KmsTiNB Smith 
61 
from which it follows that no distribution of observations other than those arrived 
at consisting of two equally big groups can give /j,-^, fx^ and fj.^ the values required. 
We accordingly reach the result that : lohen observing a function of the first degree 
for which the standard deviation of the observations is ct(1 + ax^), symmetrical about 
the middle of the range, we get the best function for a',, bij taking two equally big growps 
of observations, at the ends of the range if a ^ \ and at v = ± h \/ 1 + ^ — 1 if 
a > 1. 
(3) According to (70) the maximum of for this distribution is 
cn, = ^{l+av^f(l + ^, 
V being equal to 1 for a ^ -j and v being determined by (75) for a > ^. 
We shall next consider the distributions (i) for which (x) is constant from — 1 
N 
to 1 and (ii) for which 0 (x) consists of ^ observations uniformly distributed from 
N . 
— 1 to 1 and — into two clusters. 
z 
(i) For a uniform distribution from — 1 to 1 we have ^t2 = i) 1^4 = 1 and, 
according to (67), i 
Jc 
N 
the actual distribution is hence, as (f){x) = , 
1 + fa + \a\ 
iA(x) = 
(1+ ax 
.2^2 
2 1 + fa + 
1 ^2 ' 
and the maximum d\j as given by (70) for a; = ± 1, 
a;=±l fJl 
al =^(l+|a + ia2).4. 
.. N N 
(ii) When <f){x) = — with the additional clusters at ± a we have 
H-2 = Ii +4"^ and /X4 = yL -j- 1 
According to (70) the maximum a; is then 
= ^ [1 + a (i + ^^^) + aMi^ + i«^)] (1 + 3^ 
We shall now determine u so as to make this a minimum. We find that 
x = l 
da'i 1 
du^-N-^l^^^"^^^^^ "^'^^ - (3^^ + 1? + ^^'^ \ = 0 
requires 
45a2tt6 + 15a (3 + 5a) + 5a (6 + 7a) u^ - (90 - 5a + Qa^) = 0 ...(76), 
the root u^ of which is > 1 for a < -5576. 
x = l 
For a ^ -5576 we hence get the minimum a?, by taking the clusters at « = dz 1 
and for a > -5576 at the places ± u determined by (76). 
