280 
On the Systematic Fitting of Curves 
We want to determine the values of the constants a, b, c, d, when a curve of 
type 
y = a + bx + cx'^ + dx^ 
is fitted to the above data. 
In using the method of moments we require to evaluate S{yx"') up to n = 3 
from a knowledge of its value at a number of isolated points. In order to do this 
we require to use a quadrature formula, and the exactness of our results will increase 
as we use better formulie. The object of this illustration is to show the increasing 
accuracy of different quadrature formulae. The actual values of a, h, c, d are given 
in terms of the moments in the second part of this paper. In calculating the 
moments x = b was taken as origin, and in each case the same quadrature formula 
was used for the area and all the moments. The following methods were used, — 
R. M. s. stands for root mean square of the error of ordinate at the 11 given 
points : — 
I. The curve was taken through four selected points. This method was 
adopted by Dr Chree, and I have merely transferred the result obtained by him 
to the centre of the range : 
y = 1-269,100 + •024,000a; - -027,3200;= + •000,969a;^ 
R. M. s. = -0126. 
II. The area and moments were evaluated by treating y as if it were A : 
y = 1-270,290 + -033,402^ - •02G,806a'2 + -000,3279^^3, 
R. M. s. = -0094. 
III. The area and moments were evaluated by Parmentier's Rule, or (v) with 
2p/{2p — I) put unity : 
?/= 1-263,808 + •032,311« - •026,380a.-= + "000,4 113««, 
R. M. s. = -0089. 
IV. The area and moments were evaluated by Simpson's Rule (7): 
y = 1-270,130 + -027,046a; - -027,180a;= + •000,7326a;3, 
R. M. s. = -0070. 
V. The area and moments were evaluated by Sheppard's Rule (X) : 
y = 1-268,898 + -029,388a; - -026,853a;2 + 000,57640;*, 
R.M. S. = 0057. 
VI. The curve was fitted by the method of least squares : 
y = 1-268,800 + -028,700a; - 026,8800;= + •000,6167ar', 
R. M. s. = -0055. 
Now these results show us at once that with (X) we have a fit by the method 
of moments sensibly as good as that obtained by the method of least squares. Had 
