6. Give, with instances real or invented, all the uses of ut, quin, ὡς, ὅπως.

7. Trace the effect of Greek idioms on the syntax of Latin writers.

8. Explain ανάκρισις—ἀντιγραφή - προβολή-πρόκλησις -unciarium faenus-ex iure in iudicium-fiscus

et aerarium.

SCHOOL OF MATHEMATICS AND NATURAL PHILOSOPHY.

MATHEMATICS.-PAPER I.

Professor Nanson.

1. Find the equation to the evolute of a parabola.

From any point three normals are drawn to a parabola. Shew that the centres of curvature of the parabola at the feet of the normals lie on a hyperbola one of whose axes is parallel to the axis of the parabola and which touches the axis in a fixed point.

2. Find the condition that a straight line whose equation is given in homogeneous coordinates should touch a conic whose equation is given in the same coordinates.

A conic is described through the vertices of a given triangle and cutting a given conic in P,

Q, R, S. Prove that if PQ passes through a fixed point RS will envelop a conic inscribed in the given triangle.

What does this theorem become when the given fixed point lies at infinity and two of the vertices of the triangle are the circular points at infinity?

3. Prove that a force defined by the rectangular components X, Y, Z at x, y, z is equivalent to an equal force at the origin together with couples yZzY, zX-xZ, xY-yX about those

axes.

If the axes are oblique then the couples about the axes are L, M, N where

L+M cos y + N cos ẞ= V (yZ −zY)

ß:

&c. &c.,

=

where a, ß, y are the angles y Oz, zOx, x Oy, and V is the volume of a parallelopiped with unit edges parallel to the axes.

If forces Ea, nb, e act in BC, CA, AB where a, b, c are the respective lengths of these lines, and if OA, OB, OC be the axes of reference and a, b, c the respective lengths of these intercepts, then shew that the system is equivalent to the component forces

(n − 5)a', (5 — §)b′, (§ — n)c'

in the axes and to component couples given by L+ M cos y + N cos ẞ= V¿ b'c'

&c. = &c.

Hence or otherwise shew that if the system be equivalent to a single force then §, î, ¿ must not

all be equal, and that in this event the line of action of the single force is given by

4. Find the tension and pressure when a uniform string rests in contact with a smooth surface under forces that have a potential.

Prove that the equations of equilibrium of a

string resting on a smooth sphere of radius a are

where 0, are the co-latitude and longitude and

→, are the corresponding forces.

5. State the principle of virtual work, and deduce the ordinary equations of equilibrium of a rigid body.

Three rods AB, BC, CD of the same length, AB and CD being of equal weight, are connected by smooth hinges at BC, and hang on two pegs in the same horizontal line. Find the positions of equilibrium, and find whether they are stable or not.

6. Give Weber's transformation of the equations of motion of a fluid, and apply them to find the equations of impulsive motion of a liquid.

7. Obtain the general equations of steady motion of a liquid in two dimensions.

An elliptic cylinder is filled with liquid which has molecular rotation at every point, and whose particles move in planes perpendicular to the axis. Prove that the stream lines are similar ellipses described in periodic time (a2 + b2) abw.

π

8. Define the velocity and stream functions where a liquid moves irrotationally and symmetrically about the axis of z, and find the differential equations which must be satisfied by these functions.

The resolved attractions of a body symmetrical about the axis of z are ƒ (z, p) and 4 (z, p) respectively perpendicular and parallel to that axis. The equation of a solid of revolution is pf (z, p) = a p2 + b, where a and b are two constants and p is the distance of any point from the axis of z. Prove that if this solid be made to move parallel to its axis in an infinite mass of liquid the stream lines are given by equating the left side of this equation to any constant, and the velocity function is (z, p) multiplied by a

constant.

9. Find the expression for the velocity of propagation of a series of simple periodic waves in water of uniform depth, the motion being small and in two dimensions.

If two such series of equal amplitude and nearly equal wave length travel in the same direction, so as to form alternate lulls and roughnesses, prove that in deep water these are propagated

with half the velocities of the waves; and that as the ratio of the depth to the wave length decreases from co to 0, the ratio of the two velocities of propagation increases from to 1.

10. Give a general account of the plan of successive approximation employed in solving the differential equations of the moon's motion, and explain how it is that the first approximate solution requires modification before it can be employed to determine the second approximate solution.

MATHEMATICS.-PAPER II.

Professor Nanson.

1. Find the attraction at any point of a fine uniform straight rod.

An infinite right circular cylinder attracts a point on the rim of one of its ends; shew that the attraction is equivalent to two components Ma and 4Mra respectively perpendicular and parallel to the axis, where a is the radius and M the mass per unit of length of the cylinder.

2. Prove that the potential at an external point xyz of an ellipsoidal shell of mass M is