ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 407 



r dx'dxi dxlP,' \ 2 J ,^ , „ 



f '■ '—^ ^, = -L-jl— ir...(l6). 



-^ \{x,'-x,y+{x,'-x,y+... + {x:-x.y+u'\— r(^^) 



For under the present form both its members evidently satisfy the 

 equation (2), the condition (9), and give V" = 0. Moreover, when the 

 condition (3) is satisfied, the same members are equal in consequence 

 of (15). Hence by what has before been proved (No. 4), they are 

 necessarily equal in general. 



By comparing the equation (16) with the formula (1), it will become 

 evident, that whenever we can by any means obtain a value of V satis- 

 fying the foregoing conditions, we shall always be able to asSgn a value 

 of p which substituted in (1) shall reproduce this value of V. In fact, 

 by omitting the unit at the foot of P", which only serves to indicate 



the limits of the integral, we readily see that the required value of p is 

 p'= \ P' {c). 



r^ 'r. fn~S+l\ ' 



7. The foregoing results being obtained, it will now be convenient 

 to introduce other independent variables in the place of the original 

 ones, such that . 



^1 = «i?i» «a = 02^2j x, = as^„ u = hv, 



Oj, ttj, flj being functions of h, one of the new independent variables, 



determined by 



a,' = «;* + h', a,- = (h' + h\ a/ = aj' + /^^ 



and V a function of the remaining new variables, f,, ^2, ^3, ^s satis- 

 fying the equation 



1 = v' + |;^ + e/+ + U; 



a,', a/, Os', 0/ being the same constant quantities as in the equation 



(a), No 1. Moreover, Oi, a.^, a, will take the values belonging to 



the extreme limit before marked with two accents, by simply assigning 

 to h an infinite value. 



