# Examination papers by Alexander Macfarlane

Alexander Macfarlane was appointed an Examiner in Mathematics at the University of Edinburgh in 1881 for three years. We give below four examination papers which he set, one as sole setter, the other three jointly with either George Chrystal or P G Tait, all from the 1881-82 diet.

1. M.A. Pass Examinations, 1881-82.

MATHEMATICS

PAPER I

Tuesday, 18th October 1881. - 2.30 to 4.30 o'clock.

Examiners - Professor Chrystal and Dr Macfarlane.

1. If two triangles have one side of the one equal to one side of the other, and the angles adjacent to these sides equal, they are congruent (equal in every respect).

A point lies between two parallel straight lines, and is such that one straight line drawn through it terminated by the parallels is bisected, prove that every straight line so drawn is bisected.

2. In any right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the sides which contain the right angle.

A rectangular strip of paper, 3 inches long by 2 inches broad, has a square one inch in the side cut off from one corner. Show how to cut the remainder into three pieces that will make up a square.

3. In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

The locus of the middle points of all the cords of a circle which are drawn through a given point is a circle.

4. Define the tangent to a circle.

To describe a circle touching three given straight lines. Point out how many such circles there are in general.

If two of the given straight lines remain fixed, and the third move, what will be the locus of the centre of the various circles that touch the three lines?

5. In a right-angled triangle, if a perpendicular be drawn from the right angle to the hypotenuse, the two triangles on each side of the perpendicular are similar to the whole triangle and to one another.

Every chord of a circle is a mean proportional between the diameter drawn from one of its extremities and its projection on that diameter.

6. Prove that similar triangles are to each other in the duplicate ratio of their homologous sides.

Indicate, without detailed proof, how this proposition is extended to similar figures generally.

State the corresponding proposition for similar solids.

7. If a straight line is perpendicular to each of two intersecting straight lines at their point of intersection, it will be perpendicular to the plane which contains them.

Find the locus of points in space equally distant from two given points.

8. The tangent at a point in a parabola bisects the angle between the focal distance of the point and the perpendicular from the point on the directrix.

The locus of the foot of the perpendicular from the focus on the tangent to a parabola is the tangent at the vertex.

9. The tangents drawn from any external point to an ellipse subtend equal angles at either focus.

When two tangents to an ellipse and one focus are given, what is the locus of the other focus?

Or as an alternative -

Calculate the area of that zone of the earths's surface which lies between Lat. 30º and Lat. 45º, assuming the earth's radius to be 4000 miles, and π to be 3.1416.

10. Prove directly that

$\sec^{2}A = 1 + \tan^{2}A$;

and hence deduce, by means of the values of sec A and tan A in terms of sin A and cos A, that

$\sin^{2}A + \cos^{2}A = l$.

Why is $\cos (-A) = \cos A$, but $\sin (-A) = -\sin A$?

11. Prove geometrically that

$\sin 2A = 2 \sin A \cos A$,

drawing your figure to suit the case where $A < 45°$.

Prove that

$\cos 2A = \Large\frac{1 - \tan^{2}A}{1 + \tan^{2}A)}$

and

$\Large\frac{\cos A + \sin A}{\cos A - \sin A}\normalsize - \Large\frac{\cos A - \sin A}{\cos A + \sin A}\normalsize = 2 \tan 2A$.

12. Prove that

$\tan \large\frac{1}{2}\normalsize (B - C) = \Large\frac{b - c}{b + c}\normalsize \cot A$;

and explain the use of this formula.

Express $\Large\frac{1 - \cos A}{1 - \cos B}$ in terms of the sides of the triangle.
2. M.A. Pass Examinations, 1881-82.

NATURAL PHILOSOPHY.

PAPER I

Thursday, 20th October 1881. 1 to 4 o'clock.

Examiners - Professor Tait and Dr Macfarlane.

Not more than eighteen questions to be answered, of which six must be taken from questions 1 - 8 inclusive.
1. Define Velocity, Acceleration, Momentum, and Force. Which of these ideas are kinematical, and which dynamical ?

2. Define Mass and Weight, explaining carefully how each is to be determined. How is it proved experimentally that, in any given place, the weights of all bodies are as their masses?

3. Define Average Velocity.

What is the average velocity of a point executing a Simple Harmonic Motion for the time occupied in moving from the one to the other extremity of its range, its maximum velocity being 5 feet per second ?

4. Prove that the path of an unresisted projectile is a parabola; and show how to find, when the initial velocity is given, the directions of projection from one given point so that another given point may be struck.

5. State the laws of Statical Friction.

A 28 lb. weight is suspended by means of a ring on a straight rod which revolves in a horizontal plane about a fixed axis. The coefficient of friction between the ring and the rod is $\large\frac{1}{5}\normalsize$ and the distance of the weight from the axis is 5 feet. Find the rate of revolution when the weight will begin to move outwards.

6. A body rests on a horizontal plane whose coefficient of friction is 1/2; at what inclination must a force equal to the weight of the body be applied so that it may be just on the point of moving the body.

7. Explain the nature and use of Contour Lines.

How will the following physical features be indicated, (1) a waterfall, (2) a lake with outlet, (3) a lake without outlet, (4) the junction of two streams, (5) an overhanging precipice?

8. Describe the circumstances of the direct impact of two elastic spheres. If the coefficient of restitution is 1, and the spheres equal in mass, show that the velocities are simply interchanged.

9. What is the logical nature of a Physical Series or Scale, as for example, the Scale of Hardness ? Give other examples of such a Series.

10. Give a general explanation of the rise of water in a clean capillary tube of glass.

How can the surface tension of a liquid be detected by direct experiment ?

11. Describe an experiment which proves that the boiling-point of water depends upon the pressure.

How may this principle be applied to determine the height of a station above the level of the sea?

12. Explain one method of determining the compressibility of water?

13. State briefly the principal advances in the Science of Electricity associated with the names of Volta, Coulomb, Ampere, Faraday, and Ohm.

14. State Kepler's Laws, and the immediate consequences of each as deduced by Newton.

15. Describe the different electric or magnetic phenomena which go by the name of induction.

16. State, in any of its forms, the second law of Thermodynamics; and point out the main novelties introduced by Carnot into physical reasoning with regard to heat.

17. State the fundamental laws of Geometrical Optics.

Explain the sharpness of the shadows produced by the arc electric light.

18. The thermal conductivity of iron is about 0·0133, the units being the foot, minute, and degree C.

Find how much heat, per hour, is lost by a boiler of $\large\frac{1}{4}\normalsize$ inch plate whose surface is 10 square yards, and which contains water at 110º C, the external surface of the boiler being kept at 100° C.

19. State the laws of the Refraction of Light, and show how they necessarily lead to the conception of a critical angle.

20. Define Specific Heat, and give instances of the utility of the great specific heat of water.

21. Find the formula for correcting the error in the time published by the Castle Gun due to distance from the Castle.

22. Describe the conditions of distinct vision, and explain the action of a simple convex lens used as a microscope.

23. Give an outline of the facts and reasoning by which it is concluded that certain elementary substances, such as sodium, iron, hydrogen, exist in the Sun.

24. Explain the origin of the waves which are seen on a shelving shore, even when the sea appears absolutely calm. Why do we sometimes find every ninth or tenth, etc., wave considerably higher than the others? State, concisely, the origin of a breaker, and of a bore.

Candidates for the Neil-Arnott Prize will answer the following, in addition to not more than 12 of the preceding questions.

(a) How do we judge of the position of a source of sound?

(b) What is the nature of ordinary Colour-Blindness?

(c) Explain how a forged bank-note can be detected by means of the stereoscope.

(d) How can the rate of propagation of a nerve-disturbance be measured?

(e) What is the cause of the difference between the light of the electric arc, and that of a solid incandescent black body as in Swan's lamp?

(f) State, approximately, how the rate of flow of water through very fine tubes depends upon -

(i) Temperature.
(ii) Pressure,
(iii) Diameter of tube.
(iv) Length of tube.

How are these modified in wide tubes, such as those which supply a town with water?
3. M.A. Pass Examinations, 1881-82.

NATURAL PHILOSOPHY.

PAPER II

Wednesday, 12th April 1882. - 1 to 4 o'clock.

Examiners - Professor Tait and Dr Macfarlane.
1. Define Velocity and Average Velocity.

Two passenger trains having equal velocities, and consisting each of 12 carriages, are observed to take 9 seconds to pass one another; what is the velocity, estimating the length of a carriage at 23 feet ?

2. Show that the Velocity of a point is fully determined, in magnitude and direction, by means of its Components parallel to any set of three lines at right angles to one another.

3. Define Acceleration. What is its relation to Force ?

A man steps on to an elevator, which thereupon descends with a uniform acceleration of 20 feet per second; what sensation will he experience, and calculate its amount?

What sensation will he experience when the elevator strikes the ground ?

4. Enunciate Newton's Second Law of Motion, and show how to find the resultant of two forces acting at one point.

A string has its ends fixed at two points, and a mass of 10 lbs. is attached to an assigned point in it. Find the tensions produced in the two parts of the string.

5. Define Kinetic Energy and Work.

Calculate the Kinetic Energy of a tram-car weighing 2.5 tons, when it is moving at the rate of 6 miles an hour, and is laden with 36 passengers averaging 9 stones each in weight.

If the co-efficient of Kinetic Friction for a tram-car moving on its rails is $\large\frac{1}{8}\normalsize$; find how much work is done, when the above car, loaded as stated, is pulled 3 miles along a level road.

6. Compare the amounts of Momentum and of Kinetic Energy in (a) a pillow of 20 lbs. which has fallen through one foot vertically, and (b) an ounce bullet moving at 200 feet per second.

7. Define Centre of Inertia, and Centre of Gravity.

Show that when the latter Centre exists it coincides with the former.

Two labourers carry a rectangular block of stone on a hand-barrow up a scaffolding stair; show how the weight of the stone will be distributed between them.

8. Define Simple Harmonic Motion; and show that the resultant of two S. H. M., of the same period, and in one line, is another S. H. M. of the same period. Apply this to the indications of a Tide-gauge.

9. State the principle of the Common Balance. What conditions must a Balance satisfy in order to be good?

A shopkeeper, having a faulty balance, proposed to weigh a commodity by suspending it from the long arm; but the customer, interposing, proposed that it should be weighed from the short arm; eventually they agreed to take the arithmetical mean of the two values. Did this compromise favour either party, and if so, by how much?

10. Explain, according to the accepted theory, the rise of water in a capillary glass tube. Point out the various assumptions required by the theory, and show how each may be experimentally verified.

11. Compare the characters of Frictional Electricity with those of Voltaic Electricity.

What is the principle involved in the construction of a Magneto-electric machine?

12. Define Contour Lines; and explain, by an example, the use of the principle they involve as regards applications in Physical Science.

13. Define Electric Conductivity, and point out its analogy to Thermal
Conductivity.

Arrange the following substances in the order of their Electric Conductivity, beginning with the best conductor:-

14. Define Elasticity, and show how the elastic properties of bodies enable us to distinguish clearly between solids and fluids.

How is the distinction of fluids into gases and vapours determined?

15. What tests can be applied to determine -

(1.) Whether or not a current of electricity is flowing round a circuit ?

(2.) Whether or not a bar of steel, such as a sewing-needle, has been magnetised?

16. Define a Musical Note in terms of the corresponding agitation of the air.

In what separate respects may two notes differ?

What are the physical causes of these differences; and what is the relative sensibility of the ear to them?

17. Define Specific Heat, and Latent Heat.

Calculate the cooling effect of a cube of ice 2 feet in the side, taken at 0º C, and reaching 27° C, when its cooling power has been exhausted. (The coefficient of the expansion of water on changing into ice is $\large\frac{1}{11}\normalsize$, and the number of pounds in 1 cubic foot of water is 62.4.)

18. Explain the so-called Aberration of Light; and show how it affects the apparent position of (a) a fixed star, (b) a planet.

19. State the chief principle involved in the construction of a Spectroscope.

Certain dark lines in the Solar Spectrum are attributed to absorption of the Sun's light by the Earth's atmosphere, and certain others are attributed to absorption by the Sun's own atmosphere. How can this discrimination be made?

What is the Rain-band?

20. Explain the action of a lens of short focal length when used as a hand magnifier, and show how to calculate its magnifying power.

When two such lenses are placed close together, what is the magnifying power of the system in terms of those of the components?

21. Explain the action of Oil in destroying waves.

22. On what properties of matter do the following contrivances
depend:-

(a) Achromatic Lenses.

(b) Compensation Pendulums.

(c) The production of Green paints by mixing Blue and Yellow.

(d) The Governor of a Steam-engine?

23. State the principal advances in Physical Science associated with the names of Torricelli, Pascal, Boyle, Fresnel, and Young respectively.

24. Show how the Earth's Magnetic Force at any place can be fully determined in direction and magnitude; pointing out in general terms the nature and use of each of the instruments employed.

25. What is meant in Thermodynamics by a Cycle of Operations, and what by a Reversible Cycle? Show, from the modern view of the nature of heat, that a reversible engine is perfect.

26. Show how the idea of Temperature is suggested, how temperature is commonly measured, and how it can be measured absolutely.
4. M.A. Examinations for Honours 1882.

APPLIED MATHEMATICS.

Saturday, 15th April 1882. - 9 to 12 o'clock.

Examiner - Dr Macfarlane.

Not more than nine questions to be selected.
1. A point moves in a plane curve ; find the expressions for the components of the velocity and of the acceleration along and perpendicular to the radius vector respectively.

Show that the investigation can be simplified by taking $\rho = r• \theta$, here $r$ denotes the Tensor and $\theta$ the Versor of the radius vector.

2. Compound two Simple Harmonic Motions having their amplitudes unequal and in different directions, the ratio of whose periods is 3 to 1.

Show what the equation becomes, when the difference of epoch of the two components is zero; and trace the curve in this case.

3. Find the locus of the vertex of a jet of water issuing under constant pressure, when the direction of emission is varied,

4. Find the polar differential equation to the path described by a body moving under the influence of a central force.

Determine the law of the force when the body describes the curve

$r = a\cos^{-2} \Large\frac{\theta}{2}$,

the centre of force being at the pole.

5. At a straight incline on a railway a descending train is attached by a rope passing round a pulley to an ascending train, and so that the rope is on either side parallel to the rails. Find, taking into account the friction of the trains on the rails, and of the rope on the pulley, the following:-

(1) The condition that the system may be able to move under the action of gravity alone.

(2) The velocity which would be acquired when the descending train reached the bottom, supposing that gravity were sufficient and unrestrained.

(3) The amount of work performed by an engine in making the transfer, the action of gravity not being sufficient.

(4) The Tension of the rope.

6. Form the equations of equilibrium of a flexible chain under the action of forces which are in one plane:

What must be the law of the line-density, in order that the chain may, gravity being the only applied force, assume the form of a curve in which the tangent of the angle which the tangent line makes with the horizontal is proportional to the length of the curve measured from
its lowest point?

7. Analyse the homogeneous strain

$\begin{vmatrix} A & b & c\\ d & E & f\\ g & h & I \end{vmatrix}$

into the pure strain and the rotational strain of which it is composed.

8. Find the attraction of a circular disc of uniform density on a particle situated in its axis.

Two circular plates are placed parallel and opposite to one another, and so that the distance between them is small compared with the radius of either. The inner surfaces are charged uniformly, the one with positive, the other with negative electricity. Find the attraction on a small charge of electricity situated between the plates.

9. Find the form of the equipotential surfaces for charges of electricity 1 and $-\large\frac{1}{2}\normalsize$ situated at a distance of 3 centimetres apart. Show what the surface becomes in the case of the potential having the value zero.

Find the law of the density in the region where the law of the potential is

(1) $V = \Large\frac{C}{\sqrt{x^{2} + y^{2} + z^{2}}}$.

(2) $V = C(x^{2} + y^{2} + z^{2})$.

(3) $V = \log (xyz)$.

10. A sheet of paper, folded in the form of a cone but so as to have a small aperture at the apex, and supported so that the axis of the cone is vertical, is filled with mercury; find the time required for the mercury to empty out.

11. A globule of mercury, immersed in an insulating liquid, is charged with electricity; find the pressure at a point in its interior.

12. State and prove Fourier's Theorem.

Apply the theorem to expand $\theta^{2}$ in a series of cosines or sines of multiples of $\theta$.

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