Prof. Clausius on the Conduction of Heat by Gases. 525 



place equally in all directions, and then supposing a small addi- 

 tional component velocity in the direction of positive x, which we 

 represented by pe, to be imparted to all the molecules. It fol- 

 lows thence that, if M.dx denotes the number of molecules emitted 

 in a unit of time, their collective positive momentum will be 

 expressed by 



dxmMpe. 



Comparing this expression with that previously arrived at, we 

 have 



dxmMpe = jdxmMqe, 

 and hence 



P=k (49) 



Having obtained this result, let us return to equation (19), 

 which is as follows : — 



du 



and by means of the foregoing equation may be transformed into 



5 du , 



1=~i C dx < 50 ) 



The magnitude c which here occurs may be deduced from what 

 has gone before in the following manner. According to equa- 

 tion (15), 



1 



— =ce, 



"o 



where <x Q denotes the particular value possessed by a, in the case 

 of those molecules which move perpendicularly to the axis of x, — 

 a value which is obtained by putting 8 and cos rj equal to in 

 equation (XL), namely, 



N 



;° = S? 



This value, introduced into the foregoing equation, gives us 



N 

 Hence equation (50) becomes 



<=AT (51) 



,—5S»£* xii.) 



* If the calculations are worked out further than they have been above, 

 by taking account, that is, throughout of the next higher power of c, it will 

 be found that the expressions deduced above, for the number and momentum 

 of the molecules which impinge within the stratum, have such a degree of 

 accuracy that only a quantity of the order of e L ', as compared with unity, has 

 been disregarded throughout. 



