16 
Choice in the Distribution of Observations 
(6) Adjusting by a function of the fifth degree six equally big groups of obser- 
vations at the arguments ± 1, ± a and ± y the squared standard deviation of the 
adjusted y is 
. a2 (1 
;cr„= • 6 
1^ 
2 1 
+ 1) + 2 
1 
+ 2 
a (1 - a2) (a2 - y2) 
" (x2 - 1) {x^ - a2) ' 
_y(r-"yW"-T'")J 
The condition for maximum at x = ± a is 
9a* - 5a2y2 - 5a2 + y^ = 0, 
which together with the condition for maximum at cc = ± y 
9y« - 5a2y2 - 5y2 + = 0, 
since must be g y^ results in 
a2 + y2 = I and a^y^ = J- 
(x2 + y2) 
or 
7 ± 2V7 
21 
When these values are substituted in the expression above for a',j this may by 
somewhat lengthy algebraic operations be brought into the form 
^a; = . 6 |l - (X2 - a2)2 (x^ - y2)2 (1 _ x2)^ . 
(7) For a function of the sixth degree the observations may be supposed to 
be at ± 1, ± a, ± y and 0. 
The expression for the squared standard deviation of an adjusted y becomes 
N 
r.7 
{x^-a^) («2_y2)(^2_ 1) 
a2y2 
X (x2 - y2) (a;2 - 1) " 
_a2(a2^y2) (a2~^l) 
(x2 + a2) 
x(x2- a2) (x2- 1) 
' 2 Ly2(a2 - y2)(y2 _ 1) 
A maximum at x = ± a requires 
Ila*-7a2y2-7a2+3y2 = 0, 
and a maximum at x = ± y requires 
lly* - 7a2y2- 7y2 + 3a2 = 0, 
which added and subtracted provide 
11 (a2 + y2)2 _ 36a2y2 - 4 (a2 + y2) = 0 
and (a2 - y^) {1 1 {a^ + y2) - 10} = 0. 
Since we must have $ y2, 
a2 + y2 = and a2y2 = ^, 
'x (j;2 - a2) (x2 - y2)' 
L (a2 - 1) (y2 - Ifj 
(X2+ 1) . 
or 
a2) _ 15 ± 2V15 
y2) ^' 33 ■ 
The expression for may after rather laborious operations be brought into the 
form 
> y N ( 
33 ■ 7 ■ 112 
27 
X2 (^2 _ ci2)2 (;r2 _ ^2)2 (] _ ^.2^ 
