51* DIFFUSION 457 



within a tolerably large region, throughout which, therefore, 

 Maxwell's law may be considered to hold. 



According to a formula which we have developed before, and 

 used several times, the element dS is reached by a number 



dS cos s sin s ds dr d 



of particles, which proceed in unit time with speed w from the 

 volume-element r 2 dr sin s ds d(f> expressed in polar coordinates 

 with dS as origin and the normal to dS as axis. N, m, and B are 

 here magnitudes which have different values for the two kinds of 

 gas, and must therefore be distinguished by subscripts 1 and 2. 

 N and B are also functions of the position ; but it will be 

 sufficient in the case of B, the collision-frequency, to assume a 

 mean constant value, and consequently to take into account only 

 with respect to N that we must employ that value of it which is 

 proper for the position of the volume-element r 2 dr sin s ds dty, and 

 which should be indicated by the argument x r cos s. Since 

 r is small, the function N with this argument may be put 



dN 



N(x r cos s) = N -- r cos s, 

 ax 



where the letter N without any argument denotes the value at the 

 position x. 



We are not concerned with the whole number of particles that 

 pass through dS, but only with the difference between the numbers 

 which pass from the right and from the left ; this difference does 

 not depend on the absolute value of N, but is conditioned only 

 by its variation. Hence, on introducing into the above formula 

 the expression we have developed for N, we neglect the first term 

 and investigate only the second 



^(km/rfBe-^e-^Wdu dS cos 2 s sin s ds rdr d$, 



dx 



which we have to integrate between and oo in respect to w and 

 r, over unit area as regards dS, from to JTT in respect to s, and 

 from to 2 * in respect to <j>. We thus obtain as expression for 

 the number which pass through unit area in the direction of 

 increasing x, in consequence of the unequal distribution, 



_ 



ON/TT ax 



For the number passing in the opposite direction the same 

 expression holds, but with changed sign. 



