3. If the force vary as find the time of falling into the (Dist.) centre. 4. If a chain of great length be suspended at the top, its lower end touching the earth, and then be let fall, find the velocity of the chain. 5. Find the velocity and time of a body's descending in an evanescent ellipse towards the centre of force placed in the focus; and supposing two bodies to descend, one in an evanescent ellipse, and the other in a right line, how will each move after it has reached the centre? 6. If BDA be a parabolic arc described by a comet round the Sun in the focus C, and P equal the periodic time of the Earth, her mean distance being 1; then the time of the comet's describing the arc BDA= {(a + b + c)* − (a + b − c)*}. P 12π 7. If two bodies moving, the one in a curve round, and the other in a straight line passing through the centre of force, have equal velocities at any the same distance from the centre, they will have equal velocities at all other equal distances. 8. If the force vary as 1 (Dist.) a = distance of projection from the centre, × velocity of projection = velocity in a circle at that distance x q√2, that this is an equation to a conic section. 9. In Prop. 45 it is said, that orbits have the same figure when the forces by which they are described are made proportional at equal distances: Prove this; and shew that no curve described by a force varying as any power of the distance, can have more than two different values of the apsidal distance. 10. If the force be constant, and the eccentricity of the orbit indefinitely great, prove that the angle between the apsides = 90°. 11. Prove that the time down any arc of a hypocycloid, beginning from the highest point, is proportional to the arc of the generating circle which is cut off by the string. 12. If a body, urged by a centripetal force, move on a curve surface whose axis passes through the centre of force, and if the path described by the body be projected on a plane perpendicular to the axis, shew that the projected area is proportional to the time. 13. If a body oscillating in a very small circular arc, have a very small motion communicated to it, when it is at its highest point, in a direction perpendicular to the plane of vibration, shew that it will describe a curve differing insensibly from an ellipse, and that its oscillations will be very nearly isochronous. 14. Let P and p denote the periodic times of a planet, and its satellite respectively s the sine of the angle under which, at the planet's mean distance from the Sun, the mean radius of the satellite's orbit is seen, then the quantity of matter in the planet is equal to P2 2 P2 p2 +(s. 7) nearly, the Sun's mass being' 1. 15. Find the effect of the Moon in disturbing the motion of the Earth round the Sun, and shew that the force, urging the centre of gravity of the Earth and Moon towards the Sun, follows much more nearly the law of the inverse square of the distance than that which urges either of those bodies. 16. From the mean angular motion of the nodes of the Moon's orbit, deduce those of the satellites of Jupiter. 17. If a corpuscle be placed any where within a hollow cylinder, extended infinitely both ways in the direction of its axis, the attraction 1 to each particle varying as (Dist.)e' prove that it will remain at rest. 18. If in an oblate spheroid of small eccentricity b be the polar radius, b + c the equatoreal, and the angle which a semidiameter makes with the axis; shew that the attraction on a point at the ex 19. If a corpuscle of light, moving in a given direction, with a given velocity, be attracted towards a refracting medium terminated by a plane surface, by a force varying according to any power of the distance from it, find the equation to the curve which it describes, and shew that the sine of the angle of incidence is to the sine of the angle of refraction in a given ratio. 20. If the Earth were an homogeneous fluid mass, revolving round her axis, and gravity tended to the centre, and varied as the distance from it, prove on Newton's principle of all columns extending from the centre to the surface being in equilibrium, that her figure must be that of an oblate spheroid. TRINITY COLLEGE, 1822. 1. EXPLAIN the principles of the methods of exhaustions, indivisibles, and limits. State accurately the meaning of the expression "ultimæ quantitates," as used by Newton; and show that the results he obtains by the system of limits are not approximately but strictly true. 2. Determine 1st., By the method of Exhaustions the value of the area of a circle. 2dly., By the method of Indivisibles the ratio between a sphere and its circumscribing cylinder. 3dly., By that of Limits the ratio between a cone and its circumscribing cylinder. 3. (1). Define accurately the circle of curvature. (2). PT being the tangent to any curve PQ at P, and PV a straight line drawn, making any finite angle with PT, determine the curve FK whose intersection with PV in V, shall cut off PV a chord of the circle of curvature. (3). Show that the direction with which F cuts or touches PV determines the degree of contact between PQ and its circle of curvature:-and that there may be an indefinite number of curves touching at P, to which the circle of which PV is a chord will be the circle of curvature. (4). Prove that when PQ is a parabola, FK is a straight line. 4. The areas round a centre of force are in the same plane, and proportional to the times." Prove the proposition. Explain the difference which in the change from polygonal to curvilinear motion takes place in the effect of the centripetal force;and show that, notwithstanding this difference, BV, (the diagonal of the parallelogram AC,) may still be used as a proportionate measure of it. 5. At similar points in similar curves described round a centre of AS force similarly placed, F x P being the periodic time. and explain to what units F and V are referred in this and similar equations. 8. Prop. x. Cor. 1. "Si vis sit ut distantia movebitur corpus in ellipsi centrum habente in centro virium." Given the velocity and direction of a projectile attracted by such a centre of force, construct the curve it will describe, and show from the construction that whatever be the velocity, the curve will still be an ellipse. 9. Prop. xiii. Cor. 1. "Si corpus aliquod quâcunque cum velocitate exeat de loco P, et vi centripetâ simul agitetur quæ sit reciprocè ut quadratum distantiæ, movebitur hoc corpus in aliquâ Sectionum Conicarum umbilicum habente in centro virium." Construct the curve; and shew from the construction that according to the ve locity of projection, the curve will be an ellipse, a parabola, or an hyperbola. 10. Determine the law of variation of the angular velocity in any curve; and compare the angular velocity at any point in the ellipse, (force in the focus,) with the angular velocity in a circle at the same distance. 11. If an nth part of the Earth were taken away, what change would be produced in the Moon's orbit? and in what ratio would her periodic time be increased? the orbit before the change being supposed circular. Exemplify in the cases where n = 2, or is greater or less than 2. Conics. 1. IF any two ordinates Q Q', qq' terminated both ways by the curve of a parabola, intersect each other in M, and P, Р be respectively their parameters, prove that QM. MQ qM. Mq:: P: p. 2. In the ellipse, CD2 = SP X HP. 3. Rectangle under the abscissæ of any diameter to an ellipse : ord.:: CP2: CD. 4. Show that the hyperbola admits an asymptote :-and that, if any line Rr between the asymptotes cut the curve in R, r at a given angle, the rectangle RP X Pr will be invariable. 5. Any section of a paraboloid not perpendicular to the base is an ellipse. 6. Show that the shadow of the circular horizontal rim of a lamp traced out on a perpendicular wall is an hyperbola :—and the height of the flame above the rim, and its distance from the wall being given, construct the hyperbola, and determine its major and minor axes. TRINITY COLLEGE, 18283. 1. Ir a right cone be cut by a plane of known inclination to the axis, find the equation to the section, and shew in what cases the section will be a circle, an ellipse, an hyperbola, or a parabola. 2. A parabola and an ellipse being traced upon a plane, find (1). The axis of the former. (2). The centre and axes of the latter. |