15. If s be the space, t the time, and f the force, then

16. The force which accelerates the descent of a body in a cycloid, is proportional to the distance from the lowest points; prove this, and shew that the time of oscillation is the same both in great and small arcs.

17. Find the law of force, by which a body may describe a circle, the force tending to an eccentric point; determine likewise, from the requisite data, the periodic time.

18. If a body be projected from any point with a given velocity and attracted to a centre of force, its velocity at any other given distance from the centre is independent of the direction of projection.

19. In ellipses of small eccentricity, the greatest equation of the centre is nearly equal to twice the eccentricity.

20. An inflexible and imponderable rod, to which two equal weights are attached, is fixed to a given point, round which it vibrates; find the time of its making a small oscillation.

21. Find the focal length of a double concave lens.

22. In the modern construction of the Newtonian telescope, the small mirror is replaced by a prism; explain the principle and advantages of this substitution.

23. The water in a clepsydra, whose form is a solid of revolution round a vertical axis, descends through cqual spaces in equal times; find the equation of the generating curve.

24. Is the Sun's place affected by aberration?

25. An iron globe descends in water; find expressions for the velocity acquired, and the space described in a given time from the beginning of the motion.

26. If the equations of two planes be

Ax+ By + Cr = 1,

ax + by + cr = 1;

then the cosine of the angle between them is

Aa + Bb+Cc.

√ (A2 + B2 + Ç2 ) ; √ { q2 + b2 + c2,

TRINITY SCHOLARSHIPS, APRIL 1828.

1. Equiangular are to each other in the ratio compounded of the ratios of the sides. [Euclid 23, Book 6.]

2. Construct a regular tetrahedron, and compute the angle formed by two adjacent surfaces.

3. What is the discount of £70. 5s. 6d. due 7 months hence interest being allowed at 4 per cent?

5. Find integral and positive values of x in the equation

7x+17y= = 172.

6. Every equation has as many changes of sign from + to,

or from

to +, as it has positive and possible roots, and as many continuations of sign, from +to+, or from

negative and possible roots.

to

as it has

7. Explain the mode of transforming the equation of a curve from one system of co-ordinates to another, and apply the formulæ to

transform the equation y2 = 2ax + x to a system of rectangular co-ordinates, making angles of 45° with the former, and the coordinates of whose origin are 0 and a respectively.

O

8. Investigate the differential expression for finding the length of a curve, and apply it to find the length of the cycloid.

10. Find the equation of the curved surface traced out by one of the angles of a square, in constant contact with a spherical surface, without sliding upon its surface.

11. Find the centre of gravity of two or more bodies considered as material points.

12. A perfectly elastic ball let fall from the top of a tower meets another at half the height, of twice its weight, and which was projected upwards from the bottom at the same time; what are their motions after collision, and when will they reach the ground?

13. What is meant by the centre of percussion, and how is it found?

14. A double convex lens is placed at a given distance from a screen: to determine the position of the object, so that the image may fall upon the screen; is this always possible?

State the order of the colors in the primary and secondary rainbows, and the reason of their being the inverse of each other. 16. Determine the greatest height to which water can be raised by the common pump.

17. Investigate the precession of a star in right ascension.

18. The equation of time is Zero four times in the year, and at very unequal intervals of time.

19. Find the law of force to the focus of the logarithmic spiral, and find an expression for the whole time of descent, from a given distance, to the centre of force,

20. A body is projected from a given point in a given direction with a given velocity, and is repelled from a given point by a force varying as the distance: determine the orbit described.

21. An iron ball descends in a fluid: what is the limit of its velocity?

22. The Moon is retained in her orbit by the force of gravity.

23. Give the moment of inertion round any axis passing through the centre of gravity of a body, to find the moment round any other axis parallel to the former.

24. If d and d be the lengths of a degree of the meridian at the equator, and at latitude a respectively, and a and b the equatorial and polar diameters, then

25. To find the nature of the curve when the normal bears a constant ratio to the part of the axis intercepted between the origin and the normal.

TRINITY COLLEGE, 1829.

1. EQUIANGULAR

7s have to one another the ratio which is

compounded of the ratios of their sides.

2. If two straight lines be to the same plane, they are || to one another.

3. Required the value of the Metre of France in terms of the foot of Cremona; if 48 feet of Cremona 56 English feet and the metre = 39.371 English inches.

S

cos.√1. sin.0.

6. Small spherical may be considered rectilinear, without sensible error.

7. The roots of the reducing cubic equation in Des Cartes' solution of a biquadratic are quadruple of the roots of the reducing cubic equation in Euclid's Solution.

8. Determine the position of the centre and axes of the ellipse represented by the equation

Ay2+Cx2+ Dy + Ex + F = 0,

all the coefficients being supposed positive.

9. Investigate the differential expression for the radius of the O of curvature.

What are the analytical and what the geometrical properties of the of curvature, and how are they dependent upon each other.

10. Integrate

dx

dx

x √(1 + x2)' (1 + x)3 (from x = 0 to 1) and

i.

in a series which converges rapidly when e is nearly

11. Find the equation of the epicycloid when the radius of the generating is equal to the radius of the base; and determine its

area.

12. In the

game of whist what is the chance of the dealer having the four aces?

13. In the lever the velocity of the weight is to the velocity of the power as the power is to the weight.

15. Find the length of a pendulum which would oscillate once

in the period of a satellite at the Earth's equator.