Probabilities to the Movement of Gas-Molecules. 267 



collisions with {u 9 v, + £1...) is the same as the loss to class 

 (U', V', — Qi...) from collision between members of that 

 class and class («', c, — <)i ...) (w' reversed overtaking 

 W reversed), that is, the gain to the class of (U, V, — Qi . .), 

 Ave have — granted the postulate (31) — the condition 



AQi AQ 2 . . . Atfi A52 • • • AU A V AQ X AQ 2 . . . Au Av A'< n Aq 2 . . ., 

 W - it 1 F(Q l5 Q 2 . . . u , V, Q 1? Q 2 . . 0/fais &...«, ^ &, ? 2 . . .), 



.... (33) 



equal to the expression which is formed by retaining all the 

 symbols relating to co-ordinates and affecting all the velocities 

 with dashes. The equation may likewise be expressed in 

 terms of the components of momentum, say 



MU, MV, P l5 P 2 ... mu, ??ir, pi, p 2 .... 



This equation relates to the genus (or pair of genera) which 

 is defined by values of the co-ordinates between assigned 

 values of Q x and Qi + AQx, Q 2 and Q 2 -f AQ2 •-■ Qi and 

 q x + Ag 1 . . . ; which are the same on both sides of the equation, 

 unaltered by the collision. F and /, as before, denote fre- 

 quencies of classes, for the definition of which it is necessary 

 now to specify the coordinates as well as the velocities ; 



AUAVAQ 1 ...A^ArA^ 1 ... = AU'AY'AQ 1 '...Aw'A^AQ 1 '... 



.... (34); 



since each P (and so each Q) is a linear function of P's *, 

 and accordingly the right side of (34) = the left side multi- 

 plied by a constant —which constant cannot be other than 

 unity. Also ( W -ic)= w' -W. Thus (33) will be satisfied if 



F(Q 1 Q 2 ...UYQ l „.)J(q 1 ,q 2 ...u,v,q 1 ...) 



= F(Q 1 Q 2 ...U'V 1 'Q 1 ...)/( 7l , ?2 ...w',»', i/...); . (35) 



subject to the condition that T' the energy of the two 

 molecules after the collision equals T what it was before. 

 Whence, as before t, it is deduced that the required distri- 

 bution is of the form J exp — AT, where T is a quadratic 

 function either of the components of velocity or those of 

 momentum. For the co-ordinates it is convenient to take, 

 besides the velocity components of the centre of gravity, 



* As pointed out by Jeans, op. cit. p. 18 

 t Op. (15) and (22). 



T2 



