356 VIII. NEWTONIAN POTENTIAL FUNCTION. 



take the sum of such expressions for all dm's in K, that is, perform 

 a volume integration. 



dv d rrr Q dadbdc err z /i\ 



V* = ^JJJ " = J J J * IT* (r) 



- I I I Q rS dadbdc = I I I ~cos(rx)dadbdc. 



Let the direction cosines of E be cos A, cos J5, cos (7, and since 



2 < -2 < TV 



r z r r x 



1 1 



#> '., r -t 



~s cos (rx) > - -^ cos (rx) > - -^ cos (r a?) . 



r z - r t 



Multiplying and dividing the outside terms by cos A and integrating, 



Multiplying by jR 2 and letting J5 increase without limit, since 



,. JS 2 ,. E* ,. cos(ra?) 

 lim -T = I 5- = hm = 1, 



= -Jfcos^l, 



62) lim [jR 2 1?1 = - M cos ^, 



Therefore the first derivatives of F, and hence the parameter, 

 vanish at infinity to the second order. 



In like manner for the second derivatives, 



a 2 F a rcr^dr rrr v /i\ , 

 ^ = ^ J J J ~ = J J J ? w (7) dr 



Every element in all the integrals discussed is finite, unless 

 r = 0, hence all the integrals are finite. We might proceed in this 

 manner, and should thus find that: 



