52 Mr. GL F. C. Searle on the Impulsive 



mean-value problem may then be expressed concisely as 

 follows : — 



A sphere being given, AB being a fixed chord, and CD being 

 fixed points on the chord, the mean value of the reciprocal of the 

 product of the distances of C and D from a valuable [tangent 

 plane is log (ABCD)-hAB . CD." 



§ 9. To apply (26) to the electrical problem, we put 



B = l, ^ = 1/71! =w/«i, h 2 = l/n 2 =zv/ih, 

 It will be convenient to write 



n 1 2 + 7i 2 2 —2n 1 n 2 cosa = m' 2 = iv 2 lv 2 , . . (27) 



where w is the change of the velocity of the sphere, so that 

 u 2 is the resultant of u L and w. Then 



L 2 sin 2 a = — 9 + 



2 cos 



and 



n x n 2 



n^nJ 



(L 2 -B 2 )sin 2 a = 

 By § 6 we have 



F(wi, n 2 > a) = 



■7i, 2 w 2 2 sin 2 a 

 n x 2 n 2 2 



4-7T 



n Y n 2 



r-i 



provided 7i 1? 7j 2 and R have the values just written. Inserting 

 those values in (26) we have 



F(w l5 w 2 , a) = 



27r . 1 — n x ii 2 cos a + (^ 2 — n 2 n 2 sin 2 a)* 



(??i 2 — n^w/ sin 2 a)* ° 1 — n^ cos a — (W — w^ra/ sin 2 a)* 



. . .'(28) 



§ 10. We now proceed to give a second investigation of 

 the value of 



• "FOii, rc 2 , a) = J -— - 



7li COS ^ 2 )(1 — ?i 2 C0S ^2) 



where the integration is taken over a 

 sphere of unit radius. 



Let the plane containing the radius 

 OP and the radius Oi^ make an angle 

 <£ with the plane containing Oi^ and 

 Ou 2 , as shown in fig. 2. Then we 

 have 



cos 2 — cos 6 1 cos a -f sin 1 sin a. cos (f>, 



while 



dw = sin # x ^ : rf<£. 



Fix. 2. 



Az? 



