Kinetic Theory of Gases. 619 



and the law of equipartition can be deduced in the usual 

 ™ay. . 



This proof of the law o£ equipartition is independent of 

 the special assumptions both of Maxwell and of Boltzmann. 

 We do not assume (as Maxwell does) 



(a) That the representative point passes in turn through 

 every point of that part of the generalized space for which 

 the total energy has the assigned value. 



(b) That the total time during which the point is in any 

 element of this space is proportional to the size of the 

 element. 



Neither do we assume (as Boltzmann does) 



(c) That the gas is, at every instant, " ungeordnet," or, 

 more precisely, that Burbury's " Condition A ". is satisfied at 

 every instant. 



The precise assumptions upon which our proof rests are : — 



(d) That at any instant that part of the total energy of 

 the gas which is accounted for by the intermolecular forces 

 forms an infinitesimal fraction of the whole ; and 



(e) That the conservation of energy is maintained through- 

 out the motion of the Pas. 



V. — A Gas ivith a Mass- Velocity. 



§ 38. We have seen (§ 30) that a gas is in a state different 

 from its normal state through only an inappreciable fraction of 

 the whole of the generalized space, but at the same time (§ 31) 

 it is possible that it may be in a state different from normal 

 throughout the whole of a stream-line. For instance, the 

 normal state is such that the mass-velocity of the gas is nil, 

 but we know that if the gas starts with a mass-velocity it 

 will retain this velocity throughout its motion, and will there- 

 fore never reach a normal state. The stream-lines for which 

 the gas possesses an appreciable mass-velocity form only an 

 inappreciable fraction of the whole, and are not worthy of 

 consideration with reference to our arbitrary standard of 

 probability. With reference to the conditions of nature the 

 case is different, so that we now proceed to consider these 

 particular stream-lines. 



Let U, V, W be the components of the mass-velocity of 

 the gas, then the analysis of § 29 will apply, provided we 

 suppose / subject not only to the conditions expressed by 

 equations (36) and (37) , but also to 



fdxdydzdudvdio=JJ . . . (46) 



