Kinetic Theory of Gases. 609 



infinite quantity of the order of N, and is for this reason diffi- 

 cult to manipulate. We accordingly introduce a new quantity 

 K a defined by 



K «= s m:-:> ™ 



so that K is, in general, finite ; and we have 



6> a =e-NK a (14) 



§ 21. But now the different classes of systems may be 

 represented in a new generalized space — the subsidiary space, 

 as we may call it. This space is to have n — 1 dimensions, 

 corresponding to n variables £ x £ 2 . . . f n subject to 2f=!N\ 

 A system of class A is represented by the point 



?i = «i, £ 2 = «2> • • . Zs=a s (15) 



which we shall call the point A. 



The various classes may therefore be represented by the 

 intersections of n systems of parallel lines drawn at unit 

 distance apart. Further, we can represent the different 

 systems which occurred in our original generalized space in 

 the subsidiary space, by placing an imaginary mass 6 a at the 

 point A, 6b at B, and so on. 



We now pass to the limit and take N = *d . The values of 

 £i> ? 2 • • • ( or what is the same thing a ly a 2 . . .) will now be 

 infinitely great, and we accordingly take the unit of length 

 in our subsidiary space to be infinitely small. 



We can now replace the distribution at points by a con- 

 tinuous volume-distribution. The distribution a at point A 

 is now to be replaced by a density p spread through the 

 rectangular unit element of which A is the centre such that 

 the aggregate density in this unit element is 6 a . It is 

 obviously possible (equation 11) to take 



p=e- JS % 



as the general continuous distribution, where K may be chosen 

 so that it is continuous, and becomes identical with K a at A. 

 K b at B, &c. All the systems in our original space may in 

 this way be represented in our subsidiary space. 



The number of systems of any specified type in our original 

 generalized space (the total being taken as unity) is there- 

 fore equal to the number of the same type in the subsidiary 

 space, and this 



= s e„ =nr...p *<*&... 



A,B,... JJJ 



