estimation of the value of life contingencies. 247 
t = o to t=m, equal to — mU„+ n x T^P+ n 
T 4 - L' 1 L A , &r 4 - &c.) + &c. = — mV K+n * 
• r. 5, See. U ^ q + n n,p,r,s,& c. • _ 
t neglecting: the remaining terms in which 
^n+irrtyp.q.ryS, &c.> t> 5 
m 3 &c, not herein contained are concerned, as they will De 
small if m be small ; though, if necessary, we may pursue 
similar means to those used in the last article. 
Article 13. But if during short periods, instead of the arith- 
metical progressions, we use geometrical progressions, see 
Article 2 , Section 2 ; and x being =n-{-t we take = 
'Lp+n • 10 I’ ?+ & c -»L x+x =L,+„.Io| 
and therefore putting tt for the hyperbolical logarithm of 10, 
t L' 
T __t T' * we shall have if for the sake 
* ^+n‘ 101 ’ ; 
of brevity we put |x = Vp +n + + L , r+n + See. + L x+Wj 
Lx q, r, s, Sec. * * + w * :p, q, r, s, &c., * * ^ ’ and 
the fluent generated w'hilst x from w becomes = n-\-m is = 
L' 
L n : p, q, r, s, &c. x ' * 
: p, q, r, &C., k* 
*-f n 
(75T* — i) = — . L' xL 
+ m-.p,q, c.&r.jK 
Article 14. Because L, ;>j?jrj&c>x = L M : ^ fjr>&CtX xiol^and 
L = L lot it follows that — V . = 
Log : of L — Log of L 
; x being equal to n-\-t } 
Log . of . p, ^ r * ^ ^ Log . of L^ _ ^ r> x 
and £ any positive quantity not greater than m ; the same set 
of geometrical series being only supposed to take effect be- 
tween the intervals n and n-\-m ; but if the geometrical series 
between those limits only take effect proximatively, our 
fluent will be but an approximation, though as correct as we 
