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THE HODOGRAPH, OR A NEW METHOD OF EXPRESSING IN SYMBOLICAL LANGUAGE THE NEWTONIAN LAW OF ATTRACTION By William Rowan Hamilton (Proceedings of the Royal Irish Academy, 3 (1847), pp. 344-353.)
Edited by David R. Wilkins 2000
THE HODOGRAPH, OR A NEW METHOD OF EXPRESSING IN SYMBOLICAL LANGUAGE THE NEWTONIAN LAW OF ATTRACTION William Rowan Hamilton [Proceedings of the Royal Irish Academy, vol. 3 (1847), pp. 344–353.] Communicated December 14, 1846. Sir William R. Hamilton made a communication respecting a new mode of geometrically conceiving, and of expressing in symbolical language, the Newtonian law of attraction, and the mathematical problem of determining the orbits and perturbations of bodies which are governed in their motions by that law. Whatever may be the complication of the accelerating forces which act on any moving body, regarded as a moving point, and, therefore, however complex may be its orbit, we may always imagine a succession of straight lines, or vectors, to be drawn from some one point, as from a common origin, in such a manner as to represent, by their directions and lengths, the varying directions and degrees (or quantities) of the velocity of the moving point: and the curve which is the locus of the ends of the straight lines so drawn may be called the hodograph of the body, or of its motion, by a combination of the two Greek words, oδ´ ‘ oς, a way, and γραφω, ´ to write or describe; because the vector of this hodograph, which may also be said to be the vector of velocity of the body, and which is always parallel to the tangent at the corresponding point of the orbit, marks out or indicates at once the direction of the momentary path or way in which the body is moving, and the rapidity with which the body, at that moment, is moving in that path or way. This hodographic curve is even more immediately connected than the orbit with the forces which act upon the bo