Prof. Clausius on the Conduction of Heat by Gases, 435 



du 



q*=q* + c 



jduy 



\dx) ' 



= lPl-. c ( t-C—+c' 2 —. 

 2u dx dx dx 2 ' 



(19) 



§ 12, Having now determined the velocities of the molecules 

 which exist simultaneously in a given stratum, it remains for us 

 to investigate the distribution of the motions of these molecules 

 among the various directions. 



If the motions were directed equally towards all points, then, 

 for the same reasons as those discussed in § 6, in treating of the 

 molecules emitted from a stratum, the number of molecules 

 whose cosine lay between /& and fjb-\-d/M would be represented as 

 a fraction of the whole number present by \dyb. In the case 

 before us, however, where the motions are not equally divided 

 among all the directions, but only among such directions as form 

 the same angle with the axis of x, we will denote the number of 

 molecules whose cosine lies between p and fA + d/j, as a fraction 

 of the whole number of molecules present by \\d\i, where I 

 signifies a function of jjl. Now it is easy to convince ourselves, 

 by considerations similar to those contained in the foregoing 

 sections, that the function I must be capable of expression by a 

 series which progresses according to powers of /-te, and it may 

 therefore be written thus, 



I = i(l + ^e + ry£ 2 +...), • • ■ (iv.) 



where i, q', r', &c. are magnitudes independent of /jl. 



The magnitude i can be easily determined at once. If the 

 expression \\dyb be integrated from //-= — 1 to yu,= +l, this 

 integration will include all the molecules present, and the value 

 of the integral must therefore be 1. Working this out by put- 

 ting for I the series just established, we get 



and thence 



i=l-^r'e* + (20) 



We will leave the other magnitudes q\ r' } &c, occurring in 

 equation (IV.), for the present undetermined, as an opportunity 

 will soon offer itself of determining them as far as is necessary. 



[To be continued.] 



2G2 



