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



yirJk 3v ir^jumn 



where & 2 = n log V (m + 1) log — and k is to be taken 



positively, when, as here supposed, w falls short of ^. 



Take for example ^=1000, m = 700, and therefore w = 300, 

 the divergence 200 far exceeding the regulation limit, which 

 is the order \/500. By the first approximation we have 

 t 2 = 80, and for the qucesitum, 



1 _ 80 r 1 1 i m 



A/V 6 L2a/80 + v / 2000J ; 



that is, about 10~ 36 , or (writing '0 X for x ciphers following a 

 decimal point) *0 36 7. By the second approximation I find 

 *0 37 6. And by another method which will be explained * 

 below I find for inferior and superior limits of the qusesitum 

 •0 37 5 and «0 37 67. 



In proving this conclusion we have had the advantage of 

 knowing explicitly the probability of any degree of divergence. 

 It is instructive to consider how we should have proceeded 

 without this start. We should then have had to resort to 

 methods of approximation analogous to those which are 

 applied to the general theory of errors of observation. The 

 following is, I think, the analogue of Laplace's procedure. 

 Consider the expression 



^ cos x 20 (fM even) . 



If it is integrated between the limits it and 0, all the terms 

 other than the xtih from the centre vanish. Hence we have 



1 C" 



- I (cos &Y cos 20x dO 



for the probability of the particular divergence x. By the 



ordinary reasoning this expression is to be reduced to the 



1 J^. 

 form — - . — e /* , which is to be summed between limits x 



V ^2 

 and go ; the process of summation, 2, as contrasted with the 

 usual f, introducing a term outside the sign of integration. 



There is another form perhaps better suited to the present 

 purpose, the investigation of the limits of correctness. This 



* See p. 321. 



