4 
Choice in the Distribution of Observations 
(2) To find the standard deviation Uy^, of an adjusted it will be easiest to 
start from the equations (2). If the first be multiplied by a^^, the second by and 
so on before summing, and if we choose ap, a„ so that 
a„m„ = 1 
+ C^n''^«+l — 
a^m^ + a^mg + a^m^ + + a„m„+2 = 
we find that y,. 
1 
N 
a' 
s 
and therefore a' 
By multiplying out the square this may be written 
+ ttj [a(,«ij + aiWi., +02^3 + + a„m„+J 
•(4), 
a„ [aoW„ + aj + a5,m„+2+ + a„m2„]} , 
or applying (4) 
CL^^Xi.). 
0 . 
Hence a^^ is found by elimination of the a's between (4) and (5), which results in 
N 
1 
n 
< ■ a2 
... Xr 
1 
''«() 
m., . . 
.... m„ 
X, 
nil 
;u.> 
»h ■■ 
... m„+i 
x; 
m.. 
. . 
• • • ^n+2 
.(6). 
This determinant is of fundamental importance for all the following work and 
it will be useful at once to examine it more closely. 
(3) First however it may be pointed out that the standard deviation of any 
other linear function = b,,ao + b^a^ + b.^a.^ + b„a„ 
of the constants of the function y may be determined in quite the same way by 
N 
b, 
K 
w 
b., .. 
.... b„ 
.... w„ 
1)1., 
... TO„+i 
TO4 .. 
= 0 
.(7). 
"^i+l >?i„_f.2 . . . .. . 
