1. The determination of a curve from some property connecting points separated by finite intervals usually involves the solution of an equation of differences, pure or mixed, or more generally of a functional equation.

The particular species of this equation will depend upon the law of succession of the points under consideration, and upon the nature of the elements involved in the expression of the given connecting property.

Thus if the abscissae of the given points increase by a constant difference, and if the connecting property consist merely in some relation between the successive ordinates, the determination of the curve will depend on the integration of a pure equation of differences. But if, the abscissae still increasing by a constant difference, the connecting property consist in a relation involving such elements as the tangent, the normal, the radius of curvature, &c, the determining equation will be one of mixed differences.

If, instead of the abscissa, some other element of the curve is supposed to increase by a constant difference, it is necessary to assume that element as the independent variable. But when no obvious element of the curve increases by a constant difference, it becomes necessary to assume as independent variable the index of that operation by which we pass from point to point of the curve, i. e. some number which is supposed to measure the frequency of the operation, and which increases by unity as we pass from any point to the succeeding point.