KiRSTiNE Smith 
15 
which shows that we have succeeded in making al a maximum at x = 0 and 
obtained a standard deviation with the maximum vakie 3, as we desired. 
(4) For a function of the third degree we find from four groups of observations 
at — 1, 1, a and y that 
ix -l){x-a){x-y)^ {x + 1) {x - a) {x - y) 
y 
2(l + a)(l + y) 
2(l-a)(l-y) 
+ 
(x^ — 1) (x — y) 
I. -,,2 
and 
N 
. 4 
2(1 
1) (j: — a) {x 
2-l)(a-y) y--^\y^-l){y-a)''' 
l){x 
y) 
{x + l)(x ~ a) {x - y) 
I 2(l-a)(l-y) . 
i)(^-y)' 
The condition 
requires 
and 
requires 
from which is got 
and, since a 5 y, 
By introducing this vahie for and y- in o',, we find 
a2) (a - 
f^^' ) = 0 
3a2 - 2ay - 1 = 0, 
{lx)^^y ^ 
3y^ — 2ay — 1 
+ 
[x^ — 1) (x — a) 
(1 - 
0, 
= y^ 
{x^ - 1)'- (1 - 
I' 
4 J 1 _ 
iV 1 2* 
which has the required maxima at ± Vi • 
(5) For the functions of higher degree we shall at once assume that the dis- 
tributions sought are symmetrical, since it is pretty clear from the symmetry of 
y and with regard to the sought positions that it must be so. 
To determine a function of the fourth degree let us put groups of observations 
at ± 1, ± a and 0. The expression for can be written down at once and is such 
that the terms arising from the groups at + 1 and — 1 can be put together as well 
as the terms from + a and — a, then 
(^') ^ ^ provides the condition — ^ ^ 
with which value the squared standard deviation becomes 
^ 5n 
N ' 
5/72 
2* 
which has the required characteristics. 
