588 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII 



There is a similar solution for the case of an elliptic cylinder rotating 

 about its axis*. The result, which may be easily verified, is 



2np (a + b) 2 &quot; 



316. The problem of relative equilibrium, of which Maclaurin s 

 and Jacobi s ellipsoids are particular cases, has in recent times 

 engaged the attention of many able writers, to whose investi 

 gations we can here only refer. These are devoted either 

 to the determination, in detail, of special forms, such as the 

 annulus*^, and that of two detached masses at a greater or less 

 distance apart J, or, as in the case of Poincare s celebrated paper , 

 to the more general study of the problem, and in particular to the 

 inquiry, what forms of relative equilibrium, if any, can be obtained 

 by infinitesimal modification of known forms such as those of 

 Maclaurin and Jacobi. 



The leading idea of Poincare s research may be stated as 

 follows. With a given mass of liquid, and a given angular 

 velocity n of rotation, there may be one or more forms of relative 

 equilibrium, determined by the property that the value of VT 

 is stationary, the symbols V, T having the same meanings as in 

 Art. 195. By varying n we get one or more linear series of 

 equilibrium forms. Now consider the coefficients of stability of 

 the system (Art. 196). These may, for the present purpose, be 

 chosen in an infinite number of ways, the only essential being 

 that V TO should reduce to a sum of squares ; but, whatever 

 mode of reduction be adopted, the number of positive as well as of 

 negative coefficients is, by a theorem due to Sylvester, invariable. 

 Poincare proves that if, as we follow any linear series, one of the 

 coefficients of stability changes sign, the form in question is as it 



* Matthiessen, &quot;Neue Untersuchungen iiber frei rotirende Fliissigkeiten,&quot; 

 Schriften der Univ. zu Kiel, t. vi. (1859). This paper contains a very complete list 

 of previous writings on the subject. 



t First treated by Laplace, &quot; Memoire sur la theorie de 1 anneau de Saturne,&quot; 

 Mem. de VAcad. des Sciences, 1787 [1789]; Mecanique Celeste, Livre 3 mc , c. vi. 

 For later investigations, with or without a central attracting body, see Matthiessen, 

 1. c. ; Mine. Sophie Kowalewsky, Astron. Nachrichten, t. cxi. , p. 37 (1885) ; Poincare, 

 I. c. infra , Basset, Amer. Journ. Math., t. xi. (1888); Dyson, I. c. ante p. 166. 



t Darwin, &quot;On Figures of Equilibrium of Rotating Masses of Fluid,&quot; Phil. 

 Trans., 1887; a full account of this paper is given by Basset, Hydrodynamics, 

 c. xvi. 



&quot;Sur 1 equilibre d une masse fluide animde d un mouvement de rotation,&quot; 

 Acta Math., t. vii. (1885). 



