252 Prof. F. Y. Edgeworth on the Application of 



of the law in a less degree *. Altogether the N velocities 

 resulting from numerous collisions will not be distributed 

 so as to satisfy Prof. Pearson's criterion, if based on the 

 supposition that they purported to be N independent obser- 

 vations f . But they will form a more nearly normal group 

 than the \xv quite independent observations %. 



From the preceding verification it appears § that U and u 

 after numerous collisions will be distributed according to 

 the law of frequency ISz, where 



z= ^^exp.-(AU 2 + 6m 2 ) (3> 



7T 



The function z must fulfil the condition 



fj: 



{-M.TJ 2 + mu 2 )zdUdu = M[U 2 ]+ro(> 2 ] =2[T], (4) 



using square brackets to denote mean values. Put for A, 

 M/2[T]; and for a, wi/2[T]. These values satisfy the 

 equation M/2A + m/2a = 2[T], derived from (4) ; and also a 

 second linear equation derived from the condition that the 

 velocity-distribution of that set of molecules which comes 

 (once) into collision during the instant At should be the 

 same just after as before collision. The number of pairs 

 with velocities U and u before collision, viz. 



A* I U-wj AUAu exp -(AU 2 -f'aw 2 ), 



is the same as the number with TT' and u after. Whence 

 A\J 2 -\-au 2 = AU' 2 -\-au /2 (the other quantities on each side 

 being equal, see (13)). Giving to XT' and u' their values in 



* The imperfection would be of the kind described in Camb. Phil. 

 Trans, loc. cit. p. 127, under the head u Relaxation of the Third 

 Condition." 



t Cp. Camb. Phil. Trans, loc. cit. p. 127, and Enc. Brit. "Probability," 

 Art. 157. 



% For the purpose of this verification the finite difference AU (and 

 likewise Au), according to which the abscissa is graduated — being of the 

 order 1/ y«, where n is the number of perfectly independent elements, — 

 should here be much smaller than 1/ V /xv and much larger than 1/ VN. 

 (Cp. 'Journal of the Royal Statistical Society,' vol. lxxx. (1917- 

 p. 427.) 



§ It being assumed a priori {cp. below, § IK.) that the distributions 

 of U and u are independent (cp. Tait, Trans. Roy. Soc. Edinburgh, 

 vol. xxxiii. (1886) p. 68. Otherwise we may start with the more 

 general form , = j ex p. -(AtP-2CUw+Bw 2 ) ; (5) 



and find from the given conditions that C is null. 



