146 THE TIDAL PROBLEM. 



6;2 



For £ = and ^ ,; < 0.22467 . . . certain solutions are represented by 



points on the curves of figure 17. Suppose for co = coq the solution a;i = x/'>^ 

 is not a multiple solution. The F^ depend upon the gravitational potential 

 and the rotational energy, and are continuous functions of e. Consequently 

 the roots of (3) vary continuously with s, and we may represent their solu- 

 tion by a point in figure 17 in the plane £ = £, where as before, each linear 



series with respect to a parameter, as ^ , ; or e, carries with it a set of 



relations which completely defines the shape of the figure of equilibrium. 

 If w = a»o and e = belong to a multiple solution of (3) there is an o) near 

 (jJq such that for £ = e equations (3) have also a multiple solution. By this 

 process the curves of figure 17 become surfaces to ^every point of which 

 belongs a figure of equiUbrium. If in this figure e is set equal to a small 

 constant a new set of curves will be obtained in a general way similar to 

 the old, and possessing maxima and points of bifurcation. In general the 

 greater e the greater will be their deviation from the forms of the curves 

 in figure 17. 



Now consider the question of stabihty. The necessary and sufiicient 

 condition for complete and secular stability of the figure of equilibrium 

 is that the total energy shall be a minimum for all variations preserving 

 constant moment of momentum.* All the quantities involved in these 

 conditions are continuous functions of e. Consequently, starting from an 

 ordinary point in figure 17 whose corresponding figure of equilibrium has 

 any properties of stability, it is found that the figure obtained by varying £ 

 through a sufficiently small range will have the same properties of stability. 

 Consider the curves obtained by giving s. a constant value. At certain places 

 the figures of equilibrium will change the character of their stability ; but as 

 in the case of £ = 0, treated by Poincar^, wherever the stability changes a 

 new series of figures branches out. Since the curves for s = e are in the 

 analytic sense the continuation of those for e = 0, the figures of equilibrium 

 for £ = £ go through a series of changes of stability entirely analogous to the 

 changes in the figures for £ = 0. Of course, it is possible that two curves, 

 Co' and Co", might cross a curve C^ at a single point for £ = 0, and that the 

 corresponding curves, G/ and CJ', might cross the curve C^ in two distinct 

 points, or the opposite. For example, there might be such a definition of cr, 

 that, for a certain value of £, the point corresponding to / of figure 17 would 

 fall on the point corresponding to 6. However, in the present connection 

 such exceptional cases are trivial. The point of interest is that for £ = £>0 

 there is a line of stable figures of equilibrium corresponding to those for 

 which this parameter is zero. 



In general, for £ > 0, the point of bifurcation corresponding to h, figure 

 17, will not appear for the same value of lo as that belonging to 6. We may 

 represent these two values of o) by a>« and Wq respectively. The question 

 of interest in the present connection is which is the greater; or, in other 

 terms, whether with increasing rotation instability occurs first in the het- 

 erogeneous or in the homogeneous body. We shall not attempt a positive 



1 Thomson and Tait's Natural Phil., Part II, 778, (j) and (fc). 



