530 



PROCEEDINGS OF SECTION C4. 



these is what is usually known as the Makehain-Gompertz fonmihi, 

 but the latter is that which appears most applicable to the period of 

 infant life. Denoting the probability of surviving a year at age x by 



p,^ I zz -2+M we have fioin Makeham's second- expression 



log /^ = log X- + ■.*; log s + X- log h + C-" log g, 

 and log /^.+i — log k + («+ 1) log s + {x + ly logA + c^+Mog <7. 



i 1 cnce log i^-t? = log p_, — log s + (2x -[■\)logh + c-"' (c — 1 ) log c/ 



= (log s + log h) + 2x log h + <::■'■ (o— 1 ) log </. 



Since/* J. is necessarily always less than unity, Iog^>„ is negative. 

 It is fonsequentiy convenienf, in practice to operate with colog ^). 

 (z: — Iog/?J. 



We then have colog jt>,.=i — (log s + log h) — 2x log /i — (•'■ U'— I; logy 



=a + y.t- + /3c'', («) 



where a n — (logs 4- log h);y— — 2 log /i ; and ft — — {c ~ 1) log (j. 



The expression (a) 'covers both Gorap rtz's formula and 

 Makeham's first modification. In the former case s zz. h :=. 1, hence 

 a zz y zz 0, and hence colog ^j. — ftc'\ In the latter A = 1, hence 

 7 = 0, and colog p^, ■=. a -^ ftc^'. 



Expression (a) indicates that colog p,. may be represented by a 

 o-Gometrical progression plus a linear function, whilst by Gompertz's 

 original formula it would be represented by a geometrical progression 

 alone, and by Makeham's first modification by a geometrical progres- 

 sion plus a constant. 



7. Application to Infantile Mortaltty Expkuience. 



For the purpose of testing the applicability of a formula of the 

 Makeham-Gompertz type to the infantile mortality experience of the 

 Commonwealth for 1901-10 the values of colog jo^ have been obtained 

 for each sex, and are as follows : — 



Table V. 



