KiRSTiNE Smith 
31 
Hence the curve for 2p+i^I ^ maximum for the abscissae at which 2p(yl has 
a minimum. A comparison with the results in Section II shows that the abscissae 
of the maxima found here are the same as those of the best places of observation 
for (n + 1) equally big groups of observations of a function of the nth degree. 
These places tally with the places where was a maximum. Thus if we 
imagine that we had started the investigations with a uniform distribution of 
observations, and to lower the maxima of the curve of standard deviation had put 
clusters of observations at those maxima and at the ends of the range we should not 
get the best curve of standard deviation till all the observations of the continuous 
distribution had been distributed at the n — 1 places of maxima and at I and — 1. 
The minima of the standard deviations obtained from a uniform continuous 
distribution and the (« + 1) best groups of observations do not fall at the same 
abscissae. 
(4) The curves are very far from our ideal of a constant standard deviation 
throughout the range. To obtain the same maximum of standard deviation as 
{n + 1) groups could give us we should have to limit the part of the range used to 
the following fractions of the range : 
It is not likely that the range of values of the function which we investigate 
would only be of interest inside a range so much smaller than that within which 
we might actually observe ; further it seems likely that observations all of which 
were taken inside the smaller part of the range would give better information for 
that special interval. I shall therefore examine in the following sections if a uniform 
distribution of observations to which is added clusters of observations at the ends 
of the range will not possibly give a more satisfactory curve of standard deviations. 
V. Uniform continuous distribution of observations with additional observations 
clustered at the ends of the range; constant standard deviation of observations. 
General formulae. 
(1) Suppose we have N . observations uniformly distributed from — 1 
N a 
to 1 and besides ^ - . 3— — observations at — 1 and the same number at 1. We 
Z i + a 
then have 1 f/'^ ^x'^'' , Na ) 
for 1st degree 
•58 
•73 
•80 
•84 
•83 
•73 
„ 2nd 
„ 3rd 
„ 4th 
„ 5th 
„ 6th 
