2 34 
Mr. Gompertz’s analysis applicable to the 
p\a,b =rK -ti±±+ (rf. tL£±y+&c {r* . Y; 
« 4 h a,bJ L a, b ) 
a,b,c=rK r? V^ + &c trf.^±i) m ; 
— *, & L a,6,c / 
P 
P 
m 
and also 
/> L, , L v® L. , v w 
?[ e = r p . -{- ( rA — ^±1 ] + &c (r'-£±I^ 
L , L , ) L 2 ) 
m 
Hence, if ■ — — f - "~ , then /|« = ^ 
e a, b rn ' m 
a, 6 ; and also 
a, b, c = /> 
- — m 
e, c ; that is, according to this hypothesis, if the 
value of a periodic income of one pound to be paid at the ex- 
piration at every interval/*, for a given number of intervals, 
on two joint lives, whose present ages are a, b, be equal to a 
similar income on the single life, whose present age is e, then 
the similar income on the three joint lives of the present ages 
a, b, c, will be equal to a similar income on the lives whose 
present ages are s and c. And it also follows, that a similar 
reduction may be made for more lives, and their value thus 
obtained according to the hypothesis. And this, I imagine, 
is the foundation of the practice for determining the value of 
annuities, on many joint lives, from two tables, of which the 
one is on single lives, and the other on two joint lives ; and 
we discover, that if the same geometrical progression, or a 
proximity thereto, does not continue through the whole, or 
the certain portion in question of the lives, this method, which 
