310 Mr. F. Y. Edgeworth on the Law of Error 



of error which occur in rerum naturd may very generally be 

 regarded as Binomials. 



It suits our purpose to take as the qusesitum not as usual * 

 the probability of one of the events occurring within certain 

 limits on either side the greatest term, but the probability that 

 the divergence from the greatest term should exceed a given 

 limit in an assigned direction ; in short, one of the extremities 

 rather than the body of the curve or locus under consideration. 

 In this inquiry let us take Poisson f for our guide. And 

 according to his notation let us call the total number of 

 trials ft, the observed number of successes m, the number of 

 failures n. 



It is shown by Poisson that the probability of obtaining at 

 least m successes, at most n failures, may be represented by a 

 certain fraction, of which the numerator and denominator are 

 both integrals ; the subject of integration being the same for 

 both, but the limits different. The subject of integration is 

 of the form 



B.e- t2 (h ! + 27i"t + UV + &c), 



where h', h", &c. constitute a descending series, if, as usual, 

 fju, m, and n are large. In order that the quotient of the 

 integrals should converge, a further condition must be fulfilled 

 by the inferior limit of the numerator, which Poisson calls h. 

 This limit is found by substituting ^ for p and q in the fol- 

 lowing expression — 



ft 1°£ —? — 7~T\ + ( m + 1 ) log —7 — rr\> 



and extracting the square root. If we actually perform the 

 work of integration and division, we shall find that the result 

 consists of two portions, one under the sign of integration 

 which does not explicitly contain ~k, and one outside the sign 

 of integration affected with k after this wise, 



where V, X", X" ! . . . are of the same order respectively as 



h' h", ft!" ••• ? that is —,-,—.. . In order that the above 



expression should converge it is necessary that the order - 

 should be fractional. ^ 



h 1 



If we have - of the order -, h of the order unitv, we shall 

 A* /* 



* E. g. Todhunter, art. 993. 



t Meclierches sur la probabilite, chap. 3. 



