[ Boa ] 



LXXI. On the Conduction of Heat by Gases. Bi/Yx. Clausius. 

 [Concluded from p. 435.] 



IV. Behaviour of the Molecules which traverse a given Plane in 

 a Unit of Time. 



§ 13. "VX^E will direct our attention to any plane situated 

 ▼ ▼ perpendicularly to the axis of x, and to the mole- 

 cules which traverse this plane. Let us take, for instance, the 

 plane whose abscissa is x, and which is therefore the first limit- 

 ing plane of the infinitely thin stratum that we have been con- 

 sidering in § § 8 ei seq. ; we can then draw, from the behaviour 

 of the molecules existing simultaneously in the stratum, definite 

 conclusions as to the behaviour of those which traverse our plane 

 enuring a given time. 



Let us suppose a parr of the plane, equal in size to a unit of 

 surface, to be divided off from the rest. The cubic capacity of 

 the portion of the stratum corresponding to this extent of surface 

 will then be represented by dx, if dx is the thickness of the 

 stratum; and we will denote the number of molecules which 

 exist simultaneously in this space by ~Sdx, where X is a very 

 large number dependent upon the density of the gas at the place 

 in question. These Nife molecules move in all possible direc- 

 tions, and the number of them whose cosine lies between p and 

 fjL + dfjbis, according to § 12, the fraction \\dp of the entire num- 

 ber, and is therefore perfectly represented by the product, 



^yicixdu. 



In order from this expression, which refers to the molecules 

 simultaneously existing in the stratum, to deduce the number of 

 molecules which traverse the stratum in a unit of time, and 

 which therefore must also traverse the plane in question, we 

 must take into consideration the time which each molecule 

 requires in order to traverse the stratum from one limiting 

 plane to the other. For a molecule with the cosine fi, the 

 distance to be traversed from one plane to the other is, disre- 

 garding its sign, equal to — ; and the time required to tra- 



dx 

 verse this distance is equal to — r--, if V denotes the velocity. T\ e 



will assume provisionally that all molecules whose cosine lies 

 between fi and p+d/i have the same velocity, and therefore 

 require the same time for traversing the stratum ; the number 

 of molecules which exist simultaneously in the stratum will then 

 bear the same proportion to the number which traverse the 

 stratum in a unit of time as this small space of time bears to a 



