64 MR. IVORY ON HOMOGENEOUS ELLIPSOIDS 



Supposing X and X' to vary from this extreme, when the first increases, the other will 

 decrease ; so that, when X is infinitely great, X' will be zero ; which proves that the 

 other extreme limit is a cylinder extending indefinitely on either side of the base, 

 which is a circle having k for its radius. 



6. It remains to consider the value of ^. In the first of the equations (6.), let the 

 operation indicated be performed, and the result will be 



and from this we obtain the value of g in the extreme case when r^ = 0, or when X 

 and X' are equal, viz. 



_ /'^ p.5x^{l — oc'^)dx 



^-Jo (I +px'y ' 



which is no other than the determination of j- in a spheroid of revolution having its 

 axes equal to 



k and A;>/2-9414 = k X 17150. 

 In the other extreme case, when r^ is infinitely great, q is zero. 



It has been shown that for every given value of r^, there is only one value of p, 

 and only one ellipsoid ; and when r^ and p are both ascertained, the foregoing ex- 

 pression proves that q is fully determined. Thus there is an appropriate value of q 

 to every ellipsoid susceptible of an equilibrium. 



In the formula for q, one of the two quantities, r^ and p, increases when the other 

 decreases; and hence it may be surmised that more than one ellipsoid may answer 

 to a given value of q. Some calculation is necessary to elucidate this point. For 

 the sake of abridging expressions, put 



p = \ /(i -{-py + r^ 



Q = x/(l -^px^y^ -{-r^x^ 



M = p (1 + J9) (1 + j» .r2) + j?2 y2 ^2 



d u = 3 x^ {I — x"^) d X, 

 the variation of d u being between the limits x = and a; = 1 : then, the foregoing 

 value of q will be thus written : 



/ du.M 



and, q being considered a function of/? and t^, the fluxion with respect to p will be 



dq _ r du fc^M (I +p)M. Sx'^{\ +px^)M '\ 

 T^ —J PQ^ • I rf/? ~" P^ "~ Q^ J • 



it will be found that 



