78 Choice in the Distribution of Observations 
(6) When the standard deviation of an observation is 
Sy = o (1 -\- ax) and 0^a<l, 
we have ~ = 1 + 2a/Xi + a^/X2, 
and according to (89) we find for & function of the first degree 
a;i„ = ^ (1 + 2a/xi + a'fi,) — (100) 
1 
and af„ = (1 + ^afx^ + a^^i^) , (101). 
By differentiating (100) we find 
and 
Both of these can only be zero when 
/Xi + a/X2 = 0 (102), 
which is seen to determine a minimum of ai the value of which is The 
iV 
condition jU-a = ~ '^^''^ he fulfilled hij an infi)utij of different distributions. From 
0 ? /X2 ^ 1 
follows the condition 0 ^ fi^ ^ — a. 
We shall confine our attention to those distributions which consist of two groups 
of observations. Let there be Ny observations at and (1 — y) at v^, we then 
have 
fJ-i = + y (vi - 
from which by means of (102) is found 
y ^ ^1 - y 1 
- ^2 (1 + aVj) (1 + aVj) {Vi - Vg) {1 + a {v^ + v^)} 
1 (l+ai^iHl + aVa) 
and = i + a/xi = — . 
Thus we find that the </> (a;)-distribution consists of 
- t'2(l + av^) 
(I'l -V2){l + a {i\ + Vo)} 
^>l (1 + avj) 
{vi - V2) {1 + a («i + D2)} 
while the actual distribution 
iV at i\ 
and z^r \ , / . , X, iV at ^2. 
0 (X) = r /J'^in^''^^ . (1 + axr4> (^) 
(1 + aWj) (1 + aVa) / f v / 
