430 Prof. F. Y. Edgeworth on the 



t->^ 



such wise that the value assumed by one item does not affect 

 that of another. This is a case fully discussed by the classical 

 writers on Probabilities; under the guise of problems relating 

 to games of chance. Thus if we take several batches of balls, 

 every batch containing n balls, each of which is either black 

 or white, the probability of a white being p* ; then the fre- 

 quency of white balls is approximately represented by a curve 

 of error, of which the centre corresponds to np, the most pro- 

 bable number of white balls in a batch, and the modulus is 

 ^2np{l—p). I venture to refer to my reproduction, in a 

 former number of this Journal f, of Poisson's reasoning on 

 this problem, in order to recall a proposition which will be 

 required here : namely, that the limits, on either side of 

 up, up to which the approximation holds good are of the 

 order ^ n%. 



II. An easy transition brings us to a more general case in 

 which each item has any finite limits a r and a f (a greater than a) . 

 These limits need not be identical for each item ; provided that 

 the range of any one item, say a,. — ot n is small in comparison 

 with the sum of the ranges S(a r — « r ). Nor are we confined to 

 the supposition that each item should assume one or other of 

 two values a r and a,. ; it may assume any one of an indefinite 

 number of values indicated by the curve of distribution 

 y r =f r {x) } representing the frequency with which the rth 

 element assumes each value x ; where f t may have any form 

 whatever, continuous or not, provided that it does not extend 

 beyond a,, and a r , and that the integral between those limits 

 is unity. Moreover, it is allowable to affect each item with a 

 different factor or " weight"; provided that no weight is pre- 

 ponderant — large in comparison with the sum of all the other 

 weights. When we have thus substituted, for a " sum," a 

 linear function of independently varying items, we have 

 reached the extent of generalization to which Laplace thought 

 it necessary to carry the investigation for the purpose of 

 the theory of errors § . 



III. We euter on a less trodden path when, following the 

 lead of Mr. Glaisher||, we pass from a "linear" function of 

 items to any function whatever. To make the transition less 

 violent, let us break it up into two steps ; and first consider 



* n being a large number, and p not a very small fraction. 



t Phil. Mag. 1887, xxiv. p. 330. 



\ There is sonic approximation outside these limits ; but not of the 

 degree usually assigned. 



§ There is a good account of Laplace's analysis in Todhunter's ' 1 listory 

 of Probabilities,' art. 1001 et seqq. 



|| Memoirs of the Astronomical Society, vol. xl. p. 105. 



