9. Name some places in the Southern Hemisphere having nearly the same latitude as Dublin. 10. Draw a sketch-map of Australia, indicating the relative position of the several colonies and their capitals. ADDITIONAL EXAMINATION FOR HIGH PLACES. GEOMETRY AND ALGEBRA. MR. PANTON. 1. Prove that the sum of the sides of any quadrilateral figure is greater than the sum of the diagonals. 2. Prove that in general the sum of the squares of the sides of a quadrilateral is greater than the sum of the squares of the diagonals; and determine the nature of the quadrilateral in which these two sums are equal. 3. Inscribe a circle in a given triangle; and express in terms of the sides of the triangle the segments of any side made by the point of contact of the circle. 4. Resolve into factors at (62 – c2) +64 (c2 – a) + c* (az — 52). 5. Simplify the expression (23 – yy) (23 + y2 + x2 y + xy?) (328 – y8) { (x + y)2 – xy } 6. A can do a piece of work in 54 days, and B in 30 days ; A works at it alone for 24 days, B then works alone for 10 days, and afterwards C finishes it in 8 days. In what time could C bave done the whole work ? MR. F. PURSER. 7. On a given finite right line as hypotenuse, construct a right-angled triangle equal in area to a given triangle. (a) When will the problem be impossible ? Show that the rectangle under the segments into which the base is divided by its point of contact is equal to the area of the triangle. 9. Two points A and B are taken inside a circle: find a point Pon the circle such that the angle APB may be the greatest possible. 23 - 13x + 12. 11. Find the coefficient of x' in (1 + 3x – 4x2) 12. From the following data calculate the ratio of the meter to the inch : I cubic inch of distilled water weighs 252.5 grains. MR. W. R. ROBERTS. 13. Inscribe a square in a triangle. 14. If the middle points of the sides of a quadrilateral be joined by right lines, prove that these lines form a parallelogram. 15. Describe a circle through a given point to touch a given right line, and having its centre on another given right line. 16. If a +b+c=0, show that Translate into English 1. Beginning, Memnona si mater, mater ploravit Achillem, Ending, Restat, in Elysia valle Tibullus erit. OVID. 2. Beginning, Praetor Aetolorum Phaeneas cum eadem fere, Ending, et ne inter seria quidem risu satis temperans. MR. GRAY. Translate : Ενώ δε οι πολέμιοι έλέγοντο μεν προσιέναι, παρήσαν δε ουδέπω, εν τούτω επειράτο και Κυρος ασκείν μεν τα σώματα των μεθ' εαυτού εις ισχύν, διδάσκειν δε τα τακτικά, θήγειν δε τας ψυχάς εις τα πολεμικά. Και πρώτον μεν λαβών παρά Κυαξάρου υπηρέτας προσέταξεν εκάστους των στρατιωτων ικανώς, ών έδέοντο, πάντα πεποιημένα παρασχεϊν τούτο δε παρασκευάσας ουδέν αυτοίς άλλο έλελοίπει ή ασκεϊν τα αμφί τον πόλεμον, εκείνο δοκών καταμεμαθηκέναι, ότι ούτοι κράτιστοι έκαστα γίγνονται, οι αν αφέμενοι του πολλοίς προσέχειν τον νούν επί έν έργον τράπωνται. και αυτών δε των πολεμικών περιελών και το τόξω μελετών και ακοντίω, κατέλιπε τούτο μόνον αυτοίς το σύν μαχαίρα και γέρρω και θώρακι μάχεσθαι ώστε ευθύς αυτών παρεσκεύασε τας γνώμας, ως ομόσε ιτέον είη τοΐς πολεμίοις, ή ομολογητέον μηδενός είναι άξιους συμμάχους τούτο δε χαλεπόν ομολογήσαι, οίτινες αν είδωσιν, ότι ουδέ δι' έν άλλο τρέφονται και όπως μαχούνται υπέρ των τρεφόντων. For Greek Prose : He then told the Chaldeans, that he was not come either with a desire to destroy them, or with an inclination to make war upon them ; but with a wish to make peace between the Armenians and Chaldeans. “ Before your mountains were occupied,” said he, "I know that you had no desire for peace : your own possessions were in safety; those of the Armenians you plundered and ravaged. But you now see in what condition you are placed. Those of you, therefore, that have been taken I dismiss to your homes, and allow you, together with the rest of the Chaldeans, to consult amongst yourselves, whether you are inclined to make war with us, or to be our friends : if war be your choice, come no more hither without arms, if you are wise; if you resolve to prefer peace, come without arms. And, if you become our friends, it shall be my care that your affairs be established upon the best footing.' The Chaldeans hearing these assurances, and bestowing many praises upon Cyrus, and giving him many pledges of friendship, went home. HISTORY, GRAMMAR, AND GEOGRAPHY. MR. MAHAFFY. 1. Enumerate the Conservative politicians known in Athenian history (with dates). 2. Sketch the life of Epaminondas. 4. Enumerate the Gallic wars in Roman history up to the wars of Cesar. 5. Give an account of Clodius (the tribune). 6. What wars did the Romans wage in Britain ? 7. Draw a map of the East coast of Greece from Concyra to Pylos. 8. Ρarse ενεμέσων, ενεμέθην, νήσω, νεύσω, ενήθην, μνάσθω, μνησθώ, επλήσθην, επλήχθην, δαισθείς, δασθείς. 9. Parse situs, scitus, satus, sessus, sensus, factus, fatus, fetus, fessus, fissus. 10. What do you know about Greek athletic meetings ? SCIENCE SIZARSHIP EXAMINATION. GEOMETRY. MR. W. R. ROBERTS. 1. If two lines be divided homographically in two systems of points a, b, c, and a', 6', 6, &c., find the locus of the points of intersection of ab', a'b, ac', a'c, &c. 2. Given base and sum of sides of a triangle, find the locus of the foot of the perpendicular from one extremity of the base on the external bisector of the vertical angle. 3. A transversal passes through a fixed point 0 and meets a series of circles in points A1, Bi; A2, B2; A3, B3, &c. ; find the locus of a point P taken on it so that OP= (A + OB1 + 042 + 0B2 + 0 A3 + OB3 + &c. 4. A line A B of given length moves between two fixed right lines, if P be a fixed point on AB, find when the perpendicular from P on a given line is a minimum. 5. Two vertices of a triangle move on fixed right lines, and the third on the circumference of a given circle ; find when its perimeter is a minimum. 6. If a triangle be self-conjugate with respect to a given circle, prove that the circle described about the triangle is orthogonal to a given circle. 7. Given three pairs of corresponding constituents of two homographic systems of points on a common axis, give Chasles' construction for determining the double points. (a). What problem does this method solve by inversion ? 8. If A and B be two points each of which lies on a given circle, find the locus of intersection of tangents at A and B, being given that the line AB subtends a right angle at a limiting point of the two circles. 9. Prove that the “Triplicate Ratio" Circle divides symmetrically the sides of the original triangle. 10. Through a given point 0 on a fixed circle a line is drawn meeting the circle again in a point A, if a point P is taken on this line so that OP=0A + constant length; find the two points in which the locus of the inverse of P with regard to 0 meets a given right line. 11. Find the locus of a point from which two circles can be inverted into equal circles. 12. Given the “ Brocard Circle” of a triangle find the locus of its circumcentre. y 2. Solve the equation 8.x+ – 10x3 – 15x2 + 5x + 2 = 0, whose roots are in geometrical progression. 3. Find the commensurable roots of the equation 205 – 32x4 + 116x3 11622 + 1152 – 84 = 0, and solve the equation completely. 4. Prove the general expressions by means of determinants for the solution of a system of linear equations in four variables, and apply them to find the value of x from the following equations : 2x 30 + y - x + 38 = 11, - Y + 4% + 38 = 27, 382 – 39w = o. 5. Apply Sturm's method to the analysis of the following equation, so as to determine the number and position of its real roots : 24 – 1422 – 242 – 8 = 0, k |