KiRSTiNE Smith 
3 
are practically found from the former by multiplying their frequencies by the squared 
standard deviations of the observations at the corresponding place. But in cases 
where extrapolation is of advantage, and the whole range therefore not to be used, 
the law of the frequencies has to be examined anew. 
In Section VIII we find for the same two cases of varying error of observa- 
tion the distributions which make each single constant of a function of the first 
and of the second degree a minimum. 
I. Adjustment of a polynomial function of one variable ; general distribution 
of observations. 
(1) Let y^, 2/2 yjy ?/ v be ^ observations of a function of nth degree 
taken at the points x^, Xj, x^, 
y ^ aQ + a^x + a^x'^ + + a„a:;" (1). 
Let us assume that from earlier experience we know the standard deviation of an 
observation of ?/ to be a V/ {x). The method of least squares will then give us the 
following system of normal equations in which the sums are to be extended over 
all the observations : 
S 
fix.). 
fix,)] 
y v^j) 
fix,). 
I X, ] 
fix,)] 
•2. ^ 
fix,). 
I x; 1 
fix,)] 
fix,)] 
X,. \ 
fix,)] 
tto + S 
a^ + S 
«'0 + S 
ao + S 
fix,] 
fiXt 
Xp 
fM 
fix,] 
ai + S 
-h S 
+ S 
+ S 
Xp 
fix,) 
:i 
Xp 
fix,) 
fix,) 
fix,)\ 
ffo + 
(1-2 + 
+ s 
+ S 
+ S 
Xp 
fix,) 
fix,) 
\fix,) 
, -ill 
I fix,) 
('r 
(2). 
If / (x) is 1 the sums are the moment coefficients of the places of observations 
multiplied by N, and in the general case we shall for brevity put 
x'' 
S 
fix,) 
N . m, . 
By eHmination of the «'s between (1) and (2) we find 
N.y 
fix,) 
y,x, 
fix,) 
y ,Xp 
fix,) 
n 
fix,) 
1 
m^ 
m„ 
X 
m^ 
m. 
X" 
m. 
X" 
m„ 
m.,. 
.(3), 
which determines the adjusted y corresponding to the variable x. 
1—2 
