Prof. Clausius on the Conduction of Heat by Gases. 513 



unit of time; we must therefore divide the former number by 



the small time in order to obtain the latter. 



To apply this to the case before us, we must divide the mag- 



dx 

 nitude ^Nldxd/A by — rr, and wo thus obtain for the number of 



molecules which traverse our unit of surface during a unit of time, 

 in such directions that their cosines lie between p, and ^ 4- dp } the 

 expression tmVpdfi. 



It must, however, be further remarked that the difference of 

 sign in this expression, resulting from the circumstance that the 

 cosine ft may be either positive or negative, corresponds to an 

 essential difference in the manner of traversing the stratum. If 

 fi is positive, the molecules pass through the plane from the 

 negative to the positive side ; if p is negative, they traverse in 

 the contrary direction. 



§ 14. Before extending the expression just arrived at, which 

 refers only to an infinitely small interval of the cosine /x, and pre- 

 supposes equal velocities, we will first deduce two other corre- 

 sponding expressions, 



If we denote by m the mass of a molecule moving with the 

 velocity V, its momentum is mV, and the product mVp repre- 

 sents that component of the momentum which falls in the direc- 

 tion of x, so that a positive value of this product corresponds to 

 the case in which the component falls in the direction of positive x. 

 We will accordingly call the product shortly the positive momen- 

 tum of the molecule. Hence the collective positive momentum 

 of the above mentioned ^NlV/wf/ii molecules which traverse our 

 plane will be represented by 



|mNIVV^. 

 Farther, the vis viva of a molecule whose mass is m } and whose 

 velocity is V, will be represented by ±mV 2 . If, in addition to 

 the motion of translation with the velocity V, the constituents of 

 the molecule have also a rotatory or a vibratory motion, the col- 

 lective vis viva exceeds that product. I have spoken of these 

 additional motions, which may occur independently of the motion 

 of translation, in my memoir " On the kind of Motion which 

 we call Heat" (Phil. Mag. August 1857, p. 108), and have 

 pointed out that, for a given kind of molecules, a constant rela- 

 tion must on an average prevail between the various simulta- 

 neously occurring motions, in such sort that the ins viva of the 

 motion of translation forms a constant aliquot part of the total 

 vis viva. We will accordingly denote the mean value of the total 

 vis viva of a molecule by }jkmV~ 2 , where k is a factor whose value 



* In my earlier paper, quoted in the text, I have shown how this factor 

 may be calculated by aid of the two specific heats. For such simple gases 



Phil Mag. S. 4. No. 157. Suppl. Vol. 23. 2 M 



