THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 37 



the equation becomes 



f 3 ,\ / 2 ,\ fA fr> o o 



O C0 \ / O C/(7) \ I . / \ \ Zl V^q 



-r^ __ I I rr* ._. I *^. ( A ) [ I I w^ 



But from equation (28) 



so that on subtraction we find as the equation to be satisfied by <j>, 



The equation is too complicated to be attacked directly, but can be effectively 

 broken up by assuming a solution 



</, = w + fc, 



in which u and v are general functions of x, y, z, and X, while / is given by 

 equation (27). On substituting this value for </>, equation (30) reduces, after 

 considerable simplification, to 



4 Ci '- 



ox 



and this will be satisfied if we satisfy separately the equations 



v = 0, ...... (31) 



*+/^ =0. . . (32) 

 c 9aj/ 



On substituting for /and ^, and writing ABC = A 2 , equation (31) becomes 



A" Jo I " AT Sx % ' \A B (3^] A 



in which 



-[*o /! l-jl^-Ti ~(1 



Jo \A B O/A = Jo axU 



