2 
Choice in the Distributioti of Observations 
When we deal with, for example, a linear function which it is possible to ob- 
serve with the same accuracy for all values of the indefinite variate we should 
not hesitate to put the observations in two equally big groups as far apart from 
each other as feasible. Bxit if the standard deviation of the observations be a 
function of the indefinite variate and increases with the distance from the middle 
of the range, where is then the point in which the advantage of removing the two 
groups of observations from each other just counterbalances the disadvantages of 
increasing the error of observations? The problem becomes very complicated for 
functions of higher degrees. 
We shall in this memoir try to contribute to the solution in the case of poly- 
nomial functions by examining the standard deviations of the adjusted and more 
especially the interpolated values of such functions for different distributions of 
observations. Those values inside the working range of observations may be 
considered the sum of knowledge acquired by the experiments. The adjusted 
values outside the working range may probably in exceptional cases be of interest, 
but as only by some other type of experiment we can make sure that the form of 
function holds outside the range they are in ordinary cases without great value. 
We shall therefore aim at finding the distribution of observations which within 
the selected range gives the most satisfactory standard deviations of the adjusted 
values of the function. 
To consider the standard deviations satisfactory we must of course demand 
that they shall be as small as possible, and since a greater accuracy in one part 
may be expected to be accompanied by a smaller accuracy in another part we 
want them in addition to be as near constant as possible. In other words the 
curve of standard deviation with the lowest possible maximum value within the 
working range of observations is what we shall attempt to find. It appears that 
the distribution of observations which fulfils this demand consists of specially placed 
groups in number just sufficient to determine the constants of the function. We 
shall accordingly pay attention also to the desirability usually present of ascer- 
taining the form of function by means of the observations. As might be expected 
we find that the standard deviations obtained from a uniform continuous distri- 
bution of observations increase towards the ends of the range. By choosing a 
uniform continuous distribution with additional clusters at the ends of the range 
we shall try to find a compromise between the two desiderata of a low maximum 
of standard deviation and of a uniform distribution. 
The indefinite variate is supposed to have a vanishing error of observation 
compared with that of the principal variate. This error may be constant or varying 
with the indefinite variate, but in either case it is supposed to follow the typical 
law so closely that the method of least squares may satisfactorily be applied to the 
observations. After having found first the most advantageous distributions for 
observations of functions up to the sixth degree with constant standard devia- 
tions we examine the case for observations of functions of the first and of the 
second degree which have standard deviations of the form a (1 + ax) and ct (1 + ax^). 
If it is profitable to use the whole of the working range the latter distributions 
