THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 53 



and IAA is given by equation (80) as soon as the values of I AB and I AC are known. 

 Substituting for J A , J B , Jc, from equations (83) to (85), we obtain 



2 c 3 20 



-n, AA = -- 



(Jtr - 



a i c 



IAB = , IAO = -j -,n, IAA = -i-ip - 2 n-- 2 . . . (89) 



Finally we bave 



\d\ 



f \d\ J_r/I_l\^_ _J __ /T _T \ 



' " Jo AABC ~ c 2 -6 2 Jo \B CJ A A ~ (c 2 -fc 2 ) ^ 



IAC), C 3 = 2 (I A B IAA)- 



c 2 - 2V a*-b 



With this material it ought to be possible to find the points of bifurcation from 

 equation (80). As, however, DARWIN'S results are available, it will be sufficient to 

 make use of his results and merely verify that his quantities satisfy equation (80), as 

 they are in point of fact found to do. 



23. DARWIN'S values, calculated for an ellipsoid such that al>c = 1, are 



a = 1-885827, b = 0'814975, c = 0'650659, 

 n = -- = 0-1419990, 



2-TTp 



whence, by equations (86) and (89), 



= 0-2068037 



a 



2 b 2 



I AB = 0-1419990, I AC = 0-1611871, 1 AA = 0'0711382, 



Cl = G'07967602, c 2 = 0'02874219, c 3 = 0'2450100. 



With the use of these values, equations (76), (77) and (78) become 



0-01216184a' + 0-02450100/3' + > 02874219y / = ..... (90) 

 0'07350300a' + G'1970290/3' +0'07967G02 y ' = ..... (91) 

 0'0862266a' +0-07967602/3' + 0-3835217y' =0 ..... (92) 



The values of a', ft, y, are, of course, indeterminate to within a common multiplier. 

 The simplest set of values, obtained by cross multiplication of the coefficients of 

 equations (90) and (91), is 



a! = - 0'003710945, j3' = O'OOll 43630 / = 0'000595338. 



