272 
On the Systematic Fitting of Curves 
(b) We must know how to find the moments of any system of observations. 
(c) The expressions for the moments in terms of Ci, Cj, Cg, ... c„ must be 
such that we can easily solve the system of equations (vi). 
I propose now to consider these points in some detail, starting v^ith the second. 
(2) On the discovery of the area and moments of a curve given by a series of 
isolated observations. 
The isolated observations may be of two kinds : 
(a) Actual measurements may have been made of the ordinates of the curve 
at p points. 
This is the most common case in physical investigations, but it is not infrequent 
in economic and actuarial enquiries, e.g. the mean age of bridegroom for brides of 
a given age, or the mean number of years of insurance of those that die at a certain 
age. 
(6) The actual measurements may represent the areas for certain base 
elements, p in number, of a given curve. 
This latter is the usual case of frequency observations. We determine the 
number of individual cases which fall into each of a small series of ranges of some 
vital or economic variable, e.g. the number of deaths, which under certain circum- 
stances occur in each year of life, the number of individuals which fall into each 
small range of a particular organ or character, etc. This is the type of data on 
which "frequency curves "are based. Actually (6) would sensibly coincide with 
{a) if we took our elementary ranges for classification, extremely small. This, 
owing either to roughness and paucity of data, or to the immense labour in- 
volved, is very often practically impossible. Not uncommonly (a) is used for {b), 
and for a great majority of cases the work is close enough for the value of the 
observations. But for fine and important work it is desirable to keep the two 
classes of cases essentially distinct. 
Case (i). p ordinates of a curve are observed or measured, to find its area and 
moments. 
What we require is clearly 
from a knowledge of 2^ = ya," at p points. The answer to this problem is familiar, 
and consists in the choice of a good quadrature formula. Whether we are dealing 
with the ordinates y simply, or the more complex moments, yx''^, will make 
theoretically no difference, except that in the latter case we may have to go to 
higher differences for the purpose of reaching accuracy. 
