KiKSTiNE Smith 77 
therefore the same for a function of the second degree as for a function of the 
first degree. That is, when a > 1, a;,, is a minimiDn and equal to . ia fur two 
equally big groups of ohservalions at x = ± ^ , or for any distribution with the same 
/U.2 and ft.4, and when a ^ 1, a';;, is a minimum, and equal to^ {1 + a)'- for two equallij 
big groups of observations at x = ± 1. 
With the variates y and v (98) talves the form 
1 
(1 + 2ayv~ + a^yv' 
By differentiating with regard to we get 
dv^ iVy(l-y)v«^ '''^ 
which is negative for any a, v and y within our limits. 
For constant y, a'f, , is therefore least when v'^ = 1 and the minimum value is 
<=^(^ + 2a + a^)l^ (99). 
This is again a minimum when 
dy'-N-yHl^^^'^^y'+'y-'^ = '^' 
that is for y = — — which gives a minimum both for positive and negative a. 
-J ~{~ (X 
Thus the distribution that makes a';,, a minimum has a 0 (x)-distribution 
^ , J^- , observations at — 1 and 1 and \ 
2 (2 + a) 2 + a 
1 
consisting of ^ 7;-,--- , observations at — 1 and 1 and ,~ N observations at 0. 
We have 
2 + a 
and ^ = (1 -f- a). 
The relation ifj (x) = kc/) {x)f{x) 
N 
then gives us ijj (0) 
2 
a 
and '^(±1) = 2T2T^-^- 
From (99) we find the minimum value 
(^'L = ]y- + 
Our result is thus that &f-,^ is a minimum and equal to (2 + a)- for a distribution 
consisting of - ^ observations at 0 and 7^4^— N at ± I. 
^ + a Z (Z + a) 
