THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 71 



On collecting the values of Kj, K 2 , K 3 , K 4 , K 5 , from pp. 62, 63, and 66, and inserting 

 the numerical values already obtained, we find 



y'+^'+ m '\ -19-91885 = 28-26823, 

 a a b 2 c 2 / 



K 2 = _(+SL+I) +0-1102214 = -0-2624723, 

 \a b cv 



K 3 =_(5L'+!l+ 2 L)+O i 04221664 = -0'0986790, 

 \ct o c / 



K 4 = -2.-2__^l +0-3603665 = -0'900332-19-85984w", 



0^ C 



7 - 2 +3-556845 = -2770998. 

 a be 



To evaluate s' we have to examine the form assumed by the potential at infinity. 

 The additional terms in the potential, as far as terms in - , produced by the distortion 

 are readily found to be 



and if Sm is the additional mass produced by the distortion, this must be identical 



with . Hence, if s' is determined by the condition of constancy of mass, we have 

 r 



s' = K 4 - 1 2 n K 5 = -0-230755-13-239893 n" 

 giving (cf. equation (153)), 



s = -0-1586814-13-239893 n". 



Discussion of the Figure. 



35. The boundary of the pear-shaped figure, as far as the second order of small 

 quantities has been found to be 



a 2 



a* My 4 Nz* 2foV 2mzV 2na?y> 2px 2 2qi? 2rz 2 \ 

 K "" T5 I -- r-+ 7 . , I -- r^~ "I -- TTT- + * , + i" H -- 7- + S = 

 \ a 8 6 8 c 8 Z> 4 c 4 cV a 4 6 4 a 4 i 4 c 4 / 



In this equation all the coefficients have been determined ; the coefficients p, q, r and 

 have been found to involve n", 

 the remainder are pure numbers: 



2 



s have been found to involve n", defined in S 25 by the equation -^- = n + e 2 n", while 



2-jrp 



