770 
MR. A. McAULAY ON THE MATHEMATICAL 
so that L may be called the time flux of intrinsic energy. By Proposition YIII., 
§ 10, 
{4 > + ) d% — x (4* + < A/) 
Hence we see by equation (25), § 49, that 
L =—(<£ + 3>/) XP + vC ~ V (A + a) H/477 - (<9©V\ + 0 & Vx) . . (2). 
Now, (‘Phil. Trans.,’ 1884, Part II., pp. 343 to 349) Professor Poynting's result 
expressed in similar notation would be (calling the time flux of energy P) 
P = - {4> + d>/) XP - V {Vv + (A + a)} H/477 - (0 & V\ + O^x) . . (3). 
It is scarcely necessary to remark that we have here generalised his expression* by 
the insertion of the terms — (4> + <fy) XP ~ YaH/477 — (0 e V\ + d^Vx) and have 
substituted for his E what he means by it, namely, — (A + Vv). It might be thought 
at first that this is not quite what he means by E since he incorporates in it terms 
depending on the motion of the body with reference to the lines of magnetic induction. 
Remembering, however, that equation (6), § 60, and the equation E 0 = — (A fi- Vv) 
are identical, it will be seen that these terms have been here incorporated. 
104. The direct interpretation of equations (2) and (3) is, of course, widely different. 
Let us see if they have the same physical significance, that is, whether they lead to 
the same rate of increase of intrinsic energy in any finite space. 
For this purpose it must be asked whether or not jj 6 S (L — P) is zero. Now 
477 (L - P) = &7tvC + YVAH = YV (AH).(4). 
Hence 47rjj 6 S (L — P) d% = jj 6 SdSV (AH), or by equation (3), § 5, 
4tt jj S (L - P) dt = [ vSdpJL .(5), 
where the line integral! is to be taken over all lines of discontinuity on the true 
* I only say—generalised lais expression —sinee some such terms as liave been added in the text would, 
on Professor Poynting’s own theory, he included in the vector L, defined by equation (1), as the time 
flux of intrinsic energy. The result of the present paper is, however, in all strictness much more 
geueral than his, since it has not among other things been assumed that all the bodies in space are 
isotropic with reference to specific inductive capacity, resistance, and magnetic permeability. 
t It may he well to notice that by the conventions of § 5, above, if the closed curve be regarded as 
bounding, not the regions of the true boundary, but the part of the surface of discontinuity in the region 
of space under consideration, the sign of the line integral must be changed. 
