Probabilities to the Movement of Gas-Molecules. 257 



type, in the time t, will be compensated by the gain to class 

 U through collisions of the second type, it' 



A/(U-»)AUA«F(U) f(u) 



= &t(u'-U')&U'Au'F(V')f(u'). . (13) 



Of the factors on each side (U — u)= (u' — U ; ), the colliding 

 bodies being perfectly elastic; and AU'Au' is equatable to 

 ADA;/ independently of the variables U and u *, Accord- 

 ingly, the equation will be satisfied if 



F(U)/(») = F(\r )/(„'), .... (14) 



consistently with the conservation of energy. This func- 

 tional equation may be solved by putting cp = l g F, 



<D(U)+#» s<S>(U') +*("'), • • • (15) 

 subject to the condition 



MU 2 + «m 2 = M.U' 2 + mu' 2 . 



The condition supplies the solution f, viz. the left-hand 

 side of the equation multiplied by a constant, evidently 

 negative, say —h. That is, not taking account of the 

 conservation of momentum ; otherwise it is proper to add 

 a term of the form /.■(MU + mw). The constants may be 

 determined by well-known considerations. 



This reasoning has no claim to novelty. It is re-stated here 

 only in order to draw attention to two points. One is the 

 relation between this third argument and our first. Whereas 

 the third argument does not prove that the normal dis- 

 tribution will be set up in a molecular medley, but only 

 that if it is set up it will be maintained — this method of 

 approach has hitherto been completed by the " H " theorem 

 of Boltzmann. The error theory of Laplace is now suggested 

 as an alternative complement. 



Another point to which attention is called is the implied 

 use of that sort of Probability which has been called in 

 the i Philosophical Magazine ' " a priori" % and elsewhere 



* As may be shown by differentiating with respect to U' and u' the 

 expression for U (1) and that for u ; and more generally from the 

 consideration adduced below (34). 



t Cp. Jeans, 'Dynamical Theory of Gases/ p. 25, ed. 2. If (13) is 

 obtained by the reasoning employed in the first argument to determine 

 the constants for the normal function ((4) et seq.) the solution of (13) 

 avails to prove that the function is necessary. 



X See Phil. Mag. 1884, article by the present writer. The term 

 " a priori" is infelicitous so far as it is sometimes used with a different 

 connotation in the calculus of probabilities. As to the alternative 

 designations, see article on " Probability," Encyc. Brit. (ed. 2) p. 377 



