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



length of excursion is, that the molecules do not move equally 

 in all directions within each individual stratum considered sepa- 

 rately, and that therefore the probability of one molecule striking 

 another during the element ds of its excursion varies at the same 

 place with the various directions which ds may possess. In order 

 to bring this circumstance into calculation, let all the coefficients 

 c, c*, A, A', B be again replaced by magnitudes which are 

 dependent on the direction. Now we have already seen that 

 the magnitudes U and H, which determine the unequal motions 

 in various directions of the molecules emitted from any given 

 stratum, vary only slightly with the variations of fi, — so, indeed, 

 that they can be represented by series which progress according 

 to powers of fie. It may be inferred hence that the coefficients, 

 as modified for these unequal motions, can likewise be repre- 

 sented by similar series ; so that we may substitute for c 



c + c^e + c^e 2 -^ .. ., 



and so on for the other coefficients. By introducing these series 

 into the equations (17) and arranging the expression according 

 to fie, we again obtain for s and s 2 series which progress accord- 

 ing to powers of /xe, and which differ from the former series only 

 in the coefficients of the higher terms. If we denote these 

 coefficients by new letters, the ultimate expression which we 

 obtain by taking account of both circumstances, takes the form 



^=e(e+C^ + C'A 2 +...)/l (18) 



s 2 = 2e 2 (c 2 -f-D^e -}-...) I 



It may be further remarked that, of the coefficients of these 

 series, only c will be used in the sequel ; the higher terms, where 

 they occur, are only added for the sake of greater completeness. 



§ 11. These expressions for s and s 2 must now be introduced 

 into the equations (10) and (11) in § 8. If at the same time the 

 series given in equation (I.) be substituted for U, we obtain for 

 V, V 2 , V 3 , &c. series which progress according to powers of ju,c, 

 and which, when a few new symbols are introduced, take the 

 following forms : — • 





V*=u* + 2uq fJ ,6 + (2ur + q*)f J ?6>+ . . . , I (m ) 



V 3 =u 3 + Qu*qpe + 3 (wV + uqflftW + 



The letters q, q* } and r here introduced have the following 

 meanings : — - 



