LLOYD EXHIBITION EXAMINATION.

MR. BURNSIDE.

1. If three quadratics, U, V, W, are connected in pairs by the relations

Prove that their squares are connected by an identical relation of the form,

a U2 + ẞV2 + yW2 = o.

2. Prove that the most general binary cubic, which has the same Hessian H as a given cubic U, is a U + BG, where G is the Jacobian of

U and H.

3. Denoting by a, B, y, & the roots of a binary quartic, form the equations which have the several values of (a - ẞ) (y − d) and (a + B) (y + d)

for roots.

Prove that the roots of the equation (λ) =o are real.

5. If three normals to an ellipse at points whose eccentric angles are a, B, y, d, meet in a point, prove the relation

(a). Prove that this relation in the case of the parabola y2 - px=0

becomes

6. The tangent to the evolute of a parabola at the point where it cuts the parabola is also a normal to the evolute.

7. If a quadrilateral circumscribe a conic, the circles on the diagonals as diameters, and the director circle have the same radical axis.

8. Find the invariant condition that two points should exist; that any line drawn through either point, should be cut harmonically by two given conics.

9. Perform the integration

do

(a2 cos2 + b2 sin2)*

11. Determine the general expression for the radius of curvature of a

curve u = 0.

12. If u be a solution of the differential equation

1. If four forces are in equilibrium, the volume of the tetrahedron constructed on the representative lines of any two, is equal to the volume of the tetrahedron similarly constructed on the remaining two?

2. A system of forces acts on a rigid body: prove that the six condi- · tions of equilibrium may be obtained by equating to zero the moments of all the forces about the six edges of any finite tetrahedron.

3. From the principle of virtual volocities, deduce the conditions of equilibrium of a system of forces.

4. If a system of forces be replaced by two forces, and the line of action of one be given, determine the line of action of the other.

5. Determine the attractiou of a spherical surface on an external point, the law of force being the inverse square of the distance, and the density varying inversely as the cube of the distance from an internal point.

6. In the case of the common catenary if the string be slightly extensible, prove that the whole extension will be proportional to the product of its length, and the heighth of the centre of gravity above the directrix.

7. Prove that a parabola may be described freely by a particle under the combined action of a constant force parallel to the axis, and a force varying inversely as the square of the distance from the forces.

8. Determine the equation of a curve so that a particle starting from rest may describe any arc under the action of a force, varying as the distance from a fixed point, in the same time as the chord would be described, the same force acting.

9. A material particle is attached to one end of a cord without weight, resting on a smooth horizontal plane; if the other end of the cord describes a circle with uniform motion, determine the motion of the particle while the string is kept stretched.

Io. In a uniform fluid under the action of two forces tending to fixed points, and varying inversely as the square of the distance, one attrac

tive and the other repulsive, prove that one surface of equal pressure is a sphere.

ASTRONOMY AND OPTICS.

PROFESSOR R. S. BALL.

1. Show that, geographical considerations apart, the second of a pair of Transits of Venus at an interval of eight years is generally more favourable for Halley's method than the first of the pair.

2. Determine the effect of Diurnal aberration upon the apparent right ascension of a star.

3. If p be the parallax in altitude of a heavenly body whose zenith distance is z, and horizontal parallax P, then

p sin P sin z + sin 2P sin 2z + sin 3P sin 3z, &c.

4. Find the effect of annual parallax upon the north polar distance of a fixed star.

5. Show how the coefficient of refraction may be determined by the solstitial meridian zenith distances of the sun.

6. The declination axis of an equatorial is inclined to the polar axis at an angle 90° e where is small. What correction must be applied to the observed hour angle of a celestial body whose declination is &?

7. If 0 in the last question were unknown, by what observations could it be determined?

8. Determine the circle of least aberration when parallel rays are reflected by a spherical surface.

9. Being given the dispersive powers of flint and crownn-glass, and the focal length of an achromatic object glass; find the expressions for the focal lengths of the two component lenses.

10. Why is the eye-piece of an astronomical telescope generally made of two lenses instead of a single lens of the same power?

EXAMINATION FOR SENIOR EXHIBITION.

LATIN

PROSE.

MR. TYRRELL.

For that reason, in the happier times of India, a number, almost incredible, of reservoirs have been made in chosen places throughout the whole country; they are formed for the greater part of mounds of earth and stones, with walls of solid masonry; the whole constructed with admirable skill and labour, and maintained at a mighty charge.

In the territory contained in that map alone, I have been at the trouble of reckoning the reservoirs, and they amount to upwards of eleven hundred, from the extent of two or three acres to five miles in circuit. From these reservoirs currents are occasionally drawn over the fields, and these water-courses again call for a considerable expense to keep them properly scoured and duly levelled. These are not the enterprises of your power, nor in a style of magnificence suited to the taste of your minister. These are the monuments of real kings, who were the fathers

of their people.

EXAMINATION FOR JUNIOR EXHIBITIONS.

MR.

ARITHMETIC.

GALBRAITH.

1. Find the G. C. M. of the numbers 6061 and 8239.

2. Reduce the following series of fractions to their lowest common denominator:

18 21 7 11.

3. Find to four places, by Contracted Multiplication, the product of the numbers 23456, 11342, 0*7342, and 001465.

4. Round the world is forty million meters; 1 meter = #39 371 inches. Find the diameter in miles, the ratio of diameter to circumference being

113: 355.

5. If 4 cubic inches of water weigh 1010 grains, find the number of inches in the side of a cubical vessel which when full holds just one ton of water.

6. Find by Practice, true to an exact fraction of a penny in its lowest term, the price of 23 tons, 18 cwt. 3 qrs. 22 lbs. at £11 38. 10d. per ton.

7. Find by Practice the rent of 140 acres, 3 roods, 23 perches, at £4 138. 10d. per acre.

8. Find the interest to the nearest penny on £423 68. 8d., at 4 per cent., for 237 days.

9. Standard gold is coined in the British Mint at the rate of 44 guineas to the pound Troy, calculate to an exact fraction of a penny the value of 14 lbs. Avoirdupois of pure gold, standard gold being an alloy of which is pure gold.

10. If £1000 cash be invested in 3 per cent. Stock when the price is 921, find the annual interest on the investment. When the price rises to 93, let the Stock be sold, and the proceeds invested in 5 per cent. Stock at 105 find the annual interest.