186 Mr. J. C. Maxwell on the Dynamical Theory of Gases. 



the molecules is completely defined if we know BA the relative 

 velocity before the encounter, 20 the angle through which BA 

 is turned during the encounter, and <£ the angle which defines 

 the direction of the plane in which BA and B'A' lie. All en- 

 counters in which the magnitude and direction of BA, and also 

 6 and <£, lie within certain almost contiguous limits, we shall 

 class as encounters of the given kind. The number of such en- 

 counters in unit of time will be 



V^de, (17) 



where n t and n 2 are the numbers of molecules of each kind under 

 consideration, and F is a function of the relative velocity and of 

 the angle 6, and de depends on the limits of variation within 

 which we class encounters as of the same kind. 



Now let A describe the boundary of an element of volume 

 dY while AB and A'B' move parallel to themselves, then B, A' 5 

 and B' will also describe equal and similar elements of volume. 



The number of molecules of the first kind, the lines repre- 

 senting the velocities of which terminate in the element dY at 

 A, will be 



n^f^dY (18) 



The number of molecules of the second kind which have veloci- 

 ties corresponding to OB will be 



» 2 =/ 2 (i)JV; (19) 



and the number of encounters of the given kind between these 

 two sets of molecules will be 



Ua)flb)<W*Vde (20) 



The lines representing the velocities of these molecules after en- 

 counters of the given kind will terminate within elements of 

 volume at A' and B', each equal to ^V. 



In like manner we should find for the number of encounters 

 between molecules whose original velocities corresponded to ele- 

 ments equal to dY described about A' and B', and whose subse- 

 quent velocities correspond to elements equal to dY described 

 about A and B, 



A(a%{V)d\^ck, (21) 



where F' is the same function of B'A' and A'GA that F is of BA 

 and AGA'. F is therefore equal to F'. 



When the number of pairs of molecules which change their 

 velocities from OA, OB to OA' OB' is equal to the number 

 which change from OA', OB' to OA, OB, then the final distri- 

 bution of velocity will be obtained, which will not be altered by 



