416 VIII. NEWTONIAN POTENTIAL FUNCTION. 



For every value of # there is one value of A, given by the 

 cubic 21). 



Let us now change the variable 5 to t, where, # being constant, 

 s = & 2 t, ds = & 2 dt, and put A = & 2 u. 



Then 



1 oo 



/"* /* dt 



23) V= 2xabc I &d& I g 9 > 



x X 



w 



where u is defined by 



24) ^ - 



Since &* is thus given as a uniform function of u, we will now 

 change the variable from # to u. 

 Differentiating 24) by #, 



25) 



When ^ = 0, w = oo, and when ^ = 1, u has a value which we 

 will call 6j defined by 



X 2 ?/ 2 2 2 



^r G + ^f^ + ^qr^ = 



Accordingly, changing the variable, 



27) F=^a&c / \^^ + ^^ + ^^^ du C-^=^=^^ 



The three double integrals above are of the form 



28) 

 where 



This may be integrated by parts. 

 Call 



