518 Mr. Gompertz on the nature of the function 
the contrary appears to take place at certain periods) it would 
follow that the number of living out of a given number of 
persons at a given age, at equal successive increments of 
age, would decrease in a greater ratio than the geometrical 
progression, and then the chances against the knowledge of 
any one having arrived to certain defined terms of old age 
might increase in a much faster progression, notwithstanding 
there might still be no limit to the age of man. 
Art, 5. If the average exhaustions of a man^s power to 
avoid death were such that at the end of equal infinitely 
small intervals of time, he lost equal portions of his remain- 
ing power to oppose destruction which he had at the com- 
mencement of those intervals, then at the age .x his power to 
avoid death, or the intensity of his mortality might be denoted 
by aq"", a and q being constant quantities ; and if be the 
number of living at the age x, we shall have ^ x q-^ for 
the fl uxion of the number of deaths = — (LJ ; /. abq*=— -f- > 
ahq* = — h y p . log. of 6 x h y p . log. of L^, , and putting 
the common logarithm of ^ x square of the hyperbolic loga- 
rithm of 10 = r, we have c.q"" = common logarithm of 
L 
_f ; d being a constant quantity, and therefore or the 
d ^ ^ 
number of persons living at the age of x=d.g\^ ; g being 
put for the number whose common logarithm is c. The 
reader should he aware that I mean " to represent g raised 
to the power q * and not g ? raised to the x power ; which 
latter I should have expressed by g^ , and which would 
evidently be equal to g^^ . I take this opportunity to make 
this observation, as algebraists are sometimes not sufficiently 
precise in their notation of exponentials. 
