44 



THE DYNAMICAL THEORY OF GASES. 



BA will be the velocity of A relative to B; and if we divide AB in G 

 invenelj aa the masses of the molecules, and join OG, OG will be the velocity 

 of the centre of gravity of the two molecules. 



Now let OA' = tf' and 0/T=*5' be the velocities of the two molecules after 

 the encounter, GA^GA' and GB = GB f , and A'GR is a straight line not 

 necessarily in the plane of OAB. Also AGA' = 20 is the angle through which 

 the relative velocity is turned in the encounter in question. The relative motion 

 of the molecules is completely defined if we know BA the relative velocity 

 before the encounter, 26 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 encounters in which the magnitude and direction of BA, 

 .UK! also B and <^, lie within certain almost contiguous limits, we shall class 

 as encounters of the given kind. The number of such encounters in unit of 

 time will be 



njifde ....................................... (17), 



where n, and n, 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 dV while AB 

 and A'B move parallel to themselves, then B, A', and B' will also describe 

 equal and similar elements of volume. 



The number of molecules of the first kind, the lines representing the velo- 

 cities of which terminate in the element dV at A, will be 



i=/i(a)dr .................................... (18). 



The number of molecules of the second kind which have velocities corresponding 

 to OB will be 



(19); 



and the number of encounters of the given kind between these two sets of 

 molecules will be 



(20). 



The lines representing the velocities of these molecules after encounters of the 

 given kind will terminate within elements of volume at A' and R, each equal 

 to dV. 



