404 Mr. S. H. Burbury on Boltzmann's Laic of 



The question now is what is the form of the function A. 

 In stationary motion Ae~ hQ - cannot vary with the time. It 

 follows that 



t /^d *frf\ At _n^ =0l . . . (1) 



\dt dx at duj 



with corresponding equations for y, v, and z, iv. Also for 

 each molecule m 



d 4- = w, 4- e- h *™ 2 = - 2hmue- 2h * mu \ 

 at du 



and by definition of %, 



m ZJ7 = 

 dt dx 



We have then 



2u(— +2hA&\e->*™ 2 = 0, &c, . . (2) 



\ax ax j 



which is satisfied if for each molecule 



dx dx ' 



with corresponding equations for y and z. 



The solution is then A = e~ 2 ^, or, as 1 shall write it, 



A = e- 2h *x, (3) 



% now denoting for any molecule the potential of all the 

 forces acting on that molecule. 



3. These forces may be either external forces, independent 

 of the positions of the other molecules, or intermolecular forces. 

 With regard to the latter, we must assume, until we can find 

 a more excellent way, that the force between two molecules, 

 m p and m q , is a function of the distance, r pq , between them, 

 and acts in the line r, so that ^ is evidently a function of 

 the r's. In fact we assume here instantaneous action at a 

 distance, although in other branches of physics we have 

 discarded that assumption. Boltzmann does the same thing 



with his force varying as — 5 . 



4. The proof above given applies formally to all forces 

 which have a potential ^, and therefore to the iutermolecular 

 as well as to the external forces. As, however, some writers 

 hesitate to apply the theorem to intermolecular forces, the 

 following considerations may remove a difficulty, though 

 in my opinion the proof is complete without them. Suppose 

 a single centre of force of mass tw', at first fixed at a!y l z\ 



