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



3. Analogy of the law-of-error in general statistics. — The 

 legitimacy of the argument is confirmed by its use outside 

 molecular dynamics to prove that the normal error-function 

 is the ultimate form approached by the continued super- 

 position of independently fluctuating statistics ; subject to 

 the condition that the mean square of the compound is 

 constant (/. c. p. 256 ; and cp. p. 269). The analogy throws 

 light on the implicated question whether there be an ultimate 

 form and its connexion with Liouville's theorem. The 

 existence of such a form is presumable on grounds of common 

 sense. But hesitation may be felt when we deal with an 

 unfamiliar case. Suppose MU to be the amount of money 

 possessed by a person of one class, and mu by one of another 

 class ; and that on a deal between them U becomes XT' and 

 w, ti , in virtue of the conditions 



MU 4 -fmM 4 = MU'Hm^ and MU + mw = MU' + mw' 

 (M and m constants). 



Let there occur an immense number of such transactions ; 

 while the sum-total of MU + wra, and likewise of MU 4 + ?nw 4 , 

 in the society remains constant. May we apply the second 

 argument and conclude that the money will be ultimately 

 distributed among the individuals according to a law of 

 frequency of which the logarithm is 



Constant — \(MU 4 + mi« 4 ) + /A(MU + ?nu) ? 



The answer is supplied by Liouville's theorem. If a stable 

 state is attained, the frequency-distribution for U and u must 

 be the same as for U' and u', the values into which U and u 

 pass by a transaction of the kind supposed. Now the 

 frequency-distribution of U' and u' (given functions of U 

 and u) is obtainable from that of U and w, viz. (hypothetical ly) 



AU Aw Const, exp — X(MU 4 + mi* 4 ) + /*(MU + mw), 



by substituting in the integral part of the above expression 

 for U and u their values in terms of U 7 and w', and for the 

 differential factor 



a tt a a tt; a ; fdTJ du dJJ du \ ,_, ON 



AUA«, AU'A^^-^^). . (13) 



The first substitution leaves the integral value unaltered. 

 Accordingly it is necessary to stability that the bracketed 

 factor of AIT Aw' in (13) should be identically equal to 

 unity : that is, that the theorem proved by Liouville and his 

 followers for a conservative dynamical system should hold 

 good. 



To apply this criterion to the proposed problem ; let us 



