KiRSTiNE Smith 
29 
(2) Since ^af, = „-icr';, + S„ the curve for ^al is entirely above the „-ict;, curve 
except where /S„ = 0. 
Solving the equations *S„ = 0 the following roots are found : 
For ^1 = 0 X = 0 
„ 8^ = 0 x = ±V^.= + -5773 
„ ^3 = 0 x = 0 x = ±V^= ± -7746 
„ / 15 ± 2V36 f-8611 
S, = 0 x=±x/ 35 = ± |.34^o 
on A / 35 ± 2V7O f-OOSO 
63 (•5438 
p2386 
„ = 0 x = ± ] -6612 
i-9325 
Since all the roots are rational and all lie between — 1 and + 1, „a- therefore 
equals ^^jct';; for >i values of x all of which are inside the range of the observations. 
The adjusted values of the functions at these abscissae appear to be of special 
interest since they are uncorrelated as was shown in Section I under (9). 
(3) Looking at Diagram 2, representing the curves of „ay up to « = 6, it is seen, 
as was also clear from the formula for al and a'i given in the last section, that 
while the standard deviation in the middle of the range increases slowly with the 
degree of function it increases very rapidly at the ends of the range. At a; = 0 
the curve has a minimum when the degree of function is odd and a maximum when 
it is even. Besides that the curve has (2n — 2) maxima and minima between 
— 1 and 1. As the curve for n<^l is of the 2Hth degree, ^a;, is therefore increasing 
for X increasing above 1 or for x decreasing below — 1 . 
The abscissae of the maxima and minima are given in the following table. 
Degree of 
function Abscissae of maxima Abscissae of minima 
1 0 
2 0 ±a/i=±-4472 
3 ±V\=±-U72 
0 
±Vf=± -6547 
j _ 0 ^ / l ±2\/7 _ |-7651 
I ± a/|= ± -6547 ^ 21 ~ (•2852 
0 
21 ^(-2852 i^V 33 -^(-4689 
0 
8718 
/15±2V15 (•8302 i'Z]^^ 
