74 PROFESSOR TAIT ON THE 



average path for each speed by the probability of that speed, and~take the 

 sum of the products. Since the probability of speed v to v + dv is 



/A 3 

 the above definition gives for the length of the mean free path, 







or, by the expression for e above, 



1 



mrs* 



^ , t^+i-^T , +^> 



This may without trouble (see § 9) be transformed into the simpler expression 



4afit~ xt d% 



n-n-s- I 



J § 



x e- j:i + (2x 2 + l)^''e- xi dx 



which admits of easy numerical approximation. The numerical work would be 

 simplified by dividing above and below by e~ r2 , but we prefer to keep the 

 present form on account of its direct applicability to the case of mixed 

 systems. And it is curious to note that 4«~* 8 is the third differential coefficient 

 of the denominator. 



The value of the definite integral (as will be shown by direct computation 

 in an Appendix to the paper) is about 



0-677 ; 



and this is the ratio in which the mean path is diminished in consequence of 

 the motion of the particles of the medium. For it is obvious, from what 

 precedes, that the mean path (at any speed) if the particles were quiescent 



would be 



1 



717TS 2 



The factor by which the mean path is reduced in consequence of the 

 " special" state is usually given, after Clerk-Maxwell, as I/*/ 2 or 0707. 



But this appears to be based on an erroneous definition. For if n v be 

 the fraction of the whole particles which have speed v, p v their free path; we 

 have taken the mean free path as 



according to the usual definition of a " mean." 



