221 
estimation of the value of life contingencies . 
before. The present value of a periodic income of one pound 
payable on the joint lives of persons of the present ages a , b, 
c, Sec. at equally distant intervals p , the first payment being 
due in the time n, and the last in the time m, the present value 
of one pound certain due in one year being r. 
If there be not any character in the place of the r, in the 
last symbolic form, the interest is supposed to be the same as 
in some previous consideration, which shall be evident from 
the text ; thus, if we had been conversing of an interest of 
5 per cent, per annum, we should write » a, b, c, to express 
the present value of an annuity of one pound (that is one 
pound payable yearly) on the joint lives of the present ages 
a, b, See. the first payment to be made in one year, because 
in this case/>=i, ?i=i, and m is infinite, or extends to the 
longest possible duration of life. 
Art. 5. If the income is on the present ages a, b, c, See. be- 
tween the terms &c., on the contingency of v or more of them 
being living, we write as a symbol of its present value 
a, b, c, &c. , and according to this notation we should have in 
the following particular cases for the present value of the in- 
a > b, c ^when the income 
p 
n 
a, b, c, 
P 
same as n 
W 
c 9 « 
a , 
m 
3 
m 
m 
z 
for the term depends on two or more of the three persons 
whose present ages are a , b, c , being living ; « 
m 
a > b> c . when it 
depends on one or more of the three being living ; that is, 
r 
'w 
p 
what is termed on the longest of the lives ; and £ 
mdcccxx. G g 
in 
a, b, c 
