2 Mr. T. R. Edmonds on the Law of Human Mortality 

 of decrement in the (£-f l)th year of age is 

 2( P,-P«+i) = 2AP, 

 P* + P*+i P*+P*+i 



AP 



which may also be represented by -~ — '. The mean annual rate 



AP 



of decrement in the (£+l)th year being ^ — -, the mean infinite- 



simal rate of decrement during the same year of age will be 



- x =5 — , if q be taken to represent the infinitely great 



Q "t+k 



number of equal parts into which a unit of age is supposed to be 

 divided so that the infinitely small given time dt multiplied by 

 q is equal to unity. 



The infinitesimal rate of decrement (or ratio of dying to living) 

 will at the precise age (t) be represented by 



?«=£±* = ^£ =< UogP ( 

 L t r t 



(since -^- = d . log P, according to a well-known property of hy- 

 perbolic logarithms) . That is to say, the infinitesimal rate of 

 decrement at any age is identical with the differential of the 

 hyperbolic logarithm of the number living or surviving at that 

 age. Consequently, if the infinitesimal rate of decrement is a 

 known function of the age, then the logarithm of the number 

 living (loggP^), being the sum of such infinitesimals, will also be 

 a function of the age, which may be found by integration. 

 The infinitesimal rate of decrement at the precise age (t -f ^) 



dP i 



years is-^-^ 17 , which is the differential of log V t+ ±. This may 



be taken to represent very nearly the mean infinitesimal rate of 



AP 1 



the (t + 1) th year of age, for which the expression x ^ — 



q /#+! 

 has already been found. On equating these two approximate 

 values of the mean infinitesimal rate of decrement in the (t + l)th 



AP 



year of age, it will ensue that dP* + £= very nearly; i. e., the 



differential of the number living at the age (t + J) years is equal 

 to the number of deaths in the (/ + l)th year of age divided by 

 q, the infinitely great number of equal parts into which the 

 unit one year is supposed to be divided. 



When P is a function of the age (known or unknown), it will 

 be found that all three of the following important quantities are 

 equal to one another, very nearly, for any annual or quinquen- 



