Appendix 303 



which will be referred to as a Perks curve since this is the name by 

 which it is generally known by actuaries (Perks, 1932; Beard, 1936, 

 1939a, 1951a, 1952a; Registrar General, 1951; Mortality of Assured 

 Lives, 1956). 



4. Now [Xg is the ratio of the ordinate at age x of the curve of 

 deaths to the area under the curve above age x. We may look upon 

 the curve of deaths as a frequency distribution of deaths by age at 

 death and for many types of frequency curves it will be found that 

 this ratio has a sigmoid form. It is not apparent whether the satis- 

 factory representation of /x^ by a Perks curve is because the formula 

 has a theoretical significance or because the formula does provide a 

 good approximation to the particular function of a family of fre- 

 quency curves which can be used to represent the distribution of 

 deaths by age (Perks, 1953). 



5. What evidence is available tends to support the idea that the 

 force of mortality does not continue to increase indefinitely with age. 

 The concept of a limiting age by which all individuals must be 

 dead (i.e. a maximum lifespan) does not seem to be in accordance 

 with the facts — the use of a limiting age as a mathematical device 

 to cut off" a long slender tail has nothing to do with the present dis- 

 cussion. Formula (1) leads to an upper limit of BID for fx^ and it is 

 not without interest to note that the numerical values of B/D 

 obtained from the graduation of human mortality data are of the 

 same order as the force of mortality which can be deduced from select 

 mortality tables as being appropriate to "damaged lives", i.e. about 

 0-57 (Beard, 19516). 



6. If the rapidly decreasing mortality associated with the infantile 

 and growth period be ignored the pattern of human mortality then 

 exhibits a basic sigmoid form on which are superimposed waves and 

 other disturbances. The waves appear to be due largely to secular 

 effects (e.g. selective effect of war deaths); the main disturbances 

 are those arising from accidental deaths and the (rapidly disappear- 

 ing) hump at the early adult ages from deaths from tuberculosis. 



7. For a broad mathematical approach we will be concerned with 



(a) accidental deaths (assumed to be at a constant rate at all ages), 



(b) an upper limit to the rate of mortality, and (c) a progression in 

 time. 



Gompertz' law arises by using condition (c) only, 



i.e. djjijdx = A/x^ whence fi^ = B e^' (2) 



Makeham's law arises by using conditions (a) and (c), 



i.e. dfxjdx = X{ix^—A) whence ii^ = A+Be^ (3) 



