358 VIII. NEWTONIAN POTENTIAL FUNCTION. 



65) \r | 



if $ m is the greatest value of Q in K'. 



As we make the radius e diminish indefinitely this vanishes, 

 hence the limit 



is finite. 



In like manner for the derivative 



d = 

 dx~ 



Separate oft K' from X". The part of the integral from K" 

 is finite. In the other K' introduce polar coordinates, putting & = (rx), 



66 ) 



dV 



dx 



I dr I I \ sin -9- cos # | d&d<p, 



' / 



o oo 



dV 

 which also vanishes with g. Therefore ^ is everywhere finite, and 



dV 3V 



in like manner o > -^- 

 oy oz 



3 2 V 

 If we attempt this process for the second derivatives -~-^i 



it fails on account of > which gives a logarithm becoming infinite 



in the limit. 



BV 

 We will give another proof of the finiteness of We have 



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 



