20 General Dynamical Principles [CH. u 



In this solution the quantities 1? 2 , ... will be functions of the constants 

 which enter into the function W as given by equation (3). But in problems 

 of cosmogony in which changes of a secular or evolutionary nature occur, 

 these constants must themselves be supposed to vary ; they are better spoken 

 of as parameters than as constants. WJien equations such as (5) are satisfied, 

 an astronomical mass has assumed a position of equilibrium for the moment, 

 but with the course of time the physical conditions will change, and the con- 

 figuration of equilibrium will give place to another. Analytically this process 

 is represented by slow changes in the parameters which occur in the speci- 

 fication of W by equation (3). 



STATICAL SYSTEMS 

 Linear Series 



18. Let us consider in detail the changes produced in (3 l5 <s) 2 , , the 

 coordinates of a configuration of equilibrium, as one of the variable parameters, 

 say p, is allowed slowly to vary. 



A slight change in the value of /u,, say from //, to yu, + dp, will alter the 

 values of 1} 2 , ... by quantities which will in general be small quantities 

 of the same order of magnitude as dp. Thus on making this small change 

 in //,, a configuration of equilibrium such as that given by equations (7) 

 gives place to an adjacent configuration of equilibrium. On continually 

 varying p we pass through a whole series of continuous configurations of 

 equilibrium, and these form what Poincare has called a "linear series*." 



We may in imagination construct a generalised space having 



as coordinates. Any one plane //, = cons, will be suitable for the representation 

 of all the configurations which are possible for one value of /-t, and therefore 

 for all which are possible for one definite physical state of the system. The 

 particular points in this plane determined by equations such as (7) will 

 represent the configurations of equilibrium in this physical state. 



The function W must, from its meaning, be a single valued function of 

 0,, 2 , and /Lt, so that the surfaces W = cons, in the (n + l)-dimensional 

 space are necessarily non-intersecting surfaces. The condition that a con- 

 figuration shall be one of equilibrium, as expressed by equations (6), is exactly 

 identical with the condition that the tangent to the surface W = cons, shall 

 be perpendicular to the axis of p. Thus if for convenience we think of the 

 axis of /* as being vertical, the configurations of equilibrium are represented 

 by points at which the tangents to the surfaces W = cons, are horizontal ; 



* Poincare, Ada Math. 1 (1885), p. 259, or Figures d'Equttibre d'une Masse fiuide. (Paris, 

 1902.) See also Lamb, Hydrodynamics, p. 680. 



