ARISING FROM INEQUALITIES OF TEMPERATURE. 691 



We shall suppose, however, with Boltzmann, that in a medium in which there 

 are inequalities of temperature and of velocity 



where F is a rational function of 77, , which we shall suppose not to 

 contain terms of more than three dimensions, and / is the same function as 

 in equation (9). 



Now consider two groups of molecules, each defined by the velocity- 

 components, and let the two groups be distinguished by the suffixes (,) and 

 (,). We have to estimate the number of encounters of a given kind between 

 these two groups in a unit of volume in the time St, those encounters only 

 being considered for which the limits of b and < are b^db and (j>^d<j). 



Let us first suppose that both groups consist of mere geometrical points 

 which do not interfere with each other's motion. The group d^ is moving 

 through the group dN t with the relative velocity V, and we have to find 

 how many molecules of the first group approach a molecule of the second group 

 in a manner which would, if the molecules acted on each other, produce an 

 encounter of the given kind. This will be the case for every molecule of the 

 first group which passes through the area bdbd<j> in the time St. The number 

 of such molecules is d^Vbdbd^St for every molecule of the second group, 

 BO that the whole number of pairs which pass each other within the given 

 limits is 



and if we take the time St small enough, this will be the number of encounters 

 of the real molecules in the time St. 



(3) Effect of the Encounters. 



We have next to estimate the effect of these encounters on the average 

 values of different functions of the velocity-components. The effect of an indi- 

 vidual encounter on these functions for the pair of molecules concerned is 

 given in equations (3), (4), (5), (6), (7), each of which is of the form 



SP = Q sin*0 cos 5 6 (13) 



where P and Q are functions of the velocity-components of the two molecules, 



872 



