the Variation of Entropy. 253 



Let us call <ji'...y'^ the upper, and ^/...^V the lower, 

 limit. Then the number o£ systems which in time dt pass 

 the lower limit by variation of y:>i alone is (p. 7) 



Dp/dtclp2 ... dqn, 

 dp2 being now written for pJ' —p2 &c., and p^ written for 



-~^ at the lower limit, and thev ixiss in or out accordino- as 

 dt . I n 



pi is positive or negative. In like manner the number 

 of systems which in time dt pass the upper limit by variation 

 of pi is 



and they pass out or in according as pi^ is positive or 

 negative. Xow D and j'l have slightly different values at 

 the two limits respectively. Therefore the increase in time 

 dt of the number of systems between the limits due to the 

 variation of pi alone is 



— ;,— (D/.^i) dpi...dqn dt. 



api 



This is of course on the assumption that pY —p\. is in- 

 finitesimalj and may be represented by dpy. 



6. ^Yhat has been proved for /;^, is equally true for each of 



the other /^'s and (^^s. Let then ^-- denote the rate of in- 

 crease of D per unit of time, for the particular extension in 

 phase considered^ due to this passing in and out of systems. 

 Then 



But by LagrangVs equations, for each q and the corre- 

 sponding p 



g + |=^ (^) 



And therefore 



7. Again, if we follow the motion of a chosen system, and 



B.=-^^;i-AV:) (^) 



