THEORY OF ELECTROMAGNETISM. 
771 
boundary of the region considered, that is, over the trace on that surface of surfaces 
of discontinuity. The element dp is, of course, taken twice, namely, once for each of 
the two regions of the true boundary which it bounds. Since dp is in the surface of 
discontinuity, and the component of H parallel to that surface is not discontinuous, 
we see that [ScZpH]„ + & = 0. Hence the part contributed by dp to the line integral 
may be written \y] a _ b [Sc?pH]„ or 
v tt _ b Sdp 0 K .(6). 
If then v is continuous, the line integral is zero. It has already appeared [equation (2), 
§57] that if l s is everywhere zero this is the case. Hence, unless l s exist, the 
physical results of supposing P to be the time flux of intrinsic energy are identical 
with those of supposing L. 
If l s exist, we have at present no right to say that on the present theory L may be 
taken as the time flux; but in §111, below, this will be proved. The conclusion is 
then that, to explain the rate of variation of energy, Professor Poynting’s flux P 
must be supplemented by a finite flux P* along surfaces of discontinuity in the 
potential, where 
4irP, = [uVUVH]„ + 6 = v 0 _jVUV c ,H.(7). 
[In verifying the sign of this expression attention must be paid to the caution in the 
last foot-note.] This of course is rather an unnatural, though by no means absurd, 
result, and therefore I think it better to regard L as, more probably than P, repre¬ 
senting the true time flux of intrinsic energy. Another reason for preferring L to P 
is that for a field at rest, i.e., such that p , 6, ©, C and c are everywhere zero, L is zero, 
whereas P = YHVu/47r. 
In now contrasting Professor Poynting’s result with that of the present paper, we 
will suppose v continuous. 
105. Very shortly after the first publication of Professor Poynting’s paper, 
Professor J. J. Thomson in criticising it remarked (‘ B. A. Reports,’ 1885, p. 151). 
“ This [Professor Poynting’s] interpretation of the expression for the variation in the 
energy seems open to question. In the first place it would seem impossible, d priori, 
to determine the way in which the energy flows from one part of the field to another 
by merely differentiating a general expression for the energy in any region, with 
respect to the time, without having any knowledge of the mechanism which produces 
the phenomena which occur in the electromagnetic field; for although we can, by 
means of Hamilton’s or Lagrange’s equations, deduce from the expression for the 
energy the forces present in any dynamical system, and therefore the way in which 
the energy will move, yet for this purpose we require the energy to be expressed in 
terms of the coordinates fixing the system, and it will not do to take any expression 
which happens to be equal to it. The problem of finding the way in which the 
energy is transmitted in a system whose mechanism is unknown, seems to be an 
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