86 The Law of Distribution [OH. v 



When the spheres of two or more molecules intersect, an "encounter" is 

 said to take place, lasting until the spheres again become clear of one another. 

 The individual molecules are now to be the systems under discussion, and 

 the " encounters " are to play the part of the fortuitous agencies which 

 disturb their motion. Each molecule is to have n degrees of freedom, in 

 addition to the three degrees of freedom represented by the motion of its 

 centre of gravity in space, and the possible states of a single molecule are 

 to be represented in a space of Zn + 3 dimensions, of which 2n represent the 

 internal coordinates and momenta of the molecule, and the remaining three 

 represent the velocities of the centre of gravity. 



Binary Encounters. 



97. We shall begin by considering binary encounters only. That is to 

 say, we work on the hypothesis that the event of a sphere being intersected 

 by two other spheres simultaneously is so rare that it may be neglected. In 

 this case the spheres become identical with the "spheres of molecular action" 

 of 82. 



We treat this case as follows. As soon as an encounter begins between 

 two molecules their existence as single molecules is supposed to be abruptly 

 terminated, and their representative points are removed from our generalised 

 space of 2w + 3 dimensions. During the progress of the encounter the two 

 molecules together will be supposed to form a new dynamical system a 

 double molecule. This system will be specified by 4n + 9 independent co- 

 ordinates, 2w for the internal coordinates of each constituent molecule, six 

 for the velocity and position of the centre of gravity of one molecule rela- 

 tively to that of the other, and three for the velocity of the centre of gravity 

 of the whole system in space. Hence any such system can be represented 

 by a point in a space of 4w + 9 dimensions. We shall not, however, require 

 the whole of this 4w + 9 dimensional space. If x, y, z t x , y , zf are the co- 

 ordinates of the centres of the two molecules, the condition that an encounter 

 is beginning or ending is 



( gs -. a fy + (y-y f y + ( z -^= tr (204). 



In the 4n + 9 dimensional space this equation will be the equation of a 

 certain " surface " 8 (of dimensions 4>n + 8), and the representative points of 

 all double molecules will be inside 8. We shall find it convenient to denote 

 each double molecule by two representative points, since the roles of first and 

 second molecule can be allotted in two different ways. 



Let p 2 be the density of representative points in any small element of 

 volume in this new space, and p l the density in the original space of 2n + 3 



