Heat through Rarefied Gases. 13 



the plates, and consequently the influence of the mutual 

 encounters of the molecules to be negligibly small. Then 

 the whole number of molecules in unit volume n will be 

 composed of four parts : 



n=« 1 4-w 2 + w 3 4-w 4 , (1) 



viz.: 



n l molecules moving upwards with mean velocity ci, corresponding- to temperature fli, 

 n i » „ downwards „ „ „ ,, „ 



n 2 ,, ,, upwards with mean velocity c 2 , corresponding- to temperature Z , 



n./ „ „ downwards „ „ „ „ „ 



These four kinds, each with velocities distributed according 

 to Maxwell's law, do not undergo any mutual influence, they 

 are subjected only to the impacts on the walls. 



The number of impacts, for unit of time and surface, is 

 given by 



2nc 



if n denotes the number of molecules for unit of volume, 

 moving only upwards or only downwards. 



Now considering the process at the lower plate, we see 

 that the molecules n l are made up of the fraction (1—/) of 

 the incident molecules n/ and of the fraction f of the whole 

 number of molecules which are impinging on the lower 

 plate, whence 



n 1 c 1 =(l-f)7i l / c 1 +f(n l 'c l -{-n 2 'G 2 ), ... (2) 



and similarly 



n 2 c 2 — (l — f)n 2 'c 2 (3) 



By adding these two equations we get an equation expressing 

 the fact that no one-sided current will originate : 



n x Ci + n 2 c 2 = n x 'ci + n 2 c 2 , 



This equation and equation (3) and a similar one for the 

 molecules moving in reverse direction take the form 



(n x — nj)ci = (nj — n%) ^2 



whence follows 



n l 'c l — n. 2 c 2 — (\.—f) 



"""^ , \ (4) 



