The idea of a function of an infinite number of variables, like so many other mathematical ideas, has its origin in the study of problems of applied mathematics. Such problems, as is usual, preceded by some considerable time the explicit formulation of the idea. As an introduction to this discussion it will be sufficient to consider briefly two of them.

The Problem of the Brachistochrone.—Given two points P, Q in a vertical plane, what is the shape of a smooth curve through P, Q for which the time required to descend from P to Q under gravity is a mimimum.