ox OUR KNOWLEDGE OF THERMODYNAMICS. 103 



the beginning of the first of these parts the time zero ; the beginning of 

 the second <,, the beginning of the third t^, &c. ; the end of the last of the 

 n parts t„. After the whole time has elapsed, let another series of times 

 of length 5 begin. Denote the end of the first part after t„ by <„+i ; the 

 end of the next following part t„^.^, Ac. Assume for a moment that we 

 liave n separate vessels, all exactly similar to the one containing the gas ; 

 that each of these n vessels contains the same gas, and that the motion of 

 the gas is the same in each. The beginning, however, is different. For 

 example, let the gas in the second vessel at time zero be in the same con- 

 dition in which the gas of the first vessel is at the time t^ ; in the third 

 vessel let the gas at the time zero be in exactly the same condition as it is 

 in the first vessel at the time <.,) a,nd so on. We have now in the dift'erent 

 vessels all the different states of the gas existing simultaneously which in 

 the first vessel exist successively during the whole time interval 0. 



The probability div that the co-ordinates and momenta of a molecule 

 may lie between the limits 



a and a + da, b and b + db . . . j) a-nd p -\- dp, q and q + dq ... . (1) 



can be defined in two ways. If we consider a single vessel containing gas, 

 we must observe it for a long time © ; if r be the fraction of the time 

 during which the co-ordinates and momenta of a molecule lie between the 

 limits (1) — which we shall call the condition (1) — then r/©is the probability 

 required. The limits (1) differ only infinitesimally from one another. No 

 two molecules of the same gas can be in the condition (1) at the same 

 time. On the other hand, if we consider the above series of n vessels at 

 any single instant of time, we can define the probability dw to be dz/71, where 

 dz is the number of vessels in which a molecule is in the condition (1). 

 Evidently dio will have different values for different values of the co-ordi- 

 nates and momenta. It will also be proportional to the differentials 

 da, db . . . We may therefore put 



■^=. — z=dw=/ (a, b, . . . p, q, . . .)dadb . . . dpdq ... . (2) 

 n 



To find the condition for a stationary state we may consider one gas 

 .at successive instants, or the series of vessels at one instant. In the 

 first case the values of dw for the stationary state will be the same, 

 whether we consider the gas from time to time t„, or from time ty to 

 time <„_,_i, or in genei'al from t,; to <„+,.. Evidently the converse is true ; 

 that is, if dw has the same values for all these cases, the state is 

 stationary. By the second method we must remember that at the time 

 we have in our n vessels all the states which appear in the first case from 

 time to time <„ ; at the time <, we have in these vessels all the states 

 which appear in the first case from /;, to <„+i. . . . The above statement, 

 thatr/0 has the same values in all cases, whether we consider the time 

 from zero to t,„ or from <, to <„+i, or from t,. to ^,,+j, becomes in this second 

 case identical with the statement that dzjn has the same value, whether 

 we consider the n vessels at time zero, or time <|, or time <2) *^c. That 

 is, since the difference between <, and zero, t.2 and <, . . . can be made 

 infinitely small, the above statement amounts to saying that for the 

 stationary state dzjn has the same values at all times. We shall next prove 

 that dzjn has this property under the following conditions : — 



We define a free molecule to be one which is not acted upon by any 

 other molecule. For each free molecule let the values of _/' {a,b . . . 2^, q • . •) 



