2. Find the area of a triangle in terms of the co-ordinates of its vertices, the axes being rectangular.

Find the area of the triangle, the equations of whose sides are respectively :

x cos ay sin a=p, x cos a'+y sin a'=p',

x cos a"+y sin a′′=p′′.

3. Being given a finite arc of a parabola, determine its vertex, focus, and principal parameter.

Determine also the centre and axes of an ellipse, of which a finite arc is given.

4. Find the polar equation of the ellipse or hyperbola, the focus being pole.

Prove that any focal chord is a third proportional to the transverse axis and the parallel diameter.

5. Find the locus of the centre of a conic passing through four given points. Find also the locus of the centre of a conic touching four given lines.

7. The roots of the equation ax3 + px2 + qx + r = 0 are a,, y; find the equation whose roots are

βγ-α? γα - β' αβ- γ.

8. Explain any method of solving a biquadratic. Solve x42x3 + 4x − 2 = 0.

9. Apply the Integral Calculus to find

(1.) The area of the sector of an ellipse bounded by two focal radii and the curve.

(2.) The volume of the solid generated by the revolution of this sector round the major axis.

Professor NIVEN, M.A.

11. State and prove Gregorie's Series for expanding tan-1 in powers of x.

12. Investigate the solution of right-angled spherical triangles, in which two sides are given.

If great circles AO, BO, CO bisect the angles of a triangle ABC and meet the opposite sides in D, E, F ; show that sin AO sin a

sin DO sin 26 + sin 2c + 2 cos a sin 6 sin c 13. Investigate the values of

xn

14. Tangents are drawn to a parabola; find that for which the intercept between the axis and latus-rectum is a minimum.

15. Define the curvature of a plane curve and find an expression for it in terms of rectangular co-ordinates.

The two circles which pass through any point of a lemniscate, have diameters coinciding with the tangent there, and pass through a focus of the curve, are equal. Show also that the chord of curvature of the double pointed lemniscate through the centre is one-third of the corresponding diameter of the curve.

MATHEMATICAL PHYSICS.

7th October, 1874.-Afternoon.

Professor ENGLAND, M.A.

1. Three pressures, P, Q, and R, of 30 lbs., 40 lbs., and 50 lbs., respectively, act at a point; Q makes an angle of 30° with P, and R an angle of 120° with Q, calculate their resultant.

2. Find the line of quickest descent from a point within a circle to its circumference, the circle being in a vertical plane.

3. A body weighs 45 grains in air of density 0012, and 40 grains in alcohol of density 0.8, what is the specific gravity of the body?

4. Investigate a formula for the focal length of two lenses in contact.

5. Given the sun's declination and latitude of a place, show how to find the interval from sunrise to sunset.

Professor CURTis, ll.d.

6. Determine the condition of equilibrium when a heavy body is placed on an inclined plane and is acted on by a force inclined to the plane at a given angle(a.) If the plane be smooth;

(b.) If the plane be rough.

7. A body is projected horizontally with a certain velocity v, after a certain interval of time the spaces moved over horizontally and vertically are to one another as m:n; find the interval of time and the spaces moved over.

8. If a triangular area be immersed in a homogeneous liquid, the vertex being in the surface of the liquid, prove that the distance of the centre of pressure from the base is the fourth part of the altitude.

9. Define the centre of a lens, and give the construction for determining it.

10. How would you determine by calculation and by means of a celestial globe the interval between day-break and sunrise on a given day at a given place?

Professor EVERETT, D.C.L.

11. Prove that the algebraic sum of the moments of two forces about a point in their plane is equal to the moment of their resultant.

12. A uniform cube has a spherical hollow excavated in it, of diameter equal to half the edge of the cube; and the centre of the hollow is midway between the centre of the cube and the centre of one face. Find the centre of gravity of the remaining solid.

13. Investigate the formula for "centrifugal force"; and show that this name is objectionable.

14. Define sidereal noon, apparent solar noon, and mean solar noon; and compute the ratio of a sidereal

hour to a mean solar hour.

15. Trace the pencil of rays by which the eye (regarded as a point) sees the image of an image of an object by means of two plane mirrors (arranged as you please).

FIRST EXAMINATION IN ARTS.

GREEK.

October, 1874.

Professor MacDOUALL, LL.D.

Translate the following passage from XENOPHON'S Memoirs of Socrates:

καὶ ἔλεγε μὲν ὡς τὸ πολύ, τοῖς δὲ βουλομένοις ἐξῆν ἀκούειν. οὐδεὶς δὲ πώποτε Σωκράτους οὐδὲν ἀσεβὶς οὐδὲ ἀνόσιον οὔτε πράττοντος εἶδεν οὔτε λέγοντος ἤκουσεν. οὐδὲ γὰρ περὶ τῆς τῶν πάντων φύσεως ᾗπερ τῶν ἄλλων οἱ πλεῖστοι-διελέγετο, σκοπῶν ὅπως ὁ καλούμενος ὑπὸ τῶν σοφιστῶν ὁ κόσμος ̓ ἔχει καὶ τίσιν ἀνάγκαις ἕκαστα γίγνεται τῶν οὐρανίων, ἀλλὰ καὶ τοὺς φροντίζοντας τὰ τοιαῦτα μωραίνοντας ἀπεδείκνυει καὶ πρῶτον μὲν αὐτῶν ἐσκόπει, πότερά ποτε νομίσαντες ἱκανῶς ἤδη τἀνθρώπειαν εἰδέναι ἔρχονται ἐπὶ τὸ περὶ τῶν τοιούτων φροντίζειν, ἢ τὰ μὲν ἀνθρώπεια παρέντεςὶ τὰ δαιμόνια δὲ σκοποῦντες ἡγοῦνται τὰ προσήκοντα πράττειν. ἐθαύμαζε δ' εἰ μὴ φανερὸν αὐτοῖς ἐστιν ὅτι ταῦτα οὐ δυνατόν ἐστιν ἀνθρώποις εὑρεῖν· ἐπεὶ καὶ τοὺς μέγιστον φρονοῦντας ἐπὶ τῷ περὶ τούτων λέγειν οὐ ταὐτὰ δοξάζειν ἀλλήλοις, ἀλλὰ τοῖς μαινομένοις ὁμοίως διακεῖσθαιὶ πρὸς ἀλλήλους. τῶν τε γὰρ μαινομένων τοὺς μὲν οὐδὲ τὰ δεινὰ δεδιέναι, τοὺς δὲ καὶ τὰ μὴ φοβερὰ φοβεῖσθαι, καὶ τοῖς μὲν οὐδ ̓ ἐν ὄχλῳ δοκεῖν αἰσχρὸν εἶναι λέγειν ἢ ποιεῖν ὁτιοῦν, τοῖς δὲ οὐδ ̓ ἐξιτητέον εἰς ἀνθρώπους εἶναι δοκεῖν, καὶ τοὺς μὲν οὔθ ̓ ἱερὸν οὔτε βωμὸν οὔτ ̓ ἄλλο τῶν θείων οὐδὲν τιμᾶν, τοὺς δὲ καὶ λίθους καὶ ξύλα τὰ τυχόντας καὶ θηρία σέβεσθαι· τῶν τε περὶ τῆς τῶν πάντων φύσεως μεριμνώντων τοῖς μὲν δοκεῖν ἓν μόνον τὸ ὂν εἶναι, τοῖς δ ̓ ἄπειρα· τὸ πλῆθος, καὶ τοῖς μὲν ἀεὶ κινεῖσθαι πάντα, τοῖς δ ̓ οὐδὲν ἄν ποτε κινηθῆναι, καὶ τοῖς μὲν πάντα γίγνεσθαί τε καὶ ἀπόλλυσθαι, τοῖς δὲ οὔτ ̓ ἂν γενέσθαι ποτὲ οὐδὲν οὔτ ̓ ἀπολέσθαι.

1. Parse fully and accurately every word to which the figure 1 is attached.

2. Derive or decompound every word to which the figure 2 is attached.

3, 4. Explain the usage of the genitive or the dative case for every word numbered 3, and, in regard to άπειρα, the usage of the neuter plural.

Professor D'ARCY THOMPSON, M.A.

EURIPIDES-Medea.

Translate the following passages:—
(α.) ΚΡ. ἥκιστα τοὐμὸν λῆμ ̓ ἔφυ τυραννικὸν,
αἰδούμενος δὲ πολλὰ δὴ διέφθορα·

καὶ νῦν ὁρῶ μὲν ἐξαμαρτάνων, γύναι,
ὅμως δὲ τεύξει τοῦδε· προυννέπω δέ σοι,
εἴ σ ̓ ἡ 'πιοῦσα λαμπὰς ὄψεται θεοῦ
καὶ παῖδας ἐντὸς τῆσδε τερμόνων χθονός,
θανεῖ· λέλεκται μῦθος ἀψευδὴς ὅδε.

νῦν δ', εἰ μένειν δεῖ, μίμν' ἔθ' ἡμέραν μίαν

οὐ γάρ τι δράσεις δεινὸν ὧν φόβος μ' έχει.-(348-356.)

(6.) ΜΗ. ή πολλὰ πολλοῖς εἰμὶ διάφορος βροτῶν.

ἐμοὶ γὰρ ὅστις ἄδικος ὢν σοφὸς λέγειν
πέφυκε, πλείστην ζημίαν ὀφλισκάνει
γλώσσῃ γὰρ αὐχῶν τάδικ ̓ εὖ περιστελεῖν,
τολμᾷ πανουργεῖν· ἔστι δ' οὐκ ἄγαν σοφός.
ὣς καὶ σὺ μὴ νῦν εἰς ἔμ' ευσχήμων γένη .
λέγειν τε δεινός· ἓν γὰρ ἐκτενεῖ σ' ἔπος.

χρὴν σ', εἴπερ ἦσθα μὴ κακός, πείσαντά με γαμεῖν γάμον τόνδ', ἀλλὰ μὴ σιγῇ φίλων.—(579-587.) (c.) ΜΗ. συλλήψομαι δὲ τοῦδέ σοι κἀγὼ πόνου·

πέμψω γὰρ αὐτῷ δῶρ ̓ ἃ καλλιστεύεται
τῶν νῦν ἐν ἀνθρώποισιν, οἶδ ̓ ἐγώ, πολύ,
λεπτόν τε πέπλον καὶ πλόκον χρυσήλατον
παῖδας φέροντας. ἀλλ' ὅσον τάχος χρεων
κόσμον κομίζειν δεῦρο προσπόλων τινά.
εὐδαιμονήσει δ ̓ οὐχ ἓν ἀλλὰ μυρία,
ἀνδρός τ' ἀρίστου σοῦ τυχοῦσ ̓ ὁμευνέτου
κεκτημένη τε κόσμον ὅν ποθ' "Ηλιος
πατρὸς πατὴρ δίδωσιν ἐκγόνοισιν οἷς.

λάζυσθε φερνὰς τάσδε, παῖδες, εἰς χέρας,

καὶ τῇ τυράννῳ μακαρίᾳ νύμφη δότε

φέροντες· οὗτοι δῶρα μεμπτὰ δέξεται. (946-958.)

(d.) Parse fully the following words :

(1.) From the first passage above: διέφθορα—τεύξει

θανεῖ.

(2.) From the second: διάφορος—όφλισκάνει—λέγειν

—ἐκτενεῖ.