GRADUATION OF A SICKNESS TABLE BY 
MAKEHAM'S HYPOTHESIS. 
By JOHN SPENCER, Actuary. 
Some time ago an examination of certain sickness tables suggested to me the 
possibility that for a considerable period of life a law analogous to the well-known 
law of Mortality of Makeham might be found to apply to Sickness, i.e., that the 
force of sickness at any age x or the proportion of persons sick out of the 
number living at the precise moment of attaining that age might admit of being 
expressed in the form A + Bd^, and I made various experiments to test the theory. 
Since that time Mr W. Palin Elderton, who had also investigated the same point, 
has referred to the matter in Biometrika* , and in a paper read in September, 1903, 
before the International Congress of Actuaries in New York, and has given the 
values of logio c deduced by him from various sickness experiences. 
The idea of a mathematical law governing Sickness is by no means a novel 
one. In a paper submitted to the International Statistical Congress in 1860 f 
Gompertz suggested that for ages between 20 and 60, 8^, the rate of sickness at 
age X or the average number of weeks' sickness experienced during a year by 
persons of that age, might be written kg'^, from which would follow the relation 
\og8:,= A, + B, c^^. 
Again Makeham himself dealt with the subject in 1871 when he read a paper 
" On the laws of Sickness and Invalidism " before the Institute of Actuariesj. He 
concluded from his investigation of data then available that the quantity con- 
sisting of the number of healthy, as distinct from sick, persons aged x out of the 
number living on the usual form of mortality table, was a function of the form 
ks^cf^ , this being the shape assumed by the l^. column when Makeham's law of 
mortality applies. This theory of Makeham's regardiog sickness possesses one or 
two advantages of some importance and it differs from the other hypotheses 
* Vol. II. p. 504. 
+ Beisrinted in Journal of Institute of Actuaries, Vol. xvi. p. 329. See also Phil. Trans. Vol. 152, 
p. 554. 
t J. I. A. Vol. XVI. p. 408. 
