522 
ME. B. GOMPEETZ ON THE SCIENCE 
and consequently 
-A^A^x^ +4A4^r® +&c.' 
+ AiB,a7+2A2Bi.37^+3A3Bi^® &c. 
+ AiB2'^^ + 2A2B2A’^ &c. 
+ AiBs-^*^'* &:c. 
&c. 
— B,— 2B2.2’— SBg^^— 4 B 4 ir® &c. 
Consequently 
Ai = B,; A 2 =B 2 — ;^A,B, ; A 3 =B 3 — ^ (2A2B1+A1B2), 
A4=B4 — j{3A3Bi+ 2A2B2+ AjBg} ; A5=B5 — ^ {4A4B1 + 3A3B2+2A2B3+A1B4}, &c. 
for turning natural expressions into Napierian logarithms; and we also have for the 
reverse, namely, turning expressions of Napierian into anti-Napierian logarithms, 
B,=A,; B,=A,+iA,B,; B.=A,+h2A3.+A,B.) ; B.=A.+i(3A,B,+2A,B,+A,B,), ^ 
Art. 9 . And now supposing the Napierian logarithm of the chances of persons now of 
the ages a, / 3 , y, &c. being living in x years be turned into anti-Napierian logarithms by 
the theorem at the end of art. 8, that is to say, into analytical natural expressions 
corresponding to those Napierian logarithms, and that the value of those chances be 
respectively deducted from unity, they will represent the respective chances of their 
being extinct, and may be expressed by 
and supposing the Napierian logarithms of these several multipliers 
l + l-\-%X + ^]Q,iX\ &C., 
be taken by the same theorem of art. 8, and that there be n persons of the above said 
ages a, / 3 , y, &c., and that the sum of these Napierian logarithms thus taken be repre- 
sented by &c., and that the product of the n quantities Pa:, &c., 
be represented by P ; then the present value of unity to be received in x years, pro\ided 
all the persons before mentioned to be living are extinct, will be P^”, multiplied by the 
anti-Napierian logarithm of (r+’P)-^+^P^'^+^P^’^? &c. ; and if the anti-Napierian loga- 
rithm be represented by &:c., the said value will be 
P. P. -1-P- &c. 
We may execute the required calculations in this manner instead of adopting rules 
laid down by authors of solving such questions by aid of Tables of the values of annuities 
on single lives and two joint lives, and obtaining by an inaccurate interpolation the 
values of annuities of a larger number of joint lives, as there may be required many 
values of annuities on three joint lives, on four joint lives, and on more joint lives, which 
