QUEEN'S COLLEGE, MAY 1830.

1. EXTRACT the fifth root of the binomial 76 + 44 √3.

2. Reduce 3-1 +5/-1 to the form of a ±ß√ −1.

3.

Determine the quantity which multiplied into 3/3 + 1/4 will make it rational.

4. If in the expansion of (1 + x)m, K and L be any two conse cutive coefficients; prove that the coefficients after L will be

5. There is a series of numbers, each of which when divided by p, 2p, 3p, &c. in succession, gives remainders p-1, 2p — 1, 3p-1, &c. respectively; find the sum of n terms of this series, beginning with the least.

6. There are two light-houses, the position and the magnitude of whose lights are known; find the path which a vessel must describe so that the quantities of light it receives from each may always be in the same ratio.

7. A given sector of a circle is immersed in a fluid, with its axis vertical, and its vertex coincident with the surface: divide it by a horizontal line into two parts which shall sustain equal pressures.

8. Compare the times of emptying the upper half of a paraboloid whose axis is horizontal, through a small orifice in its vertex, with that of emptying the lower half through a small orifice in its lowest point.

9. A chain fastened by two tacks in the same horizontal line, forms itself into a common parabola; find the law of its thickness.

10. When one curve surface rests upon another, shew under what circumstances the equilibrium is stable, unstable, and indifferent; and when one paraboloid of revolution, cut off by a plane perpendicular to its axis, rests with its vertex upon that of another of equal parameter, the axis of both being vertical; find the length of the axis of the upper paraboloid so that the the equilibrium may be that of indifference.

11. A uniform beam is connected at one of its extremities by a hinge, to a horizontal rod, which is at liberty to move in a horizontal plane about a given point; find the velocity of revolution, so that the beam may rest inclined at a given angle to the rod.

12. A body acted on by gravity slides down the convex side of a vertical elliptic quadrant; find the point at which it will leave the curve, and also the point at which it will strike the horizontal plane.

13. A small ring slides down a straight rod, while the rod describes a conical surface, with a uniform velocity; find the path described by the ring and the time of moving down at a given length.

14. Find the density of the air at any point below the Earth's surface, the temperature being supposed the same, and the force of gravity varying as the distance; and explain the nature of the correction to be applied if the temperature vary.

15. Two stars are observed to rise at the same time in a given latitude, and when one has reached the prime vertical the other is observed to be on the six o'clock hour line; having given the right ascension and declination of the one, find the right ascension and declination of the other.

16. Find the equation to the orbit described, the angle between the apsides, and the points in which the orbit intersects itself, when a body is projected about a given centre of force, varying as

m

D

n

+ with a velocity due to a finite space.

ᎠᏰ

17. Apply the differential equations of motion to determine the time of vibration of a chord of given length (a) stretched by a given weight, the equation to the initial curve being

and shew that it is the same as that determined from the general equation of vibrating chords.

18. Find generally the content of a solid polyhedron, and apply the expression to a dodecahedron.

sin.pdo

19. Integrate

1 + e. sin.20

; and shew the use of integration

in determining the mean annual motion of the Moon's nodes.

x

20. In the expansion of ee determine the coefficient of a".

23. Apply Lagrange's theorem to determine y from the equation

y=1+m. .ey.

24. Find the curve surface in which the distance of any point measured along the normal from the plane of x and ≈, is equal to m times the distance of the same point from the plane of y and z, measured along the same line.

CAIUS COLLEGE, DEC. 1828.

1. FIND the time in which a circular basin, half a mile in diameter, and 10 feet deep, will be half emptied through a flood-gate 15 feet wide.

2. A candle is placed on a circular table; find the nature of its shadow on the floor.

3. Find the inclination of that diameter of a circle to the horizon, the time through which 2 time down the vertical diameter.

4. Explain the principle on which achromatic telescopes are constructed.

5. A perfectly elastic ball is projected at a given angle, from a given point in an horizontal plane, and after rebounding, strikes a given object; find the velocity of projection.

6. Determine the length of that chord of a circle passing through the lowest point, down which a body acquires th of the velocity in falling down the diameter.

7. The specific gravity of air being a, and that of water unity: if W be the weight of a body in air, and W1 in water, prove that its

a

. (W

1

weight in vacuo=W+ ____. (w− w1).

1 -a

8. Explain the method of measuring altitudes by means of the barometer and thermometer.

9. If two weights acting upon a wheel and axle without weight put the machine in motion, find the pressure on the axis.

10. Find the distance of the point of suspension from the centre of a given sphere, that the time of oscillation may be a minimum.

11. Find the image of the arc of a circle placed before a double convex lens, and not concentric with it.

12. Find the time of oscillation of a body in a cycloid, in a medium whose resistance is constant.

13. The radii and distance of the centres of two side wheels of a coach being given, a particle driven from the highest point of the hind wheel falls on the highest point of the fore wheel: what is the velocity of the coach?

14. The specific gravity of gold is 18000, of silver 11000, of air 1; the radius of the Earth is 4000 miles, the height of an homogeneous atmosphere 5 miles, and the whole weight of the atmosphere is equal to the weight of a sphere composed of a mixture of gold and silver whose diameter is 55 miles: required the proportion of gold to silver.

1.

CAIUS COLLEGE, DEC. 1828.

IF I be the length of a straight line, drawn from the vertex C of a triangle bisecting the base (2c); shew that the area of the tri

sin.3° = {}. {(1 − √√3). √(5 + √5) + (1 + √3). √(3−√5)},

3. Draw through a given point in the diameter of a circle a chord which shall form with the lines joining its extremities with either extremity of the diameter, the greatest possible triangle.

4. Eliminate a and b from the equations ax2 + by2 = ax + by. ax2 + by2 = bz. (y-x).

Solve the equation (9+5 √3). x2 + (15+7 √3).x+6=0, and exhibit the roots under the form of binomial surds.

7. Find by the method of continued fractions, a series of fractions converging to √(17).

8. Sum the series

(cos.a+-1.sin.a) + (cos.a+-1. sin.a)2+...... to n terms, and thence deduce the sums of

cos.a+cos.2a+· and sin.a + sin.2a+

...

...... to n terms.

9. A person having a capital of £a, spends annually a sum b, greater than the interest of his capital: in how many years will it be exhausted?

10. Find the value of sin.x in the equation

sin. (x — a) = sin.(x — 2a).