222 
Mr. Gompertz’s analysis applicable to the 
when it depends on none, or any of them being living, and 
is therefore the income certain and quite independent of 
the ages a, b,c, and therefore this case may be written with- 
■, and will be expressive of the pre- 
sent value of an income certain for the term, See . ; o inserted in 
the angle signifying that it does not involve any contingency; 
and this presents us with a ready notation for the purpose of 
expressing an income for a term, a rate of interest, and a 
periodic interval of payment given on any nominated contin- 
gency specified by any particular character, by placing that 
character in the angular point ; thus if the contingency in- 
volved were designated by the character C, then the income 
for the term, rate of interest, &c. depending on that contin- 
ent those letters, thus 71 
gency, we should denote by 
r 
Art. 6. Moreover, as 
expresses the value of the in- 
p 
come certain for the term, and « 
m 
a ’ b> c ’ &c ~ expresses the value 
of the income for the term on the contingency of v, or more 
of the persons of the present ages a, b , c , Sec. being living ; the 
excess of the former above the latter, will express the value 
of the income for the term on the contingency of ix — v + l, or 
more of the persons being dead ; if [x be the number of persons 
in all : “ because it is certain that out of p persons living at any 
“ time, there will after that time either be v or more of them 
“living, or v — a or less of them living including o ; butv— i 
“ or less of them being living, is the same as p — K-f* 1 or more 
