301 
the decrement of human life. 
cent, interest, an annuity payable half yearly is the same, 
throughout the middle of life, that would be granted on the 
life of a person a year older, if payable annually/’ 
If it is required to ascertain the value of a reversionary 
annuity payable half yearly or quarterly, the calculation be- 
comes in appearance a little paradoxical ; for since the true 
value of a reversionary annuity for the life of one person, 
for example, after the death of another, is the difference be- 
tween the values of two annuities on the single life and the 
joint lives, and since an equal addition must be made to these 
values in consideration of the period of payment being short- 
ened, it follows that the reversionary annuity must be of 
equal value in either form. This conclusion would indeed 
be strictly true if the periodical times of payment remained 
unaltered, according to the supposition from which the value 
of the annuities is deduced ; while in fact it is usual to grant 
such an annuity to commence at the first quarterly, half yearly, 
or annual period after the contingent event: a variation 
which would have no sensible effect in the case of daily pay- 
ments, but which lessens the value of reversionary annuities 
at other periods by that of half a payment for the given 
period, reduced to the present time in the same manner as 
any other sum assured as payable upon the same contingency 
of survivorship. 
The simplicity observable in the progression of the values 
of annuities, calculated according to the values of lives here 
supposed, and at 3 per cent, interest, leads us to inquire what 
would be the exact law of mortality required to make that 
progression strictly uniform throughout life ; and it will appear 
on investigation, that in order to have the value 24.45 — 
R r 
MDCCCXXVI. 
