398 Prof. A. Gray on the Attraction of 



18. For a solid homogeneous ellipsoid, which has 



2 = 1 



a 2 

 for the equation of its surface, (15) becomes 



Y^pahcCC • = 'gL,—^, . (19) 

 J a Jf(u) V (a a + u)(b* + w)(c 2 + u) } 



where X [the positive root of the equation in u 



t 2 

 a- + a 



is the lower limit of the integration with respect to u, and 

 f(u) is the value of k that fulfils equation (7) for any one 

 particular value that u may have in the range of integration. 

 This value of h is given by 



. f 2 

 h — ^^-1 



a z -\-u 

 Thus 



rx ( i—2-p— V" 



\=7rpa^ -. . . - . (20) 



J A v/(tt 8 + t0(&* + «) («* + ") 



There is no difficulty in the application of this method of 

 integration by shells to the formal determination of the 

 potential of solid ellipsoids or of thick shells of varying density, 

 if each homoeoidal film is of uniform density throughout. If 

 the density p vary from shell to shell then for the ellipsoid 

 by (16) 



Y — 7rabc 



Jo J, 



\/(a 2 + u)(b tz + u){c 2 + u) 



where \' is again the positive root of (7) for any given value 

 ■oik. But 



dk=-\ t f /* y2 \du 

 L (a 2 +uy J 



and when k = 0, u = cc ; likewise when k = 1 u=\, where X 

 is the positive root of 



2-4- = l 



a- + u 

 regarded as a cubic in u. Therefore 



V = TrahrCp{Zj-.;^}du f%- ? -l^ j^ - (20'; 



