for the Valuation of Life-annuities. r *3 
half-yearly or quarterly payment which, in confequence 
of being laid up and improved at compound intereft, 
will in a year amount to the lum that makes the yeaily 
intereft. It is obvious that this cannot be the propel me- 
thod of computing theie values. But not to infill on this ; 
I will next ftate the different values of the fecond for t ot 
annuities ; or of life-annuities, according as they aie fup- 
pofed to be payable yearly, half-yearly, quarterly, or mo- 
mently. 
Let r as before be the intereft of i £• for a year; n the 
complement of a given lifers; y, h , <7, and 7;/, the values 
refpedtively of an annuity certain for n years payable 
yearly, half-yearly, quarterly, or momently ; r the per- 
petuity ; y the prefent value of an annuity on a life whofe 
complement is payable yearly ; h the value of the fame 
annuity payable half-yearly ; and Q^and M the values of 
the fame annuity payable quarterly and momently. 
I • 
(a) The complement of a life is, in Mr. de moivre’s hypotheCs, the num - 
ber of years it wants of 86. In all other cafes, it is double the expeaatton of 
a life ; that is, it is double the quotient (diminilhed by i unity) anfing from di- 
vidin'* the fum of all the living in a table of obfervations from the age (inclufive) 
of the given life to the extremity of life, by the number of the living at that age. 
See Effay I. in my Treadle on Reverlionary Payments. 
Vol. LXVI. 
Q 
Then, 
