R', 2', P' &c. integers. Required 7. Given the specific gravities of two bodies, to find the ratio of their quantities of matter, when they balance each other in a fluid whose specific gravity is also given. 8. A given cylindric vessel is supplied with water by a cock at a given rate. Then suppose when the vessel is full, that an aperture of given area is made in its bottom; what is the lowest point to which the surface of the water in the vessel will descend, and also the time of its descent; the influx of the water by the cock being supposed slower than the efflux when the vessel is full? 9. A given weight, suspended by a thread of a given length, oscillates, in the manner of a pendulum, through a semi-circle. Compare the weight of the body with the force which stretches the string in any position. 10. Find the magnitude of that part of the disturbing force of the sun upon the noon, which acts in a direction perpendicular to the plane of the moon's orbit. 11. Required the fluents of x 3 x × a2 + x2 } } › dx z 4 n-1% a+bzn√c+ezni and zn xa+bz 12. Find the relation of x to y in the equation ry —ÿ x = ¿ √ x2+ y2. 13. Required the sum of n terms of the series 1.2.4+2.3.5 +3.4.6+ &c.; and the sum of the 1 222 &c. equal to (2 being the length of the qua drantal arc to the radius=1); to find the sum of the 1 1 series + + &c. ad inf. 15. The fluent of yử=y× x Ў x2 X3 1.2.3 X + X 1 X 1.2 22 -&c. ( being constant); required the inves tigation. 16 At what distance from its extremities must a slender rod, of uniform thickness and given weight, be suspended, that it may oscillate in a given time? 17. Suppose the proportion of the chances for the happening of an event, be to that of its failing in any one trial as a to b; what is the probability that the event will happen precisely p times in î trials? 19. If a body revolve in an ellipse round a center of force situated in the focus; investigate the relation between the mean angular velocity of the body, and its angular velocity round the upper focus; and shew from this relation that when the ellipse is of small excentricity, the angular velocity round the upper focus is nearly uniform. 20. The greatest velocity which can be acquired by a spherical body descending in a fluid, is equal to that which would be acquired by it in descending from rest in vacuo, by the constant force of its comparative gravity, through a space which is to of the diameter, as the density of the sphere to the density of the fluid. 21. Suppose the force of gravity to be uniform, and to tend perpendicular to the horizon; what will be the path of a projectile in a medium, the resistance of which is proportional to the velocity of the body? 22. If the centripetal force vary reciprocally as the distance from the centre; and the density of the fluid be proportional to the compressing force; then if distances from the centre be taken in geometrical progression, the corresponding densities of the fluid will be also in geometrical progression. 23. Bradley observed that every star passed the meridian farthest south when it came about six in the morning, whatever were its position with respect to the cardinal points of the ecliptic. Suppose then the place of a star to be given; on what day will it pass the meridian of London farthest to the south; and at what hour will it pass it on that day? 24. A perfectly flexible chain is wound round a cylinder supported with its axis parallel to the horizon. Then, if the weight and dimensions of the cylinder be given, and also the length and weight of the chain; it is required to determine the time in which the chain, impelled by the force of gravity, will unwind itself; a given length being unwound at the commencement of the motion. Second Morning-Mr. Barnes. 1. Find the value of the decimal 726363 &c. in a vulgar fraction. 2. Required the value of x in the equation 2x √ 3. If x2-pxn--+qxn--2-&c.±W=A, and a be any root of the equation "—pxn--1 +qxn--?— &c.± W=0; Prove that x-a is a divisor of the expression *— pcn+q^__ &c.+W. 4. Suppose the arms of a bent lever, inclined to each other at a given angle at the fulcrum, to be also given: Required the position of the lever, when two given weights, suspended freely from its extremities, balance each other. 5. A given orifice is opened in the bottom of a vessel of given altitude filled with water; What must be the nature and dimensions of the vessel, that the surface of the water may descend with a given uniform velocity? 6. Required the nature of the caustic, when the reflecting curve is the common cycloid, and the rays are incident parallel to the axis. 7. Two stars whose right ascensions and declinations are known, were observed to rise at the same moment: Required the latitude of the place of observation. 8. Prove that the periodic times of bodies, describing different ellipses round different centres of force situated in their foci, are in the sesquiplicate ratio of the najor axes directly, and in the sub-duplicate ratio of the absolute forces inversely. 10. Required the sum of the first n terms of the 11. Find the sum of the first n terms of the series 12. Compare the mean quantity of the force by which the sun disturbs the motion of the moon, with the force of gravity at the earth's surface. 13. Compare the force which accelerates the centre of gravity of a cylinder, while it rolls down a given inclined plane, with the weight of the cylinder. 14. Given the latitude of the place, and the right ascension and declination of a star, to determine the time of the year when the star rises with the sun. Second Afternoon.—Mr. Barnes. FOR THE THIRD, FOURTH, FIFTH, AND SIXTH CLASSES. 1. What is the interest of £400. for 9 months at 4 per cent, per annum? 2. What is the discount of the same sum for the same time, at the same rate of interest? 3. Reduce 1.6363 &c. of a crown to the fraction of a guinea. 1 4. Given Sx2-3x+6=5, to find r. 5. Required the sum of n terms of the series, 1 1 1 1 + + 2 4 8 16 &c.; and also the sum of the series continued ad infinitum. |