21* MAXWELL'S LAW 397 



and similarly with the others ; whence we see that the magnitudes 

 a, b, C must in general be constant in respect to u, v, w also. 



The integration of the equations is now easy to carry out, and 

 we obtain 



F = Ae ~ *+du dv dw du dv dw 

 where 



$ = ra{(w-a) 2 + (v /3) 2 + (w y) 2 } 



+ m{(u - a) 2 + (t> - b) 2 + (n> - c) 2 } + 20, 



and A denotes a constant which may be different for each kind of 

 atom. This function F may, as before, be broken up into the 

 product of several simple functions, for we may put 



of which the first three have the form 

 U(u)=Be- km 

 V(v) =j3 e - * 



where by B may be understood a constant which is the same for 

 all molecules and atoms, and the fourth is 



, t>, w) = S3e~* x 



where x = {( ~ <*) 2 + (*> &) 2 + (w c) 2 } 4- 20 



and S3 denotes a constant which may have a special value for each 

 kind of atoms. 



The three former functions have the same meaning as before in 

 14*, so that, for instance, U(u)du denotes the probability that the 

 atoms of a molecule move parallel to the #-axis with a velocity 

 between u and u + du. The determination of the constants can 

 therefore be carried out exactly as before ; in this case, too, we 

 have 



00 



duue' 



with similar equations for the components v and w. Hence follow 



B = (Tr-^m)* 

 a = a, fi = b, y = c. 



We obtain, therefore, exactly the same equations as before, 



