PROP. II. To draw the circle of curvature at any point of a conic. Draw the tangent at P, of which methods have been given above. Draw PQ equalFy inclined to the axis, cutting the conic again in Q Draw PO at ri<.ht angles to the tangent, and make the angle PQO equal to the angle QPO. This gives the centre of the circle of curvature. PROP. III. The focal chord of curvature in the parabola is equal to 4SP. Let the common chord of the circle of curvature and the parabola be PQ, cutting the axis at F (fig. 48). Draw the double ordinate PNP , cutting the axis at N ; the tangents at P, P will meet the axis in the same point T. Then angle PFT = angle PTF = angle P TF. . . TP , PQ are parallel. Now let PS produced cut the circle again in U ; join UQ. Then angle UQP- angle TPU(EucL iii. 32) = angle PTS = angle PFT ; Fig. 48. therefore UQ is parallel to SF .-. PU :PS = PQ : PF = 4 :1, or PU = 4SP. PROP. IV. To find an expression for the chord of curvature at any point of a parabola drawn in any direction. Using the same construction as in Prop, iii., let PW (fig. 49) be the chord required. Draw SY parallel to the given direction to meet the tangent at Pin Y. Then angle PWU- angle SPY (Eucl. iii. 32). and angle UPV = angle YSP. Therefore the triangles UWP, YSP are similar, and PW : PU = SP : SY pw= 4Sp2 "SY Fte. 49. COROLLARY. The diameter of 4SP 3 curvature = --where SY is the perpendicular on the tangent. PROP. T. If the chord of intersection PQ (fig. 50) of an ellipse or hyperbola with the circle of curvature at P meet CD the semi-diameter conjugate to CP in K, then PQ. PK Draw the double ordinate P,NP ; the tangents at P, P meet in the axis at T, and the tangent at P is parallel to PQ, and therefore CP bisects PQ in V . Let PQ meet the axes in F, F , then PV : PF -FV : P C = TF : TC = PF : PK since CD is parallel to PT. Therefore PV . PK = PP . PF = PT.PT -CD 9 Fig 50 and (Ellipse, Prop. xv. and Hyperbola, Prop, xxi.) .-. PQ.PK = 2CD 2 . PROP. VI. If the chord of curvature PQ (fig. 51) of an ellipse or hyperbola in any direction meet CD in K , then PQ .PK = 2CD 2 . The angle Q QP = angle TPK = angle PKK ; therefore the triangles 1 KK , PQ Q are similar, and PQ :PQ = PK:PK .-. PQ .PK -PQ.PK = Fig. 51. If PQ" be the chord of curvature through the focus, then PK"=CA and PQ". CA = 2CD. If PQ " be the chord of curvature through the centre PK "=CP and PQ ". CP=2CD 2 . If PQ"" be the diameter of curvature PK"". CD = CA . CB and PQ"". PK""=2CD 2 . . . PQ"". CA . CB = 2CD 3 -.
For other powerful methods of investigating the properties of the conic sections which have been much developed of late reference is made to Geometry and other headings.
of the " Corpus " chair of Latin literature in the University of Oxford, was born on the 10th August 1825 at Bost6n in Lincolnshire, his father, the Rev. Richard Conington, being incumbent of the chapel of ease in that town. He was a remarkably precocious child, knowing his letters when fourteen months old, and being able to read well at three and a half. After two years training at Beverley grammar school, ho was sent in 1838 to Rugby, where his " remark able memory and very good scholarship" drew special commendation from Dr Arnold. In 1843 he went to Oxford, matriculating at University College at midsummer, but entering upon residence in the October term ah Magdalen, where in the interval he had been nominated to a demyship. His university distinctions were numerous. He was Ireland and Hertford scholar in 1814; in March 184G he was elected to a scholarship at University College ; in December of the same year he obtained a first-class in classics, graduating B.A. soon afterwards ; and in February 1848 he became a fellow of University College. Finding no career open to him at the university, and having obtained the Eldon scholarship in 1849, he proceeded to London in fulfilment of its conditions to keep his terms at Lincoln s Inn. The profession of law, however, proved eminently distasteful to him, and after six months lie resigned the scholarship, and returned to more congenial work at Oxford. During his brief residence in London he formed a connection with the Morning Chronicle, which was maintained for some time. He showed no special aptitude for journalism, but a series of articles on University Reform (1849-50) are noteworthy as the first public expression of his views on a subject that always deeply interested him. In 1854 his appointment to the chair of Latin literature, newly founded by Corpus Christi College, gave him a position which exactly suited him. He had published, in 1848, an edition of the Agamemnon of ^Eschylus with notes and a translation into English verse, and he had devoted much study to the other plays of yEschylus, of which the only published result is the very valuable edition of the Choephori (1857). From the time that he became professor, however, he confined himself with characteristic conscientiousness almost exclusively to Latin literature. The only important exception was the translation of the last twelve books of the Iliad in the Spenserian stanza in completion of the work of Worslcy,
and this was undertaken as a labour of love, in fulfilment