188 Non-Spherical Masses—Dynamical Principles [ch. vii 
motion is so slow that their kinetic energy is negligible. For such configura 
tions we may put T— 0, so that equations (169*5) reduce to 
d 0 1 ’ 
dW 
d 0 2 
dW 
= 0, ^- = 0, etc (169*6). 
do. 
These may be regarded either as equations of equilibrium or as equations 
determining the configurations of a very slowly changing mass. Regarded as 
equations in 0 lt 0 2 , 0 3 , ..., there will be a number of solutions of which a 
typical one may be taken to be 
0! = ©j, 0 2 = © 2 , etc (169*7). 
In this solution the quantities © 1} © 2 ,... depend on the constants which 
specify the potential energy W in terms of 0 lf 0 2 , ..., as given by equation 
(169*3). In problems of cosmogony in which changes of a secular or evolu 
tionary nature occur, these constants must themselves be supposed to vary; 
they are better spoken of as parameters than as constants. When equations 
such as (169*6) are satisfied, an astronomical mass is momentarily in a 
position of equilibrium, but if the physical conditions change in the course 
of time, this particular configuration of equilibrium will give place to another. 
We may represent this process by supposing slow changes to occur in the 
parameters which enter into the specification of W by equation (169*3). 
Linear Series. 
170. Let us consider in detail the changes produced in ©j, © 2 , ..., the 
co-ordinates of a configuration of equilibrium, as one of the variable para 
meters, say jx, is allowed slowly to vary. 
A slight change from /x to fx + dfi in the value of fx will alter the values 
of ©j, © 2 , ... by quantities which will in general be small quantities of the 
same order of magnitude as d/x. On making such a small change in a 
configuration of equilibrium such as that given by equations (169*7) gives 
place to an adjacent configuration of equilibrium. On continually varying /x 
we pass through a whole series of continuous configurations of equilibrium, 
which form what Poincare has called a “linear series*.” 
Suppose for the moment that our dynamical system is specified by only 
two co-ordinates 0 1} 0 2 , as, for instance, the co-ordinates of a particle moving 
on a curved surface. We may in imagination construct a three-dimensional 
space having 
0 i, 0 z, 
as co-ordinates. Any one plane ¡x = cons, will be suitable for the representation 
of all the configurations which are possible for one value of /a, and therefore 
for all which are possible for one definite physical state of the system. The 
* Poincaré, Acta Math. vii. (1885), p. 259, or Figures d'équilibre d'une masse fluide. (Paris, 
1902.) See also Lamb, Hydrodynamics, p. 680. 
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