. Methods of Integrating Partial Differential Equations. By Avcustus
De Morean, of Trinity College, Cambridge, Secretary of the Royal Astro-
nomical Society, and Professor of Mathematics in University College, London.

[Read June 5, 1848.]

Tue following methods for the treatment of certain cases of partial differential equations of two
independent variables will be interesting, both as having something new, and as combining and
bringing together some isolated instances given by different writers.

FIRST METHOD.
Let the differential equation be

p (2, Ys Ps q= 0,

d d
p and qg meaning and rt Contrive that this equation, @ = 0, shall be the result of elimi-

a

nation between two others, 4 = 0, B=0, or, at full length,

A (2, Ys P» v) = 0, B (a, Y, P> UW v) = 0.

Accordingly, v is an implicit function of w and y. Let 7, s, and ¢, as usual, be the second
differential coefficients of x, and form the four additional equations
d

d
A, + A,r + dys + 4y 5 =0 Bs + Byr + Bys + By = 0,

dy dv
eee ese oa WY ana ah ile eT aka

From the six equations* eliminate p, q, 7, 8,¢; there will result an equation between

dv dv : :
— Pa which will often be more tractable than @=0. When, after integration, v is
y

found in terms of # and y, p and q can be found in the same terms from 4=0, B= 0, and
then x from dx = pda + qdy.

By Ys Vy

This method was derived from the suggestions afforded by a previous treatment of the equation
Apqg+ Bp + Cq+ D=0,

A, &c. being functions of # and y; which occurs in the process of developing any surface which
admits it upon a plane. Reduce the preceding to the form

(p+ P)(q+Q)=R.

* With regard to the notation, I must state that by such a | however useful it may be as an abbreviation, almost as useful in
symbol as A, 1 mean the partial differential coefficient of 4 with | the way of distinction. It points out the ultimate and elementary
respect to a, as obtained from an equation in which A is explicitly | process, on one or more of which the implicit differential coefli-
given in the form d= (@,....05 ). I have found this notation, | cients depend.

Mr. DE MORGAN,

Let p+P=WMv, Géwoe

MN being any convenient resolution of R into two factors. We have then

dv
ot Ey aM, it My,
y

Ndv N,
s+Q,=-=—4+—.

v dx v
dv Ndv N.
M—+—-—= - —_
aie Py-Q. + : M,,

which depends on ordinary differential equations.

ON METHODS OF INTEGRATING, ETC.

But it must be observed that the integration

of this subsidiary equation frequently leads to a form from which v cannot be directly exhibited as
a function of « and y. Where this happens, we must obtain a particular form which contains one
arbitary constant; another will be introduced in the integration by which z is obtained; and

Lagrange’s process may then be applied to the primary form so obtained.

For example, let pg= px +qy, or, (p — y) (4-2) = zy.

dv
Let -y= iS —-1=2 —
et p—y=av, 8 oy?
y y dv dv y dv
-—-trt=- —_ == = = 35 it 0)
Se ee : vw da "dy * de j

or av?—y°=f(v). Let fv=av*, and we have

a
v= y z

ee = I =Uty/a-a,
WiGec Vine re eae

av

s=aytyV/u—-at+b.

Let b = pa:

x= wyt+y/(a -a)+ oa,

y ,
2= Tera a + H*
But if we take p—y =v, we find

and we ultimately obtain the same form.
We may also obtain as the primary solution
= 4/.(a° —a) (y’ — pa) + ay t+ ye.

If we apply the whole process to pq = a. yy, we find for a primary solution

w= 2/ipio(Wiy —4)} + fa,

where piv=fpoda, Wy = fyydy.
Next, take the instance (p + q) (pa + qy) =1-
412

then the general solution of pq = px + qy may be obtained by eliminating a from

608 Mr. DE MORGAN, ON METHODS OF INTEGRATING

1
Let pr+qy=, Rey
Form the four other equations and eliminate, which gives

: d d
CD ee bia

v
Let fv = av; then v = 2a ate
l-a
gt a 1 2 1 I
V (1 =a) 4/(y-ae)’ E uaAlt =i) J/(y — az)’
Bes AY Poe
1-a

For -—a (1-a)~‘ write a: then we deduce the general solution by eliminating a between

z= 2y/fa(w -y) +y} + ha,

a-y ;
0. = + a.
Via@-n +7?
Let Ap? + Bpq+Cq@=D
which can be resolved into (p + Kq) (p + Lq) = MN.
Let p+ Kq= Mv, p= (AS - rate) 1),
N
p+lq=~, = (Mo -2) ae -

The two values of s thence derived, equated to each other, give the equation for determining v.
Accordingly, since A &c. may be any functions of w and y, the general equation of the second
degree is reducible to ordinary differential equations, provided that x do not appear in it.

In these examples, I have chosen, merely for simplicity, cases in which p and q are explicitly
found, and the values of s equated. This amounts to exhibiting @ =0 under the form of
A=0 and B=0, and determining v so that pda + qdy may be a complete differential. And
in like manner as every particular value of v leads to a particular value of x, so does each
value of = lead to one of v. And in this way a particular solution of one partial differential
equation may lead to a particular solution for another and a more difficult one. Thus, if p=o
be derived from A=0, B=0, leading to the new partial equation Y=0; and if it also be
derived from d’=0, B’= 0, leading to U'=0: by means of a solution of U = 0, leading to a
solution of @=0, one solution of U’=0 may be found.

Take the instance VP t+ r/q = 28s

or p= (v-»)’, q=(v+ »)’,

dv dv
(@ +0) 5, + @-0) F =-@ +0)

PARTIAL DIFFERENTIAL EQUATIONS.

609
O@
Vitter OF @)s Bay =a,
ee a
u+y-—a
2 r = a2 4
den eeeey =) gen :
u+y—a (w+y-a)? ”
per tk 2 aft

Sb HL
3 @+y-a

Make b= ga, and proceed as before.

Another form of solution is derived from v + =a, or
dz = (2m —a)’da + a’dy,

1
s=-(2ae-a)+ay+ da.
ns ) y+
Resolve the same equation into

nels dv d
giving

2
From the solution of the original take a=( a

th} =; +
. @+y — 2a

This ought to be a solution of the differential equation last written, and it will be found
to be so on trial.

SECOND METHOD.
Let there be given the equation

P(x, Y, % DP» qT 8 t=O.

Interchange p and w, q and y, x and pa+qy-—x, rand

“Ss t -8s r
giving o(r% PpEet+qdy — ®% VY; Téi— 6° Spite? =5)=0

If either of these equations can be integrated, say by
then the solution of the other is obtained by eliminating Y and Y from

Y wa fa X, Y),

dZ dZ
, Dae 2 y* OY’

zseawX +yY — Z.

610 Mr. DE MORGAN, ON METHODS OF INTEGRATING
The root of this theorem lies in the following, that the interchanges above mentioned do
not alter the truth of the equations
dz = pdx + qdy, dp=rdxa+sdy, dq =sda + tdy,
so that we have

d(px + qy — #) = wdp + ydq,

— PRA?

dy =— dq.

8 ?
——d
-—s* Pt pag

Let w, y, # be considered as functions of p and q, derived from the equations

SF (@s Ys %) = 0; Fat f.p =%, Sy th. d = 93
but remark that there is a case of exception, namely, when the second and third equations
give simultaneous elimination of w, y, and x, or lead to \y(p, q)=0. Since
s=pa+qy— {(wdp +ydq),
xdp + ydq must be a complete differential. Let it be dv, then we have, v being a function
of p and q,

dv dv
ap’ a Te Swe gy — 0.
Let the second differential coefficients of v be p, o, tT, we have then
T eo
dz = pdp+cadq, dp = —_—,, da - 4 dy;
pT Go PT oO
de sedpe sag | dg= = —— de + —. dy,
pT—-oa ig 0;
whence r= - 3? = —-_, t=-—f 2
prs preg P'S

Hence, in order to make p and q the independent variables instead of w and y, we must
assume a function v, of p and q, such that

‘ dv _ dv J dv dv
dp’ Ye Pap Taq >
and then we must find v by integrating
ie dv dv dv T -o p ) = 0
\dp” dq’ Pap ‘dq 9 Ps pt-oa pt-a° pt-a Fs

The manner in which I first stated the theorem changes the meaning of the letters # and y
without changing the letters themselves.

Of this method, I find one instance. Legendre (see Lacroix, Vol. 11. p. 622) has employed it
as a casual artifice for the reduction of

fi (ae @ +7 +fe(p: 9-8 + fs (Pq) +t = %
to fila, y) 1 -fi(@y) -8 + fy (a, y)-t=0.

PARTIAL DIFFERENTIAL EQUATIONS. 611

But I am not able to find that it has ever been applied to any equation of the jirst order.
Lacroix (Vol. 11. p. 558) gives something as near to the whole method as can well be imagined.
He sees everything except the completely interchangeable character of x and px +qy— x; that
he did not see this last may be suspected from his making the restriction that x must only enter
in pe +qy-2%.

It is to be noted that, so far as equations of the first order are concerned, the solution takes
exactly the same form, even though we can only integrate the transformed equation by reducing
it to

yw (X, Y, Z, A) = 0, a = 0,

dZ dZ
for the forms of ix and aY

one more quantity, A, to eliminate.

are unaltered. There is now one equation more, Pris 0, and

Let the first instance be
Auv+ By +Czx+D=0,

where each of the four, 4, &c. is any function whatever of p,q, and pa +qy—z. The transformed
equation is obviously of the form Pp + Qq +R =0, where P, Q, # are all functions of x,y, z.

Lagrange has given a laborious method for the integration of x =pqg, and Lacroix (Vol. 1.
p- 565) does not refer to p. 558, I suppose for the reason just given. The transformed equation is
px+qy —z = xy, of which the integral is

ie y
s—-avay=uf (%) :
We may therefore find the general solution of x= pq from
4 seine Yy
Z-xY+ Xz), 2=¥-3r(z)+7(z).
y=X+f'(5). 2 = XY.

Generally, however, the most convenient method is to select an appropriate primary solution, and
then to use Lagrange’s process. This may be done, if we please, from the common differential
equations which integrate the transformed partial ones. These are, in the present case,

z= 2y + ba, y = au.
-The retransformed equations are
pe +qy —% = pq + bp, q = ap.
With these, and x = pq, eliminate p and q, which gives

“2+ayr- b)* f Fs, °
hessiag = » so that we have the general solution by eliminating a from

y=
L=

4a
(7+ ay+qa) dz
w= Wt.a9 +29) » and — =0.
4a da
But we may often, most often I think, procure the primary solution in an easier manner from the

result of the complete method. Let fx =az +6, and we then have

a=Y+b, yo X+a, z= XV,

612 Mr. DE MORGAN, ON METHODS OF INTEGRATING

or = (a —b)(y—a), a very obvious solution. Hence we must eliminate a from
z=(@-a(y-a), 0=(@-ga)+¥y-a) ga

In my Differential Calculus, p.717, I gave a method for the general case x = (p,q); but
the following, derived from the present method, is preferable.

es f(% 0) = fo (X, aX). Xa.
Then Z=xXf(x,5)+XF (5).

with which proceed as before.
Of the instances which I have tried by the other method,
pa=px+qy gives pa+qy= xy, from which

pestres(®), ent Zell)
y= 4x+5F(F)> <=-4.x7-4(5)-
In this case we may conveniently take the retransformed equations
= ap, pet+qy—zx=4pq+6, which with pq= pa +qy,
give 2a(z+6)=(e@+ay)*, say 2ax+ b =(w# + ay)’.
Again, (p + 9) (pa2 +qy) =1 transforms into

1 y
d =] a =].
Ree a dy PE eg r(%)

+6, which will show that the general

a
Treat this by the method, and assume fx = *
+2

solution can be obtained by eliminating a between

a — a
z= he y +a,

0=-- + —— ‘a.
a + wa

The equation ,/p+4/q = 2a transforms into
Sutr/y=2p, or x=tabthayi + fy,
w=3XI4+7 ¥!,
bxYlas'y,
1 xt_lyxyi.ry'y - fy.

This is not an easy form. But if we take the retransformed equations

Wi

q=a, px+gy—z=pi+tpgi+b,
and join p?+q)= 2 with them, we find

« = 1 (2a -a)'+a°y+b, a primary solution, being the one already obtained.

PARTIAL DIFFERENTIAL EQUATIONS. 613

The equation ar+bs+ct=(rt—s°).f (p,q)
transforms into at —bs + er =f(a,y), which, a, b,c, being constants, is integrable.

Since r¢ —s° transforms into (r¢—s*)-', the equations rt—s*= f(p, q), and rf—s* =

4f(#,y)}~' depend each upon the other.

The failure of this method in the case of developable surfaces may be illustrated geometrically,
as follows. Let the equation @=0 be that of a surface, and for each point (x,y, x) of that
surface, take another point having for its co-ordinates p, q, pa+qy-—sx. The surface which has
the second point for its locus is conjugate with the first; that is, what properties soever connect the
first with the second, the same connect the second with the first. This conjugation cannot exhibit
any absolute geometrical properties, for the conjugate surface depends, as to what it shall be, not
only on the primitive surface, but on the position of the axes of co-ordinates, and also on the linear
unit chosen, Thus it will be found that the conjugate surface of a given sphere is a double
hyperboloid of revolution, having for its real axis the diameter of the sphere which is parallel to
the axis of x, and for its imaginary semiaxis the linear unit. Now when the first surface is
developable, its conjugate surface becomes a cylinder described by a straight line parallel to x,
guided by the curve f (p,q) =0 on the plane of wy. There is then no relation which involves all
the three co-ordinates.

It may be worth while to notice, that we can at pleasure obtain forms for elimination which
reproduce the function originally given, by assuming an equation which is its own tranformation.

A. DE MORGAN.

University Cottece, Lonpon,
April 27, 1848.

June 1, 1847, I had finished the foregoing Paper, as here written and dated, and it was in the hands of
a friend for transmission to the Society, when I happened to have occasion to turn over all the Notes of M.
Chasles’s Apergu Historique ....des méthodes en Géométrie, that I might collect all that has reference to the history
of Arithmetic. To my surprise, at Note xxx. p. 376, under the head Sur les Courbes et Surfaces réciproques
de Monge, being an account of an wnpublished memoir of Monge in possession of the Institute, I found the
second of these methods fully described. But to judge from all elementary writings, as well as from the apparent
resources of those who have had to use modes of integration, this method is not known; and therefore I do not

abandon my intention of communicating it to the Society.
A. DE MORGAN.

Vou. VIII. Parr V. 4K

XLIV. Second Memoir on the Fundamental Antithesis of Philosophy. By
W. WheEwe tt, D.D., Master of Trinity College, and Professor of Moral
Philosophy. ©

[Read November 13, 1848.]

31. In the course of 1844 I had the honour of reading before the Philosophical Society a
Memoir On the Fundamental Antithesis of Philosophy ; and this Memoir has since been printed in
the Society’s Transactions. 'The Fundamental Antithesis of which I then treated, is that which
is expressed in various ways :—for instance, by speaking of Things and Thoughts; of Sensations
and Ideas; of Fact and Theory ; of Experience and Necessary Truth; of the Objective and
Subjective Elements of our Knowledge. I endeavoured to make it apparent that all these are,
at bottom, the same antithesis, and that this antithesis is an antithesis of inseparable Elements ;—so
inseparable, that the opposed terms cannot, either of them, be applied absolutely and exclusively in
any case.

32. To give value to the exposition of this antithesis, it must be used in the expression of
philosophical truth. The antithesis may be looked upon in the light of a Definition by which we
are to enunciate one or more Propositions. In this, as in other cases, the Definition gives meaning
to the Proposition, the Proposition gives reality to the Definition. The Definition saves the
Proposition from being vague or ambiguous; the Proposition saves the Definition from being
arbitrary or empty.

In the Memoir just referred to, I have already used the fundamental antithesis in stating views
respecting the reality and the developement of human knowledge. But I would wish to be allowed
to pursue the subject a step further, and to express in a more general and distinct form than I have
there done, a general truth in the history of science, which I have there stated in a partial and
imperfect manner.

33. The general Truth of which I speak may be thus expressed:—that the Progress of
Science consists in a perpetual reduction of Facts to Ideas. Portions are perpetually trans-
ferred from one side to another of the Fundamental Antithesis: namely, from the Objective to
the Subjective side. The Center or Fulcrum of the Antithesis is shifted by every movement
which is made in the adyance of science, and is shifted so that the ideal side gains something from

the real side.

34. I will proceed to illustrate this Proposition a little further. Necessary Truths belong to
the Subjective, Observed Facts, to the Objective side of our knowledge. Now in the progress of
that exact speculative knowledge which we call Science, Facts which were at a previous period
merely Observed Facts, come to be known as Necessary Truths; and the attempts at new advances
in science generally introduce the representation of known truths of fact, as included in higher and
wider truths, and therefore, so far, necessary.

35. We may exemplify this progress in the history of the science of Mechanics. Thus the

property of the lever, the inverse proportion of the weights and arms, was known as a fact before
the time of Aristotle, and known as no more; for he gives many fantastical and inapplicable reasons

Dr. WHEWELL’S SECOND MEMOIR, ETC. 615

for the fact. But in the writings of Archimedes we find this fact brought within the domain
of necessary truth. It was there transferred from the empirical to the ideal side of the Fundamental
Antithesis; and thus a progressive step was made in science. In like manner, it was at first taken
by Galileo as a mere fact of experience, that in a falling body, the velocity increases in proportion
to the time ; but his followers have seen in this the necessary effect of the uniform force of gravity.
In like manner, Kepler’s empirical Laws were shewn by Newton to be necessary results of a central
force attracting inversely as the square of the distance. And if it be doubtful whether this is
the necessary law of a central force, as some philosophers have maintained that it is, we cannot
doubt that if those philosophers could establish their doctrine as certain, they would make an
important step in science, in addition to those already made.

And thus, such steps in science are made, whenever empirical facts are discerned to be neces-
sary laws; or, if I may be allowed to use a briefer expression, whenever facts are idealized.

36. In order to shew how widely this statement is applicable, I will exemplify it in some of
the other sciences.

In Chemistry, not to speak of earlier steps in the science, which might be presented as instances
of the same general process, we may remark that the analyses of various compounds into their
elements, according to the quantity of the elements, form a vast multitude of facts, which were
previously empirical only, but which are reduced to a law, and therefore to a certain kind of ideal
necessity, by the discovery of their being compounded according to definite and multiple propor-
tions. And again, this very law of definite proportions, which may at first be taken as a law given
by experience only, it has been attempted to make into a necessary truth, by asserting that bodies
must necessarily consist in atoms, and atoms must necessarily combine in definite small numbers.
And however doubtful this Atomic Theory may at present be, it will not be questioned that any
chemical philosopher who could establish it, or any other Theory which would produce an equiva-
lent change in the aspect of the science, would make a great scientific advance. And thus, in this
Science also, the Progress of Science consists in the transfer of facts from the empirical to the neces-
sary side of the antithesis; or, as it was before expressed, in the idealization of facts.

37. We may illustrate the same process in the Natural History Sciences. The discovery of
the principle of Morphology in plants, was the reduction of a vast mass of Facts to an Idea; as
Schiller said to Géthe when he explained the discovery; although the latter, cherishing a horrour
of the term Idea, which perhaps is quite as common in England as in Germany, was extremely
vexed at being told that he possessed such furniture in his mind. The applications of this Principle
to special cases, for instance, to Euphorbia by Brown, to Reseda by Lindley, have been attempts to
idealize the facts of these special cases.

38. We may apply the same view to steps in Science which are still under discussion ;—the
question being, whether an advance has really been made in science or not. For instance, in Astro-
nomy, the Nebular Hypothesis has been propounded, as an explanation of many of the observed
phenomena of the Universe. If this Hypothesis could be conceived ever to be established as a true
Theory, this must be done by its taking into itself, as necessary parts of the whole Idea, many
Facts which have already been observed ; such as the various form of nebula ; many Facts which
it must require a long course of years to observe, such as the changes of nebule from one form to
another ; and many facts which, so far as we can at present judge, are utterly at variance with the
Idea, such as the motions of satellites, the relations of the elements of planets, the existence of vege-
table and animal life upon their surfaces. But if all these Facts, when fully studied, should appear
to be included in the general Idea of Nebular Condensation according to the Laws of Nature, the
Facts so idealized would undoubtedly constitute a very remarkable advance in science. But then,
we are to recollect that we are not to suppose that the Iacts will agree with the Idea, merely
because the Idea, considered by itself, and without carefully attending to the Facts, is a large and

striking Idea. And we are also to recollect that the Facts may be compared with another Idea, no
4K 2

616 Dr. WHEWELL’S SECOND MEMOIR ON THE

less large and striking; and that if we take into our account, (as, in forming an Idea of the Course
of the Universe, we must do,) not only vegetable and animal, but also human life, this other Idea
appears likely to take into it a far larger portion of the known Facts, than the Idea of the Nebular
Hypothesis. The other Idea which I speak of is the Idea of Man as the principal Object in the
Creation ; to whose sustenance and developement the other parts of the Universe are subservient as
means to an end; and although, in our attempts to include all known Facts in this Idea, we again
meet with many difficulties, and find many trains of Facts which have no apparent congruity with
the Idea; yet we may say that, taking into account the Facts of man’s intellectual and moral con-
dition, and his history, as well as the mere Facts of the material world, the difficulties and apparent
incongruities are far less when we attempt to idealize the Facts by reference to this Idea, of Man as
the End of Creation, than according to the other Idea, of the World as the result of Nebular Con-
densation, without any conceivable End or Purpose. I am now, of course, merely comparing these
two views of the Universe, as supposed steps in science, according to the general notion which I
have just been endeavouring to explain, that a step in science is some Idealization of Facts.

39. Perhaps it will be objected, that what I have said of the Idealization of Facts, as the
manner in which the progress of science goes on, amounts to no more than the usual expres-
sions, that the progress of science consists in reducing Facts to Theories. And to this I reply,
that the advantage at which I aim, by the expression which I have used, is this, to remind
the reader—that Fact and Theory, in every subject, are not marked by separate and promi-
nent features of difference, but only by their present opposition, which is a transient rela-
tion. They are related to each other no otherwise than as the poles of the fundamental anti-
thesis; the point which separate those poles shifts with every advance of science; and then,
what was Theory becomes Fact. As I have already said, elsewhere, a true Theory is a Fact;
a Fact is a familiar Theory. If we bear this in mind, we express the view on which I am
now insisting when we say that the progress of science consists in reducing Facts to Theories.
But I think that speaking of Ideas as opposed to Facts, we express more pointedly the original
Antithesis, and the subsequent identification of the Facts with the Idea. The expression appears
to be simple and apt, when we say, for instance, that the Facts of Geography are identified with
the Idea of the globular Earth; the Facts of Planetary Astronomy with the Idea of the Helio-
centric system; and ultimately, with the Idea of universal Gravitation.

40. We may further remark, that though by successive steps in science, successive Facts
are reduced to Ideas, this process can never be complete. However the point may shift which
separates the two poles, the two poles will always remain. However far the ideal element may
extend, there will always be something beyond it. However far the phenomena may be ideal-
ized, there will always remain a portion which are not idealized, and which are mere pheno-
mena. This also is implied by making our expressions refer to the fundamental antithesis :
for because the antithesis is fundamental, its two elements will always be present; the objective
as well as the subjective. And thus, in the contemplation of the universe, however much
we understand, there must always be something which we do not understand; however far
we may trace necessary truths, there must always be things which are to our apprehension
arbitrary : however far we may extend the sphere of our internal world, in which we feel
power and see light, it must always be surrounded by our external world, in which we see
no light, and only feel resistance. Our subjective being is inclosed in an objective shell, which,
though it seems to yield to our efforts, continues entire and impenetrable beyond our reach, and
even enlarges in its extent while it appears to give up to us a portion of its substance,

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY, 617

ADDITIONAL NOTE TO TWO MEMOIRS «ON THE FUNDAMENTAL
ANTITHESIS OF PHILOSOPHY.”

Of certain Modern Systems of Philosophy.

I am desirous of adding, as a note to this and the preceding Memoir, some very brief remarks
relative to certain philosophical systems which have been much spoken of in modern times, especially
those of the celebrated German philosophers, Kant, Fichte, Schelling, and Hegel.

Every system of philosophy offers to us a special and characteristic mode of criticizing preceding
systems: and since every new system aspires to be true, it includes that which was true in the
preceding systems, and is therefore able to point out where the true part of each is. The doctrine
which I have endeavoured to explain in the two preceding Memoirs is, that there is a Fundamental
Antithesis of two elements, of which the union is involved in all knowledge, and of which the
separation is the task of all philosophy. This doctrine naturally directs us to consider how far
each preceding system of philosophy has performed this task; and the survey of such systems from
this point of view, may enable us to characterize them by a few sentences, at least so far as they
regard one leading point of such systems, the account which they give of the nature and foundations
of human knowledge.

The doctrine of the Fundamental Antithesis, which I haye endeavoured to expound in the
above Memoirs, and in other places, is briefly this:

That in every act of knowledge (1) there are two opposite elements which we may call Ideas
and Perceptions ; but of which the opposition appears in various other antitheses ; as Thoughts
and Things, Theories and Facts, Necessary Truths and Eaperiential Truths ; and the like: (2) that
our knowledge derives from the former of these elements, namely our Ideas, its form and character
as knowledge, our Ideas of space and time being the necessary forms, for instance, of our geome-
trical and arithmetical knowledge ; (3) and in like manner, all our other knowledge involving a
developement of the ideal conditions of knowledge ewisting in our minds : (4) but that though ideas
and perceptions are thus separate elements in our philosophy, they cannot, in fact, be distinguished
and separated, but are different aspects of the same thing; (5) that the only way in which we can
approach to truth is by gradually and successively, in one instance after another, advancing from
the perception to the idea; from the fact to the theory; from the apprehension of truths as
actual to the apprehension of them as necessary. (6) This successive and various progress from
fact to theory constitutes the history of science; (7) and this progress, though always leading us
nearer to that central unity of which both the idea and the fact are emanations, can never lead us
to that point, nor to any measurable prowimity to it, or definite comprehension of its place and
nature.

Now the doctrine of the Fundamental Antithesis being thus stated, the successive sentences
of the statement contain the successive steps of German philosophy, as it has appeared in the
series of great authors whom I have named.

Ideas, and Perceptions or Sensations, being regarded as the two elements of our knowledge,
Locke, or at least the successors of Locke, had rejected the former element, Ideas, and professed to
resolve all our knowledge into Sensation. After this philosophy had prevailed for a time, Kant
exposed, to the entire conviction of the great body of German speculators, the untenable nature of
this account of our knowledge. He taught (one of the first sentences of the above statement) that
(2) Our knowledge derives from our Ideas its form and character as knowledge ; our Ideas of
space and time being, for instance, the necessary forms of our geometrical and arithmetical
knowledge. FYichte carried still further this view of our knowledge, as derived from our Ideas, or

618 Dr. WHEWELL’S SECOND MEMOIR ON THE

from its nature as knowledge; and held that (3) all owr knowledge is a developement of the ideal
conditions of knowledge ewisting in our minds, (one of our next following sentences). But when
the ideal element of our knowledge was thus exclusively dwelt upon, it was soon seen that this ideal
system no more gave a complete explanation of the real nature of knowledge, than the old sensa-
tional doctrine had done. Both elements, Ideas and Sensations, must be taken into account. And
this was attempted by Schelling, who, in his earlier works, taught (as we have also stated above) that
(4) Ideas and Facts are different aspects of the same thing :—this thing, the original basis of truth
in which both elements are involved and identified, being, in Schelling’s language, the Absolute,
while each of the separate elements is subjected to conditions arising from their union, But this
Absolute, being a point inaccessible to us, and inconceivable by us, as owr philosophy teaches
(as above), cannot to any purpose be made the basis of our philosophy: and accordingly this
Philosophy of the Absolute has not been more permanent than its predecessors. Yet the philosophy
of Hegel, which still has a wide and powerful sway in Germany, is, in the main, a developement
of the same principle as that of Schelling;—the identity of the idea and the fact; and Hegel’s
Identity System, is rather a more methodical and technical exposition of Schelling’s Philosophy
of the Absolute than a new system. But Hegel traces the manifestation of the identity of the idea
and fact in the progress of human knowledge ; and thus in some measure approaches to our doctrine
(above stated), that (5) the way in which we approach to truth is by gradually and successively,
in one instance after another, that is, historically, advancing from the perception to the idea, from
the fact to the theory: while at the same time Hegel has not carried out this view in any compre-
hensive or complete manner, so as to show that (6) this process constitutes the history of science :
and alike with Schelling, his system shews an entire want of the conviction (above expressed as part
of our doctrine), (7) that we can never, in our speculations reach or approach to the central unity
of which both idea and fact are emanations.

This view of the relation of the Sensational School, Kant, Fichte, Schelling, and Hegel, and of
the fundamental defects of all, may be further illustrated. It will, of course, be understood that
our illustration is given only as a slight and imperfect sketch of their philosophies; but their
relation may perhaps become more apparent by the very brevity with which it is stated ; and the
object of the present note is not detailed criticism, but this very relation of systems to each other.

The actual and the ideal, the external and the internal elements of knowledge, were called by
the Germans the objective and the subjective elements respectively. The forms of knowledge and
especially space and time, were pronounced by Kant to be essentially swhjective ; and this view of
the nature of knowledge more fully unfolded and extended to all knowledge, became the subjective
ideality of Fichte. But the subjective and the objective are, as we have said, in their ultimate and
supreme form, one; and hence we are told of the subjective-objective, a phrase which has also
been employed by Mr. Coleridge. Fichte had spoken of the subjective element as the Me, (das
Ich) ; and of the objective element as the Not-me, (das Nicht-Ich) ; and has deduced the Not-me
from the Me. Schelling, on the contrary, laboured with great subtlety to deduce the Me from the
Absolute which includes both. And this Absolute, or Subjective-objective, is spoken of by Schelling as
unfolding itself into endless other antitheses. It was held that from the assumption of such a prin-
ciple might be deduced and explained the oppositions which, in the contemplation of nature, present
themselves at every step, as leading points of general philosophy :—for example, the opposition of
matter as passive and active, as dead and organized, as wnconscious or conscious ; the opposition of
individual and species, of will and moral rule. And this antithetical developement was carried
further by Hegel, who taught that the absolute idea developes itself so as to assume qualities,
limitations, and seeming oppositions, and thus completes the cycle of its developement by returning
into unity.

That there is, in the history of Science, much which easily lends itself to such a formula, the views
which I have endeavoured to expound, show and exemplify in detail. But yet the attempts to carry

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 619

this view into detail by conjecture, by a sort of divination, with little or no attention to the histori-
cal progress and actual condition of knowledge, (and such are those which have been made by the
philosophers whom I have mentioned,) have led to arbitrary and baseless views of almost every
branch of knowledge. Such oppositions and differences as are found to exist in nature, are assumed
as the representatives of the elements of necessary antitheses, in a manner in which scientific truth
and inductive reasoning are altogether slighted. Thus, this peculiar and necessary antithetical
character is assumed to be displayed in attraction and repulsion, in centripetal and centrifugal forces,
in a supposed positive and negative electricity, in a supposed positive and negative magnetism; in
still more doubtful positive and negative elements of light and heat; in the different elements of the
atmosphere which are, quite groundlessly, assumed to have a peculiar antithetical character: in
animal and vegetable life: in the two sexes: in gravity and light. These and many others, are
given by Schelling, as instances of the radical opposition of forces and elements which necessarily
pervades all nature. I conceive that the heterogeneous and erroneous principles involved in these
views of the material world show us how unsafe and misleading is the philosophical assumption on
which they rest. And the triads of Hegel, consisting of thesis, antithesis, and union, are still more
at variance with all sound science. Thus we are told that matter and motion are determined as
inertia, impulsion, fall; that absolute Mechanics determines itself as centripetal force, centrifugal
force, universal gravitation. Light, it is taught, is a secondary determination of matter. Light
is the most intimate element of nature, and might be called the Me of nature: it is limited by what
we may call negative light, which is darkness.

In these rash and blind attempts to construct physical science a priori, we may see how imper-
fect the Hegelian doctrines are, as a complete philosophy. In the views of moral and political sub-
jects the results of such a scheme are naturally less obviously absurd, and may often be for a
moment striking and attractive, as is usually the case with attempts to reduce history to a formula.
Thus we are told that the State appears under the following determinations :—first, as one, sub-
stantial, self-included: next, varied, individual, active, disengaging itself from the substantial and
motionless unity: next, as two principles, altogether distinct, and placed front to front in a marked
and active opposition: then, arising out of the ruins of the preceding, the idea appears afresh, one,
identical, harmonious. And the East, Greece, Rome, Germany, are declared to be the historical
forms of these successive determinations. Whatever amount of real historical colour there may be
for this representation, it will hardly, I think, be accepted as evidence of a profound political philo-
sophy ; but on such parts of the subject I shall not here dwell.

I may observe that in the series of philosophical systems now described, the two elements of the
Fundamental Antithesis are, alternately dwelt upon in an exaggerated degree, and then confounded.
The Sensational School could see in human knowledge nothing but facts: Kant and Fichte fixed
their attention almost entirely upon ideas: Schelling and Hegel assume the identity of the two,
(a point which we never can reach,) as the origin of their philosophy. The external world in
Locke’s school was all in all. In the speculations of Kant this external world became a dim and
unknown region. Things were acknowledged to be something in themselves, but what, the philoso-
pher could not tell. Besides the phenomenon which we see, Kant acknowledged a noumenon
which we think of; but this assumption, for such it is, exercises no influence upon his philosophy.
Things in themselves, are in his Drama, merely a kind of mute personages, kwpa mpoowma, which
stand on the stage to be pointed at and talked about, but which do not tell us anything, or enter
into the action of the piece. Fichte carries this further, and if we go on with the same illustration,
we may say that he makes the whole drama into a kind of monologue; in which the author tells the
story, and merely names the persons who appear. If we would still carry on the image, we may say
that Schelling, going upon the principle that the whole of the drama is merely a progress to the
denoument, which denoument contains the result of all the preceding scenes and events, starts
with the last scene of the piece, and bringing all the characters on the stage in their final attitudes,
would elicit the story from this. While the true mode of proceeding is, to follow the drama

620 Dr. WHEWELL’S SECOND MEMOIR, ETC.

scene by scene, learning as much as we can of the action and the characters, but knowing that we shall
not be allowed to see the denoument, and that to do so is probably not the lot of our species on earth.
So far as any philosopher has thus followed the historical progress of the grand spectacle offered to
the eyes of speculative man, in which the Phenomena of Nature are the Scenes, and the Theory of
them the Plot, he has taken the course by which knowledge really has made its advances. But
those who have partially done this, have often, like Hegel, assumed that they had divined the
whole course and end of the story, and have thus criticized the scenes and the characters in a spirit
quite at variance with that by which any real insight into the import of the representation can be
obtained*.

I will only offer one more illustration of the relative position of these successive philosophies.
Kant compares the change which he introduced into philosophy to the change which Copernicus
introduced into astronomical theory. When Copernicus found that nothing could be made of
the phenomena of the heavens so long as everything was made to turn about the spectator, he
tried whether the matter might not be better explained if he made the spectator turn, and
left the stars at rest. So Kant conceives that our experience is regulated by our own faculties,
as the phenomena of the heavens are regulated by our own motions. But accepting and
carrying out this illustration, we may say that Kant, in explaining the phenomena of the
heavens by means of the motions of the earth, has almost forgotten that the planets have
their own proper motions, and has given us a system which hardly explains anything besides
broadest appearances, such as the annual and daily motions of the sun; and that Fichte
appears as if he wished to deduce all the motions of the planets, as well as of the sun,
from the conditions of the spectator;—while Schelling goes to the origin of the system like
Descartes, and is not content to shew how the bodies move, without also proving, that from
some assumed original condition, also the movements and relations of the system must neces-
sarily be what they are. It may be that a theory which explains how the planets with
their orbits and accompaniments haye come into being may offer itself to bold speculators,
like those who have framed and produced the nebular hypothesis, But I need not here re-
mind my hearers either how precarious such a hypothesis is, or that if it be capable of being
considered probable, its proofs must gradually dawn upon us, step by step, age after age:
and that a system of doctrine which requires such a scheme as a certain and fundamental
truth, and deduces the whole of astronomy from it, must needs be arbitrary, and liable
to the gravest error at every step. Such a precarious and premature philosophy, at best, is
that of Schelling and Hegel; especially as applied to those sciences in which, by the past pro-
gress of all sure knowledge, we are taught what the real cause and progress of knowledge
is: while at the same time we may allow that all these forms of philosophy, since they do
recognize the condition and motion of the spectator, as a necessary element in the explanation
of the phenomena, are a large advance upon the Ptolemaic scheme, the view of those who
appeal to phenomena as the source of our knowledge, and say that the sun, the moon, and
the planets move as we see them move, and that all further theory is imaginary and fan-

tastical.

W. WHEWELL,

* If it be asked which position we can assign, in this dramatic | say that they look on with a belief that the drama has no plot, and
illustration, to those who hold that all our knowledge is derived that these scenes are improvised without connexion or purpose.
from facts only, and who reject the supposition of ideas ; we may

————

XLV. Observations of the Aurora Borealis of November 17, 1848, made at the
Cambridge Observatory, By the Rev. J. Cuauuis, M.A., F.R.S., F.R.AS.,

Plumian Professor of Astronomy and Experimental Philosophy in the University of
Cambridge.

[Read November 27, 1848.]

Tue observations I am about to lay before the Society, relate principally to the position of
the Corona in the splendid display of Aurora Borealis which occurred on the night of Nov. 17.
During thirteen years that I have resided at the Cambridge Observatory, there certainly has not
been so favourable an opportunity of observing the position of this critical point of the pheno-
menon as on the recent occasion: and as the observations I took have enabled me to make a
comparison of the position of the Corona with the Magnetic Declination and Dip at the place of
observation more accurately than in any former instance that I am acquainted with, I have thought
them worthy, with their results, of being formally submitted to the notice of the Society.

The observations were made partly by estimation of the position of the Corona with reference
to neighbouring stars, and partly by means of a small altitude and azimuth instrument, which was
constructed by Mr. Simms (Fleet Street, London), according to my directions, expressly for taking
observations of this kind, I call the instrument a Meteoroscope. It has a graduated azimuth
circle of four inches radius, and a portion of an altitude circle of the same radius graduated from
0° to 120°... An arm somewhat longer than the radius of the altitude circle, and turning about a
horizontal axis passing through the centre of the graduation of that circle, carries a bar eighteen
inches long, by means of which the observations are taken. ‘To that extremity of the bar which
is turned towards the object observed, a rectangular piece is attached having one side horizontal,
and consequently the other movable in a vertical plane. The other end carries a plate in which
is made an eye-let hole one-sixth of an inch in diameter. An altitude is taken by observing through
the eye-let hole the coincidence of the object with the horizontal side of the rectangular piece, and
an azimuth by observing its coincidence with the vertical side. Both are taken simultaneously by
observing the coincidence with the angular point. The bar is set obliquely on the arm which
carries it, for the purpose of observing altitudes a few degrees beyond the zenith, for which purpose
also the graduation of the altitude circle extends beyond 90°. When the object is near the zenith,
for convenience it is looked at through another eye-let hole made in a small plate standing at right-
angles to the larger plate, the object being seen by reflexion at a small mirror, the plane of which
is inclined at an angle of 45° to the direction of the bar. In this case the other angular point of
the rectangular piece is brought into coincidence with the object, care having been taken by the
maker of the instrument that the direction of collimation should in the two cases be the same. ‘The
bar and altitude circle may be readily turned together about the vertical axis, and the bar separately
about its horizontal axis of motion, and both may be quickly clamped as soon as the observation is
taken. he graduations are read off by verniers to single minutes. ‘The instrument has a tripod
stand, furnished with adjusting screws for the purpose of placing the axis of motion vertical by
means of a small spirit-level, which is carried round with the vertical circle. ‘The feet of the
screws rest in three angular grooves formed each by two plane faces, by applying the feet to which,
the instrument is instantly in position, the vertical adjustment of the axis having been previously

Vox. VIII. Panr V. 4L

622 PROFESSOR CHALLIS’S OBSERVATIONS OF THE

made. It is proper on each’ occasion of using it to determine the index errors by observations
of stars.

I proceed now to give the observations just as they were set down in the memorandum-book,
inclusive of those for index errors, premising only that the instrumental azimuths are reckoned
from East towards South, and that the noted times were taken from a solar chronometer, which by
a comparison with the transit clock immediately after the observations was found to be 1™. 52°.
fast on Greenwich Mean Time.

(2.2 gt. 49™. 0° Corona 1° South and 3° East of 8 Andromede.

(@)ashoo: SID 19 0 ference em D uth een eo. pasewewet.ceeern=ecareee

@) Pan 22 Sb Com Ness-ese> MUSOU ARGer yt Unsnteslscusds sen cceirecoenlcms

(4)..02 GLMietOenti gee: onmth andi pee. tills oe ceusetvees Uo.

©) nes-09. 4). 25 3° South and 20 pac dee adoqasbion

(6)....-. 9.9. 0 ‘Altitude of Corona by Micibroscope 63°. i Azimuth 438°, 44’
(Gi Sano Oya). B35 sectereee snes 8 Andromede ............ fo LU erprerecb els GIS
(8)....-- OV rat ieg TES Selse sno oacidon Corona me ta cBestece ss OTe Al NesnsceceshiOlee OO
(@)\sagn3¢ 9). 19).120) Wee es eee 5-20) COTO: ee chen TORT: Setceaset 42 . 28
(dO) Seren Qt 20s Lei aeteeeer ase ton SPAN TOMED seeses aniee'sess AO” Ssacsoue> 77-10
(11) eoee- 9.32. 0 Corona 1° South, and 4° East of 8 Trianguli.

(12) eeeeee 9.35.15 Altitude of Corona by Meteoroscope 69.20 Azimuth 387 . 24
(13). 52 <5 OV AA FILO! Setteweivetiias: WICOLONA MT UMN ee icons ews 1) SEE BoSeraous 41. 21
(14)...26.9. 49.40 ..... B Andromede ..........++ TU ROM sae wav eos ai
(a5) Sects 9.56. 0 Gansu “9? South, and 12° East of B Trianguli

(16)..... 10.6.0 ...... 2° South, and 3 Sst cosets Eeele's

(CYP )osn055 10.10.11 Altitude of Corona by Meteoroscope 69. 28 ear 42.12
(Us) anes LOA SOS A B Trianguli ............ Te pas eee AO 65. 30
(19)....-. 10.20.80 ............ Corona Metesseser OUR SOc Gs, dacrateo lena
(Ch) Beaeee LOW B et roetee cleeiesis sles Corona BanoaGonpocon., “hls UE} MeGceantou 40. 0
(QI) Ree O20 ee et Corona SpdbcHionatoo Midt MOWUNVEAR Seccen 62. 35
(22)ccae SUDHIOTSG eset Corona pases LOO Oras emescose 55 . 30
(23)...... 110250) eee. eer ee OOFON Pn aeens ceniens WOOD te ienterespic: 52. 4
(24) eases UD) A) LON ee eee Woronaes Wal eens. Seige GOlro0 rs ee sbeeees 47. 6
(C35) snoon 11.16. 0 The star e Persei appears in the middle of the Corona.
(26)...... 11.17.30 Altitude of Corona by Meteoroscope 67 . 30 Azimuth 49 , 40
(21) ease. Sea iN el ceca (CHRCFSEl@t aaa: wereatersees) O9'- 10) waviersnss 46. 14

The Corona seemed coincident with ¢ Persei
(28)......11.26. 0 Altitude of Corona by Meteoroscope 70.10 Azimuth 43. 6

Of the above observations Nos. (19) and (21) were marked ‘doubtful.’ Nos. (23) and (26)
were reckoned good.

The position of the Corona was calculated from these observations in the following manner.
When the observation was made by reference to a star, from the noted time corrected for error of
the chronometer, and the known longitude of the place of observation (viz. 23°,5 East), the sidereal
time was calculated in the usual way, and then from the known Right Ascension of the Star, the
hour angle (4) Eastward was deduced. The co-latitude of the Observatory (viz. 37° . 47’) being
represented by ), and 6 being the North Polar distance of the Star, its distance (m) from the
meridian, and its distance (*) from the astronomical zenith, were calculated by the following
formule :

sin m= sind sinh, tan @=tandcosh, cos = cos (p —X) cos m.

Let » and ¢ represent the estimated distances of the Corona from the star Eastward and South-

AURORA BOREALIS OF NOVEMBER 17, 1848. 623

ward. Since these were made in the directions of the arcs m and x, the distance (M) of the
Corona from the meridian is m + ,, and its distance (Z) from the zenith is = + ¢

When the Meteoroscope was used, the recorded altitude A, and azimuth B from East
towards South, were first corrected for index error by a and 8 respectively, and Z and M were
then obtained by the formule,

Z=90°—(A+a), sin M = cos (B + ) sin Z.

The index corrections were deduced from Nos. (7), (10), (14), (18), and (27). The calculated
altitudes and azimuths of the stars, compared with the instrumental readings, gave the following
values of a and £3.

No. (a) ; (B)
(I) eS 40 8
(10) ... 1
(KD) coseeoncbbed 29
(18) Weed. ese) 6
(Qi) ese: Beopeci — 53 . 48
Mean — 16 .18

Respecting these values it is to be remarked, that the discordances between them are much
greater than might have been expected. From subsequent trials of the Meteoroscope I have
found that, without taking particular care in making the observation, the error in an arc of a
great circle may amount to 12’. Whether the discordances above arose from unsteadiness in the
support, the observations being made on the roof of a small out-building, on which several persons
were standing; or from incautiously bending the collimating bar in the act of observing ; or, in
short, from inexperience in the use of the instrument, this being the first occasion of my using it
in a series of observations, I am unable to say. On replacing the instrument (Nov. 24), I ob-
tained from much more consistent values, the mean results — 51’ and + 23°. 34’. I have, how-
ever, considered it best to adopt the first determinations.

I have now to explain in what manner the point of the heavens to which the South end of the
Dipping Needle was directed, which for the sake of brevity I call the Magnetic Zenith, was ascer-
tained. As we have no Magnetic Observatory here, this was done inferentially. I have assumed
that for any place in England, Scotland, and Ireland, the Westerly Declination of the needle (V) and
the Dip (D) may be given approximately by the formule,

Y= Vt ake be

(OH IDK eX STATE
V, and D, being the Declination and Dip at the Greenwich Observatory, X the Longitude of the place
Westward of Greenwich, / the excess of its Latitude above that of Greenwich, and a, b, a’, 6’ certain
constants, which may be calculated by knowing the simultaneous values of V and D at Greenwich
and two other positions. From the published results of magnetic observations made in the year 1843
at Greenwich, and at the Observatory of Sir Thomas M. Brisbane (Makerstoun) ; and from a com-
munication, kindly made to me by Professor Lloyd, of the mean Declination at Dublin for the same
year as determined by 3600 observations, (viz. 27°. 9’, 87,) and the Dip at Dublin as determined by
an elaborate series of observations in September of 1843, (viz. 70°. 41,3), I have deduced very accu-
rate contemporaneous values of V and D, which with the Latitudes and Longitudes of the three
positions are here subjoined.

Lat. Long. West. Declination. Dip.
Greenwich ...... 519. 28/6 ... 0™.0°,0 ... 23°. 17,59 ... 69°. 1',9
Makerstoun ...... 55. $4,7 ... 10. 3,5 ... 25. 22,85 ... 71 . 25,0
Dublin Seer Pee DO ieee O en AAO won Rie OSU aesn lO» Wise
From these data, were derived the following formule, which probably may be applied at the present

time and for several years to come, with considerable accuracy to any place in the United Kingdom:
41L2

624 PROFESSOR CHALLIS’S OBSERVATIONS OF THE

KV =¥%
D-—- Dy

0,142518A + 0,159548/
0,027713A + 0,5135231.

Il

These formule give V-V, and D-D, in minutes, X being expressed in seconds of time, and J
in minutes.

For the Cambridge Observatory, V — V, = + 3',7, and D — D, = + 22’,0.

In order to make the proposed comparison of the position of the Corona with the Magnetic Zenith,
it is now only necessary to obtain the Magnetic Declination and Dip at the respective times of obser-
vation. These I have derived from observations made at the Greenwich Observatory during the
prevalence of the Aurora, which, on my preferring a request, were promptly forwarded to me with
all the requisite data, by James Glaisher, Esq., who is at the head of the Magnetical Department in
that Institution, and which the Astronomer Royal has allowed me to publish with this communica-
tion. For this favour I beg here to express my thanks. The observations are given at length in
Tables I, II, and III, at the end of this Paper, as well because they are used in the calculations, as
because they present so striking an instance of great magnetic disturbances occurring simultaneously
with an extraordinary display of the Aurora Borealis, that the connexion in some way of the two
kinds of phenomena must be regarded as a physical fact.

The Westerly Declinations at Cambridge at the times of observation were inferred from those at
Greenwich at the same times by merely applying the value of V—V, already obtained, viz. + 3',7.
The latter were deduced from the declinations recorded in Table I. by simple interpolation, it being
understood that the motion of the magnet was uniform in the intervals between the times there given.
The Greenwich observations were made by the admirable photographic process, which has been
brought to so great perfection by C. Brooke, Esq., of St. John’s College in this University. Between
g". 25" and 9". 44™, the disturbance was so great that the magnet passed the limits of the photo-
graphic paper. The same thing took place in the contrary direction between 10", 10™ and 10", 40".
As Mr, Glaisher states that the motions at these times were smooth and without checks, I have
ventured to deduce the maximum elongation between gh, 25" and 9", 44™ on the supposition that the
magnet continued to move after 9". 25" in the same manner as from 9". 20™ to 9°, 25™, till it attained
the maximum, and then that it immediately returned by the same motion that it had from 9". 44™ to
10". 10", The maximum elongation between 10". 10™ and 10", 40™ was inferred on the same principle.

Mr. Glaisher furnished me with the following values of the Dip at Greenwich :

1848. Nov. 12. 21°.
16>) Me Saee e. 608). 20155
1G NOT ae OB OSE
Baie Saha Ies O8).155 05

Hence it is inferred that the Dip, if undisturbed, would have been 68°. 55,0 during the Aurora.
The disturbed Dip was calculated in the manner I am about to explain. In the Greenwich obser-
vations (Tables II. and III.), the readings for the horizontal force variations are given in terms of the
whole horizontal force; but the vertical force readings are given in divisions of the scale, which
require to be converted into parts of the whole vertical force. The factor for this purpose is 0,00067,
which is the value of one division.

The scale reading of the vertical force magnet at Nov. 17, 0°, was 21%%-,7, and at Nov. 18,
o", 21°%,5, at which times there appears to have been no disturbance. The undisturbed read-
ing is consequently assumed to be 21% ",6,

The reading of the horizontal force magnet in parts of the whole horizontal force, was 0,1099
at November 17, 0", and 0,1074, at November 18, 0", the latter of which Mr. Glaisher states to be
somewhat below the average value for the season and time of day. The undisturbed reading
during the Aurora is assumed to be the mean between those two readings, viz. 0,1086.

_—

ee ee

AURORA BOREALIS OF NOVEMBER 17, 1848. 625

The variation of horizontal force was not registered from 10°. 2™ to 12", the disturbance carry-
ing the magnet out of the limits of the photographic paper. From 12" the observations were made
independently of the self-registering process. I have assumed that from 10°.2™ to 12" the dis-
turbance followed the same law as from 12”. 39™ to 14>, 5™, when the phenomenon reappeared in
a similar phase, and accordingly have taken 0,0890 to be the mean horizontal force reading in the
former interval.

Let now X and ¥ be the undisturbed horizontal and vertical forces respectively, X’, Y’ their
disturbed values at any given time, and a, y, the horizontal and vertical force readings at that time,
deduced by interpolation from Tables II. and III, the former divided by 10,000. Then

X'= X— (0,1086 — aw) X, Y'= Y - (21,6 — y) 0,00067 Y.

Y V4
Hence since xy tan of the actual Dip, and x7 tan 68°. 55’, it is readily shewn that

the actual Dip = 70°. 43’,6 — [3,06215]« + [9,88822] y,

the numbers in brackets being the Logs of the coefficients of w andy. The Dip at Cambridge is
assumed to be the value given by this formula, increased by D-—D,, or + 22’,0.

From the Declination (V) and Dip (D), the distance Z’ of the Magnetic Zenith from the
Astronomical Zenith, and its distance M’ from the meridian are given by the expressions,

Z'=90°— D, sin M’= sin V cos D.

The following are the results of the calculations which have been now explained.

cos

: . 3-5
a a a
s*. 4771 18°, 59’ 20°. 35' = 1°. 36) Grass 7°, 48° SM OABr s
49,1 19. 10 20. 39 =i,2 5. 36 7.48 —2.12 s
5351 18 . 44 20 . 38 —1.54 4.47 7.46 — 2.59 s

8 . 59,8 20. 3 20 . 37 0. 34 3.55 7.50 = 35-55 s
9. 2,6 20 . 30 20. 36 = 04,6 4. 6 7.51 —3.45 s
71 22. 5 20. 38 epee ei 7. 44, 7.49 —0. 5 m
11,4 23.12 20. 36 +2. 36 10.13 7.36 +2.387 m
17,5 18 . 29 20. 42 —2.18 6.53 7.44 —0.51 m
30,1 20 . 27 20 . 26 Onaat Ons 8 . 23 +0. 45 8
33,4 20 . 56 20 . 26 + 0.30 9. 26 8.19 +1. 7 m
42,3 21.21 20 . 32 +0. 49 8.18 S019) +0. 9 m

9 . 5451 20. 14 20. 42 — 0.28 5.10 7.55 — 2.45 8
10. 41 20. 0 20 . 34 —0. 34 4. 36 7. 88 —-3. 2 8
yh ss 20. 48 20 . 28 + 0.20 7.49 7.30 +0.19 m
18,6 20. 46 20 . 30 + 0.16 4.52 7.22 —2.30 m

10 . 21,2 19 . 48 20. 31 — 0.43 Sees 7.20 +0. 48 m
11. 4,6 (19 . 16 20 . 23 = Wt 0. 42 7.85 — 6.53) m
951 21.11 20 . 24 + 0.47 3.19 7. 36 —4.17 m
11,0 19 . 21 20 . 24 —'..8 4.9 7. 87 — 3.28 m
12,3 20. 26 20 . 24 +0. 2 6. 4 7.87 —1. 83 m
14,1 Bian 7 20 . 24 +1. 48 8 . 52 7.387 lea o 8
15,6 22. 46 20 . 23 +2.28 5.46 7.87 -1.51 m
21.43 20 . 24 +1.19 7.85 7. 89 -0O. 4 8

oe {a . 6 20 . 24 +0. 42 6. 33 7.39 -1. 6 n|

11 . 24,1 20. 6 20 . 24 —0.18 7.16 7.40 —0. 24 m

Means 20. 35,8 20 . 30,9 +0. 4,9 6. 30,8 7. 44,6 Sa

626 PROFESSOR CHALLIS’S OBSERVATIONS OF THE

The mode of observing by reference to a star is indicated by the letter s, and that by the Meteor-
oscope by the letter m. In taking the means, the observation at 11".4™,6 is excluded, the Corona
at that time being seen very obscurely after an interval of total disappearance. The great westerly
deviation given by that observation is, however, supported by the two that follow.

The observations by stars taken separately, give —0°.21',8 for the value of Z—Z’, and —1° . 50,7
for that of M-—M’. The observations by the Meteoroscope give + 23',9 for the former, and
— 0°. 47',5 for the latter.

The discordances in the positions of the Corona deduced from observation, are no doubt partly
owing to errors of estimation, or instrumental errors, and partly to the extreme difficulty of fixing
with precision on the centre of convergence of the Auroral streamers. But if these were the only
sources of discordance the distances from the zenith and from the meridian would be equally affected,
whereas the latter appear to be the more discordant. The fact seems to be, that the centre of the
Corona is continually shifting its position. This may be owing to several causes. The formation
of the Corona is merely an effect of perspective, the apparent convergence of the streamers being due
to the immense height to which they rise. If the streamers were all parallel to a fixed straight line,
they would apparently converge to a fixed point. But the foregoing discussion, and facts that will
be hereafter mentioned, shew that they take, at least very approximately, the direction of the dipping
needle at the locality from which they ascend. Consequently the point of convergence will be
different for streamers. rising from different quarters. Again, the directions of the streamers may
vary by the same causes which produce the disturbances of the position of the dipping needle : and
this change of direction would of course alter the position of the Corona. Lastly, the course of the
streamers may not be rectilinear. The foregoing comparison appears to prove that the Corona is
decidedly more Westward than the Magnetic Zenith, being less distant from the meridian than the
latter by 19.14. This is accounted for by saying that the streamers on rising from the Earth are
bent in a westerly direction. The apparent point of convergence would thus depend on the height
to which they rise, and would be continually varying. It is quite possible that streamers rising from
different quarters and to different heights, might apparently cross each other, and so form a fictitious
point of convergence. This explanation will, I think, sufficiently account for the discordances
observable in the foregoing results, and will serve also to shew why they exhibit no decided agree-
ment between the changes of position of the Corona and the changes of position of the Magnetic
Zenith. Such agreement may very well be veiled by the causes just mentioned. It seems to me,
however, that a general accordance of this nature is perceptible. As when the needle was most dis-
turbed, a large Easterly deviation of the South End was succeeded by a large Westerly deviation,
so a large deviation of the Corona to the East of its mean position was succeeded by a large Westerly
deviation: and as the changes of M’ are more marked than those of Z’, so the changes of M are
greater those of Z.

For the purpose of farther illustrating the subject, I propose to add a discussion of a few
observations of the position of the Corona, made in the iostance before us, and in one or two

others, in different parts of England. I have selected those of which the data seemed to be most
precise.

Mr. Boreham of Haverhill informed me by letter that he found the Right Ascension of the
Corona of the Aurora Borealis of Nov. 17, to be 1". 58™. 3°; and its declination + 31°. 18’, at 9°. 15™
Greenwich mean time, the latitude of the place of observation being 52°. 5, and the longitude
1", 46°, East. Hence I find by calculating as already described,

Z Z' Z-Z M MM’ M—M
2°56’ = 20°. 4’ 2°. 11" Tigre Mase! se ehiy

is)

The differences in this instance are large, but not very different from those resulting from the
observation made at Cambridge at Qe e 4),

ne

AURORA BOREALIS OF NOVEMBER 17, 1848. 627

An anonymous observer at Darlington, states in the Durham Advertiser of Nov. 24, 1848, that
at Nov. 17, 11". 27", which I suppose to be Darlington time, Persei was exactly in the centre cf
the Corona. The latitude of Darlington is 54°. 32’, and the longitude 6™. 12%. West. Hence I have

deduced,
v4 Z
20°. 3

Tate M mM M— M’
USroU ee ee arenes (Goes © Biz o gq tes SBGle a3!

From observations of the Aurora of November 17, made at Lansdowne Crescent, Bath, by
H. Lawson, Esq., and E. J. Lowe, Esq. ; (1) “ At 10". 20". (Bath time), the Corona was situated at
21 Persei.” (2) At 11",20™. the centre of the Cupola was ¢ Persei.” The assumed latitude of
Bath is 51°. 22’, and the assumed longitude 9™. 28°. West; and the results of calculation are

Z Zz Z-Z M aU M—- M
(Cl) ae 2s 2OVz0seGnuN | (O%Nadi! VE° as? a7 AEighay’
(2) 20) 2m 20RIS2 Servo Ol 95 24a) 9's <9 — 2.28
Means 20.56 20.34 40.22 7.5 7.58 —0.58

A remarkable Aurora occurred on Oct. 18, 1848, which was not seen at Cambridge, on account
of clouds. A description of it was sent to me by J. F. Miller, Esq., of Whitehaven. The most
precise observations of the position of the Corona contained in the account are the following :—

(1). 10". 7". G. M. T., the centre of the Corona is about 5° above a Andromedz, and nearly
in a line with yy Pegasi.” It had consequently nearly the same Right Ascension as a Andromede.

(2). 10". 24". G.M.T. The whole hemisphere is covered with streamers converging around
a Andromede.”

(3). ‘10. 52".G.M.T. £3 or « Andromede, appears to be the centre of convergence.” I
have taken the mean position between the two stars.

(4). “115. 17".G.M.T. TI have watched the Corona very attentively some time, and I
think 8 Andromedz as nearly as possible marks its centre.”

(5). 115.37". G.M.T. The coronal centre seems now to be about mid-way between vy
Andromede, and (3 Trianguli.”

I have compared these observations with the mean Magnetic Declination and Dip, deduced
from those of Greenwich, which for Oct. 18, are assumed to be 22°. 53’, and 68°. 55’. The latitude
of Whitehaven is 54”. 33’, and the longitude 14™. 128. West. The following are the results of the
calculations.

Z Zz’ Z-—Z' M M' M— M'
(Cats) 0". 76 4 2098 Br ae 8h as 4°, 22!
(2) 22. 14 10). 1 a O\. k | Olu 8. 4 —2.0
(8) “18.58 “19.7 ~-0.14 5.25 8.4 —-2.39
(4) 19 . 47 LON. 7 +0. 40 1,42 8.4 — 6.22
(5) Se 0° Or” 061 846) 8.4 0), 42
Means 20.8 19.7 +1.1 5.8 8.4 -—2.56

In the instance of the Aurora Borealis of Oct, 24, 1847, I observed that at 10", 10™,, Cambridge
Mean Time, the centre of the Corona was at a point of less R.A. than $8 Andromedex by 10", and
of greater N. P. D, by 2’. I am able to compare this observation with the actual Declination and
Dip at the noted time, by means of Greenwich Magnetical Observations inserted in the published
account of this Aurora drawn up by Mr. Morgan and Mr, Barber. From these data I find that
the Declination at Cambridge was 23°, 5’, and the Dip 69°, 24’. Hence the result of the com-
parison is,

Z Z Z-Z' M M M— M'
20°, 10’ 20°. 36° — 0°, 26 6°, 24! 7°, 56° Se

On this occasion the Auroral light descended but a few degrees southward of the Corona, and the

628 PROFESSOR CHALLIS’S OBSERVATIONS OF THE

streamers forming the Corona did not meet in a point, but left a circular dark space, which seemed
to be constant in its position, and the centre of which it was easy to fix upon. On this account I
consider the above results, though derived from a single observation, to be worthy of confidence.

From a consideration of all the results derived from the foregoing discussion of observations made
on different occasions and at different places, the following conclusions seem to be established :—

First, that the Corona of an Aurora Borealis is formed near the Magnetic Zenith of the place of
observation.

Secondly, that the observations, while they indicate no decided difference of altitude between the
two points, shew with great probability that the Corona is situated between 1° and 2° more to the
West than the Magnetic Zenith,

The Aurora Borealis which gave rise to the present communication, was more remarkable in its
features and more extensively seen than any that have occurred for a long period, having been
visible, as appears by authentic accounts, in France, Italy, Spain, Portugal, and the Azores. I have
therefore thought it would not be out of place to add here a description of it which I derived from
memoranda made very soon after its occurrence, and which was communicated to the Cambridge
Chronicle of Nov. 25, 1848.

“¢ Shortly after eight o’clock on the evening of Friday, Nov. 17, my attention was called to an
unusual appearance of light stretching from N. to W., which gave indication of a coming Aurora.
There was no arch, but the light was diffused and of considerable brilliancy. The maximum of the
brightness was at a position a few degrees N. of W., at an altitude of about 20°, which appeared to be
a stationary centre of luminosity during the whole of the display. The diffused light increased by
degrees in intensity, and spread upwards till it reached the Zenith; but during this time there were
no streamers. ‘The principal features of the phrenomenon were, frequent pulsations, and sudden
appearances and disappearances of streaks and large patches of light, so much resembling white
clouds that but for their rapid changes of form and brightness, it would have been difficult to dis-
tinguish them from the latter. The streaks darted in various quarters and different directions,
waning as quickly as they formed, and auroral clouds of all imaginable shapes were continually
bursting forth and vanishing, so as to present a spectacle of the utmost bizarrerie, till at length
greater order began to prevail. Streamers of some degree of definiteness arose, and in a short time
surrounded the magnetic Zenith. I then first observed the appearance of a corona or central point
towards which the streamers converged, and estimated its position at 8". 47" Greenwich mean time, to
be one degree South and half a degree East of 8 Andromede.

‘« A large red patch due West and rising about 20°, was observed to retain its position from 8". 35™
to 8", 51™, At 8h. 56™ a broad red band stretched from the Corona through Capella, and in a few
seconds changed to an auroral cloud of great brilliancy having Capella at its centre. At 8". 58™ an
extraordinary red band of irregular width was formed extending across the heavens from a little S. of
W.to N.E. These two azimuths were the prevailing positions of the red light during the whole of
the phenomenon. The band seemed to be a kind of junction of two red clouds. Its general course
was through a and $ Andromede to the Corona, and from thence its axis passed through Capella.

«* At 9". 15™ the phenomenon was at its greatest height of beauty and perfection. Streamers
reached the Corona or Magnetic Zenith from all points of the Compass. The towt ensemble was a
canopy of drapery, having the Corona for the point of divergence of the folds, and extending rather
more Northward than Southward of the Astronomical Zenith ; while its boundary all round was con
siderably elevated above the horizon. The outline was very irregular, but sharply defined, giving
irresistibly the idea of the lower boundary of a suspended curtain. This feature was in greatest per-
fection towards the N.W., where a broad space appeared so dark by contrast with the bright curtain
above it, that it might have been mistaken for a cloud had not stars shone throughit. The predomi-
nating colour of the streamers was white, but about W.S. W. and N. E., the peculiar ruddy tint of
the Aurora was remarkably intense, and in other quarters the streamers were tinged with green and

t

AURORA BOREALIS OF NOVEMBER 17, 1848. 629

blue. Altogether, both as to form and colour, the spectacle at this time was so singular and so beau-
tiful, that those who witnessed it here could not forbear giving repeated expression to their feelings of
wonder and delight.

‘“* The heavens were then partially covered with light clouds, through which the brightness of the
Aurora seemed to penetrate. At 9*. 58™,a red patch covered the constellation of Orion. At 10",
when the clouds had dispersed, the general light resembled that of a night in midsummer, or the dawn
of morning. Birds were heard to chirp in several quarters.

* At 10".15™, I saw a meteor, as bright as a star of the second magnitude, move slowly in a
westerly direction, and disappear at an altitude of about 53°, and at an azimuth of about 28° from
W. towards S. Flashes, supposed to be of lightning, were twice noticed. One occurred in the S. W. at
10". 23™, At this time the Aurora had much declined in brightness; but at 11" it broke out afresh,
and the Corona was again formed, not however with the same distinctness as before. At 11.18™, a
meteor, equal in brightness to a star of the second magnitude, was seen to cross the heavens slowly
from E. to W. N. W., leaving a train behind it. Shortly after 11". 24™ the Corona became invisible,
and the Aurora generally declined. I saw it, however, again between 14" and 15" in great brilliancy :
a tolerably regular arch was formed in the N. W., from which very definite streamers rose, but did
not reach the zenith: the red light also re-appeared in the West.”

With reference to the above particulars I have two remarks to make. First, having in repeated
instances of the Aurora observed the red light to prevail in the same azimuths, I made a comparison
of the azimuths noted here in the instance of November 17 with statements respecting the prevailing
direction of the red light given in descriptions of the same phanomenon as seen at other places, and
it seems to me probable that the red auroral clouds are formed over the Atlantic and German oceans.
Secondly, the occurrence of meteors during an Aurora has been so frequently remarked, that one can
hardly avoid suspecting some connexion between the two kinds of phenomenon.

Vou. VIII. Panr V. 4M

630 PROFESSOR CHALLIS’S OBSERVATIONS OF THE

The following are the Tables of Magnetic Observations referred to in the foregoing com-
munication.

Taste I. The Westerly Declinations of the Declination Magnet about the time of the Aurora
Borealis of November 17, 1848, as observed at the Royal Observatory of Greenwich.

Greenwich
Westerly Mean Time, Westerly Mean
Declination. 1848 Declination. 18

Greenwich
Mean Time,
1848.

Greenwich
Time,
48,

Westerly
Declination.

h 5 5 A ‘ “
Nov. 17. 0. . 21.54. 3 Nov. 17.

22. 25. 30

Wis ° 48 . 30
33 . 30

44. 30

34. 50

To these observations the following remarks were attached :
*‘ The magnet began to be disturbed at 0". 50™.”
* Rapid changes occurred from 6" to 8.”

‘It is not known to what extent the Magnet moved between 9".25™ and 9>. 44™ on one side,
and between 10". 10™ and 10". 40™ on the other. ‘The motion at these times was smooth and
certainly without any checked motion whatever.”

Taste II.

of force.

Greenwich
Mean Time,
1848,

AURORA BOREALIS OF NOVEMBER 17, 1848.

631

Readings of the Horizontal Force Magnet in parts of the whole horizontal force,
about the time of the Aurora Borealis of November 17, 1848, as observed at the Royal Observatory
of Greenwich, the whole force being reckoned 10,000, and increasing numbers denoting an increase

Horizon!.
Force
Reading.

Greenwich
Mean Time,
1848.

Horizon!,
Force
Reading.

Greenwich
Mean Time,
1848.

Horizon!.
Force
Reading.

Greenwich
Mean Time,
1848.

Horizon!,
Force
Reading.

Nov. 17.

1099

h, om

Nov. 17, 8 . 40

958 —

h.
Nov.17. 15.

979

1119 45| 958 — 1002
1155 1001 992
1083 5 967 989
1121 1032 985
1133 - 968 983
1111 958 990
1153 - 910 981
1109 937 991
1114 956 1000
1131 956 1011
1103 965 1025
1140 953 1044,
1068 ; 829 1042
1065 902 1019
1096 862 1002
1111 871 982
1079 877 969
1115 906 950
1096 750 A 969
1101 781 . 8| 928
1084 846 , 926
1046 710 4 931
1100 885 924
1046 948 938
1017 948 998
1043 ! 985 928
1096 937 960
1028 943 1076
958 903 1075
958 892 1074
997 900
972 916
992 935

Nov. 18. 0.

These observations were accompanied by the following remarks :

“« Between 7". 23™ and 7". 40™ small changes were constant. Between 8".14™ and 8", 24™, and
again between 8". 40™ and 8". 45™, the registering pencil of light went beyond the photographic
paper. he same thing happened at 10".2™, and the light remaining off the paper, the remaining
observations, commencing at 12". 0™, were made independently of the self-registering apparatus.”

4™M 2

632 OBSERVATIONS OF THE AURORA BOREALIS OF NOVEMBER 17, 1848.

* At 13". 45™ the force was at its lowest value, being at that time below its usual value by about
one twenty-fifth part of the whole horizontal force. The force at 14". 40™ was but little above the
minimum, after which it increased very gradually till about noon of November 18 it nearly reached
the average value for the season and time of day.”

Taste III. Approximate scale divisions of the Vertical Force Magnet about the time of the
Aurora Borealis of November 17, 1848, as observed at the Royal Observatory of Greenwich.

Increasing readings denote an increase of force. One scale division is equal to the fractional
part 0,00067 of the whole vertical force.

Vertical
Force
Scale

Readings. Readings.

Greenwich pie tee! Greenwich

Mean Time Mean Time.
1848,” Scale 1848. 7

Vertical * Vertical
Force Greenwich Force

Scale iii Scale
Readings. x Readings.

Greenwich
Mean Time,
1848.

h. om div. he om Rey Dna div. he om.
Nov.17.0. 0| 21,7 | Nov.17. 9.14 Nov.17. 10.37] 17,5 | Nov.17. 13.49
1.20] 20,3 16 f 39} 18,8 51
2.20| 20,7 20 40| 17,2 13.59
3.20| 25,5 27 45 | 20,9 14.28
4.20] 24,6 37 58) ays 37
5.20] 22,2 20,3 14,49
45
6.18] 26,5 ; 16,3 154 0
30| 25,7 48 18,1 20
35| 26,8 49 15,8 82
26,2 50 17,0
26,8 15,8
16,8 21,1
4,0 16,0
14,3 20,7
9,2 3 19,0
15,1 20,8
13,2 14,7
16,8 16,1
13,5 17,0
15,3 9,3
14,0 | 16,8
16,3 | 17,0
14,0 11,0
16,0 15,0

“ Between 9h. 27" and 9". 37™, the reading was somewhat greater, the light being off the paper.
Where two readings are included by a bracket, it is to be understood that the motion of the Magnet
was so rapid that both took place nearly simultaneously. Between 17". 35™ and 20". 5™ the reading
increased very gradually.”

A comparison of Tables II. and III. shews that the disturbance of the Horizontal Force was
much more considerable than that of the Vertical Force, and that both were generally below their
average values.

CamBRIDGE OBSERVATORY, J. CHALLIS.
November 27, 1848.

—

XLVI. On Clock Escapements. By EvmuNp BecKxetT DEntson, Ese, M.A.,
of Trinity College, Cambridge.

[Read November 27, 1848.]

In the year 1827 the present Astronomer Royal wrote a paper in the Cambridge Phil. T'rans.,
Vol. 111. p. 105, “On the Disturbances of Pendulums and the Theory of Escapements,” in which he
investigated the effects produced on a free pendulum by connecting it with each of the three classes
of escapements ; and from the amount of the disturbance in each case, he inferred the relative merits
of the escapements. He added: “* The Theory of Escapements is by no means complete, but I hope
it will be found that the principal points have been touched on, and that enough has been said to
enable any one else to pursue the subject as far as he may wish.”

I know of no work in which the subject has been pursued further; and therefore I propose to
exhibit a few of the results which are to be obtained by following up Mr. Airy’s calculations, and
which I arrived at in investigating the merits of an improved remontoir or gravity escapement, in-
vented and constructed by a friend of mine*, avoiding certain mechanical objections to which such
escapements have hitherto been liable; and it will be seen, from the following remarks, that they may
be made, by a particular arrangement of the parts, free from the mathematical objection which Mr.
Airy says renders them almost as bad as the common recoil escapement. Mr. Bloxam has had some
communication respecting his clock with the Astronomer Royal; and I shall be glad if he is thereby
induced to complete his Theory of Escapements. In the mean time the following remarks may be
of some use. I shall take the mathematical results, though not the practical conclusions, of Mr. Airy’s
paper for granted, as they are sufficient for my purpose ; and his method of obtaining them may be

seen either in the volume referred to, or in Pratt’s Mechanics, into which the substance of his paper
has been copied.

I shall presume that every one who is at all acquainted with clocks understands the construction
of the Dead Escapement, as it has superseded all others in clocks that are expected (as the clock-
makers say) to perform correctly ; though it does not appear to be generally known from what the
accuracy of its performance really arises. I shall follow Mr. Airy in assuming the maintaining force
to be constant, although it is not quite so, since the inclination of the tooth of the escape-wheel to
the face of the pallet is greater at the end of the impulse than at the beginning, by nearly the angle
which the wheel moves through in one beat. Let 3 be the angle which the faces of the pallets make
with their dead or circular part ; then, since the tooth ought to be a tangent to the dead part, #3 will
also be the inclination of the tooth to the face of the pallet at the beginning of the impulse ; and we
shall assume it to remain the same throughout the impulse.

Let Pg be the moving force of the clock-weight referred to the extremity of the escape-wheel’s
teeth: p the length of the pallet measured from the axis of suspension of the pendulum: M the
mass of the pendulum, and / its length: @ its angle with the vertical. Then the equation of
motion is

£8 = (042% ans) --E(0)

say, (neglecting the moment of inertia of the wheel, and putting @ for sin @ as usual).

* J.M. Bloxam, Esq., of Lincoln's Inn, Barrister-at- Law.

634 Mr. DENISON, ON CLOCK ESCAPEMENTS.

Then if a be the extreme value of 0, y the angle at which the impulse begins, and yy’ on the
other side of zero at which it ends, Mr. Airy shews that A, the increase of the time of an oscil-
lation due to the escapement,

= 2 Wane - VERT
?

27a

(y+ y) (y' = y) nearly,

4
if ry and +’ are so small, that ~ may be neglected.

Mr. Airy remarks: ‘This is a quantity extremely minute; for -y and yy’ are generally small,
and y — ry may be made almost as small as we please. It cannot, however, be made absolutely 0;
for the wheel must be so adapted to the pallets, that when it is disengaged from one it may strike
the other not on the acting surface, but a little above it; therefore y must be greater than vy ; but
the difference may be made so small that the effect on the clock’s rate shall be almost impercep-
tible. This escapement therefore approaches nearly to absolute perfection ; and in this respect theory
and practice are in exact agreement.”

Since A is only the increase in the time of one vibration, and there are 86,400 vibrations in a
day, (assuming the clock to have a second’s pendulum,) and a second a day is a large error, it is
worth while to see what A really is. If Wg be the clock-weight, and h its fall in a day ; then, since
p (y +) tan B is the thickness of the pallets, or the drop of a tooth in one heat,

Pp ed ee
ari (1 +7) 120 Bs oF $ Cy + 7) = FET eGa003

and this quantity (which we may call F), will be the same for all clocks of the same kind, whatever
B or y + yy’ may be; and
F

27a

A=

5 (y' - y)-

Now a weight of Qlbs. falling 9 inches a day will keep a well-made clock of this kind vibrating
29 on each side of zero. Jis 39 inches, and M is usually about 14lbs. Therefore (allowing nothing
for the friction of the train),

F 2x9 .033
~ 14 x 39 x 86400 86400
005 +’ — :
and 86400 A = —— pune since a = 2° = .035.
.001 a

I understand from clockmakers that yy’ --y can hardly be made less than 20’, and is seldom so
, es 1 ;

little; .. Yo ee and 86400 A =.8 of a second, nearly. This is the amount of A in a day
a

But it is mot the error of the clock, being only the difference between the rate of a free pendulum

and one disturbed by this escapement. The error, or, as it is called, the ‘‘ rate,” of the clock,

with the sign changed from what it would naturally have, is the variation of A, which depends

on the variation of a and of F, according to the friction of the train and the pallets.

Differentiating A with regard both to a and F,

Mr. DENISON, ON CLOCK ESCAPEMENTS. 635

: dF 3d
or the “daily rate” = — .8° (Ge - -<).
F a
We see, therefore, that the real merit of this escapement arises from the two causes of error tending
to counteract each other ; for, though no exact relation can be determined between the changes of
the arc and of the force, since they depend on the changes in the friction of different parts of the
clock, yet it is easy to see that a will diminish when F does, under the influence of increasing
ee ; da. #
friction as the clock gets dirty. It appears that —* is not generally so much as bs = , and
a 3
therefore the clock gains as the are diminishes. Moreover, the circular error, which is never com-
pletely corrected by the pendulum-spring, I understand, tends to make the clock gain as the are

tare 5 2 ada
diminishes; since dA for the circular error = a as may be seen from any book on pendulums.

I have in one instance seen the contrary effect take place, where a church-clock, soon after it was
put up, spontaneously increased its are by more than a degree, from the pallets polishing themselves
more perfectly than had been done by the maker, and at the same time it gained considerably, as
we see it ought to have done. The tendency to gain as the are diminishes has led to the practice
of making turret-clocks, which are liable to great changes both in the force and the arc, with a
slight recoil in the place of the dead part of the pallets, as the effect of the recoil is to diminish
the time as the are increases.

The principle of nearly all the gravity or remontoir escapements is this: There are two small
arms three or four inches long on each side of the pendulum suspended separately on an axis coinci-
dent with that of the pendulum and moving in the same plane with it: these arms carry a small
weight at their lower ends, and also a detent to stop a tooth of the escape-wheel, and a pallet of some
kind by means of which the arms are alternately raised by the wheel at every beat. The pendulum
in ascending, at an angle vy from the vertical, impinges on one of the arms, unlocks the wheel and
carries the arm with it as far as it swings ; the arm then descends with the pendulum, not only to ry,
but farther to an angle (3, less than y. The maintaining force of the arms therefore acts on the
pendulum through y—, and the work which the clock has to do is raising the arms from # to ry.
This is the way in which these escapements have been usually made, I suppose with the view of keep-
ing the pendulum free during as much of its are as possible ; but we shall see that it is much better
to make 3 = — +, or one arm to be taken up by the pendulum just when the other is left; and as
it is also more simple, I shall consider that case first.

In order to find the errors of such an escapement, let p be the length of the arms supposed to
be without weight; Pg the weight they carry at their lower end; 6 the angle which the arm in con-
tact makes with the pendulum when it is vertical. Then the equation of motion will be

P
26 “8 + Hh sin (6 + }

— =).
de” 1 ree:
Mr
2
We may, in considering the error in the going of the clock, neglect the denominator 1 + a not

only because it very nearly =1, but because it only causes a permanent change in the effective

“It must be remembered that experiments of altering the | clock is left to itself the are varies probably more from the varying
clock-weight, to find what effect is produced on the arc, do not | friction and state of the oil on the pallets than from the change of
represent what takes place in the clock naturally ; for when the | force in the train.

636 Mr. DENISON, ON CLOCK ESCAPEMENTS.

length of the pendulum, since one of the arms is, in this form of escapement, always acting on the
pendulum. And expanding sin (6 + @) we may put 1 for cos 0, and @ for sin @ as before,

@O0 g Pp cos § Pp sind
at (a+ Mi )e+ MI }

= My

55 ot &@ g
which is of the form qe 7 (m0 + p) =0;
r

l
* 9 = ] = a 9 g K
. if @ were = 0 the time would = 7 /- , or for a second’s pendulum 7 must = — = 7” say,

which is a very little less than a”. The only part of the force which produces an effect involving
the are is 7, and it is a constant force, Therefore we may apply to it Mr. Airy’s expression for
the increase of time due to such a force acting from a down to — y; and we have

1 (7 ~ed0 = —
eerie wana. a

and it is the same from — y to the other extremity of the are; therefore for the whole oscillation

2p =
Aix mera Sie ie

This is, in fact, Mr. Airy’s result for a recoil escapement ; and if the pallets of a recoil escape-
ment were made of any regular form, so that we could separate the force into one part varying as the
are and the other part constant, it would be the same thing as a gravity escapement, only with
much greater friction, and the important difference, that the force depends upon the train, whereas
in a gravity escapement it is independent, and therefore uniform. Mr. Airy proceeds to remark,
that “the differential coefficient of this quantity with respect to a is

2p a’ — 2" dA

= a/a — > dvs

Hence it appears that the vibrations are quicker “ than they would be without the maintaining force ;
but if the are be increased while the maintaining force remains the same, the vibrations are slower.

If while the arc remains the same the force be increased, the vibrations are quicker.”
: A . : dA
But something else appears also: viz. the important fact, that if y be made =e PF 0,
2 da

provided the force remains the same, as it does in a gravity escapement. And luckily this is a per-
fectly practicable value for yy, though it is larger than a clockmaker would probable make it
without knowing anything of this result; for if a = 120, y will = .7 x 120’ = 84’, and a — y,
or the space in which the unlocking has to take place = 36, which with p = 4 or 5 inches will do
very well for a clock which is liable to such small changes of are as these are. Therefore a gravity
escapement may be made, in which the error will be nothing for a small alteration of the arc; and
in such an escapement there is no such variation in either force or friction as can cause any material

change in the are,
In a recoil escapement we should have to differentiate A with respect both to a and @;

Mr. DENISON, ON CLOCK ESCAPEMENTS. 637

This is always negative; for it cannot be + unless

dp

a ae aie ieee
da a» a
a

which is impossible, since @ always increases faster than a. Therefore the recoil escapement always
gains as the arc increases, as is well known; and the cause of its inferiority to either of the others
is evident.

But still we want to ascertain what the error of a gravity escapement with yy of the proper value
will amount to, for some definite value of da, which the clock is not likely to exceed. Therefore we
must find the value of @.

Now the work done by the clock-weight is raising the weight P through
picos (0 — ry) — cos (6 ++y)} ‘= 2p sin 6 sin ry, 86400 times a day.

Then assuming V and p the same as before (though this clock evidently does not require the same
maintaining power as the dead escapement with its large amount of friction, I believe not half as
much),

Wp 2x9
2Pp sind sin y = —— = ae
ee eo 86400 86400
_ 2p _ 2Ppsind 4x 9 Olja:
Ml 14 x 89 x 864007 864007"
012 SS = ary
*, 86400A = — — {Ja = yt = — 20sec., if — be made = 2, and a= 2°
ays Y
= 0 when vy = a.
as a clockmaker would probably make it, in ignorance of the fact that * should = 4/2;
Y
a’ 2
2 2 _ Qe? af Did
snap am yee ot OSA, ie a 2

oie Venp Ja,
=

,

= .577 sec., for the last mentioned value of St da 5
a

This then is the daily error of a gravity escapement made, as we may say, at random, for an increase
of the arc of 5’, remembering that we have taken @ twice as large as it need be.

4 A F . :
But if % is made of the proper magnitude, so as to make —— = 0, we must differentiate again,
a a

and put a’ = 2+*, in order to find the actual error for a small increase of a: then we have

!

O12 2ad d f
ea = of a second, if da =5.

or the daily rate = —- ——$———— —— =
2 2
i w/e Le

Vou. VIII. Parr V. 4N

=} fh

638 Mr. DENISON, ON CLOCK ESCAPEMENTS.

And we see that the clock will gain if a be either increased or diminished from yV/2. Therefore if
the pendulum be adjusted when the clock is clean to vibrate 2’ or 3’ more than y/2, the are may
diminish (and it will never spontaneously increase) as much as 5’ or 6’ with even a much less error
than that above deduced, which it is to be remembered was already too large in consequence of our
assuming the same maintaining force as in the dead escapement.

I have omitted in this calculation the effect of the impact of the pendulum against the arms, and
the small friction at unlocking, as I found in the calculations which I made retaining them, that
they only introduced a very small term of a lower order than .

There is another form of the gravity escapement, in which, instead of one arm being taken
up just as the other is left, the pendulum is free for some space in the middle of its arc, This
is evidently inferior to the other, for if the are varies, the proportion between the time during
which the pendulum has only its own moment of inertia, and that during which it has that of
one of the arms also, will vary. And it will be seen that the inferiority is still greater from
another cause.

For if we put y for the angle at which each arm is met by the pendulum, and #3 for that to
which it descends; then for the two portions of the are in which the pendulum is acted on by the
arms, we may integrate the same expression as before, only from y to a, and down again to 8;

af ia B+ e— 7's
Ta
dA # [oF 7}
da 7a Va — 2? Va

One value of @ and + that will make this = 0 is evidently B = + y = wr but if B = y there

A=

and

is no maintaining power; and if = — + it becomes the former kind of clock. In order to find
other values,

Leta’ —- B= 0°; «a —2 B=20'-’,
ay =; Wa 2 = 24? - a’,
then we shall find that

dA . 2 a na [oi Dae eee
—— = 0, if 2 = ay = Va - 8 a = >

Since the value of for 8 and + is useless, let us take the highest value for 8 that will leave
a — (3 of a sufficient size to secure the unlocking always taking place; which can hardly be less
than 30’ with arms of moderate length: then 6 will be 90’ and y = 78’. And this leaves only 12’
fcr the maintaining power to act through. An escapement of this sort is therefore barely practi-
cable; and in it the weight of the arms, and consequently the errors of the clock, must be much
larger than in one where the action takes place through 84’ on each side of zero. This kind of
escapement, however, would do for such a clock in which the force acts on the bob of the pendulum
for a short distance at each extremity of the arc—the worst possible place, unless the are through
which it acts satisfies the above condition. However, the object of this paper was to shew that,
mathematically speaking, gravity escapements may be made very superior to the dead escapement
with its large amount of friction and variation of arc, and to remove the cloud which has hitherto
lain over them in consequence of it being supposed that whatever mechanical improvements might
be made in them, they must remain liable to an insuperable mathematical objection.

E. B. DENISON.

er

XLVII. (SuppLement.) On Turret-Clock Remontoirs. By E. B. Denison, M.A.,
of Trinity College, Cambridge.

[Read February 26, 1849.]

I wave given above a general description of a remontoir escapement, and shewn its advantages
when properly made. But a remontoir apparatus may be introduced below the escape-wheel of a
common escapement, and will have the same effect as a remontoir escapement, except that it will
not remove the variable friction of the pallets, and will generally introduce some friction of its own.
For astronomical clocks, probably a remontoir escapement is the best construction. But there is
another class of clocks on which some attention has at last begun to be bestowed in this country,
and which, from the great length and weight that may be given to their pendulums, are capable,
when properly made, of excelling the performance of most astronomical clocks: I mean turret-
clocks. And these clocks require a remontoir more than all others, on account of the great in-
equality in the force of the train, arising from the varying friction of the very heavy machinery,
and the occasional exposure of the oil to a freezing temperature, and the action of the wind on
the hands.

Now any remontoir escapement, to satisfy the condition which I have shewn ought to be
satisfied by them, will require great accuracy in its construction, and will be too expensive to
have any chance of being generally adopted. Moreover, there are two other conditions which a
turret-clock must satisfy, in order to be of any use as a public regulator of other clocks; viz. that
of striking the first blow at exactly the proper second, and that of enabling people to distinguish
every twentieth or thirtieth second by a quick and visible motion of the minute-hand only at those
intervals. These conditions were laid down by the Astronomer-Royal for the Royal Exchange
Clock, and are also proposed by him for the great Clock for the Houses of Parliament. And these
conditions, especially the second of them, can only be satisfied by introducing a remontoir into the
train somewhere between the dial-work and the escapement. In the Exchange Clock a small weight
is raised by a wheel with internal teeth at every twentieth second: in some French turret-clocks
the weight is raised in a somewhat similar manner by two bevelled wheels: in a clock put up
in Edinburgh in the last century, mentioned by Reid in his book on Clock-making, there was
an endless chain remontoir (which has also been attempted again in France) ; but it was removed
on account of the rapid wearing both of the chain and the letting-off pins. But for this, and
the variable friction of such a chain, that kind of remontoir is probably the most tempting, as
it is the most simple, of the gravity remontoirs. Both the other constructions are complicated
and expensive, and have a good deal of friction of their own; and though I think a different
kind of gravity remontoir may still be made, more simple and quite as effective (which however
I shall not stay to describe), I am inclined to propose a spring remontoir as superior to any
gravity one, on account of the greater facility of its construction, and the unusual cireumstance
of its being possible totally to exclude friction in its application; and I may also mention, as an
incidental advantage, that it possesses a sort of natural compensation, the spring being stronger
in cold weather, when the oil on the pallets is less fluid, and therefore a greater maintaining
force is required. I find indeed that a spring remontoir is not new, having been tried in France,

4N2

640 Mr. DENISON ON TURRET-CLOCK REMONTOIRS.

but without success, from evident defects in its construction, the clock sometimes failing to wind
it up, which of course need no more happen with a spring than a gravity remontoir. It seems
also to have been applied to a peculiar kind of escapement; which was trying two experiments
at once,—always an unscientific proceeding.

The obvious mode of applying a spring remontoir is to make the pinion of the escape-wheel
ride upon the axis instead of being fixed to it, and to connect the pinion and the wheel by a
spiral spring. Then if the pinion is turned (say) a quarter round by the train at intervals,
the wheel will be driven through a quarter of a revolution by the force of the spring only.
This is the plan I find described for the French spring remontoir, and a similar one has been
proposed by Mr. Airy, only to be wound up at every beat by means of a double escape-wheel and
pallets, and the principle of it was applied to a chronometer many years ago. But if nothing
more than this is done, the escape-wheel axis will have to turn within the pinion, as in a socket
with considerable pressure and friction upon it, which will probably be worse than the ordinary
friction of the train. The method I propose therefore is, to make the pinion (a brass lantern-
pinion, having the inner end of the spiral spring attached to it) ride upon a steel pin fixed to
the frame in the same line as the axis of the escape-wheel, and having in its end (or rather
in a piece of brass screwed on to its end) the pivot hole for the escape-wheel axis. The outer
end of the spring is to take hold of what I believe is called a dog (the shape of which will be
best described by the drawing), which screws on to the escape-wheel axis (the screw also acting
as a connterpoise), so that the tension of the spring can be adjusted to make the pendulum swing
as far as is required. It is evident that the wheel will thus be driven by the spring without
any friction,

The mode of letting off the train at intervals, adopted in the
Exchange Clock and the above-mentioned Clock by Reid, is by fixing
two or more sets of long teeth on spikes in two or more planes on the
broad rim of a wheel on the same axis as the wheel which drives the
escape-Wheel pinion ; and notches are cut nearly half through the escape-
wheel axis over each set of spikes, which will let a spike pass through
whenever the corresponding notch is in its lowest position; and the
driving wheel is then stopped by one of the other set of spikes coming
against the axis in a place where the corresponding notch is not yet in a
position to let that spike pass. The objection to this is, that the spikes
strike the axis with considerable force, and also press on it pretty heavily
when at rest, which causes additional friction and requires a stronger
maintaining power than would otherwise be necessary. The blow against
the axis it is already proposed to diminish by a fly, to restrain the
velocity of the train when it is let off; and a fly is now used in the
French remontoirs ; where however it is much less needed, for they are
let off by pins raising a lever just like a common striking part; and it
does not signify how hard the lever is struck. But this plan also is
objectionable because of the friction and loss of power in the escape-
wheel in raising such a lever, which is much more than would be due
to the pressure of the same lever, if only exerting a dead pressure on the
axis until it slips through a notch.

I propose to use a fly, but more for the sake of diminishing the pres-
sure on the escape-wheel axis than of diminishing the velocity of the
train; which is immaterial, except so far as it effects the escape-wheel.
The two letting-off pins are to be one at each end of the fly ; and if the
radius of the fly is equal to the diameter of the driving wheel, and the

End view of Cylinder.

Mr. DENISON ON TURRET-CLOCK REMONTOIRS. 641

fly makes twelve revolutions for one of the driving wheel, the fly-pins will only exert 3th of the
pressure on the axis that the spikes on the driving wheel exert. The fly is very light and made of
thin brass, and is of itself a spring; and so its axis will not be stopped with a sudden shock, and
the impact of the end of the fly on the escape-wheel axis may be made inconsiderable. This axis
must be prolonged beyond the frame, to allow a fly of larger radius than the driving wheel to be
used, and must end in a cylinder about half an inch thick. If the remontoir is to be let off every
thirty seconds, which is a better interval for observation than twenty, the projecting cylinder may
have two notches cut in it as before described, if the escape-wheel revolves in a minute. But
it is generally made to revolve in two minutes, in order to save a wheel in the train; and in that
case the letting off may be done better than with a four-armed fly, by making two notches across the
end of the cylinder, at right angles to each other, one broad, and the other narrow and deeper, so
that a broad pin will pass through one of the notches only, and a narrow and long pin through the
other only. These pins are of course to be parallel to the axis of the fly; and the fly pinion must
have half the number of leaves that the escape-wheel pinion has, whatever may be the number of
teeth of the driving wheel. A Church-clock is now making on this plan. I have added a
drawing of the material parts, placed in the way most convenient for shewing their action.

~

E. B. DENISON.

XLVIII. On the Formation of the Central Spot of Newton's Rings beyond the Critical
Angle. By G. G. Sroxss, M.A., Fellow of Pembroke College, Cambridge.

[Read December 11, 1848.]

Wuen Newton’s Rings are formed between the under surface of a prism and the upper surface
of a lens, or of another prism with a slightly convex face, there is no difficulty in increasing the angle
of incidence on the under surface of the first prism till it exceeds the critical angle. On viewing the
rings formed in this manner, it is found that they disappear on passing the critical angle, but that the
central black spot remains. ‘The most obvious way of accounting for the formation of the spot under
these circumstances is, perhaps, to suppose that the forces which the material particles exert on the
ether extend to a small, but sensible distance from the surface of a refracting medium; so that in the
case under consideration the two pieces of glass are, in the immediate neighbourhood of the point of
contact, as good as a single uninterrupted medium, and therefore no reflection takes place at the
surfaces, This mode of explanation is however liable to one serious objection. So long as the angle
of incidence falls short of the critical angle, the central spot is perfectly explained, along with the rest
of the system of which it forms a part, by ordinary reflection and refraction. As the angle of inci-
dence gradually increases, passing through the critical angle, the appearance of the central spot changes
gradually, and but slightly. To account then for the existence of this spot by ordinary reflection
and refraction so long as the angle of incidence falls short of the critical angle, but by the finite
extent of the sphere of action of the molecular forces when the angle of incidence exceeds the critical
angle, would be to give a discontinuous explation to a continuous phenomenon. If we adopt the
latter mode of explanation in the one case we must adopt it in the other, and thus separate the theory
of the central spot from that of the rings, which to all appearance belong to the same system ; although
the admitted theory of the rings fully accounts likewise for the existence of the spot, and not only for
its existence, but also for some remarkable modifications which it undergoes in certain circumstances*.

Accordingly the existence of the central spot beyond the critical angle has been attributed by
Dr. Lloyd, without hesitation, to the disturbance in the second medium which takes the place of that
which, when the angle of incidence is less than the critical angle, constitutes the refracted light F.
The expression for the intensity of the light, whether reflected or transmitted, has not however been
hitherto given, so far as I am aware. The object of the present paper is to supply this deficiency.

In explaining on dynamical principles the total internal reflection of light, mathematicians have
been led to an expression for the disturbance in the second medium involving an exponential, which
contains in its index the perpendicular distance of the point considered from the surface. It follows
from this expression that the disturbance is insensible at the distance of a small multiple of the
length of a wave from the surface. This circumstance is all that need be attended to, so far as the
refracted light is concerned, in explaining total internal reflection ; but in considering the theory of
the central spot in Newton’s Rings, it is precisely the superficial disturbance just mentioned that must
be taken into account. In the present paper I have not adopted any special dynamical theory: I
have preferred deducing my results from Fresnel’s formule for the intensities of reflected and re-

* L allude especially to the phenomena described by Mr. Airy + Report on the present state of Physical Optics. Reports of
in a paper printed in the fourth Volume of the Cambridge Philoso- | the British Association, Vol. 111. p.310.
phical Transactions, p. 409,

Mr. STOKES ON THE FORMATION OF THE CENTRAL SPOT, ETC. 643

fracted polarized light, which in the case considered became imaginary, interpreting these imaginary
expressions, as has been done by Professor O’Brien*, in the way in which general dynamical con-
siderations show that they ought to be interpreted.

By means of these expressions, it is easy to calculate the intensity of the central spot. I have
only considered the case in which the first and third media are of the same nature: the more general
case does not seem to be of any particular interest. Some conclusions follow from the expression
for the intensity, relative to a slight tinge of colour about the edge of the spot, and toa difference in
the size of the spot ascending as it is seen by light polarized in, or by light polarized perpendicularly
to the plane of incidence, which agree with experiment.

1. Let a plane wave of light be incident, either externally or internally, on the surface of an
ordinary refracting medium, suppose glass. Regard the surface as plane, and take it for the plane
wy; and refer the media to the rectangular axes of w, y, x, the positive part of the latter being
situated in the second medium, or that into which the refraction takes place. Let J, m,m be the
cosines of the angles at which the normal to the incident wave, measured in the direction of
propagation, is inclined to the axes; so that m=0 if we take, as we are at liberty to do, the axis
of y parallel to the trace of the incident wave on the reflecting surface. Let V,V, V’ denote the
incident, reflected, and refracted vibrations, estimated either by displacements or by velocities, it
does not signify which; and let a, a, a’ denote the coefficients of vibration. Then we have the
following possible system of vibrations :

2
V=acos — (vt —lv — nz),
r
2a
Vi= pos Or ta ns), Shoreeposdnanson “)}:
2 , ,
V' =a! cos (v't - Ua —n'z),

In these expressions v, v’ are the velocities of propagation, and ), )’ the lengths of wave, in the
first and second media; so that v, v’, and the velocity of propagation in vacuum, are proportional
to X, A’, and the length of wave in vacuum :'/ is the sine, and 7 the cosine of the angle of incidence,
U the sine, and m’ the cosine of the angle of refraction, these quantities being connected by
the equations

+25, TAIOV eS WEI ces 5 EB oer (1).

2. The system of vibrations (A) is supposed to satisfy certain linear differential equations of
motion belonging to the two media, and likewise certain linear equations of condition at the surface of
separation, for which x = 0. These equations lead to certain relations between a, @,, and a’, by virtue
of which the ratios of a and a’ toa are certain functions of /, v, and v’, and it might be also
of X. The equations, being satisfied identically, will continue to be satisfied when l’ becomes
greater than 1, and consequently 7’ imaginary, which may happen, provided v' >v; but the
interpretation before given to the equations (A) and (1) fails.

When 7’ becomes imaginary, and equal to v\/-1, v’ being equal to / 1 — 1, x, instead of
appearing under a circular function in the third of equations (A), appears in one of the exponentials

9 =
e**”*, k’ being equal to = . By changing the sign of \/-—1 we should get a second system

of equations (A), satisfying, like the first system, all the equations of the problem ; and we should

* Cambridge Philosophical Transactions, Vol, vi11. p. 20

644 Mr. STOKES, ON THE FORMATION OF THE CENTRAL SPOT

ve r — 36
get two new systems by writing vé + 4 for vt. By combining these four systems by addition and

subtraction, which is allowable on account of the linearity of our equations, we should be able to get
rid of the imaginary quantities, and likewise of the exponential e**’*, which does not correspond to
the problem, inasmuch as it relates to a disturbance which increases indefinitely in going from the
surface of separation into the second medium, and which could only be produced by a disturbing
cause existing in the second medium, whereas none such is supposed to exist.

3. The analytical process will be a good deal simplified by replacing the expressions (4) by
2
the following symbolical expressions for the disturbance, where & is put for a so that kv =k'v’';
v= aghet—le—n2V/ 1

| AE pe ical deine tes CB):

, 1K (vt-Tx—n'2)N=1 |
V' =a'ke ene) >

In these expressions, if each exponential of the form eV be replaced by cos P + \/—1 sin P, the
real part of the expressions will agree with (4), and therefore will satisfy the equations of the pro-
blem. The coefficients of y/— 1 in the imaginary part will be derived from the real part by writing

r : Aerts : :
t+ re for ¢, and therefore will form a system satisfying the same equations, since the form of these
v

equations is supposed in no way to depend on the origin of the time; and since the equations are
linear they will be satisfied by the complete expressions (B).

Suppose now 1’ to become greater than 1, so that m’ becomes +yv\/—1. Whichever sign we
take, the real and imaginary parts of the expressions (B), which must separately satisfy the equations
of motion and the equations of condition, will represent two possible systems of waves; but the
upper sign does not correspond to the problem, for the reason already mentioned, so that we must
use the lower sign. At the same time that 2’ becomes v'\/ — 1, let a, a, a’ become

pe, per, pen,

then we have the symbolical system

respectively :
V= pegs _ gklut—le—nz) v1,
Vip es Ns eet ree (2)
ae pe v2 enue. ek Wt-t)V-7,
of which the real part
V = pcos {k(vt — lw - nx) - 0}
V = p,cos {k(vt — lw + nz) - 0}, ee. (D).
V’ = p'e~** cos {k'(v't -I'a) - 6, |
forms the system required.

As I shall frequently have occasion to allude to a disturbance of the kind expressed by the
last of equations (D), it will be convenient to have a name for it, and I shall accordingly call it
a superficial undulation.

4, The interpretation of our results is not yet complete, inasmuch as it remains to consider
what is meant by V’, When the vibrations are perpendicular to the plane of incidence there is no

OF NEWTON'S RINGS BEYOND THE CRITICAL ANGLE. 645

difficulty. In this case, whether the angle of incidence be greater or less than the critical angle,
V’ denotes a displacement, or else a velocity, perpendicular to the plane of incidence. When the
vibrations are in the plane of incidence, and the angle of incidence is less than the critical angle,
V’ denotes a displacement or velocity in the direction of a line lying in the plane az, and inclined

” T * 7 : °
at angles 7 —7, — (= - i) to the axes of w, x, # being the angle of refraction. But when the

angle of incidence exceeds the critical angle there is no such thing as an angle of refraction, and
the preceding ankerpretation fails. Instead therefore of considering the whole vibration V’, consider
its resolved parts V,’, V.' in the direction of the axes of w, x. Then when the angle of incidence is less
than the critical angle, we have

V,=-nV' =-cosi?.V’; V{=IUV’ =sini’.V’,

V’ being given by (A), and being reckoned positive in that direction which makes an acute angle
with the positive part of the axis a z. When the angle of incidence exceeds the critical angle, we

must first replace the coefficient of V’ in V2, namely — n’, by v 1 aval 1, and then, retaining v’

£ Tv si : .
for the coefficient, add 3 the phase, according to what was explained in the preceding article.

Hence, when the vibrations take place in the plane of incidence, and the angle of incidence
exceeds the critical angle, V’ in (D) must be interpreted to mean an expression from which the
vibrations in the directions of a, x may be obtained by multiplying by v’, 7’, respectively, and

‘ ° : T
increasing the phase in the former case by at Consequently, so far as depends on the third

of equations (D), the particles of ether in the second medium describe small ellipses lying in
the plane of incidence, the semi-axes of the ellipses being in the directions of w, x, and being pro-
portional to v’, 7’, and the direction of revolution being the same as that in which the incident ray
would have to revolve in order to diminish the angle of incidence.

Although the elliptic paths of the particles lie in the plane of incidence, that does not prevent
the superficial vibration just considered from being of the nature of transversal vibrations. For it
is easy to see that the equation

Cee Uh ee
+

— =0
da dz

is satisfied ; and this equation expresses the condition that there is no change of density, which is the
distinguishing characteristic of transversal vibrations.

5. When the vibrations of the incident light take place in the plane of incidence, it appears
from investigation that the equations of condition relative to the surface of separation of the two
media cannot be satisfied by means of a system of incident, reflected, and refracted waves, in which
the vibrations are transversal. If the media be capable of transmitting normal vibrations with
velocities comparable with those of transversal vibrations, there will be produced, in addition to
the waves already mentioned, a series of reflected and a series of refracted waves in which the
vibrations are normal, provided the angle of incidence be less than either of the two critical angles
corresponding to the reflected and refracted normal vibrations respectively. It has been shewn
however by Green, in a most satisfactory manner, that it is necessary to suppose the velocities of
propagation of normal vibrations to be incomparably greater than those of transversal vibrations,
which comes to the same thing as regarding the ether as sensibly incompressible; so that the two
critical angles mentioned above must be considered evanescent*. Consequently the reflected and

* Cambridge Philosophical Transactions, Vol. vit. p. 2.

Vor. VIII; Pant V. 40

646 Mr. STOKES, ON THE FORMATION OF THE CENTRAL SPOT

refracted normal waves are replaced by undulations of the kind which I have called superficial.
Now the existence of these superficial undulations does not affect the interpretation which has been
given ‘to the expressions (4) when the angle of incidence becomes greater than the critical angle
corresponding to the refracted transversal wave; in fact, so far as regards that interpretation, it
is immaterial whether the expressions (4) satisfy the linear equations of motion and condition
alone, or in conjunction with other terms referring to the normal waves, or rather to the superficial
undulations which are their representatives. The expressions (D) however will not represent the
whole of the disturbance in the two media, but only that part of it which relates to the transversal
waves, and to the superficial undulation which is the representative of the refracted transversal wave.

6. Suppose now that in the expressions (4) m becomes imaginary, »’ remaining real, or that
nm and n’ both become imaginary. The former case occurs in the theory of Newton’s Rings when
the angle of incidence on the surface of the second medium becomes greater than the critical angle,
and we are considering the superficial undulation incident on the third medium: the latter case
would occur if the third medium as well as the second were of lower refractive power than the
first, and the angle of incidence on the surface of the second were greater than either of the critical
angles corresponding to refraction out of the first into the second, or out of the first into the third.
Consider the case in which » becomes imaginary, ”’ remaining real; and let /-1=2v. Then

it may be shewn as before that we must put — v»/—1, and not v \/ —1, for n; and using p, 0
in the same sense as before, we get the symbolical system,

v= pe“, en hie, gk(vt— tz 7,
-0V 1

pe kvz _k(vt —lx)\/=1
Vi= pe .€ ;

-€
V= Rene ef t-le—n2V 1
to which corresponds the real system
V = pe~"* cos fk (vt — lw) - 6},
V = pel” cos {k (vt — la) — 0}, (-+---(P).
V'= p'cos$k'(v't-l'v-n'z) - 6}, |

When the vibrations take place in the plane of incidence, V and V, in these expressions must
be interpreted in the same way as before. As far as regards the incident and reflected superficial
undulations, the particles of ether in the first medium will describe small ellipses lying in the plane
of incidence. The ellipses will be similar and similarly situated in the two cases ; but the direction
of revolution will be in the case of the incident undulation the same as that in which the refracted
ray would have to turn in order to diminish the angle of refraction, whereas in the reflected
undulation it will be the opposite.

It is unnecessary to write down the formule which apply to the case in which » and n’ both
become imaginary.

4. If we choose to employ real expressions, such as (D) and (F), we have this general rule.
When any one of the undulations, incident, reflected, or refracted, becomes superficial, remove z
from under the circular function, and insert the exponential ¢~*”*, e*”*, or « “”*, according as the
incident, reflected, or refracted undulation is considered. At the same time put the coefficients,
which become imaginary, under the form p (cos 0 + / —1 sin @), the double sign corresponding
to the substitution of 4y./— 1, or +'\/—1 form or m’, retain the modulus p for coefficient,
and subtract @ from the phase.

It will however be far more convenient to employ symbolical expressions such as (B). These
expressions will remain applicable without any change when m or n’ becomes imaginary: it will

OF NEWTON’S RINGS BEYOND THE CRITICAL ANGLE. 647

only be necessary to. observe to take + » V Aner or + p'\/—1 with the negative sign. If we had
chosen to employ the expressions (B) with the opposite sign in the index, which would have done
equally well, it would then have been necessary to take the positive sign.

8. We are now prepared to enter on the regular calculation of the intensity of the central
spot; but before doing so it will be proper to consider how far we are justified in omitting the
consideration of the superficial undulations which, when the vibrations are in the plane of incidence,
are the representatives of normal vibrations. These undulations may conveniently be called normal
superficial undulations, to distinguish them from the superficial undulations expressed by the third
of equations (D), or the first and second of equations (#'), which may be called transversal. The
former name however might, without warning, be calculated to carry a false impression; for the
undulations spoken of are not propagated by way of condensation and rarefaction; the disturbance
is in fact precisely the same as that which exists near the surface of deep water when a series of
oscillatory waves is propagated along it, although the cause of the propagation is extremely
different in the two cases.

Now in the ordinary theory of Newton’s Rings, no account is taken of the normal superficial
undulations which may be supposed to exist ; and the result so obtained from theory agrees very well
with observation. When the angle of incidence passes through the critical angle, although a material
change takes place in the nature of the refracted transversal undulation, no such change takes place
in the case of the normal superficial undulations: the critical angle is in fact nothing particular as
regards these undulations. Consequently, we should expect the result obtained from theory when
the normal superficial undulations are left out of consideration to agree as well with experiment
beyond the critical angle as within it.

9. It is however one thing to show why we are justified in expecting a near accordance between
the simplified theory and experiment, beyond the critical angle, in consequence of the observed
accordance within that angle; it is another thing to show why a near accordance ought to be expected
both in the one case and in the other. The following considerations will show that the effect of the
normal superficial undulations on the observed phenomena is most probably very slight.

At the point of contact of the first and third media, the reflection and refraction will take place
as if the second medium were removed, so that the first and third were in contact throughout. Now
Fresnel’s expressions satisfy the condition of giving the same intensity for the reflected and refracted
light whether we suppose the refraction to take place directly out of the first medium into the
third, or take into account the infinite number of reflections which take place when the second
medium is interposed, and then suppose the thickness of the interposed medium to vanish. Conse-
quently the expression we shall obtain for the intensity by neglecting the normal superficial undu-
lations will be strictly correct for the point of contact, Fresnel’s expressions being supposed correct,
and of course will be sensibly correct for some distance round that point. Again, the expression for
the refracted normal superficial undulation will contain in the index of the exponential — k / x, in

2
place of -k V/ F- as x, which occurs in the expression for the refracted transversal superficial
undulation; and therefore the former kind of undulation will decrease much more rapidly, in receding
from the surface, than the latter, so that the effect of the former will be insensible at a distance from
the point of contact at which the effect of the latter is still important. If we combine these two
considerations, we can hardly suppose the effect of the normal superficial undulations at intermediate
points to be of any material importance.

10. The phenomenon of Newton’s Rings is the only one in which T see at present any chance
of rendering these undulations sensible in experiment: for the only way in which I can conceive
them to be rendered sensible is, by their again producing transversal vibrations ; and in consequence
of the rapid diminution of the disturbance on receding from the surface, this can only happen when

402

648 Mr. STOKES, ON THE FORMATION OF THE CENTRAL SPOT

there exists a second reflecting surface in close proximity with the first. It is not my intention to
pursue the subject further at present, but merely to do for angles of incidence greater than the
critical angle what has long ago been done for smaller angles, in which case light is refracted in the
ordinary way. Before quitting the subject however I would observe that, for the reasons already
mentioned, the near accordance of observation with the expression for the intensity obtained when
the normal superficial undulations are not taken into consideration cannot be regarded as any valid
argument against the existence of such undulations.

11. Let Newton’s Rings be formed between a prism and a lens, or a second prism, of the same
kind of glass. Suppose the incident light polarized, either in the plane of incidence, or in a plane
perpendicular to the plane of incidence. Let the coefficient of vibration in the incident light be
taken for unity ; and, according to the notation employed in Airy’s J'racf, let the coefficient be mul-
tiplied by 6 for reflection and by ¢ for refraction when light passes from glass into air, and by e for
reflection and f for refraction when light passes from air into glass. In the case contemplated 8, c,
e, f become imaginary, but that will be taken into account further on. Then the incident vibration
will be represented symbolically by

ehtvt lene =1

according to the notation already employed ; and the reflected and refracted vibrations will be repre-
sented by

k(vt—la+ =
be (vt —lr+nz)J ;
ce *'=, eh Wt-taV 1

Consider a point at which the distance of the pieces of glass is D; and, as in the usual investi-
gation, regard the plate of air about that point as bounded by parallel planes. When the superficial
undulation represented by the last of the preceding expressions is incident on the second surface, the
coefficient of vibration will become eq, q being put for shortness in place of ¢~*”?; and the reflected
and refracted vibrations will be represented by

eqee*””, rps dah lan

cgf htm - I
# being now measured from the lower surface. It is evident that each time that the undulation
passes from one surface to the other the coefficient of vibration will be multiplied by q, while the
phase will remain the same. Taking account of the infinite series of reflections, we get for the sym-
bolical expression for the reflected vibration

fb ae cefg (1 ms eg st eq’ + er)t eh(et—betne\/=1
Summing the geometric series, we get for the coefficient of the exponential

cefy?
(45 —
‘i 1-é¢
Now it follows from Fresnel’s expressions that
b=-e, ef =1-e*.

These substitutions being made in the coefficient, we get for the symbolical expression for the
reflected vibration

(UH 07) © pet-tetner
fee tml svevecsees- (GQ).

* Ihave proved these equations ina very simple manner, without any reference to Fresnel’s formule, in a paper which will appear in
the next number of the Cambridge and Dublin Mathematical Journal,

OF NEWTON’S RINGS BEYOND THE CRITICAL’ ANGLE. 649
Let the coefficient, which is imaginary, be put under the form p (cosy + /—1sin ); then
the real part of the whole expression, namely
pcos }k (vt — law + nz) + Wt,

will represent the vibration in the reflected light, so that p* is the intensity, and \ the acceleration
of phase.

12. Let i be the angle of incidence on the first surface of the plate of air, « the refractive
index of glass; and let \ now denote the length of wave in air. Then in the expression for q

25 p
kiy'= <a V v2 sin®é — 1.

In the expression for b we must, according to Art. 2, take the imaginary expression for cos?
with the negative sign. We thus get for light polarized in the plane of incidence (Airy’s T'ract,

p- 362, 2nd edition*), changing the sign of wile
b =cos20 + \f —1 sin20,

Sie sin’? — 1

pp COS a

where

tan@ = nonnscoooned (EL

Putting C for the coefficient in the expression (G), we have
1-¢ 1-¢°
bt- qb (1 —q*)cos20 — \/—1 (1 + q’) sin 20
Ps (1-9) S(1 — 7) cos20 + Cy Ae (1 + q’) sin 20} ;

C=

(1 - ¢)? + 4¢’ sin’ 20 i
whence
1 2
tan yy = ; a HELL!) sco paooannecspecceoonne (a)
G=o)
pe
p= G4)! a Aglaiaek Cae (4),
where

2rD

| aera ey
aKa V pe sinti-l

q=e ~poondiccciocen Oy
If we take p positive, as it will be convenient to do, we must take yy so that cos yy and cos 20
may have the same sign. Hence from (3) sin yy must be positive, since sin 20 is positive, inasmuch

as @ lies between 0 and zu Hence, of the two angles lying between — a and 7 which satisfy (2),

we must take that which lies between 0 and z.

For light polarized perpendicularly to the plane of incidence, we have merely to substitute @ for
@ in the equations (3) and (4), where

tan p

The value of g does not depend on the nature of the polarization.

p. v/ sin’? — 1
cost

RED

* Mr. Airy speaks of “ vibrations perpendicular to the plane of | which requires us to enter into the question whether the vibrations
incidence,” and “‘ vibrations parallel to the plane of incidence,” | in plane polarized light are in or perpendicular to the plane of
adopting the theory of Fresnel ; but there is nothing in this paper | polarization,

650 Mr. STOKES, ON THE FORMATION OF THE CENTRAL SPOT

13. For the transmitted light we have an expression similar to (@), with — mz in place of nz,
and a different coefficient C,, where

= 1 Pg? a) en ee eee
C= cgf(i+ eg +eg'+...) Te 5 ee pea:

When the light is polarized in the plane of incidence we have
—/—1.2q sin 20
(1 = q°) cos 20 — »/—1(1 + q’) sin20

_ 2qsin 20 $(1 + q°) sin26 — /-1(1 — q) cos 20%

‘

(1 — 4) + 4q°sin’20 eins
so that if yy, and p, refer to the transmitted light we have
tan y= — 3 cot DO eath srs scacccs (8);
4q° sin?20
p2 iE hee ae (9).

7 (1 — q)? + 4q° sin*20

If we take p, positive, as it will be supposed to be, we must take yy, such that cosy, may be
positive; and therefore, of the two angles lying between — + and a which satisfy (8), we must

choose that which lies between -= and +2. Hence, since from (3) and (8) y, is of the form
T : , .
Wt a +n, n being an integer, we must take vW=W- a

For light polarized perpendicularly to the plane of incidence we have only to put @ for @. It
follows from (4) and (9) that the sum of the intensities of the reflected and transmitted light is equal
to unity, as of course ought to be the case. This renders it unnecessary to discuss the expression
for the intensity of the transmitted light.

14. Taking the expression (4) for the intensity of the reflected light, consider first how it varies
on receding from the point of contact.

As the point of contact D = 0, and therefore from (5) q=1, and therefore p = 0, or there is
absolute darkness. On receding from the point of contact q decreases, but slowly at first, inasmuch
as D varies as 7°, r being the distance from the point of contact. It follows from (4) that the
intensity p” varies ultimately as 7‘, so that it increases at first with extreme slowness. Consequently
the darkness is, as far as sense can decide, perfect for some distance round the point of contact.
Further on q decreases more rapidly, and soon becomes insensible. Consequently the intensity de-
creases, at first rapidly, and then slowly again as it approaches its limiting value 1, to which it soon
becomes sensibly equal. All this agrees with observation.

15. Consider next the variation of intensity as depending on the colour. The change in @ and
in passing from one colour to another is but small, and need not here be taken into account: the
quantity whose variation it is important to consider is g. Now it follows from (5) that g changes
the more rapidly in receding from the point of contact the smaller be A. Consequently the spot
must be smaller for blue light than for red; and therefore towards the edge of the spot seen by
reflection, that is beyond the edge of the central portion of it, which is black, there is a predominance
of the colours at the blue end of the spectrum; and towards the edge of the bright spot seen by
transmission the colours at the red end predominate. The tint is more conspicuous in the trans-

od

OF NEWTON’S RINGS BEYOND THE CRITICAL ANGLE. 651

mitted, than in the reflected light, in consequence of the quantity of white light reflected about the
edge of the spot. The separation of colours is however but slight, compared with what takes place
in dispersion or diffraction, for two reasons. First, the point of minimum intensity is the same for
all the colours, and the only reason why there is any tint produced is, that the intensity approaches
more rapidly to its limiting value 1 in the case of the blue than in the case of the red. Secondly,
the same fraction of the incident light is reflected at points for which D < \, and therefore 7 « 4/2;
and therefore, on this account also, the separation of colours is less than in diffraction, where the
colours are arranged according to the values of ), or in dispersion, where they are arranged according
to values of \~? nearly. These conclusions agree with observation. A faint blueish tint may be
perceived about the dark spot seen by reflection ; and the fainter portions of the bright spot seen
by transmission are of a decided reddish brown.

16. Let us now consider the dependance of the size of the spot on the nature of the polarization.
Let s be the ratio of the intensity of the transmitted light to that of the reflected; s,, s., the par-
ticular values of s belonging to light polarized in the plane of incidence and to light polarized
perpendicularly to the plane of incidence respectively ; then

4q° sin°20 4q° sin°2 pb
Sir eae -eNa de) |) Sian ae ana 8
(Ga (henge)
Ss, sin 20° et
ae asd) = {(u’ + 1) sin?é — 1}*...... (10).

Now according as s is greater or less, the spot is more or less conspicuous; that is, conspicuous
in regard to extent, and intensity at some distance from the point of contact; for in the immediate
‘neighbourhood of that point the light is in all cases wholly transmitted. Very near the critical angle
we have from (10) s.=u‘s,, and therefore the spot is much more conspicuous for light polarized
perptndicularly to the plane of incidence than for light polarized in that plane. As i increases the
spots seen in the two cases become more and more nearly equal in magnitude: they become exactly
alike when i=1, where

sin®, =

1+ yp
When 7 becomes greater than « the order of magnitude is reversed ; and the spots become more
and more unequal as 7 increases, When i=90° we have s,=y's,, so that the inequality becomes very
great. This however must be understood with reference to relative, not absolute magnitude; for
when the angle of incidence becomes very great both spots become very small.

I have verified these conclusions by viewing the spot through a rhomb of Iceland spar, with its
principal plane either parallel or perpendicular to the plane of incidence, as well as by using a
doubly refracting prism; but I have not attempted to determine experimentally the angle of inci-
dence at which the spots are exactly equal. Indeed, it could not be determined in this way with
any precision, because the difference between the spots is insensible through a considerable range of
incidence.

17. It is worthy of remark that the angle of incidence « at which the spots are equal, is exactly
that at which the difference of acceleration of phase of the oppositely polarized pencils, which arises
from total internal reflection, is a maximum.

When i =. we have

9
“ 5 2u
sin 29 = sin2@ = —>- er whence cot @ = tan p = ..-(11) 5
w+

(re wy = 9)? ee Sed bars (12
G+)’ 0 — ¢) + 16y?q"” where q = 6 ++ (12),

and p =

652 Mr. STOKES, ON THE FORMATION OF THE CENTRAL SPOT

If we determine in succession the angles 0, ¢, 7 from the equations cot 0 =p, tan (= q, tan 7=
sin 26 tan2 G we have p? = 1— p?=4 versin 27. The expression for the intensity may be adapted
to numerical computation in the same way for any angle of incidence, except that 6 or @ must be

determined by (2) or (6) instead of (11), and g by (5) instead of (12).
18. When light is incident at the critical angle, which I shall denote by ry, the expression for

: : 0 i
the intensity takes the form a: Putting for shortness 4/,,? sin’i — 1 = w, we have ultimately

27D w
q=1-——w, tan@=@= sed ="_ >) p=pd;

cost wl
(=) ie 4
x i oe
P= (ey) 1 2 a pt
* a joa Xr 1 age

r=————-...... (13),
according as the light is polarized in or perpendicularly to the plane of incidence. The same for-
mule may be obtained from the expression given at page 304 of Airy’s T’ract, which gives the inten-

and we get in the limit

0
sity when i <-y, and which like (4) takes the form 7 when 7 becomes equal to y, in which case

e becomes equal to — 1.

19. When z becomes equal to +, the infinite series of Art. 11 ceases to be convergent: in fact, its
several terms become ultimately equal to each other, while at the same time the coefficient by which
the series is multiplied vanishes, so that the whole takes the form 0xc. The same remark applies
to the series at page 303 of Airy’s Tract. If we had included the coefficient in each term of the
series, we should have got series which ceased to be convergent at the same time that their several
terms vanished. Now the sum of such a series may depend altogether on the point of view in which
it is regarded asa limit. Take for example the convergent infinite series

: : ; 2a sin
f (a, y) = wsiny + Fa*sin 3y + 20° sin5y +... =} tan™! nected

1—a*’

where w is less than 1, and may be supposed positive. When # becomes 1 and y vanishes
f (x,y) becomes indeterminate, and its limiting value depends altogether upon the order in which we
suppose w and y to receive their limiting values, or more generally upon the arbitrary relation which
we conceive imposed upon the otherwise independent variables w and y as they approach their limit-
ing values together. Thus, if we suppose y first to vanish, and then a to become 1, we have

Ff (#,y)=0; but if we suppose first to become 1, and then y to vanish, f(v,y) becomes + ip +or—

according as y vanishes positively or negatively. Hence in the case of such a series a mode of
approximating to the value of # or y, which in general was pefectly legitimate, might become inad-
missible in the extreme case in which a=1, or nearly =1. Consequently, in the case of Newton’s
Rings when i ~+y is extremely small, it is no longer safe to neglect the defect of parallelism of the
surfaces. Nevertheless, inasmuch as the expression (4), which applies to the case in which i>y,
and the ordinary expression which applies when i<-y, alter continuously as é alters, and agree with (13)
when i=-y, we may employ the latter expression in so far as the phenomenon to be explained alters
continuously as i alters. Consequently we may apply the expression (13) to the central spot when
i=ry, or nearly =r, at least if we do not push the expression beyond values of D corresponding to
the limits of the central spot as seen at other angles of incidence. To explain however the precise

OF NEWTON’S RINGS BEYOND THE CRITICAL ANGLE. 653

mode of disappearance of the rings, and to determine their greatest dilatation, we should have to
enter on a special investigation in which the inclination of the surfaces should be taken into
account.

20. I have calculated the following Table of the intensity of the transmitted light, taking the
intensity of the incident light at 100. The Table is calculated for values of D increasing by +),
and for three angles of incidence, namely, the critical angle, the angle « before mentioned, and a
considerable angle, for which I have taken 60°. I have supposed » =1°63, which is about the refrac-
tive index for the brightest part of the spectrum in the case of flint glass. This value of u gives
ry =37°51, . = 42°18’. The numerals I., II. refer to light polarized in and perpendicularly to the
plane of incidence respectively.

io

I.and II.

AS

_

0
1
2
3
4
5
6
vi
8
9
0)
1

_

_
3°
— Ss mm mH WO OH BHO

So meee BO & DH

21. A Table such as this would enable us to draw the curve of intensity, or the curve in which
the abscissa is proportional to the distance of the point considered from the point of contact, and the
ordinate proportional to the intensity. Tor this purpose it would only be requisite to lay down on
the axis of the abscisss, on the positive and negative sides of the origin, distances proportional to the
squares of the numbers in the first column, and to take for ordinates lengths proportional to the
numbers in one of the succeeding columns. To draw the curve of intensity for i= or for i= 60°,
the table ought to have been calculated with smaller intervals between the values of D; but the law
of the decrease of the intensity cannot be accurately observed.

22. From the expression (13) compared with (4), it will be seen that the intansity decreases
much more rapidly, at some distance from the point of contact, when i is considerably greater than y
than when i= nearly. This agrees with observation. What may be called the ragged edge of
the bright spot seen by transmission is in fact much broader in the latter case than in the former.

When i becomes equal to 90° there is no particular change in the value of q, but the angles 6
and @ become equal to 90°, and therefore sin 20 and sin 2q vanish, so that the spot vanishes. Ob-
servation shows that the spot becomes very small when é becomes nearly equal to 90°.

Vor. VIII. Parr V. 4P

654 Mr. STOKES, ON THE FORMATION OF THE CENTRAL SPOT

23. Suppose the incident light to be polarized in a plane making an angle a with the plane
of incidence. Then at the point of contact the light, being transmitted as if the first and third
media formed one uninterrupted medium, will be plane polarized, the plane of polarization being the
same as at first. At a sufficient distance from the point of contact there is no sensible quantity of
light transmitted. At intermediate distances the transmitted light is in general elliptically polarized,
since it follows from (8) and the expression thence derived by writing @ for @ that the two streams
of light, polarized in and perpendicularly to the plane of incidence respectively, into which the inci-
dent light may be conceived to be decomposed, are unequally accelerated or retarded. At the point
of contact, where g=1, these two expressions agree in giving y,,=0. Suppose now that the trans-
mitted light is analyzed, so as to extinguish the light which passes through close to the point of
contact. Then the centre of the spot will be dark, and beyond a certain distance all round there
will be darkness, because no sensible quantity of light was incident on the analyzer ; but at inter-
mediate distances a portion of the light incident on the analyzer will be visible. Consequently the
appearance will be that of a luminous ring with a perfectly dark centre.

24. Let the coefficient of vibration in the incident light be taken for unity; then the incident
vibration may be resolved into two, whose coefficients are cosa, sina, belonging to light polarized
in and perpendicularly to the plane of incidence respectively. The phases of vibration will be
accelerated by the angles y,,, ,, and the coefficients of vibration will be multiplied by p, p,, if w,,
p,, are what \/, p, in Art. (13) become when @ is put for 6. Hence we may take

2a
p, COS a. COs ee (vt — wa’) + v! 5

4 Qa '
ss (vt —pw') +,

to represent the vibrations which compounded together make up the transmitted light, 2’ being mea-
sured in the direction of propagation. The light being analyzed in the way above mentioned, it is
only the resolved parts of these vibrations in a direction perpendicular to that of the vibrations in

the incident light which are preserved. We thus get, to express the vibration with which we are
concerned,

sin @ cos | cos ( ) = (vt = wv’) ) r t 4
a ‘ + §
Pi ia 5 P,, COS ( (v pa) + v.) 5)

which gives for the intensity (7) at any point of the ring

I = isin’ 2a }(p, cos Wy — p,, cos Wy)? + (p, sin, — p, sin W,)°} --- (14),

= Fsin’2a tp; +p, — 2p,p,, c08 (Wy, — y)}-
Let P,, Q, be _Tespectively the real part of the expression at the second side of (7) and the
coefficient of »/— 1, and let P,; Q, be what P,, Q, become when p is put for 8. Then we may
if we please replace (14) by
T= sin? 2a} (Py — Py)? + (Q, — Qy)*h. vesccesseeeeere (15):

The ring is brightest, for a given angle of incidence, when a = 45°. When i=., the two kinds
of polarized light are transmitted in the same proportion ; but it does not therefore follow that the
ring vanishes, inasmuch as the change of phase is different in the two cases, In fact, in this case
the angles @, 0 are complementary ; so that cot 2@, cot 20 are equal in magnitude but opposite in
sign, and therefore from (8) the phase in the one case is accelerated and in the other case retarded
by the angle

2

1- 1-¢gyw-1
tan" 7 cot 26) 2.05 tan-1(—f “—) c
g Lg: see

OF NEWTON’S RINGS BEYOND THE CRITICAL ANGLE. 655

It follows from (14) that the ring cannot vanish unless p, coswy,=p,, cosyy,, and p, sin =
p, sin vy. This requires that p?=p,*, which is satisfied only when i=., in which case as we have
seen the ring does not vanish. Consequently a ring is formed at all angles of incidence; but it
should be remembered that the spot, and consequently the ring, vanishes when i becomes 90°.

= . - (0)
25. When i=+, the expressions for P,, Q,, take the form a2 and we find, putting for

aD
shortness ee?

3 (uw? — 1)7? be pi (uw? — 1)"
t) p uy (uw = 1)7) ? Ce pr Fs wi (ua aia ’
p(w -1y3 PACE

Q,=- Qs =

p+ (v2 -1)7"’ p? + ut (uw? = 197)

If we take two subsidiary angles y, w, determined by the equations

aD
xe p? —1 = tany = p’ tana,
we get
P, = cos” x Ps= cos’w, @Q,= —sin xeos yxy, Q, = — sinw cos w.

Substituting in (15) and reducing we get, supposing a = 45°,
T= versin (2y — 2w) .-.--.,-205-0-« (16).

When i = 1, cos 2 = — cos 26, sin 2 = sin 20; and therefore P,= P,, Q,= — Q,, which when
a = 45° reduces (15) to I = Q,”.

If we determine the angle @ from the equation

1 —q° = 2qsin 20 tan w, or tan w = cot 2¢. cosec 20,
we get
I= sin*2@ .cos*20,........... (17).

In these equations

log, tan ¢ = — ——

26. The following Table gives the intensity of the ring for the two angles of incidence i =-y and i=:,
and for values of D increasing by 4A. The intensity is calculated by the formule (16) and (17).
The intensity of the incident polarized light is taken at 100, and » is supposed equal to 1°63,
as before.

4P2

656 Mr. STOKES, ON THE FORMATION OF THE CENTRAL SPOT

>|v

oer nee UOeK NO

eh OOH WH HO HO EF ee FOF OH Oo

SCOMADANAEwWHNHNH OW WMAAMXEHWANHHOUOAAARKEWNHE GS
SOA aAdnawwrwonowoornve anor eP Or ODonroranw o

1:
1
1
1
1
1
1
1
1:
1:
2
2
2
Qe
Qa:
2
Q°
2
2°
2
3°

The column for i=-y may be continued with sufficient accuracy, by taking J to vary inversely as
the square of the number in the first column.

27. Ihave seen the ring very distinctly by viewing the light transmitted at an angle of inci-
dence a little greater than the critical angle. In what follows, in speaking of angles of position, E
shall consider those positive which are measured in the direction of motion of the hands of a watch,
to a person looking at the light. The plane of incidence being about 45° to the positive side of the
plane of primitive polarization, the appearance presented as the analyzer, (a Nicol’s prism,) was
turned, in the positive direction, through the position in which the light from the centre was extin-
guished, was as follows. On approaching that position, in addition to the general darkening of the
spot, a dark ring was observed to separate itself from the dark field about the spot, and to move
towards the centre, where it formed a broad dark patch, surrounded by a rather faint ring of light.
On continuing to turn, the ring got brighter, and the central patch ceased to be quite black. The

OF NEWTON'S RINGS BEYOND THE CRITICAL ANGLE. 657

light transmitted near the centre increased in intensity till the dark patch disappeared: the patch
did not break up into a dark ring travelling outwards.

On making the analyzer revolve in the contrary direction, the same appearances were of course
repeated in a reverse order: a dull central patch was seen, which became darker and darker till it
appeared quite black, after which it broke up into a dark ring which travelled outwards till it was
lost in the dark field surrounding the spot. The appearance was a good deal disturbed by the
imperfect annealing of the prisms. When the plane of incidence was inclined at an angle of about
—45° to the plane of primitive polarization, the same appearance as before was presented on revers-
ing the direction of rotation of the analyzer.

28. Although the complete theoretical investigation of the moving dark ring would require a great
deal of numerical calculation, a general explanation may very easily be given. At the point of contact
the transmitted light is plane polarized, the plane of polarization being the same as at first*. At some
distance from the point of contact, although strictly speaking the light is elliptically polarized, it
may be represented in a general way by plane polarized light with its plane of polarization further
removed than at first from the plane of incidence, in consequence of the larger proportion in which
light polarized perpendicularly to the plane of incidence is transmitted, than light polarized in that
plane. Consequently the transmitted light may be represented in a general way by plane polarized,
with its plane of polarization receding from the plane of incidence on going from the centre
outwards. If therefore we suppose the position of the plane of incidence, and the direction of
rotation of the analyzer, to be those first mentioned, the plane of polarization of light transmitted by
the analyzer will become perpendicular to the plane of polarization of the transmitted light of the
spot sooner towards the edge of the spot than in the middle. The locus of the point where the two
planes are perpendicular to each other will in fact be a circle, whose radius will contract as the ana-
lyzer turns round. When the analyzer has passed the position in which its plane of polarization is
perpendicular to that of the light at the centre of the spot, the inclination of the planes of polar-
ization of the analyzer and of the transmitted light of the spot decreases, for a given position of the
analyzer, in passing from the centre outwards; and therefore there is formed, not a dark ring
travelling outwards as the analyzer turns round, but a dark patch, darkest in the centre, and
becoming brighter, and therefore less and less conspicuous, as the analyzer turns round. The
appearance will of course be the same when the plane of incidence is turned through 90°, so as to
be equally inclined to the plane of polarization on the opposite side, provided the direction of
rotation of the analyzer be reversed.

29. The investigation of the intensity of the spot formed beyond the critical angle when the
third medium is of a different nature from the first, does not seem likely to lead to results of any
particular interest. Perhaps the most remarkable case is that in which the second and third media
are both of lower refractive power than the first, and the angle of incidence is greater than either of
the critical angles for refraction out of the first medium into the second, or out of the first into the
third. In this case the light must be wholly reflected; but the acceleration of phase due to the
total internal reflection will alter in the neighbourhood of the point of contact. At that point it will
be the same as if the third medium occupied the place of the second as well as its own ; at a distance
sufficient to render the influence of the third medium insensible, it will be the same as if the second
medium occupied the place of the third as well as its own, The law of the variation of the accele-
ration from the one to the other of its extreme values, as the distance from the point of contact varies,
would result from the investigation. This law could be put to the test of experiment by examining
the nature of the elliptic polarization of the light reflected in the neighbourhood of the point of

* The rotation of the plane of polarization due to the refraction at the surfaces at which the light enters the first prism and quits the
second is not here mentioned, as it has nothing to do with the phenomenon discussed.

658 Mr. STOKES, ON THE FORMATION OF THE CENTRAL SPOT, ETC.

contact when the incident light is polarized at an azimuth of 45°, or thereabouts. The theoretical
investigation does not present the slightest difficulty in principle, but would lead to rather long

expressions; and as the experiment would be difficult, and is not likely to be performed, there is no
occasion to go into the investigation.

30. In viewing the spot formed between a prism and a lens, I was struck with the sudden, or
nearly sudden disappearance of the spot at a considerable angle of incidence. The cause of the
disappearance no doubt was that the lens was of lower refractive power than the prism, and that the
critical angle was reached which belongs to refraction out of the prism into the lens. Before disap-
pearing, the spot became of a bright sky blue, which shows that the ratio of the refractive index of
the prism to that of the lens was greater for the blue rays than for the red. As the disappearance
of the spot can be observed with a good deal of precision, it may be possible to determine in this
way the refractive index of a substance of which only a very minute quantity can be obtained. The
examination of the refractive index of the globule obtained from a small fragment of a fusible
mineral might afford the mineralogist a means of discriminating between one mineral and another.
For this purpose a plate, which is what a prism becomes when each base angle becomes 90°, would
probably be more convenient than a prism. Of course the observation is possible only when the
refractive index of the substance to be examined is less than that of the prism or plate.

G. G. STOKES.

XLIX. Of the Intrinsic Equation of a Curve*, and its Application.
By W. Wuewe 1, D.D., Master of Trinity College, Cambridge.

[Read February 12, 1849.]

1, Maruemaricrans are aware how complex and intractable are generally the expressions
for the lengths of curves referred to rectilinear coordinates, and also the determinations of their
involutes and evolutes. It appears a natural reflexion to make, that this complexity arises in a
considerable degree from the introduction into the investigation of the reference to the rectilinear
coordinates (which are extrinsic lines); the properties of the curve lines with relation to these
straight lines are something entirely extraneous, and additional with respect to the properties of
the curves themselves, their involutes and evolutes; and the algebraical representation of the former
class of properties may be very intricate and cumbrous, while there may exist some very simple
and manageable expression of the properties of the curves when freed from these extraneous append-
ages. These considerations have led me to consider what would be the result if curves were
expressed by means of a relation between two simple and intrinsic elements, the length of the curve
and the angle through which it bends: and as this mode of expressing a curve much simplifies the
solution of several problems, I shall state some of its consequences.

2. Let s=f(q), any function of d, when s is the length of the curve, and @ the angle of
deviation of the tangent from the tangent at the origin.

Then using the common notation we have ds =f’ (p).d@. The curve may be constructed
approximately by taking small jinite differences of , and determining the corresponding rectilinear
elements of the polygon or approximate curve, by the equation

As =f’ (p). Ag. See Fig. 1.

gd gaits ds . ; ,
3. It is evident that qo 8 P the radius of curvature. Hence p =f’ (#), and the curve may

also be contracted approximately by taking small finite differences of @, drawing a line perpen-
dicular to the curve at first, setting off the value of p, drawing a circular are to radius p for Ad,
then setting off p,, and drawing a circular arc with radius p, for A@,, and so on. See Fig. 2.

4. The evolute of a curve is easily found from this equation, For if (Fig. 3) AP be the
curve, BQ the evolute, AP = s, BQ=s’, it is evident that is the same for both s and s', if s in

BQ, p be measured from BA, perpendicular to Aw.
, , ds 4
And QP = QB - BA, or p=s' +C. Hence s =f eae —C.

If the curves have the forms represented in Fig. 3a, the formule are nearly similar.

* After writing this paper, 1 found that Kuler had, in the solu- | tion being the integral of Euler’s, has, of course, one more arbitrary
tion of a particular problem, expressed curves by means of an | constant than his. There may very possibly be other modes of
equation between the are and the radius of curvature. This equa- | expressing curves which may be fitly described as ** intrinsic equa-
tion is, as is shown in the paper, the differential of my ‘intrinsic | tions’? to the curves. I was not able to find any other name for the
equation,” and has an equally good right to the name. My equa- | equation which I have employed.

660 Dr. WHEWELL, ON THE INTRINSIC EQUATION OF

5. Hence s’, s”, &c. indicating the successive evolutes, p, p’, p’, &c. the successive radii
of curvature, we have,
ds , das’ ds”
s'=— -C, s"=— -C,
do dp ~’ do
ds ase see

P~ ag ? ag” sare aah

ZA Cr. &e.

6. Also we may in like manner find the successive involutes s,, s,, s,, &c. For we have

ds

Giedeamiaae ehetenre ka.S fs

d¢
So ds,
d

Gute, tulle sede edi os

8 =fi(p) + GeO, p-
Hence s is known in function of @, and therefore the curve known. And in like manner s,, s,, &e.,
if these be the arcs of the successive involutes.
In Fig. 4, CR, BQ, AP are successive involutes of DS.

7. It is evident that the intrinsic equation to the circle is
8 = ag, a being the radius.
Also for the equiangular spiral, since the curve from its origin is everywhere similar to itself,
the radius of curvature is proportional to the whole are. Hence
ds
dp

If s and @ vanish at the same time, s = a (e”? — 1).

= ms; whence s = a”, if s be measured from the pole.

We shall afterwards give general formulz for obtaining the intrinsic equation from the ordinary
coordinate equation, and reversely. But the operation of our method will be better seen by first
taking some special cases.

Of Cycloids, Epicycloids, and Hypocycloids.

8. In the Cycloid, if VB, Fig. 5, be the diameter of the generating circle, rolling on the
straight line DB from the initial position AD, when it is perpendicular to DB, and P the describ-
ing point at that time, PQ being the diameter, by the mode of description, the are BQ = BD.
But the curve at P is perpendicular to PB; and if @ be the angle of deflexion, @ = VBP, and
2@=VCP. Hence chord VP = 2bsin , if 6 be the radius of the circle. And the arc AP
= 2chord VP. Hence the intrinsic equation to the cycloid is

s=45 sin p-

When ¢ becomes a right angle, s becomes a maximum. At this point there is a cusp (Z), and the
added part of s after this is negative; and so continues, till @ = 8 right angles, where there is
another cusp (Z), and the added part of s becomes positive ; and so on.

A CURVE, AND ITS APPLICATION. 661

9. In the Epicycloid, if we take Newton’s construction (Princip. B. 1v. Sect. 10, Prop. 49),
Fig. 6, CB the radius of the globe, VB the diameter of the wheel when the describing point is at
P, E its center, we have, by Newton’s proposition, only measuring the are from A the vertex,
instead of Z the cusp, the are at P perpendicular to the chord BP; and

CB : 2CE :: chord VP : arc AP.

Let a be the radius of the globe, 6 the radius of the wheel: @ the angle DCB, through which
the wheel has rolled upon the globe. Then (PQ being a diameter) by the mode of description, arc

(7)
BQ=BD. Therefore angle BEQ = =: therefore chord VP = 2bsin = and

. a0 4
a: 2(a+b) :: aban as spliced g =e ee ae
2b a 2b

0
But VBP = = , and DCB = 0. Hence BP makes with C4 an angle = — +@. And since the
a

curve at A is perpendicular to CD, and at P, to BP, it is evident that if @ be the angle through

which the curve has deflected at P, @ = = Oia — 0.

2b

Hence 0 =
a+2h

p; and the intrinsic equation to the epicycloid is

~ 3 Gere e p-
a a+2b

This may coincide with any curve of which the equation is
s=/sin md, where m is less than 1.

a 4(a+b)b
, ia;
a+2b a

In this case, m =

l1—m 2(1+m)b
, =

b
whence — =
a 2m m

10. In the same manner we shall find that the intrinsic equation to the hypocycloid is

4(a—b)b. a
ere ar

And this may coincide with any curve of which the equation is

s=lsinm@, where m is greater than 1, by making

11. It is evident from the equation s = sin mq, that the curves represented by that equation
will be of such forms as are seen to result from the epicycloidal mode of description. Thus the

equation 8 = Tene gives a curve in which s continues to increase from A, where p=0, till
2
p=, after which it decreases, Hence there will be a cusp when the curve has deflected
through two right angles, as at Z, Fig. 7. After this point the curve goes on in an identical
inverted course, till q = 27, as at A’, when s = 0, the negative part having destroyed the positive
Vor. VIII. Parr V. 4Q

662 Dr. WHEWELL, ON THE INTRINSIC EQUATION OF

part. The negative value of s goes on increasing till @ = 37, at Z’, when there is another cusp
Afterwards the are becomes positive, and the curve returns to A, having deflected through 47*.

The curve is an epicycloid in which b = 4 a.

12. Again, if s=/sin2q@, s increases from A, where ¢ = 0, till e. a, when it is a maxi-

mum, and there is a cusp, Z, Fig. 8. After this the are (from Z) is negative till @ = ey when
4

: 1 . eas : 5
there is a second cusp, Z’. Then the arc is positive, till @ = = (at Z”). Then it is negative

till p= = (at Z’”). When @ = 27, the curve returns to 4.

The curve is a hypocycloid in which 6 = da.

13. If s=vsin®, e=tsin®, sa lsin®, &e.

: ay te Mee (i A
we have epicycloids in which — is respectively
a
1, 3, 2, &e.
The radius of the wheel in these latter cases is greater than that of the globe, and the curve is
deflected through more than a whole circumference before it comes to a cusp. ‘Thus in the case

. Tv
s=sin , the curve deflects through 27 + eS to come toa cusp. See Fig. 9.

14. In the same way, if we have
s=/sin 3q, = lsin4q, s=Isin5d¢@, Xe.

Lie ?
we have a series of hypocycloids, in which = is respectively

; b
As m becomes larger and larger, = approaches more and more nearly to 4, but never attains
that magnitude. As is well known, for that supposition, the hypocycloid is a straight line.

15. It is evident that the ordinary properties of epicycloids and hypocycloids, as to their
lengths, radii of curvature, involutes, evolutes, &c., all follow with great facility from the use of our
equation. Thus the length of the epicycloid from the vertex d to the cusp Z is had by making

4(a +b)b
a

the angle —$ =<, which gives the length of that half of the curve = , and the
a+ 2

from cusp to cusp, the known values.

8 b)b
whole length —

Also the radius of curvature of the epicycloid
ds 4(a+b)b a
ee COS
dp a+ 2b a+2b

gp, the known value.

; $ ; ; ‘ ds : A
© hat there will be a cusp when s isa maximum, appears also by considerin, that in that case —— =0, that is, the radius of cur-
Pp P y ig dp

vature vanishes.

A CURVE, AND ITS APPLICATION. 663

16. Again, for the evolute of the epicycloid, let, in Fig. 6, ZO =’ be the arc of the evolute,
Therefore

, 4(a+b)b a
eS a+2b jug prays

But at Z, where s’=0,

a

And the deflexion of the evolute beginning from Z and going to 0 is the excess of this value

of @ above the value at P, because at every point the evolute is perpendicular to the curve.
: . ae, 2b

Therefore if @’ be the deflection of s’, On ee a

-. -—@3 1 EH a ts ie
2 a ? a aon 2 eae

4(a+b)b , a
sin
a+2b a+2b

; , 4(av4+5)0 a
equation agrees with s’= ; sin -o, if —=
4 gr a a’ +20 ?> fe

Therefore s’ =

Q. This is an epicycloid similar to the first; for the

Of Running-pattern Curves.

17. By Running-pattern Curves I mean curves in which a certain form of curve is repeated
over and over again in the progress of the whole curve. For example, let @ = sins; as s increases

i * : : 7 3a 5 3 4
from 0 to infinity, it becomes successively 0, ie ge? 2m,» &c., and the corresponding values

of @ are 0, 1, 0, —1, 0, 1, &e.: and the curve is evidently a sinuous curve, as represented in Fig.
10, in which the same form is constantly repeated every time that s goes through the value 2 7.

The greatest angle which the curve makes with the original direction is 1 and —1; that is,
the angle of which the are = 1, to the one side and to the other.

18 If p=m. sin s, we shall in like manner have a sinuous curve in which the greatest
angles of deflexion to one side and the other are = m.

Th : ; c ae
If @ = @ ass these deflexions become right angles, and the curve is as represented in Fig. 11.

19. If @ = 7 sins, the curve from s = 0tos = “is of the form CA, Fig. 12; A being
2

2 7 : ds 1 , Beal
behind C. For in this case, —— = . Hence the radius of curvature is— at C, where s = 0,
dp mrcoss T

and increases to A, where it is infinite. The evolute is of the form BD, and has for its asymp-
tote the line 4, perpendicular to the original direction. And hence the general form of the curve
CA is manifest. At A there will be a point of inflexion; and after A the curve will be repeated
in inverse position, as 4C’, and then continued reversely from C’ to A’, and so on, as in the Figure.

. . . . < Tv
20. If p@ = 2m sins, it will be seen, in like manner, that the curve from s = 0 to s =~

is of the form CA, Fig. 13, and by the repetition of this, we have the curve as represented.

21. The pattern in the above curves is symmetrical with regard to a line transverse to the
line @ = 0. But we may have patterns which are not thus symmetrical.
sin @ dy m + cos v
Let y= — —, whence - Bet 3+ (m<1).
1 +m cos w dua (1 +m cosa)
4Q2

664 Dr. WHEWELL, ON THE INTRINSIC EQUATION OF

If wv, y, be ordinary coordinates, these equations represent a curve sinuous, but each sinus
not symmetrical. The angles at which the curve cuts the axis are alternately those of which the
tangents are

Hence the descending side is more inclined than the ascending.

We shall obviously have a curve nearly resembling this, if we take the intrinsic equation

= aieeete which differs from the former by putting p for tan Q; and s for x.

?

The curve will be a sinuous curve, inclined to the original line @ = 0, at maximum angles

1 1 5 ‘ A
p= fut? ees on one side, and on the other, when coss= +1. Andif a be the are in

the first quadrant for which cosa = m, gp =0 whens =7r-a,7+a, 37—a, 347+ a, &e.; and
the curve will be as represented in Fig. 14.

j 1 On .
For example, if m ar the curve deflects alternately on the positive side, so that the angle of

Bs ate : : = ae ; :
deflexion is re and on the negative side, so that the angle is a that is, the angles are respectively

43° and 86° nearly.

ds (1 +m cos s)*
dp (m’-1+meooss) sins’

We have, in this case,

This is the radius of curvature, which becomes infinite when s = 0, 7, 27, &.; that is, at 4,
A’, A”, &c., when there are points of inflexion.

m+ COS §

} h = p.—_—_______
22. If we have @ =p (Caaicn a

we shall still have a sinuous curve, and the greatest deflexions will be 7 > and — : Ls
+m —m
° Tw 1
Thus if p = = and m = 2? the greatest angles are
2 or. Tv P
— ~, and —2.—; that is 60° and — 180°.
8 2 2
Hence the curve will be of the form represented in Fig. 15, making at A an angle of 60° with
the line @ = 0, and at B, where cos s = — 1, the curve being parallel to @ = 0, but in the opposite
direction.

The radius of curvature is infinite at 4 and at B, and has a minimum value at some intermediate
point, nearer to B.

23. Itis easy to construct running patterns curves of this kind which have any given angles
for their extreme deflexions. Thus let Fig. 16 represent a pattern curve which sinuates between the
angles 60° one way and 3x 90° = 270° the other. Then,

7 p 3a Pp l+m 9
8 t4+m"’ 2 1=—m" “1—-m 8

A CURVE, AND ITS APPLICATION. 665

And the curve is @ = Se adres ( = “)

11 (4+ m cos s)?’

m + COs § 5
The curve @ = e —— m =)
2 (1 +m cos s)* 8

3 : 87, 8 E
gives the angles of deflection = ea and shag that is, 554° and 240°, which nearly resembles
the last, and may also be represented by Fig. 16.

The curve may deflect through more than a circumference, Thus if p= ke Ries :
27 (l1+m cos s)°

7 ° 7 87 4 +6 :
(m = ‘) , the greatest deflexions are 3 and “3 that is 60° positive, and 360° + 60° negative.

Hence the curve at A and B, Fig. 17, is parallel, at both points making an angle of 60 with @ = 0.

Such a curve has a loop; C being the place of minimum radius of curvature, the curve opens
both ways from C.

Of Diminishing Running-pattern Curves.

24, If @=sin s’a, where a is a quadrant, we shall have a sinuous curve ;

‘
And if we make s=1, 4/3, 4/5, 4/7, \/9, 4/11, &c.,

we shall have a series of points of inflexion in the curve. And since these values of s have

for their differences

V3-1, Wf5-V3 AYI- V5 VI-V7 Vl -f9 &e-
which are a decreasing series, it is evident that we shall have such a curve as Fig. 18, in which the
lengths of the curve between the points of inflexion, 4 4’, A” A”, A” A", &c. constantly decrease.

The same will be the case if @ =p sin s°a.
If p be large enough, such curves will have loops, like those represented by @ = p sin s.

Thus if d -= sin s?a, we shall have a curve such as Fig. 19. (See Fig. 12, which

A :
represents @ = 3 Sim a as to its general form).

The lengths of the alternate loops will constantly diminish, and the whole curve will occupy a
triangular space, like a writing-master’s flourish.

25. We may have a similar flourish, but unsymmetrical, by taking, instead of sin s*a, the

‘ m + cos $°a
expression ———————__..
P (1 + m cos 8*a)*

This will give a figure like Fig. 20.

Of Circularly-running Pattern Curves.

; Raed;
26. If we take the equation @ = p sin ae

666 Dr. WHEWELL, ON THE INTRINSIC EQUATION OF

- : - 8 :
we shall have the figure in which a curve such as @ = p sin 2 rune along the circumference of a

circle.
And in the same manner, by adding to the value of @, in any of the other cases previously given,

s : P
a term —, we have the equation to the pattern curve there considered, made to run round the
a
circumference of a circle.
cakags® Sue Cc
Thus p = sin 5 ta Bives such a figure as Fig. 21;
a

m + COS §

7 s
=-—- ——— +-— such a figure as Fig. 22
? 2 (4+mecoss)? «a 8 5 ae

1
m being about a 3 in Fig. 16.

27. The radius of the circle round which the pattern runs is less than a. When @ has gone
through all its values, so that s = 27a, the curve has not been laid along the circumference of the
circle, but has, besides, followed all the sinuosities of the pattern.

Of the Catenary and Tractria.

28. The intrinsic equations simplify the properties of these curves. ‘

Fig. 23. Let CO be any are of a Catenary from C the lowest point; OS, CS, tangents, OV
vertical, meeting CS’; therefore OS'V is the triangle of the forces which support the weight of
CO; and if O be the tension at C, expressed in length of the curve,

~ = oe tan OSV, and if OSV = ¢,

a
s=atan @,
the equation to the catenary.

29. For the Tractrix, let PZ’ be the tangent, A7' the fixed line, PN, perpendicular on
AT =a, tan NPT =p. Then PT = @ 4/1 4 p® = c, a constant, by hypothesis.

c d d ae
Hence @ a aeet ae -a AL also (s being now CP,) +. =—-V1tp.
d ;
Therefore a =
dp 1+p

But if @ be the angle of deflexion, beginning when the curve is perpendicular to A7’, p = tan @;

d
therefore a = sec? pb = 1+ p”.

d bate
Hence — =cp=ctang=ce——>

dp cos ~

This is the equation to the evolute. Integrating

s
c

1
s = cl——-; or cos @ = e~
cos dp’ ? :

the equation to the tractrix.

a

A CURVE, AND ITS APPLICATION. 667

30. It appears in this investigation, that the evolute of the tractrix is the catenary, a well-
known property.
General Properties of the Intrinsic Equation.

31. Given the equation of a curve to rectangular coordinates 2, y, to find the intrinsic

equation.

Let y = f («): hence, f” (@) = = rr ae » p being 0 for y.

Hence w is known in terms of tan d@. Let w = F (tan p);

du Ud 2
Then wa = F'(tan d) x see* o.

d
Also - = cosec dp.

Hence ve = F' (tan ) . sec? p. cosec @ = pHeAenisp
dp sin p.cos*p

32. Exampies. 1. The Common Parabola.

on.

d
y= saa; Jt. ~ = api ; hence # = a tan’ @.
dx 2asin
Hence — = 2a tan p. sec’ = - Fe
dp cos’ p
il ds 2
And ai =-——. Hence — ee :
dx sin o) dp cos’ p

zleap 2a
Hence the equation to the evolute of the parabola is s’ = er

‘ 2a
The length of the parabola may be found by integrating coai

2. The Semicubical Parabola,.

d 2 as 1 S8atan®
ee. = ; hence 7 = %
dw 3. tan Q7
dt 8a eee 1 h ds 8a sing
— = — tan? se anc ——; whence — = —.—,—
dp hat ee dv sing dp 9 “cos ‘Dp’
the intrinsic equation to the curve.
Int ti h He 1
ntegrating, we have s = — ———I1>,
8 B» 27 |cos'

3. The car,

2
a

«Va ; a dy b ¢ , Hence a = ——————
ean dz.) as /a— oe tang Ja + b*tan® p

668 Dr. WHEWELL, ON THE INTRINSIC EQUATION OF

dx —s @b' tang. sec? h | a’ B sind
dp (a+b? tan’)? — (a* cos* p +B sin® p)?
1 222
And de =- Hence ce

ds . . . . .
Fs na i = @org+ Pan’ oP the intrinsic equation to the ellipse.

ab

Hence the radius of curvature is : 3
(a cos’ @ + B°sin® p)?

2

When ¢ = 0, this radius is = @ beginning at the extremity of the major axis, which was the

supposition made.

The intrinsic equation to the evolute of the ellipse is
; ab? b?
if s’ begin from a cusp, where @ = 0.

33. Given the intrinsic equation, to find the equation to rectangular coordinates.

. . age Tv
Let the coordinates «, y, be in the positions @ = 0, @ = =.

Then it is evident that w = fds .cos gy = fds.sin p:

and the equation being given, these coordinates are found by integration.
Thus in the cycloid, s=asin g@. Hence ds = a cos p.do;

v=facos p.dd, y= fasing cos ddd. Hence, integrating,
o =~ sin p cos p + e dg, y= © sin? @: the equation to the cycloid,
Q ue 2

34, In the running-pattern curves (Art. 17, &c.) of which the equation is

: dp : Yas y
p =msin s, we have = =m cos 8 =/(m? — ¢°): ag a Heaoas
Hence igo cospdp | sin pd

J (m? am ¢’) Bat a/ (m* = ~)

If these could be integrated, we could find the dimensions of the loops in Figures 11, 12, 13.

There is one case for which a taken from @ = 0 to P=m is=0. In this case the

curve neither runs forward as in Fig. 11, nor backward as in Fig. 12, but is simply two loops, Fig. 24.

35. The following proposition, enunciated by John Bernoulli and proved by Euler, may
easily be proved by means of the intrinsic equation.

If AB be any curve, AB’ its involute beginning from A, B’ d’ the involute of AB’ beginning
from B’, 4’ B” the involute of 4’ B’ beginning from 4’; and so on alternately and indefinitely :
the successive involutes approach indefinitely to the form of the common cycloid, provided the tan-
gents at A and B in the original curve are perpendicular to each other.

(The proof of this is here omitted, being included in the proof of the extended propositions
given in the Additional Note.)

A CURVE, AND ITS APPLICATION. 669

36. The following proposition may be proved by the intrinsic equation.

Fig. 25. Let any curve be evolved, and its involute evolved, and the involute of that evolved, and
so on, beginning always the evolution with a rectilinear tail, Ad’, extending beyond the curve,
and all these tails being equal. The curve tends continually to the form of the equiangular spiral.

7

Let s, s’, 8”, s’”, &c. be the successive curves, @ the angle, which is the same for all, be-
ginning from 0 for each. And let each of the tails 44’, 4’A”, A’A'', &e. =a.

Let s= apt a, + a,° + &c., which may express any curve.
Th , Gee lets Gag
en f= f(ats)dp= apt ~ P+ P+ p+ &e.

a,

a a.
ee , d = 2 3 a: 4 x
8” =f(a+s') do ap+ rere a rey ey + &c

a a a, ¢
Rae a+s" d a + 2 3 iP &e.
I( ) P ae ache Sees :

And as the operation goes on, the terms in @,, 2, a, &c. being divided by the factorials 2.3.4,
&e. indefinitely, may be neglected as to their influence on the curve. Therefore ultimately
a a
s=aphp+— + °+ &e. =a[e*-1
© 1.2? meat Ee: h

which is (Art. 7) the equation to the equiangular spiral.

Of course, from the nature of the construction, the curvature of the original curve is throughout
towards the same side*.

Additional Note to a Memoir on the Intrinsic Equation of Curves.

Trinity Cotuece, April 12, 1849.

In the Memoir on the Intrinsic Equation of Curves, I gave a proof of the following Proposi-
tion, which was enunciated by John Bernoulli, and demonstrated by Euler. (Novi Comm. Petrop.
Tom. x.)

Fig. 26 and 27. If AB be any curve, 4B’ its involute beginning from A, Bd’ the involute
of AB’ beginning from B’, A’B" the involute of 4’B’, beginning from A’; and so on, alternately

. and indefinitely : the successive involutes approach indefinitely to the form of the common cycloid,
provided the tangents at A and B in the original curve are perpendicular to each other.

The question naturally offers itself, What is the curve to which the successive involutes tend, if
the original curve do not conform to the condition above stated, that the total deflexion is a right
angle ?

I am now able to state that in that case the curve will be an epicycloid or a hypocycloid as the
total deflexion is greater or less than a right angle.

* Also it is necessary, as has been remarked to me, that the | ginal curve, or any of its evolutes in infinitum. For if it were,
point where =0, is not a point of contrary flexure from the ori- | some of the quantities a,, dy, &e. would be infinite,

Vor. VIII. Part V. 4h

670 ADDITIONAL NOTE TO Dr. WHEWELL’S MEMOIR ON THE

The proof of this extension of Bernoulli’s proposition easily follows from the mode of repre-
senting curves by their Intrinsic Equation, namely, the equation between the tangent and the
deflexion.

Let AOB be any curve, and let the tangents at A and at B make with each other any angle
ma, a being a right angle.

Let APB’, B’O'A’, A’P’B", B’O" A", &c. be the successive involutes, beginning alternately at
opposite ends. if

Let AB’=b,, A’B’= b., &c. the whole ares of the alternate involutes.

Let the intrinsic equation to 4OB be

a. a. ‘
s=agot = ae rae gp + &c., which may express any curve.
a, Qs ds
Hence AP = ¢, = — P+ $4 ——-__ fi + &e.
i ae ?—amenee 1.2.3.4?

aq

Wr fen fee 2 — e —
And B'P = b, ra ae 1.2.8.4?

«. AO =8,= [PO.dp = [BP.dd¢, beginning from gp =O0at A’;

* — &e.;

e pr a
ae 1.2.3 ie” nee

In like manner, if 4’P’=¢,, AO" =s,, &e.

b, : a, 3
eG ares Pty ee th a antigen Se:

Ox, Be 9;
5, = yp — 2. p+ =? pt = &e, + —_"_ ___ g"*! 4 Be,
1.2.3 1...” 1... (2n + 1)

Now as 7 becomes larger, the terms in a), a,, &c. which have for denominators the factorials
1.2.3...(m —1) &c. become smaller and smaller, and thus the arc s, depends less and less upon
the form of the original are 4OB. Hence we may ultimately omit those terms.

Of the arcs ¢,, f,, ... t,, each vanishes when g =0; and when @ = ma, they become respectively
b,, b,,.+-.b,- Hence, by the expression for ¢,, ultimately

m? a m4 a’ m> a?
+ b,_3

ina Veen AeA6

— &e.

b, = Da-1

This expresses a relation among the successive ares, b,, by, bs... b,, which relation, it appears,
is ultimately independent of the form of the curve AB. But since a is a quadrant, and cosa = 0,

we have

INTRINSIC EQUATION OF A CURVE, AND ITS APPLICATION.

2 a’ 6
ee = + &e.
Le2h Ay S1225)6:
whence
2 n 6
a a a
6,=56 = ob
eae’ Vl 8. earner <

Hence the necessary relation among 6,, 6,, 63, &c. is satisfied if

Gyan Oy = 9090s = Os eo aces = 19010,

1 1 1
b > b =—45,, b,-3 = — by, &e.
mM

that is, if 6,..= — 6, ae
m m

Hence we have, by the expression for s,, ultimately,

1 3
PN eds gr =e
mae G2 Sem nee tS ene

p

= mb, sin— .
m

This is the equation to an epicycloid, if m>1; and to an hypocycloid, if m <1.
If A and B be the radius of the globe and wheel of the epicycloid ;

4B(4+B). A
Fema cee

4+2B B m-1
Hence m= 3 = =——.
A A 2

: 4B(A-B). A
For the h = . ;
or the hypocycloid, s A sin Via co)

q
oO
=|
8
3
i}
hm
Ry
vo

Tn the figures, the angle ACB = (m —1)a, when m>1:

ACB =(1—m)a, when m <1.

tn2

671

L. On the Variation of Gravity at the Surface of the Earth. By G. G. Sroxes, M.A.,
Fellow of Pembroke College, Cambridge.

[Read April 23, 1849.]

On adopting the hypothesis of the earth’s original fluidity, it has been shewn that the surface
ought to be perpendicular to the direction of gravity, that it ought to be of the form of an oblate
spheroid of small ellipticity, having its axis of figure coincident with the axis of rotation, and that
gravity ought to vary along the surface according to a simple law, leading to the numerical relation
between the ellipticity and the ratio between polar and equatoreal gravity which is known by the
name of Clairaut’s Theorem. Without assuming the earth’s original fluidity, but merely supposing
that it consists of nearly spherical strata of equal density, and observing that its surface may be
regarded as covered by a fluid, inasmuch as all observations relating to the earth’s figure are reduced
to the level of the sea, Laplace has established a connexion between the form of the surface and
the variation of gravity, which in the particular case of an oblate spheroid agrees with the connexion
which is found on the hypothesis of original fluidity. The object of the first portion of this paper
is to establish this general connexion without making any hypothesis whatsoever respecting the
distribution of matter in the interior of the earth, but merely assuming the theory of universal
gravitation, It appears that if the form of the surface be given, gravity is determined throughout
the whole surface, except so far as regards one arbitrary constant which is contained in its com-
plete expression, and which may be determined by the value of gravity at one place. Moreover
the attraction of the earth at all external points of space is determined at the same time; so that
the earth’s attraction on the moon, including that part of it which is due to the earth’s oblateness,
and the moments of the forces of the sun and moon tending to turn the earth about an equatoreal
axis, are found quite independently of the distribution of matter within the earth.

The near coincidence between the numerical values of the earth’s ellipticity deduced independ-
ently from measures of arcs, from the lunar inequalities which depend on the earth’s oblateness,
and, by means of Clairaut’s Theorem, from pendulum experiments, is sometimes regarded as a
confirmation of the hypothesis of original fluidity. It appears, however, that the form of the surface
(which is supposed to be a surface of equilibrium,) suffices to determine both the variation of gravity
and the attraction of the earth on an external particle*, and therefore the coincidence in question,
being a result of the law of gravitation, is no confirmation of the hypothesis of original fluidity.
The evidence in favour of this hypothesis which is derived from the figure and attraction of the
earth consists in the perpendicularity of the surface to the direction of gravity, and in the circum-
stance that the surface is so nearly represented by an oblate spheroid having for its axis the axis
of rotation. A certain degree of additional evidence is afforded by the near agreement between

* It has been remarked by Professor O’Brien, (Mathematical If we have given the component of the attraction of any mass,
Tracts, p. 56) that if we have given the form of the earth’s sur- | however irregular as to its form and interior constitution, in a di-

face and the variation of gravity, we have data for determining
the attraction of the earth on an external particle, the earth being
supposed to consist of nearly spherical strata of equal density ; so
that the motion of the moon furnishes no additional confirmation
of the hypothesis of original fluidity.

rection perpendicular to the surface, throughout the whole of the
surface, we have data for determining the attraction at every ex-
ternal point, as well as the components of the attraction at the
surface in two directions perpendicular to the normal, The corre-

sponding proposition in Fluid Motion is self-evident.

—— ee as

Mr. STOKES, ON THE VARIATION OF GRAVITY, ETC. 673

the observed ellipticity and that calculated with an assumed law of density which is likely @ priori
to be not far from the truth, and which is confirmed, as to its general correctness, by leading to a
value for the annual precession which does not much differ from the observed value.

Since the earth’s actual surface is not strictly a surface of equilibrium, on account of the ele-
vation of the continents and islands above the sea level, it is necessary to consider in the first
instance in what manner observations would have to be reduced in order to render the preceding
theory applicable. It is shewn in Art. 13 that the earth may be regarded as bounded by a surface
of equilibrium, and therefore the expressions previously investigated may be applied, provided the
sea level be regarded as the bounding surface, and observed gravity be reduced to the level of the
sea by taking account only of the change of distance from the earth’s centre. Gravity reduced in
this manner would, however, be liable to vary irregularly from one place to another, in consequence
of the attraction of the land between the station and the surface of the sea, supposed to be prolonged
underground, since this attraction would be greater or less according to the height of the station
above the sea level. In order therefore to render the observations taken at different places com-
parable with one another, it seems best to correct for this attraction in reducing to the level of the
sea; but since this additional correction is introduced in violation of the theory in which the earth’s
surface is regarded as one of equilibrium, it is necessary to consider what effect the habitual neglect
of the small attraction above mentioned produces on the values of mean gravity and of the ellipticity
deduced from observations taken at a number of stations. These effects are considered in Arts.
17, 18.

Besides the consideration of the mode of determining the values of mean gravity, and
thereby the mass of the earth, and of the ellipticity, and thereby the effect of the earth’s
oblateness on the motion of the moon, it is an interesting question to consider whether the
observed anomalies in the variation of gravity may be attributed wholly or mainly to the
irregular distribution of land and sea at the surface of the earth, or whether they must be
referred to more deeply seated causes. In Arts. 19, 20, I have considered the effect of the excess of
matter in islands and continents, consisting of the matter which is there situated above the actual
sea level, and of the defect of matter in the sea, consisting of the difference between the mass
of the sea, and the mass of an equal bulk of rock or clay. It appears that besides the attrac-
tion of the land lying immediately underneath a continental station, between it and the level of
the sea, the more distant portions of the continent cause an increase in gravity, since the attraction
which they exert is not wholly horizontal, on account of the curvature of the earth. But besides
this direct effect, a continent produces an indirect effect on the magnitude of apparent gravity.
For the horizontal attraction causes the verticals to point more inwards, that is, the zeniths to
be situated further outwards, than if the continent did not exist; and since a level surface is
everywhere perpendicular to the vertical, it follows that the sea level on a continent is higher than
it would be at the same place if the continent did not exist. Hence, in reducing an observation
taken at a continental station to the level of the sea, we reduce it to a point more distant from
the centre of the earth than if the continent were away; and therefore, on this account alone,
gravity is less on a continent than on an island. It appears that this latter effect more than
counterbalances the former, so that on the whole, gravity is less on a continent than on an island,
especially if the island be situated in the middle of an ocean. This circumstance has already
been noticed as the result of observation. In consequence of the inequality to which gravity is
subject, depending on the character of the station, it is probable that the value of the ellipticity
which Mr, Airy has deduced from his discussion of pendulum observations is a little too great, on
account of the decided preponderance of oceanic stations in low latitudes among the group of
stations where the observations were taken.

The alteration of attraction produced by the excess and defect of matter mentioned in the

674 Mr. STOKES, ON THE VARIATION OF GRAVITY

preceding paragraph does not constitute the whole effect of the irregular distribution of land and
sea, since if the continents were cut off at the actual sea level, and the sea were replaced by rock
and clay, the surface so formed would no longer be a surface of equilibrium, in consequence of
the change produced in the attraction. In Arts 25—27, I have investigated an expression for the
reduction of observed gravity to what would be observed if the elevated solid portions of the
earth were to become fluid, and to run down, so as to form a level bottom for the sea, which in
that case would cover the whole earth. The expressions would be very laborious to work out
numerically, and besides, they require data, such as the depth of the sea in a great many places,
&c., which we do not at present possess; but from a consideration of the general character of the
correction, and from the estimation given in Art. 21 of the magnitude which such corrections are
likely to attain, it appears probable that the observed anomalies in the variation of gravity are
mainly due to the irregular distribution of land and sea at the surface of the earth.

1. Conceive a mass whose particles attract each other according to the law of gravitation, and
are besides acted on by a given force f, which is such that if X, Y, Z be its components along
three rectangular axes, Xdw + Ydy+Zdz is the exact differential of a function U of the co-
ordinates. Call the surface of the mass §, and let V be the potential of the attraction, that is
to say, the function obtained by dividing the mass of each attracting particle by its distance from
the point of space considered, and taking the sum of all such quotients. Suppose § to be a
surface of equilibrium. The general equation to such surfaces is

UES ORR Ras Spam (B))

where ¢ is an arbitrary constant; and since § in included among these surfaces, equation (1)
must be satisfied at all points of the surface S', when some one particular value is assigned to c.
For any point external to §, the potential V satisfies, as is well known, the partial differential
equation

av av adv

na a
Hate dy ato @)

and evidently V cannot become infinite at any such point, and must vanish at an infinite distance
from §. Now these conditions are sufficient for the complete determination of the value of V for
every point external to S, the quantities U and ec being supposed known. The mathematical
problem is exactly the same as that of determining the permanent temperature in a homogeneous
solid, which extends infinitely around a closed space .§, on the conditions, (1) that the temperature
at the surface iS shall be equal to c— U, (2) that it shall vanish at an infinite distance. This
problem is evidently possible and determinate. The possibility has moreover been demonstrated
mathematically.

If U alone be given, and not c, the general value of V will contain one arbitrary constant,
which may be determined if we know the value of V, or of one of its differential coefficients, at
one point situated either in the surface S or outside it. When V is known, the components
of the force of attraction will be obtained by mere differentiation.

Nevertheless, although we know that the problem is always determinate, it is only for a very
limited number of forms of the surface S that the solution has hitherto been effected. The
most important of these forms is the sphere. When § has very nearly one of these forms the
problem may be solved by approximation.

2. Let us pass now to the particular case of the earth. Although the earth is really
revolving about its axis, so that the bodies on its surface are really describing circular orbits

AT THE SURFACE OF THE EARTH. 675

about the axis of rotation, we know that the relative equilibrium of the earth itself, or at least
its crust, and the bodies on its surface, would not be affected by supposing the crust at rest,
provided that we introduce, in addition to the attraction, that fictitious force which we call the
centrifugal force. The vertical at any place is determined by the plumb-line, or by the surface
of standing fluid, and its determination is therefore strictly a question of relative equilibrium.
The intensity of gravity is determined by the pendulum; but although the result is not
mathematically the same as if the earth were at rest and acted on by the centrifugal force, the
difference is altogether insensible. It is only in consequence of its influence on the direction
and magnitude of the force of gravity that the earth’s actual motion need be considered at all in
this investigation: the mere question of attraction has nothing to do with motion; and the results
arrived at will be equally true whether the earth be solid throughout or fluid towards the centre,
even though, on the latter supposition, the fluid portions should be in motion relatively to the
crust.

We know, as a matter of observation, that the earth’s surface is a surface of equilibrium, if
the elevation of islands and continents above the level of the sea be neglected. Consequently the
law of the variation of gravity along the surface is determinate, if the form of the surface be
given, the force f of Art. 1 being in this case the centrifugal force. The nearly spherical form
of the surface renders the determination of the variation easy.

3. Let the earth be referred to polar co-ordinates, the origin being situated in the axis of
rotation, and coinciding with the centre of a sphere which nearly represents the external surface.
Let r be the radius vector of any point, @ the angle between the radius vector and the northern
direction of the axis, @ the angle which the plane passing through these two lines makes with a
plane fixed in the earth and passing through the axis. Then the equation (2) which V has to
satisfy at any external point becomes by a common transformation

d.rV 1 5 (x oo) jb CRUE
dr *sinodd\” dd) * sim d¢e

- = OWkencaiterno see (3)

2
Let w be the angular velocity of the earth; then U = > r’sin’@, and equation (1) becomes

w .
V+ 37 an'é= Gah diags a siz.» digs othr neta cles aaeeeeetcn (4)

which has to be satisfied at the surface of the earth.

For a given value of r, greater than the radius of the least sphere which can be described
about the origin as centre so as to lie wholly without the earth, V can be expanded in a series
of Laplace’s coefficients

Vit Vi + Vo + 2005
and therefore in general, provided r be greater than the radius of the sphere above mentioned,
V can be expanded in such a series, but the general term V,, will be a function of 7, as well as of
@ and @. Substituting the above series in equation (3), and observing that from the nature of
Laplace’s coefficients

1 d dV. 1 dake
a Hes sium : en , Van nett kee aeeupea\e.
Ganda (sin =a) + ae dg n(n + 1)V,, (5)

we get

fr en —n(n +1) rs} = 0,

where all integral values of m from 0 to x are to be taken.

676 Mr. STOKES, ON THE VARIATION OF GRAVITY

Now the differential coefficients of V,, with respect to r are Laplace's coefficients of the n*
order as well as J”, itself; and since a series of Laplace’s coefficients cannot be equal to zero unless
the Laplace’s coefficients of the same order are separately equal to zero, we must have

ad .rV,
a —= $7 (AM) SHLO ot Shetek ses eete (6)

The integral of this equation is

Yr

where Y,, and Z, are arbitrary constants so far as r is concerned, but contain 6 and g. Since these
functions are multiplied by different powers of r, V,, cannot be a Laplace’s coefficient of the n™
order unless the same be true of Y, and Z,. We have for the complete value of V

K,,. Yan~ Fe

—— + a

ies + ee wet Zot Zr t+...

Now V vanishes when r = ~, which requires that Z)=0, Z, =.0, &c.; and therefore

Vine ere Y.
Ve= = Nes = Hin hacia seta. deh deet wee (7)

re

4. The preceding equation will not give the value of the potential throughout the surface of
a sphere which lies partly within the earth, because although V, as well as any arbitrary but finite
function of @ and @, can be expanded in a series of Laplace’s coefficients, the second member of
equation (3) is not equal to zero in the case of an internal particle, but to — 427°, where p is the
density. Nevertheless we may employ equation (7) for values of 7 corresponding to spheres which
lie partly within the earth, provided that in speaking of an internal particle we slightly change the
signification of V, and interpret it to mean, not the actual potential, but what would be the poten-
tial if the protuberant matter were distributed within the least sphere which cuts the surface, in
such a manner as to leave the potential unchanged throughout the actual surface. The possibility
of such a distribution will be justified by the result, provided the series to which we are led prove
convergent. Indeed, it might easily be shewn that the potential at any internal point near the
surface differs from what would be given by (7) by a small quantity of the second order only ;
but its differential coefficient with respect to r, which gives the component of the attraction along
the radius vector, differs by a small quantity of the first order. We do not, however, want the
potential at any point of the interior, and in fact it cannot be found without making some hypo-
thesis as to the distribution of the matter within the earth.

5. It remains now to satisfy equation (4). Let » = a (1 + w) be the equation to the earth’s
surface, where w is a small quantity of the first order, a function of 8 and g. Let w be expanded
in a series of Laplace’s coefficients «+ wu, +.... The term w will vanish provided we take for a the
mean radius, or the radius of a sphere of equal volume. We may, therefore, take for the equation
to the surface

oman (eet eye tee 2)! sesseelssalealecde sacee'ees'(8)

If the surface were spherical, and the earth had no motion of rotation, V would be independent
of @ and @, and the second member of equation (7) would be reduced to its first term. Hence,
since the centrifugal force is a small quantity of the first order, as well as w, the succeeding terms
must be small quantities of the first order; so that in substituting in (7) the value of r given by
(8) it will be sufficient to put » =a in these terms. Since the second term in equation (4) is a
small quantity of the first order, it will be sufficient in that term likewise to put »=a. We thus
get from (4), (7), and (8), omitting the squares of small quantities,

EEE

NE EEE EEE’!

AT THE SURFACE OF THE EARTH. 677
YOY. hg

ia (1 — uw — uw, — Us...) + a+ a +... t ie S10) 0'—= esp eseeseee (9)

a

The most general Laplace’s coefficient of the order 0 is a constant; and we have
sin°@ = 2 + (2 — cos*6),
of which expression the two parts are Laplace’s coefficients of the orders 0, 2, respectively. We
thus get from (9), by equating to zero Laplace’s coefficients of the same order,
Y, = ae - 40° a’,
Y,=aY,u,
Y¥, = a°Y.u, — $w°a5 (4 — cos? @),
Y; = a Y,uz, &e.
The first of these equations merely gives a relation between the arbitrary constants Y, and e;
the others determine Y,, Y., &c.; and we get by substituting in (7)

V Y 1 a a wa
= (+ ome Gu +-.) =

on (Ge (0H8) ies oadderne (10)

6. Let g be the force of gravity at any point of the surface of the earth, dm an element of
the normal drawn outwards at that point; then g = — (V+U). Let W be the angle between
the normal and the radius vector; then gcosyy is the resolved part of gravity along the radius
vector, and this resolved part is equal to — = (V+U). Now w is a small quantity of the first
order, and therefore we may put cos Wy = 1, which gives

d

where, after differentiation, 7 is to be replaced by the radius vector of the surface, which is given by
(8). We thus get
Y,

Mi
g= #3 (1 — 2u, — 2u, — 2Uy...) + ey (2m, + 3u, + 4us...) — Zw*a (4 — cos’) — w*a (3 + 4 — cos°6),

which gives, on putting

and neglecting squares of small quantities,
& = G {1 — Sm (4 — cos’O) + wy + 2Uy + BAUy eed. veevecees (12)
In this equation G is the mean value of @ taken throughout the whole surface, since we know

wn pin
that [ i u, sin 0 d0dq@ = 0, if nm be different from zero. The second of equations (11) shews
0 0

that m is the ratio of the centrifugal force at a distance from the axis equal to the mean distance to
mean gravity, or, which is the same, since the squares of small quantities are neglected, the ratio
of the centrifugal force to gravity at the equator. Equation (12) makes known the variation of
gravity when the form of the surface is given, the surface being supposed to be one of equilibrium ;
and, conversely, equation (8) gives the form of the surface if the variation of gravity be known,
It may be observed that on the latter supposition there is nothing to determine w,. The most
general form of w, is
asin 0 cosh + B sin @ sin p + ¥ cos @,

Vou, VIII. Parr V. 45

678 Mr. STOKES, ON THE VARIATION OF GRAVITY

where a, 3, y are arbitrary constants; and it is very easy to prove that the co-ordinates of the
centre of gravity of the volume are equal to aa, af, ay respectively, the line from which @ is
measured being taken for the axis of x, and the plane from which @ is measured for the plane of
az. Hence the term wz, in (8) may be made to disappear by taking for origin the centre of gravity
of the volume. It is allowable to do this even should the centre of gravity fall a little out of the
axis of rotation, because the term involving the centrifugal force, being already a small quantity
of the first order, would not be affected by supposing the origin to be situated a little out of
the axis.

Since the variation of gravity from one point of the surface to another is a small quantity of
the first order, its expression will remain the same whether the earth be referred to one origin or
another nearly coinciding with the centre, and therefore a knowledge of the variation will not
inform us what point has been taken for the origin to which the surface has been referred.

7. Since the angle between the vertical at any point and the radius vector drawn from the
origin is a small quantity of the first order, and the angles 0, @ occur in the small terms only of
equations (8), (10), and (12), these angles may be taken to refer to the direction of the vertical,
instead of the radius vector.

8. If E be the mass of the earth, the potential of its attraction at a very great distance 7 is
ultimately equal to —. Comparing this with (10), we get ¥,=, and therefore, from the first
r

of equations (11),
E=Ga'+30'a@ = Ga’ (1+3m),......(13)
which determines the mass of the earth from the value of G determined by pendulum experiments.

9. If we suppose that the surface of the earth may be represented with sufficient accuracy by
an oblate spheroid of small ellipticity, having its axis of figure coincident with the axis of rotation,
equation (8) becomes

r=a{1+e(4 — cos’6)},......... Pate (14)

where ¢ is a constant which may be considered equal to the ellipticity. We have therefore in this
case U, = 0, Uy = 4 — cos’, uw, = 0 when 2 > 2; so that (12) becomes

g= G51 — (Sm —c) (4 —cos’)},.0. ee eeeee ees (15)
which equation contains Clairaut’s Theorem. It appears also from this equation that the value of
G which must be employed in (13) is equal to gravity at a place the square of the sine of whose
latitude is 4.

10. Retaining the same supposition as to the form of the surface, we get from (10), on
replacing Y, by E, and putting in the small term at the end w°a> = m Ga‘ = m Ea’,

7 2

E Ea :
View Sse > $11) 3 (4 — cos’ 8).--.--eee eee (16)

Consider now the effect of the earth’s attraction on the moon. ‘The attraction of any particle
of the earth on the moon, and therefore the resultant attraction of the whole earth, will be very
nearly the same as if the moon were collected at her centre. Let therefore be the distance of the
centre of the moon from that of the earth, @ the moon’s North Polar Distance, P the accelerating
force of the earth on the moon resolved along the radius vector, Q the force perpendicular to the
radius vector, which acts evidently in a plane passing through the earth’s axis; then

dV dV

AT THE SURFACE OF THE EARTH. 679

whence we get from (16)

2

Ea E
+ 3 (e-—3m) = (4 - cos’ 6), Q = 2 (c= 4m) — sin cos 6.......(17)
ia NE 7

E

eS

a

The moving forces arising from the attraction of the earth on the moon will be obtained by

multiplying by MM, where M denotes the mass of the moon; and these are equal and opposite to

the moving forces arising from the attraction of the moon on the earth. The component MQ of

the whole moving force is equivalent to an equal and parallel force acting at the centre of the earth

and a couple. The accelerating forces acting on the earth will be obtained by dividing by EF; and

since we only want to determine the relative motions of the moon and earth, we may conceive equal

and opposite accelerating forces applied both to the earth and to the moon, which comes to the

same thing as replacing E by E + M in (17). If K be the moment of the couple arising from the
attraction of the moon, which tends to turn the earth about an equatoreal axis, K = MQr, whence

MEa

7 Sin @ cos @,......(18)
ms

K =2(e—14m)

The same formula will of course apply, mutatis mutandis, to the attraction of the sun.

11. The spheroidal form of the earth’s surface, and the circumstance of its being a surface of
equilibrium, will afford us some information respecting the distribution of matter in the interior.
Denoting by a’, y’, s' the co-ordinates of an internal particle whose density is p, and by a, y, =
those of the external point of space to which V refers, we have

‘de’ dy dz’
AN ome ere cee em cory itt

the integrals extending throughout the interior of the earth. Writing dm' for p' da’ dy’ dz’,
putting A, 4, v for the direction-cosines of the radius vector drawn to the point (2, y, x), so that
v= X17, y= pr, x = vr, and expanding the radical according to inverse powers of r, we get

1 | r phe decnuetel 5 mote = ibs ai
V=—fffdm re Se dm + Fa 2 (8N - 1) fffa* dm + <5 Edm fffa'y dm’ + ...(19)

= denoting the sum of the three expressions necessary to form a symmetrical function. Comparing
this expression for V with that given by (10), which in the present case reduces itself to (16), we
get Y, = [[{dm' = E, as before remarked, and

Pfifardm! = 0; fifify) dm!’ = 0, fiffix dim =0;....c.000c0-.00- pease (20)
45 (8d? - 1) ff[fa? dm’ +3 Edu fffa'y'dm’ = (e - 4m) Ea’ (4 - cos’ );......(21)
together with other equations, not written down, obtained by equating to zero the coefficients of

35 &e. in (19).

Equations (20) shew that the centre of gravity of the mass coincides with the centre of gravity
of the volume. In treating equation (21), it is to be remarked that A, »,v are not independent, but
connected by the equation \2 + »? + »°=1. If now we insert * + »° + v” as a coefficient in each
term of (21) which does not contain , u, or v, the equation will become homogeneous with respect to
d, 4, , and will therefore only involve the two independent ratios which exist between these three

quantities, and consequently we shall have to equate to zero the coefficients of corresponding powers

of X,n,v. By the transformation just mentioned, equation (21) becomes, since cos @ = »,
D(X? — 4p? - dv’) [ffo'? dm! + 3EAp fffa'y'dm!' = (e- 4m) Ea’ (§N + $u° - $v);
and we get
[[fa'ydm' = 0, fffy'x'dm' =0, fffx'a'dm’ = 0,...... nog obivaaeeeras steve Rismaeds Sonal (22)

Ueto’ — hiya ~ bisfehdn = fran’ ~ bffferaes — Aflaaal) ay

= 4 fffetdm! +} fffal'dm’ + ffftdml = Yo ~ hm) Bar
482

680 Mr. STOKES, ON THE VARIATION OF GRAVITY
Equations (22) shew that the co-ordinate axes are principal axes. Equations (23) give in

the first place
[[fu''am'=[fy"am

which shews that the moments of inertia about the axes of w and y are equal to each other, as might
have been seen at once from (22), since the principal axes of w and y are any two rectangular axes
in the plane of the equator. The two remaining equations of the system (23) reduce themselves to
one, which is

[ffe'*dm' — fffzx*dm’ = 2(e — 3m) Ea’.
If we denote the principal moments of inertia by A, A, C, this equation becomes
C-A=8(e-—Jm) EA, .....1.0e eee eee ccmicemeaice (24)
which reconciles the expression for the couple K given by (18) with the expression usually given,

which involves moments of inertia, and which, like (18), is independent of any hypothesis as to the
distribution of the matter within the earth,

It should be observed that in case the earth be not solid to the centre the quantities 4, C must
be taken to mean what would be the moments of inertia if the several particles of which the earth
is composed were rigidly connected.

12. In the preceding article the surface has been supposed spheroidal. In the general case of
an arbitrary form we should have to compare the expressions for V given by (10) and (19). In the
first place it may be observed that the term w, can always be got rid of by taking for origin the
centre of gravity of the volume. Equations (20) shew that in the general case, as well as in the
particular case considered in the last article, the centre of gravity of the mass coincides with the
centre of gravity of the volume.

Now suppress the term w, in w, and let w= wu! + w’, where wu” =4m(4—-cos’@). Then wu’ may
be expanded in a series of Laplace’s coefficients w’, + wv’, +.-.; and since Y, = E, equation (10) will
be reduced to

2 3
ID Ee
v= B (45 w+ W's.) eeeces eeecese os eencceves (25)

If the mass were collected at the centre of gravity, the second member of this equation would
be reduced to its first term, which requires that w’, = 0, u';=0, &c. Hence (8) would be reduced
to r=a(1+w’"), and therefore aw” is the alteration of the surface due to the centrifugal force, and
au’ the alteration due to the difference between the actual attraction and the attraction of a sphere
composed of spherical strata. Consider at present only the term w’, of w’. From the general form
of Laplace’s coefficients it follows that aw’, is the excess of the radius vector of an ellipsoid not much
differing from a sphere over that of a sphere having a radius equal to the mean radius of the ellipsoid.
If we take the principal axes of this ellipsoid for the axes of co-ordinates, we shall have

uw’, = (4 — sin* 0 cos’ p) + &” (4 — sin’ Asin? p) + (4 — cos’),
e,e ,¢” being three arbitrary constants, and @, @ denoting angles related to the new axes of a, y, x
in the same way that the angles before denoted by 0, @ were related to the old axes. Substituting
the preceding expression for w’, in (25), and comparing the result with (19), we shall again obtain
equations (22). Consequently the principal axes of the mass passing through the centre of gravity
coincide with the principal axes of the ellipsoid. It will be found that the three equations which
replace (23) are equivalent to but two, which are
A-2¢Ea = B - Se Ea = C- ge" Ea’,

where A, B, C denote the principal moments.

The permanence of the earth’s axis of rotation shews however that one of the principal axes of
the ellipsoid coincides, at least very nearly, with the axis of rotation; although, strictly speaking, this

—s”

fs

-

PP yrs

AT THE SURFACE OF THE EARTH. 681

conclusion cannot be drawn without further consideration except on the supposition that the earth
is solid to the centre. If we assume this coincidence, the term e”(4 — cos’ @) will unite with the
term w” due to the centrifugal force. Thus the most general value of u is that which belongs to an
ellipsoid having one of its ae axes coincident with the axis of rotation, added to a quantity
which, if expanded in a series of Laplace’s coefficients, would furnish no terms of the order 0, 1, or 2.

It appears from this and the preceding article that the coincidence of the centres of gravity of the
mass and volume, and that of the axis of rotation and one of the principal axes of the ellipsoid whose
equation is r=a(1+2,), which was established by Laplace on the supposition that the earth consists of
nearly spherical strata of equal density, holds good whatever be the distribution of matter in the interior.

13. Hitherto the surface of the earth has been regarded as a surface of equilibrium. This we
know is not strictly true, on account of the elevation of the land above the level of the sea. he
question now arises, By what imaginary alteration shall we reduce the surface to one of equilibrium ?

Now with respect to the greater portion of the earth’s surface, which is covered with water, we
have a surface of equilibrium ready formed. The expression level of the sea has a perfectly definite
meaning as applied to a place in the middle of a continent, if it be defined to mean the level at
which the sea-water would stand if introduced by a canal. The surface of the sea, supposed to be
prolonged in the manner just considered, forms indeed a surface of equilibrium, but the preceding
investigation does not apply directly to this surface, inasmuch as a portion of the attracting matter
lies outside it. Conceive however the land which lies above the level of the sea to be depressed till
it gets below it, or, which is the same, conceive the land cut off at the level of the sea produced,
and suppose the density of the earth or rock which lies immediately below the sea-level to be in-
creased, till the increase of mass immediately below each superficial element is equal to the mass
which has been removed from above it. The whole of the attracting matter will thus be brought
inside the original sea-level; and it is easy to see that the attraction at a point of space external to
the earth, even though it be close to the surface, will not be sensibly affected. Neither will the
sea-level be sensibly changed, even in the middle of a continent. For, suppose the sea-water intro-
duced by a pipe, and conceive the land lying above the sea-level condensed into an infinitely thin
layer coinciding with the sea-level. The attraction of an infinite plane on an external particle does
not depend on the distance of the particle from the plane; and if a line be drawn through the
particle inclined at an angle a to the perpendicular let fall on the plane, and be then made to revolve
around the perpendicular, the resultant attraction of the portion of the plane contained within the
cone thus formed will be to that of the whole plane as versina to 1. Hence the attraction of a
piece of table-land on a particle close to it will be sensibly the same as that of a solid of equal
thickness and density comprised between two parallel infinite planes, and that, even though the
lateral extent of the table-land be inconsiderable, only equal, suppose. to a small multiple of the
length of a perpendicular let fall from the attracted particle on the further bounding plane. Hence
the attraction of the land on the water in the tube will not be sensibly altered by the condensation we
have supposed, and therefore we are fully justified in regarding the level of the sea as unchanged.

The surface of equilibrium which by the imaginary displacement of matter just considered has
also become the bounding surface, is that surface which at the same time coincides with the surface
of the actual sea, where the earth is covered by water, and belongs to the system of surfaces of
equilibrium which lie wholly outside the earth. ‘To reduce observed gravity to what would have
been observed just above this imaginary surface, we must evidently increase it in the inverse ratio
of the square of the distance from the centre of the earth, without taking account of the attraction
of the table-land which lies between the level of the station and the level of the sea. ‘The question
now arises, How shall we best determine the numerical value of the earth’s ellipticity, and how
best compare the form which results from observation with the spheroid which results from theory
on the hypothesis of original fluidity ?

682 Mr. STOKES, ON THE VARIATION OF GRAVITY

14. Before we consider how the numerical value of the earth’s ellipticity is to be determined,
it is absolutely necessary that we define what we mean by ellipticity ; for, when the irregularities of
the surface are taken into account, the term must be to a certain extent conventional.

Now the attraction of the earth on an external body, such as the moon, is determined by the
function V, which is given by (10). In this equation, the term containing r~* will disappear if r
be measured from the centre of gravity; the term containing r~*, and the succeeding terms, will
be insensible in the case of the moon, or a more distant body. The only terms, therefore, after
the first, which need be considered, are those which contain r~°. Now the most general value of wu,
contains five terms, multiplied by as many arbitrary constants, and of these terms one is $ — cos? 6,
and the others contain as a factor the sine or cosine of @ or of 2 @. The terms containing sin @ or
cos @ will disappear for the reason mentioned in Art. 12; but even if they did not disappear their
effect would be wholly insensible, inasmuch as the corresponding forces go through their period in
a day, a lunar day if the moon be the body considered. These terms therefore, even if they ex-
isted, need not be considered; and for the same reason the terms containing sin 2@ or cos 2 may
be neglected ; so that nothing remains but a term which unites with the last term in equation (10).
Let ¢ be the coefficient of the term } — cos’ @ in the expansion of w: then ¢ is the constant which
determines the effect of the earth’s oblateness on the motion of the moon, and which enters into
the expression for the moment of the attractions of the sun and moon on the earth; and in the
particular case in which the earth’s surface is an oblate spheroid, having its axis coincident with
the axis of rotation, ¢ is the ellipticity. Hence the constant ¢ seems of sufficient dignity to deserve
a name, and it may be called in any case the ellipticity.

Let r be the radius vector of the earth’s surface, regarded as coincident with the level of the
sea; and take for shortness m { f(0,@)} to denote the mean value of the function f (8, p)

1
throughout all angular space, or zo f?"f (@,) sin@d@dq. Then it follows from the theory
7
of Laplace’s coefficients that
e=im fo Sin? 2), 6 5 anise da ecn'a o00(26)
1 being the latitude, or the compliment of @. To obtain this equation it is sufficient to multiply
: 1
both sides of (8) by a (4 - cos’ 6) sin 0d@ dq, and to integrate from 6 = 0 to @ = 7, and from

p=0top=27. Since + — cos’ @ is a Laplace’s coefficient of the second order, none of the
terms at the second side of (8) will furnish any result except aw, and even in the case of w, the
terms involving the sine or cosine of @ or of 2@ will disappear.

15. Let g be gravity reduced to the level of the sea by taking account only of the height of
the station. Then this is the quantity to which equation (12) is applicable; and putting for w, its
value we get by means of the properties of Laplace’s coefficients

45
G= m(g), G(Bm -e)=- ice mt 44 = sin? 1) gh. ..0.0eeee0-0+(27)

If we were possessed of the values of g at an immense number of stations scattered over
the surface of the whole earth, we might by combining the results of observation in the
manner indicated by equations (27) obtain the numerical values of Gand e«, We cannot, however,
obtain by observation the values of g at the surface of the sea, and the stations on land where
the observations have been made from which the results are to be obtained are not very numerous.
We must consider therefore in what way the variations of gravity due to merely local causes are
to be got rid of, when we know the causes of disturbance; for otherwise a local irregularity,
which would be lost in the mean of an immense number of observations, would require undue
importance in the result.

Al

—

AT THE SURFACE OF THE EARTH. 683

16. Now the most obvious cause of irregularity consists in the attraction of the land lying
between the level of the station and the level of the sea. This attraction would render the
values of g sensibly different, which would be obtained at two stations only a mile or two apart,
but situated at different elevations. To render our observations comparable with one another, it
seems best to correct for the attraction of the land which lies underneath the pendulum ; but then
we must consider whether the habitual neglect of this attraction may not affect the mean values
from which G and e are to be found.

Let g=2,+g, where g’ is the attraction just mentioned, so that g, is the result obtained by
reducing the observed value of gravity to the level of the sea by means of Dr. Young's formula*.
Let h be the height of the station above the level of the sea, o the superficial density of
the earth where not covered by water; then by the formula for the attraction of an infinite plane
we have g’ =2ach. To make an observation, conceived to be taken at the surface of the sea,
comparable with one taken on land, the correction for local attraction would be additive, instead of
subtractive; we should have in fact to add the excess of the attraction of a layer of earth or rock,
of a thickness equal to the depth of the sea at that place, over the attraction of so much water.
The formula g’ = 27ch will evidently apply to the surface of the sea, provided we regard h as a
negative quantity, equal to the depth of the sea, and replace a by o — 1, the density of water being
taken for the unit of density; or we may retain o as the coefficient, and diminish the depth in
the ratio of c to o—1.

Let p be the mean density of the earth, then

, 2aroh 3ah
g=2ach=G>— =G—_.
3 Tpa 2 pa
If we suppose c =24, p=54, a= 4000 miles, and suppose hk expressed in miles, with the

understanding that in the case of the sea h is a negative quantity equal to 2ths of the actual
depth, we have g’ = .00017 Gh nearly.

17. Consider first the value of G. We have by the preceding formula, and the first of
equations (27),

G =M(g,) + G x .00017 m (A).

According to Professor Rigaud’s determination, the quantity of land on the surface of the
earth is to that of water as 100 to 276+. If we suppose the mean elevation of the land 1th
of a mile, and the mean depth of the sea 34 miles, we shall have
Sy 31» g76—-1
m(h) = -° eee 5 ee Seng nearly ; -
so that the value of G determined by g, would be too great by about .000253 of the whole. Hence
the mass of the earth determined by the pendulum would be too great by about the one four-
thousandth of the whole; and therefore the mass of the moon, obtained by subtracting from
the sum of the masses of the earth and moon, as determined by means of the coefficient of lunar
parallax, the mass of the earth alone, as determined by means of the pendulum, would be too
small by about the one four-thousandth of the mass of the earth, or about the one fiftieth of
the whole.

18. Consider next the value of e. Let e, be the value which would be determined by sub-
stituting g, for g in (27), and let

* Phil. Trans. for 1819. Dr. Young’s formula is based on the
principle of taking into account the attraction of the table-land
existing between the station and the level of the sea, in reducing | 7=24, p=64. Mr Airy, observing that the value o
the observation to the sea level. On account of this attraction, the | little too small, and p=54 a little too great, has employed the

2h A , F - factor .6, instead of . 16,
multiplier (=) which gives the correction for elevation alone t Cambridge Philosophical Transactions, Vol, v1. p. 297.

hile Ba a ;
must be reduced in the ratio of | to L- vad 1 to .66 nearly, if
4p

2h is «

684 Mr. STOKES, ON THE VARIATION OF GRAVITY

45 :
= m {(sin?Z - L)g't = Gq.

In considering the value of q we may attend only to the land, provided we transfer the defect of
density of the sea with an opposite sign to the land, because if g’ were constant, g would vanish.
This of course proceeds on the supposition that the depth of the sea is constant. Since e =e, —q,
if q were positive, the ellipticity determined by the pendulum would appear too great in con-
sequence of the omission of the force g’. I have made a sort of rough integration by means of
a map of the world, by counting the quadrilaterals of land bounded each by two meridians
distant 10°, and by two parallels of latitude distant 10°, estimating the fraction of a broken
quadrilateral which was partly occupied by sea. The number of quadrilaterals of land between
two consecutive parallels, as for example 50° and 60°, was multiplied by 12 (4 — sin*/) cos 7, or
3cos 31 + cos/, where for J was taken the mean latitude, (55° in the example,) the sum of the
results was taken for the whole surface, and multiplied by the proper coefficient. The north pole
was supposed to be surrounded by water, and the south pole by land, as far as latitude 80”. It
appeared that the land lying beyond the parallels for which sin*/ = 4, that is, beyond the
parallels 35° N. and 35°S. nearly, was almost exactly neutralized by that which lay within those
parallels. On the whole, g appeared to have a very small positive value, which on the same
suppositions as before respecting the height of the land and the depth of the sea, was .0000012.
It appears, therefore, that the omission of the force g’ will produce no sensible increase in the
value of ©, unless the land be on the whole higher, or the sea shallower, in high latitudes than in
low. If the land had been collected in a great circular continent around one pole, the value of
q would have been .000268; if it had been collected in a belt about the equator, we should have
had g = — .000362. The difference between these values of q is about one fifth of the whole
ellipticity.

19. The attraction g’ is not the only irregularity in the magnitude of the force of gravity
which arises from the irregularity in the distribution of land and sea, and in the height of the
land and depth of the sea, although it is the only irregularity, arising from that cause, which is
liable to vary suddenly from one point at the surface to another not far off. The irregular coating
of the earth will produce an irregular attraction besides that produced by the part of this coating
which lies under and in the immediate neighbourhood of the station considered, and it will
moreover cause an irregular elevation or depression in the level of the sea, and thereby cause a
diminution or increase in the value of g,.

Consider the attraction arising from the land which lies above the level of the sea, and from
the defect of attracting matter in the sea. Call this excess or defect of matter the coating of the
earth: conceive the coating condensed into a surface coinciding with the level of the sea, and
let 46 be the mass contained in a small element A of this surface. Then 6 =ch in the case of the
land, and 6 = —(a—1)h in the case of the sea, h being in that case the depth of the sea. Let
V. be the potential of the coating, V’, V” the values of V, outside and inside the surface respec-
tively. Conceive 6 expanded in a series of Laplace’s coefficients ot 6) + ..-, then it is easily
proved that

a

; 1 a 1 r
Vi=4ra (- 6 +—6h+—8+ END V" = 47a? (—6, + — 6+ We poeeosees|(25)
r sr br a7]

3a’

r being the distance of the point considered from the centre. These equations give

dV’ Tl aN ey i us ee
———— = so = 4 - jt tet eee ene weneee weenie.
7 and ai (*) 8; op TZ a (: 6 (29)

Consider two points, one external, and the other internal, situated along the same radius vector

AT THE SURFACE OF THE EARTH. 685

very close to the surface. Let E be an element of this surface lying around the radius vector, an
element which for clear ideas we may suppose to be a small circle of radius s, and let s be at the
same time infinitely small compared with a, and infinitely great compared with the distance between

, ”

: seer d
the points. Then the limiting values of aa and er will differ by the attraction of the element
Z

£, an attraction which, as follows from what was observed in Art. 13, will be ultimately the same

- ve
as that of an infinite plane of the same density, or 26*. The mean of the values of == and a

r r
will express the attraction of the general coating in the direction of the radius vector, the
general coating being understood to mean the whole coating, with the exception of a superficial
element lying adjacent to the points where the attraction is considered. Denoting this mean by

c

gare WE get, on putting r=a,

SN ss 6 .
dr 2141

This equation becomes by virtue of either of the equations (28)

dV, Ve
BS pt > Ne eetecmaeeeneecn (Er
dr 2a eo

which is a known equation. Let either member of this equation be denoted by -g”’. Then
gravity will be increased by g”, in consequence of the attraction of the general coating.

20. But besides its direct effect, the attraction of the coating will produce an indirect effect by
altering the sea-level. Since the potential at any place is increased by V, in consequence of the
coating, in passing from what would be a surface of equilibrium if the coating were removed, to the
actual surface of equilibrium corresponding to the same parameter, Sthat is, the same value of the
constant ¢ in equation (1),{ we must ascend till the labouring force expended in raising a unit of

j Viet &
mass is equal to V,, that is, we must ascend through a space re or G nearly. In consequence of
this ascent, gravity will be diminished by the quantity corresponding to the height G~! V,, or h’
suppose. If we take account only of the alteration of the distance from the centre of the earth,

,

fre 2 if ” . .
this diminution will be equal to Ge or saat or 4g", and therefore the combined direct and
indirect effects of the general coating will be to diminish gravity by 3g”.

But the attraction of that portion of the stratum whose thickness is h’, which lies immediately
about the station considered, will be a quantity which involves h’ as a factor, and to include this
attraction we must correct for the change of distance h’ by Dr. Young’s rule, instead of correcting
merely according to the square of the distance. In this way we shall get for the diminution of

” 8 ” ” ”
gravity due to the general coating, not 3g”, but only 4 (1 =e - g, or kg” suppose. If
P
o:p::5:11, we have k =1. 64 nearly.

* This result readily follows from equations (28), which give, | the attraction of the interposed clement of surface, which, being

; DEER Nad Side Sada a Dhin difference af ultimately plane, will act equally at both points; and, therefore,
1 Ueiaael SO a lc alla us cierence OF | the attraction will be in each case 27rd, and will act outwards in

attraction at points infinitely close can evidently only arise from | the first case, and inwards in the second,

Vou. VIII. Parr V. 4T

686 Mr. STOKES, ON THE VARIATION OF GRAVITY

If we cared to leave the mean value of gravity unaltered, we should have to use, instead of 0,
its excess over its mean value 6,. In considering, however, only the variation of gravity from one
place to another, this is a point of no consequence.

21. In order to estimate the magnitude which the quantity 3g” is likely to attain, conceive
two stations, of which the first is surrounded by land, and the second by sea, to the distance of
1000 miles, the distribution of land and sea beyond that distance being on the average the same at
the two stations. Then, by hypothesis, the potential due to the land and sea at a distance greater
than 1000 miles is the same at the two stations; and as we only care for the difference between the
values of the potential of the earth’s coating at the two stations, we may transfer the potential due
to the defect of density at the second station with an opposite sign to the first station. We shall
thus have around the first station, taking h’ for the depth of the sea around the second station,
d=oh+(c-1)h’. In finding the difference V of the potentials of the coating, it will be amply
sufficient to regard the attracting matter as spread over a plane disk, with a radius s equal to 1000
miles. On this supposition we get

V = fis-'. 2adsds = 270s.

Now G = 4 pa, and therefore ye = Ad = OPE es z ar lan Ie Ge Making the same

2a Apa 4 pa a

suppositions as before with regard to the numerical values of oc, p, h, h’, and a, we get

3g” =-000147G. This corresponds to a difference of 6-35 vibrations a day in a seconds’ pendulum.

Now a circle with a radius of 1000 miles looks but small on a map of the world, so that we may

readily conceive that the difference depending on this cause. between the number of vibrations

observed at two stations might amount to 15 or 20, that is 7.5 or 10 on each side of the mean, or

even more if the height of the land or the depth of the sea be under-estimated. This difference
will however be much reduced by using kg” in place of 3g”*.

22. The value of V, at any station is expressed by a double integral, which is known if 6 be
known, and which may be calculated numerically with sufficient accuracy by dividing the surface
into small portions and performing a summation. Theoretically speaking, V7, could be expressed
for the whole surface at once by means of a series of Laplace’s coefficients ; the constants in this
series could be determined by integration, or at least the approximate integration obtained by
summation, and then the value of V, could be obtained by substituting in the series the latitude and
longitude of the given station for the general latitude and longitude. But the number of terms
which would have to be retained in order to represent with tolerable accuracy the actual state of the
earth’s surface would be so great that the method, I apprehend, would be practically useless ;
although the leading terms of the series would represent the effect of the actual distribution of land
and sea in its broad features. It seems better to form directly the expression for V, at any station.
This expression may be calculated numerically for each station by using the value of 4 most likely
to be correct, if the result be thought worth the trouble; but even if it be not calculated
numerically, it will enable us to form a good estimation of the variation of the quantity 3g” or kg”
from one place to another.

Let the surface be referred to polar co-ordinates originating at the centre, and let the angles
\, x be with reference to the station considered what 0, were with reference to the north pole.
The mass of a superficial element is equal to da sin ydydy, and its distance from the station is

V

2a sin or Hence we have

V.=affo cos dydy. asaasa(el)

© The effect of the irregularity of the earth’s surface is greater than what is represented by kg", for a reason which will be explained
further on (Art. 25).

AT THE SURFACE OF THE EARTH. 687

Let 6,, be the mean value of § throughout a circle with an angular radius y,, then the part of

V, which is due to an annulus having a given infinitely small angular breadth dy, is proportional to
v

9 >

6 COS or to 6,, nearly when y is not large. If we regard the depth of the sea as uniform, we

may suppose 6 = 0 for the sea, and transfer the defect of density of the sea with an opposite sign to
the land. We have seen that if we set a circle of land + mile high of 1000 miles radius surrounding
one station against a circle of sea 34 miles deep, and of the same radius, surrounding another, we get
a difference of about 4 x 1.64 x 6.35, or sh nearly, in the number of vibrations performed in one
day by a seconds’ pendulum. It is hardly necessary to remark that high table-land will produce
considerably more effect than land only just raised above the level of the sea, but it should be
observed that the principal part of the correction is due to the depth of the sea. Thus it would
require a uniform elevation of about 2.1 miles, in order that the land elevated above the level of the
sea should produce as much effect as is produced by the difference between a stratum of land
34 miles thick and an equal stratum of water.

23. These considerations seem sufficient to account, at least in a great measure, for the
apparent anomalies which Mr. Airy has noticed in his discussion of pendulum experiments*. The
first table at p. 230 contains a comparison between the observations which Mr. Airy considers first-
rate and theory. The column headed “ Error in Vibrations” gives the number of vibrations per
diem in a seconds’ pendulum corresponding to the excess of observed gravity over calculated
gravity. With respect to the errors Mr. Airy expressly remarks ‘upon scrutinizing the errors of
the first-rate observations, it would seem that, ceteris paribus, gravity is greater on islands than on
continents.” ‘This circumstance appears to be fully accounted for by the preceding theory. The
greatest positive errors appear to belong to oceanic stations, which is just what might be expected.
Thus the only errors with the sign + which amount to 5 are, Isle of France + 7.0; Marian
Islands + 6.8; Sandwich Islands + 5.2; Pulo Gaunsah Lout (a small island near new Guinea and
almost on the equator,) + 5.0. The largest negative errors are, California — 6.0; Maranham — 5.6;
Trinidad — 5.2. These stations are to be regarded as continental, because generally speaking the
stations which are the most continental in character are but on the coasts of continents, and Trinidad
may be regarded as a coast station. That the negative errors just quoted are larger than those that
stand opposite to more truly continental stations such as Clermont, Milan, &c. is no objection,
because the errors in such different latitudes cannot be compared except on the supposition that the
value of the ellipticity used in the comparison is correct.

Now if we divide the 49 stations compared into two groups, an equatoreal group containing the
stations lying between latitudes 35°N. and 35°S., and a polar group containing the rest, it will
be found that most if not all of the oceanic stations are contained in the former group, while the
stations belonging to the latter are of a more continental character. Hence the observations will
make gravity appear too great about the equator and too small towards the poles, that is, they will
on the whole make gravity vary too little from the equator to the poles; and since the variation
depends upon 3m —e, the observations will be best satisfied by a value of e which is too great.
This is in fact precisely the result of the discussion, the value of « which Mr. Airy has obtained
from the pendulum experiments (.003535) being greater than that which resulted from the dis-
cussion of geodetic measures (.003352), or than any of the values (.003370, .003360, and .003407),
obtained from the two lunar inequalities which depend upon the earth’s oblateness.

Mr. Airy has remarked that in the high north latitudes the greater number of errors have the
sign +, and that those about the latitude 45° have the sign — ; those about the equator being

* Encyclopaedia Metropolitana, Art. Figure of the Earth,
472

688 Mr. STOKES, ON THE VARIATION OF GRAVITY

nearly balanced. To destroy the errors in high and mean latitudes without altering the others, he
has proposed to add a term — 4 sin?) cos’), where ) is the latitude. But a consideration of the
character of the stations seems sufficient, with the aid of the previous theory, to account for the
apparent anomaly. About latitude 45° the stations are all continental; in fact, ten consecutive
stations including this latitude are Paris, Clermont, Milan, Padua, Fiume, Bordeaux, Figeac, Toulon,
Barcelona, New York. These stations owght, as a group, to appear with considerable negative errors.
Mr. Airy remarks “If we increased the multiplier of sin*),” and consequently diminished the
ellipticity, ‘‘we might make the errors at high latitudes as nearly balanced as those at the equator:
but then those about latitude 45° would be still greater than at present.”

The largeness of the ellipticity used in the comparison accounts for the circumstance that the
stations California, Maranham, Trinidad, appear with larger negative errors than any of the stations
about latitude 45°, although some of the latter appear more truly continental than the former. On
the whole it would seem that the best value of the ellipticity is one which, supposing it left the errors
in high latitudes nearly balanced, would give a decided preponderance to the negative errors about
latitude 45° N. and a certain preponderance to the positive errors about the equator, on account of
the number of oceanic stations which occur in low latitudes.

If we follow a chain of stations from the sea inland, or from the interior to the coast, it is
remarkable how the errors decrease algebraically from the sea inwards. The chain should not extend
over too large a portion of the earth’s surface, as otherwise a small error in the assumed ellipticity
might affect the result. Thus for example, Spitzbergen + 4.3, Hammerfest — 0.4, Drontheim ~ 2.7.
In comparing Hammerfest with Drontheim, we may regard the former as situated at the vertex of a
slightly obtuse angle, and the latter as situated at the edge of astraight coast. Again, Dunkirk — 0,1,
Paris — 1.9, Clermont — 3.9, Figeac — 3.8, Toulon — 0.1, Barcelona 0.0, Fomentera + 0.2. Again,
Padua + 0.7, Milan — 2.8. Again, Jamaica — 0.8, Trinidad — 5.2.

24, Conceive the correction kg" calculated, and suppose it applied, as well as the correction
—g’, to observed gravity reduced to the level of the sea, or to g, and let the result be 2: Leetige,
be the ellipticity which would be determined by means of g,, «+ Ac, the true ellipticity, Since
g§,=8-g + k’g’, and therefore g=g +g’ — kg", we get by (27)

45
Ac, = 5m {CG - sin?) (g — hg”) }. ceseseseeene (82)
Now g’ = 2ach = 270 = 2736,; and we get from (30) and (28)

}
kr =— .
dr 24 2i+1

All the terms 6; will disappear from the second side of (32) except 6,, and we therefore get
45 A k
Ac, = Gh §(4- sin®/) (: = =\an dy.

Hence the correction Ae, is less than that considered in Art. 18, in the ratio of 5 — k to 5, and is

therefore probably insensible on account of the actual distribution of land and water at the surface of
the earth.

25. Conceive the islands and continents cut off at the level of the sea, and the water of the sea
replaced by matter having the same density as the land. Suppose gravity to be observed at the
surface which would be thus formed, and to be reduced by Dr Young’s rule to the level of what
would in the altered state of the earth be a surface of equilibrium. It is evident that g,, expresses
the gravity which would be thus obtained,

The irregularities of the earth’s coating would still not be wholly allowed for, because the surface
which would be formed in the manner just explained would no longer be a surface of equilibrium,

AT THE SURFACE OF THE EARTH. 689

in consequence of the fresh distribution of attracting matter. The surface would thus preserve traces
of its original irregularity. A repetition of the same process would give a surface still more regular,
and so on indefinitely. It is easy to see the general nature of the correction which still remains.
Where a small island was cut off, there was previously no material elevation of the sea-level, and
therefore the surface obtained by cutting off the island and replacing the surrounding sea by land
will be very nearly a surface of equilibrium, except in so far as that may be prevented by alterations
which take place on a large scale. But where a continent is cut off there was a considerable elevation
in the sea-level, and therefore the surface which is left will be materially raised above the surface of
equilibrium which most nearly represents the earth’s surface in its altered state. Hence the general
effect of the additional correction will be to increase that part of g” which is due to causes which act
on a larger scale, and to leave nearly unaffected that part which is due to causes which are more
local.

The form of the surface of equilibrium which would be finally obtained depends on the new
distribution of matter, and conversely, the necessary distribution of matter depends on the form of
the final surface. The determination of this surface is however easy by means of Laplace’s analysis.

26. Conceive the sea replaced by solid matter, of density o, having a height from the bottom
upwards which is to the depth of the seaas1 toc. Let h be the height of the land above the actual
sea-level, being negative in the case of the sea, and equal to the depth of the sea multiplied by
1—oa~'. Let a be the unknown thickness of the stratum which must be removed in order to leave
the surface a surface of equilibrium, and suppose the mean value of w to be zero, so that on the whole
matter is neither added nor taken away. The surface of equilibrium which would be thus obtained
is evidently the same as that which would be formed if the elevated portions of the irregular surface

were to become fluid and to run down.

Let V be the potential of the whole mass in its first state, V, the potential of the
stratum removed. ‘The removal of this stratum will depress the surface of equilibrium by the
space G-'V,; and the condition to be satisfied is, that this new surface of equilibrium, or else a
surface of equilibrium belonging to the same system, and therefore derived from the former by
further diminishing the radius vector by the small quantity ec’, shall coincide with the actual
surface. We must therefore have

GOV, + ¢ = a =f. .oscecsee counon (EE):

Let # and a be expanded in series of Laplace’s coefficients h, +h, +... and a+, +... Then
the value of V, at the surface will be obtained from either of equations (28) by replacing 0 by ox
and putting r =a. We have therefore

Vy = Amo (ay + Fay + Pale + o0n)e soveseeeeveeree (34)
After substituting in (33) the preceding expressions for V,, h, and #, we must equate to zero
Laplace’s coefficients of the same order. The condition that w, =0 may be satisfied by means
of the constant e’, and we shall have

@;
G"'.4raa Tat =a, —hy
Qi +

. ; 3a
which gives, on replacing G~!.4aca by its equivalent — ,

= Coa 1) p m=) tp dat SEO
(21 + 1) p — 30 (21+ 1)p—-30
We see that for terms of a high order ; is very nearly equal to h,, but for terms of a low order,
whereby the distribution of land and sea would be expressed as to its broad features, w, is sensibly
greater than h,.

a;

} hi vee seness(90)

690 Mr. STOKES, ON THE VARIATION OF GRAVITY

27. Let it be required to reduce gravity g to the gravity which would be observed, in the
altered state of the surface, along what would then be a surface of equilibrium. Let the correction
be denoted by g’—3g"”, where g’ is the same as before. The correction due to the alteration of the
coating in the manner considered in Art. 20 has been shewn to be equal to

8,
Qa8 = 67> 7 - 2
20+1
and the required correction will evidently be obtained by replacing d by cw. Putting for a, its
value got from (35) we have
(27 — 2) p 8p —30
= 240 > ——_—"—__ h;= Qro X41 — ————__/h,,
(2+ 1) p-3¢ (27 +1) p— 3a
which gives, since 270 Zh, =2rah =g', and G = 4rpa,
3 3p —3 h;
3g" = Cy a —e
20 (2t+1)p—3c0a
a = 4000, and suppose / expressed in miles, we get

m

g -3g

If we put o = 23, p= 5h,

9)
15 9h;
se" =G. = — + = G x .00017 (= 4.5hg + hy + Adh2 + .290h3 + .214h, + ...). «..(87

e 88000 11i— 2 i ual a il F : hei
Had we treated the approximate correction 3g” in the same manner we should have had

u 3a Sh;

8g = G—_=— = G x .00017 (8h) + ky + Ghz + 429K; + .333h, +...)
2pa 2¢4+1

whereas, since k = 3 (1 -*), we get
p

” 3 3p — 3a)h;
gt ees ( 2 a) h;
2pa (2i+1)p

The general expressions for 3g”, 3g”, and kg” shew that the approximate correction kg” agrees
with the true correction 3g” so far as regards terms of a high order, whereas the leading terms,
beginning with the first variable term, are decidedly too small; so that, as far as regards these
terms, 3g’” is better represented by 3g” than by kg”. This agrees with what has been already
remarked in Art, 25.

If we put g—g'+3g"’=g , and suppose G and e¢ determined by means of g_, small corrections
similar to those already investigated will have to be applied in consequence of the omission of the
quantity g’ — 3g"” in the value of g. The correction to ¢ would probably be insensible for the
reason mentioned in Art. 18. If we are considering only the variation of gravity, we may of course
leave out the term hy.

The series (87) would probably be too slowly convergent to be of much use. A more
convergent series may be obtained by subtracting kg” from 3g”’, since the terms of a high order in
3g"” are ultimately equal to those in kg”. We thus get

3g” = kg” + G x .00017 (— 6.136hy + .455hy + .128h2 + .056hy + .032h, + ...) «2200. (39)
which gives g’” if g” be known by quadratures for the station considered.

= G x .00017 (1.636) + .545hy + 327h2 + .234h3 + .182h, + ..).-.(38)

Although for facility of calculation it has been supposed that the sea was first replaced by a
stratum of rock or earth of less thickness, and then that the elevated portions of the earth’s
surface became fluid and ran down, it may be readily seen that it would come to the same thing if
we supposed the water to remain as it is, and the land to become fluid and run down, so as to
form for the bottom of the sea a surface of equilibrium, The gravity g,, would apply to the
earth so altered.

AT THE SURFACE OF THE EARTH. 691

28. Let us return to the quantity V, of Art. 19, and consider how the attraction of the earth’s
irregular coating affects the direction of the vertical. Let J be the latitude of the station, which
for the sake of clear ideas may be supposed to be situated in the northern hemisphere, @ its
longitude west of a given place, & the displacement of the zenith towards the south produced by
the attraction of the coating, » its displacement towards the east. Then

ee dV, secl dV,
GEPaie ML PN Ga (die
TI 6 1 dV.
because — Th and — aE are the horizontal components of the attraction towards the north
a

and towards the west respectively, and G may be put for g on account of the smallness of the
displacements.

Suppose the angle y of Art. 22 measured from the meridian, so as to represent the north
azimuth of the elementary mass da*sinydyydy. On passing to a place on the same meridian
whose latitude is 7 + d/, the angular distance of the elementary mass is shortened by cos x. di, and

ee wae

Q 2

— @ COS

therefore its linear distance, which was a chord y, or 2a sin ¥ , becomes 2asin cos x . di.

il
Hence the reciprocal of the linear distance is increased by aa 2 Core cos y.d/, and therefore
omeee
Ne ceee oe d\ydydl. Hence we hay
= —cosy.dydydl. ave

the part of V, due to this element is increased by 40a cos” ;

ov

cos" — Cos x

eee i - Ody dy. 21.0002 (40)

sin) 2
2

Although the quantity under the integral sign in this expression becomes infinite when y
vanishes, the integral itself has a finite value, at least if we suppose 6 to vary continuously in the
immediate neighbourhood of the station. For if 3 becomes 0’ when y becomes xX + 7, we may
replace 6 under the integral sign by o- 0, and integrate from y = 0 to y = 7, instead of integrating

vo
= : usando asi d
from xX =0 to x = 27, and the limiting value of ——— when yy vanishes is 4——, which is finite.
dy
sin +
2
To get the easterly displacement of the zenith, we have only to measure x from the west
. Tv .
instead of from the north, or, which comes to the same, to write y Gai for x, and continue to
measure y from the north. We get
dV, a We
ecl * = —— [fcos? = cosee © sin y.ddwWdy...-...++-(41
da 2 I 2 g@° % vax (4)
29. The expressions (40) and (41) are not to be applied to points very near the station if 6
vary abruptly, or even very rapidly, about such points. Recourse must in such a case be had to
direct triple integration, because it is not allowable to consider the attracting matter as condensed

into a surface. If however 6 vary gradually in the neighbourhood of the station, the expression
(40) or (41) may be used without further change. For if we modify (40) in the way explained in

¥

J, Js
the preceding article, or else by putting the integral under the form /," {,’" cost cosec cos x

(8 = 8) d\dx, where 6, denotes the value of 8 at the station, we see that the part of the integral

692 Mr. STOKES, ON THE VARIATION OF GRAVITY

due to a very small area surrounding the station is very small. If § vary abruptly, in consequence
suppose of the occurrence of a cliff, we may employ the expressions (40), (41), provided the distance
of the cliff from the station be as much as three or four times its height.

These expressions shew that the vertical is liable to very irregular deviations depending on
attractions which are quite local. For it is only in consequence of the opposition of attractions in
opposite quarters that the value of the integral is not considerable, and it is of course larger in

9

proportion as that opposition is less complete. Since sin — is but small even at the distance of two

or three hundred miles, a distant coast, or on the other hand a distant tract of high land of con-
siderable extent, may produce a sensible effect; although of course in measuring an are of the
meridian those attractions may be neglected which arise from masses which are so distant as to affect
both extremities of the arc in nearly the same way.

If we compare (40) or (41) with the expression for g” or g'”, we shall sce that the direction of
the vertical is liable to far more irreguiar fluctuations on account of the inequalities in the earth’s
coating than the force of gravity, except that part of the force which has been denoted by g’, and
which is easily allowed for. It has been supposed by some that the force of gravity alters irregularly
along the earth’s surface, and so it does, if we compare only distant stations. But it has been
already remarked with what apparent regularity gravity when corrected for the inequality g’ appears
to alter, in the direction in which we should expect, in passing from one station to another in a
chain of neighbouring stations,

30. There is one case in which the deviation of the vertical may become unusually large,
which seems worthy of special consideration.

For simplicity, suppose 6 to be constant for the land, and equal to zero for the sea, which
comes to regarding the land as of constant height, the sea as of uniform depth, and transferring
the defect of density of the sea with an opposite sign to the land. Apply the integral (40) to
those parts only of the earth’s surface which are at no great distance from the station considered,

so that we may put cos =1, sin v . = = , if s be the distance of the element, measured along
a great circle. In going from the station in the direction determined by the angle y, suppose that
we pass from land to sea at distances $,, 8, 8,,---and from sea to land at the intermediate distances
8) 5y..-.On going in the opposite direction suppose that we pass from land to sea at the distances
3_\) $_g5 8_5, «-. and from sea to land at the distances s_,, s_,.... Then we get from (40),

dV,

Tie a6 f {log s, — log s_, — (log s, — log s_s) + log s, — log s_, - nea cos y.dy.

If the station be near the coast, one of the terms logs,, log s_, will be large, and the zenith
will be sensibly displaced towards the sea by the irregular attraction. On account of the shelving
of the coast, the preceding expression, which has been formed on the supposition that 6 vanished
suddenly, would give too great a displacement ; but the object of this article is not to perform any
precise calculation, but merely to shew how the analysis indicates a case in which there would be
unusual disturbance. A cliff bounding a tract of table-land would have the same sort of effect as
a coast, and indeed the effect might be greater, on account of the more sudden variation of 6, The
effect would be nearly the same at equal distances from the edge above and below, that distance
being supposed as great as a small multiple of the height of the cliff, in order to render the
expression (40) applicable without modification.

31. Let us return now to the force of gravity, and leaving the consideration of the connexion
between the irregularities of gravity and the irregularities of the earth’s coating, and of the

a

a

=>

AT THE SURFACE OF THE EARTH. 693

possibility of destroying the former by making allowance for the latter, let us take the earth such
as we find it, and consider further the connexion between the variations of gravity and the
irregularities of the surface of equilibrium which constitutes the sea-level.

Equation (12) gives the variation of gravity if the form of the surface be known, and conversely,
(8) gives the form of the surface if the variation of gravity be known. Suppose the variation of
gravity known by means of pendulum-experiments performed at a great many stations scattered
over the surface of the earth; and let it be required from the result of the observations to deduce
the form of the surface. According to what has been already remarked, a series of Laplace's coefficients
would most likely be practically useless for this purpose, unless we are content with merely the
leading terms in the expression for the radius vector; and the leading character of those terms
depends, not necessarily upon their magnitude, but only on the wide extent of the inequalities
which they represent. We must endeavour therefore to reduce the determination of the radius
vector to quadratures.

For the sake of having to deal with small terms, let g be represented, as well as may be, by
the formula which applies to an oblate spheroid, and let the variable term in the radius vector be
calculated by Clairaut’s Theorem. Let g, be calculated gravity, 7, the calculated radius vector,
and put g=g,+ Ag,r=r,+aAwu. Suppose Ag and Aw expanded in series of Laplace’s
coefficients. It follows from (12) that Ag will have no term of the order 1; indeed, if this were not
the case, it might be shewn that the mutual forces of attraction of the earth’s particles would have a
resultant. Moreover the constant term in Ag may be got rid of by using a different value of G.
No constant term need be taken in the expansion of Aw, because such a term might be got rid of
by using a different value of a, and a@ of course cannot be determined by pendulum-experiments.
The term of the first order will disappear if + be measured from the common centre of gravity of
the mass and volume. The remaining terms in the expansion of Aw will be determined from those
in the expansion of Ag by means of equations (8) and (12).

Let Niemi G (Bait 03,1 Ug tejnns)) 5) pi aewesensinee nets ea ep's (42

and we shall have
AU = Vy + EUs + HU, + one. cosececeescnceeseeres (48)

Suppose Ag = GF(6, ). Let Wy be the angle between the directions determined by the angular
co-ordinates 0, p and 0’, p’. Let (1 — 2¢ cosy) + ¢’)} be denoted by R, and let Q; be the coefficient
of in the expansion of R-' in a series according to ascending powers of & Then

2i+1

v= he FO P)Qsin FaG'ay’,

4a

and therefore if ¢ be supposed to be less than 1, and to become 1 in the limit, we shall have

%+1 C19, + ...)sin O'dO'dg. ... (44)

ra 7 for , \ (rr 7 2
4a Au = limit of [7 /(?"F (0, p') (56 Q. +5 0Qs... + reac

Now assume

Q2i+1

heal
: Piece
17-1 g @ f

Ui
y= 5CQ + SCQs:+ 9
and we shall have

PY 5 Qe + TC Qyene + (26+ 1) OQ, +005

¢
[ae aie = CQ $LiQy oo CHG, tee G(R! -Q - (Q)i

—s

whence we get, putting Z for R~' — Q,-(Qy y=2f Cd. OZ.
Vor. VIII, Parr V. 4U

694 Mr. STOKES, ON THE VARIATION OF GRAVITY

Integrating by parts, and observing that + vanishes with ¢, we get

y =207'Z + 86 C-7Zde.
The last integral may be obtained by rationalization. If we assume R = w — ¢, and observe
that Q, =1, @, = cos Ws and that 2 = 1 when « vanishes, we shall find

ee w — cos vr w—-1 w+i
RE a Se eee a citar oe ee — 2cos \y,. log Sie
When G1 we have Z = (2 — 2 cos y)~4 — (1 + cosy), w =1 resin, antl
Re-“dC= ~2sin © (1 - sin) ~ eos ylog fain ¥( + sin ait

Putting f(\/) for the value of yy when ¢ = 1, we have

\
SO) = cosee ¥ +1— 6 sin — 5 cosy, — 3 cosy, log {sin ¥ (14 sin ali desu cee (45)
In the expression for Aw, we may suppose the line from which 6’ is measured to be the radius
vector of the station considered. We thus get, on replacing F(6’, ¢') by G~'Ag, and employing
the notation of Art. 22,

Au = ag hi ae- FY) BINA Con\ten ots Gul detd = deiea vlvesle' ade Bs sot (40)

32. Let Ag=g'+A'g. Then A’g is the excess of observed gravity reduced to the level of
the sea by Dr. Young’s rule over calculated gravity; and of the two parts g’ and A’g of which
Ag consists, the former is liable to vary irregularly and abruptly from one place to another, the
latter varies gradually. Hence, for the sake of interpolating between the observations taken at
different stations, it will be proper to separate Ag into these two parts, or, which comes to the
same, to separate the whole integral into two parts, involving g’ and A’g respectively, so as to get
the part of Aw which is due to g’ by our knowledge of the height of the land and the depth of
the sea, and the part which depends on A’g by the result of pendulum-experiments. It may be
observed that a constant error, or a slowly varying error, in the height of the land would be of no
consequence, because it would enter with opposite signs into g’ and A’g.

It appears, then, that the results of pendulum-experiments furnish sufficient data for the
determination of the variable part of the radius vector of the earth’s surface, and consequently for
the determination of the particular value which is to be employed at any observatory in correcting
for the lunar parallax, subject however to a constant error depending on an error in the assumed
value of a.

33. The expression for g’” in Art. 27 might be reduced to quadratures by the method of
Art. 31, but in this case the integration with respect to ¢ could not be performed in finite terms,
and it would be necessary in the first instance to tabulate, once for all, an integral of the form
Joi f (G cosy) d@ for \values of y, which need not be numerous, from 0 to 7. This table being
made, the tabulated function would take the place of f(\/) in (46), and the rest of the process
would be of the same degree of difficulty as the quadratures expressed by the equations (31)
and (46).

34. Suppose Aw known approximately, either as to its general features, by means of the
leading terms of the series (43), or in more detail from the formula (46), applied in succession to a
great many points on the earth’s surface. By interpolating between neighbouring places for which
Aw has been calculated, find a number of points where Aw has one of the constant values

AT THE SURFACE OF THE EARTH. 695

— 28, —B, 0,8, 28-..., mark these points on a map of the world, and join by a curve those
which belong to the same value of Aw. We shall thus have a series of contour lines representing
the elevation or depression of the actual sea-level above or below the surface of the oblate spheroid,
which has been employed as most nearly representing it. If we suppose these lines traced on a
globe, the reciprocal of the perpendicular distance between two consecutive contour lines will
represent in magnitude, and the perpendicular itself in direction, the deviation of the vertical from
the normal to the surface of the spheroid, or rather that part of the deviation which takes place on
an extended scale: for sensible deviations may be produced by attractions which are merely local,
and which would not produce a sensible elevation or depression of the sea-level; although of course,
as to the merely mathematical question, if the contour lines could be drawn sufficiently close and
exact, even local deviations of the vertical would be represented.

Similarly, by joining points at which the quantity denoted in Art. 19 by V has a constant
value, contour lines would be formed representing the elevation of the actual sea-level above what
would be a surface of equilibrium if the earth’s irregular coating were removed. By treating V,
in the same way, contour lines would be formed corresponding to the elevation of the actual sea-
level above what would be the sea-level if the solid portions of the earth’s crust which are
elevated were to become fluid and to run down, so as to form a level bottom for the sea, which
would in that case cover the whole earth.

These points of the theory are noticed more for the sake of the ideas than on account of any
application which is likely to be made of them; for the calculations indicated, though possible with
a sufficient collection of data, would be very laborious, at least if we wished to get the results
with any detail.

35. The squares of the ellipticity, and of quantities of the same order, have been neglected
in the investigation. Mr. Airy, in the Treatise already quoted, has examined the consequence, on
the hypothesis of fluidity, of retaining the square of the ellipticity, in the two extreme cases of a
uniform density, and of a density infinitely great at the centre and evanescent elsewhere, and has
found the correction to the form of the surface and the variation of gravity to be insensible, or
all but insensible. As the connexion between the form of the surface and the variation of gravity
follows independently of the hypothesis of fluidity, we may infer that the terms depending on the
square of the ellipticity which would appear in the equations which express that connexion would
be insensible. It may be worth while, however, just to indicate the mode of proceeding when the
square of the ellipticity is retained. ,

By the result of the first approximation, equation (1) is satisfied at the surface of the earth,
as far as regards quantities of the first order, but not necessarily further, so that the value of
V+ U at the surface is not strictly constant, but only of the form e + H, where # is a small
variable quantity of the second order. It is to be observed that V satisfies equation (8) exactly,
not approximately only. Hence we have merely to add to V a potential V’ which satisfies equation
(3), outside the earth, vanishes at an infinite distance, and is equal to H at the surface. Now if
we suppose V’ to have the value H at the surface of a sphere whose radius is a, instead of the
actual surface of the earth, we shall only commit an error which is a small quantity of the first
order compared with 7, and H is itself of the second order, and therefore the error will be only
of the third order. But by this modification of one of the conditions which V’ is to satisfy, we
are enabled to find V’ just as V was found, and we shall thus have a solution which is correct to
the second order of approximation. A repetition of the same process would give a solution
which would be correct to the third order, and so on. It need hardly be remarked that in going
beyond the first order of approximation, we must distinguish in the small terms between the
direction of the vertical, and that of the radius vector.

G. G. STOKES,
4U2

LI. On Hegel's Criticism of Newton's Principia. By W. WuEwE.L, D.D.,
Master of Trinity College, Cambridge.

[Read May 21, 1849.]

Tur Newtonian doctrine of universal gravitation, as the cause of the motions which take place
in the solar system, is so entirely established in our minds, and the fallacy of all the ordinary
arguments against it is so clearly understood among us, that it would undoubtedly be deemed a
waste of time to argue such questions in this place, so far as physical truth is concerned. But
since in other parts of Europe, there are teachers of philosophy whose reputation and influence
are very great, and who are sometimes referred to among our own countrymen as the authors of
new and valuable views of truth, and who yet reject the Newtonian opinions, and deny the validity
of the proofs commonly given of them, it may be worth while to attend for a few minutes to the
declarations of such teachers, as a feature in the present condition of European philosophy. I the
more readily assume that the Cambridge Philosophical Society will not think a communication on
such a subject devoid of interest, in consequence of the favourable reception which it has given to
philosophical speculations still more abstract, which I have on previous occasions offered to it.
I will therefore proceed to make some remarks on the opinions concerning the Newtonian doctrine
of gravitation, delivered by the celebrated Hegel, of Berlin, than whom no philosopher in modern,
and perhaps hardly any even in ancient times, has had his teaching received with more reverential
submission by his disciples, or been followed by a more numerous and zealous band of scholars
bent upon diffusing and applying his principles.

The passages to which I shall principally refer are taken from one of his works which is called
the Encyclopedia (Encyklopidie), of which the First Part is the Science of Logic, the Second, the
Philosophy of Nature, the Third, the Philosophy of Spirit. The Second Part, with which T am
here concerned, has for an aliter title, Lectwres on Natural Philosophy (Vorlesungen tiber Natur-
philosophie), and would through its whole extent offer abundant material for criticism, by referring
it to principles with which we are here familiar: but I shall for the present confine myself to that
part which refers to the subject which I have mentioned, the Newtonian Doctrine of Gravitation,
§ 269, 270, of the work. Nor shall I, with regard to this part, think it necessary to give a con-
tinuous and complete criticism of all the passages bearing upon the subject ; but only such speci-
mens, and such remarks thereon, as may suffice to show in a general manner the value and the
character of Hegel's declarations on such questions. I do not pretend to offer here any opinion
upon the value and character of Hegel’s philosophy in general: but I think it not unlikely that
some impression on that head may be suggested by the examination, here offered, of some points in
which we can have no doubt where the truth lies; and I am not at all persuaded that a like
examination of many other parts of the Hegelian Encyclopedia would not confirm the impression
which we shall receive from the parts now to be considered.

Hegel both criticises the Newtonian doctrines, or what he states as such; and also, not deny-
ing the truth of the laws of phenomena which he refers to, for instance Kepler’s laws, offers his
own proof of these laws, I shall make a few brief remarks on each of these portions of the pages
before me. And I would beg it to be understood that where I may happen to put my remarks in
a short, and what may seem a peremptory form, I do so for the sake of saving time; knowing

————————— er

-

see

Dr. WHEWELL, ON HEGEL’S CRITICISM OF NEWTON'S PRINCIPIA. 697

that among us, upon subjects so familiar, a few words will suffice. For the same reason, I shall
take passages from Hegel, not in the order in which they occur, but in the order in which they
best illustrate what I have to say. I shall do Hegel no injustice by this mode of proceeding : for
I will annex a faithful translation, so far as I can make one, of the whole of the passages referred
to, with the context.

No one will be surprised that a German, or indeed any lover of science, should speak with
admiration of the discovery of Kepler’s laws, as a great event in the history of Astronomy, and a
glorious distinction to the discoverer. But to say that the glory of the discovery of the proof of
these laws has been unjustly transferred from Kepler to Newton, is quite another matter. This is
what Hegel says (a*). And we have to consider the reasons which he assigns for saying so.

He says (6) that ‘it is allowed by mathematicians that the Newtonian Formula may be derived
from the Keplerian laws,” and hence he seems to infer that the Newtonian law is not an additional
truth. That is, he does not allow that the discovery of the cause which produces a certain phe-
nomenal law is anything additional to the discovery of the law itself.

“The Newtonian formula may be derived from the Keplerian law.” It was professedly so
derived; but derived by introducing the Idea of Force, which Idea and its consequences were not
introduced and developed till after Kepler’s time.

“The Newtonian formula may be derived from the Keplerian law.” And the Keplerian law
may be derived, and was derived, from the observations of the Greek astronomers and_ their

‘successors; but was not the less a new and great discovery on that account.

But let us see what he says further of this derivation of the Newtonian “ formula” from the
Keplerian Law. It is evident that by calling it a formula, he means to imply, what he also asserts,
that it is no new law, but only a new form (and a bad one) of a previously known truth.

How is the Newtonian ‘‘ formula,” that is, the law of the inverse squares of the central force,
derived from the Keplerian law of the cubes of the distances proportional to the squares of the

times? This, says Hegel, is the “‘immediate derivation.” (c).—By Kepler's law, A being the dis-
j : A’ Zale ise
tance and 7’ the periodic time, Te is constant. But Newton calls Te universal gravitation; whence

it easily follows that gravitation is inversely as A’.
This is Hegel’s way of representing Newton’s proof. Reading it, any one who had never read

a ee A
the Principia might suppose that Newton defined gravitation to be 7 We, who have read the
Principia, know that Newton proves that in circles, the central force (not the universal gravitation)

is as Ss that he proves this, by setting out from the idea of force, as that which deflects a body

from the tangent, and makes it describe a curve line: and that in this way, he passes from Kepler’s
laws of mere motion to his own law of Force.
‘But Hegel does not see any value in this. Such a mode of treating the subject he ‘says (é)
“ offers to us a tangled web, formed of the Lines of the mere geometrical construction, to which a
physical meaning of independent forces is given.” That a measure of forces is found in such lines
as the sagitta of the are described in a given time, (not such a meaning arbitrarily given to them,)
is certainly true, and is very distinctly proved in Newton, and in all our elementary books,
But, says Hegel, as further shewing the artificial nature of the Newtonian formule, (h) “ Analy-
sis has long been able to derive the Newtonian expression and the laws therewith connected out of
the Vorm of the Keplerian Laws ;” an assertion, to verify which he refers to Poisson's Mécanique.

* These letters refer to passages in the Translation annexed to this Memoir.

698 Dr. WHEWELL, ON HEGEL’S CRITICISM

This is apparently in order to shew that the “lines” of the Newtonian construction are superfluous.
We know very well that analysis does not always refer to visible representations of such lines: but
we know too, (and Francceur would testify to this also,) that the analytical proofs contain equiva-
lents to the Newtonian lines. We, in this place, are too familiar with the substitution of analytical
for geometrical proofs, to be led to suppose that such a substitution affects the substance of the truth
proved. The conversion of Newton’s geometrical proofs of his discoveries into analytical processes
by succeeding writers, has not made them cease to be discoveries: and accordingly, those who have
taken the most prominent share in such a conversion, have been the most ardent admirers of New-
ton’s genius and good fortune.

So much for Newton’s comparison of the Forces in different circular orbits, and for Hegel’s power
of understanding and criticising it. Now let us look at the motion in different parts of the same
elliptical orbit, as a further illustration of the value of Hegel’s criticism. In an elliptical orbit the
velocity alternately increases and diminishes. This follows necessarily from Kepler’s law of the
equal description of the areas, and so Newton explains it. Hegel, however, treats of this acceleration
and retardation as a separate fact, and talks of another explanation of it, founded upon Centripetal
and Centrifugal Force (0). Where he finds this explanation, I know not; certainly not in Newton,
who in the second and third section of the Principia explains the variation of the velocity in a quite
different manner, as I have said; and nowhere, I think, employs centrifugal force in his explana-
tions. However, the notion of centrifugal as acting along with centripetal force is introduced in
some treatises, and may undoubtedly be used with perfect truth and propriety. How far Hegel
can judge when it is so used, we may see from what he says of the confusion produced by such an
explanation, which is, he says, a maximum. In the first place, he speaks of the motion being wni-
formly accelerated and retarded in an elliptical orbit, which, in any exact use of the word wniformly,
it is not. But passing by this, he proceeds to criticise an explanation, not of the variable velocity
of the body in its orbit, but of the alternate access and recess of the body to and from the center.
Let us overlook this confusion also, and see what is the value of his criticism on the explanation.
He says (p), “according to this explanation, in the motion of a planet from the aphelion to the
perihelion, the centrifugal is less than the centripetal force ; and in the perihelion itself the centri-
petal force is supposed suddenly to become greater than the centrifugal;” and so, of course, the
body re-ascends to the aphelion,

Now I will not say that this explanation has never been given in a book professing to be scien-
tific; but I have never seen it given; and it never can have been given but by a very ignorant and
foolish person. It goes upon the utterly unmechanical supposition that the approach of a body to the
center at any moment depends solely upon the excess of the centripetal over the centrifugal force ; and
reversely. But the most elementary knowledge of mechanics shews us that when a body is moving
obliquely to the distance from the center, it approaches to or recedes from the center in virtue of this
obliquity, even if no force at all act. And the total approach to the center is the approach due to
this cause, plus the approach due to the centripetal force, minus the recess due to the centrifugal
force. At the aphelion, the centripetal is greater than the centrifugal force; and hence the motion
becomes oblique; and then, the body approaches to the center on both accounts, and approaches on
account of the obliquity of the path even when the centrifugal has become greater than the centri-
petal force, which it becomes before the body reaches the perihelion. This reasoning is so elemen-
tary, that when a person who cannot see this, writes on the subject with an air of authority, I do
not see what can be done but to point out the oversight and leave it.

But there is, says Hegel (q), another way of explaining the motion by means of centripetal
and centrifugal forces. The two forces are supposed to increase and decrease gradually, according
to different laws. In this case, there must be a point where they are equal, and in equilibrio; and

this being the case, they will always continue equal, for there will be no reason for their going
out of equilibrium.

CC — ——

OF NEWTON'S PRINCIPIA. 699

This, which is put as another mode of explanation, is, in fact, the same mode; for, as I have
already said, the centrifugal force, which is less than the centripetal at the aphelion, becomes the
greater of the two before the perihelion; and there is an intermediate position, at which the two forces
are equal. But at this point, is there no reason why, being equal, the forces should become unequal ?
Reason abundant : for the body, being there, moves in a line oblique to the distance, and so changes
its distance ; and the centripetal and centrifugal force, depending upon the distance by different
laws, they forthwith become unequal.

But these modes of explanation, by means of the centripetal and centrifugal forces and their
relation, are not necessary to Newton’s doctrine, and are nowhere used by Newton; and undoubtedly
much confusion has been produced in other minds, as well as Hegel’s, by speaking of the centrifugal
force, which is a mere intrinsic geometrical result of a body’s curvilinear motion round a center, in
conjunction with centripetal force, which is an extrinsic force, acting upon the body and urging it to
the center. Neither Newton, nor any intelligent Newtonian, ever spoke of the centripetal and centri-
fugal force as two distinct forces both extrinsic to the motion, which Hegel accuses them of doing. (n)

I have spoken of the third and second of Kepler's laws; of Newton’s explanations of them,
and of Hegel’s criticism. Let us now, in the same manner, consider the first law, that the planets
move in ellipses. Newton’s proof that this was the result of a central force varying inversely as
the square of the distance, was the solution of a problem at which his contemporaries had laboured
in vain, and is commonly looked upon as an important step. “ But,” says Hegel, (d) “the proof
gives a conic section generally, whereas the main point which ought to be proved is, that the path
of the body is an ellipse only, not a circle or any other conic section.” Certainly if Newton had
proved that a planet cannot move ina circle, (which Hegel says he ought to have done), his
system would have perplexed astronomers, since there are planets which move in orbits hardly
distinguishable from circles, and the variation of the extremity from planet to planet shews that
there is nothing to prevent the excentricity vanishing and the orbit becoming a circle.

“ But,” says Hegel again, (e) “the conditions which make the path to be an ellipse rather than
any other conic section, are empirical and extraneous ;—the supposed casual strength of the im-
pulsion originally received.” Certainly the circumstances which determine the amount of excen-
tricity of a planet’s orbit are derived from experience, or rather, observation, It is not a part of
Newton’s system to determine @ priori what the excentricity of a planet’s orbit must be. A system
that professes to do this will undoubtedly be one very different from his. And as our knowledge
of the excentricity is derived from observation, it is, in that sense, empirical and casual. The
strength of the original impulsion is a hypothetical and impartial way of expressing this result of
observation. And as we see no reason why the excentricity should be of any certain magnitude,
we see none why the fraction which expresses the excentricity should not become as large as unity,
that is, why the orbit should not become a parabola; and accordingly, some of the bodies which
revolve about the same appear to move in orbits of this form: so little is the motion in an ellipse,
as Hegel says, (f) “‘ the only thing to be proved.”

But Hegel himself lias offered proof of Kepler’s laws, to which, considering his objections to
Newton’s proofs, we cannot help turning with some curiosity.

And first, let us look at the proof of the Proposition which we have been considering, that the
path of a planet is necessarily an ellipse. I will translate Hegel’s language as well as I can; but
without answering for the correctness of my translation, since it does not appear to me to conform
to the first condition of translation, of being intelligible. ‘The translation however, such as it is,
may help us to form some opinion of the validity and value of Hegel’s proofs as compared with
Newton’s. (7)

“ For absolutely uniform motion, the circle is the only path..,.The circle is the line returning

700 Dr. WHEWELL, ON HEGEL’S CRITICISM

into itself in which all the radii are equal; there is, for it, only one determining quantity, the
radius.

**But in free motion, the determination according to space and to time come into view with
differences. ‘There must be a difference in the spatial aspect in itself, and therefore the form requires
two determining quantities. Hence the form of the path returning into itself is an ellipse.”

Now even if we could regard this as reasoning, the conclusion does not in the smallest degree
follow. A curve returning into itself and determined by two quantities, may have innumerable
forms besides the ellipse; for instance, any oval form whatever, besides that of the conic section.

But why must the curve be a curve returning into itself? Hegel has professed to prove this
previously (m) from ‘the determination of particularity and individuality of the bodies in general,
so that they have partly a center in themselves, and partly at the same time their center in another.”
Without seeking to find any precise meaning in this, we may ask whether it proves the impossi-
bility of the orbits with moveable apses, (which do not return into themselves,) such as the planets
(affected by perturbations) really do describe, and such as we know that bodies must describe in all
cases, except when the force varies exactly as the square of the distance? It appears to do so: and
it proves this impossibility of known facts at least as much as it proves anything.

Let us now look at Hegel’s proof of Kepler’s second law, that the elliptical sectors swept by
the radius vector are proportional to the time. It is this: (s).

‘In the circle, the are or angle which is included by the two radii is independent of them. But
in the motion [of a planet] as determined by the conception, the distance from the center and the
arc run over in a certain time must be compounded in one determination, and must make out a whole.
This whole is the sector, a space of two dimensions. And hence the are is essentially a Function
of the radius vector; and the former (the arc) being unequal, brings with it the inequality of the
radii.”

As was said in the former case, if we could regard this as reasoning, it would not prove the
conclusion, but only, that the arc is some function or other of the radii.

’ Hegel indeed offers (t) a reason why there must be an are involved. This arises, he says, from
‘the determinateness [of the nature of motion], at one while as time in the root, at another while
as space in the square. But here the quadratic character of the space is, by the returning of the line
of motion into itself, limited to a sector.”

Probably my readers have had a sufficient specimen of Hegel’s mode of dealing with these
matters. I will however add his proof of Mepler’s third law, that the cubes of the distances are as
the squares of the times.

Hegel’s proof in this case (w) has a reference to a previous doctrine concerning falling bodies,
in which time and space have, he says, a relation to each other as root and square. Falling bodies
however are the case of only half-free motion, and the determination is incomplete.

‘* But in the case of absolute motion, the domain of free masses, the determination attains its
totality. The time as the root is a mere empirical magnitude: but as a component of the deve-
loped Totality, it is a Totality in itself: it produces itself, and therein has a reference to itself.
And in this process, Time, being itself the dimensionless element, only comes to a formal identity
with itself and reaches the square : Space, on the other hand, as a positive external relation, comes
to the full dimensions of the conception of space, that is, the cube. ‘The Realization of the two
conceptions (space and time) preserves their original difference. This is the third Keplerian law,
the relation of the Cubes of the distances to the squares of the times.”

‘* And this,” he adds, (v) with remarkable complacency, ‘ represents simply and immediately the
reason of the thing :—while on the contrary, the Newtonian Formula, by means of which the Law

OF NEWTON'S PRINCIPIA. 701

is changed into a Law for the Force of Gravity, shews the distortion and inversion of Reflexion,
which stops half-way.”

I am not able to assign any precise meaning to the Reflewion, which is here used as a term of
condemnation, applicable especially to the Newtonian doctrine. It is repeatedly applied in the
same manner by Hegel. Thus he says, (g) “that what Kepler expresses in a simple and sublime
manner in the form of Laws of the Celestial Motions, Newton has metamorphosed into the Reflexion-
Form of the Force of Gravitation.”

Though Hegel thus denies Newton all merit with regard to the explanation of Kepler’s laws
by means of the gravitation of the planets to the sun, he allows that to the Keplerian Laws
Newton added the Principle of Perturbations (k). This Principle he accepts to a certain extent,
transforming the expression of it after his peculiar fashion. “It lies,” he says, (/) “in this: that
matter in general assigns a center for itself: the collective bodies of the system recognize a reference
to their sun, and all the individual bodies, according to the relative positions into which they are
brought by their motions, form a momentary relation of their gravity towards each other.”

This must appear to us a very loose and insufficient way of stating the Principle of Perturb-
ations, but loose as it is, it recognises that the Perturbations depend upon the gravity of the
planets one to another, and to the sun, And if the Perturbations depend upon these forces, one
can hardly suppose that any one who allows this will deny that the primary undisturbed motions
depend upon these forces, and must be explained by means of them ; yet this is what Hegel denies.

It is evident, on looking at Hegel’s mode of reasoning on such subjects, that his views approach
towards those of Aristotle and the Aristotelians; according to which motions were divided into
natural and unnatural ;—the celestial motions were circular and uniform in their nature ;——and
the like. Perhaps it may be worth while to shew how completely Hegel adheres to these ancient
views, by an extract from the additions to the Articles on Celestial Motions, made in the last edition

of the Encyclopedia. He says (w),

**The motion of the heavenly bodies is not a being pulled this way and that, as is imagined
(by the Newtonians). They go along, as the ancients said, like blessed gods. The celestial con-
formity is not such a one as has the principle of rest or motion external to itself. It is not right
to say because a stone is inert, and the whole earth consists of stones, and the other heavenly
bodies are of the same nature as the earth, therefore the heavenly bodies are inert. This conclusion
makes the properties of the whole the same as those of the part. Impulse, Pressure, Resistance,
Friction, Pulling, and the like, are valid only for other than celestial matter.”

- There can be no doubt that this is a very different doctrine from that of Newton.

I will only add to these specimens of Hegel’s physics, a specimen of the logic by which he
refutes the Newtonian argument which has just been adduced; namely, that the celestial bodies
are matter, and that matter, as we see in terrestrial matter, is inert. He says (wv),

“Doubtless both are matter, as a good thought and a bad thought are both thoughts; but the
bad one is not therefore good, because it is a thought.”

Trinity Loper,
May 2, 1849.

Vou. VIII. Part V. 4X

Appendix to the Memoir on Hegel's Criticism of Newton's Principia.

Hece.. Encyclopedia (2nd Ed. 1827) Part x1., p. 250.
C. Absolute Mechanics.

§ 269.

GraviTaTIon is the true and determinate conception of material Corporeity, which (Conception)
is realized to the Idea (zur Idee). General Corporeity is separable essentially into particular
Bodies, and connects itself with the Element of Individwality or subjectivity, as apparent (phe-
nomenal) presence in the Motion, which by this means is immediately a system of several Bodies.

Universal gravitation must, as to itself, be recognised as a profound thought, although it was
principally as apprehended in the sphere of Reflexion that it eminently attracted notice and con-
fidence on account of the quantitative determinations therewith connected, and was supposed to
find its confirmation in Ewperiments (Erfahrung) pursued from the Solar System down to the phe-
nomena of Capillary Tubes.—But Gravitation contradicts immediately the Law of Inertia, for in
virtue of it (Gravitation) matter tends owt of itself to the other (matter).—In the Conception of
Weight, there are, as has been shewn, involved the two elements—Self-existence, and Continuity,
which takes away self-existence. These elements of the Conception, however, experience a fate,
as particular forces, corresponding to Attractive and Repulsive Force, and are thereby apprehended
in nearer determination, as Centripetal and Centrifugal Force, which (Forces) like weight, act
upon Bodies, independent of each other, and are supposed to come in contact accidentally in a
third thing, Body. By this means, what there is of profound in the thought of universal weight
is again reduced to nothing; and Conception and Reason cannot make their way into the doctrine
of absolute motion, so long as the so highly-prized discoveries of Forces are dominant there. In
the conclusion which contains the Zdea of Weight, namely, [contains this Idea] as the Con-
ception which, in the case of motion, enters into external Reality through the particularity of the
Bodies, and at the same time into this [Reality] and into their Ideality and self-regarding Re-
flexion, (Reflexion-in-sich), the rational identity and inseparability of the elements is involved,
which at other times are represented as independent. Motion itself, as such, has only its meaning
and existence in a system of several bodies, and those, such as stand in relation to each other
according to different determinations.

§ 270.

As to what concerns bodies in which the conception of gravity (weight) is realized free by itself,
we say that they have for the determinations of their different nature the elements (momente) of
their conception. One [conception of this kind] is the wniversal center of the abstract reference
[of a body] to itself. Opposite to this [conception] stands the immediate, extrinsic, centreless
Individuality, appearing as Corporeity similarly independent. Those [Bodies] however which are
particular, which stand in the determination of extrinsic, and at the same time of intrinsic relation,
are centers for themselves, and [also] have a reference to the first as to their essential unity.

The Planetary Bodies, as the immediately concrete, are in their existence the most complete.
Men are accustomed to take the Sun as the most excellent, inasmuch as the understanding
prefers the abstract to the concrete, and in like manner the Fixed stars are esteemed higher
than the Bodies of the Solar System. Centreless Corporeity, as belonging to externality,
naturally separates itself into the oppusition of the lunar and the cometary Body. The laws
of absolutely free motion, as is well known, were discovered by Kepler ;—a discovery of

(a)

(0)
(c)

(a)

(e)

(f)

(g)

(h)

(i)

APPENDIX TO Dr. WHEWELL'S MEMOIR ON HEGEL'S CRITICISM, ETC. 703

immortal fame. Kepler has proved these laws in this sense, that for the empirical data he
found their general expression. Since then, it has become a common way of speaking to say
that Newton first found out the proof of these Laws. It has rarely happened that fame has
been more unjustly transferred from the first discoverer to another person. On this subject I
make the following remarks.

1. That it is allowed by Mathematicians that the Newtonian Formule may be derived

from the Keplerian Laws. The completely immediate derivation is this: In the third Kep-
APS. AP
lerian Law, 7s the constant quantity. This being put as 7a

Fe universal Gravitation, his expression of the effect of gravity in the reciprocal ratio of the

, and calling, with Newton,

square of the distances is obvious.

2. That the Newtonian proof of the Proposition that a body subjected to the Law of
Gravitation moves about the central body in an Ellipse, gives a Conic Section generally,
while the main Proposition which ought to be proved is that the fall of such a Body is not a
Cirele or any other Conic Section, but an Ellipse only. Moreover, there are objections which
may be made against this proof in itself; (Princ. Math. ]. 1. Sect. 1. Prop. 1.) and although
it is the foundation of the Newtonian Theory, analysis has no longer any need of it. The
conditions which in the sequel make the path of the Body to a determinate Conic Section, are
referred to an empirical circumstance, namely, a particular position of the Body at a deter-
mined moment of time, and the caswal strength of an impulsion which it is supposed to have
received originally; so that the circumstance which makes the Curve be an Ellipse, which
alone ought to be the thing proved, is extraneous to the Formula.

8. That the Newtonian Law of the so-called Force of Gravitation is in like manner only
proved from experience by Induction.

The sum of the difference is this, that what Kepler expressed in a simple and sublime
manner in the Form of Laws of the Celestial Motions, Newton has metamorphosed into the
Reflexion-Form of the Force of Gravitation. If the Newtonian Form has not only its con-
venience but its necessity in reference to the analytical method, this is only a difference of the
mathematical formule; Analysis has long been able to derive the Newtonian expression, and
the Propositions therewith connected, out of the Form of the Keplerian Laws; (on this subject
I refer to the elegant exposition in Franceur’s Traité Elém. de Mécanique, Liv. 11. Ch. xi,
n. iv.)—The old method of so-called proof is conspicuous as offering to us a tangled web,
formed of the Limes of the mere geometrical construction, to which a physical meaning of
independent Forces is given; and of empty Reflexion-determinations of the already men-
tioned Accelerating Force and Vis Inertia, and especially of the relation of the so-called
gravitation itself to the centripetal force and centrifugal force, and so on.

The remarks which are here made would undoubtedly have need of a further explica-
tion to shew how well founded they are: in a Compendium, propositions of this kind which
do not agree with that which is assumed, can only have the shape of assertions. Indeed,
since they contradict such high authorities, they must appear as something worse, as pre-
sumptuous assertions. I will not, on this subject, support myself by saying, by the bye,
that an interest in these subjects has occupied me for 25 years; but it is more precisely to
the purpose to remark, that the distinctions and determinations which Mathematical Analysis
introduces, and the course which it must take according to its method, is altogether different
from that which a physical reality must have, ‘The Presuppositions, the Course, and the
Results, which the Analysis necessarily has and gives, remain quite extraneous to the considera-
tions which determine the physical value and the signification of those determinations and of

4x2

704

(A)
(Y)

(m)

APPENDIX TO Dr. WHEWELL’S MEMOIR ON

that course. To this it is that attention should be directed. We have to do with a conscious-
ness relative to the deluging of physical Mechanics with an inconceivable (unsaglichen)
Metaphysic, which—contrary to experience and conception—has those mathematical deter-
minations alone for its source.

It is recognized that what Newton—besides the foundation of the analytical treatment,
the developement of which, by the bye, has of itself rendered superfluous, or indeed rejected
much which belonged to Newton’s essential Principles and glory—has added to the Keplerian
Laws is the Principle of Pertwrbations,—a Principle whose importance we may here accept
thus far; (hier in sofern anzufuhren ist); namely, so far as it rests upon the Proposition that
the so-called attraction is an operation of all the individual parts of bodies, as being material. It
lies in this, that matter in general assigns a center for itself, (sich das centrum setzt), and the
figure of the body is an element in the determination of its place; that collective bodies of the
system recognize a reference to their Sun, (sich ihre Sonne setzen) but also the individual
bodies themselves, according to the relative position with regard to each other into which
they come by their general motion, form a momentary relation of their gravity (schwere)
towards each other, and are related to each other not only in abstract spatial relations, but at
the same time assign to themselves a joint center, which however is again resolved [into the
general center] in the universal system.

As to what concerns the features of the path, to shew how the fundamental determina-
tions of Free Motion are connected with the Conception, cannot here be undertaken in a
satisfactory and detailed manner, and must therefore be left to its fate. The proof from reason
of the quantitative determinations of free motion can only rest upon the determinations of
Conceptions of space and time, the elements whose relation (intrinsic not extrinsic) motion is.

That, in the first place, the motion in general is a motion returning into itself, is founded
on the determination of particularity and individuality of the bodies in general ({ 269), so that
partly they have a center in themselves, and partly at the same time their center in another.
These are the determinations of Conceptions which form the basis of the false representatives

(n) of Centripetal Force and Centrifugal Force, as if each of these were self-existing, extraneous

(9)

to the other, and independent of it; and as if they only came in contact in their operations and
consequently externally. They are, as has already been mentioned, the Lines which must
be drawn for the mathematical determinations, transformed into physical realities.

Further, this motion is uniformly accelerated, (and—as returning into itself—in turn
uniformiy retarded). In motion as free, Time and Space enter as different things which are
to make themselves effective in the determination of the motion, (§ 266, note.) In the so-
called Ewplanation of the uniformly accelerated and retarded motion, by means of the
alternate decrease and increase of the magnitude of the Centripetal Force and Centrifugal
Force, the confusion which the assumption of such independent Forces produces is at its

(p) greatest height. According to this explanation, in the motion of a Planet from the Aphelion

to the Perihelion, the centrifugal is /ess than the centripetal force, and on the contrary, in
the Perihelion itself, the centrifugal force is supposed to become greater than the centripetal.
For the motion from the Perihelion to the Aphelion, this representation makes the forces pass
into the opposite relation in the same manner. It is apparent that such a sudden conversion
of the preponderance which a force has obtained over another, into an inferiority to the other,
cannot be anything taken out of the nature of Forces. On the contrary it must be concluded,
that a preponderance which one Force has obtained over another must not only be preserved,
but must go onwards to the complete annihilation of the other Force, and the motion must
either, by the Preponderance of the Centripetal Force, proceed till it ends in rest, that is, in
the Collision of the Planet with the Central Body, or till by the Preponderance of the Centri-

(9)

(7)

(s)

(4)

HEGEL’S CRITICISM OF NEWTON’S PRINCIPIA. 705

fugal Force it ends in a straight line. But now, if in place of the suddenness of the conversion,
we suppose a gradual increase of the Force in question, then, since rather the other Force ought
to be assumed as increasing, we lose the opposition which is assumed for the sake of the ex-
planation; and if the increase of the one is assumed to be different from that of the other,
(which is the case in some representations,) then there is found at the mean distance between
the apsides a point in which the Forces are in equilibrio And the transition of the Forces
out of Equilibrium is a thing just as little without any sufficient reason as the aforesaid
suddenness of inversion. And in the whole of this kind of explanation, we see that the
mede of remedying a bad mode of dealing with a subject leads to newer and greater confu-
sion.—A similar confusion makes its appearance in the explanation of the phanomenon that
the pendulum oscillates more slowly at the equator. This phenomenon is ascribed to the
Centrifugal Force, which it is asserted must then be greater; but it is easy to see that we
may just as well ascribe it to the augmented gravity, inasmuch as that holds the pendulum
more strongly to the perpendicular line of rest.

§ 240,

And now first, as to what concerns the Form of the Path, the Cirele only can be conceived
as the path of an absolutely uniform motion Conceivable, as people express it, no doubt it
is, that an increasing and diminishing motion should take place in a circle. But this con-
ceivableness or possibility means only an abstract capability of being represented, which leaves
out of sight that Determinate Thing on which the question turns.

The Circle is the line returning into itself in which all the radii are equal, that is, it is
completely determined by means of the radius. here is only one Determination, and that
is the whole Determination.

But in free motion, in which the Determinations according to space and according to time
come into view with Differences, in a qualitative relation to each other, this Relation appears
in the spatial aspect as a Difference thereof in itself, which therefore requires two Deter-
minations, Hereby the Form of the path returning into itself is essentially an Hilipse.

The abstract Determinateness which produces the circle appears also in this way, that the
are or angle which is included by two Radii is independent of them, a magnitude with regard
to them completely empirical. But since in the motion as determined by the Conception, the
distance from the center, and the are which is run over in a certain time, must be compre-
hended in one determinateness, [and] make out a ‘whole, this is the sector, a space-deter-
mination of two dimensions: in this way, the are is essentially a Function of the Radius
vector; and the former (the arc) being unequal, brings with it the inequality of the Radii,
That the determination with regard to the space by means of the time appears as a Deter-
mination of two Dimensions,—as a Superficies-Determination,—agrees with what was said
before (§ 266) respecting Falling Bodies, with regard to the exposition of the same Deter-
minateness, at one while as Time in the root, at another while as Space in the square. Here,
however, the Quadratic character of the space is, by the returning of the Line of motion into
itself, limited to a Sector. These are, as may be seen, the general principles on which the
Keplerian Law, that in equal times equal sectors are cut off, rests.

This Law becomes, as is clear, only the relation of the are to the Radius Vector, and the
Time enters there as the abstract Unity, in which the different Sectors are compared, because
as Unity it is the Determining Element. But the further relation is that of the Time, not
as Unity, but as a Quantity in general,—as the time of Revolution—to the magnitude of the
Path, or, what is the same thing, the distance from the center. As Root and Square, we saw
that Time and Space had a relation to each other, in the case of Falling Bodies, the case of
half-free motion—because that |motion| is determined on one side by the conception, on the

706

APPENDIX TO Dr. WHEWELL’S MEMOIR ON HEGEL’S CRITICISM, ETC.

other by external [conditions]. But in the case of absolute motion—the domain of free

(uw) masses—the determination attains its Totality. The Time as the Root is a mere empirical

(v)

(w)

(v)

magnitude; but as a component (moment) of the developed Totality, it is a Totality in
itself,—it produces itself, and therein has a reference to itself;—as the Dimensionless
Element in itself, it only comes to a formal identity with itself, the Square; Space, on the
other hand, as the positive Distribution (aussereinander) [comes] to the Dimension of the
Conception, the Cusr. ‘Their Realization preserves their original difference. This is the
third Keplerian Law, the relation of the Cubes of the Distances to the Squares of the
Times ;—a Law which is so great on this account, that it represents so simply and imme-
diately Reason as belonging to the thing: while on the contrary the Newtonian Formula, by
means of which the Law is changed into a Law for the Force of Gravity, shews the Distortion,
Perversion and Inversion of Reflewion which stops half-way.

Additions to new Edition. § 269.

The center has no sense without the circumference, nor the circumference without the center.
This makes all physical hypotheses vanish which sometimes proceed from the center, some-
times from the particular bodies, and sometimes assign this, sometimes that, as the original
[cause of motion]...It is silly (lippisch) to suppose that the centrifugal force, as a tendency to
fly off in a Tangent, has been produced by a lateral projection, a projectile force, an impulse
which they have retained ever since they set out on their journey (von Haus aus). Such
casualty of the motion produced by external causes belongs to inert matter; as when a stone
fastened to a thread which is thrown transversely tries to fly from the thread. We are not
to talk in this way of Forces. If we will speak of Force, there is one Force, whose elements
do not draw bodies to different sides as if they were two Forces. The motion of the heavenly
bodies is not a being pulled this way or that, such as is thus imagined ; it is free motion: they
go along, as the ancients said, as blessed Gods (sie gehen als selige Gdtter einher). The
celestial corporeity is not such a one as has the principle of rest or motion external to itself.
Because stone is inert, and all the earth consists of stones, and the other heavenly bodies are of
the same nature,—is a conclusion which makes the properties of the whole the same as those of
the part. Impulse, Pressure, Resistance, Friction, Pulling, and the like, are valid only for
an existence of matter other than the celestial. Doubtless that which is common to the two is
matter, as a good thought and a bad thought are both thoughts; but the bad one is not
therefore good, because it is a thought.

LIT. Discussion of a Differential Equation relating to the breaking of Railway Bridges.
By G. G. Sroxes, M.A., Fellow of Pembroke College, Cambridge.

[Read May 21, 1849.]

Yo explain the object of the following paper, it will be best to relate the circumstance which
gave rise toit. Some time ago Professor Willis requested my consideration of a certain differential
equation in which he was interested, at the same time explaining its object, and the mode of ob-
taining it. The equation will be found in the first article of this paper, which contains the sub-
stance of what he communicated to me. It relates to some experiments which have been
performed by a Royal Commission, of which Professor Willis is a member, appointed on the
27th of August, 1847, “ for the purpose of inquiring into the conditions to be observed by engineers
in the application of iron in structures exposed to violent concussions and vibration.” The object
of the experiments was to examine the effect of the velocity of a train in increasing or decreasing
the tendency of a girder bridge over which the train is passing to break under its weight. In order
to increase the observed effect, the bridge was purposely made as slight as possible : it consisted in
fact merely of a pair of cast or wrought iron bars, nine feet long, over which a carriage, variously
loaded in different sets of experiments, was made to pass with different velocities. The remarkable
result was obtained that the deflection of the bridge increased with the velocity of the carriage,
at least up to a certain point, and that it amounted in some cases to two or three times the central
statical deflection, or that which would be produced by the carriage placed at rest on the middle
of the bridge. It seemed highly desirable to investigate the motion mathematically, more especially
as the maximum deflection of the bridge, considered as depending on the velocity of the carriage,
had not been reached in the experiments*, in some cases because it corresponded to a velocity
greater than any at command, in others because the bridge gave way by the fracture of the bars
on increasing the velocity of the carriage. The exact calculation of the motion, or rather a cal-
culation in which none but really insignificant quantities should be omitted, would however be
extremely difficult, and would require the solution of a partial differential equation with an ordinary
differential equation for one of the equations of condition by which the arbitrary functions would
have to be determined. In fact, the forces acting on the body and on any element of the bridge
depend upon the positions and motions, or rather changes of motion, both of the body itself and
of every other element of the bridge, so that the exact solution of the problem, even when the de-
flection is supposed to be small, as it is in fact, appears almost hopeless.

In order to render the problem more manageable, Professor Willis neglected the inertia of the
bridge, and at the same time regarded the moving body as a heavy particle. Of course the masses
of bridges such as are actually used must be considerable; but the mass of the bars in the ex-
periments was small compared with that of the carriage, and it was reasonable to expect a near
accordance between the theory so simplified and experiment. This simplification of the problem
reduces the calculation to an ordinary differential equation, which is that which has been already
mentioned ; and it is to the discussion of this equation that the present paper is mainly devoted.

This equation cannot apparently be integrated in finite terms, except for an infinite number
of particular values of a certain constant involved in it; but I have investigated rapidly con-
vergent series whereby numerical results may be obtained. By merely altering the scale of the

* The details of the experiments will be found in the Report of the Commission, to which the reader is referred,

708 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION

abscissee and ordinates, the differential equation is reduced to one containing a single constant (3,
which is defined by equation (5). The meaning of the letters which appear in this equation will
be seen on referring to the beginning of Art. 1. For the present it will be sufficient to observe
that ( varies inversely as the square of the horizontal velocity of the body, so that a small value
of £8 corresponds to a high velocity, and a large value to a small velocity.

It appears from the solution of the differential equation that the trajectory of the body is
unsymmetrical with respect to the centre of the bridge, the maximum depression of the body
occurring beyond the centre. The character of the motion depends materially on the numerical
value of 8. When is not greater than 4, the tangent to the trajectory becomes more and more
inclined to the horizontal beyond the maximum ordinate, till the body gets to the second extremity
of the bridge, when the tangent becomes vertical. At the same time the expressions for the central
deflection and for the tendency of the bridge to break become infinite. When # is greater than
4, the analytical expression for the ordinate of the body at last becomes negative, and afterwards
changes an infinite number of times from negative to positive, and from positive to negative.
The expression for the reaction becomes negative at the same time with the ordinate, so that in

fact the body leaps.

The occurrence of these infinite quantities indicates one of two things: either the deflection
really becomes very large, after which of course we are no longer at liberty to neglect its square ;
or else the effect of the inertia of the bridge is really important. Since the deflection does not
really become very great, as appears from experiment, we are led to conclude that the effect of the
inertia is not insignificant, and in fact I have shewn that the value of the expression for the vis
viva neglected at last becomes infinite. Hence, however light be the bridge, the mode of approx-
imation adopted ceases to be legitimate before the body reaches the second extremity of the bridge,
although it may be sufficiently accurate for the greater part of the body’s course.

In consequence of the neglect of the inertia of the bridge, the differential equation here dis-
cussed fails to give the velocity for which 7’, the tendency to break, is a maximum. When B is
a good deal greater than 1, 7’ is a maximum at a point not very near the second extremity of the
bridge, so that we may apply the result obtained to a light bridge without very material error.
Let 7, be this maximum value. Since it is only the inertia of the bridge that keeps the tendency
to break from becoming extremely great, it appears that the general effect of that inertia is to
preserve the bridge, so that we cannot be far wrong in regarding 7’, as a superior limit to the
actual tendency to break. When # is very large, JT may be calculated to a sufficient degree of
accuracy with very little trouble.

Experiments of the nature of those which have been mentioned may be made with two distinct
objects ; the one, to analyze experimentally the laws of some particular phenomenon, the other, to
apply practically on a large scale results obtained from experiments made on a small scale. With
the former object in view, the experiments would naturally be made so as to render as conspicuous
as possible, and isolate as far as might be, the effect which it was desired to investigate; with the
latter, there are certain relations to be observed between the variations of the different quantities
which are in any way concerned in the result. These relations, in the case of the particular problem
to which the present paper refers, are considered at the end of the paper.

1. It is required to determine, in a form adapted to numerical computation, the value of y’ in
terms of a’, where y’ is a function of a defined by satisfying the differential equation
d*y’ by! (1)
eee OO ae rane eh eee e ones
du* (2ea’ — x)?’
with the particular conditions
dy’ ;
at when a’ = 0, ......... (2)

i eS pe es i

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 709

the value of y' not being wanted beyond the limits 0 and 2e of a. It will appear in the course
of the solution that the first of the conditions (2) is satisfied by the complete integral of (1), while
the second serves of itself to determine the two arbitrary constants which appear in that integral.

The equation (1) relates to the problem which has been explained in the introduction. It was
obtained by Professor Willis in the following manner. In order to simplify to the very utmost the
mathematical calculation of the motion, regard the carriage as a heavy particle, neglect the inertia
of the bridge, and suppose the deflection very small. Let a’, y’ be the co-ordinates of the moving
body, a’ being measured horizontally from the beginning of the bridge, and y’ vertically downwards.
Let M be the mass of the body, V its velocity on entering the bridge, 2c the length of the bridge,
g the force of gravity, S the deflection produced by the body placed at rest on the centre of the
bridge, & the reaction between the moving body and the bridge. Since the deflection is very
small, this reaction may be supposed to act vertically, so that the horizontal velocity of the body
will remain constant, and therefore equal to V. The bridge being regarded as an elastic bar or
plate, propped at the extremities, and supported by its own stiffness, the depth to which a weight
will sink when placed in succession at different points of the bridge will vary as the weight
multiplied by (2ea’ — *)*, as may be proved by integration, on assuming that the curvature is
proportional to the moment of the bending force. Now, since the inertia of the bridge is neglected,
the relation between the depth y’ to which the moving body has sunk at any instant and the
reaction & will be the same as if # were a weight resting at a distance «’ from the extremity of
the bridge ; and we shall therefore have

y = CR @eca = 2)’,
C being a constant, which may be determined by observing that we must have y’ = S when R = Mg
and w =c; whence

S
Cc =—_,.
Mege
We get therefore for the equation of motion of the body
d*y' gely’

Goa aS (2ca" — w”)?”

di
which becomes on observing that = =V

ay oe ge y
da® V? VW28' @ca! =a)?’

which is the same as equation (1), a and 6 being defined by the equations

& ge
= aa? b= Pagers (3)
2. To simplify equation (1) put
a’ = 2ca, y = 16c'ab-'y, b = 4¢°B,
which gives
AI, ea oo aL
dax* (v — wv’)

It is to be observed that # denotes the ratio of the distance of the body from the beginning of
the bridge to the length of the bridge; y denotes a quantity from which the depth of the body
below the horizontal plane in which it was at first moving may be obtained by multiplying by
16 c4ab-! or 16,8; and f, on the value of which depends the form of the body’s path, is a constant

defined by the equation

Vor, VIII. Part V. 4Y

710 Mr. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION
8. In order to lead to the required integral of (4), let us first suppose that # is very small.
Then the equation reduces itself to

_ By

Gee

A mG)
of which the complete integral is
2
a
points
+p
and (7) is the approximate integral of (4) for very small values of w Now the second of equations

(2) requires that A = 0, B = 0,* so that the first term in the second member of equation (7) is
the leading term in the required solution of (4).

4+ Awtt*E-8 5 Byi-“4-8, (7)

4, Assuming in equation (4) y = (w —a’)’ x, we get

So {(@- a'r} + Be = Bose ©)

Since (4) gives y = (# — a”)? when B= ow, and (5) gives 3 =o when V=0, it follows that #
is the ratio of the depression of the body to the equilibrium depression, It appears also from
Art. 3, that for the particular integral of (8) which we are seeking, x is ultimately constant when
is very small.

To integrate (8) assume then

= A, + Aja + Ane? +... = 2 Aja',......(9)

and we get
E(i+2) G+ 1) Ao! - 23 (6 +8) (6 +2) Ayait +E G4 4) +8) Aoi? + BTA,2 = B,
or
= {[@+ 1) @+ 2) +B] 4, - 2 +1) @ +2) 4, + © +1) G+ 2 A;-o} 2 = B, ... (10)

where it is to be observed that no coefficients A; with negative suffixes are to be taken.

Equating to zero the coefficients of the powers 0, 1, 2...of # in (10), we get
(2 + B) 4) = B;
(6 + B) A, —124,=0, &e.
and generally
{+ 1) @+2) +B} 4,- 2 (+1) @ +2) Ay + + 1) G+ 2) An =O... (11)

The first of these equations gives for 4, the same value which would have been got from (7).
The general equation (11), which holds good from i=1 to i= ©, if we conventionally regard
A_, as equal to zero, determines the constants 4,, 4,, 43... one after another by a simple and
uniform arithmetical process. It will be rendered more convenient for numerical computation by
putting it under the form

A,= §A,_, + AA,_2} () - B

ats} eee coerce ces ves ces veseeteccces (12)

* When >}, the last two terms in (7) take the form a? { Ccos
(q log x) + D sin(q log 2) }; and if y, denote this quantity we cannot

f ath when 2 = 0. If we

in strictness speak of the limiting value o'
give x asmall positive value, which we Me suppose to decrease

dy,
indefinitely, —* >, Will fluctuate between the constantly increasing

limits + a-*# ¥ {(AC +qD)?+(3D—qC)2}, or +x-*V { B(C?
+D*)}, since g=¥(8—}). But the body is supposed to enter
the bridge horizontally, that is, in the direction of a tangent, since
the bridge is supposed to be horizontal, so that we must clearly
have C?4D?=0, and therefore C=0, D=0. When B=} the
last two terms in (7) take the form at (EZ +F logw), and we must
evidently have =0, F =0.

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 711

for it is easy to form a table of differences as we go along ; and when z becomes considerable, the
quantity to be subtracted from 4;_, + AA,_. will consist of only a few figures.

5. When ¢ becomes indefinitely great, it follows from (11) or (12) that the relation between
the coefficients 4; is given by the equation

A; — 24;_,+ A=; 0; Se dee ain dielvincinndiceieivesiveciss (10)

of which the integral is
Asi Git O tydeusensoaets tae daxj apes ease asseee eG)

Hence the ratio of consecutive coefficients is ultimately a ratio of equality, and therefore the
ratio of the (i + 1)th term of the series (9) to the ith is ultimately equal to #. Hence the series is
convergent when a lies between the limits —1 and +13; and it is only between the limits 0 and 1
of @ that the integral of (8) is wanted. The degree of convergency of the series will be ultimately
the same as in a geometric series whose ratio is w.

6. When ~@ is moderately small, the series (9) converges so rapidly as to give with little
trouble, the coefficients 4,, 4,... being supposed to have been already calculated, as far as may be
necessary, from the formula (12). For larger values, however, it would be necessary to keep in a
good many terms, and the labour of calculation might be abridged in the following manner.

When ¢ is very large, we have seen that equation (12) reduces itself to (13), or to oA e105
or, which is the same, A*A;=0. When 3 is large, A? A, will be small; in fact, on substituting
in the small term of (12) the value of 4; given by (14), we see that A*4, is of the order i-!, Hence
A%4;, A*A, ... will be of the orders i-*, i-*..., so that the successive differences of A, will rapidly
decrease. Suppose 7 terms of the series (9) to have been calculated directly, and let it be required
to find the remainder. We get by finite integration by parts

: a agitt Abed art Spite
2 A,a' = const. + A; —— — AA; ——, + A*4, —_—. --..,
a1 (# — 1)? (w — 1)°

and taking the sum between the limits 7 and ¢ we get

EWINY (=) + A°A, (; 2) ede I. (15)

av

A,a' + A,,,@'+! + ... to inf. = a! {4,;

—Wv

# will however presently be made to depend on series so rapidly convergent that it will hardly be
worth while to employ the series (15), except in calculating the series (9) for the particular value 4
of w, which will be found necessary in order to determine a certain arbitrary constant *.

7. If the constant term in equation (4) be omitted, the equation reduces itself to

dy By Ep dactt dep <s. (16)

ee 0, cenccvccee
da (w — 2°)

The form of this equation suggests that there may be an integral of the form y = #”(1 —.)".
4q 8s MY 8 Y
Assuming this expression for trial, we get
d* ,
(a — a")? eg a™ (1 —w)" $m (m — 1) (1 — @)? — 2mnaw (1 — a) +n (nm — 1) wt
w
=y {m(m—1) —2m(m+n —-1)a + (m+n) (m+n—1) at,

The second member of this equation will be proportional to y, if

* A mode of calculating the value of # for v= 4 will presently be given, which is easier than that here mentioned, unless f be very

large. See equation (42) at the end of this paper.
4¥2

712 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION

and will be moreover equal to — By, if
Mi= Mt oO: ce ceucnoeeaessees (LS)

It appears from (17) that m, m are the two roots of the quadratic (18). We have for the
complete integral of (16)

y = Aa (1 — ax) + Ba (1 — 2). cc. cesceeseeees (19)

The complete integral of (4) may now be obtained by replacing the constants 4, B by functions
R, S of x, and employing the method of the variation of parameters. Putting for shortness

a" (1—2)"=4u, a" (1—2)"=2,

we get to determine # and S' the equations

an as _,
fa ape
dudR dvdS B
dada dvdu ™
d d
Since v —~ - w = =m—n, we get from the above equations
v aw
dR pe dS Bu
dx m-—n’ da m—n

whence we obtain for a particular integral of (4)

B fora -ay [wa - ayaa aay [orc ~ ay dals 20)

and the complete integral will be got by adding together the second members of equations (19),
(20). Now the second member of equation (20) varies ultimately as a*, when « is very small,
and therefore, as shewn in Art. 3, we must have 4 = 0, B =0, so that (20) is the integral we want.

When the roots of the quadratic (18) are real and commensurable, the integrals in (20) satisfy
the criterion of integrability, so that the integral of (4) can be expressed in finite terms without
the aid of definite integrals. The form of the integral will, however, be complicated, and y may
be readily calculated by the method which applies to general values of /.

8. Since [> ee = fi F (w) da - [)-* F (1 — 2) da, we have from (20)
yoo fa" (1 — a)" fh" (Ql — w)"da - a" (1- 2)" f) a (1 — a)" da},
B

shee 5m (1 — a)" fi-" (1 — @)"a" da — a" (1-2) fj? (1 - a)" a" da}.

If we put - a for the second member of equation (20), the equation just written is
equivalent to
F(a) =f (1 — 2) + P(A), «eer e eee ee ee eeereceees (21)
x _ fa"

where

p(a) = — av)" fiw (1 - a)"de — a" (1 - 2)" fja" (1 - a)" da}... (22)

Now since m + ” = 1,
s"ds

fo" (1 — a)" dw = fu (w=! = 1)" da = — fw-' (w — 1)" w* dw = We rnayts

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 713

At the limits 7 =0 and #=1, we have w= and w=1, s=o ands =0, whence if J denote
the definite integral,

™ds
f= ff «@'( —a)" da = Sac ca,
OS eee Vee GRAD
We get by integration by parts
3” ds gs” m g7- 1 ds

GQ+e 204e "2 J G40"
and again by a formula of reduction
Geads Eye s"— ds

(a +3)? Sirar + - m)

Devens

Now £ being essentially positive, the roots of the quadratic (18) are either real, and comprised
between 0 and 1, or else imaginary with a real part equal to 3. In either case the expressions
which are free from the integral sign vanish at the limits s = 0 and s = ~ , and we have therefore,
on replacing m (1 — m) by its value 6,

m—-1
a ds
2 Ig 1+8

The function @ («) will have different forms according as the roots of (18) are real or imaginary.

First suppose the roots real, and let m=4+47,n= 4 —r, so that

T= ST — Bu coecarcoseccscracseraoee (28)

In this case m is a real quantity lying between 0 and 1, and we have therefore by a known formula

wo s"-!d5 T oe
See Tetosteedeerer lee
Je )

ea >
1+6s SID 7 7 COS 1 7

whence we get from (22), observing that the two definite integrals in this equation are equal to each

other,
(a) = Er | Z )- ( z fil Sere

47 COS T' 7 1-@ 1-@

This result might have been obtained somewhat more readily by means of the properties of the
first and second Eulerian integrals.

When # becomes equal to 1, r vanishes, the expression for p(w) takes the form {, and we
easily find

When 8 >1, the roots of (18) become imaginary, and r becomes p V — 1, where

pismat Pix Aa setae (27)

The formula (25) becomes

i a - ‘
p(v) = abtanhe — g sin (blog ae -) Mee aco noA Ey)

If f(w) be calculated from a = 0 to # =}, equation (21) will enable us to calculate it readily
from w = } tow =1, since it is easy to calculate p().

9. A series of a simple form, which is more rapidly convergent than (9) when # approaches

the value 4, may readily be investigated.

714 Mr. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION

Let « =3(1 + w); then substituting in equation (8) we get
d 2
4 dw fa = wy? st + Bx = B. wocececcscee (29)

Assume

then substituting in (29) we get
TB, {2i(2i — 1) w**-*?— 2(2% + 2) (Qi + 1) w+ (Qi + 4) (27 + 3)w*!** + 4 Bw} = 428,
or,
= fi(gi - 1) B,- 2[i(2i- 1) - BI] B_.+ i(@t- 1) B,_.f w** = 28.
This equation leaves B, arbitrary, and gives on dividing by i (2% — 1), and putting in succession
i= 1 ti 2a ees

B, -2 (2 - es BE Gere oes sade st wanesieo (Ole

B, - 2 (: - Fz, + B, =0, &c.;
and generally when 7 >1,
Ree Pea ; Yas ee ee cay. 2 ia).

The constants B,, B,,... being thus determined, the series (30) will be an integral of equation
(29), containing one arbitrary constant. An integral of the equation derived from (29) by replacing
the second member by zero may be obtained in just the same way by assuming x = C,w + C,\w*+...
when C;, C,... will be determined in terms of C,, which remains arbitrary. The series will both be
convergent between the limits w= — 1 and w=1, that is, between the limits #=0 and v=1.
The sum of the two series will be the complete integral of (29), and will be equal to (a — a*)~*f (a)
if the constants B,, C, be properly determined. Denoting the sums of the two series by F, (w),
F, (w) respectively, and writing o (#) for (w — a)-*f (a), so that x =o (a), we get

o(w) = F,(w) + F, (w), o (1 — a) = F, (w) - Fy (w)s
and since 2 Fy (w) = o (w) — o (1 — w) = (w — a*)~* (a) by (21), we get
o(«) =F, (w) +4 (w- 2a)? p(a), o (1 -@) = F,(w) -—4 (w- 2°)? @ (a). eee (33).
To determine B, we have
[y= @ (ES hecbantiecr separa -Anceccde nb Sbesbe-ocosecacecocod ©)

which may be calculated by the series (9).

10. The series (9), (30) will ultimately be geometric series with ratios xv, w®, or v, (2a —1)’,
respectively. Equating these ratios, and taking the smaller root of the resulting quadratic, we
get w=. Hence if we use the series (9) for the calculation of o (w) from « =0 to w= 1, and
(30) for the calculation of o (w) from w = 4 to w =4, we shall have to calculate series which are

ultimately geometric series with ratios ranging from 0 to i,

Suppose that we wish to calculate o (w) or x for values of @ increasing by .02. The process
of calculation will be as follows. From the equation (2 + 8) 4)= 8 and the general formula (12),
calculate the coefficients do, 4), 4.,... asfar as may be necessary. From the series (9), or else from
the series (9) combined with the formula (15), calculate « (4) or B,, and then calculate B,, By...
from equations (31), (82). Next calculate o (~) from the series (9) for the values .02, .04,....26
of #, and F,(w) from (30) for the values .04, .08...,.44 of w, and lastly (w - a*)-*@ (w) for the
values .52, .54,.., .98 of a. Then we have (2) calculated directly from # = 0 to v = .26; equa-

:
:

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 715

tions (33) will give o(w) from # =.28 to «=.72, and lastly the equation o (w) =o (1 -2)
+ (wv — a*)~*@ (a) will give o (w) from « = .74 tow = 1.

11. The equation (21) will enable us to express in finite terms the vertical velocity of the
body at the centre of the bridge. For according to the notation of Art. 2, the horizontal and

vertical coordinates of the body are respectively 2cew and 16.Sy, and we have also mee = 7,

whence, if v be the vertical velocity, we get
d.16Sydz _ 8SV

Eda” Garay oe
But (21) gives f’ (4) = 4 ¢' (4), whence if v, be the value of v at the centre, we get
4a SV [3° jd
v= ENS or Se a et eee)
COS? 1 c (e+e P")

according as 8B <>1.

In the extreme cases in which V is infinitely great and infinitely small respectively, it is evi-
dent that v, must vanish, and therefore for some intermediate value of V, v, must be a maximum.
Since V < (3-2 when the same body is made to traverse the same bridge with different velocities,
v, will be a maximum when p or q is a minimum, where

p=2B-# cosrm, q= 6 ~3 ("+ e-*).
Putting for cos rq its expression in a continued product, and replacing x by its expression in

terms of 3, we get
1-48 1 - 4()
= 83-4(1— =
p = 8p (: a } (1 a ) soaaeak oa

whence
dlog p 1 1 1
St a Foe sonreo ane (86
d pB 26° 1.2+B2.34p" GP
d J dlogq
The same expression would have been obtained for —-2-. Call the second member of equa-

dp
tion (36) F (8), and let —- N, P be the negative and positive parts respectively of F (3). When
1 1 3 .
B=0,N=c,and P= et eee and therefore F (3) is negative. When (3 becomes
infinite, the ratio of P to N becomes infinite, and therefore F ((3) is positive when {3 is sufficiently
large; and F () alters continuously with 8. Hence the equation F (/3) = 0 must have at least
one positive root. But it cannot have more than one; for the rates of proportionate decrease of

1dN 1dP
Nap’ Pap
1 (1.2+)-?+(2.38+)*+...
A” (1.24 B74 @.84—)F..
and the several terms of the denominator of the second of these expressions are equal to those of

the numerator multiplied by 1.2 + 3, 2.3 + /3,... respectively, and therefore the denominator is
equal to the numerator multiplied by a quantity greater than 2 + 3, and therefore greater than jeje

the quantities NM, P, or — , are respectively

so that the value of the expression is less than =. Hence for a given infinitely small increment of

PB the change -dN in N bears to N a greater ratio than —dP bears to P, so that when WV is
greater than or equal to P it is decreasing more rapidly than P, and therefore after having once

716 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION, &c.

become equal to P it must remain always less than P. Hence v, admits of but one maximum
or minimum value, and this must evidently be a maximum.

1 a
When 6 = 4, N = 2, and P< ma F otaes Cilen <1, and therefore (8) has the same sign
as when 3 is indefinitely small. Hence it is g and not p which becomes a minimum. Equating

d c
ig to zero, employing (27), and putting 2p = log.¢, we find
6(¢ +1)
¢-1
The real positive root of this equation will be found by trial to be 36.3 nearly, which gives
p=-5717, B=4 + p= 5768. If V, be the velocity which gives v, a maximum, v, the maximum
value of »,, U the velocity due to the height S, we get

V, = a, U 873?

2 S
= =, and v, = =—~,— — MV, whence

ge ec
4BS 8S /3B G+Cic

V, = .4655 < U, v, = .6288U.

= log.¢ + 7° (log.¢)™.

12. Conceive a weight W placed at rest on a point of the bridge whose distance from the
first extremity is to the whole length as # to1. The reaction at this extremity produced by W will
be equal to (1 — x) W, and the moment of this reaction about a point of the bridge whose abscissa
2c, is less than 2c will be 2c(1 — w)a,W. This moment measures the tendency of the bridge to
break at the point considered, and it is evidently greatest when w,=a, in which case it becomes
2c(1-a)aW. Now, if the inertia of the bridge be neglected, the pressure # produced by the
moving body will be proportional to (# — x*)~*y, and the tendency to break under the action of a
weight equal to R placed at rest on the bridge will be proportional to (1 - w) w x (w—.«*)~*y, or to
(w —2#®)x. Call this tendency 7’, and let 7’ be so measured that it may be equal to 1 when the moving
body is placed at rest on the centre of the bridge. Then 7’ = C (w — 2"), and 1 = C (4 — 4), whence

T=4(av —a") x.

The tendency to break is actually liable to be somewhat greater than 7’, in consequence of the
state of vibration into which the bridge is thrown, in consequence of which the curvature is alternately
greater and less than the statical curvature due to the same pressure applied at the same point. In
considering the motion of the body, the vibrations of the bridge were properly neglected, in con-
formity with the supposition that the inertia of the bridge is infinitely small compared with that of
the body.

The quantities of which it will be most interesting to calculate the numerical values are zx,
which expresses the ratio of the depression of the moving body at any point to the statical depression,
T, the meaning of which has just been explained, and y’, the actual depression, When x has been
calculated in the way explained in Art. 10, 7’ will be obtained by multiplying by 4(# — a’), and

,

then 5 will be got by multiplying 7’ by 4 (w — 2%).

13. The following Table gives the values of these three quantities for each of four values of 3,
namely <%, 4,4, and }, to which correspond r =1, r=0, p=4, p=1, respectively. In performing
the calculations I have retained five decimal places in calculating the coefficients 4,, 4, 4.... and By,
B,, B,... and four in calculating the series (9) and (30). In calculating (w) I have used four-
figure logarithms, and I have retained three figures in the result. The calculations have not been

re-examined, except occasionally, when an irregularity in the numbers indicated an error.

92
94
96
98
1.00

vw
~
Rss
>
i>)
>
ve}
a

14.3 90.3 23.2
29.6 43.5 41.8

112 139 106
a a) &$ ow

Vou. VIII. Parr V-

TABLE I.

~wwnnwnnrns
o GS ¢ >
oa o >

3
=
or

2
-
>

878
-401
425
449
AT73
498
528
548
578
598
623
647
669
691
708
723
730
699
690
000

000

.883
.680
318
O14
04
.000

718 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION

14. Let us first examine the progress of the numbers. For the first two values of 8, =
increases from a small positive quantity up to «© as # increases from 0 to 1. As far as the
table goes, s is decidedly greater for the second of the two values of 3 than for the first. It
is easily proved however that before w attains the value 1, x becomes greater for the first value
of 8 than for the second. For if we suppose x very little less than 1, f(1-—.) will be ex-
tremely small compared with @(w), or, in case @(«) contain a sine, compared with the coeffi-
cient of the sine. Writing a, for 1—., and retaining only the most important term in f(@),
we get from (21), (25), (26), and (28)

Bear q-r 7 1 Bear : ( ~)
2 = 3 = 3 i ~
a) = ——— 2 =—alog—, or = ——————_ 3 sin { plo edehewscs (8
F( ) Ar costar 1 ide 39 eres ae p(c™ 5 e~P™) 1 e Ba, ( 7)

according as B<4, B=, or B>4; and x will be obtained by dividing f(v) by a,’ nearly.
Hence if 1>8,>,,>0, x is ultimately incomparably greater when 8 = 3, than when B= #,,
and when 8 = B, than when B=1. Since f(0) = 4, = B(2 + 8)“ = (28-' +1)~', f(0) increases
with £, so that f(x) is at first larger when 6 = 3, than when 6 = G,, and afterwards smaller.

When $8> 4, x vanishes for a certain value of a, after which it becomes negative, then
vanishes again and becomes positive, and so on an infinite number of times. The same will be
true of ZT. If p be small, f(w) will not greatly differ, except when « is nearly equal to 1,
from what it would be if p were equal to zero, and therefore f(w) will not vanish till w is nearly
equal to 1. On the other hand, if p be extremely large, which corresponds to a very slow velocity,
= will be sensibly equal to 1 except when a is nearly equal to 1, so that in this case also f(«)
will not vanish till w is nearly equal to 1. The table shews that when 8 = 4, f(x) first vanishes
between # =.98 and w =1, and when B= $ between v7 =.94 andw=.96. The first value of x
for which f() vanishes is probably never much less than 1, because as ( increases from 2 the
denominator p(c°* + ¢~?) in the expression for @(«) becomes rapidly large.

15. Since when B>i, T vanishes when # = 0, and again for a value of & less than 1, it must
be a maximum for some intermediate value. When = 4 the table appears to indicate a maximum
beyond #=.98. When 8 =, the maximum value of Tis about 2.61, and occurs when 2 = . 86
nearly. As # increases indefinitely, the first maximum value of 7 approaches indefinitely to 1,
and the corresponding value of w to 4. Besides the first maximum, there are an infinite number
of alternately negative and positive maxima; but these do not correspond to the problem, for a

reason which will be considered presently.

16. The following curves represent the trajectory of the body for the four values of (3 con-
tained in the preceding table. These curves, it must be remembered, correspond to the ideal
limiting case in which the inertia of the bridge is infinitely small.

In this figure the right line 4B represents the bridge in its position of equilibrium, and
at the same time represents the trajectory of the body in the ideal limiting case in which
B=0 or V=0. AeeeB represents what may be called the equilibrium trajectory, or the

ee

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 719

curve the body would describe if it moved along the bridge with an infinitely small velocity.
The trajectories corresponding to the four values of 3 contained in the above table are marked
bys laiaplenlen 22: 858.8; 4.4.4. 4 respectively. The dotted curve near B is meant to repre-
sent the parabolic are which the body really describes after it rises above the horizontal
line AB*. C is the centre of the right line AB: the curve AeeeB is symmetrical with respect
to an ordinate drawn through C.

17. The inertia of the bridge being neglected, the reaction of the bridge against the body, as

oe where C depends on the length and stiffness of
the bridge. Since this expression becomes negative with y, the preceding solution will not be
applicable beyond the value of a for which y first vanishes, unless we suppose the body held down
to the bridge by some contrivance. If it be not so held, which in fact is the case, it will quit the
bridge when y becomes negative. More properly speaking, the bridge will follow the body, in
consequence of its inertia, for at least a certain distance above the horizontal line 4B, and will exert
a positive pressure against the body: but this pressure must be neglected for the sake of con-
sistency, in consequence of the simplification adopted in Art. 1, and therefore the body may be
considered to quit the bridge as soon as it gets above the line 4B. The preceding solution shews
that when 8>4 the body will inevitably leap before it gets to the end of the bridge. The leap
need not be high; and in fact it is evident that it must be very small when # is very large. In
consequence of the change of conditions, it is only the first maximum value of 7’ which corresponds
to the problem, as has been already observed.

already observed, will be represented by

18. According to the preceding investigation, when 3 <i the body does not leap, the tangent
to its path at last becomes vertical, and 7’ becomes infinite. The occurrence of this infinite value
indicates the failure, in some respect, of the system of approximation adopted. Now the inertia
of the bridge has been neglected throughout; and, consequently, in the system of the bridge and
the moving body, that amount of labouring force which is requisite to produce the vis viva of the
bridge has been neglected. If €, » be the coordinates of any point of the bridge on the same scale
on which «, y represent those of the body, and € be less than a, it may be proved on the supposition
that the bridge may be regarded at any instant as in equilibrium, that

“2 (as AE se Gay

When w becomes very nearly equal to 1, y varies ultimately as (1 — v)'-’, and therefore » contains
2

dn \* d
terms involving (1 — #)~?~', and (2) » and consequently (7) , contains terms involving

(1 - w)~*-*", Hence the expression for the vis viva neglected at last becomes infinite; and there-
fore however light the bridge may be, the mode of approximation adopted ceases to be legitimate
before the body comes to the end of the bridge. The same result would have been arrived at if

f had been supposed equal to or greater than 4.

19. There is one practical result which seems to follow from the very imperfect solution of
the problem which is obtained when the inertia of the bridge is neglected. Since this inertia is
the main cause which prevents the tendency to break from becoming enormously great, it would
seem that of two bridges of equal length and equal strength, but unequal mass, the lighter would

* The dotted curve ought to have been drawn wholly outside the full curve. The two curves touch each other at the point
where they are cut by the line ACB, as is represented in the figure.
A 7% 2

720 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION

be the more liable to break under the action of a heavy body moving swiftly over it. The effect
of the inertia may possibly be thought worthy of experimental investigation.

20. The mass of a rail on a railroad must be so smal] compared with that of an engine, or
rather with a quarter of the mass of an engine, if we suppose the engine to be a four-wheeled one,
and the weight to be equally distributed between the four wheels, that the preceding investigation
must be nearly applicable till the wheel is very near the end of the rail on which it was moving,
except in so far as relates to regarding the wheel as a heavy point. Consider the motion of the
fore wheels, and for simplicity suppose the hind wheels moving on a rigid horizontal plane. Then
the fore wheels can only ascend or descend by the turning of the whole engine round the hind axle,
or else the line of contact of the hind wheels with the rails, which comes to nearly the same thing.
Let M be the mass of the whole engine, 7 the horizontal distance between the fore and hind axles,
h the horizontal distance of the centre of gravity from the latter axle, & the radius of gyration
about the hind axle, x, y the coordinates of the centre of one of the fore wheels, and let the rest of
the notation be as in Art. 1. Then to determine the motion of this wheel we shall have

d* ly 2Cy
pest (dh) ae HALAL IS Ses
es FET (7) Meh Gea ay”

: . M

whereas to determine the motion of a single particle whose mass is = ane should have had

M dy M Cy

4 dt 4 g (2c@ — a*)*

L :
Now fh must be nearly equal to re and k? must be a little greater than $77, say equal to 31%, so
that the two equations are very nearly the same.

Hence, 3 being the quantity defined by equation (5), where S denotes the central statical
deflection due to a weight ce , it appears that the rail ought to be made so strong, or else so
4

short, as to render 8 a good deal larger than 4. In practice, however, a rail does not rest merely
on the chairs, but is supported throughout its whole length by ballast rammed underneath.

21. In the case of a long bridge, 8 would probably be large in practice. When # is so large
2

that the coefficient eee , or 1 Bte~7F* nearly, in p(w) may be neglected, the motion of
pe €

the body is sensibly symmetrical with respect to the centre of the bridge, and consequently 7’, as

well as y, is a maximum whenw=21. For this value of w we have 4 (wv — x) =1, and therefore

2
zs=T'=y. Putting C,; for the (i + 1)™ term of the series (9), so that C;= 4;2~', we have
for w=4
T = Co + Cyt Cat .2...ccenscecorvesionncenenn (39)
B 6C
where Oe Se Cle B

and generally,

_ @+1)@+2)

* (@+1)@4+2)+8
whence 7’ is easily calculated. Thus for 8 = 5 we have 7? e~7* = .031 nearly, which is not large,
and we get from the series (39) 7’'=1.27 nearly. For $= 10, the approximate value of the

{C,-1 a 4 Cats

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 721

coefficient in @ (x) is .0048, which is very small, and we get 7'=1.14. In these calculations the
inertia of the bridge has been neglected, but the effect of the inertia would probably be rather to
diminish than to increase the greatest value of 7’.

22. The inertia of a bridge such as one of those actually in use must be considerable: the bridge
and a carriage moving over it form a dynamical system in which the inertia of all the parts ought to
be taken into account. Let it be required to construct the same dynamical system on a different
scale. For this purpose it will be necessary to attend to the dimensions of the different constants on
which the unknown quantities of the problem depend, with respect to each of the independent units
involved in the problem. Now if the thickness of the bridge be regarded as very small compared
with its length, and the moving body be regarded as a heavy particle, the only constants which enter
into the problem are M, the mass of the body, M’, the mass of the bridge, 2c, the length of the
bridge, S, the central statical deflection, V, the horizontal velocity of the body, and g, the force of
gravity. The independent units employed in dynamics are three, the unit of length, the unit of
time, and the unit of density, or, which is equivalent, and which will be somewhat more convenient
in the present case, the unit of length, the unit of time, and the unit of mass. The dimensions of
the several constants M, M’, &c., with respect to each of these units are given in the following
table.

Unit of length. Unit of time. Unit of mass,

M and M. 0 0 1
ec and S. 1 ) )
Ue, 1 -1 0
g- 1 —2 0

Now any result whatsoever concerning the problem will consist of a relation between certain
unknown quantities w’,w’,.. and the six constants just written, a relation which may be expressed by

f(a’, a”, ««. My M’, c, S, Vig) =0. ...000000--. (40)

But by the principle of homogeneity and by the preceding table this equation must be of the
form

af fal M' S V?

featee 30 al = 05) > aeaae ens cone (40)

@) @) "We cg
where (a’), (w”) ..., denote any quantities made up of the six constants in such a manner as to have
with respect to each of the independent units the same dimensions as a’, #”,.. , respectively. Thus,
if (40) be the equation which gives the maximum value 7’ of J’ in terms of the six constants, we
shall have but one unknown quantity a’, where w’ = 7’, and we may take for (a’), Meg, or else M’V®.
If (40) be the equation to the trajectory of the body, we shall have two unknown constants, a’, w”,
where a’ is the same as in Art. 1, and a” = y’, and we may take (a’) =e, (w”)=c. The equation
(41) shows that in order to keep to the same dynamical system, only on a different scale, we must
alter the quantities M, M’, &c. in such a manner that

M' «M, Sac, V? «cg,

and consequently, since gis not a quantity which we can alter at pleasure in our experiments, V
must vary as 4/c. A small system constructed with attention to the above variations forms an
ewact dynamical model of a larger system with respect to which it may be desired to obtain certain
results. It is not even necessary for the truth of this statement that the thickness of the large
bridge be small in comparison with its length, provided that the same proportionate thickness be

preserved in the model.

722 Mr. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION.

To take a numerical example, suppose that we wished, by means of a model bridge five feet
long and weighing 100 ounces, to investigate the greatest central deflection produced by an engine
weighing 20 tons, which passes with the successive velocities of 30, 40, and 50 miles an hour over
a bridge 50 feet long weighing 100 tons, the central statical deflection produced by the engine
being one inch. We must give to our model carriage a weight of 20 ounces, and make the small
bridge of such a stiffness that a weight of 20 ounces placed on the centre shall cause a deflection
of bth of an inch; and then we must give to the carriage the successive velocities of 34/10,
44/10, 54/10, or 9.49, 12.65, 15.81 miles per hour, or 13.91, 18.55, 23.19 feet per second. If
we suppose the observed central deflections in the model to be -12, .16, .18 of an inch, we may
conclude that the central deflections in the large bridge corresponding to the velocities of 30, 40, and
50 miles per hour would be 1.2, 1.6, and 1.8 inch.

G. G. STOKES.

Addition to the preceding Paper. .

Since the above was written, Professor Willis has informed me that the values of (3 are much
larger in practice than those which are contained in Table I, on which account it would be interesting
to calculate the numerical values of the functions for a few larger values of 3. I have accordingly
performed the calculations for the values 3, 5, 8,12, and 20. The results are contained in Table II,
In calculating x from w = 0 to # = .5, Temployed the formula (12), with the assistance occasionally
of (15). I worked with 4 places of decimals, of which 3 only are retained. The values of x for
w = .5, in which case the series are least convergent, have been verified by the formula (42) given
below: the results agreed within two or three units in the fourth place of decimals. The remaining

,

values of x were calculated from the expression for (« — a°) ~*b(v). The values of 7 and 5 were

deduced from those of x by merely multiplying twice in succession by 4a(1 —«). Professor Willis
has laid down in curves the numbers contained in the last five columns. In laying down these
curves several errors were detected in the latter half of the Table, that is, from # = .55 to wv .95.
These errors were corrected by re-examining the calculation ; so that I feel pretty confident that the
table as it now stands contains no errors of importance,

RELATING TO THE BREAKING OF RAILWAY BRIDGES.

TABLE IL.

T

723

B 3 | Bash | B=8 | B=12
857 -000} .000} .000] .000} .000
.886 -023) .027} .030] .082} .034
915 -089| .103} .113] .119] .123
-945 +195] .220] .237] .246] .252
-975| .989 -327| .367| .389| .399] .405
1.004| 1.016 486] .5385| .558| .565| .572
1.032 | 1.023 661} .713| .722| .728] .721
1.059 | 1.038 -843| .888] .889] .877] .859
1.081 | 1.049 1.023} 1.049/|1.026] .997| .966
1.099 | 1.056 1.190] 1.183] 1.127] 1.078} 1.035
1.111] 1.060 1.331] 1.274] 1.180] 1.111] 1.060
1.114| 1.058 1.431| 1.314) 1.179] 1.092} 1.037
1.105} 1.051 1.486| 1.281}1.108| 1.018] .968
1.081 | 1.038 1.446] 1.173] .954] .895] .860
1.039 | 1.021 1.334] .983| .781| .733] .720
-984| 1.013 1.111] .716] .564| .554| .570
.932| .989 -772| .896| .341| .382] .405
.925| .976 -335| .090} .172) .241| .254
1.013| .947 —.126}—.080} .104] .131] .123
-720| .943 —.297|+.045| .068| .026| .034

The form of the trajectory will be sufficiently perceived by comparing this table with the curves

represented in the figure.

As B increases, the first point of intersection of the trajectory with the

equilibrium trajectory eee moves towards 4. Since x =1 at this point, we get from the part of the
table headed ‘* x,” for the abscissa of the point of intersection, by taking proportional parts, .34, .29,
.26, .24, and .22, corresponding to the respective values 3, 5, 8, 12, and 20 of 3. Beyond this point
of intersection the trajectory passes below the equilibrium trajectory, and remains below it during
the greater part of the remaining course. As 3 increases, the trajectory becomes more and more
nearly symmetrical with respect to C: when (3 = 20 the deviation from symmetry may be considered
insensible, except close to the extremities A, B, where however the depression itself is insensible.
The greatest depression of the body, as appears from the column which gives y’, takes place a little
beyond the centre; the point of greatest depression approaches indefinitely to the centre as B
increases. This greatest depression of the body must be carefully distinguished from the greatest
depression of the bridge, which is decidedly larger, and occurs in a different place, and at a different
time. The numbers in the columns headed “ 7'” shew that 7' is a maximum for a value of # greater
than that which renders y’ a maximum, as in fact immediately follows from a consideration of the
mode in which y/ is derived from 7. ‘The first maximum value of 7, which according to what has
been already remarked is the only such value that we need attend to, is about 1.59 for B = 3, 1.38
for 3 = 5, 1.19 for 3 = 8, 1.11 for B = 12, and 1.06 for B = 20.

When £ is equal to or greater than 8, the maximum value of 7' occurs so nearly when w= .5

724 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION

U

that it will be sufficient to suppose w=.5. The value of x, 7, or = for wv = .5 may be readily

calculated by the method explained in Art. 21. I have also obtained the following expression for
this particular value ;

s=28 —48° fa = seat oo: =e. eer

When £ is small, or only moderately large, the series (42) appears more convenient for
numerical calculation, at least with the assistance of a table of reciprocals, than the series (39), but
when # is very large the latter is more convenient than the former. In using the series (42), it will
be best to sum the series within brackets directly to a few terms, and then find the remainder from
the formula

Ug — Usy) + Use — + = 4U,—-LAu, + tA*u,-...

The formula (42) was obtained from equation (20) by a transformation of the definite integral.
In the transformation of Art. 8, the limits of s will be 1 and o, and the definite integral on which
the result depends will be
2 g"-1 oy) wl
ds.
1 1+8
The formula (42) may be obtained by expanding the denominator, integrating, and expressing
m in terms of (3.

In practice the values of 8 are very large, and it will be convenient to expand according to
inverse powers of 3. This may be easily effected by successive substitutions, Putting for shortness
wv — a = X, equation (4) becomes by a slight transformation

y= X? - B21 X*—
aw

ax?

and we have for a first approximation y = X°, for a second y = X° — 3-'.X° da®?

result of the successive substitutions may be expressed as follows:
d’ d da
=i aH a — : -2 — * a ie eeeereeree
Y= x = GaX aa + B-?2X at ant RCs) (43)

where each term, taken positively, is derived from the preceding by differentiating twice, and then
multiplying by B-'X*,

For such large values of 3, we need attend to nothing but the value of s for w = 4, and this
may be obtained from (43) by putting # =4, after differentiation, and multiplying by 16. It will

: d
however be more convenient to replace a by 3(1+w), which gives ein Area X?=,,W,

where W=(1—w’)*. We thus get Best (43)
ap = w— Ez a Wixhsstty

= — (4
(48) = We
where we must put w = 0 after differentiation, if we wish to get the value of x forw=4. This
equation gives, on performing the differentiations and multiplications, and then putting w = 0,

=14B 14+ 3B * + 13B3+ ....ee eee masenss (44)

In practical cases this series may be reduced to 1 + ~'. The latter term is the same as would
be got by taking into account the centrifugal force, and substituting, in the small term involving
that force, the radius of curvature of the equilibrium trajectory for the radius of curvature of the

%

and soon. The

ee

os

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 725

actual trajectory. The problem has already been considered in this manner by others by whom it
has been attacked.

My attention has recently been directed by Professor Willis to an article by Mr. Cox On the
Dynamical Deflection and strain of Railway Girders, which is printed in The Civil Engineer
and Architect's Journal for September, 1848. In this article the subject is treated in a very
original and striking manner. ‘There is, however, one conclusion at which Mr. Cox has arrived
which is so directly opposed to the conclusions to which I have been led, that I feel compelled to
notice it. By reasoning founded on the principle of wis viva, Mr. Cox has arrived at the result
that the moving body cannot in any case produce a deflection greater than double the central
statical deflection, the elasticity of the bridge being supposed perfect. But among the sources of
labouring force which can be employed in deflecting the bridge, Mr. Cox has omitted to consider
the vis viva arising fiom the horizontal motion of the body. It is possible to conceive beforehand
that a portion of this vis viva should be converted into labouring force, which is expended in
deflecting the bridge. And this is, in fact, precisely what takes place. During the first part of
the motion, the horizontal component of the reaction of the bridge against the body impels the
body forwards, and therefore increases the vis viva due to the horizontal motion ; and the labouring
force which produces this increase being derived from the bridge, the bridge is less deflected than
it would have been had the horizontal velocity of the body been unchanged. But during the
latter part of the motion the horizontal component of the reaction acts backwards, and a portion
of the vis viva due to the horizontal motion of the body is continually converted into labouring
force, which is stored up in the bridge. Now, on account of the asymmetry of the motion, the
direction of the reaction is more inclined to the vertical when the body is moving over the
second half of the bridge than when it is moving over the first half, and moreover the reaction
itself is greater, and therefore, on both accounts, more vis viva depending upon the horizontal
motion is destroyed in the latter portion of the body’s course than is generated in the former
portion ; and therefore, on the whole, the bridge is more deflected than it would have been ha
the horizontal velocity of the body remained unchanged.

It is true that the change of horizontal velocity is small; but nevertheless, in this mode of
treating the subject, it must be taken into account. For, in applying to the problem the principle
of vis viva, we are concerned with the square of the vertical velocity, and we must not omit any
quantities which are comparable with that square. Now the square of the absolute velocity of
the body is equal to the sum of the squares of the horizontal and vertical velocities; and the
change in the square of the horizontal velocity depends upon the product of the horizontal velocity
and the change of horizontal velocity ; but this product is not small in comparison with the square
of the vertical velocity.

In Art. 22 I have investigated the changes which we are allowed by the general principle of
homogeneous quantities to make in the parts of a system consisting of an elastic bridge and a
travelling weight, without affecting the results, or altering anything but the scale of the system.
These changes are the most general that we are at liberty to make by virtue merely of that general
principle, and without examining the particular equations which relate to the particular problem
here considered. But when we set down these equations, we shall see that there are some further
changes which we may make without affecting our results, or at least without ceasing to be able to
infer the results which would be obtained on one system from those actually obtained on another.

In an apparatus recently constructed by Professor Willis, which will be described in detail in
the report of the commission, to which the reader has already been referred, the travelling weight
moves over a single central trial bar, and is attached to a horizontal arm which is moveable, with as
little friction as possible, about a fulerum carried by the carriage. In this form of the experiment,
the carriage serves merely to direct the weight, and moves on rails quite independent of the trial bar.

Vor. VIII. Parr V. 5A

726 Mr. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION

For the sake of greater generality I shall suppose the travelling weight, instead of being free, to
be attached in this manner to a carriage.

Let M be the mass of the weight, including the arm, & the radius of gyration of the whole
about the fulcrum, / the horizontal distance of the centre of gravity from the fulerum, / the hori-
zontal distance of the point of contact of the weight with the bridge, x, y the coordinates of that
point at the time ¢, &, » those of any element of the bridge, & the reaction of the bridge against
the weight, M’ the mass of the bridge, R’, R” the vertical pressures of the bridge at its two extremi-
ties, diminished by the statical pressures due to the weight of the bridge alone. Suppose, as
before, the deflection to be very small, and neglect its square.

By D’Alembert’s principle the effective moving forces reversed will be in statical equilibrium
with the impressed forces. Since the weight of the bridge is in equilibrium with the statical pres-
sures at the extremities, these forces may be left out in the equations of equilibrium, and the only
impressed forces we shall have to consider will be the weight of the travelling body and the reactions

M
due to the motion. ‘The mass of any element of the bridge will be aa very nearly; the

horizontal effective force of this element will be insensible, and the vertical effective force will be
-

Mt; ape and this force, being reversed, must be supposed to act vertically upwards.

2c dt

The curvature of the bridge being proportional to the moment of the bending forces, let the
reciprocal of the radius of curvature be equal to X multiplied by that moment. Let A, B be the
extremities of the bridge, P the point of contact of the bridge with the moving weight, Q any point
of the bridge between A and P. Then by considering the portion 4 Q of the bridge we get, taking
moments round Q,

Mu

~ Spr [REST TE EB) dha ceaneoe (88)

n being the same function of £ that » is of €. To determine KX, let § be the central statical
deflection produced by the weight Mg resting partly on the bridge and partly on the fulcrum, which
is equivalent to a weight ; Mg resting on the centre of the bridge. In this case we should have
: h
eee
de al

d
Integrating this equation twice, and observing that a =0 when € =c, and y = 0 when & = 0, and
that S' is the value of » when € =e, we get
61S
| Mghe*

Returning now to the bridge in its actual state, we get to determine R’, by taking moments about B,

me hie p @e-F) at miso Sap eral otinie ae sii 0)

R'.2c — R (2c -2) a oe
Eliminating R’ between (45) and (47), sete for K its value given by (46), and eliminating ¢

.. (46)

by the equation = = V, we get

da ne 318 (2
~ de ~ Mghe! 4

cf o rar + Abe fH (ee - yay |}... (48)

~~ .

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 727

This equation applies to any point of the bridge between A and P. To get the equation
which applies to any point between P and B, we should merely have to write 2¢-€ for &
2c — 2 for a.

’

If we suppose the fulcrum to be very nearly in the same horizontal plane with the point of
contact, the angle through which the travelling weight turns will be 2 very nearly; and we shall

have, to determine the motion of this weight,
2

d
Mk yi SU nh be SID vex cwrl « dbs sd «de _mase AAD

We have also the equations of condition,

7 = 0 when w = 0, for any value of £ from 0 to 2e;

7» =y when £ =a, for any value of # from 0 to 2c; (50)

n =0 when & = Oor = 2¢; y=0 and ©! =0 when @ =0,
Now the general equations (48), (or the equation answering to it which applies to the portion
PB of the bridge,) and (49), combined with the equations of condition (50), whether we can
manage them or not, are sufficient for the complete determination of the motion, it being under-
dy
a
mally to set down the equations of condition which express this cireumstance. Now the form of
the equations shews that, being once satisfied, they will continue to be satisfied provided n = y,
&«<w# ac, and

stood that » and vary continuously in passing from AP to PB, so that there is no occasion for-

y WR IlSM'V*y Aa)
LL Se a ee LG TY c RP.
So igi Meho? MEU G = Mehl RP

e
These variations give, on eliminating the variation of R,
ap Me AE Me
VS hl’ Me
Although g is of course practically constant, it has been retained in the variations because it
may be conceived to vary, and it is by no means essential to the success of the method that it
should be constant. he variations (51) shew that if we have any two systems in which the
ratio of Mk’ to M’I? is the same, and we conceive the travelling weights to move over the two
bridges respectively, with velocities ranging from 0 to «, the trajectories described in the one
case, and the deflections of the bridge, correspond exactly to the trajectories and deflections in the
other case, so that to pass from the one to the other, it will be sufficient to alter all horizontal
lines on the same scale as the length of the bridge, and all vertical lines on the same scale as the
central statical deflection. The velocity in the one system which corresponds to a given velocity in
the other is determined by the second of the variations (51).

yo S, clase ae sneeben: sono)

We may pass at once to the case of a free weight by putting h = k =/, which gives
ya S, V2 Sc ge®, Mc M’. ...ccccereeeeee (52)

The second of these variations shews that corresponding velocities in the two systems are those
which give the same value to the constant 3. When S «ce we get V* « ac, which agrees with
Art. 22.

In consequence of some recent experiments of Professor Willis’s, from which it appeared that

the deflection produced by a given weight travelling over the trial bar with a given velocity was
5Aa2

728 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION

in some cases increased by connecting a balanced lever with the centre of the bar, so as to increase
its inertia without increasing its weight, while in other cases the deflection was diminished, I have
been induced to attempt an approximate solution of the problem, taking into account the inertia of
the bridge. I find that when we replace each force acting on the bridge by a uniformly distributed
force of such an amount as to produce the same mean deflection as would be produced by the actual
force taken alone, which evidently cannot occasion any very material error, and when we moreover
neglect the difference between the pressure exerted by the travelling mass on the bridge and its
caene the equation admits of integration in finite terms.

Let the notation be the same as in the investigation which immediately precedes; only, for
simplicity’s sake, take the length of the bridge for unity, and suppose the travelling weight a heavy
particle. It will be easy in the end to restore the general unit of length if it should be desirable.
It will be requisite in the first place to investigate the relation between a force acting at a given
point of the bridge and the uniformly distributed foree which would produce the same mean de-
flection.

Let a force F act vertically downwards at a point of the bridge whose abscissa is w, and let
y be the deflection produced at that point. Then, £, » being the coordinates of any point of the

bridge, we get from (38)
av = y — on
So id Sem fe TG —}.

To obtain f;'ydé, we have only got to write 1—w in place of w Adding together the results,
: i : F
and observing that, according to a formula referred to in Art. 1, y= 16,8 ee . a (1-2), we
4

obtain
2SF "
Jondé = Me fa (1 —#) + a7 (1 = 2)"} 5.........460(53)

and this integral expresses the mean deflection produced by the force F, since the length of the
bridge is unity.

Now suppose the bridge subject to the action of a uniformly distributed force F’. In this case
we should have

~ Fer ERPE- iE E-L) FUE} - 2K E-6)

dyn ‘
Integrating this equation twice, and observing that Ein = 0 when & = 4, and » =0 when = 0,

and that (46) gives, on putting 7=h and c= 4, = dss we obtain

Mg
a {
fs PME AE ct Ene anes gs estes stn G8)

This equation gives ye the mean deflection

2SF"
: — —————— ae Sis Rade seas ees v'e'4 Se re rors ©
Jondé 5Mg' 376 (55)

and equating the mean deflections produced by the force F acting at the point whose abscissa is a,
and by the uniformly distributed force F’, we get F’ = uF, where

u = 5a (1— 0) + 5a (1 —2)?........0. Re cisi ateseee nis (56)

Putting » for the mean deflection, expressing J” in terms of », and slightly modifying the form

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 729

of the quantity within parentheses in (54), we get for the equation to the bridge when at rest
under the action of any uniformly distributed force

7= 5nfE(l -6+F0- eG) 2 ieree eee reer eee eee (57)

If D be the central deflection, 7 = D when & = 4; so that D : w :: 25: 16.
Now suppose the bridge in motion, with the mass M travelling over it, and let #, y be the
coordinates of M. As before, the bridge would be in equilibrium under the action of the force

2
M ( = = acting vertically downwards at the point whose abscissa is w, and the system of forces

d? ; 2
such as M'dé. ae acting vertically upwards at the several elements of the bridge. According
to the hypothesis adopted, the former force may be replaced by a uniformly distributed force the
value of which will be obtained by multiplying by w, and each force of the latter system may be
replaced by a uniformly distributed force obtained by multiplying by w’, where w’ is what w
becomes when E is put for w Hence if F, be the whole uniformly distributed force we have

d? (Pk
Fi=M(g- TY) u- a f Sp eee eee (58)

0

Now according to our hypothesis the bridge must always have the form which it would assume
under the action of a uniformly distributed force; and therefore, if ~ be the mean deflection at
the time ¢, (57) will be the equation to the bridge at that instant. Moreover, since the point (, y)
is a point in the bridge, we must have 7 = y when & = x, whence y=. We have also

ME hl liP all ,.iyptidtig yg ty Butlin, 155 d?
Teg Have. ae wf gat (sa rat MUS Se -
5Mg
2s

We get from (55), F, = x. Making these various substitutions in (58), and replacing

d d . : : P :
— by V—, we get for the differential equation of motion
dt dx
5Mg uu 155 Op
—~— p= Mgu - UV? u—_ —- —, M'V* —. seneeab
as he ! da* 126 polly dx? (59)

Since » is comparable with §, the several terms of this equation are comparable with
Mg, Mg, MV’S, MV’S,

respectively. If then V2.8 be small compared with g, and likewise M7 small compared with MW’,
we may neglect the third term, while we retain the others. This term, it is to be observed, ex-
presses the difference between the pressure on the bridge and the weight of the travelling mass.
Ve Shae a
g 168
Hence the conditions under which we are at liberty to neglect the difference between the pressure
on the bridge and the weight of the travelling mass are, jirst, that 8 be large, secondly, that the
mass of the travelling body be small compared with the mass of the bridge. If 6 be large, but
M be comparable with M/’, it is true that the third term in (59) will be small compared with the
leading terms; but then it will be comparable with the fourth, and the approximation adopted in
neglecting the third term alone would be faulty, in this way, that of two small terms comparable
with each other, one would be retained while the other was neglected. Hence, although the ab-
solute error of our results would be but small, it would be comparable with the difference between
the results actually obtained and those which would be obtained on the supposition that the
travelling mass moyed with an infinitely small velocity.

Since c = 1, we have , which will be small when # is large, or even moderately large.

eh)

730 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION, &c.

Neglecting the third term in equation (59), and putting for w its value, we get
d?
= + Qe = 2g) (at 22> Puts) oor sceeeo- oa ehatieaeiees (60)
where
_ 63 Mg 1008 MB
= 31 WPS = 31M’ tee cceeees

The linear equation (60) is easily integrated. Integrating, and determining the arbitrary con-

ecccniciots coc (ih)

+ d
stants by the conditions that » = 0, and = = 0, when w=0, we get
v

d

12.” 12 sin qv 24
na2s{a'— 2a8 — 7 i (: + =) (#- q ) te ae - cos ga) (62)

Ue q
and we have for the equation to the trajectory
y = 5p (7 — 20° + oo) = 5 (X +X)... 0c eee eee (63)

where as before X =a (1-—~).

When V=0, g=0, and we get from (62), (63), for the approximate equation to the equi-
librium trajectory,
Bil OW MCN HEN eseecicct su tece- ce cactvesces (0%)

whereas the true equation is

Since the forms of these equations are very different, it will be proper to verify the assertion
that (64) is in fact an approximation to (65). Since the curves represented by these equations are
both symmetrical with respect to the centre of the bridge, it will be sufficient to consider values of
x from 0 to 4, to which correspond values of X ranging from 0 to +, Denoting the error of the
formula (64), that is the excess of the y in (64) over the y in (65), by Sd, we have

8 = — 6X" + 20X3 + 10X%,
dé dX

—=4(-34+15X + 10X" —-.
da ( rd z Jes da

GB ;
Equating °F to zero, we get X=0, w=0, §8=0, a maximum; X=.1787, w =.233,
av

6 = — .067, nearly, a minimum; and w= 4, 5 = — .023, nearly, a maximum. Hence the greatest
error in the approximate value of the ordinate of the equilibrium trajectory is equal to about the

one-fifteenth of S.
Putting p=" +m, Y¥=Y + Yi. Where jy, Yo are the values of u, y for y= ©, we have

12 1 12\ . 24
fy 2s {Zo a) > 7) eka e qa cos ga) f scales (66)
y= Sat ives) he aH (ia) inj nos cone vinvdblosbedd vanies sce ob decnces (OT)

The values of 4, and y, may be calculated from these formule for different values of g, and
they are then to be added to the values of 4, yp, respectively, which have to be calculated once
for all. If instead of the mean deflection «1 we wish to employ the central deflection D, we have
only got to multiply the second sides of equations (62), (66) by #5, and those of (63), (67) by 38,
and to write D for ». The following table contains the values of the ratios of D and y to S for ten
different values of g, as well as for the limiting value g= ©, which belongs to the equilibrium
trajectory.

TABLE IIt.

9
Values of 2 when <2 is equal to
Ss 7

013} .022] .027 087) .053} .081] .117] .158 239 | .307
028! .048] .075] .108] .146 234) .327| 412] .580] .449
052] .099] .159] .931 309 | .469| .607} .696! .707 580
093] .177| .285! .406] .591 -746| .871 -884) .707 | .696
144) 1282] 451] .626| .7387 1,003 | 1.031 915] .689 | .794
214) .418] 650) .871 | 1.045 1.180/ 1.052} 845] .gi4 873
578 | .870} 1.115 | 1.265 | 1.938 967 -796 | 1.017 | .930
-757 | 1.097} 1.332 | 1.412 1.178] .859] .856 1.097 | .965
“947 | 1.310] 1.492 | 1.460] 1.036] .s19 1.004} .991| .977

1.139 | 1.491] 1.574}1.403| .870| .s6o 1.127] .862! .965
1.321 | 1.619} 1.562 | 1.250 739] .969 | 1.115} .872| .930
1.482 | 1.681} 1.454] 1.097] 682 1.054! .948] .959 | .873
1.609 | 1.663} 1.257 -769| .695 | 1.031 -718| .924) .794
1.691} 1.560} .990] .517 746] .869| .549| .707 696
1.717/1.371] .677| .803] .777 604] .499| .472| .580
1.681] 1.106] .350 149] .733| .325 -516| .384| .449
1.588} .776| .0387] .064| .579 -117/ .477| .885 | .307
1.402] .400] —.234] .025] .821] .091 -296| .276 | .156
1.158} .000} —.446} .019] .000! .001 —.001} .000]} .000

29

Values of % when =!
RY 7

is equal to

Meo? lac.) 4, |..5.(.8 8 | 10] 12] 16] »
——_ a ———— | | eee | ease,
:00].000} .000} .000] .000 .000} .000}| .000 .000 000} .000| .000
-051.001! .001} .001] .001 001) .001} .002} .003| .004! .006 025
+10} .003! .004] .007] .008 012] .017| .025 .037 050] .075| .096
-15}.008} 013] .022] .034} .050] .067]| .108 150] .190} .244] .207
-20].015| .081| .059! .095| .187] .184 279| .360] .414] .420] 844
-25 | .029| -056] .126] .203] .290] .378 532] .621|/ .630] .504] .496
-30].045| .117| .230| .366] .509| .640] .814 839] .744] 560] .646
+85 | .063| .191] .374] .581 -778 | .934|1.054] .940] .755 -727 | .780
40} .096| .285) .550| .828]| 1.062] 1.205 1.178] .921 -759| .969 | .886
451.133 | .394] .748] 1.085] 1.316 1.395 | 1.164] .849} .846] 1.084 +954
-50].169| .516| .947]| 1.310 1.492 | 1.460] 1.036] .812| 1.004 -991 | .977
-557.210| .632 | 1.126] 1.473] 1.555 1.587] .860] .850] 1.114] .859 O54
60] 244! .739] 1.258 | 1.542 1,487] 1.191] .704] .923] 1.062! .830] .886
65] .274| .816] 1.325] 1.502 | 1.300 917] .609| .942 848 | .857] .780
70] .292| .854| 1.308 | 1.352] 1.022] .626 505 | 839} .584) .752| .646
-75} .298 | .842 | 1.205) 1.111 -705} .869| .582}| .619 ‘391 | .488 | 496
80] .282| .770] 1.020] .814] .402] .180] .4G2 859} .2907| .280]| .844
85] .245| .644] 774] .509 161} .069] .887| .149 .237| .178} 207
90] .184] 463] .498] .244] .012] .020 -182| .087 -150} .121 | 096
95] .103}) .243) .224| .064] -.037] .004 051} 008 047 | .044 ] 025
1.00] 000} .000} .000] .000 000} .000) .000! .000 000} 000 | 000

732 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION

The numerical results contained in Table IIT. are represented graphically in figs. 2 and 3 of the
accompanying plate, where however some of the curves are left out, in order to prevent confusion in

A A 2
the figures. In these figures the numbers written against the several curves are the values of 22
Tv

to which the curves respectively belong, the symbol o being written against the equilibrium
curves. Fig. 2 represents the trajectory of the body for different values of q, and will be understood
without further explanation. In the curves of fig. 3, the ordinate represents the deflection of the
centre of the bridge when the moving body has travelled over a distance represented by the abscissa.
Fig. 1, which represents the trajectories described when the mass of the bridge is neglected,
is here given for the sake of comparison with fig.2. The numbers in fig. 1, refer to the
values of 8. The equilibrium curve represented in this figure is the true equilibrium trajectory
expressed by equation (65), whereas the equilibrium curve represented in fig. 2 is the approximate
equilibrium trajectory expressed by equation (64). In fig. 1, the body is represented as flying
off near the second extremity of the bridge, which is in fact the case. The numerous small
oscillations which would take place if the body were held down to the bridge could not be properly
represented in the figure without using a much larger scale. The reader is however requested
to bear in mind the existence of these oscillations, as indicated by the analysis, because, if the ratio
of M to M’ altered continuously from © to 0, they woill probably pass continuously into the
oscillations which are so conspicuous in the case of the fa ger values of q in fig. 2. Thus the
consideration of these insignificant oscillations which, strictly speaking, belong to fig. 1, aids us in
mentally filling up the gap which corresponds to the cases in which the ratio of M to M’ is neither
very small nor very large.

As everything depends on the value of q, in the approximate investigation in which the inertia
of the bridge is taken into account, it will be proper to consider further the meaning of this constant.
In the first place it is to be observed that although M appears in equation (61), q is really indepen-
dent of the mass of the travelling body. For, when M alone varies, (3 varies inversely as 9, and S
varies directly as M, so that q remains constant. To get rid of the apparent dependance of g on M,
let 8, be the central statical deflection produced by a mass equal to that of the bridge, and at the
same time restore the general unit of length. If # continue to denote the ratio of the abscissa of
the body to the length of the bridge, q will be numerical, and therefore, to restore the general unit
of length, it will be sufficient to take the general expression (5) for 3. Let moreover 7 be the
time the body takes to travel over the bridge, so that 2c = Vr; then we get

63gr*
Tests,

If we suppose 7 expressed in seconds, and §, in inches, we must put g = 32.2 x 12 = 386,
nearly, and we get,

2

287

= age ewes
Conceive the mass M removed ; suppose the bridge depressed through a small space, and then
left to itself. The equation of motion will be got from (59) by putting M = 0, where M is not

,

divided by SS, and replacing pal

M d
5 by 5 and V e by aE We thus get

Cu Gs"
dt® a s,\"

and therefore, if P be the period of the motion, or twice the time of oscillation from rest to rest,

318
Sia Nee = Eset teas (70)

Hence the numbers 1, 2, 3, &c., written at the head of Table III. and against the curves of

.

“I

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 733

figs. 2 and 3, represent the number of quarter periods of oscillation of the bridge which elapse
during the passage of the body over it. This consideration will materially assist us in under-
standing the nature of the motion. It should be remarked too that q is increased by diminishing
either the velocity of the body or the inertia of the bridge.

In the trajectory 1, fig. 2, the ordinates are small because the body passed over before there was
time to produce much deflection in the bridge, at least except towards the end of the body’s course,
where even a large deflection of the bridge would produce only a small deflection of the body.
The corresponding deflection curve, (curve J, fig. 3,) shews that the bridge was depressed, and
that its deflection was rapidly increasing, when the body left it. When the body is made to move
with velocities successively one half and one third of the former velocity, more time is allowed for
deflecting the bridge, and the trajectories marked 2, 3, are described, in which the ordinates are far
larger than in that marked 1. The deflections too, as appears from fig. 3, are much larger than
before, or at least much larger than any deflection which was produced in the first case while the
body remained on the bridge. It appears from Table III, or from fig. 3, that the greatest de-
flection occurs in the case of the third curve, nearly, and that it exceeds the central statical
deflection by about three-fourths of the whole. When the velocity is considerably diminished, the
bridge has time to make several oscillations while the body is going over it. These oscillations
may be easily observed in fig. 3, and their effect on the form of the trajectory, which may indeed
be readily understood from fig. 3, will be seen on referring to fig. 2.

When q is large, as is the case in practice, it will be sufficient in equation (66) to retain only
the term which is divided by the first power of g. With this simplification we get

D, 254 25
ee =— SNUG PR RISE. coddcnséndaecosaeasassosnas| (ii)

GMI S 4 84

so that the central deflection is liable to be alternately increased and decreased by the fraction

25 2 * .
ah of the central statical deflection. By means of the expressions (61), (69), we get

It is to be remembered that in the latter of these expressions the units of space and time are
an inch and a second respectively. Since the difference between the pressure on the bridge and
weight of the body is neglected in the investigation in which the inertia of the bridge is considered,
it is evident that the result will be sensibly the same whether the bridge in its natural position be
straight, or be slightly raised towards the centre, or, as it is technically termed, cambered. ‘The
increase of deflection in the case first investigated would be diminished by a camber.

In this paper the problem has been worked out, or worked out approximately, only in the two
extreme cases in which the mass of the travelling body is infinitely great and infinitely small respect-
ively, compared with the mass of the bridge. The causes of the increase of deflection in these two
extreme cases are quite distinct. In the former case, the increase of deflection depends entirely on the
difference between the pressure on the bridge and the weight of the body, and may be regarded as
depending on the centrifugal force. In the latter, the effect depends on the manner in which the force,
regarded as a function of the time, is applied to the bridge. In practical cases the masses of the
body and of the bridge are generally comparable with each other, and the two effects are mixed
up in the actual result. Nevertheless, if we find that each effect, taken separately, is insensible, or
so small as to be of no practical importance, we may conclude without much fear of error that

Vor. VIII. Parr V. 5B

734 Mr. STOKES’S DISCUSSION OF A DIFFERENTIAL EQUATION

the actual effect is insignificant. Now we have seen that if we take only the most important terms,

: arm, 2 1 25 - :
the increase of deflection is measured by the fractions B and Ps of §. It is only when these fractions
q

are both small that we are at liberty to neglect all but the most important terms, but in practical
cases they are actually small. The magnitude of these fractions will enable us to judge of the
amount of the actual effect.

To take a numerical example lying within practical limits, let the span of a given bridge be 44
feet, and suppose a weight equal to 4 of the weight of the bridge to cause a deflection of + inch.
These are nearly the circumstances of the Ewell bridge, mentioned in the report of the com-
missioners. In this case, S, = 3 x .2=.15; and if the velocity be 44 feet in a second, or 30 miles an
hour, we have + = 1, and therefore from the second of the formulz (72),

25 T
— =.0434, g=72.1= 45.9 x —.
8q 4

The travelling load being supposed to produce a deflection of .2 inch, we have 3 = 127, a7 .0079.

Hence in this case the deflection due to the inertia of the bridge is between 5 and 6 times as great as
that obtained by considering the bridge as infinitely light, but in neither case is the deflection
important. With a velocity of 60 miles an hour the increase of deflection .0434,8 would be
doubled.

In the case of one of the long tubes of the Britannia bridge must be extremely large; but on
account of the enormous mass of the tube it might be feared that the effect of the inertia of the tube
itself would be of importance. To make a supposition every way disadvantageous, regard the tube
as unconnected with the rest of the structure, and suppose the weight of the whole train collected at
one point, The clear span of one of the great tubes is 460 feet, and the weight of the tube 1400
tons. When the platform on which the tube had been built was removed, the centre sank 10
inches, which was very nearly what had been calculated, so that the bottom became very nearly
straight, since, in anticipation of the deflection which would be produced by the weight of the tube
itself, it had been originally built curved upwards. Since a uniformly distributed weight produces
the same deflection as 2 ths of the same weight placed at the centre, we have in this case S$, = $x 10

E ? 3 s 460
= 16; and supposing the train to be going at the rate of 30 miles an hour, we have rt = ar 10.5,

5 - 25 ‘ a
nearly, Hence in this case — = .043, or =), nearly, so that the increase of deflection due to the
y 8q 23 ys

inertia of the bridge is unimportant.

In conclusion, it will be proper to state that this ‘* Addition” has been written on two or three
different occasions, as the reader will probably have perceived. It was not until a few days after
the reading of the paper itself that I perceived that the equation (16) was integrable in finite terms,
and consequently that the variables were separable in (4). I was led to try whether this might not
be the case in consequence of a remarkable numerical coincidence. This circumstance occasioned
the complete remodelling of the paper after the first six articles. I had previously obtained for the
calculation of 3 for values of # approaching 1, in which case the series (9) becomes inconvenient, series
proceeding according to ascending powers of 1 —, and involving two arbitrary constants. The
determination of these constants, which at first appeared to require the numerical calculation of five
series, had been made to depend on that of three only, which were ultimately geometric series with a
ratio equal to 4.

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 735

The fact of the integrability of equation (4) in the form given in art. 7, to which I had myself been
led from the circumstance above mentioned, has since been communicated to me by Mr. Cooper,
Fellow of St John’s College, through Mr. Adams, and by Professors Malmsten and A, F. Svanberg of

Upsala through Professor Thomson ; and I take this opportunity of thanking these mathematicians
for the communication.

G, G. STOKES.

PEMBROKE COLLEGE,
Oct. 22, 1849.

net edi Le eae ee Cer area

tech Bary neo! 1 dtoktw (er ton Kean etg eh
‘i aM, a at 8 bie ey
Bre: ete 1 il

ate « es sacha wires <4
aha

a Peis a

| Ws: ff ia pny tn

re : D! Whewell on the Intrinsic Equation of Curves.
i

=) G 7 8

\
(s=U sin 2) (s=T sin 29)
Z ste 70 Gb)
IA

ae
LE Z A eS VAY,
c
)

. Me + COS S
7) => HS
f2 + 7 cos s/

ah ets 2
p=— si 52a
a8

Cc

19

SS,

AM how fe.

CN)
R
>
Ss
o
— 7

10

(p~=sm s } = Sos
(g=7 sins) Pd (TW cos 8
BI? (p= 27 stn s)
Vis
=
|
| .
oe
== sin sta
} Per.
Ds le | 17 18 ie
Dp
19
ys | _ UT M=OSS_ 4, 7} p=sin sta
TE TU ISS. uf o== LUE LOSES: fn==) | P= 3 Taxmoossy* 2
(= 73 2 |
2 imamate aa 2 (1+mess) ry ;

2:
20 2 a 23
m+cos s* eS nae, TT Mercos s s =
9 LE p=siv— + — a satan p
(7m cos s* a)? i ai 2/2+7 cos sJ~ a

Cc

AdMOT AL

VpBU wane eV

LA : LHI ?? haa
YY

volo toe ad sik ty 2 a:

a

f
PLEO
“2

»
.
7 t 4
. A = Ld = re.
e * : : i
on
: ae
A ay 4
aa
7 « > 1 5
a se
. _ an
7
> P a
—_ ‘ we
1 ’ v
< ’ é 7
4 i
7 ? 7 = ,
a .!

“4
f |
{
fi i |
-
_ ’
* ; Ay
*;
Ml i.
a rs r
i * . “
1) »
. “J y
1
‘
- 4
. . ‘ -
¥ a i
, & f , ‘
‘
r ‘ f 7) . ‘
Ee . .
| ‘ 4 ? ry
‘
F i
i ¢
14
- sy = Pe ”
e 7,
. '
4 / we .
i q
ae RY
‘ ‘ + ‘ La
. ‘
‘ ‘ “i
' he
"" “
' ‘ ~
.
. ‘ ‘ _
’ ‘
‘
ad .

anhhen es eee 2 ‘

A AG Ae H Ie 3
f meee ~ tenant , 33 Fy s

Ann pa an:

RAT
A
BA npRBGRRRE O © ane

nn ; nt ee 2S Ramen
Pane nanmanreneet RAN "AN NANA ae A ane AN,
ea Oa ARMANI RAR
Leanannnnnneh ON ; Mh RRR AAANAN Ae
NAN ANS we LEM Dy \ WI A fat =
an nar gps <a AK r RRA A lal re am
eeren vr A ne ee i Vater rae Sai Soke PAN RR are K AA manne =< =
AAA W027 “35> => \' ee : A Ne Poe =
Re Aaa Al eS Se TRS, aitenihManethn Y MEA
A AM NP PF PE panne anne n cn nnnanm anahAnnnne: nen ps ie 5
RY nanny Sa a pk AN. =A oot DAK ENE Al ng, P<
\"' : \ ah oN Ss ») ew, Anne Annee eel AAA sNAAAAA Ve >
* \ y e & =
¥ a5

: phe x ESP 2 = FAIA
ieee aes

<a
~ 7 AN Ane
Egil! Dan onan
’ Val A f ALIN SING Danang? ANA ali
> 7 An ANON an gnent BREN A are
BO fe ARRAS a y ANA A =A 0 aft
2 Ren eR AR, Ra AAA ‘
on ROA A RAs A Spann
‘NARAARA AA ps anne eS : pe: basti ' |
Ann ir oe = ~oAN MAR AAD N Re ee
panier pane’ 27 IT a SSRN nn pneensen
SKRRN nanan. fg Nee A
eee ae > * benneie WOE
* ~csnntengel iS > 33 west tannin RAM Ll 2
fs iN > BS ve tat A, Piers AANA AAS aaa Sle:
NSF FW Danaanttnty a NN LE
Aaya VAS Now ew, Lt Te a Af 3 2 >)
. : AR: \ = nan nO, iS = \ Oy Ae if: 2 > 2
ARAN” Se anon, ns Pa ea >»
it &, WSAAaa MY Aig” & 2»?
watt hie Rereren a & 33
Y Ne ahion Darna i oD \\ a
YEA annh Hane At aAAn (i *
0 anaes Men mannan i 3 ans
at A wt . »)

‘

‘ ay

RA REREAD on Anant Aas aN oe
AA Annnnne a NiKtep RA Nan NONE
ARRRARAT RAZA Nnninn eee

. nee AN Wenhee Aahn
Ann ON ‘ pee \ANA f Wo
Banantanannnnne? 7S. a SS ; nn AS, Ny

»ARAAAAANA, A awn Fy eae en MAA peetitts \ ; ~ i oe:
Aa la ih D \ AAA AAAAN: Nts BAAN AA NAG ‘aa tM og 8
7 2 Nutrient NN ot ann

1 2 > 5 a Wh Anne ee b>:

SP A patter once ME EH

2 ij A\ A> ~—oS )) : 4

S WE yl ail wr 7 nt ch s 3 ))

Re aaa nate as »)

wees BOESA APR: - |

ERRRARRRIN All Ws

ey: B edo

Weana aoe apaatias 2 2d AARC canes.

= ign ee AA a
Sennen pent eRane

ret 9 we ~ a \ ; qe ‘
EN een Bes a “

OES HALL egestas
f Zyl / C— Orn aS
OA Ee Wek AZZAM nnn

BNE

= \ on eee 3
ZS aN Se rena eeene PCE
Fa Mint p a scco vn het orn gti

nN Nn a
Or AAS
i wes
: A aes
zy eee p Ngires 2 AH BAY Y aaa.
A oRAt Pe Renn > 4, 4, 2 . : \ aca \ ’
a AD Paecanaeie pa RAR AAR / V7, 5 Pg > hy As a
mi eA AARON 3 Wrannns? BAS a as Keg 3 33 AP anne .
fl — 9 ; :
Phen NM Ne \ SS PP Aiea
4 acs =. Z : 2) : re a
BORO E COCR An onGue emotes Shan ru pe er eae ane
ARA RARARA RA 6 ae a” A IA PnAR Ninnd rhage ee annt rete
AAARAN (- a aS nt AQ AAR At ARN RRR NS neeaceret
Ps, Fog Poe wh “nar CANA ANA, ~~ Se AnAnane Cal (at a)
ae S-5 la $ AR A nm
<a SNogat SRARARA nn . ne ear Annals MAN : PARR RAR. eg

51/4 ~ \ Y WR,
ey = \ MA AAO Arn AMAR AR Ar

rae 4
“Ann” My We : . ep Py A Ann Raneeete pera \ Natnnnnnt’ Wa
NN MN a9, SY got MMe

a4 Apres a5
vat Pye ‘ea 39
Ji Ming nN 4
Z, a ne PW a 2
Ligh hes pat A
D Yi YAS
3 nat
Jd ar.
4 las rt ~
, Zo pannnnnns t Aine ee
: rane
an ;
~ — n ann’ sa: SAR
, ARETE AOS ARARRE” Peet
i _ FEN PLNONIEN NES penne
a“ N ,
ad
iRehinreiee, . Cg ik \ r A
‘ >> ;
SCR ARKRAE AAA > \ la k
Sat SORE CGS af Wi AA S*:> , = yee SN )h My Ahrens’ ene nee La in oe v
14
aude. “a a, => Wala neces a ae ‘Mey je

= “ eee
eran may - ei aenpeteat

nt

Cer
eae

ecg oie opin
NER aha a

— wae”
aa ane

Tania

teen