15. If two rays inclined to each other at a small given angle fall nearly perpendicularly on a convex mirror of given radius, and placed at a given distance from the point of intersection of the incident rays; what will be their mutual inclination after reflexion?

16. If a ray refracted into a sphere emerge from it after a given number of reflections, determine the deviation of the ray, and the angle contained between the directions in which it is incident and emergent.

17. What is the nature of the caustic when the reflecting surface is a sphere, and the focus of incident rays is in its surface?

18. Find the law of force parallel to the axis, by which a body would be made to describe a cycloid.

19. The corner of a rectangular piece of paper being doubled down, so that the triangle shall always be of a given area; prove that the vertex of the triangle will trace out a lemniscata, whose area equals the area of the triangle, and which may be described by

a force placed at its knot varying as

1

D7

shew that the velocity at any point of the

1

20. Force ∞

D2;

being the circular arc whose diameter is the described.

first distance and versed sine the space

21. Compare the quantity of light from the Sun which falls on a given area at the pole of the Earth on the longest day, with the quantity which falls on an equal area at the equator on the day in which the Sun's declination is nothing.

22. Investigate an expression for the tangential ablatitious force on P[Newton, Sect. 11]. Find when it is a maximum; and find the velocity of P corresponding to this maximum value, V being the velocity of P at quadrature.

23. Find generally the equation to the orbit in fixed space [Newton, Sect. 9], and from that equation shew that the difference of force in the fixed and moveable orbit varies as

1

D3

24. Explain the construction of Hadley's Sextant.

25. Give the latitude of a place, and the declination of the Sun; find the time that the Sun is above the horizon.

26. Shew how the time, duration and magnitude of a lunar eclipse may be computed.

(x-1)=

2. Shew that m. (am — 1) = Naperien logarithm a when m is infinite, and prove from this that Naperien logarithm x

3. Investigate Cardan's rule for the solution of the cubic equa

for determining x may be reduced to the form 2

4. If A, B, C be the angles of a triangle, a, b, c the opposite sides, and the angle contained between 6 and a straight line drawn from A to the bisection of the opposite side, prove that

5. In a given rectangle inscribe a parallelogram of given area

and containing an angle equal to a given angle; and point out the limitations of the problem.

6. ACB is a circle, AB a diameter, draw any chord AC and join CB, and in AC produced take CP proportional to the triangle CAB: find the locus of P.

7. Two forces p and q acting at right angles on a material point have a resultant r; find two other forces of which the sum is the same but which acting at an angle 6 have a resultant nr, and shew p + q √(p2 + q2)

that this is always possible when ǹ >

8. A hemisphere is placed on a horizontal plane with its vertex downwards: find the height of a cone of the same base which placed upon the base of the hemisphere will make the equilibrium of the whole mass that of indifference.

9. An imperfectly elastic ball is projected from a point in the circumference of a given circle, and after three rebounds at the circumference returns again to the point of projection; find the direction of projection.

10. Find the moment of inertia of a rectangular parallelopiped about one of its edges, and also the time of a small oscillation.

11. A hollow cylinder of given radius but not of inconsiderable thickness rolls down a given inclined plane in a given time; find its thickness.

12. A right cone standing on its base is filled with mercury and water; divide it by a plane parallel to the base so that the pressure on the upper and lower curve surfaces shall be equal, the quantities of mercury and water being supposed equal in weight.

13. A cubical vessel of given size and weight is placed on a fluid of given specific gravity having a small orifice at the bottom; what time will elapse before the vessel is completely immersed ?

14. In a vessel of fluid is placed a paraboloid with its vertex downwards; find its length so that the vertex of the paraboloid may just reach the bottom of the vessel.

15. Find the time of emptying the upper half of a paraboloid whose axis is parallel to the horizon through a small orifice in its

vertex.

16. A glass meniscus of given radii and inconsiderable thickness floats on the surface of a bason of water of given depth; required the focus of refracted rays diverging from the bottom of the bason and passing nearly perpendicularly through the lens.

17. Find the nature and length of the caustic, when the reflecting curve is a cycloid and the radiating point in the centre of the generating circle.

18. Compare the quantity of light from the sun which falls on a given horizontal area on the longest day at two places, the latitudes of which are given.

19. Having given the latitude of the place, the day of the year, and the angle at which the ecliptic is inclined to the horizon; find the hour of the day.

20. Find the aberratic curve on the supposition that the Earth moves in an ellipse and is acted upon by a force perpendicular to the major axis.

21. Find at what point in an ellipse described about the focus the approach towards the centre of the ellipse is twice as great as that towards the focus.

22. A given body is projected with a given velocity in a horizontal direction along the inner surface of a horizontal cylinder of given radius from a given point not more than 90° from the lowest point; required its position after a given time.

23. Compare the resistance of the atmosphere against a spherical shot with that against a shot of the same weight formed into a hollow cylinder of given length and radius moving in the direction of its axis with the same velocity.

24. A chain of given length and thickness suspended by its extremities from two given points in the same horizontal line forms itself into a semicircle; find the law of the density.

QUEEN'S COLLEGE, MAY 1828.

1. PROVE analytically and geometrically that the perpendiculars drawn from the angles of a plane triangle upon the opposite sides intersect each other in one point.

2. Find a point in a spherical triangle equally distant from each of the angular points.

3. The area of a regular polygon of n sides inscribed in a circle is to the area of another regular polygon of 3 n sides inscribed in the same circle as 1 to k; prove that cos.0=

√(3 + k),

e being the

, 24k

angle which a side of the last polygon subtends at the centre of the

circle.

4. Find all the roots of the equation x7

10 in a numerical

form, and reduce each to the form of a + B−1.

1

5. Prove that m. (xm-1)= Nap. log.x when m is infinite, and thence shew that

7. Solve the equation ux+2. Ux = ex, and find the general term of a series of which Sr = m. ux + nux-1, Sx being the sum of x terms, and ur being the rth term.

2

8. If two ellipses whose eccentricities are respectively e and be described on the same major axis, and and be the angles P