90 Prof. Clausius on Molecular Motion. 



(8.) After the above determination of the length of the mean 

 path, we still have to consider how the separate paths which 

 really occur are related to the mean path. 



The first question which presents itself is, in what proportion 

 is the number of cases in which the real path is less than the 

 mean path, to that of the cases in which it is greater. For answer- 

 ing this question, use is made of (5), in which we have only to 

 substitute the mean value I' for x in order to find what probability 

 there is that the true path is equal to or greater than the mean 

 one. If for /' we here make use of the expression in (6), and de- 

 note the corresponding value of W by W„ then 



Wi = e-'=0-3679 (11) 



From the above equation it follows, that out of N cases only 

 0"3679 N occur in which the real path is equal to, or greater 

 than the mean one, while in the 



0-6321 N 



cases the true path is the smaller one. 



If, further, it be required to know the number of cases in 

 which the true path is equal to or above the double, treble, &c. 

 of the mean one, the same process may be adopted as before. 

 Calling the probabilities in question Wg, W3, &c., we have 



^s=e- [ (13) 



&c. J 



These numbers evidently diminish very rapidly, since, for in- 

 stance, e~'" = 000045; and we gather from this that, although 

 in isolated cases a molecule may traverse a path considerably 

 longer than the mean one, such cases are comparatively rare, 

 and that in the majority of cases the actual path is smaller or 

 very little larger than the small mean value found above. 



(9.) If, now, these results be applied to the externally recog- 

 nizable behaviours of a gas, in which it is 2:)resumed that no 

 other motion common to the whole mass besides the molecular 

 one is present, it is easy to convince oneself that the theory 

 which explains the expansive force of gases does not lead to 

 the conclusion that two quantities of gas bounding one another 

 must mix with one another quickly and violently, but that only 

 a comparatively small number of atoms can arrive quickly at a 

 great distance, while the chief quantities only gradually mix at 

 the surface of their contact. 



From this it is clear why clouds of smoke only slowly lose 

 their form on quiet days. Even when the air is in motion, pro- 

 vided such motion consists of a uniform one of the entire cur- 



