414 MATHEMATICAL APPENDICES 29* 



Let us first of all investigate, not this value B, but the average 

 value T of the collision-frequencies of all possible particles. For 

 this we remark that the probability of the occurrence of the com- 

 ponents u, v, w is expressed by 



dw. 



We then obtain the probable average value of B by multiplying it 

 by v and then integrating with respect to u, v, w between the 

 limits oo and oo. The average value of the frequency of colli- 

 sion of any particle whatever is therefore, according to Max- 

 well's law, given by 



{00 fOQ fOO /-CO TOO fOO 



r = irs*N (km!*) 3 \ du\ dv\ dw \ dU \ dV\ dWre~* 



J CO "J 00 -CO J 00 'CO ^ CO 



where r and < are connected with the variables of integration by 



the relations 



r = V {(u - UY + (v- 7) 2 + (w - WY] 

 4 = km(u 2 + v 2 + iv 2 + U 2 + F 2 + TF 2 ). 



This sextuple integration assumes a much simpler form with the 

 substitutions 



U =n +u u=n -u 



for then 



r= 



and therefore T takes the form 

 T = 7r<r 



of a product of two triple integrals 



00 * 00 





which can be easily evaluated ; for P consists of three factors, of 

 which each is a simple integral of the form 



