344346] Collisions with Law of Inverse Fifth Power 289 



With this definition, the probability that a molecule of mass ra x and of 

 class A undergoes collision with a second molecule of mass m 2 and of class B 

 in time dt, the relative position of the molecules being such that p, e lie 

 within certain limits dp, de, is seen to be equal to the probability of finding a 

 molecule of mass w 2 and class B in a certain small cylinder of height Vdt and 

 of laa.se pdpde. The number of molecules of mass ra 2 and of class B per unit 

 volume may be taken to be 



1/2 f. 2 (u r , v', iu') du dv' dw' 

 so that the probability in question is 

 i> 2 / 2 (', v', w') du'dv'dw' Vdtpdpde (684). 



Let [Q] denote the change in the Q of a molecule of mass m l and of class 

 A caused by collision with a molecule of mass m 2 and of class B. Then the 

 average change in Q for molecules of mass m l and of class A caused in time 

 dt by collisions with molecules of mass m^ and of class B, is obtained by 

 multiplying the value of [Q] obtained from equations of the type of (683) 

 by expression (684) and integrating over all values of p and e. 



346. We can conveniently divide AQ into two parts, so that 



the former being caused by collisions with molecules of mass m l , and the 

 latter with molecules of mass ra 2 . 



Then in accordance with the definition of AQ already given (ex- 

 pression (647)), 



Q] ^1^2/1 (u> v, w)f z (u', v,' w')dudv*kvdu'dv'dw'Vpdpde 



(685). 



Since, however, (equation (675)), 



(m l + ra 2 ) K 

 we have 



Vpdp = J (m^ + m s) K add, 

 so that 



A 12 Q - V( Wl + m 2 ) K [Q] Wi (K, v, w)f 2 tf, w') 



JJJJJJJJ 



du dv dw du'dv'dw' ctda.de. 



If we write j (f> \ for the mean value of any quantity < averaged over all 

 molecules of type 1 and also over all molecules of type 2, so that 



80 +00 



|0|= 1 1 1 1 1 1 0/i (u, v, w)f z (u, v', w) dudvdw du'dv'dw', 



OO 00 



j. 19 



