14 Choice in the Distribution of Observations 
From this we find 
= (n + 1) S I ('^ ~ -^i) ~ ^2) ~ -^n+i ) ) ^ ^22) 
0-2 
which for a; = ccj, ... x„+i equals (n +1), the n terms of the sum being zero 
and the (n + l)st taking the value 1 as it ought to. If Xj, be the greatest of the 
x's it is hence clear that for x> Xj,, since 
{x — Xt^ {x - X2) {x — Xn+i) |2 ^ J 
{Xj, Xj) {Xp X2) {Xp 3^71+1)] 
The same applies to any x smaller than the smallest of the places of observation. 
Therefore as we want al to be 5 (w + 1) at the ends of the range we have to place 
two of our groups of observations there. 
Let us take the half of the range within which it is possible to make observations as 
the unit of x so that the range goes from — 1 to 1. 
(2) Hence for a linear function there is no choice left, the two groups of observa- 
tions must be at — 1 and 1. 
or^ ^ i{x + 1)2 , (X - 1) 
According to (22) we have 
or ,<,!_^.2{l-i{l-j.% 
which illustrate the well-known fact that by simple inter Violation between two 
equally good values of a table, ive obtain interpolated values with less probable error 
than those of the table. 
(3) Investigating a function of the second degree we have a third group to 
place besides the two at — 1 and 1, that is if we do not beforehand suppose the 
distribution to be symmetrical. Let the third group be at a, then the interpolation 
gives 
{x — 1) (x — a) (x + 1) (x — a) ^ x^ — 1 
= ~2"( 1 T aj" + "2 a - a) + a^l ^" ' 
from which 
CT" = vt • 3 
(x — 1) (x — a) 
2 
+ 
(x+l)(x-a)12 rx2-l 
2 (1 - a) J [a2 - 1 
fda". 
2 (1 + a) 
We want this to be a maximum for x = a, but ^^'j can only vanish for 
a = 0, in which case is reduced to 
a" 
N 
.ol=%.S{l-^xHl-x^)}, 
