14* MAXWELL'S LAW 379 



oo and + . The equations, then, which we may write with- 

 out the index and in the slightly altered form 



du 



dv 



= <iM! + zkm(w - y), 

 dw 



are the differential equations which determine the function 

 F(u, v, w) that depends on the arguments u, v, w, and which 

 therefore determine the probability that a molecule moving with 

 the velocities u, v, w occurs in the group in question. 

 By integration of the equations we obtain 



F(u,v,w) = Ce- km { (u - a) ^ (v ~^- +(w -^} t 



where e is the base of Napierian logarithms and C a constant of 

 integration. Instead of the latter we may introduce another 

 constant A given by the formula 



C Adudvdw, 



We are entitled to make this change, because du, dv, dw possess 

 constant values as differentials of independently varying magni- 

 tudes ; considering further that the occurrence of a component of 

 velocity of perfectly definite magnitude for instance u, or, more 

 properly, a value lying between the limits u and u + du can 

 have a probability that is only infinitely small and of the order 

 du, we see that C must be an infinitely small magnitude of the 

 order du dv dw. We may therefore understand by A a constant 

 of finite magnitude when we put for the probability-function 



F(vi,v,w)=Ae- km l (u - a) * + (v - f3}2 + (w - r ^ dudvdw. 



/ This expression exhibits an important property of the 



'function, viz. that it may be broken up into three simpler 



functions, each of which depends on one, argument only ; for we 



have 



F(u,v,w) = U(u)V(v)W(w)du dv dw, 

 if U(u)=Be- km(u ~ a) \ 



V(v)=Be- l:m(v - 0) \ 

 W(w) = Be -***-*, 

 and B* = A. 



