OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 721 



the system is in that phase or not, and also when it enters the phase and 

 when it leaves it. 



Boltzmann defines the probability of the system being in the phase (a^) 

 as the ratio of the aggregate time during which it is in that phase to the 

 whole time of the motion, the whole time being supposed to be very great. 

 I prefer to suppose that there are a great many systems the properties of 

 which are the same, and that each of these is set in motion with a different 

 set of values for the n co-ordinates and the n 1 momenta, the value of the 

 total energy E being the same in all, and to consider the number of these 

 systems which, at a given instant, are in the phase (a^). The motion of each 

 system is of course independent of the other systems. 



Let N be the whole number of systems, and let the number of these 

 which, at the time t, are in the phase (aj)) be denoted by N (a ly b, t). The 

 aim of the statistical method is to express JV (a lt b, t) as a function of N, 

 of the co-ordinates and momenta with their limits, and of t. It is manifest 

 that N can only enter the function as a factor, for the different systems do 

 not act on each other. Also any differential as da or db can only enter as 

 a factor, for the number of systems within any phase must vary in the ratio 

 of the interval between the limits of that phase. We may therefore write 



N(a 1 bt) = Nf(a,, a n , b, b n , t)da 2 da n d\ db n (22), 



where we have to determine the form of the function f. 



We shall now follow the motion of these systems from the time t', when 

 we begin to watch the motion, to the time t when we cease to watch it. 



Since the systems which at the tune t form the group N(a lt b, t) are 

 individually the same systems which at the time t' formed the group N (a{, b', t') 



we have 



N(a u b, t) = N(a;, b', t) (23), 



or Nf(a, t)da, db n = Nf(a 3 ' t')da,' db n ' (24). 



But by equation (21) 



da, db n (\Y l = da; A.'&T 1 - (25). 



Hence /(a, )W t'}b^=C (26), 



where C is a constant for all phases of the same motion, and we may write 



/(. t) = C(b 1 )-> (27), 



and N(a u b, t) = NC(b^da, db n (28). 



VOL. II. 9! 



