Mr . Morgan on Sitrvivorjhips. 
49 
there are fix fuch orders equally probable) 4. x V - 3CC - 2 CCC. 
But it appears, from the corredlion explained in Dr. Price’s 
Treatife on Reverfionary. Payments, Voh 1 . p. 34. that the 
value of a reverfionary fum is always lefs than the value 
of an equivalent reverfionary eflate in the proportion of 
1 to r. The fum being S the equivalent eftate or perpetual 
annuity is always Sxr-i ; and confequently the value of 
the lum S depending on the ceafing of three equal lives in any 
one particular order and thus determined, is the fame with 
that determined by the foregoing inveftigation, that is, 
S ~~ 
r — I 
x V — 3CC - 2CC. The inveftigation, therefore, is 
yj / 
right, and the correction and inveftigation demonftrate one 
another. 
But the foregoing expreffion for determining the value of 
the reverfion in this particular cafe is not only obtained imme- 
diately from the feries, but alio from the two different rules 
which have been given for determining the value when the 
lives are unequal ; and hence a proof arifes of the truth of 
thefe rules, as well as of the reafoning upon which they are 
founded. Thus the firft rule, fuppofing the lives all equal, 
dd 
becomes 
x. CK-CCK 
** KK-CKK 
__ x 
cc 
X 
X - 
c 
6 
V — CC .7-1 
2 r 
y. 
CK-CC 
— — 
X 
c 
3 
the firft rule 
r — 1 . 
. v-cc 
d CT-CCT 
x — — . 
cr 6 
KK-CKK 
TT - C TT r - i y CC — CCC 
6 r ' 3 
and the fecond rule becomes 
TT-CTT 
dd 
X 
T~ — t x 
cc . r 
r-i CC— CCC 
X —r — 
r 3 
d CT-CCT 
+ ~ X ' 
3 
Let the value according to 
the firft rule be denoted by L, and the fecond rule will be = 
1; 
2 r 
~ 2 L 
Vo l. LXXIX. 
r — 1 . CC - CCC x 
r ^ 
H 
/ f 
Hence 3L 
r 
