MR. TOWNSEND.

7. Given, of a spherical triangle, two sides and the contained angle ; state and prove the logarithmic formula for the remaining side and angles.

8. State and prove the formula for the direct determination of the sum of the mth powers of the roots of an equation of any order n.

9. Determine the value of the function

10. Express, in terms of the semiaxes, the area of the greatest triangle that could be inscribed in a given ellipse.

II. Find the equation of the envelope of

when μ and v are connected by the relation μ + v = C.

12. When two conics have double contact, show geometrically that a variable tangent to either divides the other homographically.

MR. LESLIE.

13. If a, b, c, d denote four points on a sphere, prove that

cos ac cos bd - cos ad cos besin ab sin cd cos ab^cd.

14. From a point a on the great circle oa a perpendicular ab is dropped n the great circle ob; find when the difference of oa and ob is a maxi

mum.

15. Find the values of

(a2 − x2) + ( a − x) when x = a, and (1+nx)* when x=0.
a3 − x3) § + ( a − x)§

16. Show that the expression for the radius of curvature in polar coordinates may be obtained geometrically; and apply it to find the radius of curvature of the curve r2 a2 cos 20.

=

17. Prove that the intersection of perpendiculars of a triangle circumscribing a parabola lies on the directrix;`and state the reciprocal theorem.

18. Apply the method of projections to prove that if two triangles be self-conjugate with regard to a conic, their six vertices lie on a conic.

B.

STATICS.

MR. STUBBS.

1. Find the resultant of two pairs of forces in planes not parallel. 2. Upon what do the stability and sensibility of a balance depend? and prove that, when one is increased, the other is diminished.

where W is the weight of the cord between the given point and the lowest point, and T and ▲ the tensions at these points, respectively.

4. A ladder is placed against a rough wall, and the lower end rests on a rough horizontal plane; determine the pressures against the wall and plane, and the limiting angle at which it may be inclined to the horizon, being given the coefficients of friction against the wall and plane.

5. A beam rests on two smooth inclined planes whose inclinations are i and '; find the pressures against the planes, and the angle made by the beam with the horizon when in equilibrium.

MR. TOWNSEND.

6. If any number of forces act in a plane, show that the sum of their moments with respect to any point in the plane is equal to the moment of their resultant with respect to the same point in the plane.

7. If forces proportional to the several lengths act perpendicularly, either from within or without, at the middle points of the several sides of any plane polygon, show that they will always equilibrate each other. 8. Investigate, by any method, the condition that a system of forces, acting along any lines of direction, should have a single resultant.

9. A flexible cord, suspended from two fixed points, sustains along its entire length a load whose horizontal distribution is uniform; determine, geometrically, the form it will assume in equilibrium.

10. A material point, acted on by any number of forces directed to fixed centres, and varying as its distances from the several centres, rests in equilibrium at every point of a smooth surface; determine, by the principle of virtual velocities, the form of the surface.

MR. LESLIE.

11. If three forces acting on a point are represented by the lines joining it to the vertices of a triangle, prove that their resultant will be represented by three times the line joining the point to the centre of gravity of the triangle.

12. A weight is placed on a triangular table; find the pressures on the legs (a) when it is placed at the centre of the inscribed, and (b) at the centre of the circumscribed circle.

13. A string of which the extremities are fixed has weights attached to it at several points; prove that the weights are proportional to the sum of the cotangents of the angles made with the vertical by the adjacent parts of the string; and hence deduce the equation of the catenary.

14. Find the condition of equilibrium in the screw, taking friction into account.

15. Being given the curve formed by a string whose extremities are attached to fixed points, show how to find the law of its density; and apply the method to the case of the curve being a semicircle whose diameter is horizontal.

C.

MR. STUBBS.

1. If a circle whose radius is a rolls upon another circle whose radius is 2a, any point upon the circumference of the first will describe a curve whose equation is

4(x2 + y2 — a2)3 = 27a4y2.

find the co-ordinates of the centre of curvature, and the evolute.

3. A triangle is inscribed in the conic x2 + y2 = 22, and two sides touch the conic ax + by2 = c22; find the envelope of the third side.

4. In an ellipse the radius of curvature of the evolute at a point corresponding to a given point on the curve 3 y. coto, where y is the radius of curvature of the ellipse at that point, and the angle which the tangent makes with the central radius vector.

MR. TOWNSEND.

5. If (a + bx + cx2 + dx3 + &c.)" = A + Bx + Сx2 + Dx3 + &c., determine by differentiation the values of A, B, C, D, &c., in terms of a, b, c, d, &c. 6. Investigate the equation of the envelope of

(λA)m + (μB)m + (vC)m = 0,

where λ, μ, v are connected by the relation

(a) + (ab) + (v) = 0,

in which a, b, c are constants.

7. State and prove the condition that the three pairs of lines whose equations are

ax2+2bxy + cy2=0, a2x2+2b′xy+c′y2=0, a′′x2+2b′′xy + c′′y2 = 0, should form a pencil in involution.

8. When two triangles either inscribed or exscribed to the same conic are in perspective, show, geometrically, that every line through the centre of perspective intersects with their six sides at six points in involution, and that every point on the axis of perspective connects with their six vertices by six rays in involution.

MR. LESLIE.

9. Prove that the equation of the lines joining a'B'y' to the intersections of two conics U, V, may be written in the form

(UV' + U'V − 2PQ)2 = 4 (UU′ − P2) (VV' — Q2).

What loci are represented, respectively, by the three factors of this equation?

10. Prove that the centre of the circle circumscribing every triangle self-conjugate with regard to a parabola lies on the directrix.

11. Explain the method of finding the asymptotes of a curve, and apply it to find the asymptote of 23+ y3 — 3mxy = 0.

12. If ax2 + by2 + cz2 + 2fyz + 2gxz +2hxy = (lx + my + nz)2 + (l'x+ m'y + n'z)2 + (1′′ x + m′′y+n"z)2,

prove that

abc + 2fgh — af2 – bg2 — ch2 = { 1 (m'n" — m′′n') + m (n'l′′ — n”l′)
+ n (l'm" — l′′m') }2.

Logics.

MR. BARLOW.

1. What are the Forms of Judgment, according to Kant? What, aecording to Mansel? State fully Mansel's reasons for rejecting Kant's classification.

2. How does Mansel decide the disputed question of the relation of Modals to Logic?

3. “The Principle of Causality, as far as its necessity is concerned, may be referred to an intermediate place between the axioms of mathematics and the generalizations of physical science." Explain this statement.

4. "Nihil est in intellectu, quod non fuerit in sensu." Explain the meaning of Leibnitz's addition to this axiom. On what grounds has the axiom been attributed to Aristotle? And why erroneously, according to Mansel? A nearer approach to the sensational tabula rasa may be found in a doctrine attributed to the Stoics?

5. What is Mansel's objection to Dr. Whewell's attempt to establish Mechanics as an à priori science upon the idea of force?

6. Logical Necessity is dependent on one negative condition, and on three positive laws?

7. The omission of hypothetical syllogisms has been blamed as a defect in Aristotle's Organon. How does Mansel justify Aristotle? The true character of hypothetical reasoning is lost sight of in the examples commonly selected by logicians?

8. Show that the Kantian distinction between Syllogisms of the Understanding and Syllogisms of the Reason is, both psychologically and logically, untenable.

9. Give some account of the "doctrina de adminiculis memoriæ." 10. Explain the nature of the “elenchi hermeniæ." Why are they so called? What did Bacon mean by the "transcendentes"?

DR. MALET.

In

1. How does M. de Biran account for the principle of Causality? what respects was he indebted to Locke? State fully Cousin's objections to his theory, and Cousin's own account of the principle.

2. Write, at some length, Cousin's criticism on Locke's theory of general Ideas.

3. What are his objections to Locke's chapter on Moral Relations? State your own opinion of the justice of his criticism.

4. How does Locke attempt to show that Morality is capable of demonstration? What difficulties have caused the opposite opinion? and what remedies does he suggest for these difficulties?

5. How do demonstrations in Geometry become general, according to Locke?

STEWART.-WHATELY.

DR. WEBB.

1. State the Nature and Object of the Philosophy of the Human Mind, and point out briefly its Utility.

2. State the difference of opinion between Bacon and Descartes as to the true method of Physical inquiry.

3. The followers of Bacon have shown an undue contempt of Hypothetical Theories? Bacon explicitly recognises the value of Hypothesis as an aid to the Inductive Philosophy.

4. D'Alembert in the Encyclopédie points out a curious Paradox with respect to colour? Condillac attempts to explain this paradox, and gives an Illustration? Is this Illustration philosophically correct?

5. State the difference between the effect of Association and that of Imagination in heightening the pleasure or pain received from External Objects.

6. How does Whately attempt to explain away the different cases in which we seem to reason from an Individual Instance without the employment of a Universal Premiss ?

7. Explain the different methods of computing the value of Probable Arguments.

Classics.

HOMER.

MR. POOLE.

:

Translate the following passages into English Prose :

I. Beginning, "Εκτορα δ ̓ ἐν πεδίῳ ἴδε κείμενον, ἀμφι δ' ἑταῖροι, κ. τ. λ. Ending, Καί μιν ἔπειτα Κόωνδ ̓ εὐναιομένην ἀπένεικας.

Iliad, lib. xv. 9-28.

2. Beginning, 'Αλλ' ὅτε δὴ πόρον ίξον ἐϋρρεῖος ποταμοῖο, κ. τ. λ. Ending, " Αορι θεινομένων, ἐρυθαίνετο δ' αἵματι ὕδωρ.

Ibid., lib. xxi. 1–21,