(6o 7 ) 
fame Sum to be f aid, provided one of t ho fe three terfons be Uvhg 
at the Expiration of that term Which proportion being year- 
ly repeated, the Sum of all thofe prefent values will be the 
value of an Annuity granted for three fuch Lives. But to 
explain this^ together with all the Cafes of Survivance in 
three tives: Let N be the Number in the Table for the 
Younger Age, n for the Second, and rfor the Elder Age j 
let T be thofe dead of the Younger Age in the term propo- 
fed, y thofe dead of the Second Age, and v thofe of the El- 
der Age ; and let R be the Remainder of the younger Age, 
r that of the middle x\ge. and g the Remainder of the El- 
der Age. Then fhall R-\-T be equal to N 9 r ~\-y to », and 
§-\-vto v, and the continual Product of the three Numbers 
Nnv fhall be equal to the continual Product of R ~\- T 
X ~r-\-T x ?-}-t/,which being the whole number of Chances for 
three Lives is compounded of the eight Products following. 
(i)Rr?, which is the number of Chances that all three of 
the Perfons are living. (2} r ? V, which is the number of 
Chances that the two Elder Perfons are living, and the youn- 
ger dead. (3) R?7 tne number of Chances that the middle 
Age is dead, and the younger and Elder living. ' (4) Rrv 
being the Chances that the two younger are living, and the 
elder dead. ()) § Ty the Chances that the two younger are 
dead, and the elder living; (6) rTv the Chances that the 
younger and elder are dead.* and the middle Age living. 
(7) Ryu, which are the Chances that the younger is living, 
and the two other dead. And Laftiy and Eightly, Ty b J 
which are the Chances that all three are dead. WhicrL latter 
fubfrra&ed from the whole number of Chances Nnv 3 leaves 
Nn v — Ty v the Sum of all the other Seven Produds 5 in all 
of which one or more of the three Perfons are furviving. 
To make this yet more evidentj have added Fig.B. wherein 
thefe Eight feveral Producers are at one view exhibited. Let 
the re&angled Parallelepipedon ABC DE FG Hbe corsftiru- 
tedof the fides A /?, GH.&'c. proportional to -N" the num- 
ber of the younger Age $ AC y BD, &c. proponionai to n $ 
and A G^C E, &c. proportional to the number of the Elder 3 
or v. And the whole Parallelepipedon fhall be as the Product 
Nnv, or our whole number of Chances. Let B P be as R 3 
and A? as Ti letCL be as r,and Ln as^; and G N a? and 
NA as u 3 and let the Plain P Re a be m lde parallel to the 
plain 
v. I 
