Hr. F. Y. Edgeworth an the Lav: of Error. 301 
as a function of an indefinite number of elements, each element 
being subject to a determinate, although not in general the 
same, law of facility. Starting from this hypothesis, I attempt, 
first, to reach the usual conclusion by a path which, slightly 
diverging from the beaten road, may afford some interesting 
views ; secondly, to show that the exceptional cases in which 
that conclusion is not reached are more important than is 
commonly supposed. 
The first step is one taken by Mr. Glaisher*, enabling us 
to regard the compound error as a linear function of the inde- 
finitely numerous elements. A second step is, after Laplace, 
to express the sought function as a particular term of a cer- 
tain multiple. Let us suppose at first that the elemental 
facility-functions are all identical and symmetrical, involving 
only even powers of one and the same variable; say y=/(^ 2 ), 
where 
I 
mdz=i. 
Then the sought expression, the ordinate of the curve under 
which the values of the compound error are ranged, say u xs 
(where x is the extent of error, and s the number of elements), 
is the -th term of the multiple 
+ &c. +f(z)t + ^]\ 
x 
Observing the formation of the coefficient of fr* in the (s + l)th 
power of the expression within the brackets, we have 
P" 
«*+!,*= I /OK+*,/k; . . • • (1) 
when only even powers of z are involved ; for otherwise the 
above integral will have to be separated into three parts. 
Assuming such a tendency towards a limiting form that the 
effect of proceeding from the sth to the (s + l)th power is 
indefinitely small (for that part of the result, those values of 
x with which we are concerned), we may write the left-hand 
member of (1), u + -yr- The right-hand member may be ex- 
* Monthly Notices Eoy. Aitron. Soc vol. xl. 
