684 Figures of Equilibrium of Rotating Fluid. 



It is not difficult to prove that 



From the coefficient of 6 in (13) we have 



6 



011= ~n «i2 sni2 « 



and substituting in (14) this value of a n and the value of 

 a n derived from (15), we find, on account of the relation 

 above, that the coefficient o£ 6 2 leads to the equation 



9 ff x Y*dx . 2 /Gf 1 P 2 P 4 2 ^ 15 f x P 6 P 4 2 ^ V. 



" lU IJo (l.+*W) 8 a V7j (l + AV) 2 -22j (l+AW)'/f 



= 0. 



I find that the coefficient of a 12 2 is 



•0043482+ sin 2 a(--0060458 + -0025975) = -00100. 



Each o£ the three constants has been evaluated by three 

 distinct processes, and there is substantial agreement between 

 the different results, so that it is not easy to believe that it 

 is merely an arithmetical error which causes this coefficient 

 not to vanish. We thus apparently arrive at the con- 

 clusion * that Kj^O. I am unable to see where there can 

 be any mistake in equations (C), and must for the present 

 content myself with merely noting the difficulty. It is 

 important that it should be cleared up, because further 

 progress is impossible until it can be shown either that 

 K x = is in reality a consequence of the equations to deter- 

 mine the coefficients in equation (12), or that the reverse is 

 true. The direct disagreement with the work of Poincare 

 and Darwin makes one hesitate to say definitely that K : =£0, 

 but it may be remarked that the whole difficulty arises from 

 the second order terms in equation (14), and that these terms 

 are not considered at all in Poincare's paper ; they do not in 

 fact affect the form of the bifurcation equations, and so long- 

 as we restrict ourselves to the consideration of first order 

 terms we are not led to any inconsistency in supposing that 

 K! = 0. It is, however, to be remembered that Darwin, in 

 his papers on The Stability of the Pear-shaped Figure of 



* It is easy to assure oneself that it will not do to suppose that 

 a lQ = a- ll = a l2 =0, and so to avoid this conclusion. For, if all the first 

 order terms disappear, the bifurcation equations "become illusory. 



It is, further, to be observed that if Kj^O the angular momentum is 

 not stationary. 



