1 66 



STATICS. 



[273- 



Let PA = r, PA 1 = r', and let ^/<o denote the solid angle of the cone 

 (i.e. the area it cuts out of a sphere of radius i described about P as 

 centre) ; then-rv/w is the area cut out of a 

 sphere of radius r with the same centre 

 P. Hence the element of mass at A is 

 p ^Vo>/cos PA C, and its attraction at P is 

 = Kpdw/cosPAC. Similarly, the attraction 

 of the mass element at A' is 



= K P r'-do>/(r'' 2 cos PA'C) = xprfu/cosPA'C. 



These attractions are equal, since for the 

 sphere %PAC=^PA'C. 



The whole sphere can thus be divided up 

 into elements exerting equal and opposite attractions at P ; the resultant 

 attraction of the whole shell at any internal point is, therefore, zero. 



273. (b) Attraction at an external point P. The investigation can 

 be made similar to that for an internal point by introducing the point 

 P 1 (Fig. 82), which is inverse to /*with respect to the sphere, i.e. the 

 point P' on CP for which CP- CP' = CA\ or putting CP=p, CP'=p', 

 CA = a, the point for which 



pp' = a\ (12) 



Any chord HH* through P' determines two pairs of similar triangles : 

 CUP 1 and CPH, CJf'P' and CPH* ; for each pair has the angle at C 



Fig. 82. 



in common, and the sides including the equal angles proportional by 

 (12-), since CH= CH'=a. It follows that C&P' = K CPH, and 

 ^ CH'P' = %. CPU'; hence, as the triangle HCH' is isosceles, the line 

 CP bisects the angle HPH' . 



With the aid of these geometrical properties it can be shown that 



