Actuarial Aspects of Human Lifespans 3 



of the division of causes of death into two kinds, those due to 

 chance and those due to deterioration. 



It was soon evident that such a relatively simple law would 

 not represent mortality experience throughout life and sub- 

 sequent developments led to the proposal of more complex 

 mathematical relationships between age and the force of 

 mortality and even of different relationships over different 

 parts of the age range. Thiele (1871) for example proposed 



pL^ = a^ e-^»^ -f a^ ^-^^(^-^>' + ^3 e^^'^ 



in which the last term is a Gompertz curve to represent old- 

 age mortality, the first a decreasing Gompertz curve to 

 represent the mortality in childhood, and the middle term a 

 normal curve. Perks (1932) introduced a new family of 

 curves in the general form 



A +5c^ 



Kc"^ + 1 + Dc' 



and rationalized this procedure with some interesting specu- 

 lations on the theory of mortality. He found an analogy 

 between the "inability to withstand destruction" of Gom- 

 pertz and the then current physical concept of entropy change 

 — the measure of the time progression of a statistical group 

 from organization to disorganization. Perks also referred to 

 the previous work of Karl Pearson who fitted overlapping 

 curves not to the force of mortality but to the curve of 

 deaths, the curves being intended to represent the mortality 

 of old age, middle life, youth, childhood, and infancy, the 

 causes of death being different in these different periods of 

 life. Perks pointed out that this search for homogeneity in the 

 pattern of causes of death might lead to endless subdivision 



and then considering what happens to rrix when the interval of time 

 becomes infinitely small. Clearly then the deaths between ages x^ 



Xi x+1 



and X2 in the life-table = J y.x.lx-dx- and dx = j [ix-lx-dx- 



Xi X 



The continuous curve of {[ix-lx) is called the curve of deaths. 



