estimation of the value of life contingencies. 289 
taking N. = (1 - ff) ■ (1 - ) and M x = the 
assurance of one pound on the contingency of that example 
\\ 
.-.1 i ~~h i fJt + P t. + f 1 + C+w + lP^ 
ill become ^ .(1 - — 1 ■ ( 1 — 1 ■ T7- f 
m — pi ' ' 
• (1 
~“b + 7 T p 
~‘c-\-tt + \p\ 
-A)('-” +> - ( 1 
m—p\ 
r p . x the income of one pound payable at every p in- 
c 
terval, the first in the time n — p, and the last in the time m — p 
on the life of the age c-\-±p, after the death of the two per- 
t‘s. it 
sons of the ages a and b, • r~ x the income of one pound 
payable at every p interval, the first in the time n, and the last 
in the time m on the life of the age c — ip, after the death of the 
ages a and b. “ Because the first term of these two expres- 
L 
“ sions is r f . 
j~t 
* n—p 
m — p 
. •* 
[r . l- 
L * L ' L , lA J 
a b c+\P / 
T 
({ p 
n—p 
m—p 
^c-pir-plp \ , i 
L““ * 1 l 7 ’ L ™ ) and the 
a b c / 
“ second term of the expression is 
~“c + \p 
it , 
fn 1 <* + n 
L b+n L c + v— fjA p 
L b ‘ L C-±p 
n — p 
m — p 
r n +P 
(r . 1 — 
J a -j- tr -^-p 
. 1- 
J b + *r+p 
'‘c-pir-plp \ ,, 
Or we may develope the expression in the form 
