270 
On the Systematic Fitting of Curves 
A 
= A'- 
A 
= ^ /"i 
= A'y.^ 
f dR 
A/J.3 
= ^'f^i 
Afi, 
= A'fi: 
A/x,i_i ^ 4>'vt-i - |h - 1 jiy- y') 
dR 
dan-i 
dx 
.(V). 
Now the integral term in these equations must clearly be small because 
(i) It involves the small factor y — y. 
(ii) R, the remainder, = |^ 0" {6x) will by hypothesis be small, if 71 is at all 
considerable. Hence neglecting the integral terms, we find 
A=A' \ 
.(vi). 
fJ-s = /^i 
Or, the constants of the theoretical curve are to be found by equating its area 
and first n — 1 moments to the area and first n — \ moments of the observed curve. 
These results having been obtained we may at once replace oio, flj, a^, ... ««_! by 
the real constants Cj, c^, C3, ... Cn of the theoretical curve, and we obtain the rule: 
To fit a good theoretical curve y = (f)(x, d, c,, c^, ... c,j) to an observed curve, 
express the area and moments of the curve for the given range of observation in 
terms ofci, c^, c^, .. Cn, and equate these to the like quantities for the observations. 
The moments may be taken about any axis parallel to y likely to simplify the 
results, e.g. the mid-vertical of the range or in other cases the centroid vertical. 
Returning to equations (v) we see that the solution (vi) is even more approxi- 
mate than might at first sight be imagined. For if we render identical the first 
