Conway — A Theorem on Moving Distributions of Electricity, 7 

 equations \^F, + V-'l^Fi'dt'' + 4:7Tpx,'{t) = 0, etc. For the latter we have 



F, = {' pr.'dr, \ V{iri)'' {xj, [u] + yM'^') + ^.h (w)) du duy { F- {t - uY 



- Qc - C'h {u) - xj,{u) - yMu) - z,k{u)Y 



- {z - z,(u) - a.vii(w) - yo'^^h{u) - z,n,{u)Y}-K 



Denoting dh{u)ldu, &c., by the notation h'{u), &c., we can easily verify 

 the following results : — 



The aj- velocity of the point x^y^z^ in the sphere 



= oh'{t) + x,k[t) + y,k{t) + z,k[t). 



The components Mi{t), hj^if), i03{t) of the angular velocity of the sphere 

 about the fixed axes are given by 



w.{t) = «i// 4 luk' + nj/ = - Uilj, - 01/ to - n/h, 



^3(0 "= h'/'ii' + t^i'iin + I'iinz = - h'Tiii - h'rii2 - Uni-M 

 the argument t being understood. 



On integrating with respect to f/a>, we have 



F, = \ 2iTp,\Hr\ du\n~^\j\v)y\-u,-,{,y-y,{iC))^^.,{z-z,{i>))\ 



Jo J 



.[r-(.-.)'-H-o)'-.']ic. ^:;;:;:i:i;:;:g: - 



On transforming to real A'ariables, and proceeding as before, we find 

 \'F, + V-'d'F/dt' + 47rp{- co^iy - y,{t)) + ^,{z-z,{t))) = 0, 

 so that ^-F + V-^d'FI dt' + ^ttI, = 0. 



Finally, another relation which follows directly is 



^ + — + — + r ' „T = 0, 



cix cy cz 01 



which involves the equation of continuity 



ail 3/2 ^ az 



dx Zy dz ' (t 



dli 3/2 3/3 dp „ 



