124 
LIFE ANNUITIES. 
at any ages of the same difference as that for which the various products 
have been found may be obtained by dividing the sum of the products 
above the ages, by the products opposite to them. 
129. If, previous to calculating the values of annuities on joint lives, 
calculations have been made of the values on single lives, the products 
opposite each age in the D column for the single lives will form part of 
the operation in finding the products for the joint lives. 
130. When one life is a male and the other a female, and the rate of 
mortality distinguishes the sexes, the number of living at the age of the 
male must be taken from the table of mortality amongst males, and the 
living at the age of the female from the table of mortality amongst 
females. 
Or, when there is no difference of sex, but it is thought proper to 
use different rates of mortality for the two lives, the number of living at 
the age of each of the individuals must be taken, in forming the products, 
from the corresponding rate of mortality. 
131. The following calculation of the value of an annuity during the 
joint existence of a male aged 85 and a female aged 90, will illustrate 
what has been said, and show the methods by which the values in 
Table 23 were calculated (Chester 5 per cent) : 
r m xtoo xtT= • 00760049 x 23 X 126 = 22.03782 
r" x4o Xl ai - .00798471 X 30x158= 37.84753 
?' 98 x4a X/ 93 =. 00838395 x 37 x190= 58.93918 
r 97 X l a7 Xl 92 = .00880315 x 44x221= 85,60183 
r 96 X/ 9C X/ 9l =.00924331 x 51 x252= 118.79502 
323.22138" 
r M X/95 x4o=.00970547 X 68x283= 186.77208 
r 9 ‘ x4* x/ C9 — .01019074 x 92X313= 293.45255 
» 93 X4a x/ U8 =.01070028x 116x343= 425.7426 
r 92 x4* X4?=-01123530x146x384= 629.8959 
r 01 X 4i X4e=.01179706x176x436= 905.2590 
2764.3435 
r 90 x4o X 4;= -01238691 X205 X 510 = 1295.0517. 
^00.95 — 
°85.90 — 
323.22138 
186.77208 
2764.3435 
1295.0517 
1.731 
132. The principle laid down for calculating annuities on two joint 
lives applies to finding the values on any number of joint lives: if the 
values were calculated on three lives when the differences of the ages 
are 5 and 1, the number in the D column opposite the ages 16, 21, and 
22, would be equal to the product of the number of living at 21 and 22