11] 



FIGURE OF THE EARTH. 



213 



9. From these expressions we can deduce the potential or attrac- 

 tion exerted at any point within the spheroid. 



First consider the solid sphere of radius OL = r'. The potential of 

 this at any point M (OM=r) may be divided into two parts, first, the 

 consecutive shell outside M, and second, the sphere of radius r within M. 

 The potential of the latter is the same as if the whole were concentrated 

 in O, that of the former is the same at M as it is at ; hence the 

 potential of the sphere at M is 



3 



+ 27T (r'* - r 2 ) = 27T r - 



3 



Next consider the spheroidal shell surrounding the inscribed sphere. 

 Employ the construction of 3, and take a point M' in OH produced 



such that 



OM: OL:: OL : OM' ; 



then MP : M'P :: r : r', 



and for a particle at P, 



Potential at M : Potential at M' : : r : r, 



therefore it is independent of the position of P, and the potential of the 

 whole shell at M may be found by multiplying the potential at M' by 

 r'jr. But if L be the point the sine of whose latitude is X, the potential 

 of the spheroidal shell at M' is 



8 n (r\>\ /3 . 1\ /8 ,8 r>\ /3 1\ 

 -T5" r WH X "-2J = {3 -15 VJ 6 (2 X 2J' 

 Multiply this by r'/r, and we have the potential of the shell at H 



I' 8 ^ 8 

 {3 ^-15 



add the potential of the sphere of radius r', and we have the potential 

 at H of the solid spheroid 



