30* MOLECULAR FREE PATHS 417 



To simplify this integral put, in like fashion as in the former 

 case, 



, = U + . l u , = U - . *'"*' u 



k l m l + & 2 ra 2 &,?/&! 



^=$8+1 ?-* 1) V 2 =%$-^ -T) 



,= SB + --*! ,, = i . 



fc l m l + tfiWa 

 Then 



r = x/ (u 2 + t) 2 + tt> 



= (fc lWl + fc 2 m 2 ) (U 2 



and by evaluation of the sextuple integral 



$foo poo poo poo poo poo 



y = (ir- 1 k l m l . rr- l t a m a ) \ du \ fa \ frfw <ill 4 



J 00 J 00 J 00 J 00 J 00 J 



we obtain 



y = 2x/ ( ^ffi- ^ 2 ~ 2> \ 

 V Tr^m^m^ J' 



or, on substitution of the mean values of the molecular speeds, 



Finally, then, the collision-frequencies of the particles of the two 

 kinds are given by 



r 2 = N 



These formulae, which, like that obtained before, were first 

 deduced by Maxwell, 1 allow of the simple interpretation that 

 the number of collisions of both kinds of particles together is just 

 as great as if the NI particles of the first kind were all moving 

 with the speed 12, in one direction, and the N 2 particles of the 

 second kind with the speed O 2 in a perpendicular direction. 2 



These formulae have been applied in 97 and 98 to the 

 theory of diffusion, and have also been taken into account in 104 

 in the investigation of heat-conductivity. 



1 Phil Mag. [4] xix. 1860, p. 27 ; Scient. Papers, 1890, i. p. 386. 



2 Stefan, Wiener Sitzungsberichte, Ixv. Abth. 2, 1872, p. 349. [This inter- 

 pretation does not apply to the first terms of the formulae. To include these 

 we may say that everything occurs as if the particles of the two kinds are all 

 moving with the speeds fl, and H 2 respectively, and that two colliding particles 

 always meet at right angles. TB.] 



E E 



