10 
Choice in the Distribution of Observations 
As Ai,„+2,w+2,i = ~ An+2,n+2,i,i> integral of the last term of (11) equals N. 
The integration of the other terms, including the first, gives the same result so that 
and as ['^t-t = Nmn, 
] f (x) 
the mean value of „or^, calculated in this special way is 
a_2 (n + 1) 
N ' Mq 
It is therefore clear either that „a^, must at all the places of observation be 
equal - ^^ ^ ^ n^^l ^^^i^t at some of these places be greater. The first case 
cannot be realised by a distribution of which any part is continuous, as „a^, is proved 
to be of the 2nth degree in x. If therefore we could find a distribution consisting 
of groups of observations for which at all the places of observation „ol was equal 
to . , and if further we could choose the places of observation so that „crj^ 
i-V 
at all other places within the range of observations was smaller than that value, 
we should know that no other distribution of observations with that value for wIq 
could provide a curve of standard deviation with a lower maximum. 
If the standard deviation of the observations be constant and equal o, f {x) 
equals 1, and so does mg. After what we have just proved the maximum of the 
curve cannot then be lower than ^y- + Now when we choose to distribute our 
N observations in (n+ 1) equally big groups the adjusted y at each of these (w+ 1) 
places will be the mean of the observations and its squared standard deviation will 
be ^ ('^ + Hence our problem is reduced to find out how to arrange a table of 
{n + 1) values of a function of the nth degree to make the squared standard deviation 
of any interpolation result inside the range smaller than the squared standard 
deviation of the values of the table. It will be seen in what follows that this can 
up to n equal 6 — that is so far as the problem here has been investigated — be 
obtained by one and only one form of grouping. 
When the standard deviation of the observations varies over the range, Wq 
varies with the different distributions, and we cannot use the same method for 
finding the best distribution. It even appears that the best distribution has not 
always its maxima at the places of observation. 
(8) A second problem which we want to consider here is the condition for two 
adjusted i/s being uncorrelated. In the beginning of this section it has been shown 
that the adjusted «/, 
