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



is equal to — ~^, Whence it follows that the modulus of the 



oc + fi 



representative probability-curve tends to be small when a : /3 

 is very small; and therefore that the range throughout which 

 the approximation holds is, cceteris paribus, less. Also for the 

 same range the degree of approximation is less. For in the 

 case of symmetry the equation above written is satisfied 



up to and inclusive of the term involving j- s by the solu- 

 tion of —~=±C-~. But in the case of asymmetry that term 

 does not vanish. Hence, even though on is within the regula- 

 tion limits, yet u Xs is accurate only up to the order — 7=, not 



1 ** s 



-. In order to obtain the latter degree of accuracy we should 



s 



have to add together the nth. terms on both sides of the 

 greatest term of the binomial, as is done in the books (e. g. 

 Mr. Todhunter). For in that case the third term of expan- 

 sion would vanish. 



But though apt to be incorrect, the received formula is in 

 the case of the binomial corrigible. With regard to the degree 

 of accuracy, it is necessary, in order to obtain the regulation 

 precision, to add to the argument of the probability-carve 

 the t* of Mr. Todhanter, the second f term of the expansion 

 of k ; and this even when the regulation extent of range is 



(where m has every integer value from to /u), 



~ l/x — 1 



= MPX2) == pm-lq^-m = ap, 



|m— 1 \fi — m 



Whence the first proposition. 

 Again, 



la 



21 - — J= pmqtK-m x m 2 



|m jh— m 



If*— 1 



=fipX ; %= pm-lgii-rh X m 



^| m— l |jn-m 



= u»V I— — pm-lqu-mx (w — 1)+M^2 , *=== pm-lqu-m 



\ m — 1 |/Lt— m | ft — 1 \ fi — n 



= ix(fx-l)p 2 +ixp = ny+np(l-p~) 



Whence the second proposition. 



* Or more exactly to Poisson's u, which is not quite the same as 

 Mr. Todhunter's r. 



t Poisson. 



