. estimation of the value of life contingencies. 223 
“ of them dead ; therefore the chance of v of them being liv— 
“ ing + the chance of [x — v -|-i of them being dead, is equal 
“ to unity, that is certainty/’ and the said excess will be writ- 
P 
ten n 
m 
a ’ b ’ fii/i , and is expressive of the chance of tt 
or more being dead, 7r being equal to [x — v -{- 1 ; or p } that is, 
the number of persons in all being — 1 ; and as the sign 
of summation and the interest indicated by r, the present va- 
lue of one pound certain to be received at the years end, affect 
both contingencies alike, we may join those contingencies and 
y tly by Cy See. tll0 
o, v 
use but one symbol, and we shall have « 
m 
value of the income for the term on the contingency, that out 
of 7r -j- v — 1 persons, there shall be tt or more of them dead. 
Accordingly 
r r r 
Would n 
by c 9 
— , a, b, c , and £ 
m 
0, 3 m 
0, 2 rn 
spectively signify the value of the income for the term on the 
contingency of one or more' being dead, of two or three be- 
ing dead, or of all being dead. And these besides being an 
appropriate abbreviation of the expressions, likewise indicate, 
that if from the value of the income certain for the term we 
subtract separately the value for the income for the term on 
the contingency of all three being living, on the contingency 
of two or more being living, and on the contingency of one 
or more being living, of three persons, we shall have respec- 
tively the value of the income for the term on the contingency 
of one or more being dead, two or more being dead, or of all 
