infinite Series ~ ^ ^ +.&cc. 901 
the ratio of 1 to 1 —x in napier’s Syftem of Logarithms, 
that is, in the prefent example, to the logarithm of the 
ratio of 1 to or of 10 to 1, or to napier’s logarithm 
of 10) being 2.302,585,092,994,04. 
5. I believe that fimilar applications of the binomial 
theorem may be made for the fummationof other flowly- 
converging infinite ferieles, whenever the generating 
fractions (by the multiplication of which the numeral 
co-efficients of the terms of fuch feriefes are produced 
from each other) are formed by the addition of a given 
number to both their numerators and denominators. In 
our endeavours, therefore, to fum fuch feriefes it will be 
proper to attend to the law of the faid generating frac- 
tions. 
V 
