36 



Lord Kelvin on the 



hundred decimal numbers from *01 to 1*00. The decimal 

 drawn, called a, shows the proportion of the whole period of 

 P from the cage-front C, to K, and back to C, still unper- 

 formed at the instant when Q crosses C. Now remark, that 



Table showing the Number of the different Velocities on the 

 Different Cards. 



Velocity. 



•l 



•2 



•3 



•4 



•5 



, 



•7 



•8 



-9 



1-0 



l-i 



1-2 



1-8 



1-4 



1-5 



1-6 



1-7 



1-8 



1-9 



2-0 



2'1 



2-2 





Card 1 



100 

















































o 



7 



93 













































» 3 





10 



90 











































„ 4 







9 



91 









































„ 5 









1 



84 



15 





































„ « 













60 



40 



































,. 7 















26 



57 



17 































» 8 



















31 



40 



29 



























„ 9 























3 



26 



19 



15 



11 



9 



6 



4 



3 



2 



1 



1 



2 J 



Sums of , 

 velocities \ 



107 



103 



99 



92 



84 



75 



66 



57 



48 



40 



32 



26 



19 



15 



11 



9 



6 



4 



3 



2 



1 



1 



900 



if Q overtakes P in the first half of its period, it gives its 

 velocity, v, to P and follows it inwards ; and therefore there 

 must be a second impact when P meets it after reflexion 

 from K and gives it back the velocity v which it had on 

 entering. If Q meets P in the second half of its period, Q 

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

 with this velocity escape from the cage. But it may be over- 

 taken 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 va ; 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 



everv 



collision involved interchange, the average velocity of P 

 would be equalized by the collisions to the average velocity 

 of Q, and the average distribution of different velocities 

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

 equalizing interchange can, as we have seen, only occur when 



