50 Mr. G. F. C. Searle on the Impulsive 



Similarly, since 



Z X PNx PN 2 cos a — ON 2 sin a 



PN 9 ~ PN 5 



we find 



— = COS a — sin a . OH 2 /£ 2 



But OHi sin a = CH 2 — CHj cos oc = h 2 — h\ cos a 



OH 2 sin a = CH 2 cos a — CHj = 7i 2 cos a — h v 

 and thus 



[-2/- j ] c = cos a + ( 7l 2 — 7i i cos a ) Ai • • ( 2 !) 

 fci/%] c = cos a + (^1 — 7*2 cos a) /£ 2 . . . (22) 



When we substitute in (18) the mean values shown in (19), 

 (20), (21), and (22), we obtain 



/T2 o\ • 2 T n 7*! — 7l 2 COS<Z 7i 2 — 7i! cos a 



(L 2 — r 2 )sm 2 a ■ — = h . 



But 7i 2 — 7i 1 cosa = OH 1 sina= sin a. (L 2 — 7*i 2 )*- . (23) 

 A 2 cosa— 7i 1 = OH 2 sina = sina(L 2 — 7i 2 2 )*, . . (24) 

 and hence 



Ui^Jc (L 2 — r 2 ) sin a (^ v t 2 ' J v y 



§ 8. We can now find the mean value of l/ziz 2 for the 

 surface of the sphere. If we measure x from the centre of 

 the sphere parallel to the line of intersection of the two fixed 

 planes, the area of the zone on the sphere defined by x and 

 w + dx is 27rRdx, and thus 



["— 1 = — L f I"— 1 27rR ^' 



L^i^ 2 Js Zirnr J Lz 1 z 2 '*c 



Now * r 2 *=R 2 — x 2 , 



and hence . t x 2 = 7i : 2 - r 2 = hf - E 2 -f x 2 , 

 t 2 2 = \*-r 2 = li 2 2 -n 2 + x\ 

 Thus we have 



U*Js Rsina ) L 2 -E 2 + a? 2 ( (A 1 , -'R , +« 8 )* 



(V-R 8 + oj 8 )* r 



