SCHOLARSHIP EXAMINATION.

Science.

Examiners.

JOHN TOLEKEN, M. D., Vice-Provost.

HUMPHREY LLOYD, D. D.

JOHN LEWIS MOORE, D. D.

THOMAS LUBY, D. D.

ANDREW SEARLE HART, LL. D.

CHARLES GRAVES, D. D., Professor of Mathematics.

JOHN H. JELLETT, M. A., Professor of Natural Philosophy.

JOSEPH A. GALBRAITH, M. A., Professor of Experimental Philosophy.

GEOMETRY.

DR. GRAVES.

1. Investigate the locus of the centre of a sphere which touches three given spheres.

2. If three small circles pass through the same point on the surface of a sphere, the sum of the angles of the curvilinear triangle, whose vertices are the three remaining points of intersection, is equal to two right angles. 3. State and prove the fundamental principles of the stereographic method of projection.

4. If two curves be constructed upon the same axis, so that the ordinates corresponding to the same abscissa are in a constant ratio; prove that

y, y' being two corresponding ordinates; p, p' the radii of the osculating circles at the corresponding points (x, y), (x, y'); and t, ť′ the portions of the tangents between those points and the common axis.

5. Find the locus of the intersection of tangents at the extremities of an arc which subtends a given angle at the focus of a conic section. Solve this question by means of the polar equations of the curve and tangent.

6. Find the equation of the conic section which passes through four given points, and is such that the anharmonic ratio of the chords drawn from these points to any fifth point on the curve is given.

7. Three parabolas whose axes are parallel cut one another; prove that the three chords joining their points of intersection two by two pass through the same point.

8. Prove that the tangent of the angle between the asymptotes of the hyperbola

Ay2+ Bxy + Cx2 + F = 0,

referred to rectangular co-ordinates, is

and show that this expression does not change when the equation is transformed by turning the axes through any angle.

DR. LUBY.

TRIGONOMETRY.

1. Cos (A-B)-cos (A+B) = {V(5)—1}, and sin (A – B). sin (A+B) = {V(5) - 1}; find sin A and sin B.

2. If two arcs of great circles be drawn from the angle C of a spherical triangle, one bisecting this angle, and the other at right angles to the side c; prove

tan2 (a - b) = tan (s — s′). tan (Σ − Σ');

s, s' being the segments of the base by one of the said arcs, and Σ, Σ' by the other.

3. State and prove Vieta's theorem for exhibiting the sums of the powers of the roots of the equation 22 - pz + 1 = o, by means of the exponential values of the sine and cosine of an arc.

4. Prove these exponential values, and thence De Moivre's theorem. 5. In any plane triangle, prove

b2

a2

a

6. If the tables only give the logarithm of a number n consisting of five digits; find the form of correction for finding thence the logarithm of a number N consisting of 5+ x digits.

7. If r and p be the circular radii of the inscribed and circumscribed

small circles to a spherical triangle, find the value of

sides of the triangle.

tan r

tanp

in terms of the

8. If the bisectors of two angles of a spherical triangle be equal, prove the sides opposite equal.

DIFFERENTIAL CALCULUS.

1. Prove Taylor's theorem, and show how it is limited; apply it to the expansion of sin1 (x + h) in powers of h.

2. If a semicircle be described on an axis of an ellipse, draw from the extremity of this axis a right line, so that the portion of it intercepted between the arcs of the circle and ellipse may be a maximum.

3. Show that the curve ay2 - x3 + bx2· o has a conjugate point and a point of inflexion, and determine them.

4. Draw two rectilinear asymptotes to the curve a2x2 — a2y2 — y2x2 = 0. 5. Find the general expression for the polar subtangent to a curve, and apply it to the curve r. (eo + e ̄0) = r'.

6. Inscribe a maximum triangle in a given ellipse, with one side parallel to an axis of the ellipse.

7. Draw two conjugate diameters to an ellipse so that the sum of the perpendiculars from their extremities on the axis major may be a max. Also the sum of squares.

ALGEBRA AND INTEGRAL CALCULUS.

DR. GRAVES.

I. Any power of a number which is the sum of two squares, may likewise be expressed as the sum of two squares.

2. In how many trials may a person undertake, for an even wager, to throw an ace with a single die? Solve the resulting equation by approximation?

5. Rectify the ordinary parabola (y2=px); and prove that, if tangents VT, PT, be drawn to the curve at its vertex V, and also at the point P, the difference between the arc VP and the tangent PT is equal to 2 log tan (45° + 10), 0 being the angle between the tangents.

6. Rectify the semicubical parabola whose equation is py2 = 23, taking the origin of co-ordinates as the origin of the arcs. What is there peculiar in the formula which you obtain ?

7. Always supposing = o, investigate the value of the definite integral

8. On a given curve, y = f(x), let arcs be taken, so that the difference between the abscissæ of their extremities is constantly equal to a. How do you determine that arc whose length is a maximum or a minimum ?

4

STATICS.

PROFESSOR JELLETT,

1. Two weights are connected by a cord passing over a smooth pulley. One of these weights hangs freely, and the other rests upon a rough inclined plane. Find the condition of equilibrium.

2. A rectangular block of wood is placed on the ground, and a beam rests on the ground and against the block, both being prevented from slipping by pegs. Determine the weight of the lightest beam which will raise one edge of the block from the ground.

3. If the block be rough, and the position of the beam be given, determine the smallest amount of roughness which will allow the weight of the latter to be augmented ad libitum without disturbing the equilibrium.

4. A beam is supported by two smooth vertical curves. In any position of the beam (not being a position of equilibrium), it is required to break a string by attaching it to the centre of gravity of the beam and to a fixed point. If the string be perfectly inextensible, show that this is done most effectually by choosing as the fixed point the centre of curvature of the locus of the centre of gravity, or some point indefinitely near it.

a. In what case does the rule fail with regard to the centre of curvature itself?

b. Exemplify this by the case of a beam resting on a smooth floor and against a smooth wall.

5. Two equal balls are placed within a hollow vertical cylinder, open at both ends, which rests upon a horizontal plane. Given the weight and radius of each ball, and the radius of the cylinder, to determine the least weight of the latter which is consistent with equilibrium.

6. Two weights are connected by a string passing over a smooth pulley, and rest upon two smooth vertical curves whose common plane passes through the pulley. If every position be one of equilibrium, show how to deduce the equation of one of the curves from the given equation of the other.

DYNAMICS.

DR. HART.

1. Compute the time of oscillation of a pendulum in terms of the length of the arc, carrying on the approximation as far as the fifth power of the

arc.

2. Compute the principal moments of inertia of a homogeneous octagonal prism; and hence find the differential equation of its motion rolling down an inclined plane, the co-efficient of friction being double the tangent of the plane's inclination, and elasticity and all other disturbances to rolling-motion being neglected.

3. A material point moves on a perfectly smooth right cone with vertical axis, its greatest velocity and greatest height above the vertex are each equal to g+a; find its least velocity and least height.

4. A particle constrained to move on the circumference of a circle is attracted by a force directed to a fixed point on the circumference, and varying inversely as the fifth power of the distance; find the pressure at any point.

5. A sphere is caused to move up a given inclined plane by an impulse parallel to the plane and passing through the centre of the sphere. Find the point at which it will cease to slide, the greatest height which it will attain, and the entire time of ascent and descent; the co-efficient of friction and inclination of the plane being given.

6. An elastic sphere, falling vertically with a given velocity, is struck by another of equal weight moving horizontally with the same velocity, the line joining the centres at the moment of impact makes equal angles with the two directions of motion, and with the line perpendicular to them. Find the motions after impact

Ist. If the spheres are smooth.

2nd. If the co-efficient of friction is 0.25.
3rd. If it is 0.75.

ASTRONOMY.

PROFESSOR JELLETT.

1. Given the right ascension and declination of a star; find the longitude; and determine the amount of the error to which the value so found is liable in consequence of small errors in the two given quantities.

2. Describe accurately the theoretic method of determining the elements of a planet's orbit: and hence show in what the difficulty of this problem consists.

3. Show that the position of the line of nodes may be ascertained simply by two observations made on the planet when in its node, if the distances of the planet from the Earth at the times of observation be known.

a. Why cannot this be found by the method in (1)?

4. Determine the obliquity of the ecliptic, by observing two declinations of the Sun and the difference between the corresponding right ascensions.

5. For the application of this method it is necessary to know the latitude of the place. If this be not very accurately known, in what positions of the Sun should the observations be made?

6. Find the time of shortest twilight.

ELECTRICITY.

DR. LLOYD.

1. State the more remarkable facts to prove that the conductibility of bodies for electricity varies with their physical nature, their chemical composition being unchanged.