for the Valuation of Life-annuities. 123 
Therefore, the fecond feries fubtradted from the firft, 
r r 
1 + - I 4 - 
j 2 2 
leaves x h~ p~ — - x h- h, agreeably to the fecond 
r nr nr ° 
theorem in p. 1 1 4. 
By reafoning in the fame way it may be eafily found, 
that q=p 
r r 
1+7 1 H 7— 
4 -j 1000, &c. m 
xq; and M = p x m~v , agree- 
nr nr nr ° 
ably to the third and fourth theorems in p. 1 1 4. 
Thefe theorems, I have faid, fuppofe that an annui- 
tant is entitled to no payment for that year, half-year, or 
quarter, in which he dies. If, on the contrary, he is to 
be intitled, when he dies, to fuch a part of the yearly, 
half-yearly, or quarterly payment as fliall bear the fame 
proportion to the faid payments refpectively, as the in- 
termediate time between the laft payment and his death 
bears to the whole year, half-year, or quarter; in this 
cafe, fuppofing the annuity payable yearly, it is evi- 
dent, ftnce there is the fame chance for his dying in one 
half of any year as in the other, that he will have an 
expectation of half a year’s payment more than he would 
be otherwile intitled to. But the value of half f. 1 to 
be paid at the death of a perfon whofe complement of 
life is n, is | x — L=+{x — i4=- +|x — 3 -, See. conti- 
ttXi+r MXl-f-rl wXi+r| 
nued to n terms w)= 
( d ) P. 1 18. 
R 2 
In 
