KiRSTiNE Smith 
43 
i4 = ^(l + 7^-^^^) (59)- 
(2) We shall now look at ^a'f, for special values of n and as a first attempt at 
a; = () x'-=l 
finding a flat curve for „ct;, try to make = „a'^ . 
For a linear function we find, since 
iVV l + 3c 
As a is jDositive it is obvious that we cannot make ^af, = ^ctj, which indeed we 
knew beforehand. This follows because we have proved that „al is of 2/)th degree 
and never lower. 
For x = 0 we find _ 
which holds for any symmetrical distribution of observations with constant 
standard deviation, a is the ratio between the number of observations at the ends 
of the range and the number uniformly distributed through the range, it may 
therefore vary from 0 to oo . As '^^^ decreases when a increases we get the 
1 + 6a 
flattest possible curve when a = oo , that is when the distribution of observations 
consists of two groups at the ends of the range. Then the curve is, as already shown 
in Section II, 
a 
To get a check on the degree of the function and at the same time a flatter curve 
of a'l than that obtained from a uniform distribution we may choose something 
between the two extreme cases and take for example observations at each end 
of the range and uniformly distributed through the range. 
Then a = 1 and, according to (59), 
with the maximum Oy 1"581. 
(3) For B,fvnc1ion nf the second degree we find, from (46), 
9 (1 +a) (1 + 5a) 
andfrom(50), ^ ^ (1 + a) + 
We want to make these equal and this requires 
3(1 + 5a) (1 + 3a) = 4 {] + 6a + 2(1 + 3a)} 
or 15a2 - 8a - 3 = 0. 
This has only one positive root a = -7873500. 
