ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 401 



«,, «2, a, and h being positive constant quantities; and such that 



we may have generally 



Ur > dr. 



In this case the multiple integral (5) wiU have two extreme limits, 

 viz. one in which the conditions 



V ^ or IT 71 



-\ + -^ + + -^ + t; = 1 and u— a. positive quantity (7) 



are satisfied; and another defined by 



% + %+ +-, /I and « = 0. 



jVIoreover, for greater distinctness, we shall mark the quantities be- 

 longing to the former with two accents, and those belonging to the 

 latter with one only. 



Let us now suppose that J^" is completely given, and likewise F,' 

 or that portion of f^' in which the condition (3) is satisfied ; then if 

 we regard F/ or the rest of T^' as quite arbitrary, and afterwards endea- 

 vour to make the quantity (5) a minimum, we shall get in the usual 

 way, by applying the Calculus of Variations, 



/7F'' 

 -fdx.dx, clx,u"-^^r,'~- (8) 



seeing that ^V" = and SFj' = 0, because the quantities V" and F,' 

 are supposed given. 



The first line of the expression immediately preceding gives generally 



= 2'+'— — ^Hf^ {^•\ 



' dxr du' u du ^ ' 



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



dV 

 = u'"'' ~7-^(^' being evanescent) (9). 



