•Jl 



•13:2 Trof. Clausius on the Conduction of Heat by Gases. 



foregoing conclusions remain applicable to this case also without 



modification, and hence the value — given in (12) must also be 



the mean value of these distances*. 



The mean value of s 2 may be obtained in a way quite similar 

 to the above if s 2 be used as the multiplier before integration 

 instead of s, and the rest of the operation be conducted as before. 

 We thus get 



i e" M ads=^s (13) 



Hence it follows that the two mean values s and s 2 are related 

 to each other as expressed by the equation 



? = 2(s) 2 (14) 



§ 10. We have now to investigate the modifications which 

 these mean values undergo if the gas has not a uniform tempe- 

 rature and density throughout, but if its temperature and den- 

 sity are functions of x. 



All the foregoing considerations remain applicable to the mo- 

 lecules whose motions, being perpendicular to the axis of w, do 

 not cause any alteration in the value of their abscissas. If, then, 

 in order to distinguish those particular values of the general 

 values which relate to this 'case, we attach to the letters con- 

 cerned the index (because in this case /u, = 0), we may write 



-_I andi r_ A. 



o — ~ ana S — „2 



a a 



The quantity — , which represents the mean length of excursion 



* It may perhaps appear surprising at first sight that the same value should 

 be found for the distances traversed between the last impacts and a given 

 moment of time, or between this moment and the next impacts, as for the 

 entire distance traversed in the gas from one impact to the next during a 

 given time. It must, however, be remembered that the mean value of all 

 the distances traversed in the gas between every two impacts during a given 

 time is not the same thing as the mean value which would be found by 

 taking into consideration the distances which all the molecules, which at 

 any given moment are simultaneously in one stratum, would traverse be- 

 tween their last previous and next following impacts. For the longer 

 distances would be of more frequent occurrence in the latter case than in 

 the former, since a molecule requires more time to move through a long 

 distance than to move through a short one; and the probability is therefore 

 greater that any given moment would occur during a longer than during a 

 shorter distance, whereas in the former case all the distances traversed in 

 the gas count equally. By making the calculation, it will be found, that 

 the latter supposition gives a mean value twice as great as the mean value 

 given by the former. The value of s, as determined above, is the half of 

 this greater mean value. 



