478 On the Diffusion of Gases. 



where s 



denotes the mean for all classes of molecules of the 



2 v 2 



function ^. We may write vl for ^, if / be the mean free 



path for a v. We have thus obtained an inferior limit for 7i x a. 

 17. We have now to ascertain the superior limit. Let us 

 suppose a to be for an instant constant. The gain of x velo- 

 city to the class v per unit of time due to variation of density 

 is in this case, as before, 



And there is a loss due to encounters. We cannot with a 

 constant make the loss equal to the gain for each class sepa- 

 rately ; but we can so choose the constant a as that, for an 

 instant, the loss to the whole gas per unit time shall be equal 

 to the gain for the whole gas. That is, 



= 1 dvf^n^iBct — a/-rJBa — aj). 

 Jo 

 Then, as before, 



«/ = 0; 



and, as before, 



f7wB(«-«/)<fo=0. 

 Our equation is thus reduced to 



1 fin f* 00 rt if* 00 r* 00 



and therefore 



- 2 drc 2 



Here we have v 2 /B, the quotient of the means, instead of ^ 

 the mean of the quotient. B 



Also for given stream, the loss of translation-velocity by 

 encounters is less when u is constant than when a is variable, 

 and increases with v. Therefore, to make the loss equal to the 

 gain for the whole gas, we must make a greater in the former 

 than in the latter case. And therefore the value of a last 

 found must be greater than the true stream-velocity. We may 



