306 On Boltzmann's Theorem on the average Distribution 



of the variables whatever, equations (12) and (13) may be at 

 once combined with each other, and give 



flXAtfi • • • 9n,2h • ■ ■PnT) = </\f(q\ . . . q'n,p' 2 . . ./„t)*. 



Since / no longer contains the time t, it is better to omit t 

 from under the functional sign and to write 



iiAqi • • • Q*>Ps • • -pn^q'ifWi • • • q'n,i'-2 • • .//*). . (14) 



Here q\ . . . q'n,p f 2 . . >p' n are any initial values whatever: 

 q x . . . q n ,'p-2 • - -V n are ^ e v a m es of coordinates and momenta 

 which a system starting from these initial values attains after 

 a time t, in other respects unfixed. 



Let us therefore imagine a system starting from any initial 

 values of coordinates and momenta ; then in course of the 

 motion it will assume continually new and new values of co- 

 ordinates and momenta. The coordinates and momenta are 

 therefore functions of the initial values and of the time. But 

 there will be in general certain functions of coordinates and 

 momenta which have constant values during the whole motion, 

 as in a free system the component velocities of the centre of gra- 

 vity, or the sums of angular momenta, are invariable. Let us 

 therefore imagine in the expression q\f(qi . . . q,„ ]>2 • • •/>«) first 

 of all those optional initial values from which each system 

 started, then continuously the values in order which coordi- 

 nates and momenta assume for that system as the time 

 increases; then for the existence of stationary distribution it 

 is necessary and sufficient that the value q\f shall remain 

 unaltered, or, in other words, q\f must contain only such 

 functions ofqi...p n as remain constant during the whole 

 motion of a system from any initial values whatever, and con- 

 seqnently are dependent on the initial values, but not on the 

 time which has elapsed. If the system is so constituted that its 

 coordinates and momenta, starting from given initial values, 

 assume in the course of a sufficiently long time all possible 

 values consistent with the ecpiation of energy, then q\f must 

 in general have the same value for all coordinates and momenta 

 consistent with the equation of energy — must therefore be a 

 constant. 



I will now mention some other terms employed by Max- 

 well. If one of the systems S starting from a given initial 

 condition moves, all the conditions through which it passes in 

 consequence of its motion as time increases, constitute the 



* This or the identical equation (14) is necessary that the distribution 

 maybe stationary. It is also sufficient; for from it and equation (12) 

 equation (13) follows at once for any q\ . . .p' n whatever, which is exactly 

 the mathematical expression for a stationary distribution. 



