476 Mr. IS. H. Burbury on the 



velocity between v and v + dv has on an average, if we include 

 encounters of both kinds, some translation-velocity in direc- 

 tion #, which we have called a' '. This will have altogether 

 different values according as the encounter is with another 

 molecule of gas I. or with a molecule of gas II. Let it be «/ 

 in the first case, and aj i R the second. Then, at the point C, 

 our equation (1) becomes 



Jg = iB(«-« 1 ') + *B( a -«/). . . . (2) 



14. We can now prove that at C, where w 1 = w 2 , a/ = 0. 

 This can be done as follows : — Let p, q be two velocities such 

 that p 2 + q'*> v 2 . Then, for an encounter from which a mole- 

 cule of gas I. issues with velocity v, the velocities before 

 encounter may be either p for the molecule of gas I., and q 

 for that of gas II., or vice versa. There are just as many 

 encounters one way as the other ; and the chance of the given 

 event happening, namely that the molecule of gas I. comes 

 out with velocity v, is the same in either case. 



Let V be the velocity of the centre of gravity of the two 

 molecules engaged. Then, taking the two groups p q and qp 

 together, V can have no average component in x. This 

 follows from the symmetry of the circumstances, n x being 

 equal to ft 2 . If cf> be the angle made by V with x, cos (/> = 0. 



In the figure let C P denote Y f , and Q C = Y denote the half 

 relative velocity, and Q P denote v. Then Q P must be 

 some radius vector of the cone described about C P as axis with 



semivertical angle QPC= cos -1 — ^-^7 . And since all 



directions of V are equally probable after encounter, Q P may 

 with equal probability be any radius vector of that cone. 

 Therefore if 6 be the angle made by Q P with a, and that 

 made by P with x, the mean value of cos 6, given <fi, is 

 cos QPC cos <j). And the mean value of cos 6 for all directions 



