586 Mr. J. H. Jeans on the Conditions 



The General Dynamical Theorem. 



§ 2. Let us begin by considering the motion o£ a very 

 great number of exactly similar dynamical systems, the 

 systems being supposed for the present not to influence the 

 motion of one another. We shall ultimately take such a 

 system to be a molecule of a gas. 



Let us suppose the configuration of this system determined 

 by n coordinates 



qn 92, • • • • g n > (!) 



and let the corresponding momentoids be 



p x , p 2 , . . . . p n (2) 



Now imagine a space of 2n dimensions, these dimensions cor- 

 responding to all possible values of the 2n independent 

 variables 



9^ 92, • • • • 9niP^P2, • • • • Pn' ■ • • ( 3 ) 



Then the configuration and rate of change of configuration 

 of any system can be represented completely and uniquely in 

 this generalized space by a single representative point *. 



Instead of saying that a system is in the phase (q^ q 2 , . . . 

 p v p 2 , . . .) , we shall say that it is at the point (g 1? q 2 , ■ . . 

 Pi, p 2 , . . .), of our generalized space. Let us suppose that 

 the number of systems of which the coordinates of the re- 

 presentative points lie between 



ft, q 2 . . . . p u P2 • • • ■ 



and ^ q 1 + dq 1 , q 2 + dq 2 .... Pi + dp^ p 2 + dp 2 .... 



1S fiqu q%, • • • pi, Pi • • d 9i, d q* • • • d P^ d P* ■ ■ • 



This is the number of representative points which occupy 

 the element of volume dq x dq 2 . . . dp x dp 2 ... of our 

 generalized space. We may, therefore, speak of / [q x q 2 . . . 

 P\p2 - - •) as the density at the point q x q 2 . . ,, and shall, for 

 the sake of convenience, denote it by p. 



§ 3. If we are given the values of the 2n coordinates of 

 scheme (3) at any instant we shall be able, from a knowledge 

 of the energy-function of the system, to calculate the values 



* It will tend to clearness of thought to imagine infinite space, so 

 that all the coordinates can range from +oo to — oo . If the coordinates 

 are not uniquely defined from the configuration (e. g. if q x is an angle so 

 that for a given configuration q x may have any of the values <f> , 2~-\-<p , 

 47r-\-<p . . . &c.) we may either suppose just sufficient of the space taken 

 to give only one possible value of each coordinate inside the space, or we 

 may suppose one representative point for eveiy possible system of values 

 of the coordinates, so that the arrangement of. points in our generalized 

 space is periodic. 



