74, 75] NEWTONIAN POTENTIAL FUNCTION. 147 



(i) 



\ / 



Differentiating by x, (dm and a, b, c being constant), we have 



l dm dr 



By39,( 7 ), 



3 ,. dm a; a 

 (2) 



* 

 r 2 r 



Now 



# a 



(3) 



where the direction of r is taken from a, 6, c to x, y, z. This being 

 the derivative for that part of the potential due to dm, we have 

 to take the sum of such expressions for all corn's in K, that is, 

 perform a volume integration 



dV 8 [[(pdadbdc [[[ d fl\ , ,.. 

 (4) -r- = ^- III ~ -- = I llp^ \-}dadbdc 



dx dxjjj r JJJ^dx\rJ 



= 1 1 Ip da db dc = 1 1 1 a cos (rx) da db dc. 

 Let the direction cosines of R be cos A, cos B, cos C, and since 





> 2 ^2 . 2 ' 



/2 T TI 



p / \ p p 



- 1 -: cos (r#) > - cos (rx) > cos (rx). 



r 2 z r 2 n 2 



Multiplying and dividing the outside terms by cos .4 and 

 integrating, 



(5) -^ 



Multiplying by jR 2 and letting R increase without limit, since 



,. jR 2 ,. R 2 ,. cos(?^) 

 lim = lim = lim ^-j- 7 = 1, 



-R =c _ _ 



102 



