72 
Choice in the Distribution of Observations 
The maximum of the corresponding a'l is hence 
t'l^-'^t 4-308. 
N y N 
From (87) we find that it occnrs at x = -125 as well as at a; = ± 1. The dis- 
tribution of observations is then 
•8380 iV at 1, 
•0023iVat-l, 
and -1597 iV at - -354. 
Comparing again with a distribution consisting of three groups of observations 
at — 1,0 and 1 with frequencies proportional to the squared standard deviations at 
these places we find that the distribution would be 
•7814 iV at L 
•0022 N at - 1, 
and •21G4iVatO, 
and the maximum of a; would be 
^'.3(l + 2a^.i + aV2) = ^.4^62. 
We thus find that by our special distribution the maximum of al was 7 per cent, 
lower, the choice of that distribution would thus permit us to reduce the total 
number of observations at the same rate without raising the maximum of crl . 
(12) The result of these investigations is that the maximum a,, obtained from 
the best three groups of observations differs so little from, that obtained from three groups 
at — \, 0 and 1 that the first grouping only in quite exceptional practice would be pre- 
ferred. 
We shall therefore in Table VT give the maximum Oy arrived at from the 
following three distributions: (1) three groups of observations at — 1, 0 and 1 in 
numbers proportional to the squared standard deviations at these places, (2) a 
N 
distribution for which (x) = - , and (3) a distribution for which d> (x) = -2797 N 
with additional clusters ^2203 iV at ± 1 (see Table II, p. 50). 
Both in Table V and in Table VI the difi^erence between the two first maxima 
as a proportion of the first decreases with increasing a so that the distribution with 
uniform ^ (x) is more profitable for a > 0 than for observations with constant 
errors. 
VIII. Best distribution of observations for determining a. single constant 
of the function. 
(1) Our choice of observations has hitherto aimed at giving within the working 
range of observations a determination of the function as accurate and uniform as 
possible. We shall now consider what is the best choice of observations for 
