stehen und staunen und schlagen die Hände zusammen und rufen; Aber, wie hat ein so grosser Mann nicht wissen können!' VORGEFÜHL. (6) Mich selber oft im Geist hab' ich gesehn, Doch allemal, wenn träumend so zu schau'n Wie rings ich auch, was Glück man nennt, geschaut: Mein Bart verwildert, und mein Haar ergraut. Wer grüsste mich? Wer nahm mir ab den Stab? 2. Translate into German :— FREILIGRATH. Now commenced the brightest part of Argyle's career. His enterprise had hitherto brought on him nothing but reproach and derision. His great error was that he did not resolutely refuse to accept the name without the power of a general. Had he remained quietly at his retreat in Friesland, he would in a few years have been recalled with honour to his country, and would have been conspicuous among the ornaments and the props of constitutional monarchy. Had he conducted his expedition according to his own views, and carried with him no followers but such as were prepared implicitly to obey all his orders, he might possibly have effected something great.— MACAULAY. 3. 'Der Schüler übersetzte seine Aufgabe;' der Feldherr setzte seine Truppen über den Fluss; do these sentences illustrate any general difference of meaning and accent in such verbs as are both separable and inseparable? 4. Shew by examples that the use of the infinitive is not so wide in German as in English. THURSDAY, JUNE 7, from 2 to 4.30 P. M. C. Mathematics. 1. Algebra to Quadratic Equations. [Candidates are reminded that in order to pass in Mathematics they must satisfy the Examiners in the first part of this Paper.] [N.B. No credit will be given for any answer, the full working of which is not shewn.] 1. Divide a1o+as+ 1 by the continued product of a2+a+1, a2-a+1, and at-a2 + I. Find the value of (x2—y2)3, when ≈ = (3/2+3/3) and y = + (3/2-3/3). 2. Express in their simplest forms: I I (3) a(a−b) (a–c) + b(b−c) (b−a) + c(c—a) (e—b) 3. Find the G. C. M. of 10x1 −25x3+13x2+5x-3 and 6x3-9x2-4x+6. 4. Find the G. C. M. of 2x1y2 — 2x3 y3 — 4x2ya, 6x(x3+y3), and 4xy(x2—y2)2; find also the L. C. M. of the same three quantities. (1) 9/2 (2) 6x2+x=2; (3) x2+y=37, x2y = 36. 6. A certain number of three digits is 76 times the sum of its digits. The third digit is double the second, and if the second is increased by 2 and then squared the result is the first digit. Find the number. 2. Higher Algebra. 8. If a :x:: by: :c: z, shew that (bc+ca+ab)2 (x2 + y2+z2) = (bx+cy+az)2 (a2 +b2+c2). 9. Prove that the square root of a rational quantity cannot be partly rational and partly a quadratic surd. Find the square roots of 9-4/2 and 7+√13. 10. Obtain an expression for the sum of n terms of an Arithmetical Progression. The sum of an Arithmetical Progression is 63, the ninth term 13, and the common difference. Find the number of terms. 11. Shew that in the expansion of (1+x)", the coefficient of the 7th term from the beginning is equal to the coefficient of the 7th term from the end. Find the middle term in the expansion of (x-x-1)2n. 12. Solve the equations : (1) (x2+∞)*+(x2−x)3 = x√2; (2) (3x-2)x= = yr. (3y—2)y = x) * FRIDAY, JUNE 8, from 2 to 4.30 P. M. C. Mathematics. [N.B. Candidates are reminded that in order to pass in this subject they must satisfy the Examiners in Euclid I—ÏV.] 1. Euclid I-IV. 1. Define-right angle, parallel straight lines, rhombus, angle of a segment. 2. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them of the other, the base of that which has the greater angle shall be greater than the base of the other triangle. 3. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the two parts. 4. Equal straight lines in a circle are equally distant from the centre; and those which are equally distant from the centre are equal to one another. 5. Inscribe an equilateral and equiangular pentagon in a given circle. 6. If the square described on one of the sides of a triangle be equal to the squares described on the other two sides, the angle contained by these two sides is a right angle. 7. In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle. 8. About a given circle describe a triangle equiangular to a given triangle. 9. Describe a circle which shall touch a given straight line and a given circle at a given point. 2. Euclid VI-XI, and Conic Sections. 10. Triangles and parallelograms of the same altitude are to one another as their bases. Triangles on equal bases are to one another as their altitudes. 11. If two straight lines meeting one another be parallel to two others that meet one another and are not in the same plane with the first two, the first two and the other two shall contain equal angles. 12. In any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. 13. Draw a straight line perpendicular to each of two straight lines which are not in the same plane. 14. A straight line drawn through the intersection of two tangents to a parabola parallel to the axis of the parabola bisects the chord of contact. 15. If the ordinate NP of an ellipse be produced to meet the auxiliary circle in Q, shew that NP: NQ:: BC: CA. 16. The line joining one focus of an ellipse to the foot of the perpendicular from the other to the tangent at any point passes through the middle point of the normal. 17. If the ordinate NP of an hyperbola be produced to meet the asymptotes in R and r, prove that RP.Pr = BC2. SATURDAY, JUNE 9, from 2 to 4.30 P.M. C. Mathematics. [N.B. No credit will be given for any answer, the full working of which is not shewn.] 1. Trigonometry. 1. Investigate an expression for all the angles which have a given tangent. = If tan (2a-3)= cot(3a-28), and tan(2a+38) π cot (3a+28), then both a and ẞ are multiples of ΙΟ 2. Prove that (1) tan 34-tan 2 A-tan A = tan 3 ▲ tan 2 ▲ tan A. 3. Solve the equations: (1) 2sinx-sin 2x = 2 ( 1 + cos x)2; (2) tan-1(ax+6)+tan−1 (ax—b) π = • 4 4. Find the radii of the escribed circles of a triangle; and prove that AO BO со + + CO3 = 1, 0, 01, 0, 0, being the centres of the inscribed and escribed circles. 5. Prove that in any triangle ABC, of which the perimeter |