88 Prof. Clausius on Molecular Motion, 



The difference of these two expressions^ namely, 



represents the number of those points which are detained be- 

 tween X and x + dx. The path traversed by these points may be 

 considered as x if we neglect infinitely small differences ; and 

 hence the above expression must be multiplied by this length 

 in order to obtain one of the products mentioned before, namely, 



If, now, it be desired to obtain the sum of all products of this 

 kind which correspond to the several layers of the thickness dx, 

 this must of course, in the case in point where the layers are in- 

 finitely thin, be effected by integration. Hence the above for- 

 mula has to be integrated from ^^=0 to a; = co , whence the fol- 

 lowing expression is obtained, 



irp 



This expression has now only to be divided by N in order to 

 arrive at the mean length of path required. If this be called /', 

 the equation is 



V= ^, (6) 



In the case where not one molecule only is in motion while 

 all the others are at rest, but where all molecules move with 

 equal velocity, the meaii length of way, as mentioned before, is 

 less than that above considered in the proportion of | to 1. 

 Hence if we put the simple letter I for this case, we have 



'=i|^ (^) 



Writing this equation in the form 



- = 1. — 3, {7 a) 



a simple law results. It follows from the manner in which we 

 determined the density, that the part of the given space filled by 

 the spheres of action of the molecules is related to the whole 

 given space as a sphere of action to a cube of the side X, that 

 is, as 



$ 7rp3 : \^. 



Accordingly the meaning of the previous equation may be so 

 put: — The mean Icnr/th of path of a molecule is in the same pro 



i 



