THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 



73 



POINCARE'S general analysis, coupled with the linearity of the equations (V 3 W = 

 &c.) which lead to figures of equilibrium. For the vanishing of the Hessian (A = in 

 the notation of POINCARE *) which expresses the condition that a point of bifurcation 

 should exist, expresses also the condition that the two linear series should be merged 

 with an area of linear series as they approach the point of bifurcation, t 



There is, of course, no question that in the neighbourhood of the point O two linear 

 series do actually exist, such as may be represented by the lines POP', ROE,' in 

 fig. 1 ; this is abundantly proved by POINCABE'S general argument. \ What is now 

 maintained is that an expansion as far as e 2 only, does not suffice to reveal the 



Jacobian Ellipsoids 

 (unstable) 



Jacobian Ellipsoids 

 (stable) 



P 



Fig. 1. 



directions in which these linear series start out from the point of bifurcation. So 

 long as our vision is limited to the interior of the rectangle ABCD in fig. 1, we can 

 know nothing of the direction in which the line OR starts out from 0. And the 

 whole difficulty is merely one introduced by the artificial method of expansion in 

 powers of a parameter ; as soon as this artificial method is abandoned the rectangle 

 ABCD shrinks to an infinitesimal size, and the curves POP' and ROR' become merely 

 two lines intersecting in the point O without any complications. An exactly 



* " Sur Pequilibre d'une masse fluide anime'e d'un mouvement de rotation," ' Acta Math.,' VII., p. 259. 

 t Cf. footnote to p. 74. 

 I Loc. tit., 2. 



VOL. CCXV. A. L 



