THE FIGUEES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 41 



Clearly since u a is of degree 4 in x, y, z, iv will be of degree 2, so that it/ will be 

 a function of X only. The term 2-r- ^ in equation (43) is therefore zero in the 

 present instance, but is inserted to maintain symmetry. We now have 



3 x 3 



- . 



A oa;/ \3\ A da? 



so that after simplification, 



Since / vanishes for the appropriate value of X, ^ will vanish for both X and 



fw' 

 provided w' is made to vanish when X = 0. Thus ' - will satisfy the condition to be 



satisfied by x in equation (41), and a solution of this equation will be 



r a = w+fw' (44) 



Since V 2 must vanish when X = ( 11) it appears that both / and w' must vanish 

 separately when X = 0. On transforming (4l) and (42) to tj, co-ordinates 

 (cf. equation (36)) and integrating, we obtain as the values of w and w' which vanish 

 when X = 0, 



rt px 



w = -i V 2 ^u,,d\; w' = -l\ V a tvS wd\ ...... (45) 



Jo Jo 



14. Let us introduce a differential operator D, defined by 



D ~ 8 + 3 ~" l " a ~ ' ..... (46) 



noticing that, as a function of X, D is purely a multiplier. We have 



3D i a 2 



~~ '~ 



and when X = 0, D = 0. 

 VOL. ccxv. A. 



