62 MR. IVORY ON HOMOGENEOUS ELLIPSOIDS 



5. Returning now to the general equations of the problem, let 



^0 V{\ + X^^). (1 +X'''^* 



V{\ + X2^). (1 + 



and it will be found that the two equations (3.) are thus expressed : 



d(t> d(r> , 



Further, put /? = X V, r'^ = (X — X')^, and we shall have 





V{1 +pxy + T^X^ 



and the two values of q in the partial differentials of ^ relatively to a and l! being ex- 

 pressed in the partial differentials relatively to p and r^, we shall obtain 



These two values of q coalesce in one when X — x' = 0, that is, in spheroids of revo- 

 lution; and we thus fall again upon the same equation that has already been dis- 

 cussed. In all other cases the two values cannot subsist together, unless 



d<p 



.= '. ^ "■' 



*y 



r d'r 



which equations apply exclusively to ellipsoids with three unequal axes, and solve the 

 problem with regard to that class. The latter of the equations (6.) expresses the re- 

 lation that the two quantities p and r^ must have to one another in every ellipsoid 

 with three unequal axes which is susceptible of an equilibrium. The fluxional ope- 

 ration indicated being performed in the same equation, the result will be, 



((I +px^f + T^x^y ' 



which is no other than a transformation of the equation (1.), and is equivalent to 

 other transformations of the same equation found by M. Jacobi and M. Liousville. 



The formula (7.) cannot be verij&ed unless p, or X X', be greater than 1 ; for if p 

 were equal to 1, or less than 1, the integral would be positive. This agrees with the 

 limitation of M. Jacobi. 



If any value be assigned to r^, it is evident that a corresponding value of p may be 

 found which will verify the formula (7.) : for, if p be made to increase continually 

 above 1, the integral, which is positive at first, will finally be negative; and it must 

 be zero, in passing from one of these states to the other. This proves that there does 

 exist an infinite number of ellipsoids not of revolution, which are susceptible of an 

 equilibrium. 



