KiRSTiNE Smith 
59 
(a) f{x) = {l + ax^Y, where a > — 1, 
for errors of observation increasing or decreasing in both directions from the middle 
of the range. 
(&) /(x) = (H-a«)2, where 1 >a SO, 
for error of observations increasing in one direction. 
These two forms will roughly cover two distinct and important types of cases, 
such as occur in practice. 
(2) When / {x) = (1 + ax^Y we find, according to (67), 
^ = 1 + 2a/i.2 + o?ii^, 
and as (66) for n== I gives 
a ^1 
we have for a function of the first degree 
^>H±pH _ 2^^^ ^ ^, 
^ (1 + 2a/X2 + aVa) |1 + . 2 f (69), 
This curve has a minimum for x = jj.^ and the maximum in the range is, if 
[j.i> 0, at X = — 1, and if /x^ < 0, at x = 1 ; it equals in both cases 
, , „2.. ^ \, , a + M)'i 
[jLtj] being the numerical value of yu.^ . 
Now (69) is a minimum for = 0 ; we therefore ought to choose that value 
for and we then get, from (68), 
ct;, = (1 + Sa/x, + aVa) j 1 + r [ (70) 
and ^2 = 5 1 
iu.2 and may vary between 0 and 1 independently of each other and are only 
bound bv the conditions that „ = 
For any set of values which satisfies these conditions we may determine a 
distribution consisting of ^ iV observations at x = ± v and (1 — y) N at x = 0, 
since from any two such values we could determine 
and y = — = 1. 
By introducing and y for fi^ and we get two quite independent variables 
and (70) then takes the form 
al^~il + 2ayv^ + a^yv^) + . 
