§68 
Mr. Gompertz’s analysis applicable to the 
L , L, , L . . L . L 
x : a, b '• ' *• 
— j-ti. . fluent of - y— — -j- - , a+ - T fluent of 
L , * L. “ " L L * L 
a, b b c a a 
and here it is evident, that when the lives are all 
^“c-f-x ' ^“b+x 
L ~7C, : 
c b 
h x . a a L 
equal, the contingency will become constant ^ ' ■ -j — j ~ ~ 
a, a « 
x constant 
X = 0 it IS 1 
-}- f x ) ; or if all the fluents commence with 
(L±iJ_ L±f + (L±fj; But in other 
cases, if we are to have all the contingencies commence with 
x—o, and we are satisfied with the approximation that when 
two persons are dead, it is an equal chance which has died 
first, see note, Art. q, Section 4, we shall have 
b X r c X U -j* X j b 4* X ■ j X . 
7 — fluent of -f— .-r-=-iT- — \ri~r 
a,b j x: b, c 
L * L 2 L l 1 2 L_~ 2 “ 
"x : a, 6, c 
J a, b 
J 6, c 
3 L 
a,b,c 
and 
CL -j- X 
fluent of 
^ ~“c-\-x ’ ^ b-\-x 
I a + x 
L L h 
C 0 
L , L L 
j j ^ • Clf O j X . CLy C | j X l Cl 
1 2 “ r “• r 2 " 
2 L » 2 L r - 3 L 1 2 L 
a a,b a, c a 
L. 
and therefore the whole becomes = — l —7 — - — l a * 
- -Lj r ■* L 
b a 
2 L 
L ¥ L L r 
x : b, c j x : a, c B x : a, b, c 
+ L 
2 L 
1, and the part of the 
by C C Cly by c 
contingency due to the interval between the times ?r and n r +P 
^6 + 7T ^ J 3 + ?r+/> . ^+7r ^7r:3,c ^ir+p : c , 
is — si — + — r; — + — jy— + 
T oac 1 n , - J t 
L sr: a,c n-\-p:a,c n : a, b, c 7i+p,a,b,c 
zL 
Hence the assur- 
a, c ~“a, b, c 
ance of one pound on the contingency, in case the event should 
happen between the times n—p and m ; to be paid at the first 
