8. (a) If O be the centre of mean distances of the points A, B, C, &c., for the system of multiples a, b, c, &c., prove

Σα Σα 102 = Σ ab AB.

(b). If o be the centre of the inscribed circle, and o', o", o""' the centres of the exscribed circles of a triangle, prove

9. If a a' be the distances of two variable points on different lines from fixed points on those lines, and if

Aaa' + Ba + Ca' + D = 0

where A, B, C, D are constants, prove that the variable points will describe homographic systems.

10. Show how to find two points at which corresponding pairs of points of two homographic rows on different axes subtend equal angles. II. Inscribe a triangle in a circle so that the sides may cut three given arcs in given anharmonic ratios.

12. Two circles and their centres of similitude subtend a pencil in involution at any point?

13. Determine a relation between six of the common tangents of four circles that are touched by a fifth circle.

14. Circles are described on two opposite sides of a square, and tangents are drawn from a point P on either diagonal; prove

(a). That the two chords of contact intersect at a point 0 on the other diagonal at the same distance as P from the centre of the square. (b). That the four tangents are harmonic.

ALGEBRA.

MR. BURNSIDE.

1. The roots of the quartic (a, b, c, d, e) (x, 1)4 = 0, being a, ß, y, d, form the equation whose roots are

substitute a + Bu for x, and determine a, ß, that the resulting equation in u be reciprocal.

3. If be one of the anharmonic ratios of the points on a right line determined by two quadratics

determine an equation connecting U, V, W.

5. Prove that the following determinant is a perfect square, and find its square root :

6. Express Earßayr, in terms of the sums of the powers of a, ß, y, d, &c.

7. The roots of the equation

ax3 + 3bx2 + 3cx + d

being a, ß, y: clear of radicals the equation

= 0,

√λ-a2 + √λ- ß2 + √ λ — y2 = 0,

and express the result in terms of the coefficients of the cubic.

8. Solve the system of equations

10. Approximate as far as three decimal places to one of the roots of the equation

(a). Show how to write the result as a determinant.

12 Find the condition that a binary quartic have a square factor in terms of the coefficients.

TRIGONOMETRY.

MR. WILLIAMSON.

1. Find the value of x, which satisfies the equatiou

tan-1 (x-2)+ tan-1 (x + 2) + tan1 (2x) = tan-1 (4x).

2. Prove the relation

sin 20° sin 40° sin 60° sin 80°

3

16.

3. If A + B√ = 1 = log(√3+ √ −1), find the values of A and B. 4. Find the sum ofn terms of the series sin 2a+ sin2 2a + sin2 3a + &c. 5. Prove the equation

= 2 sin (a + B) sin (a + y) sin (ẞ + y) sin (a – ẞ) sin (ẞ − y) sin (y− a).

6. Two sides a, b, and the contained_angle C of a triangle are given ; if a small error AC exist in the observed value of C, find the corresponding errors in the computed values of the base, the area, and the radius of the circumscribed circle, respectively.

9. In a plane, the locus of a point such that the sum of the square of its distances from a number of fixed points is constant, is a circle; and if the sum of its distances from a system of fixed right lines is constant, the locus is a right line; show that these correspond to the same theorem in spherical geometry, and explain how the locus, in the plane, is different in the two cases.

11. Apply the preceding formula to the calculations of log 2 and log 3 to 8 places of decimals; being given Mo°43429448.

12. In a spherical triangle prove that

and deduce from it the value of tan p by aid of the polar triangle where r and p are the radii of the inscribed and circumscribed circles, respectively.

1. Assuming the earth to be a sphere of 7912 miles diameter, calculate, in square miles, the area of a spherical triangle on its surface, whose spherical excess is 1'.

2. Calculate, to seven decimal places, the values of sin 1°30' and cos 1°30'.

3. The sides of a plane triangle are 264296, 37218.5, 391547, find the lengths of the radii of its inscribed and circumscribed circles.

4. The sides of a spherical triangle are 20°25'12", 30°14′29′′, and 28°18′35′′; calculate the radius of its circumscribing circle.

5. The angles of a triangle are to one another as 2: 35, and the radius of its inscribed circle is 325; calculate its sides.

6. The sides of a spherical triangle, on a sphere of 100 miles radius, are 17, 23, and 27 miles in length; calculate the area of the triangle in square miles.

CLASSICAL SIZARSHIP EXAMINATION.

MR. ABBOTT,

Translate :

1. Beginning, ἔπειτ ̓, ὦ κατάρατε καὶ γραμματοκύφων, κ. τ. λ. Ending, εἴπερ ἄξια ἐκείνων πράττειν οἴεσθε χρῆναι.

DEMOSTHENES, De Cor., xviii. 209, 210.

2. Beginning, ἵνα δὲ μηδὲ τοῦτ ̓ ἀγνοῆτε, κ. τ. λ. Ending, Παναθήναια φήσας ἀποπέμψειν.

Id., De Falsa Leg., xiv. 167, 168,

3. Beginning, Καὶ ἐμοὶ οὖν, ἔφη, συμβουλεύεις, κ.τ.λ. Ending, εἰς ταῦτα ἐμπίπτοντες συμποδίζωνται.

XENOPHON, Memor., lib. iii. cap. 11.

4. Beginning, Μετὰ τοῦτο Αἰνιᾶνες καὶ Μάγνητες, κ. τ. λ. Ending, πάντα ἐν ῥυθμῷ ἐποίει πρὸτὸν αὐλόν.

Id., Anab., lib. v. cap. 9.

HOMER AND EURIPIDES.

MR. GRAY.

Translate the following passages:—

1. Beginning, Τὸν τόθ' ὑπ ̓ ὀφρύος οὗτα κατ ̓ ὀφθαλμοῖο θέμεθλα, κ.τ.λ. Ending, Πέφραδέ τε Τρώεσσι, καὶ εὐχόμενος ἔπος ηὔδα· HOм. Il., xiv. 492–500.

/ 2. Beginning, 'Αλλ' ὥστ ̓ ἀμφ' οὔροισι δύ' ανέρε δηριάασθον, κ. τ. λ. Ending, Ισάζουσ', ἵνα παισὶν ἀεικέα μισθὸν ἄρηται.

Id., xii. 421-435.

3. Beginning, Ιδομενεὺς δ' Ερύμαντα κατὰ στόμα νηλέϊ χαλκῷ, κ.τ.λ. Ending, Δυσκελάδου μνήσαντο, λάθοντο δὲ θούριδος ἀλκῆς. Hom. Il., xvi. 345-357.

4. Beginning, "Η, καὶ ἀπ' ἀκμοθέτοιο πέλωρ αἴητον ἀνέστη, κ. τ. λ. Ending, "Εν τ' ὥρα οἱ φῦ χειρὶ, ἔπος τ' ἔφατ', ἔκ τ' ὀνόμαζε Id., xviii. 410-423.

5. Beginning, Ὣς ἔφαθ'· οἱ δ ̓ ἄρα πατρὸς ὑποδδείσαντες ὁμοκλὴν, κ.τ.λ. Ending, Εξείης κατέδησαν, ὑπὸ γλωχῖνα δ ̓ ἔκαμψαν.

İd., xxiv. 265-274.

6. Beginning, οὔπω δέδωκας, ἀλλ ̓ ἴσως δώσεις δίκην· κ. τ. λ. Ending, ὀλέθριον, ὀλέθριον κακόν.

EURIP. Hec., 1002–1007.

7. Beginning, ΜΕ. πατρὸς δὲ δὴ τί σ' ὠφελεῖ τιμωρία; κ.τ.λ. Ending, ΟΡ. οὗ γ ̓ οὐ μετῆν μοι· διὰ τριῶν δ' ἀπόλλυμαι.

Id., Orecl., 415-424.

8. Beginning, ὅταν γὰρ ἡβᾷ δῆμος, εἰς ὀργὴν πεσών, κ. τ. λ. Ending, ἔβαψεν, ἔστη δ' αὖθις, ἣν χαλᾷ πόδα.

Ibid., 685-696.

9. Beginning, Καπανεὺς δὲ πῶς εἴποιμ ̓ ἂν ὡς ἐμαίνετο ; κ.τ.λ. Ending, εἱλίσσετ' ἐς γῆν δ' ἔμπυρος πίπτει νεκρός.

Id., Phæniss., 1172-1186.

10. Beginning, δεινὰ τυράννων λήματα, καί πως, κ. τ. λ. Ending, δαίμων οἴκοις ἀπέδωκεν.

Id., Med,, 120-131.

MR. ABBOTT.

:

Translate the following passages:

1. Beginning, Equidem existimo nullum tempus esse.... Ending, eos omni populi Romani significatione decorari ? CICERO, Pro Sestio, lix. 125-127.

2. Beginning, Sed quaestiones urgent Milonem,.... Ending, dici potest integrius, quid incorruptius ?

Id., Pro Milone, xix. 5, 59.

3. Beginning, Ego quidem nulli vestrum deero, ...
Ending, adversus hos tentare magis, quam tueri, libertatem.

Livr, lib. vi. cap. 18.

4. Beginning, Siccitate eo anno plurimum laboratum est: Ending, insolita piacula pacis Deûm exposcendae.

....

Id., lib. iv. cap. 30.

MR. TYRRELL.

Translate the following passages :

1. Beginning, Gnavus mane forum et vespertinus pete tectum,...

Ending, Ne fueris hic tu.

HORACE, Ep. i. 6.