■hq 



406 Mr. S. H. Burbury on Boltzmann's Law of 



when every molecule is connected with every other by inter- 

 molecular forces. 



5. It should be noted, however, that for these inter- 

 molecular forces the solution A=€~ 2h X is not unique. If the 

 motion be stationary with A = e~ 2hx , then it is also stationary 

 with A=e~ 2h xf(r), where f{r), or /, denotes any function 

 of the rs only, and of the coordinates as these are involved in 

 the r's. 



For we have -j~(Af(r)e- hq ) 



\ dr 12 dx\ ar 12 dx 2 I 



^ u Y x x —u 2 x^ df & ^ { j^ VpXp — v q r q df 

 r 12 dr l2 ' r pq dr vq 



all the terms derived from differentiation of /(/•) being 

 reducible to a sum of pairs of this form. 



Now since A = e~ 2A * by hypothesis makes the motion 



stationarv, it makes every factor of the form -^ — 9 zero 



d pq 



on average. And therefore it makes -r (A/ (?-)e~ hq ) zero, and 



so Af(r)~ 2h x is also a solution. In the same way, if Q be 

 also a function of the r's, and of the coordinates only as 

 contained in the r's, the differentiation for x introduces new 

 terms, all of which can be resolved into pairs of the form 



^Q dr pq dQ dr pq 



(Xr pq OlXip Coi'pq ax q 



or 



ttl^J UptX/p ~~~ V q X q 



dr pq r pq 



which vanish on average. 



6. The proof of our law given in art. 2 is not subject in 

 any of its steps to any condition as to the density of the 

 system. It may, however, be said that the initial assumption 

 of Ae~ AQ , with the coordinates and velocities in separate 

 factors for the law of distribution, is true only for a very rare 

 gas, and therefore the theorem, if it depends on that assumption, 



