48* VISCOSITY OF GASES 449 



we put the value of B given in 32*, viz. 



} 



and substitute for kmu 2 a new variable, the constant k under the 

 sign of integration disappears, and r\ becomes inversely propor- 

 tional to \/ '(km), i.e. directly proportional to the mean molecular 

 speed O, and therefore directly proportional to the square root 

 of the absolute temperature. 



If, lastly, in the final formula of the foregoing paragraph we 

 replace the free path I of the particles which move with the speed 

 <o by the mean free path L given by the equation 



l/L = wT/lB = (u,/a)/(&raw 2 ), 



which results from those obtained in 37*, we see that ?? is also 

 proportional to L, and may be expressed by a formula of the 

 form 



in which K is a numeric which is indepandent of m and is thus the 

 same for all kinds of gases. 



The equation now appears in the same shape as before in the 

 approximate calculation, and therefore directly shows that the 

 law is still valid which lays down that the viscosity of a gas is 

 independent of its density. 



The value of the factor K is given by an integral which, though 

 not integrable, is easily interpretable. If, in agreement with a 

 notation already used in another calculation, we express the mean 

 value of a function of the speed, as calculated on Maxwell's 

 law of probability, by 



* du <a*Fwe-** 



= 2 5 /r- 2 a- 3 

 J 



then 



By means of this interpretation of the integral as a mean 

 value we are enabled to assign limits within which the value of 

 the numeric K must lie. 



The mean value in the formula is to be formed exactly as the 

 mean value M(l), which is introduced in 38* with the like nota- 

 tion. The arithmetical mean, therefore, of all values of the product 



G G 



