Mr. Gompertz's analysis applicable to the 
periods, &c. and for all these purposes it is necessary to have 
the value of the contingencies referring to each payment. 
And in order to this, for the sake of brevity, let E a n what- 
ever a and n may be, represent the chance that a person of 
the age a , shall be living at the expiration of the time n , and 
D be the chance of his being dead at the expiration of the 
time n. And consequently, E an -f- ^ a , n == 1 ; alsoF a )1 = 
L L 
H , and D a n — 1 YT~’ Moreover, if we introduce any 
letter x as a multiplier of E a n , if x be equal to unity, since 
E = oc . E a>n , we may under that idea of x being = unity, 
write s.E ai „ + D„_„ = l, *E a> „ + D c> „ = i, &c. ; and also 
( xE «,« + D ( 7 , «) * ( x Ej, n + D i>, « ) * UE c n + D Ci „ ) +&c. 
= l ; and if the left hand side be multiplied out at length, 
and there be p persons or p multipliers, then the coefficient 
of x 1 * will be the chance that all the p persons shall be living, 
the coefficient of x will be the chance that there shall 
be p — l and no more living, the coefficient of x ^” 2 will be 
the chance that there shall be p — 2 and no more living, and 
generally the coefficient of x^ n the chance that there shall 
be exactly p — tt of them living, and the sum of all the co- 
efficients from that of x 1 * to that of n both included, will 
be the chance that there shall be p — n or more of them liv- 
ing. If we write 1 — E w , a for D a ,n our equation will stand 
•*-1 • E„,„ + 1 . E fc . + 1 x i— -i" • E,_„ + 1. &c. to 
p terms = 1 , x being supposed equal to unity. This is an 
identical equation, and if for the sake of brevity we put P ^ j0 = 
