KiRSTiNE Smith 
65 
The lower sign requires 3 — 2 a/2 < a < and the upper sign a > 3 + 2 ^2 to 
make 0 < 1. The case a < J has no interest, as we have seen that when a < | 
extrapolation is not even for a linear function advantageous. We have therefore 
seen that for a< 3 + 2^2* af, has no minimum and we have thus proved that 
^=0 X'-=l 
the best distribution requires ol = a'-,,, that is 
2v2- 1_ (1 + w2) (1 + av2)2 
,-'-^l + '^+Vt°"' g (81). 
The maximum of the curve is ■ 
^=<'_ ct2 1 
To find the minimum of this value we differentiate (81) and get 
1 + a.tP' 
(2^2 _ 1)2 {4a?;« - a'<;2 - 2a - 3}, 
which is zero for 
48\ 
33 + ~ ) (82) 
a 
and positive for greater v^, so that we have found a minimum. 
For a = 3 we find from (82) f ^ — 1, hence for a ^ 3 we have to choose = \, 
from which, according to (81), follows 
1 1 X9 2(l + a)2 
= 1 + 2 (1 + a)2 or y ~ ^ ' 
1-y ' ^ ' ^ ' l + 2(l + a)^- 
When 3 + 2V2>a>3, v^= \ f 1 + a / 33 + — ^ is < 1, and for the corresponding 
8 V ^ a / 
y we have 
h 
{l+avY l + 5a + 4 + \/a (33a + 48) 
~ ^ (8<3). 
1-y 2(2^2-1) 8 (a + 2) 
Returning to the ^ {x) distribution, which is found from this distribution by 
dividing the frequencies by l: . (1 + ax^Y, we therefore find, when ^ iV is the number 
Li 
of observations at a; = ± f and (1 — e) iV, that, at a; = 0, 
e 
2 1 + 
1-e 2(2i)2-l) 
_x--\ 
* A further examination shows tliat for a>3 + 2v'2 (t^^ has a minimum but this is smaller than 
' x=0 a;- = l .r = ii 
ff';^ when a<6-7. Up to this value we therefore have a'y = u'^ for the best eurves. For a>6-7 the 
oc' = \ 
minimum of tr^, determines the best distribution. 
Biometrika xn ■ 5 
