### Abstract:

In this paper we use the set of all positive integers
as a sample space whose probability density function is unknown.
Then a generalization of the probability distribution
of the most significant digits of the set of all physical
constants is obtained on the strength of (i) a very general
assumption imposed on the density function of the
sample space, and (ii) a generalized invariance principle.
The assumption is quite weak in the sense that it merely
states that the occurrence of an event containing infinitely
many elementary events is not impossible. The invariance
principle, as is shown, is equivalent to another principle
and to two functional equations. A function is constructed
and, on the basis of the two foregoing stipulations that
characterize the generalization, it is shown that this function
is a unique solution, within a multiplicative positive
constant, to another functional equation. The function so constructed serves as a stepping stone in reaching our goal.
Having the generalization at our disposal, we deduce
from it some of the consequences that are of interest. As
it turns out, the deduction gives, on one hand, a proof to
two empirical formulas published previously and, on the
other, a fairly good agreement with the probabilities of
three continuous density functions established in the literatures
concerning the distribution of the leading digits
under algebraic computation. In concluding the paper, a
justification is made as to why a special case of the consequences
of our result coincides with the probability of
one of the three continuous density functions, even though
our function is discrete.