ASSURANCES ON LIVES. 
155 
the probability of the joint existence of the last v 
survivors at the end of 0 years, that is, of their being alive at the pre 
sent moment is unity, and the remaining part of the expression is 
2(1+0"+ 
(m, mi, m-2, &c.), n. (m, mi, m2, &c.) 
} 
.*.2(1+0"” p 
— (l+O 1 {l + « (m , ml) mi L.) } - a T^. 
V 
V 
A 
v 
;c.) ®(m, mi, mg, &c.) î'”(l" , î') flj 
(1+0 l +(l+0 
v 
V 
1 — i ci- 
1 +Ì 
The formulai 
may be used with equal convenience for finding the present value, or if 
the calculation he made by both methods, one will verify the other. 
The first of these formula is the one employed by Mr. Milne, the 
latter by Mr. Baily, in their valuable works on the subject. 
188. When there is oxdy one life the formula becomes 
•r—(1— r)rt„„ 
189. By Davies’s method— 
The present value of the wth year’s payment is found by mul 
tiplying the present value of £ 1 due at the end of n years by the frac 
tion which has for its numerator the number who die in the rzth year 
from this time, and for the denominator the number living at the pre 
sent age. Let us call d m the number who, according to the tables, die 
in the mth year of their age ; then 
rd m+l + r 2 .rf m+2 + r 3 .t? m+8 + r*.d m+i + &e. &c.