Conway — A Theorem on Moving Distributions of Electricity. 3 

 the scalar potential may be written then 



^ 



pr;-di 



cU 



duV\-ni)'' 



V'{t - n)' - {x - .r, - Oh {u)f - [y - y, - y, {u))' - {z - z, - z, {it)f}'\ 



where dh) is the element of solid angle, and the integrations are to be per- 

 formed in the order reverse to that in which they are written. We can 

 invert the order of integration with respect to f?w and du, provided that the 

 contour of complex integration encloses all the real zeros of the function 



V-{t - uf -{x- X, - x,{iiyf -iy-y,- yi{u)Y - {z - z, - z,{u))\ 



which are < t for all values of x^^, y^, z^^, such that x^ + y^^^ + «/ = r^. It is 

 thus necessary to find the maximum and the minimum values of u which 

 make the above expression zero, the variables being a?o, y^, ^y, such that 

 Xq' + 2/0" + 2;,,^ is constant. Hence we must have 



or [x - X, - X, {ii)] lx,^[y-y,~y, {u)] j y^ = [z-z,-z, {u)] / z„ 



where V~ (t - uf = [x-x^-Xi {tt)Y + [y - y, - y, {it) ]- = [z-z,-z, {u) ]-. 



It follows that 



X - Xi {u) _ y -y, ju) _ z - Zi{u ) _ r (tc) _ 



~ ~ ~ -, ' 



Xq y^ Zq Tf) 



X 

 .'. X - Xq- Xi [u] = " \l'[u) ± 7'o] ; 



.-. V^~{t-u)' = [T{^ii)±r,-]\ 



The roots, then, of this equation will determine the required values. It 

 will be necessary to distinguish between the cases in which the point is 

 (1) inside the sphere of radius r^, (2) outside. 



In the first case, the path of integration must surround the axis of real 

 quantities extending from u = r io u = r', where 



V{t - r) = r, + r(r), 

 V{t - /) = r, - r(ry 



In the second case, the extent of the real axis involved is from u = r" to 

 u = T, where 



V{t - r) = r(r) - r„ 



F(t-r")= r(r")-r,. 



[1*] 



