E 
REVERSIONS.. 
49 
Let i' be a quantity found by trial somewhat near the true value of i, 
and let i “ i' 4 x, then by substituting this value in the above equa 
tion, it will become 
(1 4 i' 4 z)~ d -(I + V 4 2 )- (d + B) = ^ (¿' 4 z); 
by the binomial theorem, 
{ (1 + V) 4 zj~ d = (14 i') 
d (d 4 1) 
- d (1 4 * 4 
(1 4 ¿')" (d + s) z 2 —& c. 
{(1 + V) + z} - ( d + n) = (1 4 ¿')" (i+n) - (d 4 n) (1 + 0- (rf+n+1) 5r 
(d 4 n) (d + n + 1) 
+ 
(1 + ¿')“ (d+ " +2) 2 2 — &C. 
subtracting the second series from the first and rejecting the terms 
affected with the second and higher powers of z, we obtain 
(1 4 i’)~ d - (1 + ¿')“ Cd+n) —d( 1 + ¿')~ (d + 1) 2 
4 (d 4 n) (1 4 ¿') _(d+B+1) 2 = — 4 —; 
by transposition, — 4 d( 1 -f ¿ / )“ c<, + 1) z—(rf-f-n) (l + i') -(d+ ” +1) ^ 
= (I +i')- d - (i + ¿')~ (d+n) - 
pi 
dividing each side by — -f-rf(1 -f- ¿ , )“ (d+1) — (d + n)(1 4 ¿')“ (d+ ” +1) 
Oj 
we obtain 
(1+0 
(i4f')- (<, +*> _ IL 
Z — 
4- d (1 + f') _(d + 1) — (d + n) (I + ¿ , )~ (d+n+1) 
Example. At what rate of interest will £645.174 purchase an 
annuity of £100 to be entered upon after the expiration of 8 years, and 
then continue 10 years? 
By a few trials we find the interest is between 3 per cent and 31,- per 
cent; let us then make i' — .03. 
1 4 V =1.03 d = 8 n = 10 a = 100 p — 645.174 
Table4, (1.03)~ 8 = .789409 = (1 4 i')~ d .03 
(1.03)- 18 = .587395 = (1 4 ¿')-( rf + n) 100)1973552 
.202014 
.193552 
,193552 = *-£ 
. 008462= (1 4 ¿')~ ri — (1 4 ¿')~ (d+n) — 
p 645.174 
~ -=7iT“ = 6.45174. 
a 100 
i'p