8. In a single throw with two dice, show that the number which there is the greatest chance of making is 6, but that the most frequent throw is 7. 9. In any triangle, prove the following equation: 1 − sin2 14 — sin2 B – sin2 0 – 2 sin ▲ sin †B sin }C=0. 10. Two ships are sailing uniformly in parallel directions, and a person in one of them observes the bearing of the other to be a from the North; p hours afterwards its bearing was B°; and q hours after that it was y°. Prove that the course of the vessels is 0° from the North, where O is given by the equation II. A common tangent to two circles is cut harmonically by any other circle passing through their points of intersection. 12. Determine the locus of a point such that the tangents t1, t2, tз to three circles may be connected by the relation at12 + bt22 = (a + b) t32, where a, b, c are abitrary constants. 13. Express the diagonals of a quadrilateral inscribed in a circle in terms of the sides. 2. Find the nth term, and the sum of an infinite number of terms of the series assuming x < 1. 1+ 3x + 13x2 + 45x3 + 145x4 + &c. 3. Find the expansion for az in a series of ascending powers of x, by Landen's method. 4. Find the roots of the cubic x3 + 3px + 2q = 0; and exhibit the form of the three roots when q2 + p3 is positive. 8. Assuming, as the results of observation, that ten persons will die in the next ten years out of every 62, whose present age is 30, 25, what are the odds against a person at the age of 30 living for 40 years? 9. If all the arcs which have the same sine be divided into n equal parts, show that, if n be even, there are an arcs having independent sines, and that if n be odd, the number having independent sines is n. 10. Show that the logarithm of any number can be found from the logarithms of three numbers which precede it and the series 11. Prove that the three values of x got from the equation (x2+s2) (x − r) = 4Rx2 will be the three exscribed radii of the triangle whose sides are a, b, c, where 28 = a + b + c, sin a. cos ß + sin 2a . cos 3ß + &c. + sin na . cos (2n − 1) B. MR. BURNSIDE. 13. Given cos a = cos a. cos 0, a + B = c, prove that sin c. sin20 = 1 - cos2a - cos2b - cos3c + 2 cos a cos b cos c. find the value of tan 10. tan ʼn in terms of p and q. D. MR. WILLIAMSON. 1. A, B, C, D are four points taken in order on a right line: if ◊ be the angle of intersection of semicircles described on AC and BD, prove that 2. Solve the following problem, by the method of inversion, or otherwise:-Through two given points to draw a circle so as to be divided harmonically by two fixed right lines. 3. A right line intersects two given circles in the points A, B, and C, D, respectively; if the ratio of the chords AB, CD be given, find the locus of the intersection of tangents at the points 4 and D. Show that the locus is the same whatever pair of tangents to the circles be taken. 4. If three coaxal circles touch the sides of a triangle in three points in a right line, prove that their centres, with the centres of the three coaxal circles drawn through the vertices of the triangle, form a system of points in involution. DR. TRAILL. 5. Show that a coaxal system of circles can be generated, by means of a variable circle which cuts two lines of constant length harmonically; and determine the position of the radical axis, and of the limiting points of the coaxal system. 6. Prove that the difference of the sides of a triangle is a mean proportional between the difference of the segments of the base made by the perpendicular from the vertex, and the difference of the segments of the base made by the bisector of the vertical angle. 7. Given any number of coaxal circles, prove that the polar of any fixed point with regard to any of the circles passes through another fixed point. 8. If you have a circle, and a triangle self-reciprocal with respect to the circle, prove that an infinite number of quadrilaterals can be inscribed in the circle, whose sides and diagonals would each pass through a vertex of the triangle. MR. BURNSIDE. 9. Through a given point draw a line, intersecting two given circles so that the point shall be a double point of the involution determined by two pairs of intersections. 10. Determine the locus of the centre of inversion so that two circles become two equal circles after inversion. 11. The circumscribing circle of a triangle intersects the three exscribed circles at the angles 01, 02, 03, prove that sin201 sin2102 sin2 103 = 12. Denoting by π', π' the perpendiculars from a fixed point O, on the tangents to a circle intersecting in P, and by the perpendicular from O on the polar of P; if π'π" - π? remains constant, prove that the locus of P is a circle, and determine its radius. Classics. EURIPIDES. MR. PALMER. Translate, adding notes on the first two passages : 1. Beginning, ΕΞ. ἡμεῖς μὲν ἀκτῆς κυμοδέγμονος πέλας, κ. τ. λ. Ending, αὐταῖσιν ἀρβύλαισιν ἁρμόσας πόδα. 1. Beginning, γνώμαν σώφρον ̓ ἀθάνατον ἀπροφασίστε κ. τ. λ. Ending, τὸν ἄθεον ἄνομον ἄδικον Εχίονος τόκον γηγενῆ. 3. Beginning, ΧΟ. αἰαῖ, στρ. Ending, ματέρος αἷμα σᾶς, ὅ σ' ἀναβακχεύει. Hippol., 1173-1189. Baccha, 1002–1013. κ. τ. λ. Orestes, 315-338. 1. State Professor Tyrrell's emendations of the following passages, as read by Paley and state the grounds of the change : : (α.) μέθεσθε χειρῶν τοῦδ' ἐν ἄρκυσιν γὰρ ὢν (β.) ἴθ' ὦ βάκχε τὸν θηραγρέταν βακχᾶν. κ. τ. λ. (γ.) οὐκ ἐξικνοῦμαι μαινάδων ὅποι μόθων. 2. What indirect evidence for the text of the Bacchæ have we besides MSS. ? 3. The scope of the Bacchæ has been generally misunderstood? 4. In what respects has Euripides been accused of lowering tragedy? 5. What differences in form have been observed between the earlier and later plays of Euripides ? 6. What was the nature of the Satyric drama ? What is the only extant specimen ? 7. Explain the terms: πάροδος, στάσιμον, ἐπεισοδίον, μονωδία, ἐπῴδος. 8. Scan the following lines : (α.) πόθι Νύσας ἄρα τᾶς θηροτρόφου θυρσοφορεῖς. (β.) ἴτε θοαὶ Λύσσας κύνες ἴτ ̓ εἰς ὄρος. (γ.) φονεύουσα λαιμῶν διαμπάζ. |