greatest velocity it can acquire, its specific gravity being (n) times that of the fluid. 13. (1) Of all quadrilateral figures contained by four given right lines the greatest is that which is inscriptible in a circle. (2) If a, b, c, d, be the sides of this quadrilateral, S its semi-perimeter, show that its area = √{(S—a)(S—b)(S−c)(S—d)}. 14. Find the centre of gyration of a given sphere. 15. Any two right lines intersect each other in space; having given their separate inclinations to three rectangular co-ordinates passing through the point of intersection: find their inclinations to each other. 16. (1) Trace the curve whose equation is y2(c-x)=x3+br2, and find its area when b=0. (2) The equation to a curve is y3—axy+x3 =0; find the value of the ordinate when a maximum, and the corresponding value of the abscissa. Show also that it is a maximum and not a minimum. 17. State the principle of virtual velocities; and hence show that if any system in equilibrium, acted on by gravity alone, have an indefinitely small motion communicated to its parts, its centre of gravity will neither ascend nor descend. and find the relation of (x) to (y) in the equations (1) xdy -- ydx = ydx log. Y (2) dx+x3dx=dy+ydx. 19. If two weights acting upon a wheel and axle put the machine in motion, find the pressure upon the axis without taking into account the machine's inertia. 20. If (a) and (b) denote the semi-axis of an ellipse, (0) the angle at which the radius of curvature (r) at any point cuts the axis, prove that r= a2b2 (a2 cos.20+b2 sin.20) 21. The roots of the equation -2 x” —px”―1 + qx”—2 —&c. = 0, being a, ß, y, &c. find the value as am. +ßm+y” +....... in terms of the coefficients p, q, r, &c. 22. AP is any arc of a parabola whose vertex is A and focus S; let N be the intersection of a perpendicular from S on the tangent at P with the perpendicular to the axis from A. Then if AS-a, LASN=p. Show that are AP-PN-a. l. tan. 23. If a circle whose diameter is equal to the whole tide in any given latitude be placed vertically, and so as to have the lower extremity of its diameter coincident with the level of low water, prove that the tide will rise or fall over equal arcs in equal times. TRINITY COLLEGE. Mr. Hamilton's Paper. NEWTON'S PRINCIPIA. BOOK I. Book 1. (1) The centripetal force (F) in any curve =Q.- dp (p) being the perpendicular from the centre of force on the tangent, at distance (x) Determine Q. (2) Find the value of (F) in the ellipsethe force tending to the center. 2. If a body be acted on by two forces tending to two fixed centers, it will describe, about the straight line joining those centers, equal solids in equal times. 3. A body describes a parabola about a center of force situated in the focus, (1) Find its position at any assigned time. (2) Given two distances from the focus, and the difference of anomalies, Find the true anomaly. 4. The time of a body's descent, in a right line towards a given centre of force varies as 1 (dist.) from that centre. Required the law of the variation of the force. 5. A body at P is urged by an uniformly accelerating force in the direction PS, and at the same time is impelled in the opposite 1 direction by a force varying as from S. (dist.) Find its velocity at any point N. N +P S 6. In the logarithmic spiral find an expression for the time of a body's descent from a given point to the centre, and prove that the times of successive revolutions are in geometrical progression. 7. A body acted on by a force varying as 1 from the centre, is projected from a given (dist.)" point, in a given direction, and with a given velocity. (1) Find the equation to the trajectory described (2) Determine in what cases the body will fall into the centre, or go off to infinity. 8. The force varying as 1 (dist.) shew under what restrictions of the velocity of projection, the body's approach towards the centre, and its recess towards infinity, will be limited by asymptotic circles. 9. The difference of the forces by which a body may be made to move in the quiescent and in the moveable orbit varies as centre. 1 (dist.)3 from the 10. (1) Deduce the equation to the orbit in fixed space. (2) Shew that when any one of Cotes's three last spirals is made the moveable orbit, the orbit in fixed space will be one of the same species |