29-32] Rotating Systems 33 



Various forms for the Equations of Equilibrium 



31. The preceding theory has reduced the problem of determining a 

 sequence of stable configurations to the simpler problem of mapping out all 

 configurations of equilibrium. For this latter problem the conditions of equi- 

 librium may be expressed in whatever form is found to be most convenient. 



We have already seen that possible forms are 



=0 (w = constant) .................. (47), 



= (M = constant) .................. (48). 



Another form is contained in the ordinary hydrostatic equations of equi- 

 librium 



in which V is the gravitational potential and p, p denote the pressure and 

 density respectively. 



For a mass of uniform density p, equations (49) have the common 

 integral 



? = V + > 2 (x- + ?/ 2 ) + cons. 



and so the equations reduce to the single condition that 



F + i&> 2 (# 2 + 2/ 2 ) = cons ......................... (50) 



over the boundary of the fluid. 



32. In the classical treatment of the rotational problem by Poincare* 

 and Darwin f, the equations of equilibrium are introduced in the form (48) ; 

 while Liapounoffj treats the same problem by means of equation (50). 

 The method of treatment of the present book finds it convenient to use 

 equation (50) for the incompressible mass, and equation (49) for the com- 

 pressible mass, this latter case not being discussed at all by Poincare, Darwin 

 and Liapounoff. 



Thus, so far as the treatment of the problems in the present essay is 

 concerned, it was unnecessary to introduce equations of the type (48) for the 

 discussion of figures of equilibrium, but the theory of secular stability could 



* Acta Math. I.e. ante, also "Sur la Stabilite de 1'Equilibre des Figures Pyriformes affectees 

 par une Masse Fluide en Kotatioa," Phil. Trans. 198 A (1901), p. 333. 



f "On the Pear-shaped Figure of Equilibrium of a Rotating Mass of Liquid," Phil. Trans. 

 198 A (1901), p. 301, and subsequent papers. These will be found in Vol. in of Sir George 

 Darwin's Collected Works. 



J "Sur un Probleme de Tchebychef," Memoires de VAcademie de St Petersbourg, xvii. 3 (1905), 

 and other papers published by the Academy. 



j. c. 3 



