308 Appendix 



fitting well to observed values which show a negative second dif- 

 ferential coefficient over the adult ages. 



19. As an extension of formula (12) a model can be set up in 

 which the "hits" in an interval can be single, double, etc., in known 

 proportions. The basic relation then takes the form 



J" = -plT+p^^f{r)ir (17) 



This can be integrated to 



If = e-^' J pe^' 2 f{r) Vf^ dt (18) 



and by noting that ll = e-'" /„ values of If can be obtained by succes- 

 sive integration. No experiments have been made using this form, 

 mainly because the form of c/(log fJLtQIdt seems to be unsuitable for 

 human data. The form of /(r) is also speculative. 



20. A different forward model can be devised in which the proba- 

 bility of a "hit" is dependent on the number of "hits" recorded 

 already. We then have the following 



/77a 



^ = -(^4-pa) Zf + (^+^.a-l) ir (19) 



This can be integrated to give 



with 



'te^) = -(^ + 0^^) +^^^^^- (21) 



Here again the form of equation (21) does not accord with observa- 

 tions from human data. 



21. In the attempts to fit these forward type formulae to human 

 data it was found (Beard, 1950, 1952c) that satisfactory numerical 

 results could be obtained by expressing /x, /^ in gamma function form 

 subject to a terminal age cu, i.e. the infinite tail of the curve is the 

 opposite way round to what would be considered natural. This 

 formula, after elimination of a constant element representing 

 accidental mortality, can be derived from the difference-differential 

 equation 



^^=plf-pf-' (22) 



