Mr. MoHGAN ora Survivorships. 
2 37 
second year will be 
be . a' + a" . 7ie . a' -f- a!' 
zabcr 2 * zabcr x 
In the third year, 
by following the same steps, the value of the annuity will he 
found and i n the remaining years of 
C's life the value of the annuity may be determined in a si- 
milar manner. The whole value of the annuity therefore, 
when C is the oldest of three lives, will be expressed by the 
two series — -f l±T_f i Ltf+ALJ 4. o nc i a ' md ■ «« 
zac *~ ^ 2 ter* T CXC - ailCt 2abcr ~\ -^g-T 
+ + &c - The first of these series is = and 
hence the required value in this case 
zabcr 
the second is 
C-AC 
is = 
+ 
BC— ABC 
BC— ABC 
Secondly. Let A be the oldest of the three lives, and if# denote 
the number of years between the ages of A and of the last 
person in the table, C' the value of an annuity on the life of 
C for z years, and B / C / the same value on the two joint lives 
of B and C ; the value of the annuity for the first # years will 
evidently in this case be 
C'— AC- B 
1 
-ABC 
At the expi- 
ration of this term the life of A is necessarily extinct, and con- 
sequently the value of the annuity for the remaining years of 
C's life (supposing £ s, f, v , &c. to denote the number of per- 
sons living in the table at the end of 7+7, 7+1, 7+^, &c. 
years, and <p to denote the probability of B*s surviving A^ 
&c. == p . C — C'. 
will be = ® x 1 L — i . 
r C r x - 1- 1 > cr x + z cr x + 
The whole value of the annuity therefore, when A is the oldest 
* See the table in the lAXVIIIth Vol. of the Phil. Trans, p. 337 . N. B. When 
this table is used in the present and following problems, certainty must be denoted By 
unity. 
