Motion of an Electrified Sphere. 49 



Since CO = L, we find in a similar manner 



H 1 H 2 =Lsina. 

 From the triangles MiCM 2 and HjCHg we have 

 r 2 sin 2 a = (li x — z{} 2 + (7i 2 — z 2 ) 2 —2(7^ — z 1 )(7i 2 — z 2 ) cos a, 

 L 2 sin 2 a = h x 2 + h 2 — 2hih 2 cos a ; 

 and hence 



(L 2 ~r 2 ) sin 2 a = 2z x z 2 cos a + 2£ 1 (7i 1 — 7i 2 cos a) 



+ 2^ 2 (7t 2 - 7tiCos a) — z x 2 — ^ 2 2 . 

 Dividing by z x z 2 , we obtain 



(L 2 — r 2 )s'm 2 a 2(Jii — 7i 2 cos ci) 2(7i 2 — 7iiCosa) 



; = zcos« + 1 — 



Z\Z 2 r z 2 Zi 



-7-T- • (18) 



The problem o£ finding the mean value of l/~i~ 2 f° r the 

 circle is thus reduced to finding the mean values ot l/r x , 

 l/~ 3 , z x \z 2 and ^ 2 M f° r the same circle. 



If [l/^i] c denote the mean value of \jz l for the circle of 

 radius r, and if fa denote the angle PCHi, we have 



L^iJc 7r?'J hi— rcos^j? 



since the integral from 7r to 27r is equal to that from to it. 

 Putting tan \fa=.x and therefore cos ^= (1 — ^ ,2 )/(l + x 2 ), we 

 find 



rl~l _ r °° 2^/tt 2/tt p, _! /VHH" 



UJc~Jo 7 tl -r.+ (A 1 +r)^-(V-^)*L tai1 'V fc-rj. 



- (19) 



- (J^-r 2 )i t x > 

 where t x is the length of the tangent from H^ to the circle, 



Similarly [l/^] = l/*a ( 2( >) 



Now z 2 _ PNg _ PN^osg-t-ON^ina 



z\ " PN; " ~ PNx 



But there are two positions of P for which PN X or ^ has 

 the same value, and the average value of ONx for these 

 two positions is OH^. Thus we find for the whole circle 



fel =cos« + OH 1 sin«r :: l 



= cos « + sin a . OHj/^. 

 Phil Mag. S. 6. Vol. 17. No. 97. Jan. 1909. E 



