91-93] Stability 91 



When the coefficients have these values, all the conditions for equilibrium 

 are satisfied. The value of s may still be anything we please, but only one 

 value of s will keep the volume of the figure equal to the initial volume, and 

 this value is found to be 



c\ 2 - ~-___ ___ ' 



s= -01586814 -13-239893^ ........... ..T.(248). 



93. Equations (245) (248) may be written in the form 



p=p'+p" etc ............................ (249), 



2 



where f=~ - e* .............................. (250), 



and the value of Q given by equation (236) may similarly be expressed 

 in the form 



C. = Q,' + 50.". 



The equation of the boundary (equation (234)) now becomes 





This will be a figure of equilibrium whatever the values of e and 

 provided only that they are sufficiently small. If we put e = but retain 

 the equation becomes 



+ ^ =1 -^ - (252) ' 



and this is an ellipsoid whose semi-axes a', b', c' are given by 



9-20894?; 



1 + 3-50453?. 



Clearly then, as f varies with e = 0, the figure of equilibrium coincides 

 with the various Jacobian ellipsoids near to the point of bifurcation. 



On putting f =0 but retaining e in equation (251) we obtain a series of 

 figures of equilibrium for all of which the angular velocity is the same as 

 that at the point of bifurcation. 



The two series of configurations obtained by putting e and f = in 

 equation (251) may be represented by two intersecting straight lines such 

 as POP', QOQ' in fig. 16, the point being of course the point of bifur- 

 cation. But the general figure of equilibrium represented by equation (251), 

 in which e and f are limited only by the condition that e 3 and f* shall be 



