THE DYNAMICAL THEORY OF GASES. 43 



where Q is some function of 17, , &c., already determined, and /, is the 

 function which indicates the distribution of velocity among the molecules of the 

 second kind. 



In the case in which n = 5, V disappears, and we may write the result 

 of integration QN 3 , 



where Q is the mean value of Q for all the molecules of the second kind, 

 and jVj is the number of those molecules. 



If, however, n is not equal to 5, so that V does not disappear, we should 

 require to know the form of the function f t before we could proceed further 

 with the integration. 



The only case in which I have determined the form of this function is 

 that of one or more kinds of molecules which have by their continual encoun- 

 ters brought about a distribution of velocity such that the number of molecules 

 whose velocity lies within given limits remains constant. In the Philosophical 

 Magazine for January 1860, I have given an investigation of this case, founded 

 on the assumption that the probability of a molecule having a velocity resolved 

 parallel to x lying between given limits is not in any way affected by the 

 knowledge that the molecule has a given velocity resolved parallel to y. As 

 this assumption may appear precarious, I shall now determine the form of the 

 function in a different manner. 



On the Final Distribution of Velocity among the Molecules of Two Systems acting 

 on one another according to any Law of Force. 



From a given point O let lines be drawn representing in 

 direction and magnitude the velocities of every molecule of 

 either kind in unit of volume. The extremities of these lines 

 will be distributed over space in such a way that if an ele- 

 ment of volume dV be taken anywhere, the number of such 

 lines which will terminate within dV will be f(r)dV, where 

 r is the distance of dV from O. 



Let OA = a be the velocity of a molecule of the first kind, and OB = b 

 that of a molecule of the second kind before they encounter one another, then 



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