THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 57 



With the help of (ill), equations (99) to (101) may be replaced by 



n" = -4-6 (- + + r] (U2 \ 



n 4t7 L.* + 7.4 + ^ > v 11 ^ 



a 



c 



while from equations (102) to (110) we obtain 



3L m n 



+ - + -=0 

 3_N l_ in = 



These three equations with equations (102) to (104), namely, 



^n = iff, (118) 



< = i^, (119) 



C33 = i^, (120) 



may be used to replace the group (102) to (107). 



27. We proceed to evaluate these quantities in detail. The values of u and i\ from 

 equation (47) are 



(122) 



and the value of u 2 is already given by equation (95). 



For convenience in computation we shall combine all the terms in ? 2 which are 

 independent of X in the first two lines, with the similar terms in the last line ; we 

 accordingly write 



4 3 , 2 . 2 



a' * - ' ' ' ' b" " ' c 2 7 * 



^ 2 +2(pf + q n 2 + r^)+s 



+s f 



= '(6*f) ..... ............... (123) 



VOL. CCXV. A. I 



