58 
Choice in the Distribution of Observations 
1 [ lb (x) 
where w„ = ^ \ x'" 'ttt—c dx, ih (x) dx being the number of observations between 
*^ N J f{x) ' ^ ' ' ^ 
X and x + dx and the integration being extended over the range of observations. 
It is clear that if we have found a suitable curve of squared standard deviation 
for adjusted y by taking a distribution 0 {x) of observations with constant standard 
deviations a corresponding curve can be derived for observations with varying 
standard deviations by using the distribution 
,Pix) = mx).f{x) (65). 
As ^k<f> (x) .f{x) dx = N the constant A; must be 
N 
^(l){x).f{x) dx' 
Ix'P .(f> {x) dx N/jL^ 
Hence we find 
^^{x).fix)dx l<j>{x).f{x)dx' 
where p,^ is the ^^th moment coefficient for the distribution </> {x), and as 
k 
N 
fj-p I4){x) .f{x)dx 
for any p the determinant may be written 
1 
X 
X^ 
X^ 
1 
1 
■• 
.... [In 
X 
x"^ 
P-i ■■ 
■■■ H'n+2 
.(66). 
We thus find the same determinant as the distribution c/) (x) would give for 
observations with constant error of observation except that the factor k has come 
in, that is to say the expression for aj; has been multiplied by 
1 1 
k 
^ I (f' (■^) •/(»') dx 
.(67). 
The goodness of the distribution therefore will partly depend on the value of ^ , 
fC 
and because we have found (f) (x) the best distribution for observations with constant 
standard deviation it does not follow that 
x!j{x) = k<h{x).f{x) 
is the best distribution for observations with the standard deviation oVf (x). 
But the deriving of ip {x) from cf) (x) is nevertheless useful as a means of simplifying 
the investigations and will be applied in the following special inquiries. 
We shall consider two forms of / (x) and try to find the best distributions for 
functions of the first and of the second degree. 
