Electromagnetic Theory, Vol. 2 (1899) — part 17 of 31

But although I cannot see the utility of the proposed change, which seems to be a retrograde step, I think that Mr. Macaulaj's mathematics, which is of a strong kind, may be of Talne in the electromagnetic field in other ways, especially when deared of useless and treacherous potentials.

Second Sotational Analogy: Induction Compaied witb

Velocity.

§ 156. Returning to the suppositional Ether, the conception is of an incompressible medium possessing mass, which involves translational inertia, and therefore kinetic energy, and which elastically resists rotation, and so stores potential energy ; and we have supposed that the velocity of the medium means magnetic force, and that its density means /x, so that the kinetic energy and the magnetic energy are compared. But we may equally well have this comparison of the energies com- bined with a different interpretation of /x. Take, for instance, the induction to mean velocity. Then, since the magnetic energy is ^ft~^B2, we see that it is now, not fi, but /x-i that is the density, whilst H is momentum. Also, since ficv^ = 1 = pv~^v\ we see that the new interpretation of c is vvK That is, the change of /& from density to its reciprocal involves the change of c from compliancy to its reciprocal multiplied by v^, making p^/v orp/v\ But it is better to write them out side by side, thus : —

-1

(mag. energy) J/i~*B- stands for ^pQ-,

(induction) B

{reluctivity) /*

{mag. force) H

(el. current) curl H

<e1. displ.) D

(permittivity) c

(el. force) B

<eL energy) JcE^

If

II

II

II

II

II

II

II

II

11

II

II

(kin. energy), (velocity), (density), (momentum).

II

Oi

p curl q, /> curio,

vp~^ curl G,

Jv(curl (potl energy).

250

ILBOIBOllAaNKnO THVORT.

CH. ni.

The potential energy still correctly localising the electric. ^€i'gy> ill spite of the other changes, we may expect the flux, of energy to be correct We have

VBH = v/>-i V curl a. = - I'Vq curl (J,

. as before, equation (356).

Similarly, we may assume that fi is not the density, nor its- reciprocal, but an unstated function of the density, say /i »/(/>), and work out the various correspondences that this necessitatee. We shall, for example, by the formulse for magnetic and kinetic energy, require

Of course, in thus comparing magnetic with kinetic energy, we shall always have H compared with q, or with some multiple of q, which may change its " dimensions," as, for example, in the above detailed cases, where fi is changed from velocity to momentum.

Probability of the Kinetic Nature of Magnetic Energy.

§ 157. If it be asked why, in the previous analogies, a pre- ference hasusually been shown for the representtCtion of magnetic force (or a constant multiple thereof) by velocity, the answer would be, substantially, that it has been done in order to make the magnetic energy be kinetic. Now, it is true that in a per- fectly abstract electromagnetic scheme, arranged in duplex form — in which every electric magnitude has its magnetic re- presentative— and, therefore, including a magnetic conduction- current with waste of energy, there would be a perfect balance of evidence as regards ttie kinetic nature of either the electric or the magnetic energy, when the other is to be potential energy, as of a state of strain in an elastio medium. For, if it were argued from a certain set of relations that the magnetic energy was kinetic, the force of the argument could be at once destroyed by picking out an analogous set of relations tending to show (in the same way as before) that the electric energy was kinetic.

But, as matters actually stand, with an imperfect corre» spondence between the electric and magnetic sides of the elec- tromagnetic scheme, there seems to be a considerable pre- ponderance of evidence in favour of the kinetic nature of the

ELEMENTS OF VECTORIAL ALQEBUA AND ANALYSIS. 251

magnetic energy. We may refer, in particular, to the laws of linear electric circuits, which were shown by Maxwell to be simply deducible by the ordinary equations of motion (gene- ralised) of a dynamical system, on the assumption that magnetic energy is kinetic ; there being one degree of freedom for every circuit, and the variables being such that every linear electric current is a (generalised) velocity. In this theory, the dielectric in which the conducting circuits are immersed is regarded as unyielding, so that electric dis- placement camiot occur in it| and the currents are confined entirely to the conductors.

Now, we could construct a precisely similar theory of con- ductive magnetic circuits immersed in a medium permitting displacement, but destitute of magnetic inductivity. The circuital flux of magnetic induction in the former case would now be replaced by a similar circuital flux of electric dis- placement; the former electric currents becoming magnetic currents, and the magnetic energy becoming electric energy. But this electric energy, when expressed in terms of the linear magnetic currents, would possess the property of allowing us to deduce from it the laws of the magnetic curcuits, by usuig the generalised equations of motion, on the assumption that electric energy is kinetic, in a manner resembling Maxwell's deduction of the laws of electric circuits. Therefore, sup- posing the state of things mentioned to really exist, we might become impressed with the idea that electric energy is kinetic ; just as, at present, it seems hardly possible to avoid entertain- ing the idea of the kinetic nature of the magnetic energy.

Vie do not, however, need to go to generalised dynamics to arrive at this probable conclusion. It is sufficient to start with a sound general knowledge of dynamical facts and prin- ciples, not necessarily mathematical, but such as may be ac- quired in practical experience by an intelligent and thoughtful mind, involving clear ideas about inertia, momentum, force, and work, and how they are practically connected (the Act of Parliament notwithstanding). On then proceeding to the expe- rimental study of electrokinetics, including the phenomena of self-induction in particular, the dynamical ideas will be found to come in quite naturally. Lastly, the generalised theoretical dynamics will senre to clinch the matter. Although adding

252

BLBOTBOlCAaKimO THEOBT.

CH. ni.

little that is noYel, it will oorroborate fonner oonduttona^ and 4)0-<ndiiiate the Caote in a oompaot and ^ysteiiiatic maimer, suit- Able to a dynamioal soienoe.

On the other hand, there is little that is snggeatiye of kinetic ideas in eleotrostatios, whilst there is much that is suggestiTe of the potential energy of a strained state. This fact, combined with the kinetic soggestiveness of the facts (and the equations embodying them) in which magoetio induction and electric currents are concerned, explains why the associa- tion of magnetic with kinetic, and electric with potential energy becomes natural. It is, therefore, somewhat a matter of surprise, as well as rather vexing, to find that in order to extend the last-considered rotational analogy to a conducting dielectric, we must, if we wish to do it simply, give up the com- parison of magnetic force with velocity and electric force with torque, and adopt the converse system, making the electric energy be kinetic, and the magnetic energy the potential energy of the rotation.

TJniutelligibility of the Rotational Analogue for a Conduc- ting Dielectric when Magnetic Energy is Kinetic.

§ 158. We can, perhaps, most easily see that this plan should be adopted by writing down the two circuital equations of electro- magnetism, and then, immediately under them, the correspond- ing proposed circuital equations of the rotational ether. Thus, taking H to be velocity, as before, we have, to express the first circuital law in the dielectric and its mechanical companion,

g-HcurlH = <yE, (357)

gi+curla = l/)(JS). . . . (358)

Here the terms in vertical are to be compared. Magnetic force H becoming velocity q, the electric force E bceouies iS, or half the torque ; c the permittivity is the compliancy v~\ and gj is a newly-introduced companion to g, which is the source — curl h. It will be interpreted later.

The other circuital equations are

f-curlE«f^)H, .... (359) fi - curl (iS) =pj>(i. .... (360)

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 25^

We now have the additional analogues of inductivity and density, and f, which is the source curl e, is compared with f^, which is impressed translational force per unit volume.

So far relating to a non-conducting dielectric, and giving an- intelligible djnamioal analogy, if we wish to extend it to a con- dacting dielectric, according to Maxwell's scheme (including, of oourse, a pure eonduetor which has, or is assumed to have, no- permittivity), we require to change cp in the first circuital law (357) tak + epi whilst the second circuital law needs no change. Bat the right member of (358) needs to be changed to match the modified (357). Thus,

g + cnrl H=(X- + <y)E, .... (361)

gl + curl<l = (^•^ + v-l^)(JS), . . (362)

where is the new coefficient, to match k. But what is its* interpretation in the rotational ether, and how is the latter to be modified to make as intelligible as the other constants!

Now, since we use rotational elasticity to obtain the po- tential energy, and we associate the rotation with the electric displacement) it is suggested that rotational friction should be introduced to cause the waste of energy analogous to that of Joule. But if there were frictional resistance proportional to- the spin, we should have

S = 2(»' + »'ijp) curia, .... (363)

instead of (352), connecting the torque with the rotation. But this will not harmonise at all with (362). We cannot do what we want by rotational friction, but require some special arrangement, whose nature does not appear, in order to interpret (362) intelUgibly.

The Botational Analogy, with Electric Energy Kinetic, extended to a Cktnducting Dielectric by means of Trans- lational Friction.

§ 159. But, by changing the form of the analogy, choosing the electric energy to be kinetic, the extension to a conducting dielectric can be made in a sufficiently obvious manner. Put (360) under (361), thus

g + curlH = (^ + c^)E, .... (364)

fj + curl ( -JS) = (/»!+ f)p)(i, . . . (865)

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^54

BUKnsoiiAmisno theort.

CH. in.

where we introdace fii to xnAtoh k. Alio pat (358) under (359) to get the other pair, thus

f-curl£ = /ujpH, . • • . (366) -gi-curU«i/>(-iS). . • . (367)

We have now a fit, with an intelligible meaning to be given to the new coefficient that brings in waste of energy. For (365), which is compared with the first circuital law, is the trans- lational equation of motion in the rotational ether when there 18 frictional resistance to translation, expressed by /s^q, so that Pi IB the irictionali^, and /),q- the rate of waste. The correspondencee in detail are as follows : —

A = £/p stands for Q,

(spacial displ.).

(eL force)

E

If

ft

Of

(velocity).

(permittivity)

e

ti

ft

(density).

(el. displ.)

D

f

ft

(momentum).

(el. energy)

II

II

(kin. energy).

(mag. force)

H

a

11

{-I torque).

(inductivity)

ft

ft

(compliancy).

(induction)

B

It

ft

(mag. eneigy)

ft

ft

Jv(ouria)^, (rotl. energy).

(oonduotivity)

k

ft

ft

Pv

(frictionality).

(cond. current) O^-AS

II

}>

(Joule-heat)

*}

II

PiQ'i

(rate of waste).

(true current)

ff

ft

(/>i+/>i>)<l-

These should

be studied

in

oonneotion with the cireuital

equations (''64), (366), and their matches underneath them, if it be desired to obtain "an intelligent comprehension" oi the true nature of the analogy, and to correct any errors that may have crept in.

If we keep away from the sources of energy, there is little difficulty in understanding the analogy in a broad manner. But the sources require special attention before the electro- magnetic and rotational analogues are intelligible. We know that activity is the product of two factors, a "force" and a " velocity." Newton knew that. In modern dynamics, too^

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ELSMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 256

'where the velocity is not a primitive velocity, but has a geueralised meaning — the time-rate of change of some variable — the corresponding force generalised (not a primitive force) is still such that the product of " force " and " velocity " is activity, or activity per unit volume, tfec, according to con- venience. In our electromagnetic equations, for instance, EC and HB and ED are activities (per unit volume), and we call S and H the " forces " (intensities), and the other factors the fluxes, the oorresponding '* velooities." Similarly, eO, eD, and

liB are aotivities (per unit Tolume understood), where e and h are intrinsic, communicating energy to the system.

Now, since the "variables" may be variously chosen in « dynamical system, we need not be surprised if it should some- times happen that the (generalised) force turns out to be a (primitive) velocity, and the (generalised) velocity to be a (primi- tive) force. Here is food for the scofiPer, for one thing. At any rate, we should be careful not to confound distinct ideas, aud remember the meaning of the activity product. Our pre- sent rotational analogy furnishes an illustration. We have the equation of activity,

e(0 + I)) + hB = Q + U-fT-fdivW, . (368)

where W» V(E-e)(H-h). . , (369)

W is the energy-flux, Q the rate of waste, U the electric, and T the magnetic energy per unit volume, whilst the left member represents the rate of supply of energy by the intrinsic sources e and h, being the sum of their activities.

Now what are the analogues of e and h ? Remember that only their curls appear in the circuital equations, and that f and g are the sources of disturbances. By inspection of (365), (367), we see that is the curl of a torque, and the curl of a velocity, say

2fi = curiSo, gi=-curlqo. . . (370)

Then, just as the analogue of H is - ^S, the analogue of h is -JQq. Impressed magnetic force is, therefore, represented by an impressed torque per unit volume, and its activity is

{-iSa)»'-^Jp(-iS), . . . (371)

' 256

BLBCTBOICAONBTIO THBOBT«

OH. in*

which is, of conxse, plain enough. But the other activity ia

doiPi + PPH (372)

where is the analogue of e. Here the supposed " force ''has become a velocity, and the " velocity " a force. For the factor of Qo in (372) is the analogue of electric current-density, and means the (primitive) force per unit volume in the rotational ether, partly employed in increasing momentum, partly work> ing against friction. Of course this perversion is rather extreme.

The rotational analogues of (368), (369) may be readfly written down by proper translations in accordance with the above^ remembering (370).

Mr. W. Williams, who has recently (Physical Society, 1892) published a very close study of the theoiy of the "dimensiooa'^ of physical magnitudes, on applying his views to the electro- magnetic equations and the rotational ether, has arrived at the conclusion that the representation of either E or H by velocity leads to the only two systems that are dynamically intelligible. It should be remembered, however, that his conclusion is subject to certain limitations (mentioned in his paper) regarding dimensions, for otherwise the conclusion might seem to be of too absolute a nature, as if one of E or H must be velocity. In any case, I cannot go further myself at present than regard the rotational ether as furnishing a good analogy, which may lead later to something better and more comprehensive. The pre- sent limitation to small motions (in general) is a serious one^ and there are other dif&culties.

Symmetrical Linear Operators, direct and inverse, referred

to the principal Axes.

§ 160. Linear vector functions of a vector play a very important part in vector-algebra and analysis, just as simple equations do in common algebra. Thus, in the theoiy of elasticity the strain vector is a linear function of the direction vector, and the stress vector is another linear function of the direction vector. In electromagnetism the electric displace- ment is a linear function of the electric force, and so is the conduction current-density ; whilst the magnetic induction is a linear function of the magnetic force, when its range is small

■LBXSNT8 OF VECTORIAL ALOBBIU AND AKALTSIB. 257

enongli. There are, therefore, at least five examples of linear ▼eetor fanctiona to be considered in electromagaetism. In other sciences too, the linear function often turns up, and frequently in pure geometry, when treated algebraically. Finite rotations may be treated by the method, and in the algebra of surfaces of the second order the linear connection between two vectors is prominent. It would be inexcusable not to give some account of linear operators in this chapter on vector-analysis and its application to electromagnetism. The subject, however, is such a large one that it is only possible to deal with its more elementary parts in a somewhat brief manner — not, however, so brief as to be useless.

The simplest kind of linear connection is that of Ohm's law in isotropic conductors. We have G»^E, where E is the intensity of electric force and C the current-density, whilst k is a constant, the conductivity. No specification of direction is here made. But when vectorised, we have the equation 0=klM instead, k being as before. We now assert that B and 0 are parallel, whilst their tensors are in a constant xatio^ lor all directions in space.

Suppose, however, we alter the eonduoUvitj of the body in a certain direction, say that of i (eg,, by means of compression parallel to i), making it ^, whilst its conductivity is k^ in all directions transverse to L We now have Ci^k^Ri, and also Oi = k^, when the electric force is parallel to L But if it be transverse to i, then we have 0 •=» kJEt, There is, in both oases, parallelism of B and 0, but the ratio of the tensors changes from kytokf. What, then, is the current when B is neither parallel nor transverse to i? The answer is to be obtained by decomposing E into two components, one parallel to 1, the other transverse ; then reekoning the currents to match by the above, and finally combining them by addition. Thus

0 = i.l'iEi+j./('2K, + k.^-._jE,. . . . (1)

where Ej, Eo, E,j are the i, j, k scalar components of E. We have here made use of the linear principle. It is that the sum of the currents due to any two electric forces is the current due to the sum of the electric forces. This principle (suitably expressed) applies in all cases of linear connection between two veotors, and is the ultimate source of the simplicity of treat- ment that arises.

8

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S58

ELBCTBOIIAOKBTIO THKORT.

CH. m.

More generally, let the oonduetiTity be different in the three oo-perpendicular directions of I, j, k, being k^^ kp kg re- spectively. Then we have

O^kE^'Uc^E^+yk^E^ + h.t^K^. ... (2)

The current is now coincident with the electric force in three directions only, Aftmely, along the principal axes of conduotivity. When E has any other direction than that of a principal axis, we have no longer parallelism of 0 and B, and their relation is expressed by equation (2). It is the general type of all linear relations of the symmetrical kind when referred to the principal axes, if we remoTC .the restriction which obtains in the electrical example that the k*9 must be all positive. Given, then, that 0 = ^'E, with the understanding that 0 is a symme- trical linear function of £, so that k must represent the linear operator conuecting them (the conductivity operator in the above example) the full answer to the question, What is the 0 corresponding to a given E? — is obtainable in the above manner, viz., by first finding the axes of parallelism of E and 0 and the values of the principal A;'s, and next, by adding together the three C's belonging to the three component £'s parallel to the axes.

The inverse question, Given E, find 0, is similarly answerable. For if 0 = pE, where p is the operator inverse to k (or the resistivity operator in the case of conduction-current), we know that k^Pi = 1, if p^is the constant resistivity parallel to i, and similarly ^ ^3=" ^ ^or the J and k axes, so that in full,

E=/50-i./)iCi+j.p,C, + k./)3C3, ... (3^

where C^, C2, C3 are the scalar components of current.

Precisely similar remarks apply to the permittivity operator c in D = cE, connecting the displacement D with the electric force E, and to the inductivity operator fi in B = /xH, connect- ing the induction B with the magnetic force H, when the con- nection is a linear one. It is not without importance to con- stantly bear in mind the above process of passing from E to 0 through k, and the process of inverting k to because when treated in a general manner, with reference to any axes, the relations become much more complicated, whilst their intrinsic and resultant meaning is identically the same.

BLEMENTS OF VSCTOBIAL ALGEBRA AND ANALYSIS. 259

<3eoxnetrical Illustrations. The Sphere and EllipsoicL Inverae Perpendiculars and Maccnllagh's Theorem.

§ 161. Next let us obtain some Koometrioal illustrations of tbe pnrioos. The transition from E to 0 ii like that from a sphere to an ellipsoid. If a solid be uniformly subjected to what is termed a homogeneous strain, any initially spherical portion of it becomes an ellipsoid. If this be done without totation, the state is such that all lines parallel to a certain axis, say that of I, in the unstrained solid, are lengthened or shortened in a certain ratio in passing to the strained state, without change of direction. The same is true as regards lines parallel to two other axes, say j and k, perpendicular to each other and to the first axis, with different values given to the ratio of lengthening or shortening in the three directions of preservation of parallelism.

The equation of the ellipsoidal surface itself is

^-S+^ + 'a

a (r

when refened to the centre and the principal axes. Here ath,e, .are the lengths of the principal semi-axes, whilst d;, y, s, are the scalar components of r, the radius yector from the centre to any point of the surface. It may be written

l-^if+j| + kiy = N^ say, • . (5)

•from which we see that when r belongs to the ellipsoid, N is a unit vector, the vector radius of a certain sphere from which the ellipsoid r may be obtained by homogeneous strain, the three ratios of elongation being a, c (that is, the co-ordinate jcja in the sphere becomes x in the ellipsoid, and so on). JHoWf ii we square (3) and divide by E^, we may write it

where the Cs axe the components of 0, and £ is the tensor of S. Now suppose £ is constant, and let E take all directions in succession. Its extremity will range over a spherical sur&ce, whilst the end of the corresponding 0 or ^E will fange over an ellipsoidal surface. For we may take k\E, L.^E, i^E in (6) to be a, 6, c in (4) ; when Cp C,, C3 will simulta- neously be y, 2, The semi-axes of this current-ellipsoid are

82

260

ELKTBOlIAGKBnC THBOBT.

cn. nr.

the principal oarrenta in the direotions of paraUelism of elootrii^ force and ouirent.

Similarly to (6), we haye, by squaring (2),

(pfi? w^)' ^ '

fh>m which we see that when G is oonstant, so that the Teotor 0 ranges over a spherioal sur&oe, the oorresponding B simnlta- neoosly ranges oyer the surface of an ellips(»d whose principal semi-axes are the principal E's.

There are other ways of illustration than the aboye. Thus we may see from the form of the right member of (4) that it is- the scalar product of r and another yeotor s, thus

rs-l, . (8)

where r and s are given by

r-lar+jy+ki, (9)

8-i4+j^ + ki^ (10)

Here we see that s is a linear function of r, and such that if K = </)r, then s = <^N = <^<^r = <^2r. See equation (5). The in- terpretation of the new vector 8 is easily to be found. It is the reciprocal of the vector perpendicular from the centre upon the tangent plane to the ellipsoid at the extremity of r. For if p be this perpendicular, the projection of r upon p is evidently p itself, because the three vectors r, p, and the line in the tangent plane joining their extremities form a right-angled triangle whose longest side is r. Therefore

rp-p« (11)

or, dividing by p',

rp-i-l (12)

comparing which with (8), which is true for all r's of the- ellipsoid, we see that B^p~\ as stated Now we have

EC />iEC pJiO p.,EC ^

from which we see that if we construct the ellipsoid whose principal semi-axes are (j>^BO)\ and so on, the radius yector will be £ and the corresponding reciprocal perpendicular will be 0/EO. Here EO is understood to be constant

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ELEMENTS OF VECTORIAL ALGEBRA AMD ANALYSIS. 26 1

Similarljr, we have

«o that in the eUi^Mdd with prindpal lemi-axes (ib|BO)^, &o., the tadiiis Teotor is 0 and the redprooal perpendteulair is E/BO. In both these methods of representation neither B nor 0 has oon- elant tensor, bat their prodaet, the aetiTity, is constant.

In the ease (6), where it is £ that is considered oonstant, the feciprooal of the perpendionlar on the tangent plane to the «llipeoid of 0 is

" 'w^'" w ' ' ' ' ^^^^

This is also an ellipsoid. For AE is an ellipsoid, and pE only differs in the principal constants being the reciprocals of those in the former case.

Similarly, in the case (7), where G is constant, the reciprocal {»erpeudicular is

(;i;cp+---cr + -"-02' • • • ^^^)

and again the extremity of kO ranges over an ellipsoidal enrfaoe.

Thos the sphere of B with constant tensor is associated with an

•^psoid whose reciprocal perpendicular is pE/E^ or Ar-^E/E*, another ellipsoid. It follows that the reciprocal perpendicular <of the latter gives us the former ellipsoid attain.

This reciprocal relation of two ellipsoids through their in- Terse perpendiculars is an example of Maccullagh's extraor- dinary theorem of reciprocal surfaces. It may be stated thus Let

riS^ = l, and 1282 = !, . . . . (17)

be the equations of two surfaces referred to the same origin, Ti and being the radius vectors, and s^, Sj the reciprocals of the vector perpendiculars from the origin on their tangent planes. If, then, we choose to be ij, we shall simultaneously make be ty That is, starting with the snrfsce rp ooiistruct the surface whose radius vector r, is 8|, the reciprocal peipen- dicular of ther, surface. Then the reciprocal perpendicular •of the second surface (s^ or r^) will be the radius vector <j the first surface^

262

JBLBCTROlCAGlIVnO THSORT.

CB. UL.

Thia admits of nmple Tectorial proof. For if riB, » 1 be the equation of a surface aooording to the above notation, W9 obtain, by difforentiation,

T^dB^ + B^dii = 0 (18)

if dr^ and da^ are simultaneona Tariations of and s^. But dii is in the tangent plane to rj, and therefore at right singles ta Bit 80 that Sj<frj==0. This leaves r^ibii^O also. But c^s^ or c&Tj equivalently is in the tangent plane of the surface whose vector radius is or ; therefore is parallel to the perpen- dicular on the same, or parallel to s,, say, r, firom which riSjBdSiB, or set^ But r|8|->l and r^-l* so x — l, and r J « By which completes the connections.

In terms of the perpendiculars, let ji^ and be their lengths and r^t the corresponding radius vectors, then

and is parallel to p,, and to p^.

Internal Structure of Linear Operators. Manipnlation of several when Principal Axes are ParalleL

§ 162. Leaving now the geometrical illustrations connected with the ellipsoid, consider the symmetrical linear operator by itself. If D » cE, where e is the linear operator, we know bj the above exactly how it operates through the principal We can, however, write e in such a manner that it shall state explicitly its meaning. Thus, if Cp r,, Cg be the principal c* belonging to the axes of i, j, k, we have

cE=i.CiEi+j.c,E2 + kcjE^ . . . (20)

by § 160. Now put the E*8 in terms of E, producing

cE=i.Cji£+j c^E + lLCgkE. . . . (21)

In this form, the operand may be separated from the operator,

thus —

<« = (i.Cii+j.C3j+k.C3k)B, . . . (22)

so that the expression for c itself is

c«l.Cii+j cJ+k.Cgk. .... (23)

In this form, with a vector to operate upon implied, the nature of c is fully exhibited. But the operand need not

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■LBMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 263

follow the operator. If it precedes it the result is just the ■Mue. Thus by (23),

«-C|E,.l + rjE^j + C3Ej,.k«rE. . . (24)

When, however, the operator is uasymmetrical, the vectors eB and "Be are not identical. They are said to be conjugate to one another, so tliat in the symmetrical case of identity, c is also called a self-conjugate operator. The distinction between symmetrical and skew operators will appear later.

There are six vectors concerned iu c, viz., i, j, k, andcji, cj, cJfL, which cull I, J, K. Thus

c=U+j.J+k,K (26)

is the type of a symmetrical operator referred to the principal axes. Now the general form of c referred to any axes and with the symmetrical restriction removed is obtained by turning these six vectors to any six others; thus,

^ = a.l + h.m + cii, *. (26)

so that (^E«a.l£+h.inE + e.iiE, (27)

£<^ = i;a.l + £b.m + £c.n, (23)

show the linear function and its conjugate explicitly. This is (with a changed notation, however) Prof. Gibbs's way of re- garding linear operators. The arrangement of vectors in (26) he terms a dyadic, each term of two paired vectors (a • 1, &o.,) being a dyad. Prof. Gibhs has considerably developed the theory of dyadics.

Returning to the simpler form (25) or (22), we may note the effect of the performance of the operation symbolised by e a number of times, directly or inversely. Thus by (22) or (21), or (20), we have

c'E = ccE = i.Ci2Ei + j.Cg^E, + k.c^^E,,

«1 ^2 ^3

and 80 on. Thus c" means the operator whose principals are the nth powers of those of the primitive c, with the same axes. We therefore get a succession of ellipsoida with similarly

264 BLECTBOHAGNBTIO THEORY. CIL UL

directed principal axes, only altering their lengths, starting from initial £ with constant tensor. (That is, provided the principal c's are all positive. If some be negative, we have other surfaces of the second order to consider.)

We may also manipulate in the same simple manner any number of operators, a, 6, c, <fec., which have the same prindpal axes, though they may differ in other respeots. Thua^ vmng the same kind of notation,

boE = obE = 1 . ajJi Ej + j . aJj^E^ + k . a^h^E^, I . (30)

c6a£ = abcE « i • a^bjC^E^^ + j • (^^f^^i^i + k • fjJb^c^E^ f

and BO on. That is, the snoeensive action of any number of symmetrical operators with common principal axes is equiva> lent to the action of a single operator of the same kind, whoso principals are the products of the similar principals of the set of operators. The resulting ellipsoids have their axes parallel throughout.

Next, multiply the equation (20) by any other vector F, producing

PcE^CiEiFi + c.E/. + c^EaF,. . . . (31)

By symmetry we see that this is identically the same as BcF. We might also conclude this from the fact that c£ and Ec are the same vector ; or, thirdly, by forming cF, and then multiplying it by E. We may therefore regard EcF as the scalar product of E and cF (or Fc), or, as the equal scalar pro- duct of Ec (or cE) and F. This reciprocity is tiie general cha- racteristic of symmetrical operators, by which they can be distinguished from the skew operators. The electrical mean- ing in terms of displacement and electric force is that the com- ponent disi^acement in any direction N, due to an electric force acting in any other direction M, equals the component dis- placement along M, due to an electric force with the same tensor acting along N.

Theory of Displacement in an Eolotropic Dielectric. The

Solution for a Point-Source. § 163t We see from the preceding that when it is the electric force that is given in a dielectric there is no difficulty in find- ing the displacement ; and oonveisely, when the displaDement

Digitized by (

■LDIBNTS OF VECTORIAL ALOXBRA AND AKALT8I8. 265

1b givoDi the electric force siinilarly beoomes known through the yaloea of the principal penrnttivities. Suppose^ however, that the data do not indude a knowledge of either the electrio teoe or the displaoement, both of which have to be found to «ait other date. It is then sufficient to find either, the linear •oonnection eetUing the other. Thus, to take an explicit case which has, for a reason which will appear, a special interest, suppose the dielectric medium to have uniform permittivity transverse to the axis of k, but to be dififerently permittive to displacement parallel to this axis. Further, put a point-charge -q at the origin, and enquire what is the equilibrium distribu- tion of displacement.