Prof. Clausius on the Conduction of Heat by Gases. 515 



The terms Xe 3 , X^ 2 , and X 2 e 3 , in which the factors X, X w and 

 X 2 denote functions of x which are left undetermined, are only 

 added in order to indicate of what degree are the terms which 

 would be obtained by carrying out the calculation still further. 

 It will be seen that in all three equations the second term is two 

 degrees higher than the first ; and if we content ourselves with: 

 such an approximation in our results that we neglect magnitudes 

 of the order e 2 in comparison with unity, which we may do 

 without hesitation, seeing that e is a very small quantity, we may 

 entirely disregard these additional indeterminate terms in the 

 developments which follow. r 



. On considering the degree of the first and important term, it 

 may perha'ps appear surprising that the magnitude F is of no 

 degree in respect of e, whereas E and G are of the first degree." 

 This, however, becomes intelligible when we remember that the 

 momentum behaves differently in regard to its sign from the 

 mass or vis viva. The momentum of a molecule which traverses 

 the plane in the negative direction is in itself negative ; but as 

 it must again receive a negative sign in consequence of its 

 passage in the negative direction, it thereby becomes positive 

 again ; so that the positive and negative passages are not in this 

 case, as in the other two, to be subtracted from, but added to, 

 each other. 



§ 16. In reference to the magnitudes E, F, and G, the assump- 

 tion that the gas is in a state of rest enables us to deduce at once 

 the following propositions : — 



1. The mass of gas which traverses the plane must be equal to 0. 

 For, since the whole quantity of gas is contained between two 

 fixed surfaces, if any gas passed through an intermediate plane 

 in either direction, the density must increase at one side of the 

 plane and diminish at the other, which would be in contradiction 

 of the presupposed conditions. 



2. The positive momentum which traverses our plane in a unit 

 of time must be independent of the situation of the plane, and there- 

 fore constant in regard to x. For if we suppose a stratum bounded 

 by any two parallel planes, the momentum which enters the stra- 

 tum through one plane must be equal to the momentum which 

 passes out of it through the other, for otherwise the momentum 

 present in the stratum must vary; this, however, would be 

 in contradiction of the stationary state which is one of our 

 conditions. 



3. The vis viva which traverses the plane in a unit of time must 

 be constant in regard to x, for the same reason as that given in 

 the case of the positive momentum. 



We can therefore establish the following three equations of 

 condition : — 



2M2 



