633 
expressive of the law of human mortality, &c. 
general and r differ very little from unity, m 
will not differ much from unity ; and therefore if p be not 
great, m, m^, m^, &c. will not differ much from unity ; and 
consequently, as g , g , g , &c. are small, mt^~\- ^*63+ m^ s^,„. 
m?Bp will not differ much from g^+ g^^+ £3+ + 
has been shown to be o ; consequently m&^-\- m^ + m^ 
-f- differs very little from o, or in other words is very 
small ; and consequently, the value of the annuity differs 
very little from mQ> + m^ V m^ + m^ ; and the 
same method of demonstration would apply with any one of 
the other ages, the remaining ages being supposed to possess 
the property of the accurate geometrical progression ; not- 
withstanding this, however, as none of them probably will 
contain that property, but in an approximate degree, a varia- 
tion in the above approximations may be produced of a 
small quantity of the second order ; that is, if the order of 
the product of two small quantities ; but, as in this approxi- 
mation, I was only aiming at retaining the quantities of the 
first order, I do not consider this as affecting the result as far 
as the approximation is intended to reach : thus far with 
regard to the first accommodated ratios. 
Art. 5. Moreover, on the supposition that L , L 
L, ,.»..L, , and also L . , L , , L , . , . L 
c-f 1 ’ 
d+p 
are 
series in geometrical progression, and that r. 
"c+i 
^d+i 
- 
m = n.q. Since the annuity for /> years on the three lives is 
•t If I b-\- 2 b-\- p P • r ’ll 
equal to — — . m — . w * -j- — . m it follows 
