2. A curve touches the axis of x at the origin. Show that the locus of middle points of chords parallel to the axis of x passes through the origin, and that the tangent to it there makes with the axis of x an angle

5. Show how to place two given quadrilaterals so that each may be the conical projection of the other.

6. Find the equation to the plane of that section of the surface

ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy + 2lx + 2my + 2nz + d = 0 whose centre is at a given point (X, Y, Z).

Find the locus of the centres of sections which contain a given point (x', y', ').

7. Each side of a closed polygon in space passes through a fixed point, and all the vertices but one lie in fixed planes. Prove that the locus of that one is a curve of the third degree, and that of each of the others a conic.

8. There are μ changes of sign in a given equation. The roots are increased by a positive quantity h, and the number of changes thus becomes v. Prove that v - μ exceeds the number of roots of the original equation that lie between - h and 0 by an even positive integer or zero.

9. If the roots of an equation are all real, prove that those of each of its derived equations are all real.

Hence show that, if a1a, > a2 2, the equation

аx2 + 4α1x3 + 6а2x2 + 4а3x + αş

cannot have four real roots.

= 0

10. Form the linear differential equation of the second order that is satisfied by

11. Prove that the order of a differential equation homogeneous in the dependent variable and its derivatives may be lowered by unity.

SENIOR SCHOLARSHIP-NATURAL PHILOSOPHY.

MATHEMATICAL PHYSICS.

Examiner-PROFESSOR ALEXANDER ANDERSON.

1. Forces proportional to the sides of an unclosed plane polygon act along these sides in order; show that they have a single resultant; and find its position and magnitude.

2. A uniform flexible chain whose ends are fixed is in the form of a parabola under the action of a repulsive force from the focus, find the law of force; and show that, if T1, T2 are the tensions at the extremities of a focal chord, T2+T2 const.

=

3. Prove that, if the equation of virtual moments is satisfied for external forces acting on a system of bodies for all virtual displacements consistent with the geometrical conditions of the system, the forces are in equilibrium.

4. Investigate in three-dimensional polar co-ordinates the equations of motion of a particle, and from these equations deduce the equations of angular momentum and energy for the case of a heavy particle moving on the surface of a smooth sphere.

5. Find the law of force to a point in the orbit when the free path of a particle is circular; and show that the complete orbit is made up of two circles touching at the centre of force.

6. A rigid body is set in motion by a blow applied at a given point. Show that the kinetic energy produced is Iv, where I is the blow and the component of the velocity of the point of application in the direction of the blow.

7. Prove the expression

AK2
V

for the height of the meta

centre of a floating body above the centre of buoyancy; and hence find the criterion of stability.

8. Prove that, when the total deviation of a ray of light passing through a prism is least, the path of the ray lies in the principal plane.

9. Find expressions for the longitudinal and lateral aberrations of a ray of light incident on a spherical mirror parallel to its axis.

10. Find, at a given time, the positions of all stars whose aberration in longitude is zero.

EXPERIMENTAL PHYSICS.

Examiner-PROFESSOR ALEXANDER ANDERSON.

1. What is meant by the viscosity of a gas? Give an experimental method of determining it. Of what theoretical importance is this determination?

2. Enunciate the two laws of thermodynamics, and deduce a definition of absolute temperature. Find an expression for the availability of a substance, the different parts of which are at given different absolute temperatures.

3. Describe the method ordinarily used to determine the conductivity of the earth's crust.

4. Explain how the apparent pitch of a note depends on (a) the motion of the source of sound, (b) the motion of the observer, and (c) the motion of the intervening air.

5. Show that from Maxwell's theory of the action of electrified bodies on one another may be deduced the existence of a tension along the lines of force, and an equal pressure at right angles.

6. Prove that, in a medium of permeability unity, the force due to current in a closed circuit is equal to that due to a uniform magnetic shell whose boundary is the circuit, and whose strength is numerically equal to the current. What modification must be made in this proposition when the medium is of permeability μ different from unity?

7. Describe a method of measuring the energy dissipated when a piece of soft iron is subjected to a complete cycle of varying magnetic force.

8. What do you understand by Cathode rays? Give the two leading theories as to their nature, and give arguments in support of one of them.

9. Explain the elastic-solid theory and the electromagnetic theory of light, and point out for what reasons the latter is regarded as the more satisfactory.

10. What do you understand by the elements of elliptically polarized light? and how would you determine them by experiment?

PRACTICAL PHYSICS.

Examiner-PROFESSOR ALEXANDER ANDERSON.

1. Find the angle A of the given prism.
2. Find the specific resistance of the given wire.

SENIOR SCHOLARSHIP-METAPHYSICAL AND ECONOMIC SCIENCE.

METAPHYSICS.

Examiner-THE PRESIDENT.

1. Analyse, after Hamilton, the phenomena of our Conscious life in general, and of our Knowledge in particular.

2. What, according to Hamilton, are the tests by which the primary data of consciousness are to be determined.

Does he apply them in the construction of his own system?

3. Cognitio nostra omnis a Mente primam originem, a Sensibus exordium habet primum.' Translate, and comment on this passage, and name its author.

4. State the functions of the Elaborative and Regulative Faculties in Hamilton's system.

What analogous distinction does he claim to find in Kant?