304 On Boltzmann's Theorem on the average Distribution 



p n + dp n . Then equation (5) holds good between the products 

 of differentials. 



V is a function of the coordinates determined by the con- 

 stitution of the system; so also T is a function of the momenta 

 or of the momenta and coordinates determined by the consti- 

 tution of the system. Therefore also E = V + T is a function 

 of coordinates aud momenta F(q x . . ,p n ) given by the consti- 

 tution of the system. If we imagine the variables replaced 

 by their values at the time t, then E appears also as a function 

 of these values, which we have also denoted by q x . . ,q n . E 

 may therefore be introduced in the product dq y . . . dp n of 

 equation (5) in the place of one of the variables, e. g. jh ', so 

 that we obtain 



dq x . . ldp n =d qi . . . dq n dp 2 . . . dp n dE+ dF (9i-"Pn) m ^ 



This magnitude E, the total energy of the system, is obtained 

 also by substituting in the function F for q 1 . . . p n their values 

 g\ • • ' p' n at the commencement of the time. Then E appears 

 expressed as a function of q\ . . .//„, and may be introduced 

 in the product dq\ . . . dp' n of equation (5) instead of p\, 

 which gives us 



dq\ . . . dp f n =dq J 1 . . . dq f n dp' 2 . . . dp' n dK+ dF(q )',' ' P ' n \ (6a) 



dp i 



If q t be expressed as a function of the old independents 



q\ . . . q' n r, then -p is the differential quotient of q l in the 



usual sense, by allowing the time to increase without altering 

 otherwise the initial conditions (q'. . .p'„). Maxwell denotes 

 it by q v It is of course also a function of q\ . . . p' n r. Let its 

 value when r=0 be q\ ; then, according to Hamilton, 



_<m_ dF( qi ...p n ) _ dF(q',...p' n % 



h ~ d P r d Pl ,Ql ~ dp\ 



Substituting the values (6) and (6«) in equation (5) and 

 dividing by tZE, %ve obtain 



(7) 



dq x . . . dq n dp 2 . . . dp n _ dq\ . . . dq ' n dp' 2 . . . dp'„ 



(cZE does not occur here). The equation admits of the following 

 interpretation. Let there be given an infinite number of simi- 

 larly constituted systems S. Let the time of the entire 

 motion have for all exactly the same value t, and the total 

 energy exactly the same value E. Let the values of the 



* Compare Thomson and Tait, new edition, § 318, equation ('30). 



