520 
ME. B. GOMPEETZ ON THE SCIENCE 
w years, provided the persons of the present ages a, h, c, &c. are then all living, repre- 
sented by 
(H-'A+‘B+>C, &c.>+(^A+^B+^C, &c.>HCA+^B+^C, &c.>r^, &c., 
a very converging series ; but in the use of the method it is necessary to have, instead of 
the common logarithm, the Napierian logarithm, and the Napierian logarithms of 
L 4 +,, &c. would be expressed by 
£+^^“X ^l+g_.^-f£.|-+&c.^, £+^.^*x (^l+gA’+£.|- &c.^ ; 
and if, for the sake of brevity, we represent &c.}, multiplied respectively 
1-1® 1 -'3 
by S'? 2 ^y shall have the Napierian logarithm of 
the present value of unity to be received in x years, if the persons of the present ages 
a, i, c, &c. be living, represented by rd-Ai.^+A 2 ^r^+A 3 .r®+ &c., a very converging 
series. And if the anti-Napierian logarithm of this expression be represented by 
&:c., in which the coefficients of the different powers of x, namely, 
‘S, ^S, ^S, form a very converging series, then on using the notation adopted in my paper 
of 1820, ^ x"jto express the sum of all the values of x’*, from x=w to x=z inclusNely 
of both by increasing x continually by unity, the present value of an annuity of 1 on the 
joint lives of persons now of the age of a, h, c, &c., the first payment to be made in w 
years, and the last in z years, will be represented by the series 
X X X X 
1 
1 
1 
1 
w 
z 
w 
w 
w 
which will be a converging series. 
And if the annuity were intended to be entered on immediately, so that the fii’st pay- 
ment was to be in one year, and it was intended to continue as long as the joint lives 
continued, it would be written 
X X X X 
i|a^®-|-’S il^^-h&c. ; 
and if this were to be an increasing annuity, continually increasing by the additions 
from year to year of the payment %, the ^’th payment would be 1-^nx, and the present 
value of that a^th payment would be 
and therefore the present worth of such an increasing annuity on the jomt lives would 
be represented by 
XX X X X ‘ 
1| ^“+(‘S+w) il . ’S) I| x^-{-(^S+n . ^S) I x^+(^S-\-n . ^S) I| x\ &c. ; 
and a similar mode may be adopted for more complicated regulations of the payments 
from year to year. 
