The functional equation to be resolved is this—

In this equation xo° and x′ denote any two values of the indefinite variable x, and c is a constant quantity. The object of inquiry is the forms of the function.

It is proved that there are two forms of the function, which alike satisfy the proposed equation: viz.

In the first of these functions x increases, while f(x) decreases; and in the second, x and f(x) increase together. The first of these functions is applied in the paper to the theorem of the Parallelogram of Forces, whilst the theory of Curves of Equilibration is deduced from the second function: these last, in their most general form, have two parameters. The curve of equilibration is formed by a perfectly flexible chain suspended in a vertical plane between two fixed points.