wohl polirt und an beiden Spitzen mit Goldblech beschlagen. (5) So schildert z. E. Virgil in seinem Gedichte vom Landbaue eine zur Zucht tüchtige Kuh. (6) Ein einziger unschicklicher Theil kann die übereinstimmende Wirkung vieler zur Schönheit stören. What is the meaning of Kenner, Stümper, Tonkünstler, eine spanische Wand? 9. Write a short account of Haller, Moses Mendelssohn. Whom does Lessing call "der griechische Voltaire"? 10. Translate the following: (1) (2) Hie kann nicht sein ein böser Muth, Wo da singen Gesellen gut; Hie bleibt kein Zorn, Zank, Hass noch Neid, Geiz, Sorg, und was sonst hart anleit, Es trägt ein' Schneck für und für, Wo sie hingeht, ihr Haus mit ihr. Drum meint man, dass die Leut von Schnecken -Fischart. (3) Wen er traf, der lass' ihn sitzen Und erduld' ein wenig Schmerz! Wer geprüften Rath verachtet (4) Und ihn auszureissen trachtet, Der zerfleischet ganz sein Herz.-Bürger. Thalatta! Thalatta! Sei mir gegrüsst, du ewiges Meer! (5) Sei mir gegrüsst zehntausendmal Unglück bekämpfende, heimathverlangende, Wär' ich im Bann von Mekka's Thoren, Dann führt' ein Schwert wohl diese Hand. -Freiligrath. LOWER MATHEMATICS. Professor Nanson. 1. If any number of magnitudes of the same kind are proportionals, then as one of the antecedents is to its consequent so is the sum of the antecedents to the sum of the consequents. If A A' B: B':: C: C': &c., : and l, m, n, &c., be any whole numbers, prove that 1A+mB+nC+&c. : lA'+mB' + n C+ &c.::A : A'. 2. ABC is a triangle, and AP, AQ are drawn equally inclined to AB, AC respectively, and meeting the base BC and the circumscribed circle in P, Q respectively. Prove that the rectangle AP, AQ is equal to the rectangle AB, AC. Examine the cases in which AP coincides with the bisector of the angle BAC, and the perpendicular from A on BC. 3. Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane. Prove that if the lines joining two of the vertices of a triangular pyramid to the centres of the circles inscribed in the opposite faces intersect, then the lines joining the two other vertices to the centres of the circles inscribed in their opposite faces also intersect. 4. Prove that triangular pyramids of equal bases and equal altitudes are equal, and hence shew that the volume of a triangular pyramid is one-third of its base multiplied by its altitude. Two opposite faces of a six-faced polyhedron are similar rectangles with corresponding sides parallel. Prove that the volume of the polyhedron is abc (1 + k + k2), where a, b are the sides of one of the rectangular faces, ka, kb those of the other rectangular face, and c is the distance between the rectangular faces. 6. Find, ab initio, the number of combinations of n things r at a time. A can At a certain examination there are 7 subjects. A pass paper is set in each subject, and in m of the subjects an honour paper is set also. didate may take up the pass paper or the honour paper, if any, in any subject. In how many ways can a candidate enter for n subjects? Test the accuracy of the formula obtained by considering the special cases n = 1 and m = 1. 7. Sum to n terms each of the series 9. The angles of elevation of the top of a tower at the ends and middle point of a base of length 2a measured on the horizontal plane through the foot of the tower are a, ß, y respectively. Prove that the height of the tower is a√2 ✓cot'a + cot2ß - 2 cot2y Adapt this formula for logarithmic computation in the case in which a = ß. 10. Perpendiculars are drawn from the vertices of an acute-angled triangle to the opposite sides; shew that the sides and angles of the triangle formed by joining the feet of these perpendiculars are a cos A, b cos B, c cos С, π-2A, π — 2B, π—2C, respectively. Shew that in any triangle a cos A cos 2A + b cos B cos 2B + c cos C cos 20 = 2a cos A sin 2B sin 2C. 11. If be the circular measure of a positive angle less than a right angle, prove that sin lies be 03 4 1. Prove that the ratio of the rectangles contained by the segments of any two intersecting chords of a conic is the same as for any other two chords parallel to the former, each to each. Find the point on a conic at which a given chord of the conic subtends the greatest angle. 2. Given a focus and two tangents to a conic, the envelop of the conjugate axis is a parabola. |