Mr. Morgan* on Survivorships . 2 57 
sy?nbols c', c", c"', &c. must be substituted for c — d, e, 
e — f, Sec. and the symbols b — m, m — n, n — o, Sec. for b\ 
b", b"', Sec. and the value of the given sum for the first z years, 
or during A J s life, will be = x ~~ -f s ~ -f -f- &c. 
S bsc . mtd' . 
6ubc X r + r z + 
toe"' 
, _L Rrr -L -L- v — _L ntc " i ouc "' i c r „ , » c' , c" . 
t^c.t^ x- r + 7 --f 7 -+&c. + ^ X T+F + 
-4- S gc ' i sc " 1 r o S me' ~ 
r , + .... W — — x T + ? + - + & c- - *x- + 
qq\h 
~~P~ H — — + Sec (z). Let p, p, v, J, &c. represent the 
number of persons living in the table opposite the age of B 
at the end of 2, z - j- 1, 2 + 2, &c. years, and v!, z", z’" 3 Sec. 
the decrements of life opposite the age of C at the end of those 
years respectively ; then, by reasoning as in the foregoing 
case, the value of S after the necessary extinction of the life 
S t — u. . y' . ZTv . 
of A will be 
zbcr x 
&C. 
bcr x + l 
P — ’X . y" ^ — k ■ y" 
r ~T r z 
-!•: 
&c. 
This last series is = 
S . ir .k . r — ] 
7+71+73+ Sec . 
ries may be resolved into 
cr x + l 
S.p 
x V — O. The other two se- 
P * 
ebr « X 7+P+“ + &C -“ *bc7* 
S 
7 +— + T3 + &c --7^*T + ^ + -^ + &c. The 
first of these is = 1 x V — C*; the second (supposing 
F'C' to denote the value of the joint lives of F and C for z 
years) is = 
. FC — F'C' 
BCJ — B'C' 
; the third is a continuation 
of the seventh series in this solution, and therefore the whole 
MDCCXCIV. LI 
