KiRSTiNE Smith 
81 
(10) The general minimum conditions for a„,, cannot be found without more 
elaborate investigations into the possible variations of the moment coefficients 
than are at present available and we shall limit our research to the case of three 
groups of observations. 
Let us suppose yi N, N and (1 — yi — 72) ^ observations taken at x^, Xo 
and X3, and let the corresponding means be J/i, [/^ and y^- 
We then find, when 
A = (Xj — X2) {X.2 — X^) (Xg — Xi), 
and 
«2 = ^ {l/i {^3 - ^2) + ll'i (*i - ^z) + Vz (■^a - ^1)}, 
(72 r (X3 - x^Y {l + ax^Y (Xi - x^f ( 1 + ax.,f , (x^ - x^f ( 1 + ax^f 
A2 . iV ( yi ya l-y^-y^ 
(111). 
Differentiations first with regard to y^ and then with regard to y^ give the 
minimum conditions 
Ti 72 ^ (1 - 7] - 72)^ 
{x^ - x.^f {I + ax^f {x^- x-if {I + ax^f {x^- x^f {I + ax^f 
or, when we suppose < < , 
7i - 72 _ 1 - 7i - 72 1 
(Xg — ccj (1 + ciXi) (Xj — Xj) (1 + aXj) (Xo — ^Ci) (1 + aXg) 2 (Xg — (1 + aXg) 
(112). 
With these values for 71 and 72 we get from (111) 
CTM2(x3-a;i)(l + aa;2))2 aM 2 (1 + ax^) ^ 
N ( A J iV ! (x^ — a;i) (X3 - cca) 
This for constant x^ is obviously a minimum for Xi = — 1 and Xg = 1 and is then 
equal to 
(t2 (2 (1 + aX2))2 
From this we find 
which shows that 
determines a minimum. 
The minimum value is 
N \ 1 - £c; 
da^^ _ o 2 {axii + 2^2 + a) 
dx,^ ~ VN (1 - 4f 
3^9 A / 1 
a 
< = ^(l+Vl-a2)2, 
and the frequencies found from (112) are 
1 
Vl - a (Vl + a - VT^) .N at - 1, 
Biometrika xn 
