30 ON PROBABILITY. 
least 7 years is equal to the product of the probability of his living at least 
n — q years, multiplied by the probability of an individual aged m + n — ¢ 
years living at least ¢ years, or 
Pm, -_ Pm, n -q xX Pmta -g 09 
and therefore, putting 
0 —- 1 for q, Pia, y= Een 
= Dom, , 
yyy =r Pm i 7 Pog -+ &e. 
io 7° Pu,2. + 13 Pus T SC I” 
= r Pm, 1 cae 
Emin Tha oo 
7 Pm, 1 r Pm, 1 Gg 
By means of this expression, which appears first to have been noticed by 3 
Mr. Barrett, the value of any annuity may be deduced from that which pre- oft 
cedes or follows it. al 
Thus if m=20, r= x Dini = LVL according to Table II., and pre 
303.77 5765 ° pol 
a, = 20-1428, according to Table III., fhe 
(ies, 
20-1428 ‘0 5765 Ir 
as, = 207128 x 1:03 x 5765 — 1 = 199580. Mr 
5707 om 
56. It has been seen that the values of annuities, reversionary payments, Tall 
&ec., consist of the sum of a number of separate payments. If these pay- nor 
ments are calculated accurately at certain intervals the values of those which foo. 
are intermediate may be interpolated by known methods. } ’ 
In fact, if ¥,, 9., 92. &c. are successive values of the variable y, and if tne 
AYo=Yn—Yo» D°Y=Y, + Yo, — 2¥,, &c., and 7; be any value of y regs 
intermediate between 7, and Yo k 
. Whi 
p=g+iag+ HD any 4g year 
When the sum only of the values of y is required it is not necessary, how- stane 
ever, to go through the labour of calculating each particular quantity in the 
series, it may easily be shown that this sum is equal to were 
dy mel se Yn) ow 
n—1 (r=) (n+) ( da 
dle sone ! “™ } —e ret PAY, ~ AY Ke. Say 
The problem appertains to what are called mechanical quadratures, and Clo 
this method is similar to that made use of in summations which are required aes 
in calculating the perturbations of a comet. See the Mécanique Céleste, vol. iv. lor 
p- 206. In applications of this series to the calculation of annuities, reversionary h 
payments, &c. ¥,, Ay, &c. = 0. The first term in the series of the values pose 
of y or y, is the value of a present payment = 1, if we neglect the term re ¢ 
first 
(n— an } 
Sr AA A 
12n ya Ye 
and the following, and suppose the values of the annual payments to be in 
arithmetical progression, the value of an annuity on the life of a person aged 
20, to commence at the end of the first year, supposing 7 = 10,