l'intégration des équations différentielles aux
différences finies, et sur leur usage dans la théorie des
be a sum which Paul constitutes to an
a way that the interest is
of that which is due
to him: I suppose that, for some arbitrary reasons, one keeps each year
of this interest, so
that Paul, at the end of the first year, for example, must collect only
, this put,
if one pays him every year the sum
, and, consequently, more
than is due to him, and let the surplus be used to amortize the
one asks what this capital will become in the year
Let denote this capital in the year x . At the end of the year Paul will be due . Since the sum is paid, the capital will be diminished by . We have therefore
The initial condition is . Therefore, the solution is given by
We require when the capital will vanish. That is, for what value of x is ?
If we substitute, as does Laplace, and then we find that this occurs in 53.3 years.
person owes the sum a
, and wishes to release himself at the end of
that she owes nothing in the year
the interest being
of the quantity due;
the question is to find what
must she give for this each year.
With Laplace, we let p be what she must give each year.
Laplace puts .