Motion of an Electrified Sphere. 53 



Hence 



F= CC sin 6, 010,(1$ 



JJ(1 — wi cos 6-^(1 — n 2 cos a cos ^ x — 7i 2 sin a sin #_cos(£) 



where <£ goes from to 27r and ^ goes from to ir. 

 By § 7 we see that 



I 



2n 



d(j> 



1 — n 2 cos a cos #! — ?i 2 sin a. sin #! cos <f> 



2tt 



- -j (1 — n 2 cos a cos Oy) 2 — ?i 2 2 sin 3 a sin 2 X \* 



prodded that the result is real. But 



(1 — ft 2 cosacos #i) 2 — ^ 2 2 sin 2 a sin 2 ^ 



= (cos «— « 2 cos^ 1 ) 2 + sin 2 a(l — n 2 2 ), 



and hence this quantity is always positive, since ?i 2 2 <l. 

 The integration is therefore valid. 



Replacing sin 2 1 by 1 — cos 2 ^ in the ^-integral, we find 



F _ f* 2tt sin 6_ dd x __ 



"~~ J (1 — ni cos #i)(l — >t 2 - sin 2 a — 2n 2 cos a cos 1 + r? 2 2 cos 2 0^ 



To reduce the integral to a simpler form, put 



1 — ??x cos X = l/.i?, cos #j = (#— l)/n_#. 

 Then sin #x d#x = — da/n^v 2 , 



and .£=(1 — tti) -1 when 0=0 and cr=(l + ?i 1 ) -1 when 0=7r. 

 Making these substitutions, wc easily find 



F = 27rf B X-^, 



where 



X —p 2 x 2 — 2^« 2 {n 2 — ??iX) -f ?? 2 2 



p 2 = n l 2 + n 2 2 -2n 1 7i 2 \-n 1 W + n ] 2 ?i 2 2 \ 2 ■ ■ (29) 



X = cosa (30) 



A=(l + n,)-', B=(l-n ] )" 1 . 



Hence F = — logY, 



p ° ' 



where 



B 



log Y = log {p 2 x — n_(n 2 — n{K) +pX?} 



A 



