Referring again to Art. 4 of Gompertz's treatise, let us denote by a0dx the actual probability of dying in the infinitely small time dx, at the initial age of the mortality table, and by axdx the abstract chance of death (in the time dx) at the end of x years; that is to say, the chance considered “independently of” (as Gompertz expresses it) or as “abstracted from” the deterioration resulting from increased age, then, according to Gompertz's law, the actual probability of dying, in the time dx, at the latter age will be axqxdx, in which expression q only is an arbitrary constant denoting the rate of deterioration. If, in axqxdx, we put x=0, it becomes a0dx, which coincides with the expression first assumed. Hence, Gompertz's law, which supposes that “the “vital force or recuperative power loses equal proportions in equal “times”, is expressed, in its general form, by the equation μxdx=axqxdx, or, which is the same thing, by μx=axqx.
LONG WAVES IN CHANNELS Of ARBITRARY CROSS-SECTION
