THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 45 



It will be shown later that an ellipsoid distorted in this way cannot possibly be a 

 figure of equilibrium for a rotating fluid ( 20). 

 For the solution 



i = (f+/V + yf) (56) 



we have 



and the solution is 



x / rj? ^K>y\ ^l i* 2 , ?/ , z 2 ,\/3X /3X 



= ' - 



It will be shown later that this distortion leads to the Darwin-Poincare series 

 of pear-shaped figures of equilibrium of a rotating fluid. 



n = 4. The analysis of 13, 14 was confined to the case in which ?, was 

 supposed of degree not greater than 3 in ,,, g. But if, in the solution finally 

 obtained in 15, we take u^ = 0, which involves taking also i\ = 0, we are left with 

 a solution (cf. equation (50)) 



</> = e?(u a +fw+f a u/), 



in which u. 2 , w, and w' do not vanish on account of the occurrence of the arbitrary 

 function . And since has been supposed of the fourth degree, this solution 

 gives us the solution of degree n = 4 to the first order, the parameter <? replacing 

 the former e. Thus the solution of degree 4 is 



u = a general function of degree 4 in /, (the old e 2 2 ) 



This solution is not discussed in detail in the present paper, but is classified here 

 with the other solutions for the sake of completeness. An ellipsoid distorted in 

 accordance with this solution would give rise to a series of dumb-bell shaped figures, 

 which would be figures of equilibrium for a rotating liquid. They would be unstable 

 for a homogeneous mass, but the corresponding figures might conceivably be stable 

 for a heterogeneous mass (cf. POINCARE'S remark quoted in 1 of the present paper). 



17. One point of interest must be mentioned here in connection with the potentials 

 derived from these solutions. 



In the potentials arising from the solution of degree n = 2, ^ = a^ 3 , the internal 

 or boundary potential will be of the form Ix 2 +my 2 +nz 2 , where I, m, n do not 

 involve x, y, z, or X. Since this must be a solution of LAPLACE'S equation, l + m + n 

 must vanish, and the potential must be expressible in the form m (y'^x 2 ) + n (z 2 x 2 ). 

 All the other potentials may be similarly treated. 



