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



if an indefinitely large set of series each numbering n were 

 averaged — are np and nq. For any particular series the 

 chance that f x , representing the observed number o£ black 

 and white balls, should diverge from / by one compartment 

 having an excess and the other a defect of exactly r is 



n ! 



rfip+r n np-r (1\ 



(np + r)\ (np-r)\ P 9 ' " * # ' l> 



well known to be proportional to 



exp.— r 2 /2npq, approximately* (8) 



That expression is the " $> " in this case f. Accordingly, 

 13_ = r 2 /2npq. A more complex case arises when the drawing 

 is made from a medley containing balls of several colours. 

 The (approximate) probability that a particular set of 

 numbers (of balls of each colour) will be presented has 

 for exponent a quadratic expression, ^ 2 , of which Pro- 

 fessor Pearson has made splendid use J. This expression, 

 an alias of H, is used by the present writer to determine- 

 the best values of the constants pertaining to a function of 

 known form — given the values of strips of area (subtended 1 

 by a curve) between certain points on the axis. The most 

 probable values of the constants are those which minimise 

 the expression for ^ 2 obtained from the data §. A bolder 

 stroke of inverse probability is now required : to determine 

 not, as usual, the most probable constants || for a function of 



* See, e. g., Todhunter, ' History of Probabilities,' p. 548. 



t Note that <p itself is but an index of the quantity proper to measure 

 the probability of a deviation so large as r occurring in n trials, namely 

 (twice) the integral of <p between limits r and oo , corresponding to 

 the P in Pearson's Criterion of good fit. Accordingly, H may be 

 viewed as a positive index of 1 —P. 



\ To measure, by means of a proper integration (see preceding note), 

 the probability that a given distribution,/! (in our notation), is a fortuitous 

 specimen of the frequency distribution/'. See Phil. Mag , July 1900. 



§ 'Journal of the Roval Statistical Society,' vol. Ixxvii. (1913-14) 

 p. 724 et seqq., and vol. lxxx. (1916) pp. 471 & 485. 



|| The application of inverse probability to the determination of 

 constants has been discussed at length by the present writer in the 

 'Journal of the Royal Statistical Society ' for 1908. The analogy between 

 the present argument and the more familiar reasoning there employed is 

 brought out by regarding an undetermined (frequency-) function as a 

 determinate function with very numerous undetermined constants. 



The argument here summarized is believed to be in accordance with. 

 Boltzmann's reasoning about the " H " theorem ( Vorlesungen ilber Gas- 

 theorie and We.itere Bemerkimgen ilber . . . Warmetheorie, Sitzungs. Be- 

 richte der . . . Akademie der Wissensehaften ) vol. Ixxvii. : Vienna) ; with, 

 the " considerations of probability " from which, according to Jeans 

 (' Dynamical Theory of Gases,' ed. 2, p. 12), " the law of distribution 

 of velocities could have been predicted." 



