11. Find whole positive values of x and y in the equation

and find the sum of the first twenty values of

y.

12. One thousand pounds is invested in the three per cent consols at 63 per cent, and sold out afterwards at 64: required the sum gained by the speculation.

13. If squares be described on the sides of any triangle and the angular points of the squares be joined; the sum of the squares of the sides of the hexagonal figure thus formed is equal to four times the sum of the squares of the sides of the triangle.

14. If all the angular points of a regular polygon of n sides be joined, and r be the radius of the circumscribing circle; the sum of all lines, including the periphery of the polygon, is equal to

15. If from a given point straight lines be drawn cutting an ellipse, the locus of the middle points of the chords is an ellipse similar to the given ellipse.

FOR SCHOLARSHIPS.

1.

2.

REQUIRED the charge on £573. 18s. 6d. at 12 per cent.

After deducting a charge of 83 per cent. on a certain sum, and then a charge of 6 per cent. on the remainder, the result is £.256. 15s. Required the original sum.

3. Reduce 13s. 74d. to the decimal of a pound: also of a shilling to the decimal of a crown.

4, Two clocks point to XII at the same instant; one of them gains 7", and the other loses 6" in 12 hours: after what interval will one have gained half an hour of the other? and what o'clock, exactly, will each then shew?

5. If two angles of a triangle be equal, shew that the sides subtending them are also equal.

6. Prove that, in a right-angled triangle, the square on the side subtending the right angle, is equal to the square on the sides containing the right angle.

7. If a straight line be divided into two equal, and also into two unequal parts, prove that the squares of the two unequal parts are, together, double of the square of half the line, and of the square of the line between the points of section.

8. If a straight line touch a circle, and, from the point of contact, a straight line be drawn cutting the circle, shew that the angles made by those lines will be equal to the angles in the alternate segments of the circle.

9. Demonstrate the proposition usually cited Ex æquo. [Euclid, Book 5, Prop. 22.]

10. Shew that the equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.

11. Prove that two straight lines, which are each of them parallel to the same straight lines, and not in the same plane with it, are parallel to each other.

12. From a given point without a circle, draw a straight line which will cut off a quadrantal arc of the circle.

13. In a right-angled triangle, a and b are the sides containing the right angle, and c the side subtending it: prove that the radius of its inscribed circle - = {(a + b − c).

14. Multiply

-m

a"~m + an—2mfm + an¬3mb2m... + ambn−2m + bn-m by am — bm.

Also shew that

√(20) + √(12)
√5-√3

=8+2/15.

15. Solve the following equations :

(1). a +x √(a + x) = ax√(a− x);

8

17

(2). / +2 ==

(5). — x2 + x = ; (shew that x = 6689).

16. The difference between two numbers is 24, and their arithmetic mean exceeds their geometric mean by 6: required the numbers.

and shew that if the nth term of the series be multiplied by

18. If N=ǹ × 10", shew in what respect the logarithms of N and n will differ; and having given the logarithm of 2.651 (= 0.4234079), state the logarithms of 2651, and of '002651.

19. Find the sum of the successive mean proportionals between a, ar; ar, ar2; ar2, ar3; ar3, art; ar1, ar5;

Also sum the following progressions:

20. Find an equation, each of whose roots is less by three, than the corresponding root of the equation x3- 15x2 + 71x-105=0; and from thence determine the roots of the given equation.

21. In an oblique-angled triangle, having given two sides a and b, and B the angle subtending b, state the logarithmic computation for finding A, the angle opposite to a. Explain also why, in some cases, A will have two values, in others only one.

22. Treatises of Trigonometry usually give two methods of finding the angles of a plane triangle, from the sides. Describe the two processes; and prove the leading proposition on which each method depends.

23. Prove that sin.a = √(±r2 — ¦r cos.a),

and that cos.a = √(3r2 + 1r cos.a);

also from the first expression, find the sin.22o, 30′; and from the second, the cos.15°.

24. Shew that, in the equilibrium of two weights, acting perpendicularly to the arms of a straight lever, the weights are reciprocally proportional to their distances from the fulcrum.

25. Prove that, if a body be moved from rest by the action of an uniform force, the space described will vary as the square of the

time.

26. P and Q are connected by a string passing over the top of an inclined plane: P, descending down the altitude, draws Q up the plane: given the length and inclination of the plane, find the time of Q's ascent to the top of the plane, and the velocity acquired.

27. A clock loses a second every hour; required the alteration to be made in the pendulum; the length of the true seconds' pendulum being 39.2 inches.

28. Given the velocity and direction of projection of a body, to find the horizontal range, the time of flight, and the greatest altitude.

29. Shew that, if circles be uniformly described by bodies acted

upon by forces in the centers, those forces vary as

30. Prove that, if a body describe a circle uniformly, the arc described in any time is a mean proportional between the diameter of the circle, and the space which the body would fall through in the same time, if acted upon by the force which retains the body in the circle.

TRINITY COLLEGE, 1820.

1. FIND the value of £.3.869, and 365-24215 days.

square root of 0·676, and 6·76.

2. If twenty men, in digging a canal, must pump out six tons of water daily, in order to excavate 160 cubic yards in a week, how many cubic yards can thirty men excavate in a week, supposing them to be obliged to pump out eight tons of water daily?

3. If S denote the sum of the even terms, and S" the sum of the odd terms, of the expansion of (a + b)"; then Se~S"2=(a2 ~b°)".

5. Find the roots of an equation of this form by construction,

6. Compute the numerical value of the side of a regular decagon inscribed in a circle, whose radius is ten inches.