86 Prof. L. Boltzmann on some Questions 



layer itself were at rest, but the N molecules had the velocity 



v --n/p' + C^ + B^-B^ + A 

 The time dt which each of the N molecules requires to pass 

 through the layer is d% : p. Since the number of impacts is 

 evidently dependent simply upon the relative motion, the 

 number of those of our N molecules which in the layer di; 

 come into collision is 



NKx/p + te + Ba-Bfl' + r*) • S 



This expression also gives the decrease of the number N in 

 the layer djj, and may therefore be denoted by — dS. If the 

 total velocity of the layers is very small in comparison with 

 the velocity of the molecular motion, B may be taken to be 

 very small', and the function / may be determined according 

 to Taylor's the orem. If, fu rther, we put / for /(vV + q 2 + r 2 ) 

 and /' for / {\/p* + q 2 + r 2 ), where the dash denotes the derived 

 function, we have 



N=N <? P *<&+¥+*; 



or, since B may be assumed to be very small, 



2p \/p 2 + q' 



If in this expression we put £=.?.', we obtain the number of 

 those of our N molecules winch reach the YZ-coordinate 

 plane without coming into collision. Since cadi of these 

 molecules has the velocity-component q + B.c in the direction 

 of the Y-axis, we find the total momentum estimated with 

 reference to the Y-axis carried through the unit surface 

 of the YZ-plane in unit time from right to left by multi- 

 plying by P(^+B.i-) and integrating with reference to .<■ 

 from to co , with reference to p from — x> to 0, but 

 with reference to q and r from -co to +co. Eere P is 

 the mass of a molecule. Consequently, again neglecting B 2 , 



N _ N r, yBRto-gy -I 



/' • 



Bx 2 q 2 J 

 2Px/7+q 



('^-l^k?)-<-"™">-* 



