Appendix 307 



chance that an individual is hit in an interval dt is p; this leads to a 

 difference-differential equation 



^l = -pl'^+plf-' (12) 



where Zf represents the number at time t who have been "hit" a 

 times. If l„ is the number of individuals at time o then a solution of 

 equation (12) is 



/« = le-^'(ptfloc\ (13) 



If the number of hits causing death is r, then the survivors at time t 

 are 



I, = l^e--*{l-]-{pt)ll\+ . . . +(pty-^l{r-l)\} 



and the deaths in the interval t to t + dt 



lji,l, = ke-^'pH^-'l(r-\)\ (14) 



The force of mortality at time t is 



(ptr' 



-Si/i-f;- 



+ 



(r-1)! 

 = pe-^\pty-^l r e^x'-^ dx (15) 



Formula (15) shows that the curve of deaths is an incomplete gamma 

 function, or a Pearson type III curve, ju, has the value for ^ = 

 and asymptotes to a value p at ^ = oo (Beard, 19396). 



17. A more natural function than ju,,. in the present context is to 

 use the function which bears the same relationship to /x^/^ as /z, does to 

 h, i.e. 



dt fjif dt 



and from formula (14) we find this to be 



'^<'°f-'-> = -p+'-^ (16) 



18. Attempts to use the formula of paragraph 16 on human 

 mortality data have been unsuccessful, the shape of c^(log /x^y/^^ not 



