30. OP, OQ are tangents to the parabola from O, prove

OP: 0Q :: SP: SQ.

31. The lines joining the intersections of the tangents to confocal parabolas drawn to both curves at their points of intersection pass through the common focus.

32. OP, OQ are tangents to the parabola from 0; if the chord PQ meets the directrix in F, prove that OSF is a right angle. If OK be at right angles to the directrix, prove SK is at right angles to PQ. If PQ cuts the axis in N, prove that KN is parallel to OS. If OM be drawn to meet the axis at right angles AM = AN.

33. OP, OQ are tangents to the parabola from 0. Prove that when PQ moves parallel to itself, O moves parallel to the axis : when PQ moves round a point in the axis, O moves at right angles to the axis: when PQ moves round a point in the directrix, O moves in a straight line to S.

34. A line drawn from the focus to meet the tangent at a constant angle, has its point of intersection with it, on one of two fixed tangents.

35. Given one tangent to a parabola, to draw two others which make a given angle with it. In what case is one of the tangents removed to an infinite distance?

36. BC the portion of a tangent intercepted between two other tangents AB and AC is bisected by D the point of contact. Prove that SA is a fourth proportional to AD, AB and AC.

37. The portion of any tangent between tangents that meet on the directrix, subtends a right angle at the focus.

38.

The tangent at P meets the directrix in F: from any point O in PF, and from F tangents are drawn to the curve, prove that they meet in the line through S at right angles to OS.

39. The chord PQ is a normal at P and QR is drawn parallel

to the axis to meet PP', the double ordinate through P, produced in R: prove that PP'. P'R is constant.

40. The ordinate through the middle point of NG = PG.

41. An ellipse and parabola have a common focus and directrix: diagonals of the quadrilateral formed by joining the four points where the tangents at the extremities of the axis major cut the parabola pass through the focus and through the extremities of the axis minor.

42. An hyperbola is confocal with a parabola, and has the tangent at the vertex of the parabola for its nearer directrix. Prove that the tangent to the parabola at the point of intersection passes through the further vertex of the hyperbola.

MISCELLANEOUS EXAMPLES.

1. The orthogonal projection of a parabola is a parabola.

2. The projection of a parabolic section of a cone on a plane at right angles to the axis of the cone is a parabola having for its focus the point where the axis cuts the plane on which the projection is made.

3. CS= the part of the generating line of the cone which has the same projection on the axis as CA has: this was proved for the ellipse, extend the proof to the hyperbola.

4. If an elliptic or hyperbolic section of a cone be projected on a plane through one vertex at right angles to the axis of the cone, CS is diminished by the square on the distance of C from the plane of projection, and one focus of the projected curve will lie at the intersection of the axis of the cone with this plane.

5. All parabolas cut by parallel planes from a given cone have their foci on a straight line through the vertex of the cone.

6. Given a right cone and a point within it; only two sections have this point for focus and their planes are equally inclined to the line joining the point to the vertex.

7. Given the vertex of a cone and the centre of a sphere inscribed in it: all sections made by planes at right angles to a generating line and to the plane of the paper containing the centre and vertex, will have one of their foci on a circle which touches the axis of the cone at the centre of the sphere.

8. Two cones touch the same two spheres, prove that by whatever planes the two cones are cut, the ratio of their eccentricities is constant.

9. Two cones have supplementary angles and are placed with their vertices and one generating line of each coincident. Curves are cut from them by a plane at right angles to the coincident generating lines: shew that the directrices of either curve pass through the foci of the other.

10. The intersection of a plane with a cylinder is an ellipse with foci at the points of contact of the plane and two spheres inscribed in the cylinder.

CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS.

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Eschylus.-ÆSCHYLI EUMENIDES. The Greek Text, with English Notes and English Verse, Translation, and an Introduction. By BERNARD DRAKE, M.A., late Fellow of King's College, Cambridge. 8vo. 3s. 6d.

The Greek text adopted in this Edition is based upon that of Wellauer, which may be said, in general terms, to represent that of the best manuscripts. But in correcting the Text, and in the Notes, advantage has been taken of the suggestions of Hermann, Paley, Linwood, and other commentators. In the Translation, the simple character of the Eschylean dialogues has generally enabled the author to render them without any material deviation from the construction and idioms of the original Greeck

"The Notes are judicious, and, a rare merit in English Notes, not too numerous or too long. A most useful feature in the work is the Analysis of Müller's celebrated dissertations.”—BRITISH QUARTERLY REVIEW.

Aristotle. AN INTRODUCTION ΤΟ ARISTOTLE'S RHETORIC. With Analysis, Notes, and Appendices. By E. M. COPE, Senior Fellow and Tutor of Trinity College, Cambridge. 8vo. 145.

This work is introductory to an edition of the Greek Text of Aristotle's Rhetoric, which is in course of preparation. Its object is to render that treatise thoroughly intelligible. The author has aimed to illustrate, as preparatory to the detailed explanation of the work, the general bearings and relations of the Art of Rhetoric in itself, as well as the special mode of treating it adopted by Aristotle in his peculiar system. The evidence upon obscure or doubtful questions connected with the subject is examined; and the relations which Rhetoric bears, in Aristotle's view, to the kindred art of Logic are fully considered. A connected Analysis of the work is given, sometimes in the form of paraphrase; and a few important matters are separately discussed in Appendices. There is added, as a general Appendix, by way of specimen of the antagonistic system of Isocrates and others, a complete analysis of the treatise called 'Ρητοριχὴ πρὸς ̓Αλέξανδρον, with a discussion of its authorship and of the probable results of its teaching.

ARISTOTLE ON FALLACIES; OR, THE SOPHISTICI ELENCHI. With a Translation and Notes by EDWARD Poste, M.A., Fellow of Oriel College, Oxford. 8vo. 8s. 6d.

Besides the doctrine of Fallacies, Aristotle offers, either in this treatise or in other passages quoted in the commentary, various glances over the world of science and opinion, various suggestions or problems which are still agitated, and a vivid picture of the ancient system of dialectics, which it is hoped may be found both interesting and instructive. "It is not only scholarlike and careful, it is also perspicuous.”—GUARDIAN. "It is indeed a work of great skill."-SATURDAY REVIEW.

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