278 Mr. Gompertz’s analysis applicable to the 
well as the assurance on each, has been already considered. 
But if we use immediately for the approximation of the fluents 
of 1 C j~ . ■ ■ b ^ rl and of 1 when those con- 
C 0 C a 
tingencies are meant to commence when x=o, and not to last 
longer than the possible joint continuance of life, the approxi- 
mations so often named, which are respectively — -j- -|- -T 
*4" 2 L c+X — 2 ^x:l,c and 2 + a + x~h $L C+X — f L x:a>e 
our formula, independent of the above reduction, will become 
U “f” o 
. L 
b + x ' ^a+x . j x ’ k<z+, v j : b, c 
fluent of — + , L 
J a -f* x 
b + x , ^a + x ' ^ b+x , T ^V+x 
“» 71 TTT r 2 
L 
& 
J b-\-x 
c a 
L, 
2 L 
L 
~} = 
J b 
L 
3L *, C 
L 
! x : a, c 
2 L 
a, c 
correction — •§■ — — £ 2L 
a 6 
£+* , , x:a,b t T 
IT T. . T 2 
fluent of 
L 
L , . L . 
c+x a+x 
-}- £ fluent of/ 
■^C+X ^6-f-x 
a, b 
— 4 fluent 
of ■ '' But this ^ ast 1T ^ et hod throws the approxi- 
c \ ^ a,b } 
mation on the whole term. 
I may observe that the first method will resolve itself into 
. L a + x L b + x „ , r ( L c+x / L x:a, b 
correction • — — p- — {- fluent of f — ^ — ( — — — 
^ a b \ c V b 
— contingency of Example 1 Art. 7 of this Section, and the as- 
surance will in consequence be the assurances on the single 
lives A and B together ; — the assurance of that Example ; — 
the assurance on the three joint lives A, B, C ; -{- the assurance 
of C’s life provided it fails before A and B. The last two 
