308 On Boltzmann's Theorem on the average Distribution 



Here the integration is effected with reference to all the other 

 p's before that with reference to p>- 



I will here show only how the integration with reference to 

 p n is to be effected when r is not equal to n. For q x is to be 

 substituted its value 



dT 

 dp 1 



= 7lPl =^^V B - T -¥"-^ • < 1,5 > 



If we consider that, in the integration with reference to p n , 

 the quantities q 1 . . . q?i,p 2 • • -Pn-i, and therefore also 71 . - . y n , 

 V, and p r are to be considered as constant, we may put 



E — V -^-= . . . ^ = «, and — » A > 



then, in integrating with reference to p n} all up to q x come.? 

 before the sign of the integral, and 1 -— reduces to 





Avhich, as is well known, can be easily calculated. The inte- 

 gration can be equally easily performed with reference to the 

 remaining p's, and lastly to/>,-. Since V is a given function of 

 the coordinates, the mean kinetic energy may be found simply 

 by repeated integration. 



The symmetry of the formula (16) shows at once that it has 

 the same value for all momenta, consequently also for all 

 atoms in the case of material points. The number Z x of the 

 systems for which the values of coordinates lie between q x and 

 q 1 + dq 1 . . . q n and q n + dq„, and the kinetic energy of the mo- 

 mentum p r between k and k + d/c, whilst all the other momenta 

 have all possible values, is found by integrating the expression 



(15) with reference to those other momenta, but putting \ / — 



dh 7 r 



and ._ — for p r and dp r ; the number Z 2 of the systems for 

 V 2 k 7 



which, whilst keeping the conditions for the coordinates, the 

 last momentum may be any we like, by integrating also with 

 reference to p r or Tc over all possible values. 



The integration, after using the substitution (16), offers no 

 difficulty, and gives 



