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ONS

CAMBRIDGE
PHILOSOPHICAL SOCIETY.
ESTABLISHED NOVEMBER 15, 1819,

VOLUME THE TENTH.

DEIGHTON,

BELL

CAMBRIDGE:
Printed at the Binibersity Bress ; AND SOLD BY
AND CO. AND MACMILLAN AND BELL AND DALDY, LONDON.

CO. CAMBRIDGE;

M.DCCC.LXIV.

Cambridge :
PRINTED ΒΥ C. J. CLAY, M.A. AT THE UNIVERSITY PRESS.

CONTENTS.

PART I.
PAGE
No. I. On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space, with Applications. By R. B. Haywarp, M.A., Fellow of St John’s College, Reader in Natural Philosophy in the University of Durham........0..0...64-

iF On the question, What is the Solution of a Differential Equation? A Supplement to the third section of a paper, On some points of the Integral Calculus, printed in Vol. IX. Part 11. By Aveausrus De Morean, of
Trinity College, Vice-President of the Royal Astronomical Society, and Professor of Mathematics in University College, London .......νννννννννννςς

ITI. On Faraday’s Lines of Force. By J, Currk Maxwett, B.A., Fellow of _ Primity College, Cambridge .........6ἀεκεονδνονννοννεννννοενονννννννκνννι νsnsνaeνeees 27

The Structure of the Athenian Trireme ; considered with reference to certain difficulties of interpretation. By J. W. Donaupson, D.D., late Fellow of DTsimnty College, Comrade TAC ας. δος. ETN ἐϑρλενον c cba deen deseceereesoes 84

Of the Platonic Theory of Ideas. By W. Wuewet, D.D., Master of Trametes: College, ΟΡ ge x3des Sisko ευνκεκνελον ss¥ oc¥ cn ονῷνορευξεν ὀξο νον cave ee 94

On the Discontinuity of Arbitrary Constants which appear in Divergent

Developments. By G. G. Sroxes, M.A., D.C.L., Sec. R.S., Fellow of

Pembroke College, and Lucasian Professor of Mathematics in the Univer-

BAU OF OCUNDTSAIG<nbn sigs sen ang ka ene s eee ΡΜ ΠΥ

ΡΥ

105

Vil. On the Beats of Imperfect Consonances. By Avucustus Dz Moreay,
F.R.AS., of Trinity College, Professor of Mathematics. in University Rolheges Londarar ':. .acseke Wud. ἀἀφυσοβον,ελλ, a. Navies hate IB. δοιπορον ευνζοοςς 129

VITE

On the Genwineness of the Sophista of Plato, and on some of its philosophical bearings. By W. H. Tuompson, M.A., Fellow of Trinity College, and Regius Professor of Greek .....cccccevcevevevsees eR stesatate capeteroaeatasoecdssoeae 146

IX. On the Substitution of Methods founded on Ordinary Geometry for Methods based on the General Doctrine of Proportions, in the Treatment of some Geometrical Problems. By G. B. Arry, Esq., Astronomer Royal . .........' 166

On the Syllogism, No. IIL, and. on Logie in general. By Aveustus DE Morean, F.R.A.S., of Trinity College, Professor of Mathematics in Cnitvorsie:OollegeseL odo τινάζει οὐοιουδυν νυν eVueesadsa dw εὐνρον νον δένονονοννον δος

On the Statue of Solon mentioned by Aischines and Demosthenes. By J. W. Donatpson, D.D., Vice-President of the Society ...ο.νο. νννν. ννο. νννν. ννν. ννννον

ΧΙ]. Instances of remarkable Abnormities in the Voluntary Muscles. By G. Ἐπ Paget, M.D., F.R.C.P., late Fellow of Gonville and Caius College .........-

XIII. On Organic Polarity. By H. F. Baxter, Esq., M.R.O.S.L, «ονννννννννοννεννεν

XIV.

A Proof of the Existence of a Root in every Algebraic Equation: with an examination and extension of Cauchy's Theorem on Imaginary Roots, and Remarks on the Proofs of the existence of Roots given by Argand and by Mourey. By Aveustus DE Morean, F.R.A.S., of Trinity College, Professor of Mathematics in University College, London........scecceeeseves

CONTENTS.

PART II.

NO. III.

PAGE

On a Chart and Diagram for facilitating Great-Circle Sailing. By Hucu Goprray, M.A., St John’s College .......ccssesecssereeneesereneens HER vices.

271

Suggestion of a Proof of the Theorem that every Algebraic Equation has
a Root. By G. B. Ary, Esq., Astronomer Roydl............c00.eeseeeeeees 283

On the General Principles of which the Composition or Aggregation of

Forces is a Consequence. By Avaustus Dz Morean, F.R.AS., of

Trinity College, Professor of Mathematics in University College, London

290

IV. On Plato’s Cosmical System as exhibited in the Tenth Book of “ The Republic.” By J. W. Donaupson, D.D., Trinity College; Vice-Presi-
dent of the Society ...cccccedeccssccsbesscvesacttescvecsdertesetenscceseasscnseseeease

On the Origin and Proper Use of the word Araument. By J. W. Donatpson, D.D., late Fellow of Trinity College, Cambridge ..........-.-

Supplement to a Proof of the Theorem that every Algebraic Equation has a Root. By G. B. Atry, Esq., Astronomer Royal ........:.sssesereereeeess

VII. On the Syllogism, No. 1V., and on the Logie of Relations, By Avucustus Dr Morean, F.R.A.S., of Trinity College, Professor of Mathematics in University College, London .......s0ssecceeeceeetenesenerenseceeegeseaeeeaeees 331

ὙΠ. On the Motion of Beams and thin Elastic Rods. Sy J. H. Rours, M.A., Fellow of the Cambridge Philosophical Society........++:-sssscvecerersenetenes 359
ΙΧ. On a Metrical Latin Inscription copied by Mr BuaxesLey at Cirta and published in his “ Four Months in Algeria.” By H. A. J. Munro, M.A. Fellow of Trinity College .....1:ssecsssrcessesccnscenersoceescseenacencscaeserses 374
On the Theory of Errors of Observation. By Aveustus Dz Moreay,
F.R.AS., of Trinity College, Professor of Mathematics in University
College, London .......scsrtese teeenecsreneeccerassseseseccsssceaeeeeerercseeresecess 409

XI. On the Syllogism, No. V., and on various points of the Onymatie System. By Avevustus De Moreay, F.R.A.S., of Trinity College, Professor
of Mathematics in University College, London.........:.:sscsseeeeseesernneees

ADVERTISEMENT.
Taz Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors.
_ Tue Socrery takes this opportunity of expressing its grateful acknowledgments to the Synpics of the University Press, for their liberality in taking upon themselves the expense of printing this Volume
of the Transactions,

a we ΕἾ Pe sitsbets
sews bad

TRANSACTIONS
STOR
CAMBRIDGE
PHILOSOPHICAL SOCIETY.

VOLUME X. PART 1.

CAMBRIDGE:

PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS;

AND SOLD BY

3

DEIGHTON, BELL AND CO. AND MACMILLAN AND CO. CAMBRIDGE;

BELL AND DALDY, LONDON.

M. DOCC, LVIIL.

I. On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space, with Applications. By R. B. Haywarp, M.A. Fellow of St John’s College, Reader in Natural Philosophy in the University of Durham.
[Read Feb. 25, 1856.]

*,..gardons-nous de croire qu'une science soit faite quand on l’a réduite ἃ des formules analytiques. Rien ne nous dispense d’étudier les choses en elles-mémes, et de nous bien rendre compte des idées qui font l’objet de nos spéculations.” Pornsor.
“_.¢’est une remarque que nous pouvons faire dans toutes nos recherches mathématiques; ces quantités auxiliaires, ces calculs longs et difficiles οὐ l’on se trouve entrainé, y sont presque toujours la preuve que notre esprit n'a point, dés le commencement, considéré les choses en elles-mémes et d'une vue assez directe, puisqu’il nous faut tant d'artifices et de détours pour y arriver; tandis que tout s’abrége et se simplifie sitét qu’on se place au vrai point de vue.” bid.

Tue general principles, which I have endeavoured to keep in view in the investigations of

this paper, are those contained in the above quotations from Poinsot. My object is not so

much to obtain new results, as to regard old ones from a point of view which renders all

our equations directly significant, and to develop a corresponding method, by which these

equations result directly from one central principle instead of being (as is commonly the case)

deduced by long processes of transformation and elimination from certain fundamental

equations, in which that principle has been embodied.

The frequent occurrence of exactly corresponding equations, (though this correspondence is

sometimes disguised under a different mode of expression) in many investigations of Kinematics

and Dynamics suggests the inquiry whether they do not result from some common principle,

from which they may be deduced once for all. An investigation based on this idea forms the

first part of this paper, in which it will be shewn how the variations of any magnitude,

which is capable of representation by a line of definite length in a definite direction and is

subject to the parallelogrammic law of combination, maybe simply and directly estimated

relatively to any axes whatever. The second part is devoted to the general problem of the

dynamics of a material system, treated in that form which the previous Calculus suggests,

together with a development of the solution in the case of a body of invariable form.

Since whatever novelty of view is contained in this paper consists rather in the relation

of the details to the general method than in the details themselves, much that is familiar to

every student of Dynamics must be repeated in its proper place, but it is hoped that such

repetition will in general be compensated by a new or fuller significance being obtained. As

regards the problem of rotation, M. Poinsot’s solution in the “‘ Théorie de la Rotation” is so

Vou. X, Past E

1

ῷ

Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

complete and so entirely satisfies the conditions expressed in our quotations above from that work, as to leave nothing to be desired. But it does not appear to me that his method, which depends essentially on the summation of the centrifugal forces, is so widely applicable beyond the limits of this particular problem as that by which the same results are obtained in this paper: but be this as it may, any new point of view, if,a true one (‘vrai point de vue”) has its special advantages, and on this ground may claim some attention,

SECTION I.
The Method, with some kinematical Applications.
1. As we shall here be concerned only with the directions of lines in space, and not with their absolute positions, it will be convenient to suppose them all to pass through a common origin O, and to define the inclination of two lines as OP, OQ by the are PQ of the great circle, in which the plane POQ meets a sphere whose centre is O and radius constant. We shall also suppose any linear velocity, acceleration or force, represented by a length along OP, to tend from O towards P, and any angular velocity or the like, represented in like manner, to tend in such a direction about OP that, if OP were directed to the north pole, the direction of rotation would coincide with that of the diurnal motion of the heavens.
2. Let τὸ denote any magnitude, which can be completely represented by a certain length along the line OU, and which can be combined with a similar magnitude v along OV by means of a parallelogram, like the parallelogram of forces or velocities. Then of course w may be resolved in different directions by the same principles, and thus if we adopt rectangular
resolution, the resolved part of w along OP will be wcos UP, which may be denoted by w,.
We proceed to inquire how w, varies by a change in the position of OP.

3. Suppose OP to be a line moving in any manner about OQ, and that it shifts from OP

to a consecutive position OP’ in the time d¢; and conceive that this motion arises from an

angular velocity Q about an instantaneous axis OJ. Resolve Q into its components Q cos 77

about OU and Qsin IU about a line in the plane JOU, perpendicular to OU: and farther

resolve this latter component in the plane perpendicular to OU into the components Q sin JU

.cos 10} in the plane POU and Qsin JU. sin JUP perpendicular to the same plane.

Then the component in OU and that perpendicular to it in the plane POU produce

displacements of P perpendicular to the are UP, and consequently do not ultimately alter the

length of the arc UP, so that uw, remains ultimately unchanged so far as the motion of OP is

due to these components: but the component perpendicular to the plane POU increases UP

by the are Qsin JU. sin 10}. dt, and therefore the increment of uw, from this component (being

equal to —- w.sin UP.d. UP) is

- uQsin UP. sin IU.sin UP.dt.

But the other increments being zero, this is the ¢ofal increment of w,, wherefore we have

d.u, dt

=

-- uQsin

JU.sin

UP.

sin JUP...(A).

VELOCITIES, ἄς. WITH RESPECT TO AXES MOVEABLE IN SPACE.

3

d.

ὃ

It is well to observe that = vanishes, when the three axes OJ, OU, OP lie in the same

plane, and in particular when two of them coincide, as is evident from the above equations, or from considerations similar to those by which it was obtained.

4. Inthe above investigation we have supposed w to be constant both in direction and intensity ;let us now suppose w to vary in both respects with the time (6). The change in τὸ in the time dé may be conceived to arise from its composition with the quantity fd¢ in the line OF, and f may properly be called the acceleration of τὸ at the time ¢. Now fdt may be resolved in the plane UOF into Κ΄. dt cos FU along OU and f. dt sin FU perpendicular to OU, and the components of τὸ + dw will therefore be w+ f.dtcos FU along OU and f.d¢ sin FU perpendicular to OU; whence, if ἀφ denote the angle through which OU shifts in the time αὐ towards OF, it will readily be seen that ultimately
ὩduΣ =fcosFU, ὦ ἊἀφΣ =fsin FU...(B).

If then the acceleration f be known both as to direction and intensity at every instant, the motion of OU and the variation in the intensity of τ may be determined by these last equations. In fact, the point UY on the sphere of reference continually follows the point F
d with the velocity τὸ so that the problem of determining U’s path is the same as the old
problem of the path described by a dog always running towards his master who is himself in motion, the only difference being that the path is here on a sphere instead of a plane.
5. Next for the variation of u,, when w varies with the time. It is plain that wu, varies from two causes; first, by reason of the acceleration f, and secondly, by reason of the motion of OP due to the angular velocity Q about OJ, and that the total variation will be the sum of these two partial variations. Now the latter has been calculated above, and the former is obviously the resolved part of fd¢ along OP or f.dt cos FP, therefore we obtain the equation*
“d.at = fcos FP -- uQsin JU.sin UP.sin IUP,..(C).
This equation of course contains the previous equations (B): thus, if OP and OU coincide always, UP is always zero and the second side of (C) reduces to its first term: and again if OP be always in the plane FOU and perpendicular to OU, τι, is always zero, Q = 2d , LE:
and UP are quadrants, JUP a right angle, and FP the complement of F'U, and therefore, as above,

o=fsin FU- ut,
6. We may farther illustrate the application of equation (C) by supposing OP to coincide with certain other lines specially connected with OU and OF.

ἘΤῚ should be remarked here that.the angle JUP must be | Q about OJ causes the motion of P, resolved in the arc (7 P, to considered positive or negative, according as the positive rotation | be from or towards U.
1—2

4

Mr R. Β. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

Let Ὁ, σ΄, U" and F, F’, F” be three consecutive positions of U and F respectively, and

K, K’ those poles of FF’, F’F” respectively, (considered as arcs of great circles) about which positive rotation brings 1" to F’, and F’ to F”. We know that U’ lies on the arc UF between

U and F, and U” on the are U’F" between U’ and F’, Also it is plain that F” is the pole of KK’, and

therefore that KK’ measures the angle of contingence

between the consecutive elements FF", F’F”: in

fact, the loci of K and F are so connected that the

elementary ares of the one are equal to the angles

of contingence of the other, and vice versa.
Suppose the locus of F to be defined by ele-

ments, corresponding to what Dr Whewell has

F

named in plane curves intrinsic elements, that is,
by elements a, ¢ such that the elementary arc F'¥” = da, and the angle of contingence between FF’ and F’F" = de: and suppose the locus of U defined in like manner, so that UU’= ἀφ, and the angle of contingence FU'U" - ἄη. Also let UF =n, and angle ΓΕ -- ν.
Now let OP coincide always with OF. Then willw,=wcosm, and I being taken to
d coincide with k, Q= oe and therefore equation (3) becomes

d di (ὦ cosy)

= f-w—.dasin,w.sin:

KU sijn

KUF.

But sinkU.sin kUF = sinkFU= sin (=-»p )= COs ν,

dndd—dsu= fosp Η

therefore we obtain after reduction

du da

‘

GE de oY Ἔ Sit= Oveeeee (1).

Again let OP coincide always with OK, then

Up = ucos UK = usinu.cos UF K = usin μ. sin ν,

and I may be taken to coincide with ¥” or ultimately with F, so that (3) becomes,

;

de

(2 being = =)>

qd (sin: usin: ν) τα— wd=e si. n FU.sinἾ UK. sςin FUK

or after reduction
The equations

é€ .
δος 4 = - Ὁ --- Β1ὴ “.COSyp

du

Gri εν tania. . 4:

—dt * e-, ττ -=-:)}.--t-ττaς n+y+ a @) = 0...... .

A and (2) together ne tle two equations (B) serve to determine

(after

eliminating «4 and ») —τand oo, when "4—~» ~~€ > dt

are

gi. ven,

that

iF s, when

the

iὅ ntensi.ty

and

VELOCITIES, ἄς. WITH RESPECT TO AXES MOVEABLE IN SPACE.

5

variation in direction of the acceleration of τ are given for every instant.
from the triangle FU’F” ultimately

dy ᾿

a .

ee a —. 810 μιΞΞ ----- 8581

And we have also

to determine πὶ and therefore we have equations to determine the intensity and variation in

direction of τὸ itself. Hence we have obtained a solution of the problem, ‘‘ Given the path of

F and the variable intensity of f, to determine the path of U and the intensity of w,” the whole

being referred to intrinsic elements.

;

7. It will be useful to obtain results analogous to equation (C) for three rectangular axes in a somewhat different form. Of course these might be obtained from that equation itself, but it will be better to investigate them independently by the same kind of reasoning.
Let w,, Uy, τύ, denote the resolved parts of w along the moveable rectangular axes Ow, Oy Ox, and let Q,, Q, Q, and f,, f,, f, denote in like manner the resolved parts of Q and f. Now by reason of the acceleration f, τό, receives in the time dé the increment f,d¢: also Ox changes its position by reason of the rotations Q,, Q,, the first of which shifts it in the plane of xa through the angle Q,dt from O,, and the latter in the plane of wy through the angle Q,dt towards O,; and from the first of these causes w, receives the increment

τ; COS (=+ Q, 4)+ u, cos (Q,dt) -- τι,»

or — u,Q,dt ultimately, while from the second it receives the increment

U,y COS G - 0,4t)+ Uz cos (Q,dt) -- uz,

or τὸ,ἀξ ultimately. Hence the total increment of w,, being the sum of these partial

increments, we obtain the equation

:

or =f, + uy,Q, - u, Ay

Similarly for w,, τὸ, we should obtain
το κὸν ΩΝ 0,0, Γ st

d$uF]=f, + Uz Qy — Uy Qu,

1}

8. Τὸ illustrate the applicability of these last obtained equations, we will select a few particular kinematical problems.

a. Relative velocities of a point in motion with respect to revolving axes. From the nature of the quantity u, it will be seen that it may be taken to denote the radius vector OP of a point P, and τι,» uy, u, may then be replaced by the co-ordinates, w, y, x: also Ff, denoting the acceleration of u, will in this case denote the absolute velocity of P, and /,, fy, Ff. the absolute velocities resolved in the directions of the axes, which we will denote by »,, Vy v,. Then by the equations above we have three equations, of which the type is

6

Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

ἀν 1 Ox - τῶ

and which determine the relative velocities mp Ὥς = of the point with respect to the co-ordinate axes.

If the point be fixed relatively to the axes, and a, Y, % be its co-ordinates, the above

equation becomes

Oy = Qy.πο= Ὡς.Yos

one of a set of well known equations, determining the linear velocity of a point in a body revolving with given angular velocities.
If the point lie in the axis of w, so that y, x both vanish,

—=v, O0=0,-aQ,, 0=4,+ ay

In these, if δ᾽, y, x are in the directions of the radius vector, a perpendicular to it in the

vertical plane, and a perpendicular to this plane respectively, and if r, 0, @ denote radius

vector, altitude and azimuth, then

ἀφ

dé

ὥ τεῦ, Qy = τις cos 8; Q,= 7

whence

dr

αθ

d

τ ΣΉΝ soa haere υ, τ τοοθ 2,

the common expressions for the components, relatively to polar co-ordinates, of the velocity of a point.

ὃ. Accelerations, radial, transversal in the vertical plane, and perpendicular to that plane.

In our general formule wu will now denote a velocity, and f an acceleration strictly so called. And in this case

dr

dé

dp

τι wore uy = τος 0s

ἀφ.

ἀφ

dé

Q, = ποτὶ and, Oy = — τ; “956, Q,=7

wherefore, by equations (1) radial acceleration =f, = a2, - (,: 4 2 Foca ὃ | ‘

transversal acceleration in the vertical plane = Ἢ

-5dt(\+5d)t - (-rsin6. cox odft " - τdt“dt|

1d/,d\ υοἱ

de]?

ΞΞ =eel (¢PP alen) Ὁ 7 5159 ὁ con 0— 66} ’

azi:muthal. acceleration rat Pad πὲ[τος θ a ὧς (= τ σῶς θ -- Υ si: n widτ

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE.

7

= ab Wit a (,ps cos ν9ὰ.2Op).

c. Let the axes of a, y, x be always parallel to the tangent, principal normal and normal to the osculating plane of any curve. Then

ds
Ue

Uy = 0,

u, = 0,

dr
Q,= 755°

Q, = 0,

de
Q,= 7?

where de, dr denote respectively the angle between consecutive tangents, and that between consecutive osculating planes.

Hence

tangential acceleration = f, = aF. ;

acceloeeration iCen prisnceicpaal normal 1 =f, = Ἔ ds Κde_ (eah ) edree,_ :1 (s5al):3 acceleration in normal to osculating plane =f, = 0.

SECTION II.
Dynamical Applications.
9. I propose here to consider the problem of the motion of any material system, so far as it depends on external forces only, and to develop the solution in that case in which the entire | motion is determined by these forces, namely, in the case of an invariable system.
10. This problem naturally resolves itself into two: for, since every system of forces is reducible to a single force and a single couple, we have to investigate the effects of that force, and the effects of that couple. Now we know that the resultant force determines the motion of the centre of gravity of the system, be the constitution of the system what it may. In like manner the resultant couple determines something relatively to the motion of the system about its centre of gravity, which in the case of an invariable system defines its motion of rotation about that point, but which in other cases is not usually recognised as a definite objective magnitude, and has therefore no received name. This defect will be remedied by adopting momentum as the intermediate term between force and velocity, and by regarding as distinct steps the passage from force to momentum and that from momentum to velocity. In accordance with this idea we proceed to shew that as in our first problem we shall be concerned with the magnitudes, force, linear momentum or momentum of translation, and linear velocity or velocity of translation, so in the other we shall be concerned with the corresponding magnitudes, couple, angular momentum or momentum of rotation, and angular velocity or velocity of rotation ; and that, as all these magnitudes possess the properties characteristic of the magnitude ὦ in the previous section, the Calculus there developed is applicable to them.

8

Mr R. Β. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

11. Consider a material system at any instant of its motion. Tach particle is moving with a definite momentum in a definite direction, which may be resolved into components in given directions in the same manner as a velocity or a force. Let this momentum be resolved
in the direction of a given axis OP, and its moment about that axis taken, the resolved part may be called the linear momentum, and the moment the angular momentum, of the particle
relatively to the axis OP. Let the same be done for every particle of the system, and the sums of their linear and angular momenta taken, these sums may then be called respectively the linear and angular momenta of the system relatively to the axis OP.

12, Let the linear momenta relatively to the three axes Ov, Oy, Ox be denoted by u,, uy, w,, and the corresponding angular momenta by h,, h,, h, respectively ;then it may easily be shewn that the linear momentum relatively to the axis, whose direction-cosines are /, m, m, is
lu, + mu, + NU,, and that the angular momentum relatively to the same axis is
th, + mh, + nh, The first expression will be a maximum, and equal to {u,” + u,’ + u7}3, when
LiMim Up?: Uy :πὸ} and if this be denoted by τι, it is plain that the linear momentum along any line inclined to the direction of w at an angle @ will be wcos@. Hence we may regard the whole linear momentum of the system as equivalent to the single linear momentum wu determinate in intensity and direction.
In like manner we may conclude that the whole angular momentum is reducible to a single angular momentum A determinate in intensity and direction.
13. Thus, just as a system of forces is reducible to a single force and a single couple, the momenta of the several particles of a system are reducible to a single linear and a single angular momentum, which we shall speak of as the linear and angular momenta of the system. It is to be observed that the linear momentum w is independent of the origin O both as regards direction and intensity, but the angular momentum ἢ is in both respects dependent on the
position of O, Also it may be proved, as in the case of a system of forces, that the angular momentum ἢ remains constant, while O moves along the direction of the linear momentum u, but changes, as Ὁ moves in any other direction; and finally, that its intensity will be a minimum and its direction coincident with that of u, when O lies upon a certain determinate line, which (from analogy) may be termed the central axis of momenta.

14, Now let us consider the changes in the linear and angular momenta, as the time
changes, when the system is acted on by any forces.
In the time dt any force P generates in the particle on which it acts the momentum Pdt,
and these momenta, being resolved and summed as was done above, will give rise to a linear
momentum Rdé in the direction of the resultant force R of the forces (P), and an angular momentum Gd¢ relatively to the axis of the resultant couple G of the same forces, Since however the internal forces consist of pairs of equal and opposite forces in the same straight line, by the nature of action and reaction, the momenta produced by them will vanish in the

VELOCITIES, ἃς. WITH RESPECT TO AXES MOVEABLE IN SPACE,

9

summation over the whole system; we may therefore regard R and G as the resultant force and resultant couple of the ewternal forces. Then the linear momentum w along the line OU
must be compounded with the linear momentum #d¢ in the line OR in order to obtain its value
at the time ¢ + dt: and in like manner the angular momentum hf relatively to the axis OH must be compounded with the angular momentum Gd¢ relatively to the axis OG.

15. Hence the method of the previous section applies to momenta of both kinds, replacing f in one case by R and in the other case by G. Thus the equations (B) give us

du qi = Roo RU;

d

ε

uP = Rsin RU,

where ἀφ is the are through which U moves towards # in the time dt: and

dh
αἰ F008 GH,

hoday, = Gainἡ GH,

where dy is the are through which H moves towards G in the time dé.

Also for fixed rectangular axes, with respect to which the components of R and G@ are

X, Y, Z and L, M, N respectively, it is plain from the above reasoning that we should have
diy yyy =ty

deck

we. cat

dt .

dh, "

dh, _

ee

—

dh, eo.

which are really the six fundamental equations of motion of our works on Dynamics. For rectangular axes moveable about O, the equations (Z) of the last section furnish
two sets of three equations, of which the types are

du, dt

=

Χ σοι, -- τ,»

dh, Leh

dt be]

a νῶ, =

"Ὧν.

16. If the system be acted on by no external forces, it follows that both w and h

are constant in intensity and invariable in direction. This result might by analogy be

named the principle of the Conservation of Momentum.

This principle, as applied to linear momentum, is obviously equivalent to the prin-

ciple of the conservation of motion of the centre of gravity: as applied to angular

momentum, the constancy of direction of the axis of h and therefore of a plane perpen-

dicular to it shews that there is an invariable axis or plane, while the constancy of its

intensity and therefore of its resolved part in any fixed direction is equivalent to the asser-

tion of the truth of the principle of the conservation of areas for any fixed axis.

It may also be noted that there is an infinite number of invariable axes, and that,

if the origin O be taken on the central axis of momenta, the corresponding invariable

axis will coincide with the central axis, and the angular momentum about it will then be

Vor. X. Part I.

2

10

Mr R. Β. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

a minimum: also that for any other position of the origin the direction of the invariable axis and the intensity of the momentum about it will depend upon the position of the line, parallel to the central axis, in which the origin lies, just as in the corresponding propositions relative to couples,

17. Any one of the different sets of equations in § (15) may be used to determine completely «w and h, when the forces are given or vice versa. It is to be observed that the equations involving A, refer either to a fixed origin, or to an origin, whose motion is always in the instantaneous direction of u the linear momentum, for, as we saw, a change of the origin in this direction does not produce a change in h, as its change in any other direction does. It would be easy to introduce terms depending on the motion of the origin; in the last set of equations, for instance, if a,a,ya, denote the linear velocities of the origin in the directions of the axes, the equation for h, becomes
dh, ; Ξ 7 hyQ, -- hQy + τἰγας -- τὐ,αν»
The equations involving uw, are entirely independent of the origin, and will there-
fore not be affected, however the origin be supposed to move.

18. It appears then that the linear and angular momenta are determined solely by the external forces acting on the system, and not on the system itself otherwise than the forces themselves depend on it: in fact, they are simply the accumulated effects of the forces and the initial momenta. To proceed to the determination of the actual motion of the system
from these momenta, the system must be particularised, and as one system may differ from another both as to the quantity of matter included in it, and as to its arrangement,
we may consider separately how much farther particularisation in either respect will enable us to carry our results.

19. If the quantity of matter or mass of the whole system be given, it is well known
that the linear momentum of the system is that of its whole mass collected at its centre
of gravity, so that, M denoting this mass, the velocity of the centre of gravity is a in
the direction of the linear momentum: thus the motion of a certain point definitely related to the system is obtained, and this is usually regarded as defining its motion of translation. For any other point definitely related to the system, the motion will in general depend also on h and the arrangement of its matter.

20. If then the translation of the system be referred to its centre of gravity, its motion about the centre of gravity will depend solely on ὦ and the arrangement of its mass; for the direction of motion of the centre of gravity being that of the linear momentum, h referred to that point as origin will be independent of w. Now the arrangement of a system
of matter. may be either permanent or variable. If the former, it is spoken of as a body

VELOCITIES, ἄς. WITH RESPECT TO AXES MOVEABLE IN SPACE.

11

or system of invariable form*, and the investigation of its motion about the centre of
gravity requires only the determination of its axis of rotation and the intensity of rotation about that axis.
If the arrangement be variable, the laws of its variation must be given, and according
to the number of possible laws will be the number of different solutions of the problem: here then the problem diverges into special problems; such as that of the motion of a body expanding or contracting according to a given law and the like, where the law of variation
is geometrically expressed; and such as the problems of the motion of fluids, of elastic bodies,
or of systems of bodies like the solar system, where the law of variation is mechanically
expressed by defining the nature of the internal actions and reactions of the system. We
shall confine our attention to the simpler problem of the motion of a system of invariable form, which we proceed to discuss.

21. The motion of an invariable system is always reducible to the motion of translation
of some point invariably connected with it combined with a motion of rotation about a certain
axis through that point. Let v,, v,,v, denote the resolved velocities along Oz, Oy, Ox of the point O, to which the translation is referred, and let w,, ων» w, denote the resolved angular
velocities about the same lines; then the velocity of any particle m, whose co-ordinates are ὦ, Y, ὧν ἰδ, +ωγῷ — wy in the direction of Ox, with similar expressions for the directions
Oy, Ox. Hence summing the linear and angular momenta of the several particles of the
system, we find
τ; = =(m) οὖ, + w,. =(mz) — w,=(my),

* T avoid the use of the term rigid body because of the mined. ‘This view presents Statics as a natural preparation for

mechanical notion conveyed in the term rigid. The pro- Dynamics, instead of as a science of co-ordinate rank separated

positions usually enunciated with reference to a rigid body by a gulf to be bridged over by a fictitious reduction of dy-

must, if that term be retained, be understood of a geometrically, - namical problems to problems of equilibrium through the intro-

not a mechanically, rigid body; that is, of a body the disposi- duction of fictitious forces. In several of our more recent works

tion of whose parts is by hypothesis unaltered, not of one in the terms accelerating force and centrifugal force have been

which the disposition cannot be altered or can only be insensibly rejected or explained as mere abbreviations, the one as not

altered by force applied to it. But itis difficult (and perhaps not desirable) to divest this term of its mechanical meaning,

being properly a force, the other as being a fictitious and not an actual force : this it would be well to carry out still more com-

as is seen in the modes of expression commonly adopted in the pletely, to restrict force in fact to that which is expressible by

case of flexible strings, fluids, &c., where it is frequently de- weight and to admit only actual forces (to the exclusion of cen-

manded of us to suppose our strings to become inflexible, our trifugal forces, effective forces and the like) under the two

fluids to become rigid, or to be enclosed in rigid envelops, and divisions of internal forces, or those whose opposite Reactions

the like—a process which must always stagger a beginner and are included within the system, and eaternal forces, or those

leave a certain want of confidence in his results, until this is whose opposite Reactions are not so included. If then Statics

gained by familiarity with the process, or until he learns that it and Dynamics were defined as above, one great division of

simply amounts to asserting that what has been laid down to Rational Mechanics would be formed of the Statics and Dyna-

be true of a rigid body is no less true of a non-rigid body, mics of a system of given invariable form, without the par-

while there is no change in the disposition of its parts. As ticular constitution of the system being defined and there-

another instance of a needless limitation in our current defini- fore independent of Internal Forces ; while the other great

tions, we may cite that of Statics as the science which treats of division would include the Statics and Dynamics of special

the equilibrium of forces, whereas the truer view would be to systems of defined constitutions, as flexible bodies, fluids,

regard it as treating of those relations of forces which are inde- elastic solids and the like, in which the laws of the internal

pendent of time, and thus every dynamical problem would have forces must be more or less completely known. These re-

its statical part in which the state of the system and the forces marks are thrown out as suggestions for a more natural

is considered αὐ each instant, and its truly dynamical part in system of grouping the special mechanical sciences than has

which the changes effected from instant to instant are deter- yet been commonly received.

2—2

12

Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

and

h, = Σρι(ψ οὖ, + WY το Wt το2. Vy + WH — W,8)

= (my) .v, — E(msz).v, + T(m.y? + 2°). ὦ, — X(may)wy — U(mxx) .w,

with similar expressions for u,, u, and hy, h;.
From these equations it appears that, when the linear and angular velocities of the system

are referred to an arbitrary point Ὁ, each depends in general on both the linear and the

angular momentum. If however O be the centre of gravity, the linear velocity depends on

the linear momentum only, and the angular velocity on the angular momentum only, for

in this case =(mwx), =(my), =(mz) all vanish, and the equations become those, of which the

types are

Uz = X(M) . Ves

hy, = =(my’ + 2)w, — =(may) . ὦν — =(mzxa)w,.

22. Thus the motions of translation of the centre of gravity and of rotation about it are independent, a property which is true of no other point. Also it is to be observed that the direction of motion of the centre of gravity coincides with that of the linear momentum, while that of the axis of angular velocity does not in general coincide with that of the angular momentum. This is the cause of a greater complication in the problem of rotation than in that of translation. In the former the passage from momentum to velocity involves the changing of the direction of the axis as well as division by a quantity of the dimensions of a moment of inertia, whose value depends on the position of the momental axis in the system: in the latter the corresponding step involves simply division by a constant quantity, the mass, without change of direction. If the operation by which the step is taken from
momentum to velocity, be considered as the measure of the inertia, we may express the above by stating that the measure of the inertia of a system relatively to translation (the centre
of gravity* being the point of reference) is the mass of the system, and that the measure of its inertia relatively to rotation is not a simple numerically expressible magnitude, but, in Sir W. Hamilton’s language, a quaternion, dependent on the position of the axis of angular ‘momentum or of that of angular velocity in the system.

23. Confining our attention henceforth to the problem of rotation, we must first obtain a more distinct idea of the relation between the axes of angular momentum and _velocity. We may obtain this from our previous equations for h,, h,, h., in their general form; but more simply when we consider our axes as coincident with the principal axes through the centre of gravity. If A, B, C denote the moments of inertia about these axes, the equations become (substituting 1, 2, 3 as subscripts for a, y, # respectively)
h, = Aw, hz = Bor, hz = σὰν
hence the axis of angular momentum OH, whose equation is @o ey 8

is parallel to the normal to the central ellipsoid

* It will be observed that, if the translation be referred to any other point than the centre of gravity, the measure of inertia relatively to translation is also a quaternion.

VELOCITIES, ἄς. WITH RESPECT TO AXES MOVEABLE IN SPACE.

13

Aa? + By? + Cx* = 1,

at the point, where the axis of angular velocity OJ, whose equation is

eo oY 8

?

@,

We

ως

meets it. Also reciprocally OJ is parallel to the normal to the ellipsoid, whose equation is

y”

2?

4*B'C

at the point where OH meets it.

Thus a simple geometrical construction enables us to determine OJ, when OF is given, and vice versa. If now ὦ be the angular velocity about OJ, and J the moment of inertia

about the same line, the angular momentum about it must be Jw, since w is the ¢otal angular velocity, and therefore the angular velocity about a line perpendicular to OI is zero; hence

Iw =h.cos HI,

an equation connecting h and , the quantities J and HJ being known when the above construction has been made.

24, If h be constant, and its direction OH invariable, it is plain from the above con- struction that OJ will not in general remain fixed, nor ὦ constant, for, by the motion of the
system about OJ, the position of OH in the system is altered, and to this new position of OH a new position of OJ will correspond, and then w will change by reason of the variation of
cos HI There is an exception however in the case where OH and OJ coincide, for then the

rotation does not change the position of OH in the system: this can only be the case

when the radius OJ of the central ellipsoid is also a normal, that is, when it coincides with one

of the principal axes. Hence the principal axes are the only permanent axes of rotation of a

body acted on by no forces (as is implied in our supposition of h being constant): in all

other cases the axis. of rotation moves in the body and in space, and the angular velocity

about it varies.

‘

25. If w be constant and its axis OJ fixed in the body, OH will also be fixed in the
body, and h will be constant; but OH will then in general move in space, and the system must therefore be acted on by forces, whose resultant couple has its axis perpendicular to OH and in the plane of motion of OH. Hence the plane of the couple is ΠΟ], if OJ be fixed in
space as well as in the body, and its moment is constant, since the velocity of OH is constant;
thus the constraining couple on a body revolving uniformly about a fixed axis through its centre of gravity is determined.
In the exceptional case of a principal axis, OH is also fixed in space, and there is no constraining couple.
26. Before proceeding to the solution of the problem of a body’s rotation about its centre of gravity by a method more in accordance with the plan of this paper, it will be well to shew how readily Euler’s equations may be obtained from our principles.

14

Mr R. Β. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

If the moveable rectangular axes in § (15) be supposed fixed in the body and coincident with the principal axes, we must substitute

ὧι» We, ὡς for Qy, Q,, Q,, and hy hy, hg, or Aw, Bw, Cw; for ἢ,» hy, ἢ,»

_and then we obtain three equations, of which the type is, either

dh,

ἀν (sd

dé =[D+ (ς- 5)hls

or A—ἃdt = ἢ +(B- C). 0,0
The latter is the well known form of Euler’s equations.

27. Instead of employing these equations, let us endeavour to solve our problem more directly. Our object is to determine the motion of OJ, the axis of rotation, both in the body and in space, and the variation of w, the angular velocity about it. This may be
conceived to be due to an angular acceleration of definite intensity about a definite line; and
this may be regarded as compounded of two similar accelerations, the one arising from the acceleration of momentum produced by the couple G about its axis OG, the other being the angular acceleration which would exist if no forces acted. Now the forces in the elementary time dé produce the angular momentum Gdt about OG, and this momentum gives rise to a corresponding angular velocity Kdt about an axis OK related to OG, just as OI is OH: thus the angular acceleration « due to the forces is determined as to direction and intensity. The other component of the angular acceleration is in like manner due to a corresponding accele-
ration of momentum, which it is now necessary to determine.

28. Regard any line OP fixed in the body and moving with it by reason of the velocity w about OJ; and apply equation (C) of section I., putting ἢ for wu; therefore
we = —hw.sin JH.sin HP.sin JHP,
which determines the acceleration of momentum for any line OP. This acceleration will be zero, if OP bein the plane ΠΟ, and a maximum, if OP be perpendicular to HOJ, when its value is hw sin HI: we may therefore regard the total acceleration* (f) due to the motion of the body as being about the line OF, perpendicular to HOI, and equal to + hw sin HJ, when OF is taken on that side of HOI on which a positive rotation about OF would move OH towards OI. Now to this acceleration of momentum (f) about OF will correspond an acceleration of angular velocity (A) about a line OL which is related to OF, just as OL is to OH.

29. Tosum up our results, we have shewn that, if OH be the axis of angular momentum
(h) and OJ that radius of the central ellipsoid at whose extremity the normal is parallel to OH, OF is the axis of angular velocity (w): if OG be the axis of the impressed couple (6); and OK the radius for which the normal is parallel to OG, OX is the axis of angular accele-

* This result is that which M. Poinsot states thus: “‘The | sion.’’”—M. Poinsot’s “couple d’impulsion” is our angular
axis of the couple due to the centrifugal forces is perpendicular | momentum. at once to the axis of rotation and to that of the ‘ couple d’impul-

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE.

15

ration due to the forces («): lastly, if OF be perpendicular to the plane HOJ, it is the axis of acceleration of angular momentum in the moving body, and OL, the radius for which the normal is parallel to OH, is the axis of angular acceleration due to the motion of the body (A).
Also we have the three equations for w, x, d, Iw =h cos HI,
Kr = GeosGk, ΤᾺ =f cos FL,
wherfe= hw sin HJ,
I, K, L denoting the moments of inertia about OI, OK, OL respectively. It will be observed that OJ is the direction, to which the plane through O perpendicular to OH is diametral, and
that OL is the direction to which the plane ΠΟ] is diametral, hence OL lies in the plane
perpendicular to OH. Also if the rectangular planes HOI, FOL intersect in OM, it will be seen that the axes* OJ, OL, OM are conjugate diameters of the central ellipsoid.
30. We will develop the solution in the simpler case of OG coinciding with OH and therefore OK with OZ. In this case OH remains fixed in space, and the motion of OJ is conveniently referred to its motion in the plane HOJ and the motion of that plane about OH.

LT = ἤ

2
Let the conjugate radii ΟἹ, OL, OM be denoted by r, γ΄, γ΄, then the moments of inertia
about them are “:> = ‘ aa by the property of the central ellipsoid : also let the angles HOJ,
FOL be denoted by 6, 6’: then our last equations become (1) w=hr'cos@, (2) «=Gr'cosO, (8) A= (hwsin θ). τ΄" cos.
Resolve w, x, along the axes OH, OM, OF; the component velocities are then w cos@ along OH, wsin@ along OM, and zero along OF, while the component accelerations are
«cos@ along OH, «sin @ + sin @ along OM, and ἃ cos @ along OF ; whence, by applying either the equation (C) or the equations (£),
ἕω 6080) =.«.008.0 = Gr* COS? O...005.0crercccrcvecceccees(e4)

* Hence if no forces act, the instantaneous motion of the axis of rotation OJ will be towards OZ, the radius with

respect to which the plane ΠΟΙ is diametral.

.

16

Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

w sin @.Q =) cos θ΄ = (hw sin 0) . 7 cos? 0',... 002000 νον κεν σον ..(6) where © is the angular velocity of OM (i. e. of the plane HOTZ) about OH.

Also we have

bent WR he innpaieec vas vos sec ccecccccctessesteeuernehieee

Let p, p’ denote the perpendiculars from O on the tangent planes to the central ellipsoid at I, L respectively, then p = r cos 0, p’ = γ΄ cos 6’.

Equation (4) becomes by (1) τ (hp*) = Gp, whence by (7), p is constant. This shews

that the tangent plane at J to the central ellipsoid is fixed, and that the central ellipsoid
therefore rolls on it as a fixed, plane.
Also by (4) and (5) d(tian6) -5d(/w*s=in@5\ ) = Asign’t , hp ,, =n θ.. τη 9; sicgeysuas (8)

and from (6)

Se

ee eee eee ΔΝ μενλυσάνον ἐς,τ

31. Now 7, τ΄, γ΄ being conjugate radii of the central ellipsoid, there exist three
relations between them and the conjugate axes; these are, (putting psec 0, p’sec @ for 7, στ΄ respectively and denoting the angle JOL by x)
p’ sec? 0 + p® sec? θ΄ + 7? = 5 1 .1 "1 E, suppose,

pr”2?/2 + prΩ”Σ + pp3”.,3 sec’ coc? θ secἘ ’ θ΄ς , sin’ χ BOb1eed© tὉ1Α 481 ar F,, suppose,

»» Fag (- aH = G, suppose,
and by reason of the rectangularity of the planes JOM, LOM, we have cos x = sin @ sin 6’.
Eliminating r” and x, we obtain
pr sec! O + p'* sect R+UG = Β,

G = + δ + p’p*(sec? @ + sec’ θ΄ -- 1) =F.

From these eliminating sec® θ΄, we obtain

f2o? {ly 4 (1-pLτoςpokὅς- PGὦp)cot* a,
which, (remembering what E, F, G denote, and putting a, β, Ὑ for the three quantities
ag ce 1 - 1 - 1 : respectively)

is equivalent to

p® = p°(1 + aBy cot? @);

VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE.

17

also, since ρ΄, θ΄ are involved in precisely the same manner as p, 6, it follows that

p? =p(1 + a'p'ry cot? 6’) ;

where a’, β΄,γ΄ are what a, B, Ὕ become, when p’ is put for p.

From these equations we obtain

wit
cot®

αaβpΎy

1+ acoyt?@τοῦθ᾽

but

a 7 el-—1=1- —1—.

1 -Ξα 1 + By cot? τθο

4"

Ap? τ-ὸοβγοοῦῦθ

1 + αβγ cot’ é

whence, with the corresponding expressions for β΄, +’, (1 + aBry cot? 6)?
cot? 0’ = -- cot? @. (1 + By cot’ 0)(1 + γα cot® @)(1 + αβ cot* 6)’

hence ρ΄, 6’ are known in terms of p, 0.

32. Substituting now for μ΄, θ΄ in terms of p, 0, we obtain from equation (8)

d(cot @) soi ,, ecotta8

dt

P cot

= + hp*S - (1 + By cot® 6)(1 + γα cot? O)(1 + a cot® θ)}},.....6.6 010)

and from equation (9)

Q = hp? (1 + aBy cot? 6).

If h be known by means of (7), these two equations determine completely the motion of
OI the axis of angular velocity in altitude and azimuth, since p, and therefore a, B, Ὑ, are

constants.

If @ denote the azimuth at any instant, τ =Q, and dividing the last equation by the

preceding, we obtain a relation involving @ and @ only, which will therefore be the differential
equation to the conical path of OJ in space; and it is worth notice that, this relation being independent of ἡ, the path of OJ is the same whether the body be, acted on by a couple whose axis coincides with OH, or whether it be acted on by no forces. The effect of the couple in
this case is in fact only to alter the velocities of the different lines, not the paths which they describe.
Also equation (1) gives w = hp? sec θ, from which ὦ is known when 6 is known by means of equation (10), and thus the velocity about OJ is known completely as well as its position at any time,

33. If there be no forces acting, i. 6, if G= 0, ἢ is constant, as is also ὦ 605, the resolved angular velocity of the body about OH. Also the vis viva of the body

w =],?=—=
r

h’p!
2 cos@

and is therefore constant; and hence ~ is constant, or ὦ « 7; both well known results. It may r

Vou. A. Pant ck,

3

18

Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING

also be well to note that p® = iaigncviswisveiovumaneengteummay" even if G do not vanish, and thereforeἢ that the vis viva « (angular momentum)*, when the angular momentum has a fixed direction.
It is needless to carry the solution farther by investigating the path of OJ in the body, the position of the principal axes relatively to OH, ΟἹ at any time, &c., since all these questions are discussed with the utmost completeness and elegance in M. Poinsot’s Théorie de la Rotation.

34. We will conclude this paper by solving the problems of Foucault’s Gyroscope as
applied to shew the effects of the earth’s rotation, as it will furnish a good illustration of the advantages of the methods of this paper in enabling us to form our equations immediately with respect to the most convenient axes.
The Gyroscope is essentially a body, whose central ellipsoid is an oblate spheroid by reason of its two lesser principal moments being equal, and which is capable of moving freely about its centre of gravity. In this case, if a rapid rotation be communicated to it about its axis of unequal moment, that axis will evidently retain a fixed direction in space however the centre of gravity move, and therefore relatively to a place on the surface of the earth will alter its position just like a telescope, whose axis is always directed to the same star. But there are two other remarkable cases, where the motion about the centre of gravity is partially constrained ; the first, where the axis of rotation is compelled to remain in the plane of the meridian, the second, when it is compelled to remain in the horizontal plane. These we will now consider.

σ
᾿8ὅ. When the polar axis of the central spheroid always lies in the plane of the meridian, let 9 denote the north polar distance of its extremity 4. Let OB coincide with the equatorial axis in the plane of the meridian, and OC with that perpendicular to the same plane, and
refer the motion to the axes OA, OB, OC. Now if Q denote the angular velocity of the earth
about its axis, the motions of OA, OB, OC will be due to the velocities Q cos θ, Q sin θ, =
about them respectively: also the actual velocities of the body about the same axes are respectively w, Q sin 0, d=,and the consequent angular momenta 4w, BQsin 0B, where w, τᾷ
are reckoned positive when the motion about their axes is in the same direction as the earth’s about its axis.

VELOCITIES, ἄο. WITH. RESPECT TO AXES MOVEABLE IN SPACE.

19

It is evident that in this case the constraint is equivalent to a couple, whose axis coincides

with OB, let this be denoted by G. Then the equations (£) in the first section applied to the

case before us give

ad 4”) “5:do. .5'ηθ -- BQ sian e8. =

qdi BO siμ n θ)- α + do.dé πο dé“ἂν cos 0,

(d;8_,3d0)o -- ΒΩ siPn θ. οο5θ -- 4ω. Ὡ si; n 6;

from the first equation, w is constant, and from the last

ἂν -- (Fe - Q cos 6)asin θ:
now in this case Q the velocity of the earth’s rotation is very small compared with w, neglecting therefore the second term of this equation,
@0 dt? =— < sin0, whence the motion of the axis OA is precisely similar to that of the circular pendulum, whose length is J, where ©= RA oe and therefore /= Ξ 3 the direction of the earth’s axis taki: ng the
ω
place of the direction of the force of gravity.
2,
Also since 75 = 0, when sin 9 = 0, there are two positions of equilibrium of the axis OA,
namely, when θ = 0, and@ = 7: the former is stable and the latter unstable, when w: and Q have the same sign. Hence the axis of rotation will remain at rest, if originally placed in the ‘direction of the earth’s axis, stably or unstably according as the rotation regarded from the end directed to the north pole is in the same direction, or the contrary, with the earth’s rotation regarded from the same pole. If placed originally in any other position, it. will oscillate about its position of stable equilibrium according to the same laws as a circular pendulum.
36. Next, let the polar axis OA always remain in the horizontal plane, and let @ denote its azimuth from the south towards the east. Taking OB and OC as before, the latter will now coincide with the vertical.
If ¢ denote the co-latitude, Q may be resolved into Ω cose vertical and Q sinc horizontal
in the north direction: hence the angular velocities by which the axes move, are relatively to OA, OB, OC respectively
dp -Qsinecosd, —Qsincsingd, ag + cose,
and the corresponding angular momenta are

4w, --BQsin ὁ sin ᾧ: a(t + Ω cose),

40

Mr R. B. HAYWARD, ON A DIRECT METHOD, &c.

whence as before,

d(4w)

-

-- 0,

dt

ale ΒΩ sine sin Φ) = G+ Aw (t+ cosο + B(Z + 2cose) .Qsin ecos p,

α(ἀ{. 22αἱ + Qeos¢) = BQ sine sind .Qsin 6 cos ᾧ + Aw. Qsine si: n d, dt\ dt and therefore w is constant, and
i "δ.= ΞΖ εἰ esin φ + QO? 5ἰπ"6. sinᾧ cos φ,

or approximately

α' ΤῸΝ <oO sin e. sin gs

whence, the rotation about OA being in the same direction seen from A as that of the earth seen from the north pole, it will be in a position of stable equilibrium when directed to the

north, and of unstable equilibrium in the opposite position: also if originally directed in any
other direction, it will oscillate about its position of stable equilibrium like a circular pendulum

about the vertical whose length is ———B-g——.

8

A@Q sin Ἢ

Duruam, Feb. 19, 1856.

R. B. H.

Il. On the question, What is the Solution of a Differential Equation? A Supple-
ment to the third section of a paper, On some points of the Integral Calculus,
printed in Vol. IX. Part Il. By Avaustus De Morean, of Trinity College, Vice-President of the Royal Astronomical Society, and Professor of Mathematics in University College, London.
[Read April 28, 1856.]
Trustine that it will be sufficient excuse for a very elementary paper, that writers of the
highest character are not agreed with each other on a very elementary point, I beg to offer
some remarks upon the usual solution of such an equation as dy* — ada’ = 0, to which Euler
assigns the integral form (y-aw+b) (y+av+c)=0, where ὃ and ¢ are independent constants. Most other writers insist on the condition ὦ = ὁ.
Lacroix refers only to Euler and to a paper by D’Alembert (Berl. Mem. 1748) which I © have not seen. All the reasons which have been given on the subject are reducible, so far as I have met with them, to those which I shall cite from Lacroix himself and from Cauchy.
Lacroix (ii. 280) in his explanation of this case, and in defence of the substitution of ( -- αὐ +b) ( -- αὐ -Ὁ δ) for (y—aw+b) (Ψ Ὁ αὦ Ὁ 6)» makes two remarks. The first,—
chacun de ses facteurs doit étre considéré isolément; the second, alluding to the form with two constants, is—on n’en tire pas d’autres lignes que celles qui résulteraient de l’intégrale renfermant une seule constante. M. Cauchy (Moigno, ii. 456) says—On ne restreindra pas la généralité de cette intégrale en désignant toutes les constantes arbitraires par la méme lettre...: and grounds the right to do this on the possibility of thus obtaining all the curves which can satisfy the equation.
In searching out this matter, I found it by no means clearly laid down what is meant by the solution of a differential equation: and, on looking further, I found some degree of ambiguity attaching to the word equation itself. The following remarks will sufficiently explain what I mean.
A connexion between the values of letters, by which one is inevitably determined when the
rest are given, may be called a relation. But an equation is the assertion of the equality of two expressions. Every simple explicit relation leads to an equation, to one equation: but every equation does not imply only one relation. The object of the problem being relation between y and x, the equation (y — x) (y — x*) = 0 implies power of choice between the relations y = a”, y=, The equation (y — a’) (ὦ — 1) =0 implies the relation y = αὐ with a dispensation from all relation in the case of # = 1.
Now I assert that in mathematical writings confusion between the equation and the simple relation is by no means infrequent: without dwelling on instances, I think we shall find, by.

22

Mr DE MORGAN, ON THE QUESTION,

examining approved modes of reasoning, that the confusion cannot but be seen to have existed,

so soon as the statement of what it consists in is made. It is affirmed that the primitive of a primordinal equation cannot have two arbitrary

constants: but all that can be proved is that no such differential equation can have two related

arbitrary constants in its primitive.

:

Let f(x, y, y’) = 0 involve any number of relations between «, y,y’: and let (a, y, a,b) = 0

be a relation between ὦ and ὃ, or any number of relations, Consequently, selecting one relation

by which to satisfy ᾧ = 0, values of a and b can be found to satisfy both p(w, y, a,b) =0, and

also p(a + h,y + k, a, Ὁ) =0, for any values of x, y,h,k. Hence, for any values of @# and y,

y’ may have any value whatever: and this is incompatible with f(a, y,y’)=0. But this is no

argument against any form of $(,y, a, b,) = 0, in which the constants are not in relation; as Wa, y, 4) - χίω, ψ, δ) = 0. For we cannot pretend to satisfy

Ve, Y; a). χίω, ψ, δ) =0, Ve +h,y +k, a) -x(@ +h,y +k, b) =0,
for any values of w,y,h,k, except by W(#,y,a) = 0, and y(w~+h,y+k,b) =0, or else by W(a +h, y + k, a) = 0, x(#, y,b) =0. And from neither set can we deducey’. If W(a, y, a) = 0 be a primitive of f(x,y, ν΄) = 0, there appears nothing ἃ priori to prevent our saying that V(a, y, 4). ψίω, y, b) = 0 isa primitive. This point will be presently examined.
It is affirmed that a primordinal differential equation cannot have two really different primitives with an arbitrary constant in each: but all that can be proved is that one primordinal relation cannot have two distinct primitives. If y'=/(«,y) be satisfied by different
relations (a, y, a) =0; V/(a, y, b) = 0, then, taking a and ὃ so as to satisfy both at a given point (v,y), we find, generally, two values of y’ at (wy). But y'=f(#,y) may give these two
values; irreducibly connected, as in ψ' = 1 + ./y, or reducibly, as in ψ' =14,/y*. The great point of algebraical interest, namely, that when the two values of y’ are irreducibly connected
= 0 and ψ =0 are the alternatives of an equation which can be rationalised or otherwise inverted into χ =0, where χ is of univocal form, is foreign to the present purpose. That purpose is, to make it clear that the common theorems about the singularity of the constant of integration must be transferred from differential equations to differential relations, of which one equation may contain any number.
The question whether y = #, which is certainly one relation for determination of y from
w, is to be considered as giving one or two relations for determination of # from y, ends in a
question of definition, perhaps, but ends in a question which cannot be adequately treated without a close attention to the meaning of the word continuity. And here immediately arises the distinction of permanence of form and continuity of value.
Form is expression of modus operandi: and permanence of form implies and is implied in permanence of the modus operandi through all values of the quantities to be operated on. In arithmetic, the signs + or — are of the form, and not of the value: but in algebra, the +
or — which the letter carries in its signification are of the value, so called. Accordingly, permanence of form does not necessarily give continuity of value. The immediate passage of
a
f sinav.v-'de from +4 to -- ἐπ, as w passes through 0, might be discovered by the
0

WHAT IS THE SOLUTION OF A DIFFERENTIAL EQUATION?

23

arithmetical computer, utterly ignorant of the Integral Calculus, by use of skeleton forms set up from one form of type. Nor does discontinuity of form necessarily give discontinuity of
value. The branch of y= which ends at w =0 joins the branch of y= +e-* which begins at w =0 with acontact of the order co, as order of contact is usually defined. We may even propound the question whether (— #)* and (+ «)* be not different forms ?
Let continuity of no order, or non-ordinal continuity, be when and so long as infinitely small accessions to the variable give infinitely small accessions to the function, And let the passage from - οὐ to τῷ c be counted under this term. I will not, on this point, give more than an expression of my conviction that the word continuity must, by that dictation which has turned wnity into a number, and its factor into a multiplier, be extended to contain the usual passage through infinity. Let 2-ordinal continuity be when and so long as y, Μ΄, y”,...y are of non-ordinal continuity.
These definitions being premised, we have in the passage from the positive to the negative value of w} an interminable continuity, and a change of form answering to, and indeed derived from, the change of form seen in (+ )* and (— @)*. We have, in truth, all the quantitative properties of one relation, and all the formal properties of two. The attainment of a reducible case is the loss of the quantitative properties also: thus (a + a)} is non-ordinally continuous, and not so much as primordinally, when ὦ = 0.
We are now in a condition to answer the question, What is the solution of a differential
equation ?—at least so far as having a clear view of the imperfect manner in which the question is put. We are obliged to ask in return, what requirements as to continuity are conveyed in the word solution ?
1, The word solution may require the most absolute notion of permanence of form, not granting even the passage from ( -- x)’ to(+a)*. In this case we must be compelled to satisfy the differential equation by a relation of permanence equally strict, and in so many ways as we can do this, in so many ways can we announce a solution. Thus to y* = 2,/y.y' we announce three solutions. To ψ' = 0, any parallel to the axis of # To ψ' =2 x the positive value of ,/y, the right hand branch, from # =a onwards, as figures are usually drawn, of any parabola y=(v-a)*. To oy =2x the negative value of 4/y, the left hand branch of the same up to #=a, The change from any one of these to any other is entirely forbidden: and a must be less in one case, and greater in the other, than any value of « which is to be employed. Problems are frequently stated in a manner which will admit only one branch of an ordinary solution: and the investigator, so soon as this is perceived, generally widens his enunciation, rather than narrow his notion of a solution.
2. Ina solution we may allow only such changes of form as take place in the inversions of ordinary algebra, and no others. In this case we should say, that we have y=a and y = (« — δ)", which we please, but only one, for the solution of y? =2,/y.y’. In this case and the last we satisfy Lacroix’s requirement that the factors must be considered in isolation : but it is not correct to imply that such isolation is part of the meaning of a compound relation.
From PQ=0 we only learn that one of the two factors is to vanish: the equation has no power to deny us the use of one factor for some values of «, and of the other factor for others.
The isolation of the factors is the postulation of a certain permanence of form.

24

Mr DE MORGAN, ON THE QUESTION,

3. In asolution we may allow change of form, with a given kind of continuity at the junction. If we mean to stipulate nothing whatever about continuity, we may at any value of Φ leave one curve, and proceed upon another. If we require non-ordinal continuity, we can only do this where two curves join each other. If we require ordinal continuity or continuity of the same order as the equation, we may propound as a solution of ψ' =24/y any number of parabolas with as much of the singular solution y = 0 as lies between their vertices. If we
require every degree of continuity, we have, in the case before us, what is tantamount to requiring permanence of form, in its ordinary sense.
No prepossession derived from ordinary algebra would be offended by a solution which has a continuity of no higher order than the order of the equation itself: which would allow us,
on arriving at the singular solution, or connecting curve, to break off from the curve thitherto employed, to proceed along any are of the connecting curve, and to abandon this last at any chosen point in favour of the ordinary solution which there touches it.
In the graphical method by which the possibility of a solution is established, that is, by construction of a polygon from Ay = x(#, y). Aw, with a very small value of Aw, which may
be as small as we please in the reasoning, a solution of y= x(a, y) is shewn to exist: but it may be one of the kind just alluded to. The draughtsman employed to construct such a solution, when his are of the ordinary curve comes very near the point of contact with the
singular solution, cannot undertake to remain on that ordinary curve, without reference to quantities of the second order. The accidents of paper and pencil are casualties of this order, which might divert his are of solution from the ordinary curve on to the singular solution, might keep it there for a while, and then throw it off upon another ordinary solution. In fact, the solution established ἃ ‘priori has not of necessity permanence of form, but has only continuity of the order of the equation. And this remark applies to equations of all orders. In the case of y’ = 2\/y, when once a side of the polygon ends on y = 0, the draughtsman can never leave that line again, without constructing one side by help of Ay = (A)?.
It may now be affirmed that ( -- αὦ --Ὁ) (y+aa+c) =0, ὁ and e being perfectly independent constants, is a solution of y’*—a*=0; nothing in the general theory of the primordinal differential redation in any way withstanding. It remains to examine the assertion that the generality of this solution is not restricted by the supposition ὃ = ὁ.
To a certain extent this assertion is true: no more curves are obtained or included before the limitation than after it. Beyond this point the assertion is not true. The condition ὃ = belongs to one mode of grouping a solution of y' = α with a solution of ψ' = —a: but there is an infinite number of modes in ὃ = de. If ordinal continuity be held sufficient, and if φίω, y, b) = 0, ψίω, y, 6) = 0 be independent relations satisfying f(a, y, y’) = 0, and if P =0 be the most complete singular solution, then
P. (a, y, by) « Pla, Y, bz)0-10 W(@, Ys 61). ψίω, Y, 62)... = 0 is the most general solution, where 6,, b,,...c;, 625... are in any number, and of any values, This however is but equivalent to P. f(a, y, 6). ψίω, y, 6) = 0 with the usual addition ‘for any values whatever of 6 and c’.
This point will be best illustrated by reference to the biordinal equation and its theory. A primordinal equation belongs to a group or family of curves which may be called of single

WHAT IS THE SOLUTION OF A DIFFERENTIAL EQUATION?

25

entry: a biordinal equation to a group of double entry, out of which an infinite number of groups of single entry may be collected. Thus, ὃ and ὁ being in relation in φίῳ, y, ὃ; 6) = 0, we may designate all the curves contained in (a, y, fc, 6) = 0 as the group (fe,c). Generally speaking, the curvesof the group (fc, 6) are different from those of (Fc, c). The unlimited number of cases of (fe, 6) is the key to the unlimited number of primordinal equations which give rise to one and the same biordinal equation, It is then the characteristic of the biordinal equation that it represents a group of double entry. When the constants are not in relation,

as in (2, y, ὃ). ψίω, y, ὁ) = 0, we have still groups of double entry, but the biordinal equation ceases to exist: the distinction between one group and another consists in the distinct ways in which individuals of the two groups @ =0 and Ψ = 0 are joined together. This defective

grouping—not defective in the variety of its cases, but defective in the variety of the elements
out of which cases are to be compounded—is within the compass of a primordinal equation, into which therefore the biordinal equation degenerates.
As an instance, let (Ρ -- δ) (0 -- ὁ) - R=0, P,Q, R, being each a function of w and y: and let P’ represent ἢ, + P,.y', &e.
When b = fc, the primordinal equation of the group (fe, 6) is

Ro +/(R? +4P’'QVR)

R-,/(+4RP?QR)

Gt

2P’

ae \Pτ

40’

| ;

Let R=,V, where V is a finite function, and » a constant. When μ diminishes without limit, and finally vanishes, each primordinal equation becomes either P’=0 or Q’=0, for

otherwise we have only Q = /P, the algebraic result of eliminating ¢ between (P — δ) (Q -- c) =0,
and (P — fe) Q' + (Ω -- ὁ) P’ =0, And the biordinal equation is determined by differentiating

b=Q+

R+f(R?+4P'VR)
2P’

i

Do this fully, clear the result of fractions, and write »V for R: it will then appear that y’’ is

seen only in terms multiplied by positive powers of 4; and so that μ =0 gives P’Q’ = 0 in place of a biordinal equation.
The correction which the common theory requires is as follows ;—An equation in which n constants are in relation with w and y, cannot have any differential equation clear of those

constants under the mth order; and an equation of single and irreducible relation between
X,Y, Yi.ey™ must have a primitive containing 2 constants in relation to ὦ and y. But a

primitive equation in which m constants are contained in alternative relations, m, in one relation, m in a second, &c. does not require a differential equation of the mth order;but has an equation
of alternative relations, one of the mth order, one of the nth order, &c.

From a primitive having m constants, in relation with a and y, no constants can be eliminated in favour of y’, y”, &c without one new equation of differentiation for every constant which is to disappear. But this is by no means true of constants in relation with x, y, and one or more of the set y’,y”,..., to begin with. This point is made clear enough in the section of my former paper to which these remarks form a supplement: but the whole may be illustrated

as follows. If p(w, y,a) =0 give a = (#,y,), and therefore ᾧ, + ®,.y’ = 0 for a differential equation, in which ὦ has disappeared and y’ is introduced, it is easy to give this differential

Vou. X. Parr I,

4

40

Mr DE MORGAN, ON THE QUESTION, WHAT IS THE SOLUTION &c.

equation a primitive containing any number of separate and independent constants. For A, + A, O(a, y) + 4. {P(x, y)}* + ... = 0 cannot give any relation in which one of these constants disappears in favour of y’ except ©,+,.y' = 0, in which they all disappear. But this is merely formal; for 4, +A, ®(#,y) + ... = 0 is but a transformation of some case of O(a, y) = SF(Ao, Ay...) or of Pha, y, f(Ay .4.»...)} =0. All we have done, then, amounts to no more than use of the obvious theorem that a single arbitrary constant is equivalent to an arbitrary

function of as many arbitrary constants as we please. Moreover, we may prove that ἢ - ψ' can only be a factor in the differential of one class of forms. If {F(a,y)}’ give M(P+y), nothing but ᾧψ ζω, y)}’ can give N(P + y’): and F(#,y)=const. and ψζω," )= const. are

the same equations.

:

But it is otherwise with P + γ΄, P being a function of 2,y,y’. This occurs, as previously

shewn, in the differentiations of two distinct classes of forms. Thus 0+y” is a factor in

§f(ay’ —y)}’ and in {Fy'}'. The equation

f(y -y) = 4, + A, Fy’ + A, {Fy}? +...

is one which contains in every sense, formal and quantitative, as many arbitrary constants as we

please; and an alteration in the value of one of them, is an alteration in the character of the _

relation subsisting between vy’ — y and y’. Nevertheless, it is impossible to get rid of any one constant in favour of y” in any way except one which results in y” = 0, an equation from which all the constants have disappeared.
Considerations similar to those which have been applied to primordinal equations might also be applied to equations of any order.

A, DE MORGAN.

University Cotitece, Lonpon, March 29, 1856.

Ill. On Faraday’s Lines of Force. By J. CuerK Maxwett, B.A. Fellow of Trinity College, Cambridge.
(Read Dec. 10, 1855, and Feb. 11, 1856.]
THE present state of electrical science seems peculiarly unfavourable to speculation. The laws of the distribution of electricity on the surface of conductors have been analytically deduced from experiment; some parts of the mathematical theory of magnetism are established, while in other parts the experimental data are wanting; the theory of the conduction of galvanism and that of the mutual attraction of conductors have been reduced to mathematical formule, but have not fallen into relation with the other parts of the science. No electrical theory can now be put forth, unless it shews the connexion not only between electricity at rest and current electricity, but between the attractions and inductive effects of electricity in both states. Such a theory must accurately satisfy those laws, the mathematical form of which is known, and must afford the means of calculating the effects in the limiting cases where the known formule are inapplicable. In order therefore to appreciate the requirements of the science, the student must make himself familiar with a considerable body of most intricate mathematics, the mere retention of which in the memory materially interferes with further progress. The first process therefore in the effectual study of the science, must be one of simplification and reduction of the results of previous investigation to a form in which the mind can grasp them. The results of this simplification may take the form of a purely mathematical formula or of a physical hypothesis. In the first case we entirely lose sight of the phenomena to be explained; and though we may trace out the consequences of: given laws, we can never obtain more extended views of the connexions of the subject. If, on the other hand, we adopt a physical hypothesis, we see the phenomena only through a medium, and are liable to that blindness to facts and rashness in assumption which a partial explanation encourages. We must therefore discover some method of investigation which allows the mind at every step to lay hold of a clear physical conception, without being committed to any theory founded on the physical science from which that conception is borrowed, so that it is neither drawn aside from the subject in pursuit of analytical subtleties, nor carried beyond the truthby a favourite hypothesis.
4—2

28

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

In order to obtain physical ideas without adopting a physical theory we must make ourselves familiar with the existence of physical analogies. By a physical analogy I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other. Thus all the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. Passing from the
most universal of all analogies to a very partial one, we find the same resemblance in mathematical form between two different phenomena giving rise to a physical theory of light.
The changes of direction which light undergoes in passing from one medium to another, are identical with the deviations of the path of a particle in moving through a narrow space in which intense forces act. This analogy, which extends only to the direction, and not to the velocity of motion, was long believed to be the true explanation of the refraction of light; and
we still find it useful in the solution of certain problems, in which we employ it without danger,
as an artificial method. The other analogy, between light and the vibrations of an elastic
medium, extends much farther, but, though its importance and fruitfulness cannot be overestimated, we must recollect that it is founded only on a resemblance in form between the laws of light and those of vibrations. By stripping it of its physical dress and reducing it to
a theory of “transverse alternations, ” we might obtain a system of truth strictly founded on observation, but probably deficient both in the vividness of its conceptions and the fertility of
its method. I have said thus much on the disputed questions of Optics, as a preparation for the discussion of the almost universally admitted theory of attraction at a distance.
We have all acquired the mathematical conception of these attractions. We can reason about them and determine their appropriate forms or formule. These formule have a distinct mathematical significance, and their results are found to be in accordance with natural phenomena. ‘There is no formula in applied mathematics more consistent with nature than the formula of attractions, and no theory better established in the minds of men than that of the action of bodies on one another at a distance, The laws of the conduction of heat in uniform media appear at first sight among the most different in their physical relations from those relating to attractions. The quantities which enter into them are temperature, flow of heat, conductivity. The word force is foreign to the subject. Yet we find that the mathematical laws of the uniform motion of heat in homogeneous media are identical in form with _ those of attractions varying inversely as the square of the distance. We have only to substitute source of heat for centre of attraction, flow of heat for accelerating effect of attraction at any point, and temperature for potential, and the solution of a problem in attractions is transformed into that of a problem in heat,
This analogy between the formule of heat and attraction was, I believe, first pointed out by Professor William Thomson in the Cambridge Math, Journal, Vol. III.
Now the conduction of heat is supposed to proceed by an action between contiguous
parts of a medium, while the force of attraction is a relation between distant bodies, and
yet, if we knew nothing more than is expressed in the mathematical formulz, there would be nothing to distinguish between the one set of phenomena and the other,

Mr. MAXWELL, ON FARADAY’S LINES OF FORCE.

29

It is true, that if we introduce other considerations and observe additional facts, the two subjects will assume very different aspects, but the mathematical resemblance of some of
their laws will remain, and may still be made useful in exciting appropriate mathematical ideas.
It is by the use of analogies of this kind that I have attempted to bring before the mind, in’ a convenient and manageable form, those mathematical ideas which are necessary to the study of the phenomena of electricity. 'The methods are generally those suggested by the processes of reasoning which are found in the researches of Faraday *, and which, though they have been interpreted mathematically by Prof. Thomson and others, are very
generally supposed to be of an indefinite and unmathematical character, when compared with
those employed by the professed mathematicians, By the method which I adopt, I hope to render it evident that I am not attempting to establish any physical theory of a science in which I have hardly made a single experiment, and that the limit of my design is to shew how, by a strict application of the ideas and methods of Faraday, the connexion of the very different orders of phenomena which he has discovered may be clearly placed before the mathematical mind. I shall therefore avoid as much as I can the introduction of anything
which does not serve as a direct illustration of Faraday’s methods, or of the mathematical deductions which may be made from them. In treating the simpler parts of the subject I shall use Faraday’s mathematical methods as well as his ideas, When the complexity of the
subject requires it, I shall use analytical notation, still confining myself to the development of ideas originated by the same philosopher.
I have in the first place to explain and illustrate the idea of “lines of force.” When a body is electrified in any manner, a small body charged with positive electricity,
and placed in any given position, will experience a force urging it in a certain direction. If the small body be ‘now negatively electrified, it will be urged by an equal force in a direction exactly opposite.
The same relations hold between a magnetic body and the north or south poles of a small magnet. If the north pole is urged in one direction, the south pole is urged in the opposite direction.
In this way we might find a line passing through any point of space, such that it represents the direction of the force acting on a positively electrified particle, or on an elementary north pole, and the reverse direction of the force on a negatively electrified particle or an elementary south pole. Since at every point of space such a direction may be found, if we commence at any point and draw a line so that, as we go along it, its direction at any point shall always coincide with that of the resultant force at that point, this curve will indicate the direction of that force for every point through which it passes, and might be called on that account a line of force. We might in the same way draw other lines of force, till we had filled all space with curves indicating by their direction that of the force at any assigned point.

* See especially Series XX XVIII. of the Experimental Researches, and Phil, Mag. 1852.

80

Mr. MAXWELL, ON FARADAY’S LINES OF FORCE.

We should thus obtain a geometrical model of the physical phenomena, which would tell us the direction of the force, but we should still require some method of indicating the intensity of the force at any point. If we consider these curves not as mere lines, but as fine tubes of variable section carrying an incompressible fluid, then, since the velocity of the fluid is inversely as the section of the tube, we may make the velocity vary according to any given law, by regulating the section of the tube, and in this way we might represent the intensity of the force as well as its direction by the motion of the fluid in these tubes. This method of representing the intensity of a force by the velocity of an imaginary fluid in a tube is applicable to any conceivable system of forces, but it is capable of great simplification in the case in which the forces are such as can be explained by the hypothesis
of attractions varying inversely as the square of the distance, such as those observed in elec-
trical and magnetic phenomena. In the case of a perfectly arbitrary system of forces, there
will generally be interstices between the tubes; but in the case of electric and magnetic forces it is possible to arrange the tubes so as to leave no interstices. The tubes will then be mere
surfaces, directing the motion of a fluid filling up the whole space, It has been usual to commence the investigation of the laws of these forces by at once assuming that the phenomena are due to attractive or repulsive forces acting between certain points. We may however obtain a different view of the subject, and one more suited to our more difficult inquiries, by adopting for the definition of the forces of which we treat, that they may be represented in magnitude and direction by the uniform motion of an incompressible fluid.
I propose, then, first to describe a method by which the motion of such a fluid can be clearly conceived; secondly to trace the consequences of assuming certain conditions of motion, and to point out the application of the method to some of the less complicated phenomena of electricity, magnetism, and galvanism; and lastly to shew how by an extension of these methods, and the introduction of another idea due to Faraday, the laws of the attractions and inductive actions of magnets and currents may be clearly conceived, without making any assumptions as to the physical nature of electricity, or adding anything to that which has been already proved by experiment.
By referring everything to the purely geometrical idea of the motion of an imaginary fluid, I hope to attain generality and precision, and to avoid the dangers arising from a premature theory professing to explain the cause of the phenomena, If the results of mere speculation which I have collected are found to be of any use to experimental philosophers, in arranging and interpreting their results, they will have served their purpose, and a mature theory, in which physical facts will be physically explained, will be formed by those who by interrogating Nature herself can obtain the only true solution of the questions which the mathematical theory suggests.

I. Theory of the Motion of an incompressible Fluid.
(1) The substance here treated of must not be assumed to possess any of the properties
of ordinary fluids except those of freedom of motion and resistance to compression. It is not

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

31

even a hypothetical fluid which is introduced to explain actual phenomena. It is merely a collection of imaginary properties which may be employed for establishing certain theorems in
pure mathematics in a way more intelligible to many minds and more applicable to physical problems than that in which algebraic symbols alone are used. The use of the word “ Fluid” will not lead us into error, if we remember that it denotes a purely imaginary substance with the following property :

The portion offluid which at any instant occupied a given volume, will at any succeeding instant occupy an equal volume.
This law expresses the incompressibility of the fluid, and furnishes us with a convenient measure of its quantity, namely its volume. The unit of quantity of the fluid will therefore be the unit of volume.
(2) The direction of motion of the fluid will in general be different at different points of
the space which it occupies, but since the direction is determinate for every such point, we
may conceive a line to begin at any point and to be continued so that every element of the line indicates by its direction the direction of motion at that point of space. Lines drawn in such a manner that their direction always indicates the direction of fluid motion are called lines of fluid motion.
If the motion of the fluid be what is called steady motion, that is, if the direction and velocity of the motion at any fixed point be independent of the time, these curves will represent the paths of individual particles of the fluid, but if the motion be variable this will not generally be the case, The cases of motion which will come under our notice will be those of steady motion.
(8) If upon any surface which cuts the lines of fluid motion we draw a closed: curve,
and if from every point of this curve we draw a line of motion, these lines of motion will generate a tubular surface which we may call a tube of fluid motion. Since this surface is generated: by lines in the direction of fluid motion no part of the fluid can flow across it, so
that this imaginary surface is as impermeable to the fluid as a real tube.
(4) The quantity of fluid which in unit of time crosses any fixed section of the tube is the same at whatever part of the tube the section be taken. For the fluid is incompressible, and no part runs through the sides of the tube, therefore the quantity which escapes from the second section is equal to that which enters through the first.
If the tube be such that unit of volume passes through any section in unit of time it is called a wnit tube offluid motion.
(5) In what follows, various units will be referred to, and a finite number of lines or surfaces will be drawn, representing in terms of those units the motion of the fluid. Now in order to define the motion in every part of the fluid, an infinite number of lines would have to be drawn at indefinitely small intervals; but since the description of such a system of lines would involve continual reference to the theory of limits, it has been thought better to suppose

92

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

the lines drawn at intervals depending on the assumed unit, and afterwards to assume the unit as small as we please by taking a small submultiple of the standard unit.
(6) Τὸ define the motion of the whole fluid by means of a system of unit tubes. Take any fixed surface which cuts all the lines of fluid motion, and draw upon it any system of curves not intersecting one another. On the same surface draw a second system of curves intersecting the first system, and so arranged that the quantity of fluid which crosses the surface within each of the quadrilaterals formed by the intersection of the two systems of curves shall be unity in unit of time. From every point in a curve of the first system let a line of fluid motion be drawn. These lines will form a surface through which no fluid passes. Similar impermeable surfaces may be drawn for all the curves of the first system, The curves of the second system will give rise to a second system of impermeable surfaces,
which, by their intersection with the first system, will form quadrilateral tubes, which will be
tubes of fluid motion. Since each quadrilateral of the cutting surface transmits unity of fluid in unity of time, every tube in the system will transmit unity of fluid through any of its sections in unit of time. The motion of the fluid at every part of the space it occupies is determined by this system of unit tubes; for the direction of motion is that of the tube through the point in question, and the velocity is the reciprocal of the area of the section of the unit tube at that point.

(7) We have now obtained a geometrical construction which completely defines the motion of the fluid by dividing the space it occupies into a system of unit tubes. We have next to shew how by means of these tubes we may ascertain various points relating to the motion of the fluid,
A unit tube may either return into itself, or may begin and end at different points, and these may be either in the boundary of the space in which we investigate the motion, or within
that space. In the first case there is a continual circulation of fluid in the tube, in the second the fluid enters at one end and flows out at the other. If the extremities of the tube are in the bounding surface, the fluid may be supposed to be continually supplied from without
from an unknown source, and to flow out at the other into an unknown reservoir; but if the origin of the tube or its termination be within the space under consideration, then we must conceive the fluid to be supplied by a source within that space, capable of creating and emit-
ting unity of fluid in unity of time, and to be afterwards swallowed up by a sink capable of receiving and destroying the same amount continually,
There is nothing self-contradictory in the conception of these sources where the fluid is created, and sinks where it is annihilated. The properties of the fluid are at our disposal, we have made it incompressible, and now we suppose it produced from nothing at certain points and reduced to nothing at others. The places of production will be called sources, and their numerical value will be the number of units of fluid which they produce in unit of time. The
places of reduction will, for want of a better name, be called sinks, and will be estimated by the
number of units of fluid absorbed in unit of time. Both places will sometimes be called sources, a source being understood to be a sink when its sign is negative.

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

33

(8) It is evident that the amount of fluid which passes any fixed surface is measured by the number of unit tubes which cut it, and the direction in which the fluid passes is determined
by that of its motion in the tubes. If the surface be a closed one, then any tube whose terminations lie on the same side of the surface must cross the surface as many times in the one
direction as in the other, and therefore must carry as much fluid out of the surface as it carries in. A tube which begins within the surface and ends without it will carry out unity of fluid; and one which enters the surface and terminates within it will carry in the same quantity. In order therefore to estimate the amount of fluid which flows out of the closed surface, we must subtract the number of tubes which end within the surface from the number
of tubes which begin there. If the result is negative the fluid will on the whole flow inwards,
If we call the beginning of a unit tube a unit source, and its termination a unit sink, then the quantity of fluid produced within the surface is estimated by the number of unit sources minus the number of unit sinks, and this must flow out of the surface on account of the incompressibility of the fluid.
Tn speaking of these unit tubes, sources and sinks, we must remember what was stated in
(5) as to the magnitude of the unit, and how by diminishing their size and increasing their number we may distribute them according to any law however complicated.

(9) If we know the direction and velocity of the fluid at any point in two different cases, and if we conceive a third case in which the direction and velocity of the fluid at any point is the resultant of the velocities in the two former cases at corresponding points, then the amount of fluid which passes a given fixed surface in the third case will be the algebraic sum of the quantities which pass the same surface in the two former cases. For the rate at which the fluid crosses any surface is the resolved part of the velocity normal to the surface, and the resolved part of the resultant is equal to the sum of the resolved parts of the com-
ponents.
Hence the number of unit tubes which cross the surface outwards in the third case must
be the algebraical sum of the numbers which cross it in the two former cases, and the number of sources within any closed surface will be the sum of the numbers in the two former cases. Since the closed surface may be taken as small as we please, it is evident that the distribution
of sources and sinks in the third case arises from the simple superposition of the distributions in the two former cases.

11. Theory of the uniform motion of an imponderable incompressible fluid through a resisting medium.

(10) The fluid is here supposed to have no inertia, and its motion is opposed by the action of a force which we may conceive to be due to the resistance of a medium through

which the fluid is supposed to flow. This resistance depends on the nature of the medium,

and will in general depend on the direction in which the fluid moves, as well as on its velocity. For the present we may restrict ourselves to the case of a uniform medium, whose resistance is the same in all directions, The law which we assume is as follows.

Vous ΣΧ, .Parr I,

5

34

Mr MAXWELL, ON FARADAY’S LINES OF FORCE. ©

Any portion of the fluid moving through the resisting medium is directly opposed by a retarding force proportional to its velocity.
If the velocity be represented by v, then the resistance will be a force equal to kv acting on
unit of volume of the fluid in a direction contrary to that of motion. In order, therefore, that the velocity may be kept up, there must be a greater pressure behind any portion of the fluid than there is in front of it, so that the difference of pressures may neutralise the effect of the resistance. Conceive a cubical unit of fluid (which we may make as small as we please, by (5)), and let it move ina direction perpendicular to two of its faces, Then the resistance will be kv, and therefore the difference of pressures on the first and second faces is kv, so that the pressure diminishes in the direction of motion at the rate of kv for every unit of length measured along the line of motion; so that if we measure a length equal to h units, the dif-
ference of pressure at its extremities will be kvh.

(11) Since the pressure is supposed to vary continuously in the fluid, all the points at which the pressure is equal to a given pressure p will lie on a certain surface which we may call the surface (p) of equal pressure. If a series of these surfaces be constructed in the fluid corresponding to the pressures 0, 1, 2, 3 &c., then the number of the surface will indicate the pressure belonging to it, and the surface may be referred to as the surface 0, 1,2 or 3. The unit of pressure is that pressure which is produced by unit of force acting on unit of surface. In order therefore to diminish the unit of pressure as in (5) we must diminish the unit of force in the same proportion.

(12) It is easy to see that these surfaces of equal pressure must be perpendicular to the lines of fluid motion; for if the fluid were to move in any other direction, there would be a resistance to its motion which could not be balanced by any difference of pressures. (We must remember that the fluid here considered has no inertia or mass, and that its properties are those only which are formally assigned to it, so that the resistances and pressures are the only things ‘to be considered.) There are therefore two sets of surfaces which by their intersection form the system of unit tubes, and the system of surfaces of equal pressure cuts both the others at right angles. Let h be the distance between two consecutive surfaces of equal pressure mea-
sured along a line of motion, then since the difference of pressures = 1,
kvh = 1,
which determines the relation of v to h, so that one can be found when the other is known. Let s be the sectional area of a unit tube measured on a surface of equal pressure, then since by the definition of a unit tube

we find by the last equation

8 = kh.

(13) The surfaces of equal pressure cut the unit tubes into portions whose length is h and section s. These elementary portions of unit tubes will be called wnit cells. In each of them unity of volume of fluid passes from a pressure p to a pressure (p—1) in unit of time, and’ therefore overcomes unity of resistance in that time. The work spent in overcoming resistance is therefore unity in every cell in every unit of time.

ΜΕ MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

35

(14) If the surfaces of equal pressure are known, the direction and magnitude of the velocity of the fluid at any point may be found, after which the complete system of
unit tubes may be constructed, and the beginnings and endings of these tubes ascertained and marked out as the sources whence the fluid is derived, and the sinks where it disappears.
In order to prove the converse of this, that if the distribution of sources be given, the
pressure at every point may be found, we must lay down certain preliminary propositions.

(15) If we know the pressures at every point in the fluid in two different cases, and if we take a third case in which the pressure at any point is the sum of the pressures at corresponding points in the two former cases, then the velocity at any point in the third case is the resultant of the velocities in the other two, and the distribution of sources is that due to the simple superposition of the sources in the two former cases.
For the velocity in any direction is proportional to the rate of decrease of the pressure
in that direction; so that if two systems of pressures be added together, since the rate
of decrease of pressure along any line will be the sum of the combined rates, the velocity in the new system resolved in the same direction will be the sum of the resolved parts in the two original systems. The velocity in the new system will therefore be the resultant of the velocities at corresponding points in the two former systems.
It follows from this, by (9), that the quantity of fluid which crosses any fixed surface
is, in the new system, the sum of the corresponding quantities in the old ones, and that the sources of the two original systems are simply combined to form the third.
It is evident that in the system in which the pressure is the difference of pressure in the two given systems the distribution of sources will be got by changing the sign of all the sources in the second system and adding them to those in the first.

(16) If the pressure at every point of a closed surface be the same and equal to p,
and if there be no sources or sinks within the surface, then there will be no motion of the
fluid within the surface, and the pressure within it will be uniform and equal to p. For if there be motion of the fluid within the surface there will be tubes of fluid
motion, and these tubes must either return into themselves or be terminated either within the surface or at its boundary. “Now since the fluid always flows from places of greater pressure to places of less pressure, it cannot flow in a re-entering curve; since there are
no sources or sinks within the surface, the tubes cannot begin or end except on the surface; and since the pressure at all points of the surface is the same, there can be no motion in tubes having both extremities on the surface, Hence. there is no motion within the
surface, and therefore no difference of pressure which would cause motion, and since the pressure at the bounding surface is p, the pressure at any point within it is also p.

(17) If the pressure at every point of a given closed surface be known, and the distribution of sources within the surface be also known, then only one distribution of pressures can exist within the surface.
For if two different distributions of pressures satisfying these conditions could be found, a third distribution could ‘be formed in which the pressure at any point shouldbe the
5—2

36

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

difference of the pressures in the two former distributions, In this case, since the pressures at the surface and the sources within it are the same in both distributions, the pressure at the surface in the third distribution would be zero, and all the sources within the surface would vanish, by (15).
Then by (16) the pressure at every point in the third distribution must be zero;
but this is the difference of the pressures in the two former cases, and therefore these
cases are the same, and there is only one distribution of pressure possible.

(18) Let us next determine the pressure at any point of an infinite body of fluid in the centre of which a unit source is placed, the pressure at an infinite distance from the source being supposed to be zero.
The fluid will flow out from the centre symmetrically, and since unity of volume flows out of every spherical surface surrounding the point in unit of time, the velocity at a distance r from the source will be
1
4πτ΄ The rate of decrease of pressure i:s therefore kv or i7k=’ and si, nce the pressure = 0

when ¢ is infinite, the actual pressure at any point will be SsTr The pressure is therefore inversely proportional to the distance from the source. It is evident that the pressure due to a unit sink will be negative and equal to k
4πτ΄ If we have a source formed by the coalition of S unit sources, then the resulting pressure wilF l be p = PΠ7Ὶo? a! that the pressure at a given di:stance varie5 s as the resi. stance and number of sources conjointly.

(19) Ifa number of sources and sinks coexist in the fluid, then in order to determine
the resultant pressure we have only to add the pressures which each source or sink produces. For by (15) this will be a solution of the problem, and by (17) it will be the only one, By this method we can determine the pressures due to any distribution of sources, as by the method of (14) we can determine the distribution of sources to which a given distribution of pressures is due.

(20) We have next to shew that if we conceive any imaginary surface as fixed in space and intersecting the lines of motion of the fluid, we may substitute for the fluid on one side of this surface a distribution of sources upon the surface itself without altering in any way the motion of the fluid on the other side of the surface.
For if we describe the system of unit tubes which defines the motion of the fluid, and wherever a tube enters through the surface place a unit source, and wherever a tube goes out through the surface place a unit sink, and at the same time render the surface
impermeable to the fluid, the motion of the fluid in the tubes will go on as before.

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

37

(21) If the system of pressures and the distribution of sources which produce them be known in a medium whose resistance is measured by &, then in order to produce the same system of pressures in a medium whose resistance is unity, the rate of production at each source must be multiplied by & For the pressure at any point due to a given source varies as the rate of production and the resistance conjointly; therefore if the pressure be constant, the rate of production must vary inversely as the resistance.

(22) On the conditions to be fulfilled at a surface which separates two media whose

coefficients of resistance are k and k’,

These are found from the consideration, that the quantity of fluid which flows out

of the one medium at any point flows into the other, and that the pressure varies con-

tinuously from one medium to the other. The velocity normal to the surface is the same

in both media, and therefore the rate of diminution of pressure is proportional to the

resistance. The direction of the tubes of motion and the surfaces of equal pressure will

be altered after passing through the surface, and the law of this refraction will be, that it

takes place in the plane passing through the direction of incidence and the normal to the

surface, and that the tangent of the angle of incidence is to the tangent of the angle of

refraction as k’ is to k.

|

(23) Let the space within a given closed surface be filled with a medium different from that exterior to it, and let the pressures at any point of this compound system due to a given distribution of sources within and without the surface be given; it is required
to determine a distribution of sources which would produce the same system of pressures
in a medium whose coefficient of resistance is unity. Construct the tubes of fluid motion, and wherever a unit tube enters either medium
place a unit source, and wherever it leaves it place a unit sink, Then if we make the
surface impermeable all will go on as before.
Let the resistance of the exterior medium be measured by &, and that of the interior.
by Κ΄. Then if we multiply the rate of production of all the sources in the exterior medium (including those in the surface), by k, and make the coefficient of resistance unity, the
pressures will remain as before, and the same will be true of the interior medium if we multiply all the sources in it by Xk’, including those in the surface, and make its resistance unity.
Since the pressures on both sides of the surface are now equal, we may suppose it permeable if we please.
We have now the original system of pressures produced in a uniform medium by a combination of three systems of sources. The first of these is the given external system multiplied by &, the second is the given internal system multiplied by k’, and the third is the system of sources and sinks on the surface itself. In the original case every source in the external medium had an equal sink in the internal medium on the other side of the surface, but now the source is multiplied by & and the sink by ζ΄, so that the result is for every external unit source on the surface, a source = (k — 1). By means of these three systems of sources the original system of pressures may be produced in a medium for which k = 1,

38

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

(24) Let there be no resistance in the medium within the closed surface, that is,

let Μ΄ = 0, then the pressure within the closed surface is uniform and equal to p, and the pressure at the surface itself is also p. If by assuming any distribution of pairs of sources

and sinks within the surface in addition to the given external and internal sources, and by supposing the medium the same within and without the surface, we can render the pressure

at the surface uniform, the pressures so found for the external medium, together with the uniform pressure p in the internal medium, will be the true and only distribution of pressures which is possible.

For if two such distributions could be found by taking different imaginary distributions of pairs of sources and sinks within the medium, then by taking the difference of the two

for a third distribution, we should have the pressure of the bounding surface constant in the new system and as many sources as sinks withia it, and therefore whatever fluid flows in at any point of the surface, an equal quantity must flow out at some other point.

In the external medium all the sources destroy one another, and we have an infinite
medium without sources surrounding the internal medium. The pressure at infinity is zero,

that at the surface is constant. If the pressure at the surface is positive, the motion of

the fluid must be outwards from every point of the surface; if it be negative, it must flow inwards towards the surface. But it has been shewn that neither of these cases is possible,
because if any fluid enters the surface an equal quantity must escape, and therefore the pressure at the surface is zero in the third system.

The pressure at all points in the boundary of the internal medium in the third case

is therefore zero, and there are no sources, and therefore the pressure is everywhere zero,

by (16).

:

_ The pressure in the bounding surface of the internal medium is also zero, and there

is no resistance, therefore it is zero throughout; but the pressure in the third case is the
difference of pressures in the two given cases, therefore these are equal, and there is only one distribution of pressure which is possible, namely, that due to the imaginary distribution

of sources and sinks.

(25) When the resistance is infinite in the internal medium, there can be no passage
of fluid through it or into it. The bounding surface may therefore be considered as
impermeable to the fluid, and the tubes of fluid motion will run along it without cutting it. If by assuming any arbitrary distribution of sources within the surface in addition to
the given sources in the outer medium, and by calculating the resulting pressures and velocities as in the case of a uniform medium, we can fulfil the condition of there being no velocity across the surface, the system of pressures in the outer medium will be the true one. For since no fluid passes through the surface, the tubes in the interior are independent of those outside, and may be taken away without altering the external, motion.

(26) If the extent of the internal medium be small, and if the difference of resistance in the two media be also small, then the position of the unit tubes will not be much altered. from what it would be if the external medium filled the whole space. :

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

39

On this supposition we can easily calculate the kind of alteration which the introduction

of the internal medium will produce; for wherever a unit tube enters the surface we must

,
conceive a source producing fluid at a rate τν

and wherever a tube leaves it we must

,

place a sink annihilating fluid at the rate

» then calculating pressures on the supposition

that the resistance in both media is k the same as in the external medium, we shall obtain the true distribution of pressures very approximately, and we may get a better result by repeating the process on the system of pressures thus obtained.

(27) If instead of an abrupt change from one coefficient of resistance to another we take a case in which the resistance varies continuously from point to point, we may treat the medium as if it were composed of thin shells each of which has uniform resistance. By properly assuming a distribution of sources over the surfaces of separation of the shells, we may treat the case as if the resistance were equal to unity throughout, as in (23). The sources will then be distributed continuously throughout the whole medium, and will be positive whenever the motion is from places of less to places of greater resistance, and negative when in the contrary direction.

(28) Hitherto we have supposed the resistance at a given point of the medium to be the same in whatever direction the motion of the fluid takes place; but we may conceive a case in which the resistance is different in different directions. In such cases the lines of motion will not in general be perpendicular to the surfaces of equal pressure. If a, ὃ, ὁ be the components of the velocity at any point, and a, /3, Ὑ the components of the resistance at the same point, these quantities will be connected by the following system of linear equations, which may be called ‘‘ equations of conduction,” and will be referred to by that name.

a=Pa+QB+ Ry, o I= PB + Qy+ R,a,

Pry + Qa + RB.
In these equations there are nine independent coefficients of conductivity. In order to

simplify the equations, let us put

Q+h, Ξ 95.) Q-R,=2lT,

δον θενος δῦ; φουτου τονεΟΣ

where

47" = (9, -- R,)* + (Q,- Δ)" + (Q, - κ᾿),

and 7, m, m are direction cosines of a certain fixed line in space. The equations then become
a=Pat+s8,8+S,y +(nB- my)T,

b= PB + δι + δια + (ly -- na)T,

c= ΡΟ + S,a + 8,8 — (ma -- 1B)T.

By the ordinary transformation of coordinates we may get rid of the coefficients marked S.

The equations then become

40

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

a= Ρία- (nB-m'y)T,
b= Ρ( β + (ly - n’a)T,
c = Py +(m'a - VB)T,

where 7’, m’, n’ are the direction cosines of the fixed line with reference to the new axes.

If we make

dp a =oda’

dp = d--y’

dp Ὦd Νso

the equation of continuity

da db ifde ΕἸ

becomes

da” dy .ds ’

d’p
Piss

, ap
Pay

,d’p
OF faa

and if we make then

aar/Plt, y= VPin #= VPS

op 2 ae d‘p

Pi5 g

+a =0 >

the ordinary equation of conduction. It appears therefore that the distribution of pressures is not altered by ‘the existence
of the coefficient 1. Professor Thomson has shewn how to conceive a substance in which this coefficient determines a property having reference to an axis, which unlike the axes
of P,, P,, P, is dipolar. For further information on the equations of conduction, see Professor Stokes On the
Conduction of Heat in Crystals (Cambridge and Dublin Math. Journ.), and Professor Thomson on the Dynamical Theory of Heat, Part V. (Transactions of Royal Society of
Edinburgh, Vol. X XI. Part I.) It is evident that all that has been proved in (14), (15), (16), (17), with respect to the
superposition of different distributions of pressure, and there being only one distribution of pressures corresponding to a given distribution of sources, will be true also in the case in which the resistance varies from point to point, and the resistance at the same point is different in different directions. For if we examine the proof we shall find it applicable
to such cases as well as to that of a uniform medium.

(29) We now are prepared to prove certain general propositions which are true in the most general case of a medium whose resistance is different in different directions and varies from point to point.
We may by the method of (28), when the distribution of pressures is known, construct the
surfaces of equal pressure, the tubes of fluid motion, and the sources and sinks. It is evident that since in each cell into which a unit tube is divided by the surfaces of equal pressure
unity of fluid passes from pressure p to pressure (p -- 1) in unit of time, unity of work is
done by the fluid in each cell in overcoming resistance,
The number of cells in each unit tube is determined by the number of surfaces of equal
pressure through which it passes, If the pressure at the beginning of the tube be p and at
the end ρ΄, then the number of cells in it will be p — p’. Now if the tube had extended from the

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

41

source to a place where the pressure is zero, the number of cells would have been p, and if the tube had come from the sink to zero, the number would have been p’, and the true number is the difference of these.
Therefore if we find the pressure at a source ,ϑ' from which S tubes proceed to be p, Sp is the number of cells due to the source δ: but if S’ of the tubes terminate in a sink at a pressure p’, then we must cut off Sp’ cells from the number previously obtained. Now if we denote the source of S' tubes by S, the sink of S’ tubes may be written — S", sinks always being reckoned negative, and the general expression for the number of cells in the system will be = (Sp).

(30) The same conclusion may be arrived at by observing that unity of work is done on

each cell, Now in each source S, § units of fluid are expelled against a pressure p, so that

the work done by the fluid in overcoming resistance is Sp. At each sink in which

S’ tubes terminate, ,5΄ units of fluid sink into nothing under pressure ρ΄; the work done upon

the fluid by the pressure is therefore S’p’.. The whole work done by the fluid may therefore

be expressed by

W = Sp -- 3S"p’,

or more concisely, considering sinks as negative sources,

W = X(Sp).

(31) Let S§ represent the rate of production of a source in any medium, and let p be the pressure at any given point due to that source. Then if we superpose on this another equal source, every pressure will be doubled, and thus by successive superposition we find that

a source nS would produce a pressure mp, or more generally the pressure at any point due to
a given source varies as the rate of production of the source. This may be expressed by the equation
p= RS, where R is a coefficient depending on the nature of the medium and on the positions of the

source and the given point.

In a uniform medium whose resistance is measured by k,

oe R k

PS 4πΆ

ry ° =Aa

R may be called the coefficient of resistance of the medium between the source and the given
point. By combining any number of sources we have generally p= (RS).

(32) Ina uniform medium the pressure due to a source S
ew
Pt"

At another source §” at a distance 7 we shall have

SPR ar ge TOP
if p’ be the pressure at § due to S’. If therefore there be two systems of sources
=(S’), and if the pressures due to the first be p and to the second ρ΄, then
=(S"p) = =(Sp’).
For every term S’p has a term Sp’ equal to it. Vou. X. Past I,

Σ(8) and 6

42

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

(38) Suppose that in a uniform medium the motion of the fluid is everywhere parallel to one plane, then the surfaces of equal pressure will be perpendicular to this plane. If we take two parallel planes at a distance equal to & from each other, we can divide the space between these planes into unit tubes by means of cylindric surfaces perpendicular to the planes, and these together with the surfaces of equal pressure will divide the space into cells of which the length is equal to the breadth. For if A be the distance between consecutive surfaces of
equal pressure and s the section of the unit tube, we have by (13) s = kh. But s is the product of the breadth and depth; but the depth is &, therefore the breadth
is ἃ and equal to the length. If two systems of plane curves cut each other at right angles so as to divide the plane
into little areas of which the length and breadth are equal, then by taking another plane at distance k from the first and erecting cylindric surfaces on the plane curves as bases, a system of cells will be formed which will satisfy the conditions whether we suppose the fluid to run
along the first set of cutting Jines or the second *.

Application of the Idea of Lines of Force.
I have now to shew how the idea of lines of fluid motion as described above may be modified so as to be applicable to the sciences of statical electricity, permanent magnetism, magnetism of induction, and uniform galvanic currents, reserving the laws of electro-magnetism for special consideration.
I shall assume that the phenomena of statical electricity have been already explained by the mutual action of two opposite kinds of matter. If we consider one of these as positive electricity and the other as negative, then any two particles of electricity repel one another with a force which is measured by the product of the masses of the particles divided by the square of their distance.
Now we found in (18) that the velocity of our imaginary fluid due to a source § at a distance x varies inversely as γ΄. Let us see what will be the effect of substituting such a source for every particle of positive electricity. The velocity due to each source would be proportional to the attraction due to the corresponding particle, and the resultant velocity due to all the sources would be proportional to the resultant attraction of all the particles. Now we may find the resultant pressure at any point by adding the pressures due to the given sources, and therefore we may find the resultant velocity in a given direction from the rate of decrease of pressure in that direction, and this will be proportional to the resultant attraction of the particles resolved in that direction.
Since the resultant attraction in the electrical problem is proportional to the decrease of pressure in the imaginary problem, and since we may select any values for the constants in the imaginary problem, we may assume that the resultant attraction in any direction is numerically equal to the decrease of pressure in that direction, or

* See Cambridge and Dublin Mathematical Journal, Vol, 111. p. 286.

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

43

By this assumption we find that if V be the potential, dV = Xdxv+ Ydy + Zdx = — dp,
or since at an infinite distance V = 0 and p=0, V = —p.
In the electrical problem we have
v= -Σ( 7.

Inthe Auid-pim Z (= =); 4nr 7

4
. S= = am.

If & be supposed very great, the amount of fluid produced by each source in order to keep up the pressures will be very small.
The potential of any system of electricity on itself will be

k

k

= (pdm) = ---, =(pS)=— W.

4π

4π

If =(dm), = (dm’) be two systems of electrical particles and pp’ the potentials due to them

respectively, then by (32)

Σ (pdm) =,k Σ(»5γ-k=, Σ( ,8) --Σ ’ἀπὸ,

π

4π

or the potential of the first system on the second is equal to that of the second system

on the first.

So that in the ordinary electrical problems the analogy in fluid motion is of this kind :

V=—-p,

:

Χ-d-=ku,

k dm = — 8S,
Amr
whole potential of a system = — 2Vdm = -w, where W is the work done by the fluid in over-
π
coming resistance. The lines of force are the unit tubes of fluid motion, and they may be estimated numerically
by those tubes.

Theory of Dielectrics.
The electrical induction exercised on a body at a distance depends not only on the distribution of electricity in the inductric, and the form and position of the inducteous body, but on the nature of the interposed medium, or dielectric. Faraday * expresses this by the conception

* Series XI.

6—2

44

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

of one substance having a greater inductive capacity, or conducting the lines of inductive action more freely than another. If we suppose that in our analogy of a fluid in a resisting medium the resistance is different in different media, then by making the resistance less we obtain the analogue to a dielectric which more easily conducts Faraday’s lines.
It is evident from (23) that in this case there will always be an apparent distribution of electricity on the surface of the dielectric, there being negative electricity where the lines enter and positive electricity where they emerge. In the case of the fluid there are no real sources on the surface, but we use them merely for purposes of calculation. In the dielectric there may be no real charge of electricity, but only an apparent electric action due to the surface.
If the dielectric had been of less conductivity than the surrounding medium, we should
have had precisely opposite effects, namely, positive electricity where lines enter, and negative
where they emerge. If the conduction of the dielectric is perfect or nearly so for the small quantities of elec-
tricity with which we have to do, then we have the case of (24). The dielectric is then considered as a conductor, its surface is a surface of equal potential, and the resultant attraction near the surface itself is perpendicular to it.

Theory of Permanent Magnets.
A magnet is conceived to be made up of elementary magnetized particles, each of which has its own north and south poles, the action of which upon other north and south poles is governed by laws mathematically identical with those of electricity. Hence the same application of the idea of lines of force can be made to this subject, and the same analogy of fluid motion can be employed to illustrate it.
But it may be useful to examine the way in which the polarity of the elements of a magnet may be represented by the unit cells in fluid motion. In each unit cell unity of fluid enters by one face and flows out by the opposite face, so that the first face becomes a unit sink and the second a unit source with respect to the rest of the fluid. It may therefore be compared to an elementary magnet, having an equal quantity of north and south magnetic matter distributed over two of its faces. If we now consider the cell as forming part of a system, the fluid flowing out of one cell will flow into the next, and so on, so that the source will be transferred from the end of the cell to the end of the unit tube. If all the unit tubes begin and end on the bounding surface, the sources and sinks will be distributed entirely on that surface, and in the case of a magnet which has what has been called a solenoidal or tubular distribution of magnetism, all the imaginary magnetic matter will be on the surface*.
Theory of Paramagnetic and Diamagnetic Induction.
‘
Faraday+ has shewn that the effects of paramagnetic and diamagnetic bodies in the magnetic field may be explained by supposing paramagnetic bodies to conduct the lines of force better,

* See Professor Thomson On the Mathematical gst of Magnetism, Chapters III. & V. Phil. Trans. 1851. + Experimental Researches (3292).

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

4

and diamagnetic bodies worse, than the surrounding medium. By referring to (23) and (26), and supposing sources to represent north magnetic matter, and sinks south magnetic matter, then if a paramagnetic body be in the neighbourhood of a north pole, the lines of force on entering it will produce south magnetic matter, and on leaving it they will produce an equal amount of north magnetic matter. Since the quantities of magnetic matter on the whole are
equal, but the southern matter is nearest to the north pole, the result will be attraction. If
on the other hand the body be diamagnetic, or a worse conductor of lines of force than the surrounding medium, there will be an imaginary distribution of northern magnetic matter
where the lines pass into the worse conductor, and of southern where they pass out, so that on
the whole there will be repulsion.
We may obtain a more general law from the consideration that the potential of the whole system is proportional to the amount of work done by the fluid in overcoming resistance. The introduction of a second medium increases or diminishes the work done according as the resist_ ance is greater or less than that of the first medium. The amount of this increase or diminution will vary as the square of the velocity of the fluid.
Now, by the theory of potentials, the moving force in any direction is measured by the rate of decrease of the potential of the system in passing along that direction, therefore when
κ΄, the resistance within the second medium, is greater than &, the resistance in the sur-
rounding medium, there is a force tending from places where the resultant force v is greater to
where it is less, so that a diamagnetic body moves from greater to less values of the resultant
force *.
In paramagnetic bodies k’ is less than ὦ, so that the force is now from points of less to points of greater resultant magnetic force. Since these results depend only on the relative values of k and ζ΄, it is evident that by changing the surrounding medium, the behaviour of a body may be changed from paramagnetic to diamagnetic at pleasure.
It is evident that we should obtain the same mathematical results if we had pinned that
the magnetic force had a power of exciting a polarity in bodies which is in the same direction as the lines in paramagnetic bodies, and in the reverse direction in diamagnetic bodies +. In fact we have not as yet come to any facts which would lead us to choose any one out of
these three theories, that of lines of force, that of imaginary magnetic matter, and that of induced polarity. As the theory of lines of force admits of the most precise, and at the same time least theoretic statement, we shall allow it to stand for the present.

Theory of Magnecrystallic Induction.
The theory of Faraday + with respect to the behaviour of crystals in the magnetic field may be thus stated. - In certain crystals and other substances the lines of magnetic force are

a Experimental Researches (2797), (2798). See Thom-

son, Cambridge and Dublin Mathematical Journal, May,

1847.

‘

ath

+ Exp. Res, (2429), (3320). See Weber, Poggendorff,
Ixxxvii. p. 145. Prof. Tyndall, Phil. Trans. 1856, p. 237.
+ Ezp. Res, (2836), &c.

46

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

conducted with different facility in different directions. The body when suspended in a uniform magnetic field will turn or tend to turn into such a position that the lines of force shall pass through it with least resistance. It is not difficult by means of the principles in (28)
to express the laws of this kind of action, and even to reduce them in certain cases to numerical
formule. The principles of induced polarity and of imaginary magnetic matter are here of little use; but the theory of lines of force is capable of the most perfect adaptation to this class of phenomena.

Theory of the Conduction of Current Electricity.
It is in the calculation of the laws of constant electric currents that the theory of fluid motion which we have laid down admits of the most direct application. In addition to the researches of Ohm on this subject, we have those of M. Kirchhoff, Ann. de Chim. x11. 496, and of M. Quincke, xivir. 203, on the Conduction of Electric Currents in Plates. According to the received opinions we have here a current of fluid moving uniformly in conducting circuits, which oppose a resistance to the current which has to be overcome by the application of an electro-motive force at some part of the circuit. On account of this resistance to the motion of the fluid the pressure must be different at different points in the circuit. This pressure, which is commonly called electrical tension, is found to be physically identical with the potential
in statical electricity, and thus we have the means of connecting the two sets of phenomena,
If we knew what amount of electricity, measured statically, passes along that current which
we assume as our unit of current, then the connexion of electricity of tension with current
electricity would be completed*. This has as yet been done only approximately, but we know enough to be certain that the conducting powers of different substances differ only in degree, and that the difference between glass and metal is, that the resistance is a great but finite quantity in glass, and a small but finite quantity in metal. ‘Thus the analogy between statical electricity and fluid motion turns out more perfect than we might have supposed, for there the induction goes on by conduction just as in current electricity, but the quantity conducted is insensible owing to the great resistance of the dielectrics +.

On Electro-motive Forces.
When a uniform current exists in a closed circuit it is evident that some other forces must act on the fluid besides the pressures. For if the current were due to difference of pressures, then it would flow from the point of greatest pressure in both directions to the point of least pressure, whereas in reality it circulates in one direction constantly. We

" See Exp. Res. (371).

+ Exp. Res. Vol. 111. p. 513.

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

47

must therefore admit the existence of certain forces capable of keeping up a constant current in a closed circuit. Of these the most remarkable is that which is produced by chemical action. A cell of a voltaic battery, or rather the surface of separation of the fluid of the
cell and the zinc, is the seat of an electro-motive force which can maintain a current in
opposition to the resistance of the circuit. If we adopt the usual convention in speaking
of electric currents, the positive current is from the fluid through the platinum, the conducting
circuit, and the zinc, back to the fluid again. If the electro-motive force act only in the
surface of separation of the fluid and zinc, then the tension of electricity in the fluid must
exceed that in the zinc by a quantity depending on the nature and length of the circuit
and on the strength of the current in the conductor. In order to keep up this difference of pressure there must be an electro-motive force whose intensity is measured by that difference
of pressure. If F' be the electro-motive force, J the quantity of the current or the number of electrical units delivered in unit of time, and K a quantity depending on the length and
resistance of the conducting circuit, then
F=IK=p-p,

where p is the electric tension in the fluid and p’ in the zine.

If the circuit be broken at any point, then since there is no current the tension of the part which remains attached to the platinum will be p, and that of the other will

be ρ΄. p—p’,, or F affords a measure of the intensity of the current. This distinction of

quantity and intensity is very useful *, but must be distinctly understood to mean nothing
more than this:—-The quantity of a current is the amount of electricity which it transmits

in unit of time, and is measured by J the number of unit currents which it contains.

The intensity of a current is its power of overcoming resistance, and is measured by F

or JK, where K is the resistance of the whole circuit.

The same idea of quantity and intensity may be applied to the case of magnetism t.

The quantity of magnetization in any section of a magnetic body is measured by the

number of lines of magnetic force which pass through it. The intensity of magnetization

in the section depends on the resisting power of the section, as well as on the number of lines

which pass through it. If & be the resisting power of the material, and ,ϑ' the area of

the section, and Z the number of lines of force which pass through it, then the whole

intensity throughout the section

ἘΠ 7tsk

When magnetization is produced by the influence of other magnets only, we may put p for the magnetic tension at any point, then for the whole magnetic solenoid

Pal fedo=K=p-y.

* Exp. Res, Vol. 111. p. 519.

+ Exp. Res. (2870), (3293).

48

.Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

When a solenoidal magnetized circuit returns into itself, the magnetization does not depend on difference of tensions only, but on some magnetizing force of which the intensity is #.
If i be the quantity of the magnetization at any point, or the number of lines of force passing through unit of area in the section of the solenoid, then the total quantity of magnetization in the circuit is the number of lines which pass through any section I = Sidydz, where dydz is the element of the section, and the summation is performed over the whole section.
The intensity of magnetization at any point, or the force required to keep up the magnetization, is measured by ki =f, and the total intensity of magnetization in the circuit is measured by the sum of the local intensities all round the circuit,
F = (fda),
where dz is the element of length in the circuit, and the summation is extended round
the entire circuit. In the same circuit we have always 1 = IK, where K is the total resistance of the
circuit, and depends on its form and the matter of which it is composed.

On the Action of closed Currents at a Distance.
The mathematical laws of the attractions and repulsions of conductors have been most ably investigated by Ampére, and his results have stood the test of subsequent experiments.
From the single assumption, that the action of an element of one current upon an
element of another current is an attractive or repulsive force acting in the direction of the line joining the two elements, he has determined by the simplest experiments the
mathematical form of the law of attraction, and has put this law into several most elegant
and useful forms. We must recollect however that no experiments have been made on these elements of currents except under the form of closed currents either in rigid conductors or in fluids, and that the laws of closed currents only can be deduced from such experiments. Hence if Ampére’s formule applied to closed currents give true results, their truth is not proved for elements of currents unless we assume that the action between two such elements must be along the line which joins them. Although this assumption is most warrantable and philosophical in the present state of science, it will be more conducive to freedom of investigation if we endeavour to do without it, and to assume the laws of closed currents as the ultimate datum of experiment.
Ampére has shewn that when currents are combined according to the law of the parallelogram of forces, the force due to the resultant current is the resultant of the forces due to the component currents, and that equal and opposite currents generate equal and opposite forces, and when combined neutralize each other.
He has also shewn that a closed circuit of any form has no tendency to turn a
moveable circular conductor about a fixed axis through the centre of the circle perpendicular to its plane, and that therefore the forces in the case of a closed circuit render Xda+ Ydy+Zdz a complete differential.

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

49

Finally, he has shewn that if there be two systems of circuits similar and similarly situated, the quantity of electrical current in corresponding conductors being the same, the resultant forces are equal, whatever be the absolute dimensions of the systems, which
proves that the forces are, ceteris paribus, inversely as the square of the distance. From these results it follows that the mutual action of two closed currents whose areas are
very small is the same as that of two elementary magnetic bars magnetized perpendicularly to the plane of the currents.
The direction of magnetization of the equivalent magnet may be predicted by remembering that a current travelling round the earth from east to west as the sun appears to do, would be equivalent to that magnetization which the earth actually possesses, and therefore in the reverse direction to that of a magnetic needle when pointing freely.
If a number of closed unit currents in contact exist on a surface, then at all points in which two currents are in contact there will be two equal and opposite currents which will produce no effect, but all round the boundary of the surface occupied by the currents there will be a residual current not neutralized by any other; and therefore the result will be the same as that of a single unit current round the boundary of all the currents,
From this it appears that the external attractions of a shell uniformly magnetized perpendicular to its surface are the same as those due to a current round its edge, for each of the elementary currents in the former case has the same effect as an element of the magnetic shell.
If we examine the lines of magnetic force produced by a closed current, we shall find that they form closed curves passing round the current and embracing it, and that the total intensity of the magnetizing force all along the closed line of force depends on the quantity of the electric current only. The number of unit lines* of magnetic force due to a closed current depends on the form as well as the quantity of the current, but the number of unit cells+ in each complete line of force is measured simply by the number of unit currents which embrace it. The unit cells in this case are portions of space in which unit of magnetic quantity is produced by unity of magnetizing force. The length of a cell is therefore inversely as the intensity of the magnetizing force, and its section is inversely as the quantity of magnetic induction at that point.
The whole number of cells due to a given current is therefore proportional to the strength of the current multiplied by the number of lines of force which pass through it. If by any change of the form of the conductors the number of cells can be increased, there will be a force tending to produce that change, so that there is always a force urging a conductor transverse to the lines of magnetic force, so as to cause more lines of force to pass through the closed circuit of which the conductor forms a part.
The number of cells due to two given currents is got by multiplying the number of
lines of inductive magnetic action which pass through each by the quantity of the currents respectively. Now by (9) the number of lines which pass through the first current is the sum of its own lines and those of the second current which would pass through the first if the

* Exp. Res. (3122). See Art. (6) of this paper.
Vous.) Parr dl,

+ Art. (13).
ῆ

δ0

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

second current alone were in action. Hence the whole number of cells will be increased by any motion which causes more lines of force to pass through either circuit, and therefore the resultant force will tend to produce such a motion, and the work done by this force during the motion will be measured by the number of new cells produced. All the actions of closed conductors on each other may be deduced from this principle.

On Electric Currents produced by Induction.
Faraday has shewn * that when a conductor moves transversely to the lines of magnetic force, an electro-motive force arises in the conductor, tending to produce acurrent in it. If the conductor is closed, there is a continuous current, if open, tension is the result. If a closed conductor move transversely to the lines of magnetic induction, then, if the number of lines which pass through it does not change during the motion, the electro-motive forces in the circuit will be in equilibrium, and there will be no current. Hence the electro-motive forces
depend on the number of lines which are cut by the conductor during the motion, If the
motion be such that a greater number of lines pass through the circuit formed by the conductor
after than before the motion, then the electro-motive force will be measured by the increase of the number of lines, and will generate a current the reverse of that which would have produced the additional lines. When the number of lines of inductive magnetic action through the circuit is increased, the induced current will tend to diminish the number of the lines, and when the number is diminished the induced current will tend to increase them.
That this is the true expression for the law of induced currents is shewn from the fact that, in whatever way the number of lines of magnetic induction passing through the circuit be increased, the electro-motive effect is the same, whether the increase take place by the motion
of the conductor itself,or of other conductors, or of magnets, or by the change of intensity of other currents, or by the magnetization or demagnetization of neighbouring magnetic bodies; or lastly by the change of intensity of the current itself.
In all these cases the electro-motive force depends on the change in the number of lines of inductive magnetic action which pass through the circuit f.

* Exp. Res. (3077), &c. + The electro-magnetic forces, which tend to produce motion of the material conductor, must be carefully distinguished from the electro-motive forces, which tend to produce electric
currents.
Let an electric current be passed through a mass of metal of any form. The distribution of the currents within the metal will be determined by the laws of conduction. Now let a constant electric current be passed through another conductor near the first. If the two currents are in the same direction the two conductors will be attracted towards each other, and would come nearer if not held in their positions. But though the material conductors are attracted, the currents (which are free to choose any course within the metal) will not alter their original distribution, or incline towards each other. For, since no change takes place in the system, there will be no electromotive forces to modify the original distribution of currents.

In this case we have electro-magnetic forces acting on the
material conductor, without any electro-motive forces tending
to modify the current which it carries. Let us take as another example the case of a linear con-
ductor, not forming a closed circuit, and let it be made to traverse the lines of magnetic force, either by its own motion, or by changes in the magnetic field. An electro-motive force will act in the direction of the conductor, and, as it cannot produce a current, because there is no circuit, it will produce electric tension at the extremities. There will be no electromagnetic attraction on the material conductor, for this attraction depends on the existence of the current within it, and this is
prevented by the circuit not being closed. Here then we have the opposite case of an electro-motive
force acting on the electricity in the conductor, but no attraction
on its material particles.

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

51

It is natural to suppose that a force of this kind, which depends on a change in the number of lines, is due to a change of state which is measured by the number of these lines: A closed conductor in a magnetic field may be supposed to be in a certain state arising from the magnetic action. As long as this state remains unchanged no effect takes place, but, when the state changes, electro-motive forces arise, depending as to their intensity and direction on this change of state. I cannot do better here than quote a passage from the first series of Faraday’s Experimental Researches, Art. (60).
“While the wire is subject to either volta-electric or magneto-electric induction it appears to be in a peculiar state, for it resists the formation of an electrical current in it; whereas, if in its common condition, such a current would be produced ; and when left uninfluenced it has the power of originating a current, a power which the wire does not possess under ordinary circumstances. This electrical condition of matter has not hitherto been recognised, but it probably exerts a very important influence in many if not most of the phenomena produced by currents of electricity. For reasons which will immediately appear (71) I have, after advising with several learned friends, ventured to designate it as the electro-tonic state.” Finding that all the phenomena could be otherwise explained without reference to the electrotonic state, Faraday in his second series rejected it as not necessary; but in his recent researches* he seems still to think that there may be some physical truth in his conjecture about this new state of bodies.
The conjecture of a philosopher so familiar with nature may sometimes be more pregnant with truth than the best established experimental law discovered by empirical inquirers, and though not bound to admit it as a physical truth, we may accept it as a new idea by which our mathematical conceptions may be rendered clearer.
In this outline of Faraday’s electrical theories, as they appear from a mathematical point of view, I can do no more than simply state the mathematical methods by which I believe that electrical phenomena can be best comprehended and reduced to calculation, and my aim has
been to present the mathematical ideas to the mind in an embodied form, as systems of lines or surfaces, and not as mere symbols, which neither convey the same ideas, nor readily adapt
themselves to the phenomena to be explained. The idea of the electro-tonic state, however, has not yet presented itself to my mind in such a form that its nature and properties may be clearly explained without reference to mere symbols, and therefore I propose in the following investigation to use symbols freely, and to take for granted the ordinary mathematical operations. By a careful study of the laws of elastic solids and of the motions of viscous fluids, I hope to discover a method of forming a mechanical conception of this electro-tonic state adapted to general reasoning +.

Part 11. On Faraday’s “ Electro-tonic State.”
When a conductor moves in the neighbourhood of a current of electricity, or of a magnet, or when a current or magnet near the conductor is moved, or altered in intensity, then a force

* (3172) (3269).

tion of Electric, Magnetic and Galvanic Forces. Camb. and

+ See Prof. W. Thomson On a Mechanical Representa- | Dub. Math. Jour, Jan. 1847.

7—2

52

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

acts on the conductor and produces electric tension, or a continuous current, according as the ‘circuit is open or closed. This current is produced only by changes of the electric or magnetic
phenomena surrounding the conductor, and as long as these are constant there is no observed effect on the conductor. Still the conductor is in different states when near a current or magnet, and when away from its influence, since the removal or destruction of the current or magnet occasions a current, which would not have existed if the magnet or current had not been previously in action.
Considerations of this kind led Professor Faraday to connect with his discovery of the induction of electric currents, the conception of a state into which all bodies are thrown by the presence of magnets and currents. This state does not manifest itself by any known phenomena as long as it is undisturbed, but any change in this state is indicated by a current or tendency towards a current. To this state he gave the name of the ““ Electro-tonic State,” and although he afterwards succeeded in explaining the phenomena which suggested it by means of less hypothetical conceptions, he has on several occasions hinted at the probability that some phenomena might be discovered which would render the electro-tonic state an object of legitimate induction. These speculations, into which Faraday had been led by the study of laws which he has well established, and which he abandoned only for want of experimental data for the direct proof of the unknown state, have not, I think, been made the subject of mathematical investigation. Perhaps it may be thought that the quantitative determinations of the various phenomena are not sufficiently rigorous to be made the basis of a mathematical theory; Faraday, however, has not contented himself with simply stating the numerical results of his experiments and leaving the law to be discovered by calculation. Where he has perceived a law
he has at once stated it, in terms as unambiguous as those of pure mathematics; and if the
mathematician, receiving this as a physical truth, deduces from it other laws capable of being tested by experiment, he has merely assisted the physicist in arranging his own ideas, which is confessedly a necessary step in scientific induction.
In the following investigation, therefore, the laws established by Faraday will be assumed as true, and it will be shewn that by following out his speculations other and more general laws can be deduced from them. If it should then appear that these laws, originally devised
to include one set of phenomena, may be generalized so as to extend to phenomena of a different
class, these mathematical connexions may suggest to physicists the means of establishing
physical connexions; and thus mere speculation may be turned to account in experimental
science,

On Quantity and Intensity as Properties of Electric Currents.
It is found that certain effects of an electric current are equal at whatever part of the circuit they are estimated. The quantities of water or of any other electrolyte decomposed at two different sections of the same circuit, are always found to be equal or equivalent, however different the material and form of the circuit may be at the two sections. The magnetic effect of a conducting wire is also found to be independent of the form or material of the wire

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

53

in the same circuit. There is therefore an electrical effect which is equal at every section of the

circuit. If we conceive of the conductor as the channel along which a fluid is constrained to

move, then the quantity of fluid transmitted by each section will be the same, and we may

define the quantity of an electric current to be the quantity of electricity which passes across

a complete section of the current in unit of time. We may for the present measure quantity

of electricity by the quantity of water which it would decompose in unit of time.

In order to express mathematically the electrical currents in any conductor, we must have

a definition, not only of the entire flow across a complete section, but also of the flow at a given

point in a given direction.

:

Der. The quantity of a current at a given point and in a given direction is measured, when uniform, by the quantity of electricity which flows across unit of area taken at that point perpendicular to the given direction, and when variable by the quantity which would flow across this area, supposing the flow uniformly the same as at the given point.
In the following investigation, the quantity of electric current at the point (yz) estimated in the directions of the axes w, y, x respectively will be denoted by a, ὅ; cs.
The quantity of electricity which flows in unit of time through the elementary area dS’
= dS (la, + mb, + ne),
where Jmn are the direction-cosines of the normal to dS. This flow of electricity at any point of a conductor is due to the electro-motive forces
which act at that point. These may be either external or internal. External electro-motive forces arise either from the relative motion of currents and
magnets, or from changes in their intensity, or from other causes acting at a distance. Internal electro-motive forces arise principally from difference of electric tension at points of
the conductor in the immediate neighbourhood of the point in question. The other causes are variations of caemical composition or of temperature in contiguous parts of the conductor.
Let p, represent the electric tension at any point, and X, Y, Z, the sums of the parts of all the electro-motive forces arising from other causes resolved parallel to the co-ordinate axes, then if a, B, ry, be the effective electro-motive forces
a, = X,- d“pδ, ε

dp,

phage

(A)

aw ierdp, ἢ|
Now the quantity of the current depends on the electro-motive force and on the resistance
of the medium. If the resistance of the medium be uniform in all directions and equal to k,,

ας = KA,

Be = Κ,}.»

y= kCo, (B)

but if the resistance be different in different directions, the law will be more complicated.

These quantities a, (8, Ὑ5 may be considered as representing the intensity of the electric

action in the directions of wyz.

δ4

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

The intensity measured along an element do of a curve

e=la+m3+ny,

where Zmn are the direction-cosines of the tangent. The integral fedo taken with respect to a given portion of a curve line, represents the total
intensity along that line. If the curve is a closed one, it represents the total intensity of the

electro-motive force in the closed curve. Substituting the values of aB-y from equations (A) fedo= [(Χάω + Ydy + Zdz) -—p+C.

If, therefore (Xdx + Ydy + Zdz) is a complete differential, the value of feda for a closed curve will vanish, and in all closed curves
Jedo = {(Xdx + Ydy + Zdz),

the integration being effected along the curve, so that in a closed curve the total intensity

of the effective electro-motive force is equal to the total intensity of the impressed electro-

motive force.

The total quantity of conduction through any surface is expressed by

fedS,

where

e=la+mb +n¢,

inn being the direction-cosines of the normal, fedS= ffadydz + [[bdzdx + f{edxdy,

the integrations being effected over the given surface. When the surface is a closed one, then we may find by integration by parts
fas = [2 + ἦνὉ 12)4 dy de.

If we make

feds = 40 |[fedadyde, where the integration on the right side of the equation is effected over every part of space
within the surface. In a large class of phenomena, including all cases of uniform currents,
the quantity p disappears.
Magnetic Quantity and Intensity.
From his study of the lines of magnetic force, Faraday has been led to the conclusion that
in the tubular surface* formed by a system of such lines, the quantity of magnetic induction
across any section of the tube is constant, and that the alteration of the character of these lines in passing from one substance to another, is to be explained by a difference of inductive
capacity in the two substances, which is analogous to conductive power in the theory of
electric currents.
* Exp. Res. 3271, definition of “ Sphondyloid.””

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

55

In the following investigation we shall have occasion to treat of magnetic quantity and intensity in connexion with electric. In such cases the magnetic symbols will be distinguished by the suffix 1, and the electric by the suffix 2. The equations connecting a, 6, 6, k, a, β, y; p, and p, are the same in form as those which we have just given, a, 6, ὁ are the symbols of magnetic induction with respect to quantity ; 4, denotes the resistance to magnetic induction, and may be different in different directions; a, β, ‘y, are the effective magnetizing forces, connected with a, 6, c, by equations (B); p, is the magnetic tension or potential which will be afterwards explained; p denotes the density of real magnetic matter and is connected with a, ὃ, ¢ by equations (C). As all the details of magnetic calculations will be more intelligible
after the exposition of the connexion of magnetism with electricity, it will be sufficient here to say that all the definitions of total quantity, with respect to a surface, and total intensity with respect to a curve, apply to the case of magnetism as well as to that of electricity.

Electro-magnetism.

Ampére has proved the following laws of the attractions and repulsions currents :
I. Equal and opposite currents generate equal and opposite forces,

of electric

II. A crooked current is equivalent to a straight one, provided the two currents nearly coincide throughout their whole length.
III. Equal currents traversing similar and similarly situated closed curves act with equal forces, whatever be the linear dimensions of the circuits.
IV. A closed current exerts no force tending to turn a circular conductor about its centre.
It is to be observed, that the currents with which Ampére worked were constant and therefore re-entering. All his results are therefore deduced from experiments on closed. currents, and his expressions for the mutual action of the elements of a current involve the assumption that this action is exerted in the direction of the line joining those elements. This assumption is no doubt warranted by the universal consent of men of science in treating of attractive forces considered as due to the mutual action of particles; but at present we are proceeding on a different principle, and searching for the explanation of the phenomena, not in the currents alone, but also in the surrounding medium.
The first and second Jaws shew that currents are to be combined like velocities or forces. The third law is the expression of a property of all attractions which may be conceived of as depending on the inverse square of the distance from a fixed system of points; and the fourth shews that the electro-magnetic forces may always be reduced to the attractions and repulsions of imaginary matter properly distributed. In fact, the action of a very small electric circuit on a point in its neighbourhood is identical with that of a small magnetic element on a point outside it. If we divide any given portion of a surface into elementary areas, and cause equal currents to flow in the same direction round all these little areas, the effect on a point not in the surface will be the

56

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

same as that of a shell coinciding with the surface, and uniformly magnetized normal to its surface. But by the first law all the currents forming the little circuits will destroy one another, and leave a single current running round the bounding line. So that the magnetic effect of a uniformly magnetized shell is equivalent to that of an electric current round the edge of the shell. If the direction of the current coincide with that of the apparent motion of the sun, then the direction of magnetization of the imaginary shell will be the same as that of the real magnetization of the earth*.
The total intensity of magnetizing force in a closed curve passing through and embracing the closed current is constant, and may therefore be made a measure of the quantity of the current. As this intensity is independent of the form of the closed curve and depends only on the quantity of the current which passes through it, we may consider the elementary case of the current which flows through the elementary area dyds.
Let the axis of # point towards the west, x towards the south, and y upwards. Let ayz be the position of a point in the middle of the area dydx, then the total intensity measured round the four sides of the element is

τῇ (x oF) ax

Total intensity = [d,- --oS)aydz.

The quantity of electricity conducted through the elementary area dydz is a,dydz, and therefore if we define the measure of an electric current to be the total intensity of magnetizing force in a closed curve embracing it, we shall have
πος ὐμαν .}
2 ds ~ dy’

δ: τὼ dry, _day yada? Sih eg

da,

dB,

Cy

dy da ant nee tee

These equations enable us to deduce the distribution of the currents of electricity whenever we know the values of a, β, yy, the magnetic intensities. Ifa, β, Ὑ be exact differentials of a function of wyz with respect to #, y and s respectively, then the values of a, ὃς ὁ5 disappear;

* See Experimental Researches (3265) for the relations between the electrical and magnetic circuit, considered as mutually embracing curves.

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

57

and we know that the magnetism is not produced by electric currents in that part of the field which we are investigating. It is due either to the presence of permanent magnetism within the field, or to magnetizing forces due to external causes.
We may observe that the above equations give by differentiation
da, db, de, da rfdy μὲdz which is the equation of continuity for closed currents. Our investigations are therefore for
the present limited to closed currents; and in fact we know little of the magnetic effects of any currents which are not closed.
Before entering on the calculation of these electric and magnetic states it may be advantageous to state certain general theorems, the truth of which may be established analytically.

The equation

Tueorem I.

oot eV as :

0,

+ay? * Sagick: Sp=

(where V and p are functions ενwys never she and vanishing for all points at an infinite distance,) can be satisfied by one, and only one, value of V. See Art. (17) above.

Tueorem II.
The value of V which will satisfy the above conditions is found by integrating the expression pdrdydx
ie -a'|bty- ψ + z- z' |)! where the limits of zyx are such as to include every point of space where p is finite.
The proofs of these theorems may be found in any work on attractions or electricity, and in particular in Green’s Essay on the Application of Mathematics to Electricity. See Arts. 18, 19 of this Paper. See also Gauss, on Attractions, translated in Taylx’s Scientific Memoirs.

Tueorem ITI,

Let U and V be two functions of vyz, then

ΟὟ at av

CGY dUdV dUdV

I e Se

ἐμ .5 +feg 1.5) ttyl Τ=ῊΝ { ada da * dy dy * de adzp)ἀπάγάς

.

where the integrations are supposed to at over all the space in which U and Vhave values

differing from 0.—(Green, p. 10.)

This theorem shews that if there be two attracting systems the actions between them are

equal and opposite. And by making U = V we find that the potential of a system on itself is

proportional to the integral of the square of the resultant attraction through all space; a

Vor. X. Part I.

8

58

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

result deducible from Art. (30), since the volume of each cell is inversely as the square of the velocity (Arts. 12, 13), and therefore the number of cells in a given space is directly as the square of the velocity.

Turorem IV.

Let a, 3, y, p be quantities finite through a certain space and vanishing in the space beyond, and let & be given for all parts of space as a continuous or discontinuous function of wyz, then the equation in p

d i

dp

dil

dp

ac are (6-3 vafae τίν - SE)+ 4x0 το,

has one, and only one solution, in which p is always finite and vanishes at an infinite distance.
The proof of this theorem, by Prof. W. Thomson, may be found in the Cambridge and Dublin Math. Journal, Jan. 1848.
If aBy be the electro-motive forces, p the electric tension, and & the coefficient of resist-
ance, then the above equation is identical with the equation of continuity

day Poni de, da * d dy y* dz +47p = 0;
and the theorem shews that when the electro-motive forces and the rate of production of electricity at every part of space are given, the value of the electric tension is determinate.
Since the mathematical laws of magnetism are identical with those of electricity, as far as we now consider them, we may regard α β as magnetizing forces, p as magnetic tension, and p as real magnetic density, k being the coefficient of resistance to magnetic induction.
The proof of this theorem rests on the determination of the minimum value of "

d
a= (if, («- a

avy ἢ

d

dV\2 1

d

τ} Ls(8 - ar ia) εχίν- Ἢ ε

dV
a

‘Steay de

where Vis got from the equation Vv ¢g av
ie Beedyoere * ae + 47p= 0,

and p has to be determined. The meaning of this integral in electrical language may be thus brought out. If the pre-
sence of the media in which & has various values did not affect the distribution of forces, then the “quantity” resolved i: n # would be sPimply dnVm and the i: ntensi:ty & ἼdVΣ But the actual quan-

1

d,

d

tity and intensity are τ(« - =) and a -- τῇ, and the parts due to the distribution of media

alone are therefore

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

59

Now the product of these represents the work done on account of this distribution of media, the distribution of sources being determined, and taking in the terms in y and z we get the expression Q for the total work done by that part of the whole effect at any point which is due to the distribution of conducting media, and not directly to the presence of the sources.
This quantity Q is rendered a minimum by one and only one value of p, namely, that which satisfies the original equation.

TuHeorEM V.

If a, ὃ. e be three functions of w, y, z satisfying the equation
i Backes spe
de dy dz Ὁ
it is always possible to find three functions a, 8, Ὑ which shall satisfy the equations

GePy
dz dy ᾿
dy ἀα.
ἀν dz ᾿
da ἀβ
ὩΣ

Let A = fedy, where the integration is to be performed upon ὁ considered as a function of y, treating w and κ' as constants. -Let B = Jadz, C = [bdv, A’=/bdz, B’ = fedx, C’ = (κὰν, integrated in the same way.
Then
ee Pa ghdx a

B= B-B + oA,
i y=C-C 7 } aἃ:

will satisfy the given equations; for

d d

d

db

τ -τῆς [τ ds -{¢abi pi dv + [Fay

and

om (Bde + [dos [Fass

ai 8 a dτyσ αν ἂdaν + fdaa dy + [dπa
=a.

00

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

In the same way it may be shewn that the values of a, , Ὑ satisfy the other given equations. The function y, may be considered at present as perfectly indeterminate.
The method here given is taken from Prof. W. Thomson’s memoir on Magnetism (Phil. Trans. 1851, p. 283).
As we cannot perform the required integrations when a, 6, 6 are discontinuous functions of ἃ, y, 2, the following method, which is perfectly general though more complicated, may indicate more clearly the truth of the proposition.
Let A, B, C be determined from the equations
dPAat*@αAν * aadea 7
sens TB aB ὌΣPΣΕhe dya?] aeaa b=0,
ard o.PC @&C da * dy? * ae +c=0,

by the methods of Theorems I. and II., so that 4, B, C are never infinite, and vanish when a, y, or αὶ is infinite.
Also let
"aἀβ deC edy

dC dA <&

Bs dx dz * dy’

. 44 dB LWdy

“dy da dz’ then

ἀβ dy _ Ὁ (442, ὩΣ ἘΞ +54+ 54)

ds dy daw\duw ἂν ἀξ

da dy ds

d (dA dB adc
* da (= dy * ay

If we find similar equations in y and x, and differentiate the first by , the second by y, and the third by x, remembering the equation between a, ὃ, c, we shall have

eda?s*dy’ * +dz.? )((4τ dB 2dC)" _o.;

and since A, B, C are always finite and vanish at an infinite distance, the only solution of this
equation is

and we have finally

with two similar equations, shewing that a, 8, y have been rightly determined.

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

61

The function y, is to be determined from the condition

da dB dy ee - αἵ ad

Ldaaγα * odrye + Ἔ oe Σ

ΣῊ

if the left-hand side of this equation be always zero, ψ' must be zero also.

Tueorem VI.
Let a, ὃ, ὁ be any three functions of #, Y, 2, it is possible to find three functions a, β, Ὕ and a fourth V, so that

du *dy * ds" ”

dB dy dV

and

hace hag ἣν + ao?

b dy da dV

de dz dy’

᾿

da dB dV

dy ἄν dz

Let

da db de

‘

dx * dy dz

BE?

and let V be found from the equation

¢

BV ον dav

da® + dy? * age =~ ΠΡ

then

;

dV

a ee

>

δ ἢ αὶ

|

dy

|

satisfy the condition

ee

dV

a ee

da db dé ὃ ἀφ * dy Ἢ ἀπ᾿ Ὁ

and therefore we can find three functions 4, B, C, and from these a, B, Ὑ, so as to satisfy the given equations.

TueEorEm VII.
The integral throughout infinity
Q = fff(aa, + 6,8; + evy:)dadydz,

62

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

where a,b, 6)» a; 3; Ὑι are any functions whatsoever, is capable of transformation into
Q = + [iff4mrppr — (acts + Bobo + γ.0.}} dadydz,
in which the quantities are found from the equations

ag a ee da, μὲdb, δι ie

‘

adἼeaΣ, Bdedpayt, = —aΝ—Δ] ++ 4 4701, = 0;

a, Bo yo V are determined from a, ὃ, ¢, by the last theorem, so that
ya eee πως OE
dz dy ‘dx a, b, c, are found from a, 8; y, by the equations

. and p is found from the equation

pd ς ΞῚ Ἐπ at &e
ἀξ dy

For, if we put a, in the form
dBy ἀγ. dV dz dy da’
and treat b, and 6) similarly, then we have by integration by parts thfugh infinity, remembering that all the functions vanish at the limits,
9- - Π γα τὰ+B) το φῇ) als-ἀ):» (dka>,y dip,a)[αυάγάς,

or and by Theorem ITI,
so that finally

Q=+ [ἀπ Κρ) = (a,d,+ Bobo + yote) }dadydz,
[[[Vp'dadydz = [[/ppddydz, Q = [Ifξ4πρρ — (αγας + Bibs = yr}sda)dyds.

If a,b, οἱ represent the components of magnetic quantity, and a, βι γι those of magnetic intensity, then p will represent the real magnetic density, and p the magnetic potential or
tension. ὧς ὃ; c, will be the components of quantity of electric currents, and ay By Ὑο will be three functions deduced from a, 6,¢,, which will be found to be the mathematical expression for Faraday’s Electro-tonic state.
Let us now consider the bearing of these analytical theorems on the theory of magnetism. Whenever we deal with quantities relating to magnetism, we shall distinguish them by the suffix (,). Thus a,6,¢, are the components resolved in the directions of #, y, z of the

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

63

quantity of magnetic induction acting through a given point, and a,B,y; are the resolved intensities of magnetization at the same point, or, what is the same thing, the components of the force which would be exerted on a unit south pole of a magnet placed at that point without disturbing the distribution of magnetism.
The electric currents are found from the magnetic intensities by the equations

When there are no electric currents, then
a,dx + B,dy + yidz = dp,,
a perfect differential of a function of a, y,z. On the principle of analogy we may call p, the magnetic tension.
The forces which act on a mass m of south magnetism at any point are
—m—ddpx,’,—-m—dd,py,’ and —m—dz’
in the direction of the axes, and therefore the whole work done during any displacement of a magnetic system is equal to the decrement of the integral
Q = S/ppidadyds throughout the system.
Let us now call Q the total potential of the system on itself. The increase or decrease of Q will measure the work lost or gainedby any displacement of any part of the system, and will therefore enable us to determine the forces acting on that part of the system.
By Theorem III. Q may be put under the form
1 Q = + re [[faa + b,B, + yy: )dadydz,
in which a, 8, γι are the differential coefficients of p, with respect to a, y, x respectively.
If we now assume that this expression for Q is true whatever be the values of a, βι γι» we pass from the consideration of the magnetism of permanent magnets to that of the magnetic effects of electric currents, and we have then by Theorem VII.
‘Q= [ff\pe -- = (aes+ Bib. + γι) ἡμιάγαν.
So that in the case of electric currents, the components of the currents have to be multiplied by the functions a,3,ry. respectively, and the summations of all such products throughout the system gives us the part of Q due to those currents.
We have now obtained in the functions ay 8, yo the means of avoiding the consideration of the quantity of magnetic induction which passes through the circuit. Instead of this artificial method we have the natural one of considering the current with reference to quantities existing in the same space with the current itself. ΤῸ these I give the name of Electro-tonic functions, or components of the Electro-tonic intensity.

64

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

Let us now consider the conditions of the conduction of the electric currents within the medium during changes in the electro-tonic state. The method which we shall adopt is an application of that given by Helmholtz in his memoir on the Conservation of Force*.
Let there’be some external source of electric currents which would generate in the conducting mass currents whose quantity is measured by a, 6, ¢, and their intensity by a, B. ys
Then the amount of work due to this cause in the time dé is

dt {ff(α;ας ὙΠ b3. + Cxry2)dadydz
in the form of resistance overcome, and

4BπeddtSfif(α,ας + b, By + Cory))dudydz

in the form of work done mechanically by the electro-magnetic action of these currents. If there be no external cause producing currents, then the quantity representing the whole work done by the external cause must vanish, and we have

Ὶ

dt d

dt ffjf(aya.+ ὃ.. + Cyty2)dadydz+ oe I[f/(fasa + 6,3, + Cory )dadydz,

where the integrals are taken through any arbitrary space. We must therefore have

Az

1d
+ b.B. + ΡΝ Δ = ree

a)

+ 8, a C20)

for every point of space; and it must be remembered that the variation of Q is supposed due to

variations of αὐ 3,7, and not of a,b,c,. We must therefore treat a,b,c, as constants, and
the equation becomes

α[α,e- e1)daa,) a+b b(y |βPεyεicαt1ςee1sa5.) +6,(v+e + —51

dry, x)

=0.

In order that this equation may be independent of the values of a, ὃ; ¢,, each of these coefficients must =0; and therefore we have the following expressions for the electro-motive forces due to the action of magnets and currents at a distance in terms of the electro-tonic functions,

It appears from experi:ment that the expressi2o5 n Τoα n refers to the change of electro-tonic: state

of a given particle of the conductor, whether due to change in the electro-tonic functions themselves or to the motion of the particle.
Ifa, be expressed as a function of a, y, x, and ¢, and if a, y, x be the co-ordinates of a moving article, then the electro-motive force measured in the direction of a is

1 (= dx da, dy da dz day’

Cg Ssσο
4π

ἀν dt +

dy dt” ds

dict sae}

* Translated in Taylor’s New Scientific Memoirs, Part 11.

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

65

The expressions for the electro-motive forces in y and x are similar. The distribution of currents due to these forces depends on the form and arrangement of the conducting media and on the resultant electric tension at any point.
The discussion of these functions would involve us in mathematical formule, of which this paper is already too full, It is only on account of their physical importance as the mathematical expression of one of Faraday’s conjectures that I have been induced to exhibit them at all in their present form. By a more patient consideration of their relations, and with the help of those who are engaged in physical inquiries both in this subject and in others not obviously connected with it, I hope to exhibit the theory of the electro-tonic state in a form in which all its relations may be distinctly conceived without reference to analytical calcula-
tions.

Summary of the Theory of the Electro-tonic State.
We may conceive of the electro-tonic state at any point of space as a quantity determinate in magnitude and direction, and we may represent the electro-tonic condition of a portion of space by any mechanical system which has at every point some quantity, which may be a velocity, a displacement, or a force, whose direction and magnitude correspond to those of the supposed electro-tonic state. This representation involves no physical theory, it is only a kind of artificial notation. In analytical investigations we make use of the three components of the electro-tonic state, and call them electro-tonic functions. We take the resolved part of the electro-tonic intensity at every point of a closed curve, and find by integration what we may call the entire electro-tonic intensity round the curve.
Prov. I. Jf on any surface a closed curve be drawn, and if the surface within it be divided into small areas, then the entire intensity round the closed curve is equal to the sum of the intensities round each of the small areas, all estimated in the same direction.
For, in going round the small areas, every boundary line between two of them is passed
along twice in opposite directions, and the intensity gained in the one case is lost in the other. Every effect of passing along the interior divisions is therefore neutralized, and the whole
effect is that due to the exterior closed curve.
Lawl. The entire electro-tonic intensity round the boundary of an element of surface measures the quantity of magnetic induction which passes through that surface, or, in other words, the number of lines of magnetic force which pass through that surface.
By Prop. I. it appears that what is true of elementary surfaces is true also of surfaces of finite magnitude, and therefore any two surfaces which are bounded by the same closed curve will have the same quantity of magnetic induction through them.
Law II. The magnetic intensity at any point is connected with the quantity of magnetic induction by a set of linear equations, called the equations of conduction*.

* See Art. (28).

Vor. X. Part I.

9

66

Mr MAXWELL, ON FARADAY'S LINES OF FORCE.

Law III. The entire magnetic intensity round the boundary of any surface measures the quantity of electric current which passes through that surface.
LawIV. The quantity and intensity of electric currents are connected by a system of equations of conduction.
By these four laws the magnetic and electric quantity and intensity may be deduced from the values of the electro-tonic functions. I have not discussed the values of the units, as that will be better done with reference to actual experiments. We come next to the attraction of
conductors of currents, and to the induction of currents within conductors,
Law V. The total electro-magnetic potential of a closed current is measured by the product of the quantity of the current multiplied by the entire electro-tonic intensity estimated in the same direction round the circutt.
Any displacement of the conductors which would cause an increase in the potential will be assisted by a force measured by the rate of increase of the potential, so that the mechanical work done during the displacement will be measured by the increase of potential.
Although in certain cases a displacement in direction or alteration of intensity of the
current might increase the potential, such an alteration would not itself produce work, and there will be no tendency towards this displacement, for alterations in the current are due to
electro-motive force, not to electro-magnetic attractions, which can only act on the conductor.
Law VI. The electro-motive force on any element of a conductor is measured by the instantaneous rate of change of the electro-tonic intensity on that element, whether in magnitude or direction,
The electro-motive force in a closed conductor is measured by the rate of change of the entire electro-tonic intensity round the circuit referred to unit of time. It is independent of the nature of the conductor, though the current produced varies inversely as the resistance; and it is the same in whatever way the change of electro-tonic intensity has been produced, whether by motion of the conductor or by alterations in the external circumstances.
In these six laws I have endeavoured to express the idea which I believe to be the mathematical foundation of the modes of thought indicated in the Eaperimental Researches. I do not think that it contains even the shadow of a true physical theory; in fact, its chief merit as a temporary instrument of research is that it does not, even in appearance, account for anything.
There exists however a professedly physical theory of electro-dynamics, which is so elegant, so mathematical, and so entirely different from anything in this paper, that I must state its axioms, at the risk of repeating what ought to be well_known. It is contained in M. W. Weber’s Llectro-dynamic Measurements, and may be found in the Transactions of the Leibnitz Society, and of the Royal Society of Sciences of Saxony *. The assumptions are,
(1) That two particles of electricity when in motion do not repel each other with the same force as when at rest, but that the force is altered by a quantity depending on the relative motion of the two particles, so that the expression for the repulsion at distance r is

* When this was written, I was not aware that part of M. | tal and theoretical, renders the study of his theory necessary to
Weber’s Memoir is translated in Taylor’s Scientific Memoirs, | every electrician. Vol. V. Art. x1v. The value of his researches, both experimen-

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

67

e7e’ (1atea r es br d=r) :
(2) That when electricity is moving in a conductor, the velocity of the positive fluid relatively to the matter of the conductor is equal and opposite to that of the negative fluid.
(3) The total action of one conducting element on another is the resultant of the mutual actions of the masses of electricity of both kinds which are in each.
(4) The electro-motive force at any point is the difference of the forces acting on the positive and negative fluids.
From these axioms are deducible Ampére’s laws of the attraction of conductors, and those of Neumann and others, for the induction of currents. Here then is a really physical theory, satisfying the required conditions better perhaps than any yet invented, and put forth by a philosopher whose experimental researches form an ample foundation for his mathematical investigations, What is the use then of imagining an electro-tonic state of which we have no distinctly physical conception, instead of a formula of attraction which we can readily understand? I would answer, that it is a good thing to have two waysof looking at a subject, and to admit that there are two ways of looking at it. Besides, I do not think that we have any
right at present to understand the action of electricity, and I hold that the chief merit of a temporary theory is, that it shall guide experiment, without impeding the progress of the true
theory when it appears. There are also objections to making any ultimate forces in nature depend on the velocity of the bodies between which they act. If the forces in nature are to be reduced to forces acting between particles, the principle of the Conservation of Force requires that these forces should be in the line joining the particles and functions of the distance only. The experiments of M. Weber on the reverse polarity of diamagnetics, which have been
recently repeated by Professor Tyndall, establish a fact which is equally a consequence of
M. Weber’s theory of electricity and of the theory of lines of force. With respect to the history of the present theory, I may state that the recognition of
certain mathematical functions as expressing the “ electro-tonic state” of Faraday, and the use of them in determining electro-dynamiec potentials and electro-motive forces, is, as far as I am aware, original ;but the distinct conception of the possibility of the mathematical expressions arose in my mind froin the perusal of Prof. W. Thomson’s papers ‘‘On a Mechanical Representation of Electric, Magnetic and Galvanic Forces,” Cambridge and Dublin Mathematical Journal, January, 1847, and his “‘ Mathematical Theory of Magnetism,” Philosophical Transac-
tions, Part I, 1851, ‘Art. 78, &c. As an instance of the help which may be derived from other
physical investigations, I may state that after I had investigated the ‘Theorems of this paper Professor Stokes pointed out to me the use which he had made of similar expressions in his
“Dynamical Theory of Diffraction,” Section 1, Cambridge Transactions, Vol. IX. Part 1.
Whether the theory of these functions, considered with reference to electricity, may lead to new
mathematical ideas to be employed in physical research, remains to be seen. I propose in the rest of this paper to discuss a few electrical and magnetic problems with reference to spheres. These are intended merely as concrete examples of the methods of which the theory has been given; I reserve the detailed investigation of cases chosen with special reference to experiment ‘till I have the means of testing their results,
9--

68

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

EXxamPLes.

I. Theory of Electrical Images.
The method of Electrical Images, due to Prof. W. Thomson*, by which the theory of spherical conductors has been reduced to great geometrical simplicity, becomes even more simple when we see its connexion with the methods of this paper, We have seen that the pressure at any point in a uniform medium, due to a spherical shell (radius = @) giving out fluid at the rate of 47Pa? unitFiat s iie n unit> of ti: me, iἾ s ΚΙ —ar 2 outsi:de the shell, and /Pa iΠnΡside iΒΥ)t, where r is the distance of the point from the centre of the shell.
If there be two shells, one giving out fluid at a rate 47Pa*, and the other absorbing at the rate 4a P’a’, then the expression for the pressure will be, outside the shells,

a 2

a "Ὁ

p =4rP r—-4rP’—r,

where 7 and γ΄ are the distances from the centres of the two shells.
to zero we have, as the surface of no pressure, that for which

Equating this expression

s

P 'q'2

Pa*®

Σ

Now the surface, for which the distances to two fixed points have a given ratio, is a sphere of which the centre O is in the line joining the centres of the shells CC’ produced, so that
_ Pay CO = CC χης Path

and its radius

Pa’. Pa?

Ὁ

Ree a

ae hae

Pa‘? - Pa? a

If at the centre of this sphere we place another source of the fluid, then the pressure due to this source must be added to that due to the other two; and since this additional pressure depends only on the distance from the centre, it will be constant at the surface of the sphere, where the pressure due to the two other sources is zero.
We have now the means of arranging a system of sources within a given sphere, so that when combined with a given system of sources outside the sphere, they shall produce a given constant pressure at the surface of the sphere.

* See a series of papers “On the Mathematical Theory of Electricity,” in the Cambridge and Dublin Math. Jour., beginning March, 1848.

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

69

Let a be the radius of the sphere, and p the given pressure, and let the given sources be at distances ὃ. ὃς &c. from the centre, and let their rates of production be 4aP,, 4aP, &c.
Then if at di:stances aar a &c. (measured i: n the same diBrielcitnion as b,b, &c. from the 1 %
centre) we place negative sources whase rates are

the pressure at the surface r= a will be reduced to zero. Now placi: ng a source 4 ΜaΗat
the centre, the pressure at the surface will be uniform and equal to p. The whole amount of fluid emitted by the surface r= ὦ may be found by adding the
rates of production of the sources within it. The result is

To apply this result to the case of a conducting sphere, let us suppose the external sources 4nP,, 42P, to be small electrified bodies, containing e, ¢ of positive electricity. Let us also suppose that the whole charge of the conducting sphere is = E previous to the action of the external points. Then all that is required for the complete solution of the problem is, that the surface of the sphere shall be a surface of equal potential, and that the total charge of the surface shall be E.
If by any distribution of imaginary sources within the spherical surface we can effect this, the value of the corresponding potential outside the sphere is the true and only one. The potential inside the sphere must really be constant and equal to that at the surface.

We must therefore find the images of the external electrified points, that is, for every

point at distance b from the centre we must find a point on the same radius at a distance

aξ2 and at that poiFnt we must place ἃ quanti: ty=-e ᾿aΞof i. magi:nary electriocsity.

1

1

At the centre we must put a quantity EZ’ such that

E'=E++e,0—5a

+e,

a —ὃ,+

&e.;

then if R be the distance from the centre, 7,7, &c. the distances from the electrified points, and 1’,r’, the distances from their images at any point outside the sphere, the potential at that point will be

70

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

This is the value of the potential outside the sphere. At the surface we have τ

so that at the surface

ὃ a

Rea

and

e+e

=

—, γι

ὃ a +=— &e. T] *1Ts

Pp —+ἀ-—ῶθι 4+ -6vΞ4agὧς,

and this must also be the value of p for any point within the sphere. For the application of the principle of electrical images the reader is referred to Prof.
Thomson’s papers in the Cambridge and Dublin Mathematical Journal. The only case

which we shall consider is that in which ool and 6, is infinitely distant along axis of «,
1
and ΕΞ.
The value p outside the sphere becomes then

and inside p=0.

pale(— 5ae),

11. On the effect of a paramagnetic or diamagnetic sphere in a uniform field of magnetic . force *.

The expression for the potential of a small magnet placed at the origin of co-ordinates in the direction of the axis of w is

d (m

Φ

-- (=) =-lm—.

dx \r

7

The effect of the sphere in disturbing the lines of force may be swpposed as a first hypothesis to be similar to that of a small magnet at the origin, whose strength is to

be determined. (We shall find this to be accurately true.) Let the value of the potential undisturbed by the presence of the sphere be

p=In. Let the sphere produce an additional potential, which for external points is

p= we,
and let the potential within the sphere be
P= Bu.
Let k’ be the coefficient of resistance outside, and k inside the sphere, then the conditions to be fulfilled are, that the interior and exterior potential should coincide at the

* See Prof. Thomson, on the Theory of Magnetic Induction, | induction (not the intensity) within the sphere to that without.

Phil. Mag. March, 1851. The inductive capacity of the sphere,
according to that paper, is the ratio of the guantity of magnetic

It Ἔis ΡtΙheΣrefore eqaual to 1‘° MoRll"Eeaese cceeording to our notation :

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

7?

surface, and that the induction through the surface should be the same whether deduced from the external or the internal potential. Putting «=r cos θ, we have for the external
potential
8
p= (w+ 45) cos0,

and for the internal

p= Br cos 6,

and these must be identical when r = a, or

Lf + A = iB:

The induction through the surface in the external medium is
1 dp = (I-24) c0s 6,
ki dr r=a
and that through the interior surface is

dp, 1 B cos@;
ya Je

These equations give

and .". Η(I-24) = ;B.

1- κα A= cba k

3k Be=

The effect outside the sphere is equal to that of a little magnet whose length is ἐ and moment ml, provided
. [= —2k_a——+-_kk_ ενa]

Suppose this uniform field to be that due to terrestrial magnetism, then, if & is less than Κ΄ as in paramagnetic bodies, the marked end of the equivalent magnet will be turned to the north. If k is greater than k’ as in diamagnetic bodies, the unmarked end of the equivalent magnet would be turned to the north.

III. Magnetic field of variable Intensity.
Now suppose the intensity in the undisturbed magnetic field to vary in magnitude and direction from one point to another, and that its components in zyx are represented by a, B,y;, then, if as a first approximation we regard the intensity within the sphere as sensibly equal to that at the centre, the change of potential outside the sphere arising from the presence of

72

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

the sphere, disturbing the lines of force, will be the same as that due to three small magnets at
the centre, with their axes parallel to Δ᾽, y, and x, and their moments equal to

The actual distribution of potential within and without the sphere may be conceived as the result of a distribution of imaginary magnetic matter on the surface of the sphere; but since the external effect of this superficial magnetism is exactly the same as that of the three small
magnets at the centre, the mechanical effect of external attractions will be the same as if the
three magnets really existed.
Now let three small magnets whose lengths are J, /,,, and strengths m, m, m, exist at the point 2yx with their axes parallel to the axes of wy x; then, resolving the forces on the three magnets in the direction of X, we have

Substituting the values of the moments of the imaginary magnets

,

! 43

The force impelling the sphere in the direction of w is therefore dependent on the variation of the square of the intensity or (αὐ + β᾽ + y*), as we move along the direction of #, and the same is true for y and x, so that the law is, that the force acting on diamagnetic spheres is from places of greater to places of less intensity of magnetic force, and that in similar distributions of magnetic force it varies as the mass of the sphere and the square of the intensity.
It is easy by means of Laplace’s Coefficients to extend the approximation to the value of
the potential as far as we please, and to calculate the attraction. For instance, if a north or
south magnetic pole whose strength is M, be placed at a distance b from a diamagnetic sphere, radius a, the repulsion will be

R24

8.2 @

4.3 at

R= k-¥)5 (Sy + waar e tee ate)

When τa. small, the first term giὃves a suffici:ent approximil atioan. The repulsiἐoξnόν is then as

the square of the strength of the pole and the mass of the sphere directly and the fifth power of the distance inversely, considering the pole as a point.

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

73

IV. Two Spheres in uniform field.

Let two spheres of radius ὦ be connected together so that their centres are kept at a dis-

tance b, and let them be suspended in a uniform magnetic field, then, although each sphere by

itself would have been in equilibrium at any part of the field, the disturbance of the field will produce forces tending to make the balls set in a particular direction.

Let the centre of one of the spheres be taken as origin, then the undisturbed potential is

p = Ircos0, and the potential due to the sphere is

;

k-k a

=e 4 - Κ' r οὐδ

el hae:
Pἀ ar(i-2sak+idk<τ,)£08 6,

1 dp

k-Kk a,

τ αθ ΤῸ πεν =) ἴα θ, - ----Ξ --

---------.-- —

dp
ἀφ-Ξ =0,

dp|? 1 ah

1 A

k-k αϑ

k—-¥ |'a°

& Ps

eos pease

pepe toes heoe

-- 13}1 Rea oe, na

2

ἘΣ

2

:

Te te ἘΠ dd asrsin’?@ dp εἰ + oe 7 (1 pack dey ary πα + eos]

This is the value of the square of the intensity at any point. The moment
tending to turn the combination of balls in the direction of the original force
aoe os VFiTπ my ὶ 1 τΠ ssὴ when r = 6,

of the couple

= 8. eka—-Kl|* ama8 Poe ks-oky, @ΤΥ oe 9,

5 ΤΣ ΣῊ πὶ οὗ +k 5) anda:

᾽

This expression, which must be positive, since 6 is greater than a, gives the moment of a force tending to turn the line joining the centres of the spheres towards the original lines of force.
Whether the spheres are magnetic or diamagnetic they tend to set in the axial direction, and that without distinction of north and south. If, however, one sphere be magnetic and the other diamagnetic, the line of centres will set equatoreally.. The magnitude of the force depends on the square of (%— Κ΄), and is therefore quite insensible except in iron *.

V. Two Spheres between the poles of a Magnet.
Let us next take the case of the same balls placed not in a uniform field but between a north and a south pole, + M, distant 2c from each other in the direction of «.

Vor, X.

ἈΠ See Prof. Thomson in Phil. Mag. March, 1851.
Paezr I.

10

74

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

The expression for the potential, the middle of the line joining the poles being the origin, is

M

1

Ps (FS cos @cr

1
4 c?7++2 cos i)"

From this we find as the value of 1",

ΤΣ re - 18“ γ sin 26,
and the moment to turn a pair of spheres (radius a, distance 2b) in the direction in which @ is increased is
k-k Mab? ar ara co sin 20.
This force, which tends to turn the line of centres equatoreally for diamagnetic and axially for magnetic spheres, varies directly as the square of the strength of the magnet, the cube of the radius of the spheres and the square of the distance of their centres, and inversely as the sixth power of the distance of the poles of the magnet, considered as points. As long as these poles are near each other this action of the poles will be much stronger than the mutual action of the spheres, so that as a general rule we may say that elongated bodies set axially or equatoreally between the poles of a magnet according as they are magnetic or diamagnetic. If, instead of being placed between two poles very near to each other, they had been placed in a uniform field such as that of terrestrial magnetism or that produced by a spherical electro-magnet (see Ex. VIII.), an elongated body would set axially whether magnetic or diamagnetic.
In all these cases the phenomena depend on k—K, so that the sphere conducts itself magnetically or diamagnetically according as it is more or less magnetic, or less or more diamagnetic than the medium in which it is placed.

VI. On the Magnetic Phenomena of a Sphere cut from a substance whose coefficient of resistance is different in different directions.

Let the axes of magnetic resistance be parallel throughout the sphere, and let them

be taken for the axes of a, y, x. . Let k,, ἴω» ks, be the coefficients of resistance in these three

directions, and let ζ΄ be that of the external medium, and a the radius of the sphere. Let 7 be the undisturbed magnetic intensity of the field into which the sphere is introduced, and

let its direction-cosines be J, m, n. Let us now take the case of a homogeneous sphere whose coefficient is k, placed in a
uniform magnetic field whose intensity is 11 in the direction of 2 The resultant potential

outside the sphere would be

:

k,-k αὐ

=

1 —_—_—- —

P u( toa a)”

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

75

and for internal points

3k,
Pi = 1] a—h——+k a.

So that in the interior of the sphere the magnetization is entirely in the direction of «. It is therefore quite independent of the coefficients of resistance in the directions of w and y,

which may be changed from k, into k, and &; without disturbing this distribution of magnetism.

We may therefore treat the sphere as homogeneous for each of the three components of J, but we must use a different coefficient for each. We find for external points

p arnω t my + Ns + (ρπαὰν

ey Ke See

2Ikn,-+KKenz)aN=a ;

and for internal points

oe ( 38k, a

Pi=

ΕΝ χ'

3k, ss

3k,

2k, + Kk Yy + 2hs+ i na).

The external effect is the same as that which would have been produced if the small
magnet whose moments are

kake,sn-mka’gk)”πο

kp— κ
2k,+Iere Ne (Becdadets PETaa

fekayta—eskeK ia 3

had been placed at the origin with their directions coinciding with the axes of a,y,x. The effect of the original force J in turning the sphere about the axis of # may be found by taking the moments of the components of that force on these equivalent magnets. The moment of the force in the direction of y acting on the third magnet is

and that of the force in s on the second magnet is
iy
hed 2k,+ hemni*a’*. 2
The whole couple about the axis of 2 is therefore
3k! (Key — ks) (le, + I’)(2k, +) 7":
tending to turn the sphere round from tlie axis of y towards that of =. Suppose the sphere
to be suspended so that the axis of δ is vertical, and let J be horizontal, then if @ be the angle which the axis of y makes with the direction of 7, m= cos 0, n = — sin@, and the expression for the moment becomes

8

K (k.- ks)

(2h, +k’) (2h,+ k’)

I*a?

sin 20

tending to increase 0. The axis of least resistance therefore sets axially, but with either end indifferently towards the north.
Since in all bodies, except iron, the values of & are nearly the same as in a vacuum, 10—2

70

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

the coefficient of this quantity can be but little altered by changing value in space. The expression then becomes
ne ἡ k,—k 3 733 sin 20,

the value of k’ tok, the

independent of the external medium *.

VII. Permanent magnetism in a spherical shell.
The case of a homogeneous shell of a diamagnetic or paramagnetic substance presents no difficulty. The intensity within the shell is less than what it would have been if the shell were away, whether the substance of the shell be diamagnetic or paramagnetic. When the resistance of the shell is infinite, and when it vanishes, the intensity within the shell is zero.
In the case of no resistance the entire effect of the shell on any point, internal or external, may be represented by supposing a superficial stratum of magnetic matter spread over the outer surface, the density being given by the equation
p= 81 cos 0.
Suppose the shell now to be converted into a permanent magnet, so that the distribution of imaginary magnetic matter is invariable, then the external potential due to the shell will be
3
»κ--- 1 5 cos 8,

and the internal potential

p,= — Ir cos 0.

’

Now let us investigate the effect of filling up the shell with some substance of which

the resistance is ἔφ the resistance in the external medium being k’. The thickness of the
magnetized shell may be neglected. Let the magnetic moment of the permanent magnetism

be Ja’, and that of the imaginary superficial distribution due to the medium k= Aa*, Then the potentials are

3
external p’= (I + A) = cos@, internal p, = (J + A) r cos θ.

The distribution of real magnetism is the same before and after the introduction of the

medium &, so that

Gnd fo 1

4

τιν πετῶ εν(1:4),

5- ἢ

Ὁ

The external effect of the magnetized shell is increased or diminished according as & is greater or less than ζ΄. It is therefore increased by filling up the shell with diamagnetic matter, and diminished by filling it with paramagnetic matter, such as iron.

* Taking the more general case of magnetic induction referred to in Art. (28), we find, in the expression for the moment of the magnetic forces, a constant term depending on 7’, besides those terms which depend on sines and cosines of 0. The result is, that in every complete revolution in the negative direction round the axis of 7', a certain positive amount of work is gained; but, since no inexhaustible source of work can exist

in nature, we must admit that 7'=0 in all substances, with respect to magnetic induction. This argument does not hold in the case of electric conduction, or in the case of a body
through which heat or electricity is passing, for such states are maintained by the continual expenditure of work. See Prof.
Thomson, Phil, Mag. March, 1851, p. 186.

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

77

VIII. Llectro-magnetic spherical shell.

Let us take as an example of the magnetic effects of electric currents, an electro-magnet

in the form of a thin spherical shell. Let its radius be a, and its thickness ¢, and let its external effect be that of a magnet whose moment is Ja*. Both within and without the shell

the magnetic effect may be represented by a potential, but within the substance of the shell,

where there are electric currents, the magnetic effects cannot be represented by a potential.

Let p’, p, be the external and internal potentials,

αϑ
p'= 1-r- οο5θ,

p, = Ar cos 0,

: ἄρ’ ἃ and since there is no permanent magnetism, — = =, when r = a,

Α - -- 4].
If we draw any closed curve cutting the shell at the equator, and at some other point for which @ is known, then the total magnetic intensity round this curve will be 3Ia cos @, and as this is a measure of the total electric current which flows through it, the quantity of the current at any point may be found by differentiation. The quantity which flows through the element ¢d@ is — 87. sin θάθ, so that the quantity of the current referred to unit of area of section is
- 31:a si; n 8.

If the shell be composed of a wire coiled round the sphere so that the number of coils

to the inch varies as the sine of 0, then the external effect will be nearly the same as if

the shell had been made of a uniform conducting substance, and the currents had been

distributed according to the law we have just given.

If a wire conducting a current of strength 7, be wound round a sphere of radius a

6

fe

Ξ

Fi

. 24

so that the distance between successive coils measured along the axis of w is —, then

n

there will be m coils altogether, and the value of J, for the resulting electro-magnet will be

= Gnal”

The potentials, external and internal, will be

p=; τςn aτ?ε088,

Pp, =~ 21,n=r=cos 8.

The interior of the shell is therefore a uniform magnetic field. .

IX. Effect of the core of the electro-magnet.
Now let us suppose a sphere of diamagnetic or paramagnetic matter introduced into the electro-magnetic coil. The result may be obtained as in the last case, and the potentials become
Τ. ΞΕ,,-nΥ,3k α= cos 8, di =— 21,—n ——88. —
The external effect is greater or less than before, according as # is greater or less than &, that is, according as the interior of the sphere is magnetic or diamagnetic with

78

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

respect to the external medium, and the internal effect is altered in the opposite direction, being greatest for a diamagnetic medium.
This investigation explains the effect of introducing an iron core into an electro-magnet. If the value of & for the core were to vanish altogether, the effect of the electro-magnet
would be three times that which it has without the core. As & has always a finite value, the effect of the core is less than this.
In the interior of the electro-magnet we have a uniform field of magnetic force, the intensity of which may be increased by surrounding the coil with a shell of iron. If k’ = 0, and the shell infinitely thick, the effect on internal points would be tripled.
The effect of the core is greater in the case of a cylindric magnet, and greatest of all when the core is a ring of soft iron.

X. Electro-tonic functions in spherical electro-magnet.

Let us now find the electro-tonic functions due to this electro-magnet.

They will be of the form

a = 0,

By = 8,

Yo= - ὧν.

where ὦ is some function of γ. each = 0, and this implies

Where there are no electric currents, we must have dy», ὃ» cs
d (s 73: me
dec fey”

the solution of which is

Ὁ
a= Ο᾽ + τ .

Within the shell ὦ cannot become infinite; therefore w = C, is the solution, and outside

α must vanish at an infinite distance, so that
o=—Cs
"3 is the solution outside. The magnetic quantity within the shell is found by last article to be

-21,26—na a—k—+8;k=+a4=

dB,
dr

dry
ἀν

therefore within the sphere

H=- —

Outside the sphere we must determine w so as to coincide at the surface with the internal value. The external value is therefore

"ga Ske τῇ
where the shell containing the currents is made up of m coils of wire, conducting a current of total quantity J,.
Let another wire be coiled round the shell according to the same law, and let the total
number of coils be m’; then the total electro-tonic intensity EZ, round the second coil is
found by integrating
2π
El,= f wa sin Ads,

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

79

along the whole length of the wire. The equation of the wire is
cos 9 = —φ—,
n'a where 7 is a large number; and therefore
ds = a sin 6ddq,
= — an'x sin*6d0,
“ς. Ε1,πae---4-π Weswan =~ 28—πann [3k——+1 k. E may be called the electro-tonic coefficient for the particular wire.

XI. Spherical electro-magnetic Coil-Machine.

We have now obtained the electro-tonic function which defines the action of the one coil on the other. The action of each coil on itself is found by putting n? or mn” for nm’. Let the first coil be connected with an apparatus producing a variable electro-motive force F. Let us find the effects on both wires, supposing their total resistances to be R and R’, and the quantity of the currents 1 and J’.

Let NW stand for p8 o(8nk+ok) , then the electro-motive force of the first wire on the second is dI
— Nnn Ud ae

That of the second on itself is

- Nnd” ta—r.’,

The equation of the current in the second wire is therefore — Nnn ,a al aei)acaemmEae Peeves ὙΠ ΞΟ ἢ

The equation of the current in the first wire is

dI

al’

ni Ἢ Nnn ae Fe RI

(2)

=

3 ΤΥ

ΑΞ:

=

eeeeeeveseeoe

Eliminating the differential coefficients, we get
pagy tiΤΣον
n n n

τ dN (nR>GR4)mEn’) aαἱtἘΞ RFGtten® GdaF o ἰὼ

from which to find Zand I’. For this purpose we require to know the value of F in terms
oft.
Let us first take the case in which F is constant and 7 and J’ initially = 0. This is the case of an electro-magnetic coil-machine at the moment when the connexion is made with the galvanic trough.
Ρ ἂ

80

Mr MAXWELL, ON FARADAY’S LINES OF FORCE.

n2

12

Putting 37 for N (7 +7) we find

-

F

#

T=

(1-67)

,

n’ -Ξ

ΤΑ,"

.

The pri9 mary current i. ncreases very rapi.dly from O to RF and the secondary commences at

wes F

,
~and

speedily

vanishes,

owing

to the value

of + being

generally

very

small.

The whole work done by either current in heating the wire or in any other kind of action
is found from the expression
ἐνP Rat.
0
The total quantity of current is
fo Tat.
0
For the secondary current we find

τὰPRRdiitt==T > τ—ἄ—ρα —τ΄

[ PABἄ! kτoτFarςc<e2t

The work done and the quantity of the current are therefore the same as-if a current
7,
of quantity 7’ = Rs had passed through the wire for a time 7, where

n?

n'2

roan (E+)

This method of considering a variable current of short duration is due to Weber, whose
experimental methods render the determination of the equivalent current a matter of great precision.
Now let the electro-motive force F' suddenly cease while the current in the primary wire is J, and in the secondary =0. Then we shall have for the subsequent time

at I=I,e *,

1 at a cs

The equivalent currents are1 J, and ἃ ἢἘΞ— , and their duration is τ᾿
When the communication with the source of the current is cut off, there will be a change of R. This will produce a change in the value of 7+, so that if R be suddenly increased, the strength of the secondary current will be increased, and its duration diminished. This is the case in the ordinary coil-machines. The quantity N depends on the form of the machine, and may be determined by experiment for a machine of any shape.

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

81

XII. Spherical shell revolving in magnetic field.

Let us next take the case of a revolving shell of conducting matter under the influence of a uniform field of magnetic force. The phenomena are explained by Faraday in his Experimental Researches, Series [1., and references are there given to previous experiments.
Let the axis of # be the axis of revolution, and let the angular velocity be w. Let the magnetism of the field be represented in quantity by J, inclined at an angle @ to the direction of x, in the plane of sa.
Let R be the radius of the spherical shell, and 7 the thickness. Let the quantities
αν» Bo Yn be the electro-tonic functions at any point of space; a@,, ὃ,» 6)» a, βι» γι symbols
of magnetic quantity and intensity; a, be, 62» ag, Bo, ‘2 of electric quantity and intensity.
Let p, be the electric tension at any point,

dp.

)

ag = a + kay

By = day tHe Vics atone edd

etl)

Ὕ2 =—ddpzs +ke 2 }

da, db, de,
nica ee de
ty, Bs ty

er

τς (2)3

‘de dy dz

i

The expressions for ap, 39, yo due to the magnetism of the field are

ἢ ἡ ay = Ay + 5 y cos 0,

By = By +5yaaa (x siστὸn 9 — x cos 8),

tae
As C, --ἰς ysin 0,

A,, B,, C, being constants; and the velocities of the particles of the revolving sphere are

da _
Ἢ" ci

ἂν τι dz ᾿

aa

ἀρ

We have therefore for the electro-motive forces

err 1etdrayy ΞΊ1Σ an Oe,

1 dB, 1

ae rie ree Pe

|

1 dy,

emir

pe

aa ae
τς 7m Coe

Vor. X. Parr J,

11

82

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

Returning to equations (1), we get

ἐ{15- Ξὴ _ 4β,, ty _
dz dy ds dy ’ de, da,\ dy, da, 1 1
(3 - 2)- pon ae gee hey
g (Sn) oe a,
dy da} dy ds From which with equation (2) we find
a las 1eral sin Owz,
bale Uy
o2aK 4: Α4 Ν θωω :

Po= sia Iw {(x* + y*) cos@— a sin 0}. 167

These expressions would determine completely the motion of electricity in a revolving

sphere if we neglect the action of these currents on themselves. They express a system

of circular currents about the axis of y, the quantity of current at any point being

proportional to the distance from that axis. The external magnetic effect will be that

of a small magnet whose moment is ἘἽ.ΝῚ3 wI sin θ, wictuhte!its diotrectiτοςon along the axisς of y,
7
so that the magnetism of the field would tend to turn it back to the axis of «*.

The existence of these currents will of course alter the distribution of the electro-tonic

functions, and so they will react on themselves. Let the final result of this action be a system of currents about an axis in the plane of ay inclined to the axis of # at an angle @

and producing an external effect equal to that of a magnet whose moment is J’R’.

The magnetic inductive components within the shell are

10sin -- 21’ cos in a,

- 21 sind in y,

I, cos @

. in x.

Each of these would produce its own system of currents when the sphere is in motion,

and these would give rise to new distributions of magnetism, which, when the velocity is

uniform, must be the same as the original distribution,

(ὦ, sin 6 -- 2I’ cos Φ) in # produces 2 =e (J, sin@- 2I’ cos φ) in y,
7
(- 27’ Ἐπsin φ) i: n y produces 2 aSkO” (2I’ sisnsφs) ina;

1 cos θ in produces no currents,

* The expression for pz indicates a variable electric tension in the shell, so that currents might be collected by wires touching it at the equator and poles.

Mr MAXWELL, ΟΝ FARADAY’S LINES OF FORCE.

83

We must therefore have the following equations, since the state of the shell is the same at

every instant,

΄

I, s: i6-n- 21΄ἢ cos φ = ἢ 5ἰἢ ὰθ + καΤτ' CF s:ing

whence

- af’ sing = = w(I, sin 6 — 2I' cos φ),
π.

Ἵ

cot p= - TR w, 24k

=}

Q4a=re

I, s;in0

T ᾿

γι + ——_w
24k ᾿
To understand the meaning of these expressions let us take a particular case.

Let the axis of the revolving shell be vertical, and let the revolution be from north to west. Let J be the total intensity of the terrestrial magnetism, and let the dip be θ, then I cos @ is the horizontal component in the direction of magnetic north.

The result of the rotation is to produce currents in the shell about an axis inclined at a

small angle = (αππὶ

T ao to

the

south

of magnetic

west,

and

the

external

effect

of these

24

currents is the same as that of a magnet whose moment is

1

Tw

2 /= 24k P+ Tut 5] cos 0.

The moment of the couple due to terrestrial magnetism tending to stop the rotation is

2Q4ark

Tw

2

RF cos? 0

dark ἢ + Jo"

al

and the loss of work due to this in unit of time is

ad

sie2 + 1515 cos? 0.

2 24ark |+ T*o*

This loss of work is made up by an evolution of heat in the substance of the shell, as is proved by a recent experiment of Μ, Foucault, (see Comptes Rendus, xu. p. 450).

11—2

IV. The Structure of the Athenian Trireme ; considered with reference to certain difficulties of interpretation. By J. W. Donaupson, D.D. late Fellow of Trinity College, Cambridge.
[Read November 6, 1856.]
Tuer formal recognition of philology, as one of the subjects for discussion at the meetings of the Cambridge Philosophical Society, seems to me to impose on those of the members, who have more especially devoted themselves to this branch of academic study, the duty. of suggesting as soon as possible some discussion calculated to awaken an interest in this new or rather additional department of our transactions, And as pure linguistic investigation is a sealed book to many, and eminently uninviting to all those, who are not critical scholars by profession, I have thought it best to take an application of philological research, on which I have something new to offer, and which is, or ought to be, both intelligible and interesting to all, who care for the language or the doings of the ancient Greeks.
As the Athenians, at the time when their literature assumed its distinctive form, were pre-eminently a maritime people, it was to be expected that nautical terms would take their place among the most usual figures of speech. Many of their best writers had either, as we say, “served in the navy,” or had become familiar with the language and habits of the seaports. Even if the wealthier men had not personally served as strategi or trierarchs, or had not made voyages for profit or pleasure, they had lounged in the dockyards and factories of the Pirseus, and seen the triremes put to sea on some great expedition; and if the poorer citizens had not pulled the long oar on the upper benches, they had lived in familiar intercourse with many whose hands were hardened with constant rowing, and whose ears were ringing with the never ceasing drone of the pipe to which they kept stroke in the voyage or the onset of battle. It is not at all surprising then that Attic literature is full of direct allusions to the structure of the ship of war and to all the incidents of sea-life. And in point of fact nothing is more common than the occurrence of nautical metaphors. But although this has been duly noticed, and though much has been written on the subject, there are still some phrases in common use, which have not yet received an adequate explanation, and consequently some passages, which still require to be illustrated by a more complete and accurate investigation of the Athenian trireme. It is my intention, in the present paper, to submit to you some of the conclusions at which I have arrived after a renewed survey of the ancient authorities.
It is a well-known fact that ships of war in the most glorious days of the Athenian republic were mainly, if not entirely, triremes, or galleys with three banks of oars. This convenient form of the rowing-vessel, combining, as it seems, the maximum of speed and power, was invented by Ameinocles the Corinthian about 700 s.c, The elementary form, of which it

Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN ΤΕΙΒΕΜΕ.

85

was an extension, and which kept its place by the side of the trireme, was the penteconter or single-banked galley with fifty rowers. The short flat-bottomed barges of the earliest seamen were not adapted either for rapid navigation or for warfare. And as soon as the Greek mariners put out to sea either to trade with or to plunder distant cities, they seem to have adopted the long sharp-prowed vessel with its twenty-five rowers on each side. Herodotus says expressly that the Phoceans, who navigated the Archipelago, the Adriatic, and the western Mediterranean as far as ‘T'artessus, used for this purpose ov στρογγύλῃσι νηυσί, ἀλλὰ πεντηκοντόροισι (1. 163), and the mythical Argo, which represents the first of those voyages, half piratical, half commercial, which the Thessalians made into the Black Sea, was undoubtedly regarded as a penteconter. The tradition generally reckons fifty Argonauts, and it was not without a distinct reference to this, that Pindar describes the dragon killed by Jason as ‘‘bigger in length and breadth than a penteconter, which blows of steel have perfected” (Pyth. tv. 255). In these galleys it is presumed that all the rowers were armed men, and Homer is careful to tell us this in speaking of the penteconters which Philoctetes took to Troy (Jl. 11. 227). Whether the ships of the Beeotians, to which Homer gives a complement of 120 men (Ji. 11. 16), were biremes, or large penteconters, with double crews, is a point which can hardly be decided; Pliny mentions (H. N. vit. 57), on the authority of Damastes, a contemporary of Herodotus, that the Erythrzans were the first to introduce biremes, but we do not know when this form was originally adopted, and it is clear that the galley with two banks was never very common. And Thucydides seems to have understood that the pente-
conters only were rowed by the soldiers, who in that case were bowmen, so that the other vessels would contain, beside the rowers, who served as archers, some seventy hoplites, who only pulled on an emergency. There is a special reason for coming t&this conclusion. Thucydides (1.10) speaks of the περίνεῳ or supernumeraries in the ships which went to Troy,
and limits them to the kings and their suite. But the Scholiast says that this term included all the ἐπίβαται or soldiers on board. Now in the nautical inscriptions published by Béckh,
we have a particular class of oars called by this name, αἱ περίνεῳ κῶπαι, and it is probable that these were intended to be used by the synonymous ἐπέβαται whenever additional hands
were wanted, to make head against wind or tide. All things considered, we may take the penteconter as the oldest and most permanent type
of the Greek war-ship. Both with regard to the number of the crew, and the vessel’s length and breadth of beam, it was the basis or starting-point of the trireme. The crew of the trireme consisted of about 170 rowers and 30 supernumeraries. As the length of the vessel over all from forecastle to poop was greater than that of its keel, there were more seats for rowers in the upper tier than in the two lower tiers, and the Scholiast on Aristophanes (Ran. 1074) tells us that at the stern the first thranite sat before the first zygite, and the first zygite before the first thalamite. It seems indeed that there were 62 θρανῖται; or bench-
rowers, in the highest tier, 54 ζυγῖται or cross-bit-rowers, on the second tier, and the same number of @adayirat, or main-hold-rowers, on the lowest tier. Unless then some of the
thranites were employed to work the two great oars, or πηδάλια, at the stern, they must have had four ports on each side more than the lower tiers. Supposing that the penteconter had exactly 50 rowers, it must have been nearly as long as the trireme, for it had 25 ports or

860 Dr DONALDSON, ON THE STRUCTURE OF THE ATHENIAN TRIREME. holes for the oars, whereas the corresponding or lower part of the trireme was pierced for 27 holes on each side. And as the interscalmium, or space between the ports, was two cubits (Vitruv. 1. 2), or 3 feet 6 inches, we should require a length of 105 feet above, and 91 feet below, exclusively of the steerage and bow, or parexeiresia. That the trireme and the oldest penteconter were exactly of the same breadth of beam, I will prove directly. And of course the height was not increased more than was necessary for the accommodation of the additional tiers of rowers.
Having regard then to that permanence of numerical arrangements which is so remarkable
among the ancient Greeks, we must see at once that the broad-side of the penteconter corresponded to the enomoty or triakad, a body of 25 to 30 men, sworn to act together, and constituting the basis of the Greek military system. Consequently, the whole crew of the penteconter corresponded to the pentekostys, and the crew of the trireme was a lochus, consisting, with the epibate, of four pentekostyes, which was the Lacedemonian arrangement at the first battle of Mantineia (Thuc. v. 68), or it was two locht of 100 men each, if we prefer Xenophon’s subdivision (Rep. Lac. τι. 4).
In regard to these general features all is plain enough. Our difficulty commences, when we come to speak of the arrangements for seating the three tiers of rowers, and it is here that I hope to clear up some obscurities, and throw a little new light on the subject. Dr Arnold has called this “an indiscoverable” or ““ unconquerable problem” (Rom. Hist. 111. 572 on Thucyd. rv. 32), and Mr James Smith, in his elaborate and interesting Essay On the Voyage and Shipwreck of St Paul, has proposed a solution quite at variance with the meaning of the Greek words which distinguish the classes of rowers*. Even Béckh, in his Archives of the Athenian Navy,‘can give us no definite information, and inclines to the erroneous belief that
* The following is Mr Smith’s transverse section of a trireme. (Voyage and Shipwreck of St Paul, p. 194.)
a. Oar of thalamite seated on deck. ὃ. Oar of zygite seated on stool on deck. 6. Oar of thranite seated on stool on gangway. Besides the objection stated in the text, that this arrangement will not explain the Greek names of the three tiers of rowers, it is impossible to conceive that the best rowers should have been placed on a platform within reach of the enemies’ shot.