TRINITY COLLEGE. 1820. PROBLEMS. 1. WHAT sum of money must be laid out in the 3 per cent. consols at 633 per cent. to produce an income of £400 a year? 2. The sine of any angle of a plane triangle has to the opposite side a constant ratio.-What is this ratio? 3. Find by the method of continued fractions a series of fractions converging to 19. 4. Given the three sides of a plane triangle, find 83 Its area, The radius of the inscribed circle. 5. Differentiate the following quantities, (1) u = 1 √1+sin. 2; 1-sin. x 1+ &c. ad inf. 6. Find a number which being divided by 2, 3, 5, shall leave for remainders 1, 2, 3, respectively. 7. The angles of any plane triangle being A, B, C, prove that [to radius (1),] 4 sin. A. sin. B. sin. C = sin. 2A+ sin. 2B+sin. 2C. 8. (1) Find the locus of the vertices of all the triangles described on the same base, when one of the angles at the base is always double of the other. (2) Hence trisect a given angle. 9. Find the radius of curvature at any point of the common cycloid. 10. In any spherical triangle, the arcs of great circles drawn from the three angular points perpendicular to the opposite sides intersect in the same point. 11. Sum the following series: 12. A sphere of given diameter descends in a fluid, from rest, by the action of gravity; find the 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, shew that its area={(Sa) (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+bx2, 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. Shew also that it is a maximum and not a minimum. 17. State the principle of virtual velocities; and hence shew 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 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, (4) the angle at which the radius of curvature (r) at any point cuts the axis, prove that a2b2 (a2 cos. +b sin.20) 3 21. The roots of the equation x" —px"1 + qx"−2· being a, B, y, &c. find the value as a terms of the coefficients p, q, r, &c. ጎዜ 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. 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. FOR SCHOLARSHIPS. 1. FIND the value of £3.869, and 365.24215 days. square root of 0.676, and 6.76. 2. If twenty men in digging a canal must pump out six tons of water daily, in order to excavate 160 cubic yards in a week, how many cubic yards can thirty men excavate in a week, supposing them to be obliged to pump out eight tons of water daily? 3. If S denote the sum of the even terms, and S' the sum of the odd terms, of the expansion of (a + b)"; then SS" = (a2—b3)*. 5. Find the roots of an equation of this form by construction, viz. x3 - 3r2x — q = 0. 6. Compute the numerical value of the side of a regular decagon inscribed in a circle, whose radius is ten inches. 7. The three angles of a plane triangle being given, and the distances of the three angular points from a given point within the triangle, to find the sides. 1 8. The complement of the hypothenuse of a right-angled spherical triangle cannot exceed the complement of either of the other sides. 9. If about the three angular points of a spherical triangle three great circles be described, the triangle formed by these latter will have its sides measures of the angles of the original triangle, or of the supplements of those angles. 10. In a given parabola, to draw a diameter which shall make with its ordinates an angle equal to a given rectilineal angle. 11. Two vertical straight lines being given, to place them at such a distance asunder in the same horizontal plane, that a heavy body shall be as long in falling down the greater, and then moving with its acquired velocity to the less, as it would be in falling down the less vertical, and moving with its acquired velocity to the greater. 12. When a radiant sphere shines upon an opaque sphere, the breadth of the illuminating portion of the former has for its measure the same number of degrees as the dark portion of the latter. 13. The distance being given to which a fluid spouts from a given orifice in the side of a cylindrical vessel, to find by a geometrical construction the height of the fluid's surface in the vessel. 14. The right ascensions and declinations of three stars being given, and the times between their passages over the same vertical wire of a telescope, to find the latitude of the place; one of the stars being supposed in the equator. 15. The altitude of the Sun when due west, and also at six o'clock P.M. being given, find the latitude and the Sun's declination. 16. Having the focus of incident rays upon a medium terminated by two plane sides inclined at a given angle, find the focus of emergent rays. 17. The exact quantity of the year being 365.24215 days, explain the reason of the corrections in the civil year introduced by Julius Cæsar and Pope Gregory. 18. Give the theory of the Trade Winds. 19. Prove that part of the equation of time which arises from the obliquity of the ecliptic to be a maximum when the longitude of the Sun equals the complement of its right ascension. 20. Compare the surface of a sphere with the area of its great circle, and its magnitude with that of its circumscribing cylinder. APPENDIX. TRINITY COLLEGE.. FOR SCHOLARSHIPS. 1. To find the locus of the extremities of all the straight lines that can be drawn from the circumference of a given circle, toward the same parts, each of them equal and parallel to a given finite straight line. 2. To find the centre of a given ellipse. 3. To construct the curve of which the equation is ax2 + ay3 + bx + cy + d= 0. 4. If the product of any two given numbers be a square, each of the two given numbers is the product of two factors, such that the four factors are proportionals. 5. Solve the following equations: (B.) aTM ̄¦— bTM × c2 = 0. (C.) 2xy+x+y-1950, (to find the integral values of x and y). (D.) {3/2 90000=0 lyx300y125x = 0 6. If none of the coefficients of the equation a+ax®1+bx*~2+ &c.+g0 be fractional, it cannot have a fractional root. 7. To compare the chance of throwing 7, with the chance of throwing 8, at one throw, with three common dice. 8. Sum the following series: (A.) 1+3+27+64+&c. (to n terms.) (B.) -/ 1 (C.) 9. 4 9 16 + + + + &c. (ad infinitum.) 9 27 81 (A.) Find the fluent of x2√ a2-v2 √ (x2 — a2). (b2 — x2) z being a circular arch, of which is the cosine, and the tangent; and of a+bx2 |