4 $ Mr. Morgan on Survivorjhips. 
fented by R and the fum of the foregoing expreffions for 
k a . HK—HBK AK — ABK a rrr — 
x — — — f -f , x — HbvJ, &c.) by M, then 
3 C 
a 
will the value of the fum S (when A is the oldeft of the three 
lives) be = S x R — M. Q. E. D. . 
If the three lives be equal, the value of the given fum for 
s.7^2) 5 s.7T2| * 2 .^_ “ 
the fir ft vear will be = 
+ 
2 c i . r 
Q 1,2 a* 
b X to + 6d r 6 Sr * 
3 dd 
6 . c 3 . r 
the value of the fame fum for the fecond year will be = 
S . TP S . d — e) • e 
6 c 3 r 2 
+ 
2 c*r 
3r 2 
S » </ 6 » c d • € ci — t ] • f — d n 
+ —77 + TTT = b 
2 c°r 
3r* 
<^“ 677 “ ^ + ^?’ the Value for the third ^ ear wiI1 be = 
S X 
3 ^ 
2«7 3^ 
+ 
, , , , a , * 3 , . , 2 and fo on for the other years to 
6 c 3 r 3 6rr 3 6c 3 r 3 6<rV 7 j 
the extremity of life. Let CC and CCC denote the values of 
the two equal and three equal joint lives, the fum of thefe 
3 . CC — 2CCC 
feries may then be found = -g- x -— + 2CCC — 3CC -j- 
* 
.= (fuppofing the perpetuity, or q, to be denoted by V) 
^x— xV- 3 .CC- 2CCC. 
or •*/ 
It muft be here remembered, that from other principles it is 
well known, that the number of years purchafe exprefling the 
value of an ejiate ox perpetual annuity to be entered upon at the 
failure of two out of any three equal lives is, i6 the difference 
4< between three times the values of two equal joint lives, and 
twice the values of three equal, joint lives fubt rafted from 
“ the perpetuity,” or V — 3CC — 2 CCC. The value, there- 
fore, of fuch a reverfion, fuppofing it to depend on the failure 
of the three equal lives in any one particular order, is (fince 
2 ' there 
