12* MAXWELL'S LAW 377 



the same value,, which is therefore a constant independent of n 

 and of u, v, w. 



13*. Another Method 



We may arrive at these formulas Jn__another, perhaps simpler, 

 way from the functional equation 



C = F(UI, v l} w l )F(u 2t v 2 , w z ) . . . F(u x , V N , W N ), 



by comparing together two states of the molecular system of which 

 the one immediately follows the other. A change of the state 

 occurs at every collision between two particles : we compare 

 therefore the state of the system before a collision between a/ty 

 two particles with its state immediately after the collision^/ Of 

 all the particles, then, only those Jwo which collided have changed 

 their_motipn. We may, then, in the product neglect the factors 

 that have remained unchanged, and thus conclude that 



F(u lt v } , wjFfa, v 2 , w 2 ) = F(U lt V lt WjFW, F 2 , TF 2 ), 



if u\ t v l} Wi and u 2) v 2 , tu 2 are the components of velocity of the 

 two particles before collision, and U it FJ, W l and U 2 , F 2 , TF 2 the 

 corresponding values after collision. 



We thus arrive at a form of functional equation to which other 

 methods of proof have also led. It first occurs in Maxwell' s 

 se&ond 1 proof, and then in the memoirs of Bolizmann. 2 

 L^ren_iz^ 3 and others. It occurs in these memoirs as expression 

 for the stability of Maxwell's state of distribution : the equation 

 may also be interpreted in such wise that the number of collisions 

 depending on the product F(UU v lt w\)F(u 2 , v 2 , w 2 ), in which the 

 components u, v, w are changed into U, F, TF, is exactly as great 

 as the number similarly determined by F(U lt V lt W } )F(U 2 , F 2 , TF 2 ) 

 in which the components u, v, w take the place of the values 

 U, F, TF. 



The functional equation is subject to the conditions 



= , 2 2 



tit + u 2 = U" } + U 2 

 Vl + V 2 = F! + F 2 

 Wl -H W. 2 = W l + W 2 



1 Phil Trans. 1866, p. 157 ; Scientific Papers, 1890, ii. p. 45. 



2 Wiener Sitzungsber. Iviii. 1868, p. 517 ; Ixvi. 1872, p. 275 ; xcvi. 1887, 

 p. 891 ; &c. 3 Ibid. xcv. 1887, p. 115. 



