OF ELLIPSOIDS OF VARIABLE DENSITIES. 199 

 dx^dx,' (fa.'P/ 



i < - xtf + - X,Y + . . . + (x; - X,Y + u'F 



. * 



V. (16). 



() 



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 satisfying the foregoing conditions, we shall always 

 be able to assign 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 



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 



*i = i?i> x z = ,&,*. = & u = hv, 



a l9 2 ,...a a being functions of h, one of the new independent 

 variables, determined by 



and v a function of the remaining new variables, f x , f 2 , f s , ... f, 

 satisfying the equation 



1 = u 2 + ? ! 2 + ? 2 2 + + f B ; 



o/, a/, a/, ... a/ being the same constant quantities as in the 

 equation (a), No. 1. Moreover, a d , a 2 , ... a a will take the values 



