necessary for Equipartition of Energy. 591 



begins between two molecules their existence as single mole- 

 cules is supposed to be abruptly terminated, and their repre- 

 sentative points are removed from our generalized space o£ 

 2/i 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 4;/ independent coordinates, 2n for each constituent mole- 

 cule. Hence any such system can be represented by a point 

 in a space of 4?i dimensions, one dimension corresponding to 

 each coordinate. We shall not, however, require the whole 

 of this 4rt-dimensional space. If x, y, z, x' , y\ z' are the co- 

 ordinates of the centres of the two molecules, the condition 

 that an encounter is beginning or ending is 



{x-x'f + {j/-y'f + {z-zj = m\ . . (15) 



In the 4/z-dimensional space this equation will be the equation 

 of a certain ' ; surface" S (of dimensions 4?i — 1), and the re- 

 presentative points of all double molecules will be inside S. 

 We shall find it convenient to denote each double molecule 

 by two representative points, since the roles of first and second 

 molecule can be alloted in two different ways. 



Let a be the density in this new space, then the necessary 

 and sufficient conditions for a steady state are 



J=°' (16) 



g=°< CIT) 



in the latter of which the change in p includes that caused 

 bv the formation and dissolution of double molecules. 



§ 10. Before determining the relation between p and a we 

 must make Boltzmann's assumption that the gas is in a 

 * ; molekular-ungeordnet " state. Having made this assump- 

 tion we proceed to calculate the number of encounters of a 

 given kind which occur in an interval dt. Equating this to 

 the number of representative points which cross the corre- 

 sponding element of the surface S during the same interval 

 we arrive at the equation 



°=PP', (18) 



in which <x is the density at any point on S, and p, p the 

 densities at the two points of 2«-dimensional space which are 

 determined by the coordinates of the two encountering 

 molecules. 



The analysis of § 4 applies (with obvious modifications) to 



