THEORY OF ELECTROMAGNETISM. 
711 
[eq. (4), § 5], where Y, Y* are scalars, and, as with a s , [YJ„ = [YJj. Equating now 
to zero the coefficients of SC, Sc, we get 
E = c Vx + a + VY, e = c Vx . . .... (28), 
E, = — [Yffi] (1+ j, e, = 0.(29). 
It should be noticed that b, defined by eq. (26), satisfies both the conditions of 
incompressibility 
SVb = 0, [SUrb] n + i = 0 . . .(30). 
The first condition is obvious from the equation b = VVa. The second is easily 
deduced from the equation [VUVa] a + i = 0. For this last asserts that the component 
of a parallel to the surface is the same for both regions bounded. Thus the line 
integral J S dpa,, which, by eq. (3), § 5, = JJ Sb d%, taken over any closed curve on the 
surface, is the same for both regions. It follows that [Scfcffi] a + 6 = 0. We naturally 
assume that a is an intensity and b a flux. Hence, by § 8, 
VVa' = b' , [YUn] s+ } = 0.(31) 
SV'b' = 0 , [StVb'] a + 6 = 0.(32). 
39. Next suppose that the only variation implied in equation (24) is in p , and, 
therefore, in ' V P. Thus 
j jj 6B(x/6) ds = — jjjsS'T'fk.CIxfyZs [eq. (13) of former paper] 
= — \ f{fs8<K x -v x '-w, 
where is defined by saying that 
3>=2i(Ia:.(33), 
and that <3> is a function of Class I. of § 9 above.'" Now since [former paper, eq. (39)] 
^ = x'x> we have 
8^ = S x '. x + X S X 
* What immediately follows is a particular case of a theorem required more than once below. Let 
O, x and a be as usual in this paper and let Qa> = — 2/3S<nx. Then 
Sx'QfCfcL = 2S/3Q 'ads. 
* 
More generally, if (w, w') be any function of two vectors w, to' linear in each 
(Qf> * = 2 (/3, QV) . ds. 
