MOTIONS IN A SPHERICAL CONDUCTOR. 
d_(d£ , dg 
dt\dx'dy'dzj 
523 
require, by (3) and (4), 
in the conductors, and 
V 2 </>= o.(10), 
*f=° 
■ • (II). 
in the dielectric. The superficial solenoidal condition 
7 . . 7 df . dg , dli 
--\-m — -\-n - 
CLL QjT/ clz 
requires, by (1), the continuity of 
/y 3 F+mv 3 Gr+nv 3 H, 
i.e., of 
da da\ , /-.db da\ , / dc dc\ 
but this is implied in the continuity of a, b, c. 
If dv, dv" be elements of a normal to the surface of a conductor, on the inside and 
outside respectively, we find from (3) and (4), taking account of the continuity of 
F, G, H, 
— = 4? nr(lf+mg+nh)—p{lu+mv-\-nw ); 
or, if cr denote the surface density of electricity, 
da 
dv' 1 dv"J w 
( 12 ). 
Hence it is only when the currents are steady that the relation between </> and the 
free electricity in the field is the same as in electrostatics. 4 ' 
If T be the kinetic and V the potentiai energy of the field, we have 
T=^Jj (Frt+ Gv-\-~Sliv)dxdydz 
+1 
F/"'+ G g-\- Wi)dgdyd£ 
V=2n ! J[f(/ ! +S 8 +iV« . 
• (13), 
(14), 
* Tliis peculiarity of Maxwell’s theory has been pointed out by C. Niven, loc. cii. 
