72 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



Let us put 



P = 



and similarly for q, r, and s, and let us put e 2 n" = ; then the equation may be put in 

 the form 



2n.iV 2^ 2g, )2 / 2 2r n z 2 



"IMF' ~^r v ~ 



For any values whatever of ^ and f, provided only that they are sufficiently small, 

 equation (id) will give a figure of equilibrium. If we put e = 0, but retain the 

 equation becomes 



which is an ellipsoid of semi-axes ,', //, c', given by 



or, numerically, 



f ^ = 1 + 12'71347 TV = 1-9-20894& ^ = 1-3'50453 

 fi~ l> v 



It is at once clear that as f varies this ellipsoid coincides with the various Jacobian 

 ellipsoids near to the standard ellipsoid. 



If we put f = but retain e in equation (1G1) we obtain equation (160) with 

 n" = ; i.e. we obtain a series of figures of equilibrium all having the same angular 

 velocity as the standard Jacobian ellipsoid from which they are derived. 



The two series of' figures obtained by putting e = and f = in equation (161) 

 may be represented by the two intersecting straight lines POP', QOQ' in fig. 1, the 

 point of bifurcation being of course represented by the point O. The general figure oi 

 equilibrium represented in equation (161) is, however, arrived at by assigning values 

 to both e and f, these values being limited by the condition only that e and f shall be 

 so small that e 3 and f "'- shall be negligible. Thus the figures of equilibrium given by 

 equation (161 ) are represented by all the points inside a certain rectangle ABCD 

 surrounding the point O. They do not fall into two linear series, as it seems to be 

 tacitly assumed by DARWIN and POINCARE that they will do. 



36. The circumstance that the two linear series lose their identity and become merged 

 indistinguishably into a general area seems to be predicted as a direct consequence of 



