OF ELLIPSOIDS OF VARIABLE DENSITIES. 195 



evidently be satisfied, provided we suppose, as we shall in what 

 follows, that 



n s + 1 is positive. 



If now' we could by any means determine the values of V" 

 and F/ belonging to the expression (1), the value of V would 

 be had without integration by simply satisfying (2') and (9), 

 as is evident from what precedes. But by supposing all the 

 constant quantities a lt 8 , a s ...... a, and h infinite, it is clear 



that we shall have 



= V", 



and then we have only to find F/, and thence deduce the gene- 

 ral value of F. 



6. For this purpose let us consider the quantity 



dV dU 



d_v_du dvdm 



7 ~~J T^ 7 7 r ** I A v I 



dx 8 ax a du du ) 



the limits of the multiple integral being the same as those of 

 the expression (5), and U being a function of x l9 a? 2 , ...... x s and 



u, satisfying the condition = U" when a l9 2 , ...... a a and h 



are infinite. 



But the method of integration by parts reduces the quan- 

 tity (10) to 



xt ...... dx a 



since = F" ; and as we have likewise = U", the same quan- 

 tity (10) may also be put under the form 



n-sdY 

 ^ 



132 



