75, 76] NEWTONIAN POTENTIAL FUNCTION. 149 



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 therefore follow that the integral becomes infinite ( 25). 





FIG. 33. 



Let us separate from the mass K a small sphere of radius e 

 with the centre at P. Call the part of the body within this 

 sphere K' and the rest K". Call the part of the integral due to 

 K', V, and that due to K" ', F'. Now since P is not in the mass 

 K" ', V" and its derivatives are finite at P, and we have only to 

 examine V and its derivatives. 



Let us insert polar coordinates 



<rt F'- 



so that, by 23 (5), 



/* e 



I 



Jo 



rdr 



o 



6 



if p m is the greatest value of p in K'. 



As we make the radius e diminish indefinitely, this vanishes, 

 hence the limit 



lim(F'+F") 



e = 



is finite. 



In like manner for the derivative 



Separate off K' from K". The part of the integral from K" is 

 finite. In the other ^ introduce polar coordinates, putting 



