How many possible roots are there? 3. Find the centre of gravity of a frustum of a cone. Shew the position to which it tends as the ends tend to equality. 4. If L describes about T areas proportional to the times, L is acted upon by a force which tends to T. Explain particularly the case where T is in motion. 5. If y al xm in one curve and y a 2* in another, m being greater than n, shew that the first curve always falls between the other and its tangent in the immediate neighbourhood of the vertex. 6. Corporis in datâ trajectoria parabolicâ moti, invenire locum ad tempus assignatum. 7. Shew how we may find by observation the place and time of the equinox. 8. When a body descends in a fluid, find the greatest velocity which it can acquire. Hence knowing the rate at which a powder descends in a fluid, find the size of its particles. 9. A particle moves upon a given surface acted upon by no force. Determine its motion. What will be the path in the case of a cone ? 10. By three observations of a solar spot determine the position of the Sun's axis of revolution. Il. If a ray diverging from the origin of co-ordinates x, y falls upon the curve, the distance to the point where it cuts the axis after 2(x + py) (px - y) dy reflexion will be = where p= 2px - (1 - ply dx In a triangle the continued product of the four radii of the circles of contact is equal to the square of the area of the triangle. 13. Given the length of a curve, to find its form that the centre of gravity may be the lowest possible. a TRINITY COLLEGE, SEPT. 1828. 1. If a straight line falls on two parallel straight lines, it makes the alternate angles equal to one another. What is the difficulty of proving this proposition satisfactorily? Mention any of the ways by which mathematicians have endeavoured to get over this difficulty. 2. Two points are taken on a wall and joined by a line which passes round a corner of the wall. This line is the shortest when its parts make equal angles with the edge at which the parts of the wall meet. 4. From the equation y = draw the curve. Find the lt 23 angle at which it cuts the axis, and whether the area from x = 0 to x = inf. is finite or infinite. What is in each case the last term and the number of terms ? Write down the expression for cos.wix in terms of y when m is fractional. 6. The equation ax? — by2 + 2cz = 0 belongs to a curve surface. What is the nature of this surface ? In what direction is its concavity? Draw a normal to a given point of it. 7. When three forces keep a body at rest, what are the conditions which they fulfil? Hence find the conditions for the equilibrium of the voussoirs of an arch. How are these affected by taking into account the friction of contiguous voussoirs ? 8. A bowl filled with quicksilver is placed in an inverted position on a horizontal plane : what must be its weight that the fluid may not escape ? If there be a portion of air above the fluid, how will this affect the answer? 9. Two inclined planes being at right angles, a uniform straight beam rests between them. Shew that its middle point is highest when it makes with each plane an angle equal to the inclination of the other plane. 10. A candle is placed before a convex lens at twice its focal distance. At what distance must a plane be placed to receive the image of the candle ? And what will be the form of the illuminated part at any other distance ? 11. The equation of time is nothing four times in the year. 12. S is a fixed point, P a point in a curve, PT the tangent meeting a fixed line SA in T: to determine the curve from the property SP= m X ST. What is the curve when m =1? 13. Find the path of a body projected with a given velocity and 1 acted on by a force which is as (dist.)3 Why does Newton's method not give all the cases ? 14. When P revolves about T, disturbed by S, the eccentricity of P's orbit increases from the time of its apsides being in quadratures to the time of the apsides being in syzygies. What is the name and nature of the inequality of the Moon's motion thus produced ? 15. Four equal particles placed in a straight line repel each other with forces varying inversely as any power of their distance. The two extreme ones being immoveable, the two intermediate ones perform small oscillations in the straight line. Define their motion. 16. A pendulum which oscillates n times a day at the Earth's surface, is carried to a depth z below the surface, and there gainst seconds a day. The radius of the earth being 1, shew that its mean density is 3 17. Find the difference of the asymptote and arc of an hyperbola when both are infinite. 18. If a, 6 be the semi-axes of an ellipse revolving in its own al + 62 plane, M its mass, and Mk its moment of inertia, ko = 4 and m = 19. On a spheroid, if I be the latitude, a the angle at the centre, a? — 62 we shall have a? + 62' m? ms sin.4a sin.6a, &c. 3 20. A's skill is to B's as 2:1 (m: n). A wants 6 (p) games to win and B wants 3 (9): what are their chances respectively? 21. Find the Moon's variation. What is the argument of this inequality ? 22. Explain and prove the principle of least action. TRINITY COLLEGE, 1830. 1. The price of gold is £3 178. 104d. an ounce, and a cubic inch weighs 10 ounces. What would be the cost of gilding the surface of England (a triangle of which the base and the perpendicular are each 315 miles) with gold .00019 of an inch thick, 2. A sets off from Cambridge to London, and 24 hours afterwards B starts from London to Cambridge: after 3 hours more they meet; and A reaches London 34 hours sooner than B reaches Cambridge. In what time does each perform the journey? 3. Give those geometrical definitions which are used in the proof of propositions. What other definitions would be requisite to supersede the axioms entirely? Prove Euclid, I. 4; and explain why the corresponding proposition concerning spherical triangles cannot be proved in the same manner ? 4. While sailing S.W. I observe two ships at anchor, one at N.N.W. and the other at W.N.W. After running 5 miles these ships are seen at N. by W. and N.W. respectively. Required their bearing and distance from each other? 5. A pyramid has for its base an equilateral triangle of which each side is 1 foot, and its slant edge is 3 feet. Required the angles which its faces make with each other. 6. A weight slides on a string which is without inertia, and has its extremities fixed: (1) find the position of rest when there is no friction : (2) find the friction requisite to sustain the weight in any other position: (3) when the weight slides freely, find the equations of motion. 7. Show that the centre of pressure of a plane coincides with its centre of percussion : the axis of motion being the intersection of the plane with the surface of the fluid. : 8. Find the focus of a refracting sphere, the index of refraction being 1.9. The diameter of the sphere being to of an inch, what is the breadth of lines which through it appears like lines of To of an inch breadth seen with the naked eye at 10 inches? 9. What is the use of observations of known stars made with a transit instrument: and the manner of making them? Prove the following formula for finding the deviation of a transit instrument from the meridian by means of a high and a low star: a cos.d cos.d (1 – t) cos.l sin.(8 — 80 T In which r is the true difference of right ascension, and t the observed difference; 8, 8, I the declinations and latitude. Why are a high and low star selected ? |