KiRSTiNE Smith 
13 
When one of them, for example s, equals the term is 
which is of the same form and this also holds for r = s = p when the term is 
x^ . X., . Dp-i-i, ii+i. 
The total result is thus 
or in the form of determinants 
nil 
m 
m. 
m 
m 
35 + 1 
1 
xl 
= 0, 
nil 
7)1-. 
2J)-1 
Xo m 
m 
m 
V+2 
III, 
m. 
2V-1 
Hence x^ and must be roots of 
1 
ni2 .. 
... nij,_i 
X 
ni^ 
... Ill J, 
x^ 
»h 
. . 
... iiij^+i 
"3? '^'p+l '"'p+2 '""Zp-l 
When Xl is found from this and substituted in (19) or (20) we get since the 
coefficient of x'.^ in the latter is zero an equation of the (p — l)st degree to deter- 
mine Xg. It is therefore clear that any pair of roots of (21) determine a pair of 
uncorrelated y's. 
II. The "best" grouping of observations with constant standard deviation. 
(1) It was shown in the last section under (7) that the nieati of the squared 
standard deviations of the adjusted y taken over the places of observation and weighted 
with the number of observations at each place is equal to {n + 1) and that there- 
fore the curve of squared standard deviation can never be entirely beloiv that value. And 
further, that since (n + 1) equally big groups of observations at the places of 
observations give the squared standard deviation this minimum, there is the 
possibility, „ct^, being of the 2nth degree in x, that by placing the groups at special 
positions the curve of squared standard deviation could have those values {n + 1) 
as its maxima within the range of observations. 
Mr 
= 0 
(211 
Let 00-^ y 
be the places of observations and g^. the mean of the 
observations at x^ , the interpolation formula of Lagrange is then 
Xr^ 
, (x Xj) {x X2) {x — X„+j) 
\{Xj) X-^ (Xj, X2) (x 
the sum taken over all the places of observation. 
