534 Mr. Gompertz on the nature of the function 
that if ^ 
t.n + + ^ ^ that if n be very nearly equal 
' 6+1 
'6 + 2 
to m, -ff^.n.q -J + &c which 
will be the value of the annuity on the three lives, will be 
nearly = t,n.q + t\n^.cf + &c If q were 
equal to unity, or, which is the same thing, m = n, the 
equality would be accurate ; but it may not be so when m 
differs from i ; but the nearer n is to m, at least when the 
difference does not exceed certain limited small quantities, 
the nearer will be the coincidence. It appears therefore, 
that if instead of taking the accommodated ratio for f ^ so that 
EjX (^6+1+ Lj + 3 • • • Lj+p) = f + e+f^ 
it will be preferable generally to take it so that Lx(/iL^_^ + 
^^^6 +2 ^6+3 ^ + &c.... in 
which n is between m and i , the nearer m the better generally, 
though possibly not universally so throughout the whole limit. 
And the second method I use for increasing the accuracy, is to 
adopt an accommodated ratio, or t , so that k x( 1,05 L -|- 
^ L V 6+1 
6 ^ 
1,05 Lj^+&c i,05^'Lj_|_^j=i7o^ S+i^ S3 
. . . 1,05) V. Another method which might have its peculiar 
Ip 
under the idea of using 
^h+iP 
advantage, is to assume s = 
a mean ratio. , ^ 
The General Tables.* 
Art. 6. I have had three general tables calculated for 
fixed periods, Numbers 1, 2, and 3. Number 1, for pe- 
* The chief of the arithmetical operations in the constructions of most of the 
tables were performed under my direction, by Mr. David Jones, of N°. io. King- 
street, Soho ; and, as far as my leisure would allow, I have endeavoured to assure 
myself of their accuracy by different inspections. 
