ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 429 



Hence we readily get for the equivalent of (47), 



rr vtc^ c^ ^ « + 2« + 2a)-l .M + 2e + 2a)4-l n + 2i + 4im-3 



2.4. 6 2w 



xKa'^'-'''"{-l)''a'A''a"-fdaa'-"'-'-"{a^-a"') ^ 



■ ■ GO 



In certain cases the value of V just obtained will be found more 

 convenient than the foregoing one (47). Suppose for instance we repre- 

 sent the value of f^ when h = 0, or a = a' by V^. Then we shall hence 

 get 

 r^ i»c> o o n + 2i+2a)-l .n + 2i + 2w + l » + 2i + 4ft.-3 



2.4,6 2a> 



g — l~n~Suo 



OD 



which in consequence of the well known formula 



r(,-p)r(H±f^) 



/"'a-'da (a' - a'')-" = - «''-"-^? x -J^ i , 



by reduction becomes 



fl+s — n\^[n + 2i + 4!w — l'^ 



r(l±|z^)rp±-±i^) 



2r(a,+ i)r( ^ + ^'^+^" ) 



since in the formula (5), r ought to be made equal to zero at the end 

 of the process. 



By conceiving the auxiliary variable u to vanish, it will become clear 

 from what has been advanced in the preceding number, that the values 

 of the function P within circular planes and spheres, are only particular 

 cases of the more general one, (49), which answer to * = 2 and s = 3 

 respectively. We have thus by combining the expressions (48) and 

 (49), the means of determining Vo when the density p is given, and 

 vice versa; and the present method of resolving these problems seems 

 more simple if possible than that contained in the articles (4) and (5) 

 of my former paper. 



GEORGE GREEN. 



3k2 



