14 0 Mr. Morgan on Survivorships. 
years, and A and C both die in that year: and 5 thly, If A 
should die after B in any of the preceding years, and C die in 
that year. From the different fractions ex pressing those pro- 
babilities, the value of S may be found = -7 * T + T' + 7 ^+ &c ’ 
, S da' 
+ ^T*T' 
'+ a " 
zabc 
S 
a acr 
da 
X — 
S a'm , a"n . a'" .0 , 1 ' 
— « — Z >» . S a'md . ci'ne . a 1 " of 1 o. r 
■+*?+*?+ +~+~ + & 
+ &c. + 
ed . 
x T + 
Il£± 2 - 4- &c. 4 
+ &c - IZbZr 
s 
nmd 
+ 
fl'-f a" . ne 
zabcr 
S .7 
x £^ 4 - 1 ± 4 ^+&c- The 
three first of these series are = 1 x V — A C + AC, 
and the remaining eight denote the value of S by the third 
problem, with contrary signs. If this last value be called \ , 
and the value of an annuity of £ i. on the longest of the two 
lives of A and C be called Z, the required value will be — 
s x y y y ; that is, the value of the given sum in 
this case is “ the difference between its value after the extmc- 
« tion of the lives of A and C, on the contingency of B's sur- 
“ viving A, and the whole value of the reversion after the 
« death of A and C, without any restriction." This rule is 
self-evident, and proves the truth of the foregoing investiga- 
tions. The solution of this problem may also be derived from 
the second problem in this paper, and the third problem in my 
paper communicated in the year 1788.* In other words, “ the 
« value of S in the present case is equal to the difference be- 
« tween its value after the death of A and B, provided B 
Phil. Trans. Vol. LXXVIII. p. 347- 
