We give four Mathematics papers set for the M.A. Pass degree and four for the M.A. Honours degree in Mathematics and Natural Philosophy at the University of Edinburgh. The papers were set for examinations in October 1882 and April 1883. We give only a sample of these questions, omitting the questions that contain mathematical symbols which make them hard to display on the web.

Examiners- Professor CHRYSTAL and Dr MACFARLANE.

1. Parallelograms on the same base and between the same parallels are equal.

   O is any point, and AB, AC any two straight lines; prove that the algebraic sum of the triangles OAB and OAC is equal to the triangle OAD, D being the fourth corner of the parallelogram of which AB and AC are the sides.

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.

   Prove the following Rule for obtaining three whole numbers which shall represent the sides of a right-angled triangle:-

   "Of any two numbers take twice their product, the difference of their squares, and the sum of their squares."

3. The angle at the centre of a circle is double the angle at the circumference which stands on the same arc; and two angles at the circumference standing on the same arc are equal.

   AOB is an arc of a circle (less than a semicircle), AD and BC perpendicular to the chord AB meet the circumference again in D and C; prove that the arc DC is equal to the arc AOB.

4. To divide a given finite straight line internally so that the rectangle contained by the whole and one of the parts shall be equal to the square on the other part.

   The radius of a circle is 8 feet; find to the third place of decimals the number of feet in the side of the inscribed regular decagon.

5. To draw a common tangent to two given circles.

   State the number of such tangents for each of the different ways in which the two circles may be related to one another as regards position.

6. The line bisecting the interior vertical angle of a triangle divides the base in the ratio of the sides of the triangle.

7. A straight line which divides the sides of a triangle proportionally is parallel to the base of the triangle.

   Through the point of intersection of the diagonals of a trapezium a line is drawn parallel to the parallel sides; prove that the parallel sides have the same ratio as the parts into which the line cuts the non-parallel sides.

8. The sum of any two of the facial angles of a trihedral angle is greater than the third.

   Or, as an alternative,

   Calculate the volume of a granite monument, consisting of a right cylindrical shaft 8 feet high, surmounted by a right circular cone 5 feet high, the common radius of the cone and cylinder being 2 ¹/₂ feet. (Take π = ³⁵⁵/₁₁₃.)

9. In the parabola the tangent at P bisects the angle between the focal distance of P and the perpendicular from P on the directrix.

   Enunciate the corresponding theorem for a Central Conic.

10. Define the radian or unit of circular measure; and find the formula for the number of degrees in a given number of radians, and vice versa.

    The difference of longitude between two places is 5°, and the latitude of both is 45°; find the distance between them along the parallel of latitude. (Take the radius of the earth to be 4000 miles.)

11. How is it that tan A goes through four signs, while sin A goes through two, when A changes from 0° to 360°?

    Express the other trigonometrical functions of A in terms of cos A.

    Prove that if A, B, C denote the angles of a triangle, then

    (cot B + cot C)/(tan B + tan C) + (cot C + cot A)/(tan C + tan A) + (cot A + cot B)/(tan A + tan B) = 1.

1. The population of Scotland at the beginning of 1881 was 3,735,573.

   If ¹/₄₅ of the population die or emigrate each year, and ¹/₃₀ of the population are born or immigrate each year, what will the population be at the end of 1884?

2. What is the remainder when

   (x - 1)(x - 3)(x - 5)(x - 7) is divided by (x - 2)(x - 4)?

   Calculate the coefficients in the expansion of (1 + x)¹⁰, and by means of them find the real and imaginary parts of (1 + √- 1)¹⁰.

3. Resolve into factors

   x² - 18x + 80;

   (x - x²⁾³+ (x² - 1)³ + (1 - x)³;

   x² - 2a²x/(a² + 1) + (a² - 1)/(a² + 1).

4. When is a series said to be convergent?

   Assuming the binomial theorem for positive integral exponents, prove by the method of indeterminate coefficients that

   (1 + x)^p/q = 1 + (p/q)x + higher powers of x.

   Write down the co-efficient of x⁴ in the expansion of

   (1 - 3x)/(1 + x)^3/2.

5. Define Compound Interest and True Discount.

   Show that the principal is equal to the reciprocal of the difference of the reciprocal of the discount and the reciprocal of the interest.

   A gentleman insured his life for £250 at a premium of £5 per annum; he died after n years, and the insurance office neither gained nor lost on the transaction. Find n, reckoning compound interest at the rate of 5 per cent. per annum.

   Given log 3 = 0.47712, log 5 = 0.69897, and log 7 = 0.84510.

6. Prove that the equation y = mx + c represents a straight line.

   The two sides of a right angled triangle being taken as axes, find the equations to the sides of the square described on the hypotenuse in terms of the lengths a and b of the two sides.

7. Exhibit the different ways in which four electors may vote when there are three candidates for one office, it being considered immaterial from whom the vote comes. It is supposed that all the electors vote.

8. Deduce the equation to an ellipse from its focal definition, taking the major and minor axes as axes of co-ordinates. Show from the equation that the ordinates of the ellipse are in a constant ratio to the corresponding ordinates of its auxiliary circle.

   Or, as an alternative,

   Prove geometrically that the locus of the middle points of a series of parallel chords of a conic section is a straight line.
   

1. If two triangles have the three sides of the one equal to the three sides of the other, each to each, then the triangles are identically equal, and of the angles those are equal which are opposite to equal sides.

   Find the locus of a point equidistant from two given points and also of a point equidistant from two given lines.

2. Any two sides of a triangle are together greater than the third.

   P is the perimeter of a convex polygon ABCDEA, w the perimeter of the crossed polygon ACEBDA formed by its diagonals; shew that w > P and < 2P.

3. Angles in the same segment of a circle are equal to one another.

   Prove that of all right-angled triangles upon the same hypotenuse, the isosceles one has the greatest perimeter.

4. The sum of the squares on the two sides of any triangle is double the sum of the squares on the median and on half the base.

   Calculate, in terms of the sides of a triangle, the segments into which the base is divided by the foot of the perpendicular from the vertex.

5. The areas of similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.

   By taking the decimetre as equal to four inches, what percentage of error is introduced first in linear measure, second, in square measure, third, in cubic measure. (A metre may be taken as exactly equal to 39.37 inches.)

6. On a given straight line to construct a segment of a circle similar to a given segment.

   To construct a triangle, given the base, the vertical angle, and the sum of the squares on its sides.

7. Prove that the area of a circular sector is r²θ/2, where r denotes the radius and θ the angle of the sector.

   A penny and a halfpenny have diameters of one-tenth of a foot and of an inch respectively. If a halfpenny lie wholly on the top of a penny, what amount of the upper surface of the penny will be left uncovered?

8. Assuming that pyramids of equal altitude on bases of equal area are equal in volume, show that the volume of a pyramid is one-third the volume of a prism of equal altitude on the same base.

   The interior of a building is of the form of a right circular cylinder of 30 feet radius and 12 feet altitude, surmounted by a right circular cone whose vertical angle is a right angle; how many cubic feet of air will it contain?

   Or, as an alternative,

   Prove that if P be any point on a conic section, F the focus, and G the foot of the normal FG = e FP.

   Hence calculate the length of the sub-normal in terms of the abscissa from the centre, the latus rectum, and the eccentricity.

9. The area of a parabola cut off by any chord is two-thirds of the area of the triangle formed by the chord and the tangents to the parabola at its extremities.

   An elliptic plot is described in a garden by means of a string 20 feet in length and passing round two pegs distant by 5 feet. What is the area of the plot?
   

1. What is the equivalent of compound interest at 2 ¹/₂ per cent. per quarter in terms of per cent. per annum?

   Find √((20605)² - (16484)²).

2. Solve

   x(I + √2 + √3) = 1 - √2 + √3

   (276 - x) (360 - x) = 180 360

   x^4/3 + 2x^2/3 = 24

   The solutions, when not integral, to be carried to two places of decimals.

3. What is meant by the base of a system of logarithms?

   What is the characteristic of 12345, 1.2345, 0.0012345?

   Given log 2 = 0.30103 and log 3 = 0.47712; find log √(45).

   Solve 10^(x-1)(2-x) = 1000.

4. Deduce from the Exponential Series (or find independently) the series for log_e(1 + z) (log to base e) in terms of x.

   Show that log √3 = 1/2 + 1/(3.2³) + 1/(5.2⁵) + ...

5. Find the number of permutations which can be made of the letters in MDCCCLXXXII.

   If the above letters be written at random, what is the chance that two sets of three like letters come together?

6. Find the expression for the present value of an annuity of £50 payable for 12 years, first, when the annuity begins to be paid at the end of one year hence, and second, when it begins to be paid at the end of ten years hence.

7. Show that Ax² + 2Hxy + By² = 0 represents a pair of straight lines through the origin, and find the tangent of the angle between them.

   Or, as an alternative,

   Find an expression for the surface of a spherical zone.

8. Show that the equation to any straight line through the intersection of ax + by + c = 0 and a'x + b'y + c'= 0 may be written ax + by + c + k(a'x + b'y + c') = 0.

   A and B are points on the axes of x and y such that OA = a, OB = b, P and Q are movable points on AB, such that the perpendicular PM on the axis of x is equal to the perpendicular QN on the axis of y; find the equation to the locus of the intersection of PM and QN.
   

HONOURS MATHEMATICS We give next the papers for the M.A. Honours degree in Mathematics and Natural Philosophy at the University of Edinburgh. There are two Mathematics papers, and two Applied Mathematics papers. All were set for examinations in April 1883.

Examiner - Dr MACFARLANE.

Friday, 13th April 1883.- 9 to 12 o'clock.

1. Two circles intersect at A and B. From any point P on one of them the lines PA, PB are drawn, and produced to meet the other circle again in C and D. Prove that as the position of P varies, the straight line CD envelops a circle concentric with the circle ABCD.

2. If a, b, c denote the sides, and R the radius of the circumscribed circle of a given triangle, then the radius of the circle inscribed in the pedal triangle is

   (a² + b² + c²)Δ/abc - 2R

   where Δ is the area of the triangle.

3. Find the number of homogeneous products of r dimensions which can be formed out of n symbols.

   Find the number of different possible states of the poll under the cumulative system of voting, when there are p candidates for q seats, and there are m electors in the constituency.

4. Are the following equations independent of one another ?

   x + 5y + 3z = 12.5.

   5x + z = 11.

   4x + 95y + 54z = 204.5.

   Evaluate the determinant with rows

   a+b+c a-b-c a-b+c

   a-b-c a+b+c a+b-c

   a-b+c a+b-c a+b+c

5. PN is an ordinate of a circle to the diameter passing through the fixed point A, and T is the intersection of the tangent at A with the radius produced through P. Find the locus of the intersection of AP and TN.

6. Find the proportion between the width and the depth of a furrow-slice, in order that the slice when turned over may present the maximum amount of surface to the atmosphere.

7. Into a full conical wine-glass whose depth is a, and generating angle α, there is dropped a spherical ball that causes the greatest overflow; show that the radius of the ball is

   (a sin α)/( sin α + cos 2α).

8. Evaluate-

   ∫1/(4 + 5 sin x) dx.

   ∫e^ax cos nx dx.

   ∫(1 - x^1/2)/(1 - x^1/3) dx.
   

Examiner- Professor CHRYSTAL.

1. Investigate a formula for the radius of curvature in terms of the polar coordinates of any point on a curve.

   A cardioid rolls on a straight line; find the area between this line and the curve traced out by the cusped vertex of the cardioid.

2. Shew that the locus of the middle points of a series of parallel chords of a conic section is a straight line.

   Define conjugate diameters, and give an analytical proof of their fundamental property.

   Shew that two concentric conics have in general one pair of common conjugate diameters.

3. Trace the curve

   x⁴ + y⁴ - 2px²y² = a²xy,

   p being < 1; and find its whole area.
   

Examiners- Professor TAIT and Dr MACFARLANE.

Not more than eight questions to be selected.

1. Explain what is meant by the dimensions of a physical quantity. What are the dimensions of acceleration, force, power, work, angular velocity?

2. Express, in symbols, two equal circular motions in one plane, with nearly equal angular velocities of opposite signs.

   Show that their resultant may be treated as simple harmonic motion, in a direction which rotates slowly.

   What are the chief physical phenomena which are kinematically explained by this theorem?

3. A straight rod has one of its extremities attached to an arm which moves with uniform angular velocity, while the other extremity moves along a diameter produced ; find the equations for the position, velocity, and acceleration of the latter point.

4. A mass, resting on a rough horizontal plane, is attached by equal stretched elastic strings to each of two points in the plane. Find the boundary of the region within-which it can remain at rest.

   Deduce, geometrically, a property of this boundary.

5. What must be the inclination of a roof in order that the rain may run down in the shortest time ?

6. Calculate the amount of energy lost in the direct impact of two spheres, whose coefficient of restitution is ¹/₂, whose masses are 5 and 7 Ibs., and whose initial velocities are 10 and -13 feet per second.

7. A smooth cylindric glass rod, the radius of which is small compared to the length, is placed in a smooth hemispherical mortar, one end projecting; find the position of rest.

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

   Apply your result to deduce the difference of the normal component of force at points very close to one another, but on opposite sides of an electrified shell.

9. A heavy pulley 4 Ibs. in weight hangs by a string over a fixed pulley, the other end of the string being attached to a weight of 10 lbs.; over the moveable pulley hangs a string with weights of 3 Ibs. and 1 lb. attached; find the accelerations of the three weights.

10. Calculate the law of distribution of mass in a chain which hangs in the form of a circular arc.

11. Find the law of density throughout a hemisphere, the density depending on the distance from the centre, in order that the centre of inertia of the hemisphere may be at a distance of one-nth of the radius from the centre.

12. Form the general equations of equilibrium of a rigid solid under any system of forces.

    Find the conditions that the resultant may be a force, or a couple, alone.
    

Not more than eight questions to be selected.

1. Apply Lagrange's generalised method to the motion of a self-closing door, when the displacement of the weight is proportional to the sine of half of the angle of displacement of the door.

2. Find the moment of inertia of a uniform cube about an axis through its centre. Hence derive those about an edge, and about a diagonal of one face.

3. A perfectly homogeneous sphere has an angular Velocity w about its diameter. If the sphere gradually contract, remaining constantly homogeneous, required the angular velocity when it has half its original diameter.

4. A chain is wound round a cylinder, which can rotate freely about its axis, placed horizontally. Calculate the motion as the chain gradually unwinds;-supposing that the free part remains constantly vertical.

5. Two circular plates of equal diameters, and placed parallel to and over one another, are kept charged with constant quantities of electricity. Investigate the motion of a small pith-ball placed in the axis of the plates.

6. Charges of electricity 15 units positive, 12 units negative, and 20 units positive, are placed at three points in one straight line A, B, and C respectively, such that AB = 9, BC = 16, and AC = 25. Find the equation for the equipotential surfaces, and show what it becomes in the case of the potential being unity.

7. Form the equations for the small transverse vibrations of a stretched string, neglecting external forces.

   Show how to introduce into the general integral, the conditions that
   (a) one end is fixed,
   (b) the other is forced to execute transverse simple harmonic motions of given period and range.

8. Show that the average value of the potential throughout a small sphere is

   V₀ - 2a²π/5 ,

   where V₀ denotes the potential at the centre, and the density at the place.

9. Find the conditions of equilibrium of a mass of heterogeneous incompressible liquid under given forces.

   Point out the criterion of stability.

10. The poles of a battery are placed in contact with an infinite and uniformly conducting plate; find the form of the equipotential lines, and also the form of the current lines.

11. Find the equation of the adiabatics for the ideal perfect gas, in which

    pv = Rt

    and thence calculate the velocity of sound-waves in such a gas.
    

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