Survive,} flips. 249 
That all the three lives (hould fail, and that A fbould die firft. 
2 dly, That B (hould die after A, and C live. 3<dly, That C 
fliould die after A, and B live. 4thly, That A only (hould 
die, and B and C both live. Thefe feveral contingencies being 
_ . , a' . b-m . c—d a' . b-m . d 
exprelled by - 
d . m 
2 abc 
, and 
dm 
abc 
I a'bc 
— X 
abc 3 
a md 
a' me. a'bd 
In 
3 abc 9 %dbc 
refpedtively, their fum will be 
the fecond and following years one or other of the fame events 
muft take place in order to receive the given fum ; that is, they 
mud: either all three die, A dying firft ; or only A and B muft 
die, A dying firft ; or only A and C muft die, A dying firft ; 
or only A muft die, and B and C both live. The different 
fractions exprefling thofe four contingencies fot the fecond year 
being f and ^ ; for 
• abc 
\abc 
2abc 
, o!" . n — o.e—f o"’.n-o.f a n d ’ 
the third year ^ > — ppp c > zaH > dUU abc ’ 
and fo on, the whole value of the given fum will be = 
S .a' be . mi 
jn'i . nit. , vf* , 
x - + — +-J-+ -7- d 
aber 330° 
bd .S'. cl' _ mi , ne 
me . S .a 
j/r 
ne , ru, rna . 
X — 4- {- T *7— + —7— ' 
aber 21 3 3^^ 
. a'" . 
* e + -+°i +J+&C. which are = — x V + 
en 
+ &C.+ 
3 3 6 1 6 1 ?> abc _ 
8 x c.+A- x"+?+^7&c. 
babe 
r 
s_ a nut a" • . *' 7 * _yfo Ci From the demonftration of 
1 — x 
~ 3^ 
the problem in my laft Paper * it appears, that the firft of 
thefe feries is = ~ x 
0 y. . FK-.AFK S 
be 
- - x BC - ABC ; that the fe 
y 
. S k . BK— ABK S m • PC APC *.u 0 > fin** 
cond feries is - - * 7— - & x —7-““ > that tne 
% See Phih Tranf. Vol LXXIX. 
L 1 
Vol. LX XX I. 
third 
