of Energy in a System of Material Points. 305 



variables q i . . . q n} p 2 • • •Pn at the beginning of the times be 

 between the limits 



q\ and q\ + dq\ . . . q' n and q r n + dq' n , > 



p' 2 and p' 2 + dp' 2 • • -p' n and p' n + dp r n , ) 



for all the systems, while p x is determined by the equation of 

 energy. If, further, we denote the limits between which 

 coordinates and momenta lie at the moment at which motion 

 ends by 



q ± and q x + dq x ...q n and q n + dq n , ~) . 



p 2 and 7? 2 + dp 2 ...p n and p n + dp n , J 



then again equation (7) must hold between the products of 

 differentials. 



Maxwell employs now a method which he calls the statistical. 

 He assumes we have a large number N of systems such as S 

 given, having all exactly the same energy E, but whose coor- 

 dinates and momenta at the commencement of motion have all 

 possible values. He proposes to himself the problem to inves- 

 tigate, not how coordinates and momenta change for each of 

 these systems with the time, but how many systems at a given 

 time "have the phase (pq)" — i- e. for how many the coordi- 

 nates and momenta lie between the limits (9). 



Px is always determined by the equation of energy. The 

 number of systems which at the time r " have the phase (pq) " 

 Maxwell denotes in general by 



N/(?i . . . q n , p2 • • -Put) dq x . . . dq n dp. 2 . . . dp n . . (10) 



The number of systems which, at time 0, have the phase (p'q f ), 

 i. e. for which the variables at this time lie between the limits 

 (8), will consequently be denoted by 



Nf(q' 1 ...q' n ,p f 2 .:.p' n 0)dq' 1 ...dq r ndp r 2 .:.dp' n . . (11) 



But, in accordance with the signification already given to 

 qi . . . p n and q\ . . .p' n , exactly the same systems have the phase 

 (pq) at the time r which had the phase (p'q 1 ) at the time 0. 

 The expressions (10) and (11) are therefore equal; whence, 

 referring to equation (7), we have 



qj(q x . . . qn,p 2 • • .p«r) = Yx/Wi ■ ■ ■ Q^'s • • • PnO). . (12) 

 Maxwell calls the distribution of the system stationary when 

 the number of systems having any given phase, e. g. (p'f), 

 does not change with the time — when, therefore, for any 

 q , 1 ...q f n ,p' 2 ...p' n , 



f(q f l.>.q r n,p f 2.-.pnT)=;W 1 ...q f n,p / 2 ...p r nO) . (13) 



Since in equation (12) q\ . . .p' n are also any initial values 

 PHI. Mag. S. 5. Vol. 14. No. 88. Oct. 1882. ' X 



