314 Mr. F. Y. Edgeworth on the Laic of Error 



This conclusion may be confirmed by comparing the ex- 

 plicit expressions for the height of the respective loci at any 

 point. Take the common centre of the two loci as origin of 

 coordinates ; then the binomial locus consists of a set of rect- 

 angular columns standing each on a base of unit length, the 

 tallest at the centre. The general expression for the height 



of a column is 



IM 



+ x 



2 x 



2*' 



This is to be compared with the corresponding ordinate of 

 the probability-curve, or rather what that ordinate becomes 

 when for x we put a? + J • Otherwise, though the height of 

 a column is less than the height of the ordinate, we cannot be 

 sure that the curve has swept clear of the discontinuous locus. 

 In fact it will be found that the centre of the central column 

 (fj, even, as in the accompanying figure) is less than the cen- 

 tral ordinate of the curve, but that the curve strikes the 

 column between x = and &=£. And for a long way the 



curve continues thus to hug the discontinuous locus. Put t) 

 for the logarithm of the ordinate at the point x + i, diminished 

 by the logarithm of the height of column x ; and let x = 



(9) (*<!)• Expanding we have 



a quantity which, fju being large, is apt to become positive as 

 soon as a is less than J; that is after x has attained the value ( ^ J • 

 This conclusion may be completed by observing the course of 

 -j- and j\. It will be found that when /u, is large the latter 



OjX QjX 



very early becomes positive, and the former after x has 



