THEORY OF ELECTROMAGNETISM. 
713 
where the % implies that any assigned group of independent velocities , and no others, 
are varied, and where S,;X is the increment in X due to the particular variation By. 
Now, on account of the conditions, 
4ttC = WE, [Vt/SH]„ + i = 0, SVC = 0, [Sc72C]„ + * = 0, 
it is necessary that we consider the whole group of velocities, SC, throughout space 
together. It is, then, as far as C is concerned, only necessary to prove that 
f j[SE SC ds + f f SE, SC ds = [f j(SSC c Vtc + SSH H V;r) ds, 
the integrals extending throughout space. (A.s to the sign of these terms, it must 
be remembered that the force corresponding to C is not E, hut — E). This is proved 
quite easily* by means of eq. (4) § 5. 
Similarly, with regard to 'P, it is only necessary to prove that 
- f [ jSF Bp ds' - f f SF,, Bp' ds' = (ffs Sd^Ckr £ ds, 
and this is obvious from the mode in which equations (33) (34) were established. 
42. To prove the second and third statements, let for any finite region JJj denote an 
integration taken over the true boundary of that region, and JJb an integral taken over 
both sides of any surface of discontinuity, as to physical quantities in the region, so 
that 
[HI,+11..<”>■ 
Then, if we can prove that for any finite region, 
(Kate of increase of heat + rate of doing work of frictional forces) 
= - f f {Sdt (0 e Vx + VaH/47r - YC) + S p'&dt) 
it will follow that the energy supply required to account for (l) the increment of 
heat, (2) the work (negative) done by the frictional forces, consists of three parts, (1) 
* It should, perhaps, be noticed that cC and tH are now not perfectly arbitrary. We may assume 
that 
Sv cC = 0 , 47r fC = Yv 
and from the equations [VUba]«+j = 0, [VUrH] a+ b = 0 
[SaUr 3H]« + s = 0. 
MDCCCXCII.—A. 4 Y 
