Royal Society. 41 
three numbers l<inv will be equal to the continual produfl of 
R -f- Y X r -f-jy ^ ? + ^> which, being the whole number of 
chances for three lives, is compounded of the eight products fol- 
lowing. I. R r f, which is the number of chances, that all the 
three perfons are living. 2. rfY, which is the number of 
chances, that the two elder are living, and the younger dead. 
3. Rfj/, the number of chances that the middle agc'is dead, 
and the younger and elder living. 4. R r i/, being the chances, 
that the two younger are living, and the elder dead. 5. f Yy 
the chances that the two younger are dead, and the elder living. 
6. rYv, the chances that the younger and elder are dead, and 
the rniddle age living. 7. ^yvy the chances that the younger 
is living, and'the two other dead. 8. Yy v, the chances that all 
three are dead • which latter fubitrailed from the whole number 
of chances N n f , leaves "Nnv — Yyvy the fum of all the other 
feven products 3 in all of which, one or more of the three per- 
Ions are furviving. 
To make this flill more evident, in Fig. 7. thefe eight feveral 
f>rodu£l:s are exhibited at one view : Let the rectangular paralle- 
opiped ABCDEFGH be conilituted of the fides A B, 
G H, ^c. proportional to N the number of the younger age • 
AC, BD, ^c. proportional to nj and A G, CE, ^c, propor- 
tional to the number of the elder, or r 3 and the whole parallelo- 
piped will be as the produ61: N /*i r, or the whole number of 
chances : Let B P be as R, and A P as Y, let C L be as r, and 
L ;? as y, and GN as f, and N A as 1*3 and let the plane VKea 
be made parallel to the plane A CGE 3 the plane NU^Y pa- 
rallel to A B C D 3 and the plane L X T Q^ parallel to the plane 
A B G H 3 and the fird product Kr§ will be as the folid S T W 
I F Z ^ ^ 3 the fecond, or r f Y will be as the folid E Y Z £> OS 
MI3 the third, Kry^ as the folid RHOUWIST3^ the 
fourth Rri/, as the folid Zal^DSNXlK, the fifth fYr, as 
the folid GQ^RSIMN03 the fixth rYt/, as the lolidlKL 
MGYZA3 the fevcnth Rjj,- y, as the folid IKP O BX UW 5 
and the eighth Yyu^ the produ61: of the three numbers of per- 
Ibns de:id, as the folid A 1 K L M N O P : Now to fhew in one 
cafe, how all the reft may be performed, let it be demanded, 
what is the value of the reverfion of the younger life, after the 
two elder propofed ? The proportion is, as the whole number of 
chances, or N « j/, to the product Kyv, lb is the certain prefent 
value of the fum payable after any term propoled, to the value 
due to fuch chances, as the younger has to furvive both the elder, 
by the term propoled, which therefore he is to pay for; here it 
Vol, lil. F is 
