KiRSTiNE Smith 
69 
with 
(1 
and 
|2 N 
l + fa2 - 4 
(1- 
The maximum of a'i is 
a)2 N ^ 
— at 
observations at — 1 
1 in addition. 
N 
(1 
TABLE V. 
a 
Maximum of 
Oil - — for 
0- 
best distri- 
bution. 
Maximum of <y„ — ^ 
a 
for distribution 
with 0 {,!•)=■! 
Maximum of <r„ - — for 
(7 
distribution with 
iV 
(/){.)•)=— and clusters 
at ±1 
•0 
1-4U 
2-000 
1-581 
•1 
1-421 
2-003 
1-587 
•2 
1-442 
2-013 
1-602 
•3 
1-477 
2-030 
1-628 
■4 
1-523 
2-053 
1-663 
•5 
1-581 
2-082 
1-708 
•6 
1-649 
2-117 
1-761 
•7 
1-726 
2157 
1-821 
•8 
1-811 
2-203 
1-889 
•9 
1-903 
2-254 
1-962 
(9) For ?i function of the second degree we found in (5) that when the standard 
deviation of the observations was Sy = a [1 + ax^) and a ^ 3 it was advantageous 
to use the whole working range of observations, much more must this be the 
case when Sy = a {1 + ax) and 0 ^ a < 1. We shall therefore try to find the three 
best groups of observations taken at — 1, v, and 1, supposing v unknown. We do 
not venture to assert that another form of distribution might not lead to a curve 
of standard deviation with lower maximum, but the solution of the general problem 
would involve a more elaborate investigation into the possible variations of yu-i, jx.2, 
/Xj and for distributions with limited range than seems desirable in this con- 
nection. We shall further limit our problem by assuming that the best distribution 
.i'=i .i=-i 
will be found among those which make a], = cr'^ and both also equal to a maximum 
situated between x = — \ and x = \. This would obviously be right if the 
maximum were found at x = ; this in fact is not the case, but still the maximum 
value is likely to be chiefly determined by the number of observations at a; = v and 
there is therefore every reason to believe that our assumption is justifiable. 
Let there be iVS observations at — 1, iV . y at 1 and iV (1 — S — y) at v. The 
interpolation formula of Lagrange then gives 
{x — v) [x — \) ^ I [x — v) {x -\- \) _ — 1 
y 
(! + ?.') -2 
V2- 1 
