64 
Choice in the Distribution of Observations 
N 
As e > 0 we have hence for any ix^ and /xg , remembering that being a squared 
standard deviation multiplied by the number of observations is ^ 1, 
N-S N 8 
that is, for either = 1 or — 1, 
Fs>Fo. 
We have thus proved that the maxima of are below those of . 
(5) Our problem is hence reduced to finding the best curve among those repre- 
sented by 
'^l = t ^4"^^ ^ t'^"^ {/^2/-4 + (/^4 - 3/.5) + (78). 
As was stated in (2) of this section we get all sets of possible values for fx^ 
and ^4 from three groups of observations symmetrical about x = 0, and we may 
therefore limit our search of the best distribution to these. 
Let the observations be N at x = ±v, at {1 — y) N at x = 0. The inter- 
polation formula of Lagrange gives, when represents the mean of the observations 
at a; = ^, 
3-2 _ ^2 _ X{X — V).^ X{X + V) _ 
from which we find 
+ 
.(79). 
iV ■ ( 1 — y y 
It is obvious that if for a certain distribution we have 
we can get a better distribution by taking more observations at 0. If on the other 
hand 
x=u x-=l 
CT^ < cr; , 
x-=\ 
the curve cannot be the best unless a], is a minimum for the present values of v 
and y. From (79) we find 
and 
da'y 
dy 
x-=l 
ct2 1 f(l -^;2)2 (1 + t;2) (1 + 
, 1 N-vHii-y) 
ct2 1 j 2 (1 - _ ( 2 + - av*) (1 + av^) 
.(80) 
1-y 
y 
from which we obtain the conditions for maximum or minimum 
1 ± 2Va 
and 
a 
y_ ^ 2\/a (l ±Vaf 
1-y a T 2 Va - 1 
