360 
MR. R. A. FISHER ON THE MATHEMATICAL 
or, if we write, 
e ^ , 
we have the two conditions, 
and 
V ^j7T 
S ( — Z s -z s+1 ) = 0 
Ps 
a n s x s 
b — z s 
l Pm U 
x 
» I i \ 
Z 1 I 
"j + l / 
= 0. 
As a simple example we shall take the case chosen by Iv. Smith in her investigation of 
the variation of in the neighbourhood of the moment solution (12). 
Three hundred errors in right ascension are grouped in nine classes, positive and 
negative errors being thrown together as shown in the following table :— 
2-3 
53 
3-4 
24 
0"*1 arc 0-1 1-2 
Frequency . . 114 84 
The second moment, without correction, yields the value 
0-,, = 2-282542. 
Using Sheppard's correction, we have 
O-,. = 2-264214, 
while the value obtained by making a minimum is 
4-5 5-6 
14 6 
6-7 
3 
7-8 
1 
8-9 
1 
cr x 2 = 2 • 355860. 
If the latter value were accepted we should have to conclude that Sheppard’s correc¬ 
tion, even when it is small, and applied to normal data, might be altogether of the 
wrong magnitude, and even in the wrong direction. In order to obtain the optimum 
8L 
value of <t, we tabulate the values of in the region under consideration; this may 
cer 
be done without great labour if values of <x be chosen suitable for the direct application 
of the table of the probability integral (13, Table II.). We then have the following 
values :— 
1 
(X 
(>'43 
0'44 
0'45 
016 
0L 
err 
+ 15* 135 
+ 2149 
- H’098 
-24-605 ' 
A 2 — 
3(7 
— 0'261 
- 0'260 
