The Geometric Interpretation of Monge's Differential Equation to all Conics—the Sought Found

Abstract

THE question of the true geometric interpretation of the Mongian equation has been often considered by mathematicians. In the first place, we have the late Dr. Boole's statement that “here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms” (“Diff. Equ.,” pp. 19–20). We have next two attempts to interpret the equation geometrically. The first of these, propositions, by Lieut-Colonel Cunningham, is that “the eccentricity of the osculating conic of a given conic is constant all round the latter” (Quarterly Journal, vol. xiv. 229); the second, by Prof. Sylvester, is that “the differential equation of a conic is satisfied at the sextactic points of any curve” (Amer. Journ. Math., vol. ix. p. 19). I have elsewhere considered both these interpretations in detail, and I have pointed out that both of them are irrelevant; the first of them is, in fact, the geometric interpretation, not of the Mongian equation, but of one of its five first integrals which I have actually calculated (Proc. Asiatic Soc, Bengal, 1888, pp. 74–86); the second is out of mark as failing to furnish such a property of the conic as would lead to a geometrical quantity which vanishes at every point of every conic (Journal Asiatic Soc. Bengal, 1887, Part 2, p. 143). In this note I will briefly mention the true geometric interpretation which I have recently discovered.