1 
Survivorjhips. 253 
by <^, be compared with the fecond of the pr ecedi ng cgpref. 
7 j ' y np Q 
fions, and we (hall then have b 
2K 
. cc - cec 
2 r 
-Q, and confequently Q=S ?»«' as before * * * ♦ ♦ 
Now it is well known, that when the lives are all equal, the 
value of the reverfion muff be one third the difference between 
the oerpetuity and the three joint lives, and tnerefore a demon- 
ft rat ion arifes of the truth of the whole folution. As a lull 
further proof of this, the foregoing theorem may be imme- 
diately deduced from the feries themfelves : thus, the value of the 
. . , . r , s . fsZffn 
given fum for the firft year will in this cafe be = ; tor 
the fecond year it will be = S -^ ; for the third year it will be 
S . * 3 -/ 3 , Hence the value of the whole reverfion will be= 
— 1373 
3^ 
S I d* e 
o r c*r c b r 
3 r * 
~CCC 
- & c . + | + &c * - 
L -CCC4 
r r 
S r- i . V — CCC 
— _ X - — — — - — — 
Q. E. D* 
problem XX. 
To determine the value of a given fum, payable if Afhouldi 
be the Jecond that fails of the three lives A, B, and C 
3 O L U T X O N. 
The fum S may be received in the firft year, provided either 
* This expreffion may alfo be obtained from either of the above general rules, 
independent of the two others, in like manner as in the folution of the fourth 
problem, ^ 
7 
