Prof. Clausius on the Conduction of Heat by Gases. 429 



general formula. They only eome into account in the numerical 

 calculation, since for this — if the velocities and the magnitudes 

 dependent upon them, which are expressed in the formula by 

 particular letters, have in reality] various values — those mean 

 values which correctly represent the values that really occur 

 must be calculated ; and for the calculation of these mean values, 

 the manner in which the values are distributed must be known. 

 Reserving to ourselves to return again at the end to the latter 

 point, we propose to ourselves now to determine the condition of 

 the gas, and particularly the vis viva traversing a plane, starting 

 from the assumption that the magnitudes U and H, determined 

 by the equations (I.) and (IT.), represent the real motions of the 

 molecules emitted from a stratum. 



III. Behaviour of the molecules simultaneously existing in an infi- 

 nitely thin stratum. 



§ 8. We will suppose two planes placed perpendicularly upon 

 the axis of x, and with the abscissse x and x -j- dx, whereby we 

 obtain again, as in the foregoing section, an infinitely thin stra- 

 tum ; but we will now consider, not the molecules emitted from 

 this stratum, but the molecules which exist in it simultaneously. 



If the gas had the same temperature and density throughout, 

 the motions would be such that an equal number of molecules 

 would move in all directions, and that the velocities would be 

 equal. But in the case before us, where the temperature and 

 density are functions of x, this uniformity does not occur. 



To determine the velocities of the molecules, let us choose any 

 direction which makes with the axis of x an angle whose cosine 

 is fi, and let us consider the molecules which move in this direc- 

 tion. Before such a molecule enters our infinitely thin stratum 

 with the abscissa x, it has in general traversed a certain distance 

 since its last impact. If this distance be called s, the abscissa of 

 the point where the last impact occurred will be x—/jls; which 

 expression determines the velocity of the molecule, since, accord- 

 ing to the assumptions made above, the velocity with which a 

 molecule is impelled after an impact depends only upon the 

 abscissa of the point of impact, and upon the direction of its mo- 

 tion. We have above denoted the velocity as a function of x 

 and fju, by U, and we may accordingly in this case, in which a 

 molecule is impelled from a point whose abscissa is x—fis, denote 

 its velocity by V, and write 



™-s~iS"- t») 



The distance s is not the same for all the molecules in our 

 stratum which have a determinate direction, so that their veloci- 



