298 
On the Systematic Fitting of Curves 
dealt with on pp. 279 — 281. But there was no reason for supposing a priori the 
observations to be suitable to sine curve representation, and the sine curve has one 
less constant. The fit is illustrated in the accompanying Fig. 4, and is seen 
to be by no means bad. The data were merely selected, as equally good with any 
others, to exemplify the process of fitting a sine curve. 
(8) Illustration IV. To fit Makeham's Curve to Mortalitxj Statistics. 
Given a mortality table— i.e. a table which gives the number of survivors out of 
71 people born in the same year at each year of age of the group — then if l^ denotes 
the number who attain the age of x, the table will be closely represented between 
the ages 20 — 25 to 85 — 90 by Makeham's formula, i.e. 
l^ = ks^(gy (xx), 
where k, s, g and c are constants to be determined from the data of the table. 
Now there will be some 60 to 70 corresponding values of x and Ix and it is 
a quite hopeless task to think of discovering the values of k, s, g and c for the equa- 
tion as it stands. If we take logarithms tlie equation may be written : 
Lx = K' + xS' + G'c"^, 
where the capitals are the logaritlims of the small letter quantities. The determi- 
nation of K', S', G' and c by the method of least squares is still impracticable. Of 
course four corresponding values of L^' and x would give K', S', G' and c, but 
such a selection of four arbitrary values out of 60 or 70 is \msatisfactory in the 
extreme. Accordingly Messrs G. King and G. F. Hardy have determined values of 
these constants by a process of averaging series of corresponding values of Lx and 
Xy so that the final values of the constants shall depend on as much of the table as 
possible*. The values reached for the constants are good, but no doubt better 
ones could be found, and the process from the standpoint of systematic curve fitting 
is unsatisfactory. It involves empirical trials — e.g. " various groupings were tried, 
and the best was found to be, four groups of eighteen years of life each " {Text-hook, 
p. 82) — and therefore follows no general rule for curve fitting. 
Accordingly it seems very desirable to indicate how the method of moments 
can be ajDplied to Makeham's formula. 
I shall take I for the range of the mortality table to be dealt with and ray 
origin of x at the mid-point of this range. If x^, be the age corresponding to the 
origin, I shall write 
l^.^k's'^'i^ig'Y'''^^ (xxv), 
whence we see that 
.(xxvi) 
k' - ks^'. 
* Journal of Institute of Actuaries, Vol. xxii. p. 200, or G. King: Institute of Actuaries Text-book, 
Vol. II. p. 79 et seq., especially p. 82. 
