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



unmanageable, but it reduces to a few and ultimately to a single 

 term at the extremity of the compound curve. That term is 

 (mutatis mutandis) 



1 rn__ "I"- 1 



'a) n \n-l [_2 X ] > 



(2a) 



for the equation of the curve between the points #=5 and 



n 

 %=~—l, the origin being taken at the central point of the 



curve, and n supposed even. Let us compare now the height 

 of this portion of curve with the height at the same region of 

 the ancillary probability curve, of which the equation is 



1 x2 



Vn^a 



Put for x rfl, and the extreme ordinate becomes 



L e~§ n . 

 \/n%a 



The logarithm of the reciprocal of this quantity is of the 

 order n ; while the logarithm of the compared quantity's re- 

 ciprocal is of the order (n — 1) log (n — l)+n log (2a). 

 Neglecting the second of these terms, since 2a may be sup- 

 posed of the order unity, we see that the representative 

 probability-curve is well above the real facility-locus. And 

 this being true of the particular case under consideration, the 

 rectangle, is a fortiori true for all other finite symmetrical 

 facility- curves which fulfil the usual condition of descending 

 from the centre. Hence it is safe for extreme degrees of 

 divergence to use the tail of the probability-curve as the 

 measure of the probability in favour of mere chance. 



To investigate the accuracy of the approximative form for 

 moderate degrees of divergence, we must leave the explicit 

 expression, and have recourse to the analogues of the methods 

 employed at p. 313. If the elementary facility-curve be 

 yz=f(z 2 ) between limits ±b, then the general term of the 

 compound fulfils the condition 



J + b 

 f(z) u sx dx. 



And this condition (with the other requisites of a symmetrical* 



facility-curve) is approximately satisfied by the expression 



1 _^! 

 e «*; 



v 



ITS C 



* See my paper on u Observations and Statistics," Cambridge Philoso- 

 phical Society Transactions, 1885, p. 142. 



