FLUXES OF ENERGY IN THE ELECTROMAGNETIC FIELD. 
451 
Derivation of the Electric and Magnetic Stresses and Forces from the Flux of 
Energy. 
§19. It will be observed that the convection of energy disappears by occurring 
twice oppositely signed; but as it comes necessarily into the expression for the stress 
flux of energy, I have preserved the cancelling terms in (132). A comparison of the 
stress flux with the Poynting flux is interesting. Both are of the same form, viz., 
vector products of the electric and magnetic forces with convection terms ; but 
whereas in the latter the forces in the vector product are those of the field (i.e., only 
intrinsic forces deducted from E and H), in the former we have the motional forces 
e and h combined with the complete E and H of the fluxes. Thus the stress depends 
does. For if we turn round an eolotropic portion of matter, keeping E unchanged, the value of U is 
altered by the rotation of the principal axes of c along with the matter, so that a torque is required. 
In equation (132a), then, to produce (1326), we keep E constant, and let the six vectors, i, j, k, c 1; c 2 , c 3 
rotate as a rigid body with the spin a = curl q. Bat when a vector magnitude i is turned round in 
this way, its rate of time-change cijdt is Yai. Tints, for 9 fit, we may put Ya throughout. Therefore, 
by (1326), 
3c 
EyE = E (Yai.Cj + Yaj.c 2 + Yak.c 3 )E + E (i.Yac! + j.Vac., + k.Yac.,)E. . . (132c) 
In this use the parallelepidedal transformation (12), and it becomes 
3c 
E E = VEafi.Cx + j.c 2 + k.c 3 )E + E (i.c 1 + j.c 3 + k.c 3 )VEa 
= (YEa)cE + Ec(VEa) = (D + D')YEa,. 
1132 d) 
by (132a), if D r is conjugate to D ; that is, D' = c'E = Ec. So, when c = c', as in the electrical case, we 
have 
3U, , _ 3c 
and similarly 
?f = | E g- E = DVEa = aVDE, 
^=iH ^ H = BVHa = aVBH. 
3 1 2 dt 
> 
. (132e) 
Now the torque arising from the stress is (see (139)) 
S = YDE + VBH, 
3 
so we have 
0 1 
(U c + T m ) = Sa = torque x spin. 
(132/) 
The variation allowed to i, j, k may seem to conflict with their constancy (as reference vectors) in 
general. But they merely vary for a temporary purpose, being fixed in the matter instead of in space. 
But we may, perhaps better, discard i, j, k altogether, and use any independent vectors, 1, m, n instead, 
making 
D = (l.c x + m.c 2 + n.c 3 ) E,.(132</) 
wherein the c’s are properly chosen to suit the new axes. The calculation then proceeds as before, half 
3 M 2 
