394 Lord Kelvin [April 27, 



will, by the first impact, get P's original velocity, and may with this 

 velocity escape from the cage. But it may be overtaken by P before 

 it gets out of the cage, in which case it will go away from the cage 

 with its own original velocity v unchanged. This occurs always if, 

 and never unless, u is less than v a ; P's velocity being denoted by u, 

 and Q's by v. This case of Q overtaken by P can only occur if the 

 entering velocity of Q is greater than the speed of P before collision. 

 Except in this case, P's speed is unchanged by the collision. 

 Hence we see, that it is only when P's speed is greater than Q's 

 before collision, that there can be interchange, and this interchange 

 leaves P with less speed than Q. If every collision involved inter- 

 change, the average velocity of P would be equalised by the collisions 

 to the average velocity of Q, and the average distribution of different 

 velocities would be identical for Q and P. Non-fulfilment of this 

 equalising interchange can, as we have seen, only occur when Q's 

 speed is less than P's, and therefore the average speed and the 

 average kinetic energy of P must be less than the average kinetic 

 energy of Q. 



§ 52. We might be satisfied with this, as directly negativing the 

 Boltzmann-Maxwell doctrine for this case. It is, however, interesting 

 to know, not only that the average kinetic energy of Q is greater than 

 that of the caged atom, but, further, to know how much greater it is. 

 We have therefore worked out summations for 300 collisions between 

 P and Q, beginning with ur = -5 (u = -71), being approximately 

 the mean of v 2 as given by the lottery. It would have made no 

 appreciable difference in the result if we had begun with any value of 

 u, large or small, other than zero. Thus, for example, if we had 

 taken 100 as the first value of u, this speed would have been taken by 

 Q at the first impact, and sent away along the practically infinite row, 

 never to be heard of again ; and the next value of u would have been 

 the first value drawn by lottery for v. Immediately before each of 

 the subsequent impacts, the velocity of P is that which it had from Q 

 by the preceding impact. In our work, the speeds which P actually 

 had at the first sixteen times of Q's entering the cage were -71, -5, '3, 

 •2, -2, -1, -1, -2, -2, -5, -7, -2, -3, -6, 1-5, -5— from which we 

 see how little effect the choice of • 71 for the first speed of P had on 

 those that follow. The summations were taken in successive groups 

 of ten ; in every one of these 2 v 2 exceeded 2 u 2 . For the 300 we 

 found 2 v 2 = 148-53 and 2 u 2 = 61-62, of which the former is 2-41 

 times the latter. The two ought to be equal according to the 

 Boltzmann-Maxwell doctrine. Dividing 2 v 2 by 300 we find -495, 

 which chances to more nearly the -5 we intended than the -52 which 

 is on the cards (§ 51 above). A still greater deviation (2-71 instead 

 of 2-41) was found by taking 2 v 3 and 2 u' 2 v to allow for greater 

 probability of impact with greater than with smaller values of v ; u' 

 being the velocity of P after collision with Q. 



§ 53. We have seen in § 52 that 2 u 2 must be less than 2 v 2 , 

 but it seemed interesting to find how much less it would be with 



