Sil. There, hold.

I will not look upon your master's lines:

I know, they are stuff'd with protestations,

And full of new-found oaths; which he will break
As easily as I do tear his paper.

Jul. Madam, he sends your ladyship this ring.
Sil. The more shame for him that he sends it me;
For, I have heard him say a thousand times,
His Julia gave it him at his departure:
Though his false finger hath profaned the ring,
Mine shall not do his Julia so much wrong.

SHAKSPEARE.

SENIOR FRESHMEN.

Mathematics.

A.

MR. W. ROBERTS.

1. Find the single equation of the two lines passing through the origin and the points of intersection of the right line 7x+3y+13=0, with the conic 3x2+5xy+7y2+8x + 2y +10=0.

=

2. Expressing the co-ordinates x'y' of a point on an ellipse by the equations x': a sin o, y' b cos p, and denoting by w the angle which a radius vector from one of the foci to x'y' makes with the axis major; find the value of cos w in terms of the eccentricity e, and of sin p.

3. Reduce to its simplest value the expression

cos (a – ẞ) sin (ẞ − y) sin (y − a) + cos (ß − y) sin (y — a) sin (a — ẞ) +cos (ya) sin (a — ß) sin (ẞ − y) - cos (a−ẞ) cos (ẞ − y) cos (y − a)

DR. SHAW.

7. Explain the principle on which the following series of reductions depends:

and show that the value of the determinant is - 15.

8. Give a construction for finding the five points in which a given pentagon is touched by its inscribed ellipse, and show how this construction is to be modified for the case of the four points in which a quadrilateral is touched by the exscribed parabola.

9. Show that the two foci of a hyperbola and the two points in which the asymptotes are cut by any tangent lie on a circle and hence prove that the product of the two focal radii = square of half the intercepted tangent.

10. Given four points on a parabola; show that the direction of the diameters is known, and hence construct the curve by points.

11. Calculate to four places of decimals the positive roots lying between 5 and 6 of the equation

x3 + 2x2-23x-70=0

12. Find the value of x which gives a maximum value for y, the two variables being connected by the equation

y2 - 2mxy + x2 = a2

and prove that such value is a maximum, and not a minimum.

MR. TARLETON.

13. Differentiate tan-1 {(1 + x2)- x}, eax cos rx,

x sin x

14. Calculate by Maclaurin's theorem the first four terms of the development of tan x in terms of x.

15. Find the condition that a conic given by the general equation should touch a given right line.

16. If from the point of intersection of common tangents to two conics lines be drawn cutting the conics, the anharmonic ratio of four of the points of intersection with one conic will be the same as that of the corresponding points on the other?

17. Calculate Σ.

α

for a biquadratic, a and ß being any two of the roots. β

18. Given the base of a spherical triangle, and the lengths of the arcs bisecting the external and internal vertical angles; construct the triangle.

B.

MR. W. ROBERTS.

1. Denoting the roots of the cubic equation 3 + px + q : =o by a, ẞ, y, it is required to form the equation the roots of which are aẞ+ y, By+a, ay + B.

2. Express trigonometrically the roots of the following cubic equation : 23-3(2 + 3 tan 20) x2 + 3 (3 + 2 tan 20) x - tan 20 = 0.

3. Two equilateral hyperbolas, concentric with a given ellipse, and cutting each other orthogonally, touch the ellipse respectively at the points P, Q; show that the envelope of a third equilateral hyperbola, concentric with the former, and passing through the points P, Q, is an ellipse confocal with the given one.

5. Find the value of the following determinant:

cos (a

a)

(a-B), cos (B-y), cos
cos (a + B), cos (B+y), COS (y+a)
sin(a+B), sin (B+y), sin (y+a)

2 sin(a-ẞ) sin(ẞ-y) sin (y-a)

7. Show that if an ellipse pass through the centre of a hyperbola, and have its own centre on the hyperbola, and its foci on the hyperbola's asymptotes

a. The axes of each curve are respectively tangent and normal to the other; and

b. The two axes which are also normals are equal to each other. 8. (a). From the locus of a parabola a right line is drawn to the intersection of two tangents, also right lines to the two points of contact; show that the first right line makes with each tangent an angle equal to the angle which the other tangent makes with the local vector; and hence (b). Show that the locus of the foci of the parabola which touch the three sides of a fixed triangle is the circle circumscribed to the triangle. 9. The roots of the biquadratic

x2 + px3 + qx2 + rx+8=0

being a, b, c, d, show that the cubic whose roots are ab+ cd, ad + bc, ac + bd, is

y3 — qy2 + (pr−4s) y − (r2 — 4qs + p2s)=0

10. In the lemniscate r2 = a2 cos 20 show that the angle between the tangent and radius vector equals

π

+20.

may be summed by multiplying each term by different numerical expressions for unity; by this means, or otherwise, show that the sum is

12. Find the locus of the intersection of the perpendiculars of a triangle inscribed in one conic, and circumscribed about another.

13. Find the locus of the vertex of a triangle whose base angles move along one conic, and whose sides touch another.

14. Find the envelope of a conic which is circumscribed to a given triangle, and with respect to which two given lines are conjugate.

15. If x3 +px2 + qx + r = o be a cubic equation whose roots are a, B, y, show that

dp dq dr

da da da

dp dq dr

ав авар
dp dq dr

dy dy dy

=

where A is the product of the squares of the differences of the roots.

I. Beginning, Πλάζομαι ὧδ ̓, ἐπεὶ οὔ μοι ἐπ ̓ ὄμμασι νήδυμος υπο

νος, κ. τ. λ.

Ending, Μή πως καὶ διὰ νύκτα μενοινήσωσι μάχεσθαι.

Iliad, lib. x. 91–101.

2. Beginning, Τὸν τόθ' ὑπ ̓ Ἰδομενῆϊ Ποσειδάων ἐδάμασσε, κ. τ. λ. Ending, Τρεῖς ἑνὸς ἀντὶ πεφάσθαι, ἐπεὶ σύ περ εὔχεαι οὕτως ; Ibid., lib. xiii. 434–447.

3. Beginning, Εν δὲ τίθει σταφυλῇσι μέγα βρίθουσαν ἀλωήν, κ. τ. λ. Ending, Μολπῇ τ' ἰϋγμῷ τε ποσὶ σκαίροντες ἕποντο. Ibid., lib. xviii. 561-572.

4. Beginning, Ος δὲ με κέρδεα εἰδῇ, ἐλαύνων ἥσσονας ἵππους, κ. τ. λ. Ending, Κυκλου ποιητοῖο· λίθου δ ̓ ἀλέασθαι ἐπαυρεῖν· Iliad, lib. xxiii. 322-340.

5. Beginning, τόφρα δ ̓ ἄρ ̓ Αρήτη ξείνῳ περικαλλέα χηλόν, κ. τ. λ. Ending, τόφρα δὲ οἱ κομιδή γε, θεῷ ὣς, ἔμπεδος ήεν.

Odyssey, lib. viii. 438-453.

1. Write notes on the preceding passages when required.

2.

Supply the digamma in the second and fourth passages.

3. What is the force of the Homeric reduplicated aorists ?

4. What is peculiar in the Homeric use of φράζω, μείρω, ἕπομαι ? 5. Account for the formation of εἶπον, εἱπόμην, εἶχον, ἔστειλα, ἕζομαι.

6. State what you know of the etymological relations of rog, oooa, μέροψ, βρότος, ἄεθλον, ἀπούρας, τέρμα, ήμι.

7. Write a short essay on the Homeric question.

8. Discuss briefly the treatment of the Grecian myths.

VIRGIL AND OVID.

MR. FERRAR.

Translate the following passages into English Prose :1. Beginning, "En, perfecta tibi bello discordia tristi;. Ending, Invisum numen, terras cœlumque levabat.

VIRGIL, Eneid, lib. vii. 545-571.

2. Beginning, Nec Turnum segnis retinet mora: sed rapit acer.. Ending, Alma Venus.

Ibid., lib. x. 308-332.

3. Beginning, Cerberus hæc ingens latratu regna trifauci. Ending, Conciliumque vocat, vitasque et crimina discit.

Ibid., lib. vi. 417-433

4. Beginning, Talibus atque aliis postquam convivia dictis...... Ending, Nominat, et cineri materno ducere pompam.

OVID, Metam., lib. xiii. fab. 5.

1. Trace the steps by which Augustus attained the supreme power. 2. What were the chief sources of the Roman revenue in his reign? 3. Julius Cæsar represented his age in his qualities and defects? 4. What influence had the Etruscan discipline on the Roman character? 5. How did Augustus try to raise the estimation of the Roman citizenship?

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