712 MR. G. F. C. SEARLE ON PROBLEMS IN ELECTRIC CONVECTION. 
-= m* m +r+ ** ** 
The second integral is by (30) and (31) simply 
2 j'j j fi H 3 dx dy dz = 2T. 
The system will be supposed to consist of two surfaces bearing complementary 
charges so distributed that it is only in the space between the two surfaces that E 
and H do not vanish. If the “Equilibrium Conditions ’ of § 15 are correct, these 
distributions are also equilibrium distributions. If we integrate the first integral term 
by term “ by parts ” and remember that 'P satisfies the differential equation 
dm , d 2 V , cm 
at all points between the surfaces, we find 
W= 2T 
K f J, cbP cn d'V n d'V 1 
87r J 1 dx a. dy a dz J 
where c/S is an element of one of the surfaces, and l, m. n are the direction cosines of 
the outward normal to c/S, the integration extending over both surfaces. Over the 
whole of each surface \P is constant, because the surface is the boundary of a region 
of zero disturbance. Also if cr be the surface density, we have by (81), 
— K [ j d'V vi d'V n d'l r 1 
° = 4 tt j & '*'7% ^7&j’ 
so that if q be the charge upon the surface Mq and — q that on the surface MG, we 
have 
W = 2T + ^Mq .(93). 
The quantity ^qMq — r>q'I r . 2 is evidently the mechanical work which must be spent 
in bringing the system together from a state of diffusion, in which, however, each 
part is still moving with velocity u parallel to x. It might, perhaps, have been 
