INCREASING AND DECREASING SCALE OF PREMIUMS. 195 
N«l+<—1 
— N 35 = 
21797.0406 
N m +2(-1 
.= N 49 = 
15933.8350 
1= n 45 == 
11414.2176 
m+4i—1 
“ N 50 = 
7962.2358 
57107.3290 
.002 
114.214658 
414.0198 __ 
N 3 o—29314.89 iqq 
1.412 = £l 8 3. 
258. Instead of the premiums being reduced or increased by a fixed 
sum, they may be reduced or increased arbitrarily, provided that in the 
case of increasing premiums those in the first instance be sufficient to 
cover the risk for the term during which they are payable; i. <?., not 
less than the annual premium for a temporary insurance for the same 
term. 
259. In the case of increasing premiums, the annual premium for 
the first interval should he more than the annual premium for a risk to 
he determined at the expiration of that term, as the party assured will 
have the advantage over the office of continuing or discontinuing the 
risk at his own option. 
260. If the annual premium for the first t years be p, for the second 
t years p,, for the third p /f , &c., and the premium is to be constant 
after vt years, we shall have if we call this last q, 
4»=p(l + a ( m) ) + «(».) P/ + «(m) p /y + +?«(») 
*—1| tM— 1 ¿12*-1 
from which we obtain by transposition and division 
— l+fl(«o) + fl(m) Pj~t a (vi) iP// + & C 'f 
l i—11 flf—l J 
7= 
&(»«) 
•pi—1 
« (m) denoting the value of an annuity for t years, to commence at the 
f\vt 
expiration of vt years. 
The expression above for A m may he thus written, 
M» ___ • 
D,„ ” 
— 
from which we obtain 
q — 
M m -{p(N m _ 1 -N, n+f _ l )4-y? / (N M+t _ l -N m+2 f_i)+P/ / (N m+2 (-i — N„i 4 . 3 (_ 1 )+ &«■} 
I 
261. To find the value of a policy payable by increasing or decreas 
ing premiums : 
0 2