46L> MATHEMATICAL APPENDICES 55* 



is determined by the magnitude k. Further, the number of mole- 

 cules per unit volume N must also vary from place to place, 

 because the warmer parts of the gas are more expanded than the 

 colder. And, finally, the magnitude B, or the collision-frequency, 

 is also variable with the place. 



As regards the last, magnitude, I shall again neglect its 

 variableness with place, as in our former investigations of viscosity 

 and diffusion ; for the approximation will be permissible that in 

 the short length of a molecule's free path a constant mean value 

 of B may be substituted for what is in reality a variable value. 



The two other magnitudes N and k must, however, be both 

 treated as variable. By reason of their continuity we can use 

 Taylor's theorem to find from their values at a point (x, y, z) 

 their values at any neighbouring point. This last, referred to the 

 former, we denote by the relative coordinates r, s, <f>. Since only 

 very small values of the distance r come into consideration, by 

 reason of the smallness of the molecular free paths, we can limit 

 the expansions by Taylor's theorem to the first two terms, 

 more especially as the functions are, on our assumption, to vary 

 only very slowly. Assuming therefore that the heat-condition of 

 the medium varies only in the direction of the coordinate x, we 

 may put 



and N + 

 dx ^ dx 



instead of k and N, where both functions are to be taken with 

 their values at the point (x, y, z), and the upper or lower sign is 

 to be taken according as the position (r, s, <) lies nearer to 

 or further from the origin of the ^-coordinates than the point 

 (x, y, z}. 



If we further neglect the square of r, we then obtain for 

 the kinetic energy which passes through unit area in unit time at 

 the point (x, y, z) in the direction of x the amount 



Q = %Nm(km/iry\ 27r d(f) [*"iefe sin s cos s^ dr{ du V, 

 Jo Jo Jo Jo 



wherein N and k have the meanings last defined, and is 

 given by 



s w , Be e -,., [:{$+ (P- - *.>! } r cos 



the upper sign corresponding to a flow in the direction of in- 

 creasing x, and the lower to one in the direction of decreasing x. 



