238 



Free Path Phenomena 



[CH. xi 



all directions for EF are equally probable*. Hence in fig. 17 all directions 

 of RQ', RP are equally probable so that the " expectation " of the component 

 of velocity of either molecule after impact in any direction is equal to the 

 component of OR in that direction. 



290. Let us now average over all possible directions for the velocity of 

 the second molecule, keeping the magnitude of this velocity constant. In 



FIG. 19. 



fig. 19 let OP, OQ as before, represent the velocities of the two colliding 

 molecules. We have to average the velocity OR over all positions of Q which 

 lie on a sphere having for centre. It is at once obvious that the average 

 component of OR in any direction perpendicular to OP is zero. We have, 

 therefore, only to find the component in the direction OP. We must not 

 suppose all directions for OQ to be equally likely, for (cf. Chapter II.) the 

 probability of collision with any two velocities is proportional to the relative 

 velocity. Thus the probability of the angle POQ lying between 6 and 6 + dO 

 is not simply proportional to sin dd, but is proportional to PQs'm6d0, for 

 PQ represents the relative velocity. The average value of the component 

 ON is therefore 



ON = 



{" ON.PQs 

 Jo __ 



f 



J 



smBdO 



sin 6 d6 



(551). 



To evaluate this fraction, let us write 



OP = a, OQ = b, PQ = r, 

 so that r 2 = a 2 + 6 2 - 2a& cos 6 



(552), 



This theorem was first given by Maxwell in 1859, Collected Works, i. p. 378. 



