direction making an angle with the line joining the centres; then B impinges upon A at rest and at the same angle, and goes off at an angle '. Prove that 0+0=180°. Prove also, that if the balls be imperfectly elastic, and the angles of incidence in the two cases be a, a', then 8. From a given height a perfectly elastic particle is let fall on a perfectly hard inclined plane, so as to strike it at a fixed point; prove that, whatever be the inclination of the plane to the horizon, the vertex of the parabola, which the particle describes after impact, will lie on a certain ellipse. 9. A perfectly elastic ball is projected at an angle ß to a plane inclined to the horizon at an angle a, so as to ascend it by bounds; find the inclination to the plane at which the ball rises at the nth rebound, and shew that it will rise vertically if cot B (2n+1) tan a. 10. A string charged with n+m+1 equal weights fixed at equal intervals along it, and which would rest on a smooth inclined plane, with m of the weights hanging over the top, is placed on the plane with the (m + 1)th weight just over the top; shew that, if a be the distance between each two adjacent weights, the velocity which the string will have acquired, at the instant the last weight slips off the plane, will be (nag). NEWTON. SECTIONS I. II. III. SECTION I. 1.. ABC is an isosceles triangle; P, Q are points on the straight lines CA, CB, such that PA = 2QB; find the point of ultimate intersection of PQ, AB, as P and Q move up respectively to A and B. 2. Two triangles CAB, C'AB' have a common angle A, and the sum of the sides about that angle the same in each: if CB, C'B' intersect in D, and B' move up to B, then in the limit DC : DB :: AB : AC. 3. A and B are fixed points, CD a fixed straight line, cd another straight line moving subject to the condition that the rectangle under the perpendiculars from A, B upon it is equal to the rectangle under the corresponding perpendiculars upon CD. If CD, cd intersect in P, prove that, ultimately, the angle APC will be equal to BPD. 4. A straight line is drawn from the centre of an ellipse meeting the ellipse in P, the auxiliary circle in Q, and the tangent at the vertex in T; prove that as CT moves up to and ultimately coincides with the semi-major axis, PT, QT are ultimately in the duplicate ratio of the axes. 5. The extremities of a straight line slide upon two given right lines, so that the area of the triangle formed by the three straight lines is constant; find the limiting position of the chord of intersection of two consecutive positions of the circle described about that triangle. 6. If a series of parallelograms be inscribed in a quadrant of a circle in the manner described in Lemma II., prove that, when their bases are indefinitely diminished in length, and increased in number, the perimeter of the inscribed figure : perimeter of quadrant :: 8 : π +4. 7. Apply Lemma IV. to shew that the volume of a paraboloid cut off by a plane perpendicular to the axis, is half that of the circumscribing cylinder. 8. Apply Lemma IV. to shew that the volume of a right cone is that of the cylinder of the same base and altitude. 9. Prove that the surface generated by the revolution of a semicircle round its bounding diameter: that generated by its revolution round the tangent which is parallel to that diathe length of the diameter: the length of the semi meter circle. 10. In an ellipse, if PCP' be conjugate to CD, shew that the chords DP, DP' cut off segments equal in area. 11. Similar conterminous arcs, which have a common tangent at one extremity, have parallel tangents at the other. 12. AB is the diameter of a circle, P a point contiguous to A, and the tangent at A meets BA produced in T; shew that, ultimately, the difference of BA, BP is equal to one half of TA. 13. Through any point on a circle a secant is drawn, cutting the circle again in P, and the tangent at a point A in T; shew that, when P approaches to A and ultimately coincides with it, the ratio PT: AT becomes indefinitely small. 14. PQ, pq are parallel chords of an ellipse; shew that if p move up to P, the areas PCP, QCq are ultimately equal. 15. If the velocity of a point at any instant be proportional to the square of the time, shew that the space described from rest is proportional to the cube of the time. 16. If the sum of the squares of the numerical measures of the time and velocity be constant, shew that the acceleration varies directly as the time and inversely as the velocity. 17. If the curve in the figure to Lemma x. be a portion of a parabola, of which the latus rectum represents the time; find the velocity acquired and the space described in any time. 18. Two points move from rest, in such a manner that the ratio of the times in which the same uniform acceleration would generate their respective velocities at those times, is constant. Shew that their respective accelerations at any times bearing this ratio are equal. 19. Two curves of finite curvature touch each other at the point P, and from T, a fixed point in the common tangent, a secant is drawn cutting one curve in the points A, B, and the other in the points A', B'; shew that, if the secant move up to the tangent, the angles APA', BPB' are ultimately equal. 20. An arc of continuous curvature PQQ' is bisected in Q, PT is the tangent at P; shew that, ultimately, as Q'approaches P the angle Q'PT is bisected by QP. 21. AB is an arc of finite curvature at A, and a point Pis taken in it such that AP: PB in a constant ratio :: m : n. At A, P, B tangents AT, TPR, RB are drawn; find the ultimate ratio of the area ATRB to the segment APB as B moves up to A, and prove that this ratio is a minimum and equal to 98 when P bisects AB. 22. PQR is an equilateral triangle in which P, Q are points on a parabola of which the focus lies on PQ, and PR is a diameter; shew that the circle described about PQR is the circle of curvature to the parabola at P. 23. The chord of curvature, in the direction of the centre, at any point of a rectangular hyperbola, is equal to the diameter at that point. 24. Prove that half the chord of curvature through the focus of an ellipse is a harmonic mean between the focal distance of the point. 25. If P be a point in an ellipse equidistant from the axis minor and one directrix, the circle of curvature at P will pass through one of the foci. 26. E is a chord of a given circle and S its middle point; construct the ellipse of which E is a point, S a focus and the given circle that of curvature at E. 27. Determine geometrically where the circle of curvature at any point of an ellipse again meets the ellipse. 28. If a circle touch an ellipse at the extremities of a chord parallel to the minor axis, prove that the chord of this circle, which passes through either of the foci and one of the points of contact, is equal to the minor axis. SECTION II. 1. A parabola is described with an acceleration tending to the focus, and a straight line is drawn from the focus perpendicular to the tangent, and proportional to the velocity at any point; shew that the extremity of this straight line will lie in a circle. 2. A hyperbola is described with an acceleration tending from the centre, and at any point P, PQ is taken along the tangent proportional to the velocity at P: prove that the locus of Qis a similar hyperbola. 3. A particle moves in a parabola with an acceleration tending to the vertex; shew that the time of moving from any point to the vertex varies as the cube of the distance of that point from the axis. 4. A particle describes a parabola with an acceleration tending to the focus; shew that its velocity at any point may be resolved into two equal constant velocities, respectively perpendicular to the axis and radius vector. 5. If the velocities at three points of an ellipse described by a particle, the acceleration of which tends to the focus, be in |