Prof. Clausius on the Conduction of Heat by Gases. 433 



for this particular case, is a magnitude of the same order as the 

 normal mean length of excursion denoted by e ; and to indicate 

 this, we will put 



^~ce, (15) 



whence we have 





(16) 



The mean length of excursion is somewhat different for those 

 molecules which do not move perpendicularly to the axis of x ; we 

 can express this by substituting, for the coefficients c and c 2 in 

 the foregoing equations, magnitudes dependent on the direction. 

 This dependence on the direction rests upon two circumstances > 

 each of which may be considered separately. 



The first circumstance is this — that a different temperature 

 and density prevail at the points from which the molecules start, 

 and in the strata through which they have to pass before they 

 arrive at the stratum under consideration, from those which prevail 

 in that stratum. If the cosine of the angle formed by a given 

 direction of motion with the axis of x be denoted by //., then the 

 distance of a molecule whose excursion is s, from our infinitely 

 thin stratum, is equal to fis. The differences of temperature and 

 density existing at this distance can be represented, in the man- 

 ner already known, by series which progress according to whole 

 powers of fis. Now, since the modifications which the coeffi- 

 cients c and c 2 undergo owing to the differences of temperature 

 and density must correspond to these differences themselves, we 

 may conclude that the modified coefficients can be represented 

 by similar series, containing, however, the proper mean values, 

 instead of the particular values s } s 2 , &c. We may accordingly 

 write 



s = e(c + a/AS + a'fj^s 2 -f- . . . ), 



?=:2e^ + bfi~s+ ...). 



By substituting for s and s 2 on the right of these equations the 

 values which rosult from these same equations, we obtain series 

 which progress according to powers of fie, and which, if we also 

 substitute simple symbols for the complicated coefficients of the 

 higher terms, may be written 



s_= e(c + AfMe + AVe 2 +...),! # , (1 7) 



The second circumstance which has an influence on the mean 

 Phil. Mag. S. 4. Vol. 23. No. 156. June 18G2. 2 G 



