48 



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



that li 1 >H and h 2 >R ? and thus the planes do not cut the 

 sphere. 



If we denote the mean value of \\z x z 2 for the surface of 

 the sphere by [l/s^Jg, we have 



J = 47rR : 



f— 1 



U%Ji 



(17) 



The first step is to cut the sphere in a circle by a plane 

 perpendicular to both the fixed planes and to find the mean 

 value of l/ZjZ a for this circle. The case in which the two 

 fixed planes are parallel has been discussed in § 4. We may 

 therefore now suppose that their line of intersection is at a 

 finite distance from the centre of the sphere. 



§ 7. Let the plane of the paper be normal to the line of 

 intersection of the two fixed planes and let it cut the sphere 

 in a circle of radius r, the centre of the circle being at a 

 distance L from the line of intersection. Let 0H n OH 2 

 (fig. 1) be sections of the two fixed planes by the plane of 



Fhr. 1. 



the paper, and let CH 1 = A 1 and CH 2 = h 2 be the perpendiculars 

 from Cj the centre of the circle, upon OH4 and OH 2 . Then 

 the angles HxCHg and HjOHg are each equal to a. If P be 

 any point on the circle and PN l9 PF 2 be perpendiculars from 

 P upon the lines OHj, 0H 2 , then P^ — z x and FN 2 = z 2 . 



Now draw FM 1 and PM 2 perpendicular to OH 1 and CH 2 . 

 Then CM^h.-z, and CM 2 = h 2 ~z 2 . Since M : and M 2 lie 

 on a circle of which CP is tb.3 diameter, and since M4M2 

 subtends the angle a at C, it follows that M X M 2 subtends an 

 -angle 2« at the middle point of CP. Hence 



MjM 2 = CP sin a=r sin a. 



