each time, that the whole quantity may be raised in the least time possible? 7. Of all cones containing a given quantity of matter, to find that which attracts a particle placed at its vertex with the greatest possible force. 8. Show that when a quantity is a maximum or a minimum, the first fluxion vanishes; and that the quantity is a maximum or a minimum, accordingly as the second fluxion is negative or positive. 9. An imperfectly elastic ball is projected with a given velocity against an hard horizontal plane, and being reflected, just reaches the point of projection in t". Required the distance of the plane, from the point of projection, and the elasticity of the body. 10. A cylindrical tube of given length, closed at one end, being let down in a vertical position into the sea, it was observed what part of the tube the water occupied. It is hence required to assign the depth, 33 feet of sea water being assumed to measure the weight of the atmosphere. How must this tube be graduated to be used as a gauge to measure depths in the sea? 11. Find the length of a common parabola, and deduce Cotes's construction. 13. Sum the series 12+ 3+ 72 + 15 &c. to n terms, and 1. Find the value of £12341414141 &c. 2. The amount of £500 in of a year was £520. Required the rate per cent. 3. Find the circumference of a circle whose radius is unity. 4. Sum the following series, (1). 2 + 3 + 2 + &c. to 20 terms, b 5. Solve the following equation whose roots are in arithmetic progression; x3 — 9x + 2 3x 16 0. 7. The arc of a circle which a body, acted upon by a centripetal force, uniformly describes in any given time is a mean proportional between the diameter of the circle, and the space described by a heavy body from rest in the same time when urged by the force in the circumference continued uniform. L 3 8. Show that the logarithm of (1+u) is equal to u- + 9. Given the radii of the surfaces of a double convex lens and the ratio of the sines of incidence and refraction. Find the focal length. 10. Find the latitude of the place at which the sun sets at three o'clock on the shortest day. 11. When the force varies as the distance, the periodic time in all ellipses round the same centre are equal. 12. A body (A) weighs 12lbs. in vacuo, and glbs. in water; another body (B) weighs 10lbs. in vacuo, and 8lbs. in water; compare their specific gravities. 13. If the number of mean proportionals interposed between two elastic bodies A and x be increased without limit, the velocity of a will be to the velocity communicated to x by means of the intermediate bodies :: x : √A. 14. The apparent diameter and declination of the sun being given, find the time of his transit over the meridian. 15. The plane of a circle being vertical, and any number of chords being drawn to the lower extremity of the vertical diameter; Find the locus of any number of heavy bodies falling together from the upper extremities of the diameter and the chords at any given instant of time. 16. If any number of projectiles be thrown at the same instant from the same point and with equal velocities, but in seve ral directions in the same vertical plane, they will at the expiration of any time all be found in the circumference of some circle. TUESDAY AFTERNOON.-Mr. BLAND. Third and Fourth Classes. 1. Shew from the principles of the fifth book of Euclid, that a ratio of greater inequality is diminished, and of less inequality increased by adding a quantity to both its terms. 2. The time of day at a given place determined from observations of the sun's altitude is 9h 10m 45'; and a chronometer set to Greenwich time shews 6 3 10. Required the longitude of the place of observation from Greenwich. 3. In any harmonic progression, the product of the two first terms is to the product of any two adjacent terms as the difference between the two first is to the difference between the two others, Required a proof. 4. An object is placed between two plane reflectors which are inclined to each other at an angle of 60°. Determine the whole number of images formed by the reflectors. 5. If the greatest possible rectangle be inscribed in the quadrant of a given ellipse, shew that the elliptic areas cut off by the sides of the rectangle are equal. 6. Prove that an equation of an odd number of dimensions, and an equation of an even number if its first and last terms be of different signs, must have at least one real root. 7. Find the sums of the series 3 1.2.4+2.3.5 + 3.4.6 + &c. to n terms, 3 + + + 3 + &c. to in terms. and 1.2 + 2.3.x + 3.4.x2 + &c. in inf. 8. If the force vary inversely as the square of the distance, and a body be projected at a given angle with a velocity which is to the velocity in a circle at the same distance, as √2: 1. Determine the nature of the orbit described. 9. Prove that the surface of any segment of a sphere cut off by two parallel planes is to the whole surface of the sphere as the intercepted portion of the diameter is to the whole diameter, 10. Find the fluent of (x (x-1). x 2 ; construct the fluent of where a is less than unity; and shew that the √(a2—bz') and radius 1. 11. A paraboloid whose vertex is downwards is filled with water to a given altitude. Having given the diameter of the upper surface, find what ought to be the diameter of the hole at the bottom, so that the upper surface may descend through a given space in a given time. 12. If the force varies as the distance, and two bodies fall to wards two different centres of force; compare the velocities at any point of their descent. 13. Two elastic balls beginning to descend from different points in the same vertical line, impinge on a perfectly hard plane inclined at an angle of 45°, and move along a horizontal plane with the velocities acquired. Shew that if a circle be described, passing through the two points from which the balls began their motion, and touching the horizontal plane, the point of contact will bisect the distance between the vertical line and the point where they impinge on each other. 14. Given, that the distance of the centre of gravity of an area from its vertex is an nth part of the abscissa; find the distance of the centre of gravity of the solid formed by the revolution of this area round its axis. 15. Determine the proportion between the radius of the globe and wheel, when the length of the cycloid within the globe, (Sect. 10) is a maximum. 16. If centripetal forces tend to the several points of spheres, proportional to the distances of those points from the attracted bodies, the compounded force with which two spheres will attract each other mutually is as the distance between the centres of the spheres. TUESDAY EVENING.-MR. BLAND. 1. The sum of n arithmetic means between 1 and 19 is to the sum of the first n 2 of them :: 5 3. Find the means. 2. Two equal weights are connected by a string passing over a fixed pulley. Supposing a weight to be added on one side, and the length and weight of the string, and the difference of the altitudes of the weights at the commencement of the motion to be given; determine in what part of the descent, the velocity will be neither increased nor diminished by the string's weight. 3. If the abscissa of a curve bear a finite ratio to the ordinate, prove that the abscissa will cut the curve in a finite angle. 4. The place of the node and the inclination of the moon's orbit to the plane of the ecliptic being given; find the place of the moon when her declination is the greatest possible. — — 5. Find the value of √(2a1x-x) — (ax), when x = a; a (ax) -- and find the fluxion of the hyp. log. (a + x) — √ (a — x)° √(a + x) + √(a — x) 6. If, in a circle a straight line be drawn cutting the diameter at any angle (A); prove that the difference of the segments of the diameter will be to the difference of the segments of the line as the diameter is to the chord of an arc, which measures twice the complement of A. 7. If, from the extremity of the major axis of an ellipse which is perpendicular to the horizon, chords be drawn making with it angles of 75° and 45°, and from the points where the chords meet the curve, ordinates be drawn to the axis; the square of the time down the first chord will be to twice the square of the time down the second in the subduplicate ratio of the rectangles under the segments of the axis made by the ordinates. 8. If a square be inscribed in circle and another circumscribed about both compare the pressures upon the circle and the squares when immersed vertically in a fluid, the angular point of the circumscribing square coinciding with the surface of the fluid. 9. A hollow cone whose vertical angle is 60°, is filled with water, and placed with its base downwards. It is required to determine the place where a small orifice must be made in its side, so that the issuing fluid may strike the horizontal plain in a point, the distance of which from the bottom of the vessel is to the distance of the orifice from the top as 5: 4. 10. The distance of the centre of gravity of the surface of a solid from the vertex is equal to half the abscissa; determine the nature of the curve by the revolution of which round its axis the surface was generated. 11. If two equal parabolas be placed in such a manner that they may touch each other at the vertices, and one be made to roll upon the other, its focus will describe a right line, and the vertex a cissoid, the diameter of whose generating circle is equal to half the latus rectum of the parabola. 12. If a body revolve in an orbit round a centre of force, and at the same time the orbit revolve round the same centre in such a manner that the angular velocity of the body in the orbit if fixed, may be to its angular velocity when revolving, in the ratio |