14. If from a quantity which varies as, any quantity be subtracted which varies as A, the remainder will vary in a higher inverse ratio than the inverse square of A; but if to a quantity varying as—another be added which varies as A the sum will vary in a lower inverse ratio than the inverse square of A. quired proof. Re 15. Find the law of the force tending to the centre of the logarithmic spiral. 16. Prove that when the force acts in parallel lines, the velocity in the direction perpendicular to the direction in which the force acts, is constant. 17. If the altitude of a cylinder be equal to the diameter of its base, the whole surface is six times the area of the base. 18. If a be constant, and (mx + n) x (nz + m) be a maximum; prove that a+n = bra+m MONDAY AFTERNOON.-Mr. MACFARLAN. Third and Fourth Classes. 1. Required the perpendicular from the vertex upon the base of a triangular pyramid, all the sides of which are equilateral triangles of a given area. 2. Given the difference of the times of setting of two stars whose declinations are known; find the latitude of the place.. 3. Find the centre of oscillation of a conical surface suspended by the vertex; and find the ratio between the radius of the base and the axis, when the centre of oscillation is in the base. 4. The length of a pendulum oscillating seconds on the earth's surface being given; find the length of a pendulum oscillating seconds at the distance of the earth's radius from the surface. Then determine a point below the surface where the last pendulum will vibrate in the same time. 5. Two roots of the equation x + x3—11x2 + 9x+18= 0 are of the form + a, - a. Find all the roots. 6. When the force varies as that power of the distance whose index is (n-1). Shew that the velocity of a body falling from the variable distance. And find the value of this expression when the force varies inversely as the distance. 7. If from the extremity of the diameter of a circle tangents be drawn and produced to intersect a tangent to any point in the circumference, the right lines joining the points of intersection and the centre of the circle shall form a right angle. 10. Find the attraction of a sphere on a particle of matter placed at any distance from the centre, the force of each particle varying inversely as the cube of the distance. 11. Find the equation to the curve, the length of whose tangent between any point and the axis is a constant quantity. 12. The equation to a curve is y3 Find axy + x3 = 0. the value of the ordinate when a maximum, and the corresponding value of the abscissa; and shew that the above is a maximum and not a minimum. 13. A paraboloid placed with its vertex downwards being full of water, is supplied at a given rate. There is a small hole in the vertex, which, when the vessel is full, would discharge a times the quantity supplied. Required the altitude at which the surface remains stationary, and the time elapsed before this takes place. MONDAY EVENING,-Mr. MACFARLAN. 1. A body placed in the centre of gravity of a triangle is acted upon by three forces represented in quantity and direction by the lines joining the centre of gravity with the three angles. Shew that the body will remain at rest. 2. The sides of the spherical triangle ABC are each a quadrant. D and E (any two points on the surface of the sphere) are joined by the arc of a great circle. Shew that the cosine of DE is equal to the coS AD X COS AE + COS BD X COS BE+ COS CD X COS CE. 3. If the sum of the odd digits in a number be 11me and of the even 11e, this number being divided B D successively by 11 and by 9, leaves the same remainder as m + ne when divided by 9. 4. Ina dial foragiven latitude, the plate of which ought to have been horizontal, the interval between ten and noon is less by two minutes, than the interval between noon and two o'clock. The line between north and south was found to be horizontal. Required the dip of the plate towards the east. 5. A sphere filled with water empties itself through a small hole in the bottom; find where the velocity of the surface of the descending fluid is a minimum, and where it is equal to the velocity when the sphere is half full. 6. The mean apparent diameter of the sun and moon's horizontal parallax being given, together with the length of a year and a month, find the density of the sun compared with the density of the earth; also shew how Newton finds the density of the moon. 7. From Newton's construction for the solid of least resistance, shew that in the section through the axis, the curve must make with the end an angle of 135 8. If the quiescent orbit be a circle (the centre of force in the circumference and the angular velocity in the moveable orbit is double that in the quiescent, Find the law of force in the orbit in fixed space, and investigate the ratio between the perpendicu lar and distance. 9. A wooden ball connected by a small wire with a ball of lead of the same diameter is dropt into the sea, and upon their striking the bottom, the wooden ball is disengaged and rises to the surface; the whole time elapsed, and the specific gravities and diameters of the balls being given; find the depth of the sea. 10. In any conic section, if tangents be drawn at the extremities of any diameter, and be produced to meet a tangent to any other point in the curve; Prove that the rectangle under the seg ments of the first tangents will be equal to the square of the semi, conjugate diameter. n or 2 qual to 1 × 2 × 3 × ...... X 2 1. The demonstration is required. 12. A given number (n) of similar balls being put into an urn; Required the chance of drawing an odd, to the chance of draw the variable distance. And find the value of this expression when the force varies inversely as the distance. 7. If from the extremity of the diameter of a circle tangents be drawn and produced to intersect a tangent to any point in the circumference, the right lines joining the points of intersection and the centre of the circle shall form a right angle. √(A+B terms, and 9. Find the fluents of 10. Find the attraction of a sphere on a particle of matter placed at any distance from the centre, the force of each particle varying inversely as the cube of the distance. 11. Find the equation to the curve, the length of whose tangent between any point and the axis is a constant quantity. 12. The equation to a curve is y3axy + x3 = o. Find the value of the ordinate when a maximum, and the corresponding value of the abscissa; and shew that the above is a maximum and not a minimum. 13. A paraboloid placed with its vertex downwards being full of water, is supplied at a given rate. There is a small hole in the vertex, which, when the vessel is full, would discharge a times. the quantity supplied. Required the altitude at which the surface remains stationary, and the time elapsed before this takes place. MONDAY EVENING,-Mr. MACFARLAN. 1. A body placed in the centre of gravity of a triangle is acted upon by three forces represented in quantity and direction by the lines joining the centre of gravity with the three angles. Shew that the body will remain at rest. 2. The sides of the spherical triangle ABC are each a quadrant. D and E (any two points on the surface of the sphere) are joined by the arc of a great circle. Shew that the cosine of DE is equal to the cOS AD X COS AE + COS BD X COS BE COS CD X COS CE. 3. If the sum of the odd digits in a number be 11m+e and of the even 11e, this number being divided B D E mon pack are dealt in order, one by one, stake each 1. with the condition, that he to whom the first knave is dealt, shall be the winner. What is the value of A's expectation? 21 Find the curve by the revolution of which round an axis the solid will be formed, which shall attract a particle placed at its vertex with the greatest possible force, the force of each particle varying inversely as the square of the distance. 22. A cylindrical vessel full of water is balanced by a weight P over a fixed pulley. A hole of given dimensions being made in the bottom, it is required to find how far p will descend during the time of emptying. 23. Prove that the sum of the reciprocals of the prime numbers is an infinitely great number though infinitely less than the sum of the reciprocals of the natural numbers, TUESDAY MORNING.-Mr. MACFARLAN. First and Second Classes. 1. A person borrowed pl. at interest. To discharge this he invested 27. at the end of the first year, 41. at the end of the second, and 8/. at the end of the third, and so on. How many years will elapse before this fund be large enough to discharge the debt, compound interest being allowed on both sides at a given rate? - 2. Required the length of a spherical shell of iron, which, when filled with a fluid, shall just float in water; the specific gravities of iron, of water, and of the fluid being given. 3. Compare the length of a degree of latitude at any place on the earth's surface, with the length of a degree of longitude at the equator. 4. The inclination of a small tube in the side of a vessel of water being given, and its height above the horizontal plane; it is required, from observing the point of the plane struck by the stream, to assign the altitude of the water within the vessel; and to describe the whole track of the issuing fluid. 5. If round any point within the circumference of a circle (not being the center) equal adjoining angles be described; of the circumferences on which they stand, that which is nearer the diameter passing through the point is less than the more remote. 6. In a combination of two wheels and axles, the circumference of each wheel is n inches; of each axle 1. A weight, P, is applied to the circumference of one of the wheels as a power to raise matter to a certain height. How much must be raised |