II. Prove that the conics

and

x2 + y2 + z2 — 2 yz cos A – 2 zx cos B-2 xy cos C = (x + y + z)2,

2 yz cos A - 2 zx cos B-2 xy cos C = {x cos (B-C) + y cos (C-A) +z cos (A − B) } 2

touch one another.

DR. LUBY.

TRIGONOMETRY.

1. In a right-angled spherical triangle, given the sum of hypothenuse and perpendicular on it from the right angle, and the sum of the sides, construct the triangle.

2. Prove that in a right-angled spherical triangle the squares of the sines of the sides are as the sines of twice the adjacent segments made by the perpendicular.

4. Prove a = sin a. seca. seca. seca. &c., ad inf.

5. Prove log u=(u — u ̄1) — } (u2 — u ̄2) + ¦ (u3 — u ̄3) + &c., and thence that a sin a-. sin 2a+. sin 3a- &c.

=

6. In the quadratic x2+px-q, show the forms which the roots take as you eliminate p or q by the equation p2 tan 20 = 49.

INTEGRAL CALCULUS.

1. Find f dx. cos 3r between the limits π and 1⁄2′′.

3. Find the value of the expression sin v. § Q. cos vdv—cos v. §Q sin vdv, where cos mv.

4. Show that the value of this integral from v=0, to any other value is

m2

- [

5. An ellipse revolves about its major axis; find its solid contents, and also that of its maximum inscribed cylinder.

6. Integrate the expression

(3x-5) dx x2-6x+8

and find its value from x = 10

to x = 20.

n

ASTRONOMY.

1. Given the Sun's apparent diameter, compute the time the Sun takes in rising.

2. Compute the approximate expression for refraction as given by Brinkley, viz.,

m I l

-

"

stating the hypothesis on which the computation is made, and the meanings of the separate terms.

3. If h be the horizontal parallax of the Moon, v its true zenith distance, and Ρ the parallax in altitude, prove

4. Explain fully the lunar method of finding the longitude, and deduce the formula for clearing the observed angular distance of the Moon and star of the effects of parallax and refraction.

5. From the difference of times of continuance of a known star at both sides of an approximate meridian, find a formula for correcting same. 6. Explain fully the method of finding the Sun's distance by the transit of Venus as actually practised in 1769.

ALGEBRA AND DIFFERENTIAL CALCULUS.

DR. HART.

I. Find the value of the sum of the squares of the reciprocals of the odd numbers from 1 to co.

2. Prove that the equation

x4+4bx3 + 4dx+e=0,

will have two equal roots when

(e-4bd)3 27 (d2+b2e)2,

and that it will have three equal roots when each member of this equation is cypher.

3. Approximate by Sturm's theorem to the roots of the equation

x2+4x3-3x2 - 20x-10=0,

so as to distinguish between any roots that are nearly equal.

4. Transform the equation

x4 + 2x3- 5x2 + 4x − 3=0,

into a cubic whose roots are each the sum of two products of pairs of roots of the given equation.

5. Find the values of 4/1 by the solution of quadratic equations.

6. If the eccentricity of an ellipse be Vg, find when the intercept on a tangent between the point of contact and a perpendicular from the focus is a maximum.

7. Find the asymptotes of the curve

x2y2 + ax (x + y)2 — 3a2y2 — a1 = 0.

8. Find the singular points on the curve

into one in which is the independent variable, supposing = tan 0. 10. Find the value of

1. A beam resting on a smooth horizontal plane, and against a smooth vertical wall, is supported by a string attached to given points in the beam and in the intersection of the wall and plane; find the tension on the string, and show that the equation obtained may be at once deduced from the figure.

a. Find when the expression becomes infinite or negative, and explain these results.

2. A cube placed on a smooth inclined plane is kept in equilibrium by a string attached to the middle point of its highest edge, passing over a pulley, and sustaining a weight; show that this is impossible if the inclination of the plane exceed 45°.

3. If the plane be rough and the string be parallel to it, find the least coefficient of friction which will give equilibrium.

4. A string passes over three smooth pegs in a vertical wall, no two of which are in contact with each other, and sustains at its extremities equal weights; if the pegs be of equal strength, the system will be least likely to break if the vertical angle of the triangle, formed by joining the points at which they are driven in, be 120°.

5. A small heavy ring is strung on a smooth cord whose extremities are attached to fixed points, the length of the cord being greater than the distance between the points; if the ring be laid on a smooth inclined plane, find the condition requisite in order that the position may be one of equilibrium.

6. A material particle is laid on a rough plane inclined to the horizon at the angle of friction; if it be required to drag it down the plane, determine the limits of the direction of the requisite force.

7. Find the position of equilibrium of a beam, one end of which rests against a vertical plane, and the other on the interior surface of a given hemisphere.

8. A uniform chain is suspended from two tacks in the same horizontal line; find the length of the chain so that the strain on the tacks may be a minimum.

DYNAMICS.

1. A material particle is attached by a string to a point in a smooth inclined plane, on which the particle itself rests; if it be projected from its point of rest with a velocity just sufficient to carry it to the highest point to which the string allows it to go, find the time of this motion.

2. If two equal and perfectly elastic spheres be dropped from different heights on the same horizontal plane, determine whether the centre of gravity will ever rise to its original height.

3. A material particle descends a smooth inclined plane under the influence of gravity; in what direction must a second particle be projected at a given instant, with a given velocity, from a given point in the plane, so as to strike the first?

4. A bead strung upon a curve described on a vertical plane is projected down the curve with a given velocity; if the weight of the entire be equal to the sum of the weights of the wire and the bead, the curve will be of the form

y3 = ax2.

5. A rough plane makes an angle of 45° with the horizon; a groove is cut in the plane, making an angle a with the intersection of this plane and the horizontal plane; if a heavy particle be allowed to descend this groove from a given vertical height, find the velocity with which it reaches the horizontal plane.

6. An ellipse is placed with its major axis vertical; find the radius vector by which a particle will descend in the shortest time from the upper focus to the curve.

7. Two balls of elasticity e are projected along a smooth fixed tube in the form of any closed curve, lying in a horizontal plane, from any two points in the tube; supposing u, v, to be the velocities of projection estimated in the same direction, and c to be the length of the tube, find the whole interval of time between the first and the (n+1)th collisions.

8. Two particles are connected by a string, and laid on a smooth horizontal plane at a distance from each other less than the length of the string. One of the particles receives a given impulse; determine the motion which ensues after the tightening of the string.

HEAT.-ELECTRICITY.-MAGNETISM.

1. Two vessels, each filled with a gas of known density and pressure, are made to communicate with each other; find the density and pressure of the mixture when equilibrium is fully established.

2. A vessel filled with air, having a given pressure, is reduced to onehalf its volume, and the new pressure noted; what is the increase of temperature?

3. Calculate by the method of Lavoisier and Laplace the ratio of the specific heat of any given solid substance to that of water.

4. Enunciate and prove experimentally the law of oblique radiation from a surface.

5. Describe the method of Dulong and Petit for obtaining the absolute dilatation of mercury.

6. Explain the principle of the Leyden jar; and show that the condensation of the electric fluid is not unlimited.

7. How is it shown experimentally that free electricity resides only on the surface?

8. Explain the attraction of a conducting body produced by an electrified body.

9. Enunciate the law of the action of an elementary current on a magnetic needle, and deduce from it the law of the action of an indefinite current.

10. Deduce from the same the law of the action of an angular current. 11. State the law of the mutual action of two currents crossing each other, and hence explain the rotation of a horizontal current produced by the action of the earth.

12. Assuming the truth of the electric theory of magnetism, find the position of equilibrium of a magnet acted on by a current.

LOGIC.

DR. TOLEKEN.

1. Locke, in his discussion on Infinity, notices an objection to his account of that idea which would appear to depend on the theory that Space and Time are mere abstractions? Cousin has fallen into a somewhat similar error?

2. Give a summary of Locke's discussion respecting Identity; and state the difficulties which, as he admits, seem to lie against his theory. How does Cousin explain our belief in our personality?

3. Examine Locke's definition and classification of Simple Ideas; and show how the indeterminateness of his language has led to error. Give a summary of his chapter on the names of Simple Ideas.

4. Cousin mentions an important instance in which he says that Locke has not confounded the condition of a principle with the principle itself? He instances two opposite schools which have fallen into error on this point?

5. How does Cousin, from an analysis of the steps by which we acquire knowledge, trace the process through which we attribute reality to our ideas?

6. Give an historical review of the à priori argument for the existence of a God. In the statement of the most important premiss in Locke's argument, his illustration is inapplicable?

7. Give Cousin's review of the different opinions respecting necessary principles, with his distinction between these and contingent principles, and his final classification of the former. To what faculty does Locke appear to attribute our à priori necessary truths?

8. By what analysis does Cousin arrive at the true meaning of the word "Liberty," and how does he explain the fact?