54 Graduation of a SicJiness Table hy Makehams Hypothesis 
conclusion that a Makeham curve* would not apply throughout the whole of the 
table, i.e., from age 16 to 100, I ultimately divided the data for ages 19 — 78 into 
three groups consisting respectively of ages 19 — 38, 39 — 58, and 59 — 78. The 
ungraduated rates in each of these groups were summed, the resulting totals being 
^20 _ -I 
c- 1 
20^ + Bc^i . ^-"^ , 
c - 1 
and 20^ + Bc^i . . 
c - 1 
Differencing these, we have an obvious means of deducing the value of log c-", 
and hence the values of the other constants. 
The following are the constants resulting from this groupingf : — 
=-747127. 5= 00680912. log^o c = '0462118. 
The values of A and B are here given in weeks, and would have to be divided 
by 52i to furnish the corresponding yearly values. 
At the outset, in considering to what extent of the table a Makeham curve 
could be fitted, one or two points presented themselves which made it clear that 
the hypothesis would not apply for the whole of life. At the beginning of the 
table the evidence points in my view to a fall for a few years in the rate of 
sickness, a feature which is not singular to this section of the Society's experience 
but is reflected in the observations relating to other occupation groups. Again 
towards the later years of life, i.e., from about age 75, there is an unmistakable 
decline in the ratio at which the rate of sickness increases and a tendency towards 
a constant rate of sickness of about 40 weeks per annum as will be seen from the 
following average unadjusted rates : — 
Age group 
Average rate of 
sickness in weeks 
68—72 
13-78 
73—77 
21-24 
78—82 
29-02 
83—87 
34-72 
88—92 
37-25 
93—97 
39-73 
Considerable weight may I think be attached to these quinquennial rates since 
they are based upon a relatively large number of observations, one of the distinctive 
* In speaking here, and in wbat follows, of a Makeham curve or graduation I refer to the hypothesis 
formulated at the beginning of this paper and not to Makeham's own theory of sickness. 
t [Mr W. Palin Elderton has at my suggestion worked out logj„ c by the method indicated by him in 
Biometrika, Vol. ii. p. 50?>. To fit Mr Spencer's range from 19 to 78, he took the origin of his exposed 
to risk normal curve at 42-5 years, and he found logj„c= -046,0043. Considering the complete difference 
of method, Mr Elderton's result closely confirms Mr Spencer's value. K. P.] 
