262 Prof. F. Y. Edge worth on ike Application of 



say (24) Ult—p't—p ; where I is negative when p't<pt. 

 One more equation connecting R with the P's and p's is 

 obtained from the principle that the relative velocity of the 

 two points of impact is the same (in absolute magnitude) 

 before and after collision : say (25) W — to = w' — W', both 

 sides positive. Now W = L^ + L 2 Q 2 + ... + L wl Q TO = 

 L 1 (B 11 P 1 + B ia P s +...) + L,(B a iP 1 + B„P s +. ..) + ... ; when 

 for each of the Q's is substituted its value in terms of 

 the p's *. And W is an analogous linear function of the 

 dashed P's. Rearranging and adding, we have W+W' = 

 (P 1 + P' 1 )(L 1 B 11 + L 2 B 12 +...) + (P 2 + P / 2 )(L 1 B 21 + L 2 B 22 + ...). 

 Likewise (iv + w') is expressible in terms of p's and b's. 

 But by (25) {W + W)-(w + w') = Q. And by (23) and (24) 

 P't = Pf— • RLf, p ! t=pt + IW«. Whence R is found as a linear 

 function of the P's and p's. Accordingly P'* v what P t becomes 

 through the collision, is a linear function of the P's and p's. 

 Likewise P"* (the result of a fresh collision) is a linear func- 

 tion of those P's and p's, and of fresh p's themselves | the 

 result of numerous previous collisions. There is evidenced 

 that plurality of components which, the composition being- 

 linear, tends to generate the law of error. 



But a scruple may be raised with respect to the condition 

 of independence. That condition is threatened by the cir- 

 cumstance that the coefficients of the P's in the expressions 

 for P 1? P 2 ... (and likewise the coefficients of the p's in 

 p 1? p 2 ...) are not, as usual in the statement of the law 

 of error, constants, but functions of the co-ordinates which 

 vary between successive collisions. It is not necessary, how- 

 ever, for the genesis of the normal law that the " constants " 

 in the linear function should be rigorously constant. It 

 suffices that they should be random specimens of a medley if , 

 as the coefficients here may be regarded. 



The distribution resulting from this jumble will be in a 

 form which may be written Z = Jexp iiT/[T] ; where J is 

 a constant securing that the integral of Z between extreme 

 limits is unity ; X T is a quadratic function of the P's and p's 

 with numerical coefficients § ; [T] is the mean energy of 

 all the molecules (in a large unit of area). To show the 

 relation of X T to T, the expression in terms of P's and p's for 

 the mean energy of molecules having the same P's and p's but 



* Cp. Watson and Burbnry, ' Generalised Co-ordinates,' § 7. 



t Cp. note to p. 251 above. 



+ "Law of Error," Camb. Phil. Trans. 1906, p. 128. 



§ As in the simpler cases, Cartesian (x and y) co-ordinates do not 

 appear in the expression for P x , P 2 , etc., nor in the expression for the 

 energy. 



