## University of Glasgow Examinations 1882-1883

The Department of Mathematics set four Honours papers, two in Pure Mathematics and two in Natural Philosophy, the first of these two being on Dynamics, the second being on Physics. |

### DEPARTMENT OF MATHEMATICS

MATHEMATICS

GEOMETRY

*D*is a point within the triangle*ABC*. Prove as Euclid does that the sum of*BD*and*DC*is less than the sum of*BA*and*AC*.- Show how to describe a triangle when one of the sides,, a side adjacent to this angle and the side opposite are given. When is the problem impossible.
- Show how to divide a straight line into two parts so that the rectangle contained by the whole and one of the parts may bc equal to the square on the other part.
If the length of the larger segment be 10 in., what is the length of the smaller?

- Of all the triangles having a fixed area and standing on a fixed base, find that which has the least perimeter.
- 0n the hypotenuse
*AC*of a right-angled triangle*ABC*the perpendicular*BD*is let fall. Show without using the theory of proportion that the square on*BD*is equal to the rectangle*AD*,*DC*. - Show how to draw a tangent to a circle from all external point.
- One circle touches another internally at
*A*, and*BCDE*is a chord of the larger circle cutting the circumference of the smaller in*C*and*D*. Prove that the angles*BAC*,*DAE*are equal. - Show how to cut off a segment from a given circle so that the angle in the segment may be equal to a given angle.
*PQ*is a given straight line. Give Euclid's construction for obtaining a triangle having*PQ*and a line equal to it for two of the sides, and the angle between these two sides half of each of the other angles.- Show how to inscribe a circle in a given regular polygon of any number of sides and prove that the same construction will enable you also to describe a circle about it.
- In a circle a square is inscribed, and the middle points of its sides are joined so as to form another square. What is the ratio of the radius of the circle to the side of the second square
- Prove that if two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportional, the triangles are similar.
- Show how to describe a rectilinear figure similar to another rectilinear figure and equal in area to a third.
- One of the parallel sides of a trapezoid is double the other. Prove that the diagonals cut each other in one of their points of trisection.

### ALGEBRA AND TRIGONOMETRY

- Simplify the expression

(and give the value of it when*x*-*y*/2)^{4}+ 6(*x*^{2}-*y*^{2}/4)^{2}+ (*x*-*y*/2)^{4},

*x*= 0.25 =*y*/2. - Find the highest common factor of

*x*^{3}*y*- 4*x*^{2}*y*^{2}- 5*xy*^{3},*x*^{3}-*xy*^{2},*x*^{4}+ 2*x*^{3}*y*+*x*^{2}*y*^{2}.

- Resolve 8
*x*^{8}*y*- 63*x*^{5}*y*- 8*x*^{2}*y*into factors. - A certain number of two digits is seven times the sum of its digits. How often does the number contain the difference of the digits ?
- One of the two parallel sides of a trapezoid exceeds the other in length by 4 feet, and if the former were made equal to the latter, the area would be increased by 8 square feet and in the ratio of 6 : 7. Find the lengths of the parallel sides.
- Given
*a*^{2}:*a*^{2}-*b*^{2}:: 36 : 35. Find*a*^{3}:*a*^{3}-*b*^{3}. - What term of the equidifferent progression 1 + 1.5 + ... is equal to the seventh term of the equal rate progression .5 + 1 + ... ?
- Find the
*r*th term and the middle term of (*x*- 1/*x*)^{2r}. - The cosecant of an angle is 5.05. Give the tangents of the complement and supplement of the angle.
- Find the value of sec
^{4}*A*(*I*- sin^{4}*A*) - 2tan^{2}*A*. - Find the integral parts of log 100 to base 2, and log 64 to base 3.5.
- Log 2 = .3010300. Find log sin 45° and log cos 60°.
- Two angles and the included side of a triangle are 30°, 45°, 160 ft respectively. Find the area of the triangle.

### NATURAL PHILOSOPHY.

DYNAMICS.

- A point moves in the circumference of a circle of
*r*feet radius with an angular velocity of*ω*radians per second. What time does it take to go over*s*feet of its path? - Enunciate the theorem known as the " triangle of velocities."
A body has simultaneously given to it three velocities, viz., 60 feet per sec. E 30° N, 70 feet per sec. W 30° N, and 80 feet per second S. Find the magnitude of the resultant velocity.

- Define acceleration. If the unit of length were
*m*feet, and the unit of time*n*seconds, what number would represent an acceleration of a feet per second per second. - A body uniformly accelerates from rest and passes over
*b*feet in*p*seconds. How long will it take to pass over the next*b*feet? - A force applied to a mass of 8 lb. produces an acceleration of 32 feet per second per second. What acceleration would it produce in a mass of
^{1}/_{2}oz? - A spring balance is graduated for a place where
*g*= 32.2, and indicates 16 lb. where*g*= 32. What is the correct mass of the body tested? - Two parallel forces, each of 8 poundals, act in opposite directions, and in a line midway between their lines of action, a third force of 16 poundals acts. Find the magnitude and line of action of the resultant.
- Three forces of 1.5, 2, 2.5 unit act on a on a particle, and are in equilibrium. Find the angle between the directions of the first two forces.
- One force is double another, but the moment of the latter round a certain point is double that of the former. Compare the distances of the point from the two lines of action.
- From an equilateral triangle whose side is a inches long a triangular piece is cut off by a line drawn from the middle point of one of the sides parallel to another side. Find the centre of inertia of the remainder.
- The mass of a gun is 8 lb., and of the shot 1
^{1}/_{2}oz. When the gun is fired the initial velocity of the recoil is 16 feet per second. Find the initial velocity of the shot. - Define the various units of work, and show how they are related.
A man weighing 12 stones does 360,000 foot-pounds of work against gravity in ascending a hill. Find the height ascended.

- Investigate the relation between the length of a simple pendulum and the time of oscillation. A "second's" pendulum is lengthened 1 per cent. How much will it lose in a day? (
*g*= 32.)

### ASTRONOMY.

- Explain what measurements are necessary for specifying the position of a celestial object, the plane of the ecliptic being taken as the plane of reference.
- Describe shortly the instruments used for obtaining the zenith distance of a celestial body and the time of its meridian passage.
- Explain the principle of the for finding Greenwich time by means of lunar observations. What other methods may be followed?
- What is meant by diurnal libration of the moon? What is its cause?
- What is the lowest latitude at which the phenomenon of a non-setting sun may be observed.
- How is it known that the moon turns on her axis in the same time as she revolves round the earth?
- Show how the distance of Mercury from the sun may be obtained from observation of his greatest and least apparent diameters.
- Give a brief explanation of the theory of periodic shooting stars.

### PHYSICS.

- Find the specific gravity of a specimen of wood from the following data :-

Weight in air of specimen, .......................................... 28 1/2 gms.

Weight in air of brass sinker attached to specimen, .... 25 gms.

Weight in water of specimen and sinker, ................... 1.9 gms.

Specific gravity of brass, ............................................ 7.9 gms.

- Describe Atwood's machine. From the following data find the numerical value of g, the acceleration due to gravity, at a certain place :-

Mass of each box, ............................................. 30 oz.

Equivalent for inertia of wheel work, ............... 10 oz.

Mass added to one of the boxes, ...................... 2 oz.

Space described in rest from 3 secs., ................. 4.0 ft.

- Write down the pendulum formula. Explain how the pendulum may be used to determine the force of gravity at a given place.
According to experiments of Sabine, the length of the seconds pendulum at London is 39.139 inches. Find the force of gravity.

- Find the pressure in grammes weight per square centimetre at the bottom of a vessel one and three-quarter metres deep, full of mercury. Sp. gr. of mercury: 13.596.
- Describe the vibrations in air which will cause a musical note to be heard, and contrast them with the vibrations which constitute light. Find also the wave-length for the note C (256 vibrations per second), and compare it with the wavelength for light of any particular colour.
- Describe any mode of obtaining plane polarized light. Describe the vibrations in plane polarized light, in circularly polarized l light, and in ordinary unpolarized light.
- Describe experiments to show the phenomenon of "interference" in light, and explain the phenomenon. A tuning fork which is vibrating is held upright by the handle and turned round the handle as an axis. Describe and explain what is heard by an ear placed at any particular place on a level with the top of the fork.
- Five hundred cubic centimetres of mercury at 56° C. are put into a hollow in a block of ice, and it is found that 159 grammes of the ice are liquefied. Find the specific heat of mercury.
- Find the weight of 1000 cubic centimetres of air half saturated with moisture at 26° C. and 74 cm. pressure:-

Elastic force of vapour of water at 26° C., ......... 2.5 cms. mercury.

Density of vapour of water reduced to standard pressure and temperature, ........... 0.622.

- A bullet weighing
^{1}/_{16}lb, and moving with velocity of 1500 feet per second strikes a target and is stopped. Find the heat generated. - What is meant by electrostatic capacity of a conductor?
Explain clearly the action of the outer coating of a Leyden jar in increasing the capacity of the inner coating.

- State Coulomb's law of electrostatic attraction and repulsion, and give a sketch of the experiments by which he arrived at them.
- Give an account of the modern "contact" theory of the Voltaic cell. How does How does it differ from the so-called "chemical theory," and from the contact theory given by Volta himself?

*Candidates for the Degree of B.Sc. in Engineering Science will take the foregoing paper, but will substitute the following for the questions numbered*1, 2, 3, 8.

A sphere of radius *r* is placed with its centre at O, and is uniformly electrified with a charge of density *ρ*. Find the work done in carrying a unit of positive electricity from a point *P* to a point *P*'.

2a. State Ohm's law. Describe Wheatstone's bridge or balance; and prove the relation between the resistance of the parts when there is "a balance."

3a. State Joule's law for the generation of heat in a conductor.

A cell of electro-motlve force *E* and resistance *r* is used to transmit a current through an external resistance *R*. Find the heat generated per second in each of the resistances.

8a. Give a brief account of Prevost's theory of "heat exchanges," and describe experiments to illustrate some of the most important phenomena of radiant heat.

### University of Glasgow Honours examinations 1883

### Honours Pure Mathematics First Paper

- From the series 1, 2, 3, 4, ... all numbers of the form
*m*(*m*+1)/2 are struck out, and those remaining are separated into lots, the first lot containing only the first number, the second lot the next two numbers, the third lot the three numbers following these two, and so on. Find the sum of the numbers in the*n*th lot. - If the radii of the escribed circles of a triangle be taken, and the sums of pairs of them in order round the triangle be denoted by
*s*_{1},*s*_{2},*s*_{3}and the corresponding differences by*d*_{1},*d*_{2},*d*_{3}, show that

*d*_{1}.*d*_{2}.*d*_{3}+*d*_{1}.*s*_{2}.*s*_{3}+*d*_{2}.*s*_{3}.*s*_{1}+*d*_{3}.*s*_{1}.*s*_{2}= 0.

- In the parabola
*y*^{2}= 4*ax*find the area of the segment which is cut off by the focal chord through the point (*am*^{2}, 2*am*). - A straight line
*AB*is bisected in*C*. Find the locus of a point*P*such that*PC*may be a mean proportional between*PA*and*PB*. - Find the equation to a conic passing through four given points, determine the arbitrary constant so that it may represent straight lines, and show what relation must hold between the co-ordinates of a point that it may be a circle.
- One fixed circle touches another internally. Prove that the locus of a point whose perpendicular distances from the circumferences are equal is an ellipse with its semi-axis major the arithmetic mean between the radii, and its semi-axis minor the geometric mean.
- Find in their simplest form the differential coefficients of

(a) (tan*x*)/2 + √2/4 tan^{-1}(tan*x*√2).(b) log(

*x*+ √(*x*-1)) - 2√3/3 tan^{-1(2/3}√(3*x*-3) + √3/3).

- Given
*x*sin*θ*+*y*cos*θ*=*x*cosec*θ*+*y*sec*θ*: find^{dy}/_{dx}in terms of*x*and*y*. - Find the first five terms of the expansion of
*e*^{(ex - 1)}according toascending powers of

*x*. - If cosh
*x*= (*e*^{x}+*e*^{-x})/2 express cosh^{-1}*x*as a logarithm. - If
*P*denotes (*d*-*c*)(*d*-*b*)(*d*-*a*)(*c*-*b*)(*c*-*a*)(*b*-*a*), express

(in terms of*b*+*c*+*d*) ∂*P*/∂*a*+ (*c*+*d*+*a*) ∂*P*/∂*b*+ (*d*+*a*+*b*) ∂*P*/∂*c*+ (*a*+*b*+*c*) ∂*P*/∂*d*

*P*. *ABC*being a spherical triangle, establish the relation

cosand thence show that*c*= cos*a*cos*b*+ sin*a*sin*b*cos*C*,

Δwhere Δ*c*= cos*B*Δ*a*+ cos*A*Δ*b*+ sin*a*sin*B*Δ*C*

*c*is the change in*c*due to changes Δ*a*in*a*, Δ*b*in*b*and Δ*C*in*C*.

What are the corresponding results in the case of a plane triangle

### Honours Pure Mathematics Second Paper

*A*and*a*being constant in the triangle*ABC*, whose area is Δ and semiperimeter*s*, show that*d*Δ/*ds*= (*b*+*c*)tan(*A*/2)- Find the greatest circle inscribable in a triangle of which one side and the opposite angle are fixed.
- Define the Gamma function. Show that by substituting
*a*-*b*√-1 for*c*in the integral which is equal to Γ(*m*)/*c*^{m}, we may express two other integrals in terms of Γ(*m*). - Find the conditions that the straight line
*x*/*l*=*y*/*m*=*z*/*n*may be perpendicular to the plane*Ax*+*By*+*Cz*=*D*. - Show that by transformation of co-ordinates the equation
*xy*+*yz*+*zx*=*a*^{2}may be put in the form 2*x*^{2}-*y*^{2}-*z*^{2}= 2*a*^{2}. - Find the locus of the point where the tangent plane to the ellipsoid

*b*^{2}*c*^{2}*x*^{2}+*a*^{2}*b*^{2}*z*^{2}+*a*^{2}*b*^{2}*z*^{2}=*a*^{2}*b*^{2}*c*^{2}

- A triangle passes through three given points. Show that the planes perpendicular to its sides, and passing through their middle points, intersect in a straight line.
- Integrate the equation
*d*^{2}*y*/*dx*^{2}+ 2/*x**dy*/*dx*+*y*/*a*^{2}= 0. - Integrate the equation
*y*(*dy*/*dx*)^{2}+ (2*x*- 1)*dy*/*dx*-*y*= 0.

### NATURAL PHILOSOPHY.

DYNAMICS.

- Explain generally what is meant by degrees of freedom and restraint.
Illustrate by considering the case of a rigid system.

- Define the moment and axis of a couple. Enunciate and prove the theorem which shows that the laws of the composition of couples and of forces are similar.
- Show how to find analytically the centre of inertia of a solid of revolution.
- The area between the curves
*y*^{2}=*x*,*x*^{2}- 2*x*+*y*^{2}= 0 and above the axis of*x*is caused to revolve round the axis of*y*. Find the centre of inertia of the solid thus generated. - Define and illustrate the terms of moment of inertia, radius of gyration, and momental ellipsoid.
- Enunciate and prove the theorem regarding the difference between the moments of inertia with respect to an axis through the centre of inertia and an axis parallel to this.
- Find the moment of inertia of an ellipsoid of mass
*m*with respect to a diameter whose inclinations to the axes are*α*,*β*,*γ*. - Prove that the attraction of a uniform spherical surface on an external point is the same as if the whole mass were collected at the centre.
- In the triangle
*ABC*the sides*AB*,*AC*are fixed in direction, and the perpendicular from*A*on*BC*is fixed in magnitude. Show that the attraction of*BC*on a particle of*A*is constant. - Two bullets are projected from the same point at the same instant and with the same elevation
*a*, but with different velocities. Find the of the line joining them*t*seconds afterwards. - A particle is projected in a plane, and is acted on by a central force
*F*. Find the polar differential equation of the orbit. - At a certain point in a given elliptic orbit, whose semi axis major is
*a*, the velocity of a body is the same, whether it describes the orbit in a time*t*about one focus, or in a time*t*' about the other. Find the focal distances of the point.

### PHYSICS.

- Find
*A*_{1},*A*_{2},*A*_{3},*etc*., so that*A*_{1}sin*x*+*A*_{2}sin 2*x*+*etc*. =*f**x*, where*f**x*is a function of*x*, arbitrary between certain limits. What are these limits? How are the values of*f**x*beyond these limits related to the values within them? Examples (1)*f**x*=*c*; (2)*f**x*= cos*x*. - Find the temperature at any point of a thin solid ring, and at any time, supposing the initial temperature to have been uniformly of a constant value
*c*through one half and the other half. - Prove that in the uniform motion of heat, with concentric spherical surfaces isothermal, in a homogeneous solid, the temperatures are inversely as the radii.
- Find the distribution of electricity on an insulated spherical conductor subject to the influence of an external point.
- Find the potential and the components of force at any point in the neighbourhood of an infinitely small magnet.
- Prove the formula for central forces:
*P*=*h*^{2}*ap*/*p*^{3}*ar*. - A particle describes an ellipse under the action of a force always directed to the centre. Find the law of the force.
- Describe and explain Joule's experiment by means of which the specific heat of air under constant volume has been obtained.
- Give an account of Prevost's theory of "heat exchanges."
- Define "a modulus of elasticity"; and define "the Young's modulus." Find the formula for calculating the modulus of rigidity of a wire from experiments made with the torsion vibrator.

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