AND TIIK M.r.i I..M; 



ir, , 



Near KO, = 0, ?< = r-^ ; the value of w, is cot *a, and successive u's satisfy 



itflt ~|~ \. ^ ' 



the difference-equation 



.T- ' ' W 



K<t 



With the help of these properties it is easy to draw approximate graphs of the 

 curves y = u n . Such a graph, for values of KO, up to the first zero of , ( = 4'49) is 

 represented in fig. 1, in which the vertical scale is 2^ times the horizontal scale. 



Fig. 1. Graphs of the functions u,,. 



In terms of these functions, the points of bifurcation are given by = 

 once evident that there is no root of this equation for values of*, less than , 

 therefore (cf. equation (19)) no point of bifurcation at all so long as , is nrtric 

 being always positive. It follows that the spherical configura 

 displacements. 



Small Rotation. 



12 When the fluid experiences a slight rotation w, the spherical configuration is 

 of course slightly flattened. The appropriate solution of equal 



-M.(r)P,. ( 29 ) 



x = 



VOL. CCXI1I. A. 



3 o 



