ago Mr. Morgan on 
The preceding theorem exprefl'es the value of the given 
fum whether C be the oldeft, or B one of the two lives to be 
ftlrvived, and will therefore be lufficient in all cafes. In order, 
however, ftill more fully to prove this, let A be fuppofed the 
oldeft life, and inftead of b — m , u. — n , n— o, &c. and a\ a" , 
a'", &c. let b\ b", b'", &c. and <T-7, 777, t- u, &c. be 
fubftituted ; then will the value of the reverfion be found = 
+ + + &c. + 
2bc r r ? 6 
S x^+^' + ^-+ &C. + 
babe r 
S csV , dUb’' eub‘ 
T + ~ + ~? 
X - 
2b c 
— + • 
r 
r z + A 
S 
^ 
— 4- 
ft . vTP' 
2- ber 
r 
r~ 
S 
X 
eu . b' b" 
4 - &c. — 
+ &c. — 
+ & c . + i.<: + & c . - 
babe r r‘ i 3 2 aber r r 
iix"'+ +^ 3 — + &c. x 111' + A^±i' + &c. 
^abc r r r 3 r r 
Let S denote the value of S on the contingency of C’s fur- 
viving B, and the general rule deduced from the preceding 
fenes will become = 2 — X — f Alv-xicK + 
xHC-HBC-il^x AC-AAC-^ r , xNC-NBC + 
X AT — ABT -f ■- ; - N V - n „ NBT ? which appears to be exaftly 
2 a 
S . « 
6 a 
S . d 
3 cr 
the fame rule with the foregoing, if the fymbols of A and B 
be only exchanged for each other. 
If the three lives be of the fame age, both thofe rules will 
feverallv become == S into ~ — - x V — CC — - 1 X KK — CKK — 
J 2 r bee 
1 x CK - CCK - x CC - CCC + ± x CT- CCT + £ x 
dd 
but 
TT- 
2 
