531 
expressive of the law of human mortality, &c. 
sent value of the p^^ payment will be ( ^ j ; hence, if 
^ . \ b, c, d ^ 
^ L- ^ b0 put == a, the value of the annuity will be 
bi c, d /> + !/> 
■ Qi _6 (I o, I— -a 
a,^a 4 -^^+^ ,.,a ^ — — — a~L_i * 
Art. 2. Consequently, let a general table be formed of the 
logarithm of for every value of the log. of and also 
let a particular table be formed for every value of the log. of 
L 
according to the particular table of mortality to be 
X L L 
adopted ; from the last table take the log. of , 
L c 
; and also from a table constructed for the purpose, 
d 
take the log. of r^, add these four logs, together, and the 
sum will be the log. of which being sought for in the 
general table, will give the log. of | 
the log. of the annuity sought for the term/>, on supposition 
of the geometrical progression being sufficiently near. Here 
I remark, that were it not for more general questions than 
the above, it would be preferable to have general tables 
formed for the values of ^ instead of the log. of such 
values ; but from the consideration that for most purposes a 
table of the logs, of will be found most convenient, I 
^ a I ^ 
have had them calculated in preference. 
Art. 3‘ The shorter the periods are, the nearer does the 
series of the number of persons living at the equal intervals 
of successive ages approximate to the geometrical progres- 
sion ; and consequently this mode, by the assumption of 
sufficiently short periods, and frequent repetitions, will answer 
MDCCCXXV. 3 Z 
