Motion of an Electrified Sphere. 51 



The integration can be effected by aid of the formula 



f ( p>-cf)Ux = 1_ , p(g> + *ty + *(p>-<ry 



J (p 2 -f ^ 2 ) (^ 2 + ^? 2 )^ 2p & p(jf + X 2 )* — x(p 2 — (f)i 



i f we put p 2 = L 2 — B 2 , ^ 2 = h 2 — B 2 , 



and ^2__^2 = L 2_ A 2 ? 



where h stands for either h x or h 2 . We thus obtain 



ty.= 2R S in<L*-R')* [ l0 « A ~ l0 S B ]/ 

 where 



■ " _ (L'-By(V-B» + * , )»+g(L»--V)* 



Thus 



|~i a! r i Ai(L 2 -K 2 )l + R(L 2 -V)^ 

 For B we have only to substitute ho for h^ Hence 



, {A 1 (L 2 ~By + B(L 2 -A 1 2 )H{A2(L 2 -R 2 )^-B(L 2 -7i 2 2 )l} 



We now substitute for (L 2 — 7i 1 2 )^ and (L 2 — A 2 2 )^ the values 

 given in (23) and (24), and then carry out the multiplications 

 in the numerator and denominator of the quantity under the 

 logarithm. Replacing h L * + h 2 2 — 2/i x A 2 cosa by L 2 sin 2 a ; we 

 then easily find 



bi*J«" 2R sin a(L 2 - B 2 )* L l0g BJ 



1 , 7^2 -R 2 cos a+ B .(L 2 — B 2 )hin a 



~ 2B sin a(L*-B 2 )* ° S A,A,-R 2 cos a-B(L*-B2)*sina 



. . . (26) 

 where L 2 sin 2 a = Z^ 2 + 7i 2 2 — 21^1^ cos a. 



We have thus found the mean value of 1/z^ for points on 

 the surface of the sphere. 



Mr. Bennett has supplied the following remarks : — "The 

 quantity under the logarithm in (26) is the cross-ratio of the 

 pencil formed by the two planes together with the two tangent 

 planes to the sphere through their line of intersection ; and 

 the result takes a simpler form if reciprocated with respect 

 to the sphere itself. The perpendiculars from a variable point 

 of the sphere to fixed planes are then replaced by (constant 

 multiples of) perpendiculars drawn from fixed points within 

 the sphere to a variable tangent plane. The result of the 



E 2" 



