Appendix 305 



9. The differences between formulae (4), (5) and (6) will become 

 apparent only at the old or very old ages and unless the data were 

 extensive the differences would be unlikely to be significant for many 

 numerical processes. From a scientific point of view the models are, 

 of course, quite different. 



10. An alternative approach to the question, but still based upon 

 rates of mortality, is to determine the conditions necessary for /x^ to 

 be a Perks (logistic) curve, given that the population can be stratified 

 according to a longevity factor and that the basic mortality law is 

 Makeham in form (Beard, 19526). Thus letfx'f, be the force of mortality 

 at time ( = age) k for the group having longevity factor s and let (f)(s) ds 

 be the proportion of the initial population having factor s. Then the 

 survivors of (j){s) ds at time k are 



<l>{s) ds . exp ( - r fit dt\ (7) 



and the total survivors at time k 



h = j <l>{s) exp (- J* fJLtdt^ds (8) 



where the integral is taken over the whole range of s. 

 The force of mortality at time k {= —d log Ikjdk) is then 



(j>{s) jLtfc exp ( — jLt* dt) ds 

 <^(5) exp ( — yi,ldt\ ds 



11. From formula (9) it will be noted that fx^ is a weighted mean 

 oi ix\ (= /x^ say). Since the number of lives with heavier mortality 

 will diminish more rapidly than those with lighter mortality, s will 

 decrease with increasing k. If the basic mortality is Makeham in 

 form, then dfij^/dk will show a slackening off at the higher ages, i.e. 

 the sigmoid feature shown by a logistic curve. In order to meet 

 practical conditions some limitations are necessary on the form of 

 <j){s) ; the lower limit must be ^ 0, but the upper limit can be oo. 



12. If it be assumed that </>(5) is a gamma function such that 

 (f){s) ds = ks^ ery' ds (0 ^ s <oo) and that the mortality function for 

 <l>(s) is fil = (x + ps e^^, we have 



f ks^ €ry\<y. + ^s e^^) exp {-[ (a + jS* e^') dt) ds 

 Jo \ J / (ifW 



H-k = 1^ 7 THc ^^ \^^) 



ks" e-y exp {-\ (ol+^s e^') dt) ds 



