KiRSTiNE Smith 
19 
function, and thus a full investigation may require more groups of observations 
than merely a number equal to the assumed number of constants in the formula. 
Besides, even when we believe we know on theoretical or other grounds before- 
hand the nature of the function a priori we may consider it prudent to distribute 
the observations so that they supply us with data whereby we may control our 
hypothesis that the assumed functioii is the right one. 
It is therefore desirable to find other forms of distributions which, at the same 
time as they make the standard deviation of the adjusted function vary little 
inside the range of observations, are more uniformly spread over this range. 
(2) A uniform continuous distribution at once recommends itself as the simplest 
assumption. As we suppose the observations to have constant standard deviations 
the elements of the determinants of (15) are the moment coefficients of the x's at 
the places of observation. 
When the N observations are uniformly spread between x = — 1 and x = 1, 
1 
2r 
and jU-ar+i = 0' 
and the expression for 2j,<^',, is, according to (15), 
1+ - + 
1. 
1 
r-z 1 
+ X^ 
f^2 ■ 
H'2 
+ 
+ x^ 
1 
4 
■• l^2v-2 
2 
x^ 
•• l^2v 
y.2p-2 
h'-2v l^2v + 2 ■ ■ ■ 
■ ■ l^ip-i 
1^2 

• P-2V-2 
^2 
• M'23) 

• P-2P 
• P-2P+2 
f^2p 
P-2P+2 •• 
■ P-iP~2 
+ 
1 
f^2 
^-2 
H'2p-2 
t^2P 
H'2p H'2p+2 P'ip-2 
1 
fJ-2 ••• 
... P'2p-2 
1 
P2 •■• 
... fJ,2p 
P-i ... 
... /Xgj, 
/X4 ... 
/^2J)-2 
p2P ■■■ 
... p-ip-i 
P2P 
P'2p+2 ••• 
... Hip 
.(23). 
By this formula we may evaluate successively jct;,, ... apCv when we know 
the two general terms of which the sum consists. 
2—2 
