416 MATHEMATICAL APPENDICES 30* 



k must be taken the same for both kinds of gases, since they are at 

 the same temperature. 



The calculation is, however, no more difficult if we do not 

 introduce the last condition, but take k too as different for the 

 two gases. This generalisation is further not without advantage, 

 but will be of importance in the calculation of the conductivity 

 for heat. In a gas in which the temperature alters from place 

 to place layers of different temperatures come into contact. 

 The free path of a particle will therefore depend not only on 

 the temperature of the layer from which it proceeds, but also 

 on the temperature of the layers into which it enters. In 

 order to be able to apply our calculation to this process also, 

 which we shall investigate later on, we assume the values of the 

 magnitude k for the two kinds of gaseous particles to be different. 



Let ra t and m 2 be the molecular weights of the two sorts of 

 particles, u lt v } , and w l the velocity-components of a molecule 

 mi, and u%, V 2 , w% the velocity-components of a molecule m 2 ; also 

 let NI and N% denote the number of molecules of each kind 

 contained in unit volume, ki and & 2 the values of the constants 

 which determine the temperatures of the two gases, Qj and O 2 the 

 mean values of the molecular speeds which are given by the 



relations 



i), Oj ass 2/>/ (7r& 2 ra 2 ) ; 



finally, let s lt s 2 , an< 3- be the radii of the spheres of action or 

 the distances within which two molecules m l , two molecules w 2 , 

 or a molecule m l and a molecule ra 2 , approach each other during 

 collision. Then the mean number of collisions r x experienced by 

 a molecule m l in unit time, and the mean number F 2 for a 

 molecule w 2 have the values 



in which the first terms are formed in accordance with the 

 formula of 29* ; in the last terms, which represent in each 

 case the number of collisions with molecules of the other kind, y 

 is given by 



I TOO /-GO Ceo TOO TOO f 00 



y= (ir -1 Ar 1 m l . ir~ l ,#>,) du 1 I dv l I dw l I du 2 \ dv x \ dtr. 2 re 



J 00 J 00 J 00 J CO J CO 



where 



r = V ui % 2 + vi v 2 2 + w w 2 2 



