94 Prof. F. Y. Edgeworth on the 



small) ; and observing that 



7(?)?^=o 



i 



b 



(since f is measured from the centre of gravity of the corre- 

 sponding curve), 



J b Jb 



we have 



This expression ought to be identical with 



Here A& and A; may be regarded as independent observa- 

 tions ; it is therefore proper to equate the coefficients of Ak in 

 the two expressions for y + A?/ ; and similarly the coefficients 

 of A/. Thus we obtain two additional partial differential 



* The reasoning requires that the expansion of y+Ay should form a 

 descending series. This condition is fulfilled by our solution. For let 



the mean square of error for each element be of the order - , then k, the 



n 

 sum of these mean squares, will be of the order unity. Accordingly if y, 



as proposed, = J/m. e ~M approximately, then 



dx 2 ' y k k' 2 ' 



^M —v= ?£ _ £! 

 dx* ' y k 2 k 3 ' 



These and higher differentials may be regarded as being of the order 

 unity for values of x between limits +1. Whence it follows that the 

 terms of the expansion in the text form a descending series; since 



( 11L ^§£, &c are of the order unity, and Ah, Aj. &c, being integrals of 

 dx 2 dx 2 



£ 2 /(£), £ 3 /(£), &c. between limits separated by a very small interval, will 

 in general form a descending series. The reasoning is not affected if we 

 change the unit : e. g. suppose the range of the elements to be of the 



order unity; in which case -£-? -£— , &c. will form a descending series, 



dx 2 dx s 

 while Ak, A/, &c. will be of the same order. Also the order of 



< zM -z- y, J*M -~ y. &c. is not affected by taking into account the second 

 dx 2 ' U dx z J ' . • . 



term of approximation to the value of y given in the text ; it being 

 observed thaty-rc 3 is small, and that the formula only professes to be 

 applicable for values of x which are of the same order as c. 



