OF ELLIPSOIDS OF VARIABLE DENSITIES. 193 



are satisfied ; and another defined by 



^2 + ^%+... + 2 <1 and u = 0. 

 a l a z a s 



Moreover, for greater distinctness, we shall mark the quan- 

 tities belonging to the former with two accents, and those be- 

 longing to the latter with one only. 



Let us now suppose that V" is completely given, and like- 

 wise V[ or that portion of V in which the condition (3) is 

 satisfied; then if we regard F 2 ' or the rest of V as quite 

 arbitrary, and afterwards endeavour to make the quantity (5) 

 a minimum, we shall get in the usual way, by applying the 

 Calculus of Variations, 



n-s dV] 



y-^-j^r -- r~r 

 dx? du 2 u du ) 



(8), 



iU 



seeing that SF"=0 and 5F/ = 0, because the quantities V" and 

 F/ are supposed given. 



The first line of the expression immediately preceding gives 

 generally 



du 



, 



which is identical with the equation (2) No. 1, and the second 

 line gives 



dV 

 = u' n ' -~- (u being evanescent) ............ (9) . 



From the nature of the question de minima just resolved, 

 there can be little doubt but that the equations (2') and (9) will 

 suffice for the complete determination of F, where F" and F/ 

 are both given. But as the truth of this will be of consequence 

 in what follows, we will, before proceeding farther, give a de- 

 monstration of it ; and the more willingly because it is simple 

 and very general. 



4. Now since in the expression (5) u is always positive, 



13 



