2. Find the values of .x and y, when xy=63, and x+y2 :: 2-y?: 64 : 1. Also value of x in this equation x2.7+4+2.7.X +4=2- +4. 3. Divide a given angle into two angles, such that their sines may be in a given ratio. 4. Express the side of a regular decagon, inscribed in a circle, in terms of the radius. 5. Two bodies, A and B, are projected perpendicularly upward with velocities (a) and (b). It is required to assign the highest point to which their common centre of gravity will ascend. 6. Find the roots of the equation x3 – 13ro +50x – 56=0, two of whose roots are in the ratio of 2:1. 7. The diameter of a cylinder is 10 inches, and the diameter of an orifice in its base .025; also the height of the water in the cylinder is 84 feet. Required the time of emptying. 8. Given the apparent perpendicular depth of an ohject under the water, to find the direction in which a ball must be fired from a given point, so as to strike the object. 1 1 9. Sum the series 3 + + + &c. ad infinitum. 2 12 Also, 1.3.5 +3.5.7 +5.7.9+&c. to n terms. 10. Determine the equation of a curve by whose pevolution a solid is generated equal, at all altitudes, to ths of its circumscribing cylinder. 11. Find the centre of gravity of a bar whose density cc.x"; x being the distance from the vertex. 12 A known star rises in the north-east point; find from this circumstance the latitude of the place. 13. Prove that V’ in any curve : v2 in a circle at ур the same distance :: where y is the variable y p distance, and p the perpendicular on the tangent. 14. How does the centrifugal force vary in different curves? and how does it vary in different parts of the same curve? Fifth and Sixt? Slasses.--Mr. Turton. 1. If } yards cost £24, find the value of 56 yards both by vulgar fractions and by decimals. 2. Prove that if a straight line stand at right angles to each of two straight lines in the point of their intersection, it will be at right angles to the plane that passes through them. 3. Define a rhoinbus; and prove that the diagonals of a rhombus bisect each other at-right angles. 4. If (a) be the first term, and (b) the sum of three terms of a geometric progression, find the common ratio. 5. If the fluxion of Nar ↑ a ==0, find the value of x. 6. Given three bodies A, B, C, and their distances from a plane; find the distance of their common centre of gravity from that plane, supposing A and B to be on one side of the plane, and C on the otiser side. 7. Given the velocity and direction of projection, find the greatest height of the projectile above the inclined plane; and from the expression deduce the greatest height above the horizontal plane. 8. Shew that if a plane mirror recede from a fixed object, the image will recede twice as fast. 9. Explain the principle on which the Hydrometer is constructed, and demonstrate the proposition on which the construction depends. 10. Construct the common Astronomical Telescope, and investigate its magnifying power. 11. Given the latitude of the place, and the sun's declination; to find his azimuth at six o'clock. 12. Shew that the velocity in any conic section is to the velocity in a circle at the same distance in the subduplicate ratio of (L x SP to SY. 12. Find the fluents of the following quantities: viz. di 25.0 mi ? x'c 1 1 14. Sum the series + + + &c. 1.2.3 2.3:4 3.4.5 to n terms, and ad infinitum. 15. Investigate the assumptions by which an equation (z" - pra-'+q.xn_2 – &c. =0) may be transformed into others wanting the second or third terms. 16. Given the earth's radius and the space fallen through in one second at its surface, find the periodic, time in a circle at a given distance above the earth's surface; gravity varying inversely as the square of the distance. 17. If a body whose elasticity is to perfect elasticity as m to 1 be let fall from a given altitude (a) above. a perfectly hard horizontal plane, and rebound continually till it's whole velocity is destroyed; find the whole space described. 18. A given paraboloid floats in a fluid with its vertes downwards; compare the specific gravities of the body and the fluid, supposing half the axis to be immersed. Evening Problems.-Mr. Walter. 1. A cylindrical bar is suspended by a given point in a semi-circle, whose diameter is the bar. Find the inclination of the bar to the horizon, upon supsition that the semi-circle is devoid of weight. 2. Prove, from a property of the circle, that if four quantities are proportionals, the sum of the greatest and least is greater than the sum of the other two. 3. Given the area of any plane surface, it is required to find the content of a solid, formed by drawing lines from a given point without the plane, to every part of its surface. 4. The inclination of a perfectly smooth bank to the horizon is 30°, and a body is projected up the bank in a direction making an angle of 45°, with the intersection of the bank and horizontal plane. It is required to determine the curve described by the body, and the spot where it will again meet the horizon. 5. If two curves have a common axis, and ordinates which are always in a given ratio to each other, then tangents drawn from the extremities of any corresponding ordinates will meet the axis in the same point. 6. The direction of a bridge is from east to west, and the sun in the meridian. The arches being supposed semi-circular, it is required to find the curve terminating that part of the surface of the water which is illuminated by the sun's rays passing through any arch. 7. It is required to express the cosine of an angle of a spherical triangle in terms of the sines and cosines of the sides. 8. If a body revolves in any curve whose equation is ap=y", y being the distance from the centre of force, and y the perpendicular on the tangent; it is required to find the equation of the curve of a star's apparent aberration, as seen from this body. 9. The roots of the equation 23 - p.x? +qx – r=-0, are a, b, c,; transform it into one whose roots shall be a +b, b+c, +c. 10. Required the position of the eye in a given line perpendicular to the horizon, so that the image of a given circle on the ground may also be a circle, when projected on a plane perpendicular to the horizon by lines drawn to the eye. . 11. Find by the help of the common tables the logarithm of a number consisting of seven figures. 12. The roots of the equation x3 – p.x2 +qx – r=0, are in harmonical progression: find them. 13. Given the sun's declination, and the latitude • of the place; find the path described by the shadow of a staff on an horizontal plane. 14. Sum the series 1.2.4+3.4.6 +5.6.8+ &c. to n terms. 1 2 3 4 Also + &c. to infinity 3 475 6 1 1 1 And + + +&c. to n terms, and to infinity. 1.3 2.4' 3.5 15. A has (P) counters, and B has (9); also the N |