2 37 
estimation of the value of life contingencies. 
the product of all the (x terms E^ n , E i n , E C>M , &c., P ^,/ \ — 
the sum of the products of every 1 terms, P ^_ 2> 2 the sum 
of the product of ^x — 2 terms, &c. the equation will stand 
x — i\ 
1 
. P . .+ x 
y . — I, I I 
2 
P „o 
when x = 1 ; that is 
— l T-v , fx — I M — ’2 15 /A— I ft— -2 Jt — 3 
. P , , &c, = 1, 
2 , 2 > 
/P„, • ^~ 3 • P* to. 
+ ■ ^" 2 ■ P,-, . + t =r ■ *t- ^ p «- ■- ■ I*- 
PjLX 2 , 2 ) &C. 
&C. 
+ ^- 2 p p _,, 2 -^x '‘“ 3 
!• 
I 
+ ^ -3 P 
And the coeffi- 
cient of 3 ? w or 
the chance of 
their being at the 
expiration of the 
time. 
Just iJ. — 7T living is P^_„^— ?-Z±±. P^_„ +1 , +*-7 
^“ 3 > 3 
^ — 77-+ 2 \t 9T+ I p r, 
2 • r /x — w+2, 7r-=»2 OU -" 
Ditto p— TT-\-l . is 
Ditto {x ““7 r *-{-■ 2 . is 
Sec. 
"f“ P/x — 7 r-pi, 7 r — 1 
ft — It -J- 2 
’ W-^2 
P|t * — tr + 2 , tr—2 ^ CC> 
See. 
and the sum of these, or the chance that there shall be [x — 7 r 
or more living, is = — f . P^_*. +1 , 7r _ I + , 
P „. + 2j ^_ 2 — &c. Similarly if x, x', x", See. be each 
unity, we have the identical equations^— 1 lE, a n -\- 1 xx— 1 .E^-f* 1 
xx — 1 E Ci n -f 1 , &c. x x'—i . E a >n -\- 1 x x ' — 1 . E^ n 4- 1 x 
x'— 1 . E c , 5 n + 1 &c. X x''~ 1 . E fl , (j H -f. 1 x x"—i . E b „ t n -f 1 &c. 
= 1, and the coefficient of / xi' ff x x ,,7t will be the chance 
of there being exactly 4 living of the first set, 7/ living of the 
second set, 7/' living of the third set, &c. We remark by the 
bye, that if a,b,c , &c. be equal to each other, then the equa- 
tion x — 1 . E fl)M -f- 1 x x — 1 . E 6jK -f 1 x &c. to p terms = 1, 
will be x — 1 . E a n -j- 1) = 1, and thatch— 1 . E a>n 4 - if being 
MDCCCXX. Ii 
