474 Mr. S. H. Burbury on the 



by the amount 



or would gain x velocity ~-^8xf{v)dv~K per unit time. This 



is the effect of the variation of density of gas I. from point to 

 point along the tube. 



8. But the motion is in fact steady. Therefore this gain in 

 translation-velocity, which would accrue to the class v in the 

 absence of encounters, is exactly compensated by the loss 

 which is due to encounters. A certain number of the class 

 undergo encounter, and are knocked out of the class in a unit 

 of time. An equal number of molecules come out of encoun- 

 ters with velocity between v and dv, that is enter the class v, 

 in a unit of time. And these, as they enter the class, have 

 less average translation-velocity than the continuing members 

 of the class have. 



Let B be the average number of encounters with molecules 

 of either gas which a molecule of class v undergoes in unit 

 time. In the special case we are now treating, B is the same 

 for all directions of motion of the molecule ; because in what- 

 ever direction there are more encounters with one gas, there are 

 by the same number fewer with the other. It follows that those 

 members of the class v which undergo encounter have before 

 such encounter the same mean translation-velocity which the 

 other members of the class have. Now n-if(v)dv B molecules 

 per unit volume and unit time undergo encounters, and cease 

 to be members of the class. Were they not replaced, the ag- 

 gregate translation-velocity of the class would be diminished 

 from this cause by n{Bf(v) dv.u per unit of volume and time. 

 But an equal number of molecules come out of encounters 

 with velocity between v and v + dv ; and these have some 

 mean translation-velocity in x, which we will call a! . The 

 effect of encounters on the translation-velocity of the class v 

 is to substitute per unit volume and unit time n-fif(v) dv mole- 

 cules with mean translation-velocity a! for the same number 

 with mean translation-velocity a. 



This diminishes the translation- velocity of the class v within 

 our element of volume 8x by 



n x f(v) dv~B(u — a') Sx 

 in unit time. 



9. For steady motion we equate this expression to the gain 

 of translation-velocity which, as we have seen, arises from the 

 variation of density. We thus obtain the equation 



