ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 403 



5. Again, it is clear that the condition (4) is satisfied for the whole 

 of F"/; and it has before been observed (No. 2.) that when V is deter- 

 mined by the formula (1), it may always be expanded in a series of 

 the form 



r = ^ + J?«' + Cu' + &c. 



Hence the right side of the equation (9) is a quantity of the order 

 ?/"-'+' ; and v! being evanescent, this equation will then evidently be 

 satisfied, provided we suppose, as we shall in what follows, that 



n — s \ \ is positive. 



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

 V( 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,, «2> «3 

 a, and h infinite, it is clear that we shall have 



= V", 



and then we have only to find V^, and thence deduce the general 

 value of V. 



6. For this purpose let us consider the quantity 



w ^ ^7 n-AdVdU dVdU , dVdU dVdU\ 



jdxidx.i...dx,duvr '{-r—-j— + -f— -j— + ••• + i—n— + -i t-)\ (10^ 



{dxidx^ dx.dxi dx.dx, du du j ' ^ ' 



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

 expression (5), and U being a function of ;r,, x^, x, and u, satis- 

 fying the condition 0= U" when «,, a^, a, and h are infinite. 



But the method of integration by parts reduces the quantity (10) to 



— fdXidxi dx,—j — u'"-' . V 



du 



-/..........x..»».-.r|.,«^+^.^^} (H, 



since = V"\ and as we have likewise = U", the same quantity (10) 

 may also be put under the form 

 Vol. V. Part III. SG 



