KiRSTiNE Smith 
79 
consists of — N at i\ 
V, - V, 
.(103). 
and ~ N at v.^ 
V1-V2 J 
We thus see that for any two points and Vo of ivhich one is negative and the other 
positive ive can choose the numbers of observations so as to make cr'-;„ = as it of 
(7) By differentiating (101) we get 
,{l+af,,){ix, + apc.^ (104) 
course tvould be by taking a single group of observations at x = 0 
ating (101) we get 
dal_a^ 2^ 
dfMi N (^2 - t^i 
and = - - (1±^^' . 
dixo^ N (/X2 - ju-i)^ " 
As the latter is always negative a';,^ is for constant fi^ least when jj..^ has its 
greatest value, that is 1. 
Introducing this in (104) we get as condition for a minimum, 
fii + a = 0. 
There is only one distribution for which ju..^ = 1 and /Xj^ = — a, and it is that 
consisting of two groups of observations at — 1 and 1 included in the distributions 
examined in (6). 
From (103) we find that the actual distribution consists of — — ^ N observations 
at — 1 and iV at 1. The minimum of a;, is from (101) found to be 
2 ' N 
The minimum of o'-,^ can. thus only be obtained by taking lur) groups of observa- 
tions at the limits of the range with numbers proportional to the standard deviation 
of observations at these places. This distribution makes also a^,^^ a minimum, but it 
is not, except when a = 0, the distribution ivhich gives a'l the lowest maximum value 
within the possible range of observations. 
(8) For a, function of the second degree, 
y = Uq + ajdj + f/j*^ 
with the standard deviation 
CT„ = CT (1 + ax), 
where 0 ^ a < 1 
we have i = 1 + layi^ + ct-yUj, 
and from (89) 
CT 
I 
ai = (1 + 2a;x, + aV.) -^'^T^'t^ ^ -(lOS), 
t^2tJ'i- fJ^i- f^:i+ ■itiitl2t^3~ fJ-lt^i 
^'(l + 2a/.i + a'V2), 3 ^"h^ ^ -(106), 
~(l+2a/.i + aV2) r-^'^^ ^ ...(107). 
iV UoU^ — Lt; — Itr, -4- 11,11. — i;r//. ^ ' 
