36 



and the corresponding maximum value oi g = 0*3370, and re- 

 marks that, " with respect to spheroids of revolution, it thus ap- 

 pears that an equilibrium is impossible when g, or its value 

 in terms of the density, is greater than 0"3370. In the extreme 

 case, when g is equal to 0*3370, there is only one form of equi- 

 librium, the axes of the spheroid being 



and A ^/{l + (2*5293)2} or 2-7197 ^b; 



but when g is less than 0*3370 there are two different forms of equi- 

 librium, the equatorial radius of the one being less, and of the other 

 greater than 2*7197 k, k being the semi-axis of rotation. 



The number of the forms of equilibrium in spheroids of revolution, 

 he remarks, is purely a mathematical deduction from the expression 

 of the ratio of the centrifugal to the attractive forces ; and as this has 

 been known since the time of Maclauriu, the discussion of it was all 

 that was wanted for perfecting this part of the theory. 



Returning to the general equations of the problem, the author 

 deduces the equations 



d(3 



d <p 

 r dr 



where ^ is a definite integral, such that 

 d(p d <p' 



I? = A A' and = (X — \'y, 



which equations apply exclusively to ellipsoids with three unequal 

 axes, and solve the problem with regard to that class. From these 

 he derives another equation, which he states is no other than a trans- 

 formation of his first fundamental equation, and is equivalent to 

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

 M. Liouville. 



He also remarks that a limitation of one of the constants, which 

 the verification of this formula requires, agrees with the limitation of 

 M. Jacobi ; and further, that the relations which may subsist be- 

 tween the constants proves that there does exist an infinite number 

 of ellipsoids not of revolution, which are susceptible of an equili- 

 brium. • 



After determining the corresponding limits of these relations of 

 the constants, p being contained between the limits 1*9414 and 1, 

 while r- increases from zero to infinity, he remarks that an eUiptical 

 spheroid formed of a homogeneous fluid can be in equilibrium by 

 the action of a centrifugal force only when it revolves about the least 

 axis. 



He next deduces the general value of g (the ratio of the forces), 

 and thence its value in one extreme case, when = 0, or when 



