Kinetic Theory of Gases. 607 



the whole number of systems which are included in the 

 average consist o£ pairs of this kind, it is clear that on the 

 average dH/dt is zero. We are, in fact, integrating dH/dt 

 through the region o£ our generalized space which is given 

 by a specified/; this region can be divided into two regions 

 such that one is the image of the other, and the values 

 of dH/dt in the two regions are equal in magnitude but 

 opposite in sign. The explanation of the apparent paradox 

 will be found in a later section (§ 32). 



The Partition of the Generalized Space. 



§ 18. We now discuss a number of questions concerning 

 the distribution o£ systems of various specified kinds in our 

 generalized space. 



§ 19. The first problem which we shall discuss is concerned 

 with the distribution of density of gas inside the containing 

 vessel. 



Let us consider a class of system which we shall call 

 class A. To define it, we imagine the vessel which contains 

 the gas divided into a great number n of " cells," each of 

 equal volume co, so that nco = fl. These cells are to be denoted 

 by the numbers 1, 2, 3, ... n. Then a system of class A is 

 defined as one such that a x molecules have their centres 

 in cell 1, a 2 in cell 2, and in general a s in cell s, so that of 



course ^a s = 'N. 



Denoting individual molecules, as before, by A, B, C . . ., 

 we see that there will be a certain number of systems repre- 

 sented in our generalized space, such that the molecule A lies 

 in one specified cell, the molecule B in another, and so on 

 for all. It will be easily seen that all systems of this kind 

 occupy a small continuous element, which we may call the 

 element c, and which, since there are N molecules altogether,, 

 forms a fraction (&>/H) N or ?i~ N of the whole. This, it must 

 be noticed, is true whether we consider all systems, or only 

 systems in which the molecules have assigned velocities. 

 Obviously also it is true, if we consider systems in which the 

 kinetic energy has a certain assigned value. 



Now if the molecules are distributed in the various cells in 

 such a way that there are a x in cell 1, a 2 in cell 2, and so on, 

 then all the systems represented by points inside the element 

 c belong to class A. There will be a number of other elements 

 containing only systems of class A; and these will correspond 

 one to each way in wdiich the N molecules can be distributed 

 into the n cells so that there shall be a, in the first, a 9 in the 



