306 Appendix 



which reduces to 



which is a Perks (logistic) form. 



13. The results of the immediately preceding paragraphs are 

 interesting in that the limiting value of [jl,, arises from the manner in 

 which the "mixed" population runs off. They have a certain appeal 

 in that they are based on the assumption that the population is not 

 homogeneous in regard to a mortality (or longevity) factor and that 

 the mortality for an individual group continues to increase indefi- 

 nitely. The Hmiting value oi ^jlj^ — ol as k^co from formula (11) is 

 (2? + l)A = 4A//S1 where ft is the Pearson moment function of (^(5). 

 For human lives fx,. '^ • 6 at the limit, according to one fairly recent 

 mortality table, and A -^ 0-1 so that ft ~ 0-67, i.e. a skew distribu- 

 tion with a tail towards the higher values of s. If 5 is a heredity 

 factor, then stability of </.(5) over generations would imply fertility 

 rates negatively correlated with longevity, otherwise the shorter 

 reproductive period of those with higher values of s would lead to a 

 falling average value of s in the population. It is an interesting co- 

 incidence that the distribution of married women according to 

 number of children born has a ft coefficient of the order of 0*7 

 (Papers of Royal Commission on Population, 1950). 



14. The assumption of other forms for (f){s) in formula (9) leads to 

 other forms for ^^ which can have the appropriate shape but which 

 are not convenient mathematically, and no experiments have been 

 made in this direction. 



15. From the point of view put forward in paragraph 1 formula 

 (10) suffers from the objection that it is based on the assumption of 

 a Makeham law, and is thus basically empirical. A further approach 

 to the question is to build up models based on the so-called "shot 

 hypothesis" in which individuals are assumed to be subject to 

 random firings and are assumed to die when they have been "hit" 

 a specified number of times. Two main types of model have been 

 investigated, which are referred to below as the "forward" and 

 "backward" models respectively. In the forward model hits are 

 assumed to accumulate and death to occur when the total reaches a 

 certain figure. In the backward model the individual is assumed to 

 start with a quota of units which are progressively lost in time, 

 death occurring when the total remaining falls below a certain 

 figure. 



16. The simplest forward model is derived by assuming that the 



