426 Trof. Clausius on the Conduction of Heat by Gases. 



dit ion for each gas — for instance, where the gas is exposed to the 

 pressure of one atmosphere, and its temperature throughout is 

 zero (the freezing-point). We will call the mean length of ex- 

 cursion which corresponds to this condition of the gas, the normal 

 mean length of excursion {normale mittlere Weglange), and we 

 will denote it by e. We can then regard the component velocity 

 already mentioned as a magnitude of the order of e, and can 

 accordingly denote it by pe. 



We will now consider any molecule whatever whose direction 

 forms the angle a with the axis of x. As in what follows we 

 have generally only to consider the cosine of the angle which 

 the direction of any molecule forms with the axis of x> we will 

 for the sake of shortness call it the cosine of the molecule, and 

 denote it by a single letter, which in the case before us shall be X. 

 If now the component velocity pe in the direction of positive x 

 be imparted to the molecule, its velocity and its cosine will be 

 thereby changed, and we will denote the altered values which 

 take the places of A and X by U and /a. We have then for the 

 determination of these two magnitudes the equations 



U/jL = A\+pe, (1) 



U 2 =A 2 + 2XA^e-f/e 2 (2) 



Substituting for XA in the second equation the value JJ/jl— pe 

 derived from the first equation, we get 



U 2 =A 2 + 2//,Upe-/e 2 . 



By solving this equation we obtain two values for TJ, one posi- 

 tive and one negative, of which it is evident that we must take 

 the positive one : this is 



U =ppe+ \/A 2 -p*{l-fJL*)e 2 . . . . (3) 

 Denoting the particular value of U when jjl = hy u, that is, 



u= \/& 2 -p*e\ (4) 



the last equation becomes 



U=pfu,e+ Vu*-\-p 2 fj}e*; (5) 



and developing this expression according to /jue, we get the fbl-* 

 lowing equation, which conveniently represents the dependence 

 of the velocity U on the cosine /jl, 



U = M+W e + ^VV+. (I.) 



The magnitudes u and p which here occur may have different 

 values in different strata, and are thus to be considered as func- 

 tions of x. 



With reference to the distribution of the molecules among the 



