272 Mr. Gompertz’s analysis applicable to the 
' 2 ah 
! c'a" , r a"c'b" 
~T 2 nl r 
c a 
a" V c" , a" h" C" . 
H — ’ 
a b , 
and the corresponding value of 
~‘x: a, b 
T T 
a, b 
a " b" 
a b 
zab c 
Hence we 
have the contingency on this hypothesis due to n - {-1 years 
a" b" 
ab 
If!j. L a ' b " 
2 h 2 a “f” 2 
+ a" b" c r a' a- c a 
n n r n n h r 
b" c" 
a 
2 
■»" c" 
ab c 
za b c 
2 a b , 
+ 
a b 
a" W c" 
ab c 
, j a" ¥ , cpbp_ i_ cf a! • 
• 2 a b 2 b c 2 a c 
; and if we take from this 
the chance of the events happening in n year, the remainder 
will be the chance of its happening during the interval be- 
tween n and n- f-i years; and will come out evidently — 
2 a 
b'—b" 
zb 
a'b" + a"b'— 2a " b" . ab'- 
1 
2 a b 
■ za ’ b' + g' b—ab"—a ll b+za"b" c' 
zab c 
a' b" -\-g" V — 2a" b" c" 
zab c 
And if we consider Mr. Baily’s a\ b 1 , c ' ; 
a' 1 , b",c" to refer to the n th and n-\-i th year, and collect his 
1st, 2d, 3d, 5th, 6th, 8th, 9th, and 12th, contingencies, we 
;and 
a' — a" Ib + y 
b—b'—b" 
c" b" 
shall find them amount to — — , — 7 — 1 — . t r- 
a \ zb 1 c zb 2b c 
collecting the remaining terms which are 4th, 7th, 10th, 11th, 
1 . 1 , . b'—b"(a + a . a— a'— a" c’ «"c"\ b'— b" 
and 13th, we obtain — r — — [ 1 . ? — 
b \ za 1 za c ac I b 
, ; and if these two be added together we shall obtain 
the above expression. 
Example 2. On the contingency of the first of the deaths 
of A and B, which shall be the first or last of the three A, B, 
C. This will evidently be the sum of the contingencies, of 
the joint lives discontinuing in C’s life time; that A dies after 
B, C having died before B ; and that B dies after A, C having 
died before A ; therefore by Art 5 of this Section it is == — 
L 
fluent of 
c-\-x 
>j- fluent of • fluent of (1 c+x 
) 
