120 
Proceedings of Royal Society of Edinburgh. [sess. 
we shall have the following expressions for % and the forces per 
unit volufhe and surface respectively due to the electric system. 
S = M &=-[>Uv],, +s . .... (75). 
And, further, if <f> he self-conjugate, both these forces will he 
explained by a stress <£, as can he seen by the above work on stress. 
For proof we have by equation (3) above 
8 W = -fff^yi^ V x dq = —ffS8rjcf>XJvds + fff^y<bi V i dq 
where of course the element ds is taken twice, namely, once for the 
region on each side. But 
8 W = - (work done by the system g, of forces) 
=// mr,ds+f//S%8r,ck, 
where the element ds is now only taken once. Equating coefficients 
of the arbitrary vector 8rj we get the required equations. 
To avoid difficulty at surfaces of discontinuity, 8 when applied to 
a function of the position of a point must be thus defined. Suppose 
that by means of the displacement 8rj, any point is moved from P to P'. 
Then Q being the value of any function at P before the displacement, 
Q + 8 Q will be the value at P' after the displacement. 8 Q is thus 
in all cases a small quantity of the same order as 8rj. With this 
meaning of 8 , 8 V is not = 0. To find it observe that 
S (dp + 8dp)(V + 8 V ) = Sdp V 
but 8dp = -Sc?pV .817 
S df P 8 V =Sd P V 1 S 8 > 7l V 
whence SV = V 1 S 8 ^ 1 V (76), 
which might have been derived from the equation (27) V' = \~ 1 V. 
Assuming that the strain due to 8r) does not alter the charge of 
any portion of matter, 
0 = S(Dc?s) = 8(crds) (77). 
To find 8 W, notice that 
8d<s = — dq S V 8yj 
4ttS!D = K8 (S + 8K.(£, 
and 
