FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
31 
Since two independent sets of computations, conducted by entirely different methods, 
have been found to lead to precisely the same result, it seems highly probable that 
this result is accurate. The agreement just mentioned may reasonably be regarded as 
guaranteeing the accuracy of all the second-order computations, both of Darwin and 
myself. 
The actual criterion of stability, however, depends on the value assigned to f, and 
since this depends in turn on the third-order terms, no check by comparison with 
Darwin’s work is possible. Some support is given to my value of £ by comparison 
with a parallel investigation of the :t Equilibrium of Rotating Cylinders,” which I 
published some years ago. Adjusting the parameters so that e shall have, as closely 
as possible, the same physical interpretation in the two problems, I find for the factor 
expressing the increase of of as we pass along the series of pear-shaped figures : 
I + 0'05227e 2 for the three-dimensional problem, 
IT 0'0513e 2 for the two-dimensional problem. 
Apart from this, the checks I have used in the present paper are such that I believe 
it would have been very difficult for any error to escape detection. 
25. The main object of the paper is achieved as soon as the pear-shaped figure is 
shown to be unstable. It is at the same time of interest to examine the bearings of 
this result on the wider question of which it is a part. 
In fig. 1 let PP' represent the series of Jacobian ellipsoids, the part PO (drawn 
thick) representing the stable part of the series, and the part OP' (drawn dotted) 
representing the unstable part, so that 0 is the point of bifurcation. Let a diagram 
be drawn about this line having the angular momentum always represented by the 
