Equilibrium of Rotating Fluid. 681 



and, n >• 1, 



Oi-l)Kla,„0*+ f'p, ie/f8»-a=0j 



"- 1 Jo " 



which, on making use of: Lagrange's expansion, become 



. . . . (C) 



for all positive integral values of n, except that when n=l 

 the left-hand side is (1 + F) (K -K) + Ka n (9. 



To equations (C) must be added the condition of constant 

 mass, i. e. of constant volume, which may be expressed by 



saying that I r 3 dx must be independent of 0, i. e. 







C 1 dv w 1 d m ~ l fr» s+1 ~\ ia "\ 



for all values of #. .... (C) 



5. To determine the stability of the pear we must examine 

 the expression for the angular momentum, which must be 

 stationary, and for stability a minimum, when = 0. That 

 is to say, we must retain squares of 0, but need not retain 

 higher powers in the equations (C) and (C). They then 

 become 



(1 + *O(K O -K) + Ka n 0=0r^^ 



1 3 \ l 



+ ai ai 2 R 3 +2 a ii 2R i +2a 11 a 12 R 1 R 3 -|- ^a u 2 K?\ ldx } (13) 



K(a 12 + a,,6 2 ) = 6 I 1^1 + &V) J a 12 R 2 + %, 2 K, + a 23 R 3 



+a 11 a 12 R 1 R 2 +a 12 2 R 2 2 ) lrf# 3 . . (14) 



Ka M ^=^rP e (l + P^)»(a il R 1+ la 11 «R 1 »'l^ . . (15) 



Jo I 25 J 



1 fl + Iv)! (^o+fluRi+auRg) =0, (16) 



| 7j^j^ 2 ^ 2 \|( a 20 + « 22 R 2 +a 2 3R3+ J«10 2 + 2«lO«llRl+ !/<10«12^2 



+ jau 2 Ri 3 + ^anai 2 RiR 2 + ? a^R/^O. . . (17) 

 Phil Mag. S. 6. Vol. 28. No. 167. Nov. 1914. 2 Y 



