2. Find the discount upon £125. 10s. od. payable at the end of three years, at 4 per cent, simple interest.

3. It is required to determine the point c, in the semicircleACB, such that the three sides of the triangle ACB shall be in geometrical progression.

4. Two bodies 1, 2, moving with velocities 1, 2, whose elasticity perfect elasticity: 1 2, impinge upon each other, making the angles of 30°, and 90°, respectively, with the plane touching them at the point of contact. Required the directions in which they will move, and their velocities after impact.

5. A body is projected down an inclined plane, with the ve locity acquired in falling down its height, and describes the length of the plane in the time of falling down its height. Required the elevation of the plane.

6. In a quadrantal triangle, the angle opposite the quadrant, and one of the other angles, are given; find the remaining angle.

7. Prove that the illumined phase of Mars is the least, when he is in quadrature.

8. If an object be viewed through a glass plate of given thickness, determine how much the apparent distance is less than

the true..

9. It is required to determine the brightest part of the visible area in Galileo's telescope.

10. A circle and its inscribed hexagon, move with equal ve. locities, in directions inclined at angles of 30° and 60°, respectively, to their planes. Compare the resistances perpendicular to their motions.

11. Sum the following series to n terms,

and 1.2 + 2.5 + 3.8 + 4.11 + &c.

12. Find the fluents of

13. Having given the ratio of the periodic times in two circles, described about different centres of force situated in their centres, and also the ratio of the radii, it is required to find the ratio of the absolute forces.

14. Determine the angle between the apsides in an orbit nearly circular, the force being constant; taking an ellipse about the centre for the revolving orbit.

TUESDAY AFTERNOON.-Mr. PEACOCK.

Third and Fourth Classes.

and P1 be two consecutive terms in a series of frac

2. Explain what is meant by the conjugate points of curve

lines.

5. If a body be projected perpendicularly upwards with a velocity (a), its height (x), at the end of the time (t), is determined from the equation

4m

6. Enumerate the different practical methods of determining the latitude of a ship at sea.

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

8. A given rectilinear object is placed before a spherical reflector of given radius. Find the equation to the conic section which is its image.

9. Find an expression for the whole time of descent of a body from a distance (a) to the centre of force, when the force varies inversely as the square of the distance.

10. Mention some of the problems, upon which the trisection of an angle, by common Geometry, may be made to depend.

TUESDAY EVENING.-MR. PEACOCK.

1. Demonstrate the rule for the extraction of the square root in numbers.

2. Every prime number of the form 4 n + 1 is the sum of two squares.

3. Approximate to the value of x in the equation

x3 2 x 5 = 0,

and explain the defects of the methods of approximation, as given by Newton and Ralphson.

(2) d2 y + Ay d x2 = x dx2, where x is a function of x.

9. Given the length of the curve; required its nature when its centre of gravity is most remote from the axis.

10. If two lines intersect each other within a parabola, the ratio of the rectangles contained by their respective segments will be the same with the ratio of the rectangles made by the segments of any other two lines which intersect each other, and which are respectively parallel to the former.

11. Apply D'Alembert's principle to the determination of the distance of the centres of oscillation and suspension in a com, pound pendulum.

12. A triangular prism being immersed in a fluid of greater specific gravity than itself, it is required to determine the differ. ent positions in which it will rest in equilibrium.

13. A machine, driven by the impulse of a stream, produces the greatest effect when the wheel moves with one-third of the velocity of the water.

14. At a place whose latitude is 48°. 50'. 14", the meridian altitude of the sun's upper limb was observed to be 62°. 29′. 56"; it is required to determine the sun's declination, the refraction being 29", the sun's parallax and apparent diameter of their mean values, and the sine of 27°. 30'. 4′′ = .4617.

15. Explain the method of correcting an error in the longitude of a place, by means of the occultation of a given fixed star by the moon.

16. If r be the radius of an isosceles lens, whose focal length is equal to that of a lens whose radii are r ̧ and r; then

1

2

+

T

17. If D be the length of a degree of the meridian at a point whose latitude is λ, the length of a degree of a curve perpendicular to the meridian at that point, a the axis major of the meridian, and e the difference of the semi-axes; then

Δ -D (nearly).

a 2 A cos 22

18. The moon is retained in her orbit by the force of gravity, Newton. Lib. III. Prop. 4.

19. The sum of the sides of a right-angled triangle remaining the same, required the nature of the curve to which the hypothenuse is always a tangent.

20. Explain the method of drawing a normal to a given curve surface.

21. Give an account of the controversy between the followers of Newton and Leibnitz, concerning the measure of motion, and reconcile the experiments and results to which the latter appealed, with the measure assumed by the former.

22. If two chords of a circle intersect each other at right angles, the sum of the squares described upon the four segments is equal to the square described upon the diameter.

23. Give some account of the Analysis of the Ancient Geometers. Exemplify it in the solution of the following problem: "To bisect a triangle by a straight line drawn through a given point in one of its sides.'

THE SENATE-HOUSE PROBLEMS,

Given to the Candidates for Honors during the Examination for the degrees of B. A. in January, 1818.

BY THE TWO MODERATORS,

MONDAY, JANUARY 20, 1818.

MONDAY MORNING.-MR. FRENCH,

First and Second Classes.

1. The present value of an annuity, to continue for a term of years at a given rate of compound interest, = m × the present value of the same annuity, to be paid only during the latter half of the same term; required to find when the annuity will cease ?

2. To determine the numerical value of the arc A which will satisfy the following equation:

sin. B sin. (A — B) + sin. (2A + B) =

sin. (A+B) + sin. (2A-B).

3. Prove that the sum of all the coefficients of a binomial raised to the (2n)th power: the coefficient of its middle term :: 2.4.6. &c. to n factors: 1.3.5. &c. ton factors.

4. A body is suspended from a given point in the horizontal plane, by a string of known length, which is thrust out of its vertical position by a rod (supposed without weight) acting from a given point in the plane, against the body; shew that the tension of the string varies inversely as the tangent of the inclina tion of the rod to the horizon.

5. Two equal hollow paraboloids have a common axis, which is vertical, and such a quantity of water is poured in between them, as just to touch the lowest point of the inner figure; demonstrate that the surface of the water will be a tangent plane to this figure, in any position of the common axis.

6. In Gregory's telescope, the focal length of the larger re flector, the position and focal length of the eye-glass, and the distance between the two images of a remote object being given; required to find the position and focal length of the smaller re