3. What is the nature of a homogeneous quadrilateral if the centre of gravity of its area coincide with the intersection of diagonals?

4. If any number of forces in one plane act on a rigid body, establish accurately the rule by which their resultant moment is found.

5. A point moves in a circle with uniform velocity; find the amount and direction of the acceleration at any instant.

6. Two perfectly elastic spheres meet directly with equal velocities; find the ratio of their masses that after collision one of them may remain

at rest.

MR. BURNSIDE.

7. A mass of 2 lbs. is drawn along a smooth horizontal table by a mass of 1 lb. hanging vertically; required the space described in four seconds.

8. A mass of 20 lbs. is placed upon a horizontal plane which is made to descend with a uniform acceleration of velocity, 30 ft. per second; find the pressure on the plane.

9. Find the line of quickest descent from a point without a given circle to the circle.

10. A sphere rests against two inclined planes, having given inclinations; find the pressures of the sphere against the planes when in a state of equilibrium.

II. If the centre of a system of three parallel forces, acting through the angles of a given triangle, be the intersection of the perpendiculars of the triangle; find the relation between the forces.

12. Find the lowest point a particle can rest on a rough, horizontal circular cylinder, friction being equal to the pressure.

MR. PANTON.

13. Squares are constructed on the three sides of a right-angled triangle; find the ratio between the sides containing the right angle when the centre of gravity of the whole figure lies on the hypotheneuse.

14. At the point of intersection of diagonals of a square, two forces of 8 and 12 lbs. respectively act along the diagonals, and two forces of 10 and 2 lbs. respectively act perpendicularly to two sides; required the direction and magnitude of the resultant.

15. A body is projected vertically upwards with a velocity of ng feet per second; find its velocity and distance from point of projection after an interval of 5n seconds.

16. Determine the time a body will take to run down a rough inclined plane of given length, whose coefficient of friction is given; find also its velocity at the bottom of the plane.

17. A body attached by a string to a fixed point revolves in a vertical circle under the action of gravity; if the tension of the string at the highest point is equal to the weight of the body, find the tension at the lowest point.

18. Prove that if n forces acting at a point be represented by right lines, their resultant will be represented by ʼn times the line joining the point to the centre of gravity of equal particles placed at the extremities of the lines.

B.

MR. CATHCART.

1. Determine the centre of gravity of an arc of a circle. What is the locus of centres of gravity of all arcs which commence at the same point of the circumference?

2. What is the spherical triangle of maximum area under a given perimeter ?

3. What are the limiting positions of equilibrium of a rigid beam on two rough inclined planes?

4. What is the least velocity with which a body must be projected vertically from the Earth's surface that it may never descend? What is the relation between the space described and the time of describing it in general?

5. The axis major of an ellipse is vertical: find the radius vector by which a particle will descend in the shortest time from the upper focus to the curve.

- 8. Find the radius of curvature at any point of the curve r = ao.

9. A particle is projected from a given point in a given direction, with a given velocity, and moves under the action of a central force varying inversely as the square of the distance; determine the orbit, and find when an ellipse, parabola, or hyperbola.

10. Two smooth spheres, whose centres are moving in the same plane, impinge obliquely; required their motions after impact.

12. State and prove Euler's theorem for the differentiation of homogeneous functions of two variables; and verify it for the function

13. The co-ordinates of a point on a curve are given in terms of an angle by the equations

prove that there are two finite points of inflexion, and find the values of eat those points.

14. A particle, acted on by gravity, is projected from the vertex along a smooth parabola whose axis is vertical, and vertex upwards; determine the motion, and the pressure on the curve.

15. A heavy rigid rod AB rests upon a fixed point D, while its lower extremity A presses against a vertical plane; find the position of equilibrium, and the pressures at A and D.

C.

MR. CATHCART.

I. Find the volume and surface of the solid generated by the complete revolution of a cycloid about its base.

น

2. Investigate the maxima or minima of λ = =

subject to the condi

tion w = o, where u, v, and w are homogeneous functions of the nth degree in three variables.

(a). How is the result related to the corresponding values of

au + B'v + y' w

au + Bv + yw

subject to the condition a"u + B”v + y′′w = 0 ?

=

=

3. If o be the condition that a line Ro may touch a conic, show that in general the sign of Σ determines whether the line meets the conic in real or imaginary points.

4. Show how to find the path of a particle projected along a rough inclined plane.

5. In the motion of a particle under the action of a force varying x'dy - y' dx xx' + yy'

inversely as the square of the distance,

is a complete differential-x, y being the co-ordinates of the point; and x', y' the components of the velocity expressed by integration in terms of x and y.

MR. BURNSIDE.

6. The co-ordinates of a point are expressed as follows:

find the equation of the curve described by the point, and the area of the portion of the plane inclosed thereby.

7. Determine the area and length of the arc of the curve given by the

a

equation r = between the limits 02 and 01.

8. A particle revolves in an orbit which is nearly circular, acted on by a central force varying as a function of the distance; to determine the apsidal angle?

9. Determine the value of the integral

J cos2 e sint ede.

10. A particle moves in a resisting medium under the action of a central force P; prove the formula for the resistance R:

11. The equation of a curve is given in the form

F(1, 2, rn) = const.,

....

where 1, 2, .... n are the distances of any point P on it from n fixed points 01, 02, On. Prove that the normal to the curve is constructed

by measuring on PO1, PO2, POn distances proportional to

....

and then finding the resultant of these considered as forces acting at the point P.

12. Show that the polar equation

p2 - 2p (a - b cos 0) + c2 = 0 (a > b)

represents two ovals, one lying entirely inside the other; and find the sum of their areas.

13. Find the equation of the catenary of equal strength; and show that it may be expressed in the form

where P is the radius of curvature at any point, and s the arc between that point and the lowest point of the curve.

14. Two rings whose weights are P and Q are moveable on a rough rod inclined to the horizon at an angle i, the rings being connected by a string of given length which passes through and supports a smooth ring whose weight is W; find the greatest distance between P and Q consistent with equilibrium.

15. Find the law of force by which a particle may describe the lemniscate, the centre of force being the node.

Classics.

TACITUS.

MR. POOLE.

:

Translate the following passages into English Prose :

1. Beginning, In civitate discordi, et ob crebras principum.... Ending, quem pessimis moribus meruerat.

Hist., ii. 10.

2. Beginning, Sustinuit labantem aciem Antonius accitis...... Ending, velut ex occulto jaculantibus, incauti offerebantur.

3. Beginning, Miscebantur minis promissa. Et, concussa Ending, et noxii capitis poena poenitentiam fateantur."

4. Beginning, Proprium id Tiberio fuit, scelera nuper...... Ending, et gratia apud Tiberium viguerit.

Ibid., iii. 23.

Ibid., v. 25.

Annal., iv. 19, 20.

1. Contrast the general character of the Histories of Thucydides and Tacitus, noticing also some points of resemblance.

2. State the circumstances which led to the degradation of the aristocracy and of the people of Rome, and prepared them for the Imperial Rule.

3. How does Merivale criticise the Roman policy beyond the Rhine which led to the defeat of Varus ?