282 Mr. Gompertz’s analysis applicable to the 
any small interval/), that the living corresponding to time 
which are in arithmetical progression, are in geometrical 
progression. (See Section 3, Art. 2), ^ 
»+ p—q 
: a, b, c, See. 
a, b, c, he. will 
be = r* . 
pothesis 
L 1 — r * . T 
•tt : a, b, c, h c. 
'‘a, b, c. Sec. 
<ir: a,b, c, he. 
~V-J- y : a, 6, c, he. 
; but by the hy- 
1— r’ 
: a, c, he. 
~‘v -\-q : a, b, c, he. 
J it : a, b, c 
1 L 
ir+p:a, b,c,hc. , . r 
— ~ L —- ; consequently, if 
it', a, b, c, he. 
r 
we put r p . c> &c ‘ = 1 — k } we shall have l 
vr: a,b, c, he. n+p—q' 
a,b, c, he. 
tt Xj tt: a,b , c, he. i — k 
r . _ — x 
"V : a, b, c, he. 
Here k is generally very 
small ; and if in the developement of 1 — k\ p , we are satisfied 
with retaining only the first and second powers of k, we shall 
have 
tt+p — q 
" ft : a, b, c, he. 
a, b, c, he. = T . ““t— — X 
J a, b, c, he. 
q—q • q—p 1 q 
’p p 2 p ' 
it *r: a, b, c, he. , 
V . — X ( 1 
^ a , b, c, he. 
p—q 
2f .k) nearly; = fr: — 
p tt it: a, b, c, he. 
. __ £_ r — 
a, b, c, he. 
* [*$■ + = 
p-\-q v v: a, b, c, he. ^—<7 V -¥P 
2 P 
2q 
r . 
J a, b, c, he. 
“h 
t*? 
. r 
—fP - a > 6 > c > &c - # Hence, if we interpret n successively by n , 
a, b, c, he. 
n -\~P> nAr^-p, &c. m — p, we shall get as an approximation 
from the above value of n 
m 
a,b 3 c,hc. i n 
— m 
a,b,c,hc. . X = ( P+q 
P l 2 p 
