484 Mr. J. Prescott on the 



Hence the potential of the shell 



= K ™ 2 \^~e-(— -W-( — —\ 



"r | ,. r \ r — b r + b) r\r—b r + bj 



+ ^(^b + r^b-) + ~{( 



= 2K ff ^{i(l42*)-?J} 



The potential, at an external point P on its axis, of a homo- 

 geneous spheroid whose polar radius is /3 and eccentricity e, 



is obtained by putting 2 -r- for cr in the expression for the 



potential o£ a shell and integrating. 

 Thus the potential 



Since the potential of the spheroid at an external point is a 

 zonal harmonic, we find that the potential at any external 

 point is 



where r is the length of the radius vector from the centre 

 of the spheroid to the point, and 6 is the angle between this 

 radius vector and the axis of symmetry. Now suppose the 

 earth is so composed that layers of equal density are the 

 surfaces of spheroids of varying ellipticity, the surface of the 

 earth itself being one of these spheroids. We may consider 

 the density of any layer and its ellipticity to be functions of 

 the polar semi-axis of the layer. Let SY be the potential 

 at r, 6 of one of these layers. Then 



SV = * Kttp ; | , { I ( 1 + -2eW-^ e/3%3 cos* 6 - 1) } 8/S. 



