THE LAW OE EEEOES OF OBSEEVATIONS. 
187 
hence 
N 
(x— m) 2 
nt — € 2 ( A -t)+0 3 
J + n + 2 (A — i) + 0* 
Now we may here assume d as small as we please'*, — that is, we may assume the Error 
(15) upon which the given system was superposed, to be of as small importance as we 
please. We conclude, then, rejecting this Error altogether, that a system of very small 
Errors, when combined, give for the resulting function of Error 
as before. 
y= 
V 2 Tr[h — i 
Q 2(h—i) 
the integral of which is 
u— Tc~ 2 
To determine the arbitrary function eft, we remark that if« = 0, u=e— kxt , 
••• ^(];)= /A 
fi-kx" 
hence 
i=fc-4^4a+^ k e * S ( 4o+ 0 =(1 + 4 aJc)~ 
kx 2 
1+4 ak * 
Another proof may he obtained by employing Poisson’s ingenious transformation (Traite de Mecanique, tom. ii. 
p. 356), which gives 
e aT> -(p(x): 
V 
=J. 
"d)(.r + 2u> Y a)da>. 
* In order that we may retain the three first terms only in the expansion 
V =/(«) - a/'OO + o., 
it is necessary to show that /'"(a?) and the succeeding differential coefficients are not infinite. Now they 
generally will be infinite in the ease where y=f(x) is an infinitesimal Error, as f(x) will be of the form 
wher e e is infinitesimal ; but in the case where 
y—f(x) — — (3 8 1 
we may take 9 as small as we please, and yet retain only the three first terms above, because the differential 
coefficients of y do not here become infinite ; in fact it is easy to see that any differential coefficient 
d n y 
— will consist of a series of terms of the form 
clx n 
C-e - * 1 : 
9 r 
now by the rules in the Differential Calculus for evaluating indeterminate forms, this quantity tends to zero 
as 9 diminishes. 
