May, June, 1876, was 1.90, 0·94, 1.27 inches respectively. being in April 0.77 above, and in May and June 146 and 1.78 inches respectively below the average of those months for the previous six years: find the average for these three months, both for the year 1876 and for the whole seven years, to two places of decimals. SECTION VIII. 1. Which of the following stocks is most profitable for investment-the 3 per cents. at 91, the 3 per cents. at 1081, the 4 per cents. at 118? Find the yearly income produced by investing £5217 16s. 3d. in the most advantageous of the three. 2. A tradesman induces customers who owe £750, and otherwise would not have paid him for 6 months, by an offer of 5 per cent. discount, to pay ready money; at the end of the six months, instead of £750, he had received £947 12s. 6d. What is his profit per cent. 4. A commission agent agrees to take 5 per cent. on all sales, or 3rd of net profit; 35 of his transactions realize 10 per cent. profit, the remainder 19 per cent. What would be the difference on sales of £600,000 according as all his customers adopt one or other agreement ? SECTION IX. 1. Find the square root of 32.7142073 to four places of decimals, and the cube root of 941,192. Show that the cube root of every perfect cube less than a million can be determined by inspection. 2. If exchange be at the rate of 25.50 francs for one pound, and of 57.75 florins for 119 francs, find the value of £4,760 in florins. SECTION X. 1. Find the cost of turfing a ground 10 chains long and 5 chains broad, each turf being 15 in. long and 6 in. broad, and 100 turfs costing 18. 3d. 2. £8 11s. is spent upon the floor of a room 24 ft. long and 18 ft. wide; the centre of the room is covered with carpet 2 ft. wide, at 4s. 3d. per yard, leaving a margin of 3 ft. all round the carpet: how much per square foot does the margin cost to paint? Euclid, Algebra, and Mensuration. 123 EUCLID, ALGEBRA, AND MENSURATION. Candidates in Scotland may answer two questions out of Section IV. if they omit Section IX. With this exception Candidates are not permitted to answer more than one question in each section. (Marks are given for portions of questions.) EUCLID. Capital letters, not numbers, must be used in the diagrams. SECTION I. Define a "straight line," a 66 66 rhombus." Show that the following definitions are incomplete: "Of quadrilateral figures, a square has all its sides equal.' An acute-angled triangle is that which has two acute angles." "Parallel straight lines are such as do not meet however far they may be produced." (These form one question.) SECTION II. 1. If the equal sides of an isosceles triangle be produced, the angles on the other side of the base shall be equal. Show that this property can be proved by a method similar to that employed in the 4th Proposition. 2. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. Write out the enunciations of any previous propositions employed in this proof. 3. Straight lines, which are parallel to the same straight line, are parallel to each other. If two adjacent sides of a parallelogram be parallel to two adjacent sides of another parallelogram, the other sides will also be parallel. SECTION III. 1. Triangles upon the same base and between the same parallels are equal. Construct a triangle equal to a given triangle and having a base three times as great. 2. To a given straight line to apply a parallelogram which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle. To one of the sides of an equilateral triangle apply an equal parallelogram having one of its angles equal to that of the given 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 those parts. [It may be assumed that parallelograms about the diameter of a square, are likewise squares.] Show algebraically that the squares on the two parts are always greater than twice their rectangle except when the line is bisected. SECTION IV. 1. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls and the straight line intercepted without the triangle between the perpendicular and the obtuse angle. If the triangle be isosceles, the square on the side subtending the obtuse angle is equal to twice the rectangle contained by either side and the straight line made up of that side and the straight line intercepted as in the proposition. 2. Equal straight lines in a circle are equally distant from the centre. AB, AC are two equal chords of a circle at right angles to each other. Show that they are sides of a square inscribed in the circle. 3. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angle which this line makes with the line touching the circle shall be equal to the angles in the alternate segments of the circle. If a tangent be defined as the limiting position of a secant, show that the tangent to a circle is perpendicular, to the radius drawn to the point of contact. Euclid, Algebra, and Mensuration. 125 ALGEBRA AND MENSURATION. The solution must be given at such length as to be intelligible to the Examiner, otherwise the answer will be considered of no value. SECTION V. Find the difference in value between the arithmetical expression 57 and the algebraical expression ab when a=5, b=7. Simplify the expression (2a- {3b-ac}+{4b-3a - c}). {3b⋅ + (1 − a) (1 − b) (c-1) d to one expression consisting Reduce (1 a) + (a 1) b + (1 a) (b 1) c of factors of the form (1 3. Show that x4 +81 is never less than 18x2; and that a3 + b3=9ab - 27 if a + b + 3 =0. SECTION VII. Solve the equations 4x + 1 Бас - 3 7x 4 6х 5 (1) (2) 53 (3) 5 7 = 12 · 56=6x2 (202 = 33) (a + b) =a3b3} 13 (These three equations form one question.) SECTION VIII. 1. (a) One number is the square of another, and their sum equals six times the smaller of the two; find the numbers. (b) A and B exchange purses; A has as many halfsovereigns as B has half-crowns, and as many sixpences as B has shillings; find the ratio of the number of halfsovereigns to the number of half-crowns, that neither may lose by the exchange. 2. Two trains, A and B, are despatched from opposite termini of a railway at the same time, with speeds of 20 and 30 miles per hour respectively; a third train, C, is despatched after B three hours later, with a speed of 40 miles per hour: what is the length of the railway, if A pass C half an hour after passing B? SECTION IX. 1. The sides of a five-sided rectilineal figure ABCDE taken in order, AB, BC, etc., are 18, 24, 30, 24, 18 feet respectively; the angles ABC, AED, are right angles; find the area in feet to two places of decimals. 2. From two of the opposite angles of a parallelogram, whose sides are 75 and 120, perpendiculars can be dropped such that the two triangles so formed are together equal to the remainder of the parallelogram: find the altitude of the parallelogram. 3. Find the cost of gilding the surface of a cone, whose radius of base is 48 feet, and whose height is 36 feet, at 3d. per square foot. MALE AND FEMALE CANDIDATES. Candidates are not permitted to answer more than one question in each section. One full map only is to be drawn. GEOGRAPHY AND HISTORY. GEOGRAPHY. SECTION 1. Draw a full map— (a) of the west coast of Great Britain from Land's End to Great Orme's Head. Or (b) Of Ireland. Or (c) Of South America. SECTION II. Define the terms " estuary," "strait," “archipelago,” “"valley." Illustrate your definition of |