Prof. Clausius on the Conduction of Heat by Gases. 523 



By help of these expressions it will now be easy to make the 

 calculations necessary for our purpose. 



§ 21. We will try to determine how many molecules strike 

 each other, with our infinitely thin stratum during a unit of time, 

 and how great is the collective momentum of these molecules. 



The probability that one molecule moving within the stratum 

 will meet another during the element of time dt, will be repre- 

 sented by adt, if we put for cos 77 and 8, in the expression for a, 

 the values corresponding to the direction and velocity of the 

 molecule in question. If, therefore, we want to determine the 

 number of molecules, out of a given large number of molecules, 

 which wilh strike eaeh other during the time dt, we only need to 

 multiply the whole number of molecules by adt, employing the 

 mean value of a, if it is not the same for all the molecules. Con- 

 sidering now the molecules simultaneously existing in a portion 

 of our stratum corresponding to a superficial unit, let us direct 

 our attention, in the first instance, to those whose cosine lies 

 between //, and fi + d[jL. The number of these is ^Nldfidx ; and 

 multiplying this expression by adt, where a denotes the mean 

 value of a for these molecules, the product ^Nladfidxdt repre- 

 sents, according to what has been said above, the number of 

 them which will strike each other during the time dt. Inte- 

 grating the last expression according to /j, from — 1 to +1, we 

 obtain the total number of molecules which will strike one another 

 within the stratum during the time dt. We now only require 

 to divide this expression by dt, in order to obtain the total 

 number of molecules which will strike one another within the 

 stratum during a unit of time. Calling this Dumber Mdx, we 

 have 



M = iNJ ladfju. . : \ . . (46) 



The quantity I, which occurs here, we already know to be 



= 1 — - fie. To obtain the value of a, we must put p for cos 77 



in equation (X.), and V— u for 8, since V denotes the velocity 

 of a molecule. But as all the molecules for which the cosine fi 

 has the same value have not the same velocity, we must make- 

 use of the mean value V in order to obtain the mean value a. 

 According to (III.), V = u + qfjbe + . . . , and, disregarding higher 



the motions of the molecules emitted from an infinitely thin stratum as 

 though the value of a. were the same for all and invariable. It accidentally 

 happens that the effect of this oversight is in the opposite direction to 

 that mentioned at § 5 ; so that the two partially compensate each other, at 

 least so far as regards the calculation of the conduction of heat. . - 



