150 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



<> -E-rjx^" 



* * 27r 



sin 6 cos | 



r* c* f 27r 



I drl I I 



which also vanishes with e. Hence ^ is everywhere finite, and in 



ox 



dv dV 



like manner , . 



If we attempt this process for the second derivatives ^ , . . . 



it fails on account of , which gives a logarithm becoming oo in 



the limit. 



dV 



We will give another proof of the finiteness of . 



ox 



We have 

 (3) ^=11^^ dadbdc 



which by Green's theorem is equal to 



This is however only to be applied in case the function - is 



everywhere finite and continuous. This ceases to be the case 

 when P is in the attracting mass, hence we must exclude P by 

 drawing a small sphere about it. Applying Green's theorem to 

 the rest of the space K' 1 ', we have to add to the surface-integral 

 the integral over the surface of the small sphere. 



/* f ^7^f 

 Since cos(nx) 1, this is not greater than p^l I = 4<7rep m , 



which vanishes with e. Hence the infinite element contributes 

 nothing to the integral. 



dV 

 In the same way that was proved finite, it may be proved 



ox 



dV' dV" 

 continuous. Dividing it into two parts ^ and ^ , of which the 



dx dx 



