362 Mr. F. Y. Edgeworth on the 
the quantity of evil incurred by a simple error is not any 
power of the error, nor any definite function at all, but an 
almost arbitrary function, restricted only by the conditions 
that it should vanish when the independent variable, the error, 
vanishes, and continually increase with the increase of the 
error. The proper symbol for the disadvantage incurred in the 
long run is an integral whose elements involve as factors the 
said arbitrary function. Advancing, then, in the direction in- 
dicated by Laplace and Gauss, let us designate the disadvan- 
tage of a single error by the symbol F(V 2 ), where x is the 
amount of error and F continually increases with x. Or, if 
the detriment is not a symmetrical function of error, is not 
equal for the same extent of error, whether it be in excess or 
defect, put F(V) for the right-hand value of* as, and f(x) for 
the left-hand value of x ; x being taken as positive in both 
cases. Now suppose we have adopted some particular system 
of values for the constants y 1} <y 2 , &c Then, by the law of 
errors, if we make several sets of observations, say 
x x x. 2 x 3 &c, 
x{ x4 
x{ &c, 
x" X " 
x 3 /f &c, 
&c. 
&c. 
quantities 
yi#i 
+ y 2 %2 +73#3 
+ &C. -r- 
S7, 
7i#i' 
+ 72^2' + 73^3' 
&c. 
' + &C. -r- 
&C 
S7, 
1 ^ 
will be ranged under a probability-curve of the form*— r=. e «-, 
V7Ta 
where a is a known function of the sought quantities 7], <y 2 , 
&c. We have now to take a so that the total disadvantage in 
the long run of an indefinite number of sets of observations 
may be a minimum. This total disadvantage is 
r 00 1 
Jo ^ a 
l € -a*[F(x)+f(x)~}dx. 
V ma. 
Put.« = «f. The quantity which it is proposed to minimize 
becomes 
ri e -^[F(«|0+/(«£)>/|f; 
Jo V 73 " 
a. being regarded as variable, the first term of variation 
* Glaisher, op. cit. 
