520 Prof. F. Y. Edgeworth on the Law 



but the second forms a sum of squares ; the particular values 

 of which sum are grouped around its average value according 

 to the law of error. 



The genuine exceptions arise when the compound function 

 cannot be reduced to a number of independently fluctuating 

 elements : for example, 



F(#i, x 2 , &c.) = (x 1 —x 1 )x (# 2 — x 2 ) x &c. x (#„— x„), 



a continued product of residuals ; or a continued fraction 

 such as 



1-4-1 + l-r-(x 1 -^ 1 ) + 1-4-1 + (xx— a?,) + &c. 



The simplest class of exceptions seems to be that which has 

 been already instanced * ; where the compound function 

 involves the elements only implicitly, through a linear function 

 of those variables ; and the first differential becomes zero or 

 infinite for the average value of the linear function. The 

 function is thus of the form F(X + £), where £ fluctuates 

 in conformity with a probability- curve about the average value 

 zero, and F'(X) is either zero or infinite. 



This class may be subdivided according as one or other of 

 the last-named conditions is fulfilled. If F'(X) is zero, X is 

 a point of maximum, or at least infinite, frequency ; if F'(X) 

 is infinite, X is a point of zero frequency. This is seen by 



P uttin £ t*=F(X + £), f=F^( W )-X. 



Then the law of frequency for u is 



Error-function [F-. l (u)~\ x ^F_i(w). 



But d ^ L\ - 1 



ahi*- l{u) -wjx+$y 



Whence the proposition. 



Simple examples are (x— X) 2 and s/x — X ; which take 

 the forms f 2 and V| respectively, when for x we put X + f . 

 The law of frequency for «=£ 2 is f, as already stated, 



11-^ 



The case of w = f is analogous. Fig. 2 is designed to re- 

 present this type. When u= Vf, it should seem that only 

 half the original curve is reproduced in the transformation; 

 the other half becoming impossible. The transformed curve is 



\7TC 



* Ante, p. 436. f Ibid. 



