16. A sphere rolls down an inclined plane; find the force accelerating its descent. 17. When two or more bodies move uniformly in straight lines their centre of gravity also moves uniformly in a straight line. 18. The power of a compound lens is the sum of the powers the component lenses. of 19. What is meant by the error of collimation in a telescope? How is it corrected? 20. Find the centre of pressure of a sluice-gate in the form of an isosceles, whose base coincides with the surface of the fluid. 21. Find the precession of a star in right ascension. 22. What is the greatest number of eclipses of the Sun and Moon which can happen in a year? 23. To find the law of force to the focus of a logarithmic spiral. and let a body be projected in any 24. Let the force o 1 (dist.)s? direction from a given point, with a velocity less than that in a O at the same distance, to determine the orbit described. 25. The length of a degree to the meridian is always greater than that of the degree of the corresponding meridian. 26. At a given plane, at a given hour, and on a given day, required the point of the compass on which a rainbow would appear. TRINITY COLLEGE, APRIL 1830. 1. TRIANGLES upon equal bases and between the same parallels, are equal to one another. 2. Equal parallelograms, which have one angle of one equal to one angle of the other, have their sides about the equal angles reciprocally proportional and conversely. 3. Find the radius of a sphere described about a given regular tetrahedron. 4. Multiply 54-6 by 8.93 in the duodecimal scale: and if the units in the factors be feet, express the product in square yards, feet; and inches. 5. The sides of a triangle are 17·09 and 93·451, and the angle opposite to the greater of them is 93°.16': find the angle opposite to the less. 6. Explain the theory and use of the columns of proportional parts, in tables of logarithms. (y). sin.(x + a) + cos.(x + a)!= sin.(x − a) + cos.(x − a) 8. Eliminate x from the two equations x* — (2y2 + 1) x2 + y^ + y2 — 20 = 0 } 9. x+ — 2yx2 + y2 —49=0. In a game > of pure skill, if A's skill be to B's as 3 to 1, what is the chance of A's winning 3 games out of 4? 10. Define a differential, and from your definition find the differential of a*. 11. Draw a tangent to a cycloid and determine its area. All parallelograms which circumscribe an ellipse are equal to each other. 13. In what respect are the curve surfaces represented by the equations distinguished from each other? Find the length of a normal to the former, intercepted between the surface and any one of its principal sections. 15. Find the distance of the centre of gravity of any number of given bodies from a given plane; the distance of the centre of gravity of each body from the plane being given. 16. If P (5) draws up Q (3) by means of a string passing over a fixed pulley, find the force accelerating P's descent and the space described in ť" (10): the weight of the string and the inertia of the pulley being neglected. 17. Investigate the conditions of the equilibrium upon the screw. 18. The centres of oscillation and suspension are reciprocal. 19. Upon what do the magnifying power, brightness, distinctness, and achromatism of the astronomical telescope depend? 20. Find the deviation of a ray passing through a prism, whose refracting angle is considerable. 21. Show how to find the approximate solution of the equation, which expresses the relation between the eccentric and the mean anomaly. 22. Find the precession of a star in R. A.: when is it zero ? 23. Investigate a formula to express the acceleration of the time of rising of a given star by the horizontal refraction. 24. Prove that the pressure upon any portion of a vessel filled with fluid of uniform density, is equal to the weight of a column of fluid whose base is the area of the surface pressed, and altitude the perpendicular depth of its centre of gravity, below the surface of the fluid. 25. Find the law of force acting in parallel lines, by which a body may be made to describe à portion of a circle. 26. A body is projected from a given point, in a given direction, with a given velocity above a given plane: find the orbit described when it is repelled from the plane by a force varying as its distance from it. 27. Determine the attraction of an oblate spheroid upon a particle situated at its equator. 28. The Moon is retained in her orbit by the force of gravity. FOR FELLOWSHIPS. TRINITY COLLEGE, 1819. i. A GROCER had 150lbs. of tea, of which he sold 50lbs. at 9s. per pound, and found that he was then gaining only 7 per cent. But he wished to gain 10 per cent. on the whole. At what rate must the remaining 100lbs. be sold that he may attain his wishes? 2. Solve the following equations : 3. In what time will a sum of money placed at compound interest double itself at 4 per cent.? 4. Required the value of the infinite series (18+189) (18+ &c.) 5. Required the value of y in the following equations, when |