Hence the locus of P is a conic section whose semi-parameter is
- - . If e<l, is within the circle and the orbit is an ellipse;
if e = l, is on the circumference and the orbit is a parabola ; and if *>1, is without the circle Q and the orbit is an hyperbola, Two values of the potential, V, can readily be found from the above : V=- but r , .-. Y = ^ . TG = TH . TG. h Also, since OGH is a right angle, Y = TH.TG = the square of the tangent from T to the circle de scribed on HO as diameter. A beautiful result con- B nected with the hodograph, Fig. 5. and one which has attracted the attention of several of the ablest mathematicians, was communicated by Sir William Hamilton to the Royal Irish Academy in March 1847. It is called the theorem of hodographic isochronism, and is thus stated: If two circular hodographs, having a common chord, which passes through, or tends towards, a common centre of force, be cut perpendicularly by a third circle, the times of hodograph ically describing the intercepted arcs will be equal. A purely quaterniou proof is given of this theorem by Hamilton in his Elements, and, following the hints given by that method, he has also indicated the follow ing geometrical proof. Let TMT M , WMW M (fig. 6), be the two hodographic circles with centres H and L and common chord MM . Let P, P , on the common chord produced, be the centres of two circles WTW T and BAB A , near each other, which cut the hodographs orthogonally. Let be the centre of force, OZ perpendicular to the tangent at T, and TR, T R perpendicular to SX. Also let AT mean the arc AT, and similarly for the other small arcs. Draw PY perpen dicular to PT. But V = HT . OZ = OP . TR from similar triangles OPZ and THR ; M . PP . TR M . PP Fig. 6, From similar triangles THA and TPY we have AT PY . TH ~ PT also from similar triangles THR and PP Y TH PF . AT = PP . TR PY Til " PT T>P T R . .AT PP . TR Similarly, PT PF . T R PT If and t be the times of hodographically describing the arcs AT and A T respectively, , = M .AT OP 2 . T R~ 2 OP 2 . TR 2 . PT OP 2 . PT . TR Similarly, t = M . PP . T R OP 2 . T R 2 . PT M . IT Now OF>.PT. 2TR . T R OP 2 . PT . T R J_ 1 _ 2M . PP i TR + T R TR TR + T R T R ) OP 2 . PT ( 2TR . T R the harmonic mean between TR and T R = QL ; 2M . PP t + t OP 2 . PT . QL From this expression we see that the time of describing the two small arcs TA and T A is independent of the radius of the hodo graph and the distance of its centre from L. Hence it is equal to the time of describing the two arcs BW and B W. By continuing the process of drawing orthogonals we arrive at the conclusion that the time of describing the whole arc ATT A is equal to the time of describing the arc BWW B . A very simple analytical proof is given in Tait and Steele s Dynamics of a Particle. Others not so simple by Cayley and Droop are to be found in the Quarterly Journal of Mathematics. We can only mention that the theorem of Lambert can be deduced from the above theorem of hodographic isochronism without using any property of conic sections. Sir William Hamilton has observed that we have a good instance of a hodograph in the curve of aberration of a star, which is merely the hodograph of the earth s annual motion. The fact that this curve is a circle in a plane parallel to the earth s orbit, abstracted, however, from the general idea of the hodograph, was known long before the date of the hodograph. It will be found clearly stated and proved in Woodhouse s Astronomy, of date 1821, and from allusions there it appears to have been known even earlier. As an application of the hodograph, Thomson and Tait point out that the heat and light received by a planet from the sun in any time are proportional to the corresponding arc of the hodograph. See Proc. Roy. Irish Acad., 1846 ; Hamilton s Elements of Quaternions; Tait, Proc. R. S. E., 1867-8; Tail s Quaternions; Thomson and Tait, Nat. Phil. (J. BL.)
HODY, Humphrey (1659–1706), an English divine, was born at Odcombe in Somersetshire in 1659. In 1676 he entered Wadham College, Oxford, of which, having proceeded M.A. in 1682, he became fellow in 1684. Previously he had published in 1680 Dissertatio contra Historiam Aristece de LXX. Interpretibus, in which he showed that the so-called letter of Aristeas, containing an account of the production of the Septuagint, was the late forgery of a Hellenist Jew originally circulated to lend authority to that version. The dissertation was generally regarded as conclusive, although Vossius pub lished an angry and scurrilous reply to it in the appendix to his edition of Pomponius Mela. In 1689 Hody wrote the " Prolegomena " to the chronicle of John Malala, published at Oxford in 1691. The following year he became chaplain to Stillingfleet, bishop of Worcester, and, on account of his supporting the ruling party in a contro versy with Dodwell regarding the nonconforming bishops, he was appointed chaplain to Archbishop Tillotson, an office which he continued to hold under Tenison. In 1698 he was appointed regius professor of Greek in the university of Oxford, and in 1704 he was promoted to the archdeacon ry of Oxford. In 1701 he published History of Enrjli -h Councils and Convocations, and in 1704 in four volumes J)e Bibliorum textis originalibus, in which he included his original work on the Septuagint, and published a reply to the attack of Vossius. He died 20th January 1706. A work, De Greeds Illustribus, which he left in manuscript, was published in 1742 by Dr Jebb, who prefixed to it a Latin dissertation.
of Upper Franconia, is beautifully situated on the Saale,
on the north-eastern spurs of the Fichtelgebirge, and at the