Electromagnetic Theory, Vol. 1 (1893) — part 15 of 31

§ 139. Returning to the three single up-step formulae, it will be observed that in every case we base the proof upon the vanishing of a surface-integral, viz., 2 Npq, or 2 Npg, or 2 VN>g. Now it is obvious that these are true if the surface be taken wholly outside the region occupied by the quantity q or g. The value pq or pg is made zero all over the surface of summation. This is what occurs in the applications made. In (246) the space-summation has to include all the B2, which repre- sents q; it is sufficient, therefore, for the surface to com- pletely enclose all B2. Again, in (255), the enclosure by the surface of all 0, which represents g, will ensure the vanish- ing of the surface-integral; and in (251) the same. Now, it is usual to imagine the surface to be at infinity, so that the space- summations extend over all space. This is also most convenient, in general. But caution is sometimes necessary when the quantity to be summed extends to infinity. The surface-summation then may, or may not, vanish, according to circumstances. Even if the quantity summed becomes in- finitely small at infinity, the summation may still not vanish. To illustrate this, it is sufficient to mention the case of a single point-source. The surface-integral of the flux it produces is finite always, being the measure of the strength of the source. At infinity the flux may be infinitely attenuated, but the sur- face is simultaneously infinitely magnified. But in the appli- cation of the above processes to practical cases in electro- magnetism it is usually quite easy to see that the integral over the surface at infinity vanishes, owing either to the actual absence of anything to be summed, or else, when there is an infinitely attenuated quantity to be summed, of its being, per unit area, of lower dimensions than 1/r2. The quantity p itself attenuates to nothing, as well as q or g, if they are existent at all at infinity.

The three up-step formulae may, by inspection, be transformed so as to involve entirely different operations. Thus, in (241), put p for 1/471T. Then we have

B2=-pot(C2/r) (256)

Do the same in (240), and we obtain

C2 = pot(D2/r). (257)

216 ELECTROMAGNETIC THEORY. CH. III.

Finally, (229) gives

. . . . (258)

We may sometimes utilise these formulee for purposes of cal- culation, should the integrations to be performed be amenable to practical treatment. But there is a caution to be men- tioned. The quantities whose potentials are calculated by the last three formulae are not functions of position, which are de- finitely distributed in space, and are independent of the position of the point where the potential is reckoned, but are definite only when this point is fixed. As you pass to another point, the potential there is that of a different distribution from that belonging to the first point. These quantities, C2/r, &c., are therefore not subject to the various properties of the circuital and divergent vectors already considered. For example, C2 is purely divergent, but D2/r has curl ; or, G1 is circuital, whilst VDj/r has divergence, and so on.

Integration "by parts." Energy Equivalences in the Circuital Series.

§ 140. The transformations (244), (249), (254), or the more general ones containing the omitted surface-summations, ex- pressed by putting the sign of summation before every term in (242), (247), (252), and converting the summation on the left to a surface-summation by the introduction of N, as in (243), (248), and (253), are examples of what is, in the Cartesian mathematics, called integration " by parts." It is usually a very tedious and uninforming process, that is, in Cartesians, with its triple and double /'s, its dS and dx dy dz, and its I, m, n. The vectorial methods go straight to the mark at once, avoid a large amount of quite useless work, and enable you to keep your attention fixed upon the actual magnitudes concerned, and their essential relations, instead of being dis- tracted by a crowd of coordinates and components.

There are, of course, many other cases which arise of this " by parts " integration. One of the most important in con- nection with the circuital series of vectors, is the following : — Substitute for p in the preceding, a vector, say f, then we shall have

2fcurlg = 2gcurlf, .... (259)

the summations being throughout all space, or, at any rate

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 217

through enough of it to include all the quantities summed. The proof is that since

divVgf=fcurlg-gcurlf, . . . (260) which is (178) again, we have, by space-summation,

2fcurlg = 2gcurlf+2divVgf, . . (261)

and the third term may be at once turned into the surface-sum- mation 2 NVgf. If this vanish, as usual, then (259) follows.

Applying this result to the series of circuital vectors A15 A2, A3, &c., each of which is the curl of the preceding, we obtain various equivalences. Thus,

2D12 = 2C1E1 = 2B1F1 = 2A1G1, . . (262)

through all space. Operate on one member by curl and on the other by (curl)-1, to pass from one form to the next. Also

2 D^ = 2 0^ = 2 6^ = 2 A^, . (263)

in a similar manner ; and so on.

In any of these summations we may convert either of the circuital vectors involved to a general vector by adding any divergent vector. For we see, by (259) that if f is polar, the right summation vanishes, and therefore so does the left. So 2DX2 is the same as 2DjD, for 20^ = 0. Similarly, 20^, is the same as 2 OFlf or the same as 2 01F, since 2 OjFg = 0, and 2 CgFj = 0. It will be understood here that the suffix 1 refers to the circuital vectors entirely, and the suffix 2 to the divergent vectors, and that 0 = 0! + C2, &c.

But we cannot generalise both vectors at the same time. Take OF for example. We have,

2 OF = 2 (Cj + 02) (F! + F2) = 2 0^ + 2 02F2, . (264)

so that the summation of the scalar product of the two diver- gent vectors now enters. The series of divergent vectors and their intermediate scalars have properties similar to (262), (263).

Energy and other Equivalences in the Divergent Series.

§ 141. These properties of the space-integrals of scalar products in the divergent series are formally obtainable from the corres- ponding ones in the circuital series by changing the suffix from

218 ELECTROMAGNETIC THEORY. CH. III.

1 to 2 ; at the same time changing the type from clarendon to roman should the quantity typified be a scalar. Thus, analo- gous to (262), (263), we have

2C22 = 2B2D2 = 2A2E2=..., . . (265) 2D22 = 2C2E2 = 2B2F2 = ..., . . (266)

starting from the square of a vector in the first set, and from the square of a scalar in the second. These transformations all rest upon (249) ; that is, we pass from any form to dhe next by the exchange of — v and div between the factors. For example,

2 D22 = 2 D2 div C2, by definition of D2,

= - 2 C2VD2, by integration, using (249), = 2 02E2, by definition of E2.

Similarly in all the rest. The preceding equations conveniently summarise them.

But it will be observed that there is another way of pairing terms in the divergent series, viz., a vector with a scalar. The corresponding transformations do not work so symmetrically as the previous. For instance,

2 C2D2 = - 2 VB2.D2, by definition of C2,

• 2B2VD2,      by  (244), 
  

= - 2 B2E2, by definition of E2, = - 2 div A2.E2, by definition of B2, = 2A2V.E2, by integration. . (267)

Here the symmetry breaks down. The last transformation depends upon

2(Nf)g = 2(Vf)g = 2(gdivf+fv.g),. . (268)

which is a case of the theorem (147) or (153). Note that in the second form V has to differentiate both f and g, so that the full expression is in the third form. If the surface-integral vanishes at infinity, we have

2gdivf= -2fV.g, .... (269)

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 219

which is the transformation used in getting (267). We have also, by going the other way,

2C2D2 = 2C2divC2=-2C2V.C2, . . (270)

by using the same transformation (269). Next take D2E2. Here the vector is the slope of the scalar it is multiplied by, whereas in the former case, the divergence of the vector was the associated scalar. So it now goes quite differently. Thus,

2 D2E2 = - 2 D2VD2 = - 2 VJD22 = - 2 N|D22 . (271) The result is therefore zero, if the surface-integral at infinity vanishae.

The Isotropic Elastic Solid. Relation of Displacement to Force through the Potential.

§ 142. It usually happens that potentials present themselves in physical mathematics as auxiliary functions introduced to facilitate calculations relating to other quantities. But in the theory of the elastic solid, the potential function presents itself in a very direct and neat manner; besides that, the elastic theory presents excellent illustrations of the above transforma- tions and the general theory of v. If the solid be homo- geneous and isotropic, there are but two elastic constants, the rigidity n and the coefficient of resistance Tc to compression or expansion. Let also m = k + Jra, in Thomson and Tait's nota- tion ; then the equation of motion is

f + 7iV2G + wVdivG = /oG, . . . (272) where G is the displacement, and f the impressed force per unit volume, whilst p is the density. Whatever limitations may need to be put upon the values of m and n in treating of solids as we find them, in speculations relating to ethers they may have any values not making the stored energy negative. Thus m may, as we shall see presently, go down to the nega- tive value - n, keeping n positive.

Split f into fx + f2, and G into Gx + G2, where fx and GT are circuital, and the others divergent. Then (272) may be split into two equations, one circuital, the other divergent. Thus, remembering (193),

pGy . . . (273)

/oG2. . . . (274)

220 ELECTROMAGNETIC THEORY. CH. III.

In the circuital equation Jc does not appear, so that the propa- gation of circuital disturbances depends only upon the rigidity and density ; speed, (n/p)l. In the divergent equation both n and k appear, and the speed of disturbances of this class is {(n + m)/p}l. In another form, if we take the curl of (272), the divergence goes out, so that the curl of G, which is twice the rotation, has the curl of f for source, and is propagated inde- pendently of compressibility.

And if we take the divergence of (272), we see that the divergence of G, or the expansion, has the divergence of f for source, and the rate of its propagation depends upon both rigidity and compressibility. But when m = 0, there is but one speed of propagation, and a complete amalgamation of the two kinds of disturbances.

In equilibrium, when equilibrium is possible, the right members of the last three equations vanish, since they repre- sent "rate of acceleration of momentum" (a long-winded expression). We therefore have

(275) . . . (276)

The solutions are visible by inspection. We see that nGrl is the potential of flt and (n + m)G2 the potential of f2. That is, the displacement produced by circuital impressed force is its potential divided by n ; and the displacement produced by divergent impressed force is its potential divided by n + m ; whilst, in the general case, we must split the impressed force into circuital and divergent parts, and then re-unite them in different proportions to obtain the resultant displacement.

Symbolically,

(277)

(278) (279)

In the case of incompressibility Jc, and therefore m, is infinite, so that G2 is zero, and the displacement is pot fjn, whatever f2 may be. For f2 is balanced by difference of pressure, which is set up instantly. The corresponding speed is infinite, but there is no displacement.

ELEMENTS OP VECTORIAL ALGEBRA AND ANALYSIS. 221

If, again, m = 0, and there is but one speed, the displacement is the potential (divided by n) of the impressed force, whatever be its type.

On the other hand, if ra + w = 0, we see by (274) that there is no steady state possible due to divergent force, as f2 is then employed simply in accelerating momentum on the spot. The corresponding speed is zero. The circuital disturbances are propagated as before. This is the case of Sir W. Thomson's contractile ether, in which the wave of normal disturbance is abolished by making the speed zero.

The Stored Energy and the Stress in the Elastic Solid. The

Forceless and Torqueless Stress.

§ 143. We may also find expressions for the stored energy from the equation of motion. The work done and stored by the impressed force f is JfG, though, if f be put on suddenly, an equal amount is dissipated, since, G being the final displace- ment, fG is the work then done by f, per unit volume. Now

by (264). Calling the first part U^ and the second part Ug, we have, by (273), (274),

U1= -2|wG1V2G1 = 2Jn(curlG)2 .... (280) U2 = - 2 i(m + w)G2V2G2 = 2 (m + n) (div G)2. (281)

The transformations here used are (259) for Uj and (249) for U2. We see that the energy stored is expressed in terms of the squares of the rotation and of the expansion. In the contractile ether U2 is zero, and the stored energy is Ur We do not correctly localise the energy by the above formulae, but only express the total amounts. For correct localisation we need to know the stress and the distortion. Now the distortion is a function of the variation of displacement, so is known in terms of G. Can we, however, find the stress itself from the equation of motion ? If not, we can come very close to it. Thus, in a state of equilibrium, F + f=0, if F is the force arising from the stress. So, by (275), (276),

..... (282) (283) and F =/tV2G + wV2G2 ..... (284)

ELECTROMAGNETIC THEORY. CH. III.

Now let PN be the stress on the plane whose normal is N. We have, by consideration of the equilibrium of a unit cube,

FN = divPN, (285)

to express the relation between an irrotational stress and the force arising from it. Therefore, applying this to (284),

divPN = 7iV2GN + wV2G2N. . . . (286)

Here the divergence of PN is given ; find PN itself. The immediate answer is, by (201), or (214),

PN= -VpotdivPN, or PN = ttVGN + mVG2N, . . . (287)

provided PN is divergent. That is, we have constructed a stress-vector giving the proper force required. But, without interfering with this essential property, we might add on to the right side of (287) any circuital vector. That we must do so now, we may see by remembering that the stress must be irrotational, or produce no torque, and then by finding that (287) does give a torque. To show this, consider the first part only of the stress (287), say with m = 0. Then

NPM - MPN = 7i(NV.MG - MV.NG). Here take M=j and N = k; then

kPj - jPk = tt(V3G2 - V2G3) = - ni curl G,

by (149). This gives the i component of the torque, which is therefore - n curl G. But G2 has no curl, therefore the second part of (287) produces no torque. We require, therefore, to add to PN in (287) a stress giving no force, but a torque n curl G. Such a stress is

XN = ncurlVGN (288)

as may be tested in the above manner. This brings us from (287) to

PN = W(VGN + curl VGN) + mVG2N, . . (289)

which is a stress-vector giving the correct force and no torque. [It should be noted here that since (285) applies to an irro- tational stress, the process employed is only justified when we finally get rid of the torque, as in (289), (290). If the stress is

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 223

rotational, the divergence of PN is really the N component of the force due to the conjugate stress. Thus the force due to nv.GN is nVdivG, and the torque —ncurlGr. The force due to ?ftV.G2N is mvdivG, with no torque. The force due to n curl VGN is — ncur!2G, and the torque n curl G. Adding together the three stresses and the three forces we obtain the stress (289), with no torque and the correct force.]

But on comparison with the real stress deduced from the elastic properties of a solid, which is

PN = tt(V-GN + NV.G) + N(m-rc)divG . . (290)

we see that they do not agree. Yet they can only differ by a stress which produces neither force nor torque. And we know already, by (288), that if G = G2, the modified stress will give no force or torque. In fact, on comparing (289), (290), we find that their difference is of this nature, (290) being equivalent to

PN = rc(VGN + curl VGN) + m(VG2N - curl VG.N), (291)

where the first and third terms on the right are sufficient to give the correct force, but with a torque, which is, however, cancelled by the second term ; whilst the fourth term is apparently (so far as force and torque are concerned) like a fifth wheel to the coach, off the ground. If we inquire under what circumstances the real stress can assume this singular form, we shall find that m + n = 0 and curl G = 0 will do it. With these conditions, only the circuital part of PN is now left, and (291) reduces to

PN = 2»curlVG2N (292)

It is the case of irrotational displacement in the contractile ether, previously referred to, and is entirely remote from the real elastic solid. The noteworthy thing is that we cannot apparently conclude what the stress is, even when the force and torque to correspond are everywhere given, owing to the forceless and torqueless stress coming into the formulae. The form (292) is convenient for showing at sight that there is no force. It may, by (181), remembering the constancy of N, be expanded to

PN = 2w(NV.G2-NdivG2), , , . (293)

224 ELECTROMAGNETIC THEORY. CH. III.

where put N = i, j, k in turns to get the three stresses on the planes having these normals. One-third of the sum of the normal tractions on these planes is - (4^/3) div G2, or, which is the same, + Jc div G. It is the negative of the pressure. But in the real stress (291) itself, the fourth part, from its having the negative sign prefixed, correctly associates pressure and compression. But this forceless and torqueless stress does not contribute anything to the total energy. The amount of stored energy is

= - J2 (F^ + F2G2 + F3G3)

div P2 + Gs div ps) P3VG3), .... (294)

where the first line needs no remark, the second is got by (285), and the third by the common transformation (249), Pp etc., being the stresses on the i, j, k planes. The final form in (294) shows the correct distribution of the energy in terms of the stress and the distortion. Now, if in the stress PN be in- cluded any terms giving rise to no force, the above transforma- tion shows that they contribute nothing on the whole to the energy of distortion. This is the case with the fourth term in (293), and would also be the case with the second term, only that we have no right to ignore it, on account of the torque thereby brought in. If the solid be not unbounded, it is sufficient for its boundary to be at rest for the same principles to apply.

[The practical meaning is that in the contractile ether the energy of distortion due to any irrotational displacement is zero on the whole, the sum of the positive amounts in certain parts being equal to the sum of the negative amounts in the rest.]

Other Forms for the Displacement in terms of the Applied

Forcive.

§144. The simple solution (279) may, of course, receive many other forms. If it be desired to find the displacement due to an explicitly given impressed forcive, it is a matter of some importance to select a method of obtaining it which shall not be unnecessarily difficult in execution j for different pro-

ELEMENTS OP VECTORIAL ALGEBRA. AND ANALYSIS. 225

cesses leading to the same result may vary greatly in readiness of application. Now, if the impressed forcive be either circuital or divergent, we do not need to modify. But if of a mixed type, then it may be desirable. Put tl = t- f2, then an alter- native form is

• & J* ^

; . . , (295)

and now, when f is given, it is only the part pot f2 that needs development. We have

. . (296) as before explained. Now, if we notice that

W = 2/r, ..... (297)

which may be proved by differentiating, we can further con- clude that

. . (298)

a very curious result. The potential of a radial vector following the inverse-square law of intensity is a radial vector with a constant tensor in all space. Using this result, we have

.. (299) so that (295) becomes

. . (300)

n(

wherein f alone appears on the right side. Another form is got by using

f2 = V2fV?7, /. potf2 = Vpot2fVp. . (301) But a better one is

potf2 = 2/v^/87r, .... (302) as

where s is length measured along f, and / is the tensor of the latter. This makes

O-Ipotf— 5/?^Z/v* . . . (303)

n n

which is generally suitable for practical calculation.

Q

226 ELECTROMAGNETIC THEORY. CH. III.

A considerably more complex form is given in Thomson and Tait. It may be obtained from (303) by means of the identity

where sl is unit s, and therefore parallel to f, making

.

ds r

the same as Thomson and Tait's formulae when expanded in cartesians. But this is a gratuitous complication, as (303) is simpler in expression and in application. Of course (300) is simpler still in expression, and the practical choice may lie between it and (303), or (295).

Notice that an impressed forcive of the divergent type with a single point-source produces not merely uniform radial dis- placement, as per (298), but an infinite discontinuity, in fact, a disruption, at the source itself. It is a very extreme case, of course.

The Elastic Solid generalised to include Elastic, Dissipative, and Inertial Resistance to Translation, Rotation, Ex- pansion, and Distortion.

§ 145. The elastic solid with two elastic constants (k and n) has not been found sufficiently elastic to supply a thoroughly satisfactory analogy with Maxwell's ether, though partial analogies may readily be found. Other kinds of elasticity than resistance to compression and distortion, and other kinds of resistance than elastic resistance, present themselves to the consideration of searchers for analogies between the propa- gation of disturbances in Maxwell's ether and in the brutally simple elastic solid of theory, which is, however, known to fairly represent real solids within a certain range.

There are four distinct ideas involved in the displacement of a small portion of matter, viz., the translation as a whole, the rotation as a whole, the change of size, and the change of shape. These separate themselves from one another naturally. As regards the mere translational motion, if it be only iner- tially resisted, we have the equation of motion

ELEMENTS OP VECTOBIAL ALGEBRA AND ANALYSIS. 227

(306)

where p is the density, q. the velocity, f impressed force (per unit volume), F tfae force arising from the stress associated with the strain (including rotation, distortion, and expansion), and 8/St the time-differentiator for moving matter. This equa- tion is constructed on simple Newtonian principles. We may, however, wish to have elastic and frictional resistance to trans- lation, as well as inertial resistance. Then generalize the above to

f +F- (ft + ft* +*£)<», . . . (307)

where G is the displacement, when small departures from equilibrium are concerned. Here p2 is the p of (306), pl is the frictionality (Lord Kelvin's word for coefficient of friction), and pQ is an elastic constant. The displacement from equi- librium calls into action a back force p0G proportional to the displacement, with storage of potential energy ; a force /a1G- proportional to the velocity, with waste of energy ; and a force /o2G proportional to the acceleration, with associated kinetic energy. The potential energy is J/o0G2, the rate of waste ftG2, and the kinetic energy Jp2G2.

As regards the force F, this is given by (284), when the stress is irrotational, and the elastic constants are n and k. But, in general, we may proceed thus. First separate the rotation from the strain vector. We have

NV.G = J(NV.O- + V.NG) + J(NV.G - V.NG)

= pN-JVNcurlG (308)

Here Ny.G means the variation of displacement per unit dis- tance along N, which is any unit vector, so that by giving N all directions (practically only three) the complete state of strain is known. This strain vector is above analysed into the strain pN without rotation, and the second part depending upon rotation. But pN includes the expansion, as well as the dis- tortion, or mere change of shape. To exhibit the distortion without expansion, one-third of the expansion (vectorised) must be deducted. Thus

NV.G = (pN - IN div G) + iN div G - JVN curl G (309)

Q2

228 ELECTROMAGNETIC THEORY. CH. III.

shows the separation of the strain into a distortion (without change of size), an expansion, and a rotation, which are naturally independent,

If distortion, expansion, and rotation are all elastically resisted, three independent elastic constants (in an isotropic medium) intervene between the above strain and the corre- sponding stress PN upon the plane whose normal is N. To obtain PN, multiply the distortional part by 2n, the expansional part by 3&, and the rotational part by 2v, and add the results. Thus,

PN = 2/t(pN - JN div G) + Nfc div G - vVN curl G. (310)

Compare with (290). Here n and k are as before, whilst v is a new elastic constant connected with the rotation, and which was previously assumed to have the value zero. That is, there was assumed to be no resistance to rotation. The corres- ponding torque is

S = 2vcurlG ...... (311)

With the rotational part of the stress is also associated trans- lational force, given by

• curl JS= -v curl2 G. . . . (312)

Adding this on to the former expression for the force (or deriving the force from (310) directly), we find that

F = w(V2G + JV div G) + &V div G - v curl2 G (313)

represents the translational force to be used in the equation of motion (307). We should notice here that the quantity k div G in (310) represents a uniform tension. It equals one- third of the sum of the normal tractions on any three mutually perpendicular planes. Its negative represents the pressure, or p = - k div G. Now when there is incompressibility, k is infinite, and div G is zero. But their product usually remains finite. So in any case we may replace the k term in (310) by

• Np, and the k term in (313) by - Vp. The energy of the strain is

. . (314)

or one-half the sum of the scalar products of the stress vector and the strain vector for three perpendicular planes, Pv P2, P3

ELEMENTS OF VECTORIAL ALGEBRA AND ANALYSIS. 229

being the stresses on the i, j, k planes respectively. On reckoning up, by (310), (309), we find the energies of expan- sion, rotation, and distortion are all independent, and that U is their sum, given by

U = n[Pl2 + p22 + p32 _ l(div G)2] + J&(div G)2 + 1 v(curl G)', (315)

where the v term is the energy of rotation ( = \ torque x rotation), the Jc term is the energy of expansion, and the n term the energy of distortion.

Now the stress (310) is derived from the strain (309) by the introduction of elastic resistances only. There is, however, no reason why we should limit the nature of the resistance in this way. Consider the n term only for example, relating to distortion. There may also be dissipative or frictional resist- ance to distortion. To exhibit it, change n to nQ + n^d/dt) in (310), (313). Then nQ will be the rigidity and n± the viscosity, or coefficient of frictional resistance to distortion. For instance, if we abolish n0 and v, and retain nv and k, we have the stress in a real viscous fluid according to Stokes's theory. It may be that there is not a complete disappearance of the rigidity in a fluid ; if so, then retain both n0 and nv When the rigidity is marked, as in a solid, nQ is the important part of n ; whilst in a fluid it is the other part. It is, however, not at all to be expected that the expression of the viscosity of solids by nv if true at all, would extend beyond the small range of approxi- mately perfect elasticity. Lord Kelvin's experiments on the subsidence of the oscillations of wires tended to show that a different law was followed than that corresponding to the viscosity of fluids, and that elastic fatigue was concerned in the matter.

There may also be, conceivably, inertial resistance to dis- tortion, and not only elastic, but also frictional and inertial resistance to rotation, and the same with respect to expansion. These will be brought in by generalising the three elastic constants k, n, and v of (310) thus : —

. . (316)

230 ELECTROMAGNETIC THEORY. CH. III.

where of the nine coefficients on the right side; the first three are elastic constants, the second three dissipative constants, the last three inertial constants ; the elastic constants involving potential energy, the dissipative constants- waste of energy, and the inertial constants kinetic energy, similarly to (307) as regards translation. Out of these twelve constants only four are in use, or seven, if v0 and v,, and p-^ be included, which additional have been speculatively employed. This does not represent finality by any means ; but sufficient has been said on the subject of generalising the elastic constants to emphasize the fact that there is plenty of scope for investigation in the theory of the motion of media of unknown internal constitution, which, externally viewed, involve the four elements translation, rotation, expansion and distortion.

The activity of the stress vector PN is PNq, and the flux of energy is (see § § 68, 72 also), if q be the speed, or tensor of the velocity,

Qtf, ..... (317)

where U and T are the complete potential and kinetic energies, per unit volume, and QN is the stress conjugate to PN, to obtain which change the sign of the last (rotational) term in (310). The convergence of the energy flux represents the work done and stored or wasted in the unit volume. Consider- ing the term - Qg^ only (the rest being the convective flux of energy), its convergence is

div (Ql2l + Q2?2 + Q3?3) = VjtoQ) + V2(P2q) + V3(P3q)

• & div Qi + & div Q2 + q3 div Q3 + Qx Vft + Q2V?2 + Q3V<?3

. . (318)

where P is the force as in (313). Fq. is the. translational activity, which may, by (307), be employed in increasing kinetic and potential energy or be wasted. The rest represents the rate of increase of the sum of the three potential energies when there is merely elastic resistance • but if we use (316) we obtain also the sum of the rates of waste of energy, through &P nv vv and also the rate of increase of the sum of the kinetic energies, through &2, n2, v2.

ELEMENTS OP VECTOBIAL ALGEBRA AND ANALYSIS. 231

Thus we have, by (310), 1V1 + P2V2 + PsV3)q

= ^ ( 2n(p, - Ji div G) + M di v G - vVi curl G ) dx\ /

• Jj div G) + Jcj div G - vVj curl G

•   2»(ps  -  Jk  div  G)  +  £k  div  G  -  vVk  curl  G 
  

dz\ /

= 2w(p, VjCt + P2V2q + p3V3q) - ^ div G div a

• & div G div q + v curl G curl q . . . (319)

Take for illustration, the rotational term v curl G curl q. This is simply (dldt) {}>v (curl G)2}, or the rate of increase of the rotational energy, when v is a constant, but when we use the last of (316) it becomes

) + v^curl a)» + ^(i"2(™rl a)*), (320)

showing the rates of increase of potential and kinetic energy, and the rate of waste (the middle term) due to rotational friction. Similarly as regards the other terms in (319). The expansion terms give

), . (321)

where the middle term is the rate of waste, and the others rates of increase of potential and kinetic energies. Finally, the distortional terms of (319) give

(p,2 + pV + p'32 -

• (322)

where the first line is the rate of increase of the potential energy, the second line the frictional rate of waste, and the third line the rate of increase of kinetic energy. We thus complete the energy relations of the stress with the generalised

232 ELECTROMAGNETIC THEORY. CH. III.

elastic constants, and so far as small motions are concerned, the equations are manageable.

Electromagnetic and Elastic Solid Comparisons. First Example : Magnetic Force compared with Velocity in an incompressible Solid with Distortional Elasticity.

§ 146. Comparisons between the propagation of electromag- netic disturbances in Maxwell's ether, or in a homogeneous isotropic dielectric, whether conducting or not, and the propa- gation of motion in an elastic solid, either simple or generalised, may be made in a variety of ways. They all break down sooner or later, but are nevertheless useful as far as they go. A few cases will be now considered, based on the preceding. First take the case of a non-conducting dielectric at rest, and compare it with the regular elastic solid made incompressible by infinite resistance to compression. The incompressibility includes inexpansibility.

In the dielectric we have, if p stands for d/dt, for convenience,

curl(H-h) = cpE, curl (e - E) = /*pH, . (323)

the circuital equations connecting E and H (§§ 38, 24, 66), where e and h are impressed, and //., c are the inductivity and permittivity. Now, the equation of motion in an incompressible solid is, by (273),

. . (324)

where p and n are the density and rigidity, G the displacement, q. the velocity, and fx the circuital part of the impressed forcive, the divergent part being inoperative. Or we may take the impressed forcive to be circuital to begin with.

Now, let h = 0, and e be finite. If it has no curl, it does nothing, as before shown (§ 89). The source of disturbance is curl e. This being the case in the dielectric, and fx being the source of motion in the solid, it will be convenient to take fx and curl e as corresponding. Let the latter be (temporarily) f. Then (323) gives (remembering that div H = 0),

f = upB. + curl E = /^H + curl2 H/cp,