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



cules which occur even when the temperature and density of the 

 gas is uniform throughout. 



Accordingly we must not attribute to the quantity u (which 

 occurs in the formula} for the motions of the molecules, and repre- 

 sents their velocity for the case where no variations of tempera- 

 ture and density occur) a fixed value applicable in the case of all 

 the molecules, but different values which vary in many ways from 

 one molecule to another. The same thing holds also for other 

 magnitudes which are dependent on the velocity of the mole- 

 cules, — e. g. for the length of excursion s, which we meet with in 

 §§ 8 et seq., and whose value must be on the average somewhat 

 greater in the case of molecules whose velocity is greater, than in 

 the case of those that have a less velocity. We have then now to 

 find mean values for these quantities, so far as they occur in the 

 formula?, which must be determined in such a way that by their 

 employment the values of the formulae remain the same as those 

 which would be obtained by taking into calculation the actual 

 velocity of each molecule. 



In order to be able rightly to calculate these mean values, we 

 must know the law which regulates the various velocities which 

 occur. As I have already stated above, such a law was esta- 

 blished by Maxwell, and it might perhaps be employed for 

 calculating the mean values*. 1 prefer, however, not to discuss 

 this subject here, as a few remarks concerning this law would be 

 required which would lead us too far at present ; and I feel the 

 more justified in leaving this point, since the numerical value of 

 e is so imperfectly known that the accurate numerical calculation 

 of a formula in which it occurs is not possible. I will therefore 

 content myself, in the calculation of the conduction of heat, with 



* I must here remark that this calculation would not be quite so simple 

 as might perhaps appear at first sight. For a point must be attended to 

 which has already been remarked upon in a similar connexion above, 

 namely, that the mean value of a power of u is not the same thing as the 

 corresponding power of the mean value of u ; and the same thing is true 

 for other quantities which depend upon u, or for products into which such 

 quantities enter. The consideration of the following series of expressions, 

 for example (a horizontal stroke being used, as before, to denote the mean 

 values), 



I**, {uf, au 2 + (\-a){u)\ -JT' &c " 



plainly shows that if all the values of u which occur in them were equal, 

 they would take the common form v? ; whereas if the value of u is not 

 everywhere the same, they are not equivalent to each other. If, therefore, 

 in any formula which is deduced upon the supposition of u having always 

 the same value, u 2 should occur, we cannot be at once certain which of the 

 mean values indicated above ought to be taken, but, in order to decide, 

 we must trace the whole development of the formula. 



