SATURDAY, Jan. 5. 1...4.

1. A NUMBER of persons were engaged to do a piece of work which would have occupied them m hours if they had commenced at the same time, but instead of doing so they commenced at equal intervals and then continued to work till the whole was finished: the payment being proportional to the work done by each, the first comer received r times as much as the last; find the time occupied.

2. If ABCD be a parallelogram, and P, Q two points in a line parallel to AB, and if PA, QB meet in R and PD, QC in S, prove that RS is parallel to AD.

3. Two sides of a triangle whose perimeter is constant are given in position; shew that the third side always touches a certain circle.

n

4. Shew that the product of the terms of an arithmetical progression is greater than (a); and that the sum of the terms of a geometrical progression is less than (a + 1) —; where in both cases a, l and n denote the first and last terms and the number of terms respectively.

2

5. If from any point P of a circle, PC be drawn to the centre C, and a chord PQ be drawn parallel to the diameter ACB and bisected in R, shew that the locus of the intersection of CP and AR is a parabola.

6. From P a point in an ellipse lines are drawn to A, B the extremities of the major axis, and from A, B lines are drawn perpendicular to AP, BP; shew that the locus of their intersection will be another ellipse, and find its

axes.

7. If two ellipses, having the same major axes, can be inscribed in a parallelogram, the foci of the ellipses will lie in the corners of an equiangular parallelogram.

8. If from the extremities of any diameter of an equilateral hyperbola lines be drawn to any point in the curve, they will be equally inclined to the asymptotes.

9. A person wishing to ascertain the distances between three inaccessible objects A, B, C places himself in a line with A and B; he then measures the distances along which he must walk in a direction at right angles to AB, until A, C and B, C respectively are in a line with him, and also observes in those positions their angular bearings; shew how he can find the distances between A, B, C.

10. A heavy body is supported in a given position by means of a string which is fastened to two given points in the body, and then passes over a smooth peg; find the length of the string.

11. Two spheres are supported by strings attached to a given point and rest against one another; find the tensions of the strings.

12.

Shew that it is possible to project a ball on a smooth billiard-table from a given point in an infinite number of directions so as after striking all the sides in order once or oftener to hit another given point; but that this number is limited if it have to return to the point from which it was projected.

13. A cone of given weight W is placed with its base on a smooth inclined plane and supported by a weight W' which hangs by a string fastened to the vertex of the cone and passing over a pulley in the inclined plane at the same height as the vertex. Find the angle of the cone when the ratio of the weights is such that a small increase of W' would cause the cone to turn about the highest point of the base, as well as slide.

14. A conical vessel containing a given quantity of fluid has its axis vertical, and another cone with the same vertical angle is placed to float in the fluid with its vertex

downwards; find how much the fluid will rise in consequence.

15. A hollow cylinder containing air is fitted with an air-tight piston which when the cylinder is placed vertically is at a given height above the base; the cylinder being now inverted and placed vertically in a fluid sinks partly below the surface; find the position of equilibrium.

16. If a luminous point be seen after reflexion at a plane mirror by an eye in a given position, there is a certain space within which the image of the point can never be situated, however the position of the plane mirror be changed; find this space.

17. If a be the angle which every diameter of a circular disc subtends at a luminous point, shew that the ratio of the light which falls on the disc to the whole light emitted is as sin2

18.

α

: 1.

If any number of particles be moving in an ellipse about a force in the centre, and the force suddenly cease

to act, shew that after the lapse of

()

th

part of the period of a complete revolution all the particles will be in a similar, concentric and similarly situated ellipse.

19. Prove that all stars which rise at the same instant at a place within certain limits of latitude will, after a certain interval, lie in a vertical great circle; and determine those limits.

20.

Shew how to find the days of the year on which the light of the sun reflected by a given window which has a south aspect will be thrown into some one of the lower windows of an opposite range of buildings.

21. Two perfectly elastic balls are moving in concentric circular tubes in opposite directions and with velocities proportional to the radii: at an instant when they

are in the same diameter and on opposite sides of the centre the tubes are removed and the balls move in ellipses under the action of a force of attraction in the common centre of the circles varying inversely as the square of the distance. After one has performed in its orbit a complete revolution and the other a revolution and a half, a direct collision takes place between the balls and they interchange orbits; find the relation between the radii of the circles and between the masses of the balls.

1851.

MODERATORS.

Arthur CayleY, M.A., Trinity College.
THOMAS GASKIN, M.A., Jesus College.

EXAMINERS.

LEWIS HENSLEY, M.A., Trinity College.
JOHN SYKES, M.A., Pembroke College.

1.

TUESDAY. Dec. 31. 9...12.

TRIANGLES upon equal bases and between the same parallels are equal to one another.

Let ABC, ABD be two equal triangles upon the same base AB and on opposite sides of it; join CD meeting AB in E; shew that CE is equal to ED.

2. In any right-angled triangle, the square described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.

If ABC be a triangle whose angle A is a right angle, and BE, CF be drawn bisecting the opposite sides respectively; shew that four times the sum of the squares of BE and CF is equal to five times the square of BC.

3. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

If a polygon of an even number of sides be inscribed in a circle, the sum of the alternate angles together with two right angles is equal to as many right angles as the figure has sides.

4.

Inscribe an equilateral and equiangular quindecagon in a given circle.

In a given circle inscribe a triangle whose angles are as the numbers 2, 5 and 8.

5.

If the angle of a triangle be divided into two