Royal Society. 39 
dead of both ages, are the chances that both the perlons are dead 5 
and the two products of the remaining fums of the one age mul- 
tiplied by thofe dead of the other, (hew the chances there are 
that each party ftirvives the other 3 whence is derived the rule to 
eftimate the value of the remainder of one life after another : Now 
as the produ6t of the two numbers in the table for the two ages 
propofed, is to the difference between that produft, and the pro- 
dua of the two numbers of perlons decealed in any fpace of 
time, fo is the value of a fum of money to be paid after any time 
propofed, to the value of the chances, that the one party has, 
that he furvives the other, whole number of decealed you made 
ufe of, in the fecond term of the proportion : Thi > perhaps may 
be better underftood, by putting N for the number of the younger 
age, and n for that of the elder 5 Y, jy the decealed of both ages 
refpeclively, and R, r for the remainders, and R -j- Y in: N, 
and r -^y =: n ; then N n will be the whole number of chances ; 
N« — \y be the chances that one of the two perlons is living, 
Yy the chances that they are both dead ^ R 3; the chances that 
the elder is dead and the younger living 5 and r Y the chances 
that the elder is living and the younger dead : Thus two perfons 
of 18 and 95 are propoled, and after 8 years thele chances arc 
required; the numbers for 18 and 35 are 610 and 490, and there 
are 50 of the firft age dead in 8 years and 73 of the elder; there 
are in all 6'ioX490 or 298900 chances, of thele there are 50X73, 
or 3/5'5o that they "ire both dead ; and as 298900 is to 298900 
— 3(^50, or 295255, {o is the prelent value of a fum of money 
to be paid after 8 years, to the prelent value of a fum to be paid, 
if either of the two live; and as 56*0 x 73, fo are the chances 
that the elder is dead, leaving the younger; and as 4.17X5O, {o 
are the chances that the younger is dead, leaving the elder; 
wherefore as die x 490 is to 5^0 X 73, fo is the prefent value of 
a fum to be paid at 8 years end, to the fum to oe paid for the , 
chance of the younger's furvivance; and as (^10X490 is to 417 
X 50, ^o is the fame prelent value to the fum to be paid for the 
chance of the elder's furvivance : This pofiibly may be ftill better 
explained by expounding thele products by re61:angular parallelo- 
grams as in Fig. 5. wherein A B or CD reprelcnts the number of 
perlons of the younger age, and DE, BH thofe remaining alive 
after a certain term of years ; whence CE will anfwcr the num- 
ber of thofe dead in that time ; lb A C, B D may reprcfent the 
number of the elder age, A F, B I the lurvivors after the fame 
term ; and C F, D I thofe of that age that are dead at that time ; 
then the whole parallelogram A B C D will be N ;/, or the pro- 
dua 
