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



is constant for each individual kind of gas*. In accordance 

 herewith, we obtain for the vis viva of the ^NlV/xd//, mole- 

 cules which traverse our plane the expression 

 lkmmV 3 fjLdfM. 



§ 15. In order now so to transform the expressions arrived at 

 in the last three sections that they may also remain applicable 

 when the velocities of the individual molecules are not equal, we 

 only require to substitute for the values V, V 2 , and V 3 the mean 

 values V, V 2 ,, and V 3 . In order further to extend the expression, 

 which at present has reference only to an infinitely small interval 

 of the cosine /a, to all the molecules which traverse the plane, we 

 must also integrate it from fi=— 1 to //,= +!. In addition, we 

 will, for the sake of uniformity, multiply the first expression also 

 by m, so that, instead of the number, it may denote the mass of 

 the molecules. If, then, for the sake of shortness, we introduce 

 the following signs, E the mass, F the positive momentum, and 

 G the vis viva, which pass in the positive direction through the 

 superficial unit of our plane in a unit of time, we obtain the fol- 

 lowing equations : — 



1 f +1 — 



1 C +1 



4 



1 f + 1 _ 



-mJcmNS lY 3 fid/i. 



m 



The last of these three quantities represents the conduction of 

 heat which occurs in the gas ; it is therefore the determination 

 of this quantity which will be principally treated of in what fol- 

 lows. The two other quantities must also be taken into consi- 

 deration, because, as we shall immediately see, they aid in the 

 determination of this one. 



By substituting in the three equations the series given in (III.) 

 and (IV.) for V, V 2 , V 3 , and I, and performing the integration, 

 we get 



E = imN(g + ^)e + Xe 3 , 



E^mN^ + X^ 2 , I . . . (VI.) 



G = ^mNw 2 (3^ + V)e + X 2 e 3 .^ 



as exhibit no irregularity in regard to their volume, and for compound gases 

 which are formed without condensation, the value of this factor is approxi- 

 mately q-631 r =l'584. For gases whose formation is attended with dimi- 

 nution of volume it is larger. 



