The Singular Solution Of Differential Equations

Abstract

In the latter half of the seventeenth century Sir Isaac Newton, eminent mathematician and physicist, developed his theory of Fluxions, treating with the subject by which he is probably best known. The Calculus, as the theory is now called, was the result of his extensive researches in physics, and the differential Calculus as he developed it treated of the motion of bodies. "It was a noble and generous tribute" writes David Smith, "that Leibniz paid when he said that, taking mathematics from the beginning of the world to the time when Newton lived, what he did was much the better half." Leibniz, a contemporary of Newton's, and also a renowned figure in mathematical history, is the man to whom we are gravely indebted for the symbolism and general theory of the Calculus. He, quite independently of Newton, developed the Calculus along the lines which were most familiar to him, and the derivative, as he unfolded it, was the tangent. He laid the foundation for the theory of envelopes and hence, as we shall see, the singular solution, which subjects are to be treated rather extensively in the course of this paper. "The theory of envelopes, or of ultimate intersections, may be said to have originated with the investigations of Huygens on evolutes . . . and those of Tschirnhauser on caustics. These authors, however, merely treated geometrically a few cases of moving right lines and did not give any general method for the investigation of such problems. Leibniz was the first who gave a general process for the solution of this class of questions."