121, 122] 



DERIVATIVES OF POTENTIAL. 



357 



At points not in the attracting masses, V and all its derivatives 

 are finite and (since their derivatives are finite) continuous, as well 

 as uniform. 



Also since 



63) 



we have by addition 

 64) 



f *j V y U fJ 



that is, F satisfies Laplace's equation. 



This is also proved by applying Gauss's theorem [ 118, 42)] to 



each element 



r 



122. Points in the Attracting Mass. Let us now examine 

 the potential and its derivatives at points in the substance of the 

 attracting mass. 



If P is within the mass, the element at which the point Q, 



where dm is placed, coincides with P, becomes infinite. It does not 

 however, therefore follow that the integral 

 becomes infinite. 



Let us separate from the mass K a 

 small sphere of radius s with the centre 

 at P. Call the part of the body within 

 this sphere K' and the rest K 1 '. Call 

 the part of the integral due to jfiT', F f , 

 and that due to K", F". Now since P 

 is not in the mass K", F" and its deri- 

 vatives are finite at P, and we have only 

 to examine F f and its derivatives. 



Let us insert polar coordinates 



Fig. 127. 



000 



so that, integrating first with respect to (p and #, since the absolute 

 value of an integral is never greater than the integral of the absolute 

 value of the integrand, 



