FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
29 
It follows that the moment of momentum M is given by 
M = M„ (l — 0'067G5e 2 ). 
Thus M < M 0 , so that the moment of momentum decreases as we pass along the 
series of pear-shaped figures, and this series is therefore unstable. 
Summary and Discussion of Results. 
23. Throughout the present paper, and my previous paper on the same subject, the 
critical Jacobian ellipsoid which bifurcates into the pear-shaped series of figures of 
equilibrium, has been taken to be 
2 *> 
a; . ir 
/= V £4^-1 = 0. 
a /) c 
Any adjacent figure, whether of equilibrium or not, may be supposed to be 
f+<P = o 
where 0 is a function of x, y, z in which the coefficients are numerically small. For 
special values of </>, this figure will be one of equilibrium. So long as we consider only 
figures which differ infinitesimally from f — 0, all the possible figures of equilibrium 
form a linear series, and <p is of the form 
<p = e P,.(108) 
where P is a function of x, y, z and e is a parameter which must be so small that e 2 
can be neglected. 
In the previous paper it was shown that as soon as e 2 is taken into account, there 
must be supposed to be a doubly infinite series of figures of equilibrium. The general 
form of <p is 
<p — eV + e z Q + ^Q 1 .(109) 
where £ is a second parameter of the same order of magnitude as e 2 , but capable of 
2 
varying quite independently of e 2 . The value of —— for this figure of equilibrium is 
2-np 
greater by £ than the value for the critical Jacobian. The possible figures of 
equilibrium may be thought of as lying inside a rectangle having e, f as rectangular 
co-ordinates. 
In the present paper I have carried the investigation as far as e'\ and find that the 
value of (p as far as third-order terms must be of the form 
<p = cP + e 2 Q + ^ + e 3 (R + KP).(110) 
where R is a new function of x, y. z and K is a constant. At first sight K appears to 
be at our disposal, for if we replace the parameter e by a new parameter e+0e 3 , we can 
