12 
Choice in the Distrihiition of Observations 
(9) Returning to the formula (11) for a\ written as a sum of squares we shall 
now prove that the {p + \ )st term of this put equal to zero determines a set of p abscissae 
the adjusted y's of which are mutually uncorrelated both for a function of the pth and 
the (p — l)st degree. 
The condition for ?/j and y2 corresponding to the arguments and ccg being 
uncorrelated is for a function of the {p — l)st degree 
0 1 
1 Wq m-y 
m-. 
Mo 
m 
v-l 
m„ 
TO. 
"J) 
m 
TO 
TO, 
= 0. 
23)-2 
and for the same distribution of observations and for a function of the pth. degree 
the condition is 
0 
1 
X^ 
x\ .. 
.... x^; 
1 
m-i 
7n .2 . . 
.... mj, 
x^ 
»h 
nis .. 
.... TO.j,+i 
x' 
2 
'"'2 
TO4 . . 
TOj,_|_2 
Putting 
7n 
.... TO-. 
Mo 
m^ 
m.2 .... 
. TOj, 
TO3 .... 
TO, 
.... 
nip 
TO,j,+i 
"''j)+2 
• »«'22> 
2D 
= 0. 
these conditions may be written 
and 
S {aj, . a^2-^})+i, ji+i, r+i, s+i) — ^ 
p 
2 {a;, . a^2-^r+i, s+i}= 0 
.(19) 
.(20), 
where the sums include all combinations of powers with r and s lying between 0 
and {p — 1), and 0 and p respectively. 
Now we have for an orthosymmetrical determinant A, 
A,, .A,,,.. -A. A,,,.," = A,,'. A,,". 
If therefore (19) is multiplied by D and subtracted from (20) multiplied by 
Dp+i, p+i tbe coefficient of x[ . x^ becomes 
D . D 
p+l, v+l, r+l,s+l 
as long as both r and s are smaller than p. 
