THE FIGUKES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 63 



whence we obtain as the contributions to P 1( P 2 , P 3 , P 4 (cf. equations (128) to (131), 



P! = P 2 = P 3 = 0, 



J ' .' I ~J > ~J \ 



It now appears that p', q', ?', s' contribute nothing to the values of c n , c l2 , ... , c 23 

 (cf. equations (I34)'to (139), p. 60.) Their contributions to 4d,, 4cZ 2 , and 4c 3 are as 



follows : 



, , f dX i2p' ^ P 4 

 4fti = I -^ 4- - 



Jo A VA 3 A 



= Jo A" \ A 7 ~ a 2 P ~ 6 2 AB ~ c 2 AC/ 

 so that the contributions are 



4c?j = 2p' JAA 2 IA.\ ^!AB -T!AC, 

 ' T P' T ^ T *'' T 



2 AB 6 2 * (? 



Since this part of the potential should be harmonic, we ought to have di + d 2 + d 3 = 

 (cf. equation (ill)). This is clearly the case, in virtue of the identity 



2 2 J AA = IAA + IAB + IAC- 



I have verified that these identities are satisfied by the values in the table opposite, 

 and the contributions are found to be 



= -0141999 ) +0'5 



Ct 



id s = -0161187 ^j-0'39 



a 



Contributions from Terms in L', M', N 1; I', m', n'. 

 30. As regards these terms, we may take (cf. equations (125) and (126)), 



it_ SM; r 



a 2 6 2 c 2 



\a o c / 

 K 4 = 0, 



L/ // TT T / 7/ 



y I , Jt-l 3V J_V 



a 4 6 2 c 2 ^ a 2 a 4 6V 



