line and the extremities of the base have the same ratio which the other sides of the triangle have to one another. 7. Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides. 8. Trisect a given straight line. 9. Construct a rectangle which shall be equal to a given square (1) when the sum, and (2) when the difference of two adjacent sides is equal to a given line. 10. A tangent to a circle at the point A intersects two parallel tangents in B and C, the points of contact of which with the circle are D, E, respectively; show that if BE, CD intersect in F, AF is parallel to the tangents BD, CE. No. 2. 1. Define a circle and a rhombus, and give Euclid's 12th axiom (relating to parallel lines). 2. If from the ends of a 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. 3. What is meant by a corollary? State and prove the corollaries to the proposition in which it is proved that the three angles of a triangle are together equal to two right angles. 4. If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. 5. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts shall be equal to the square of the other part. 6. Draw a straight line from a given point, either without or in the circumference, which shall touch a given circle. 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. ALGEBRA. Set to candidates for the Admiralty who selected Algebra as a subject of Examination. (Time allowed, 3 hours.) No. 1. 1. Find (1) the sum, (2) the difference, of the two expressions 1 (x-2)(x-1)x(x+1)* 1 (x − 1)x(x+1)(x+2)' (3) the quotient when the former is divided by the latter. 2. Reduce to their simplest forms the expressions 3. Prove the rules for pointing in the multiplication and division of 2x2+11x+15=0 (x) { x ty (x2+y^=82 5. Find a number of two digits, which is three times the sum of its digits, and such that the difference between the digits is 5. 6. If any number of fractions be equal, show that each of them sum of any multiples of the numerators sum of the same multiples of the denominators 1 7. Prove that a"=1, and a"="/a. Is any assumption necessary in order that this may be true? 8. Investigate a rule for finding the sum of n terms of an arithmetical progression. 27 In the series of 3-+-&c., find s (1) when n=5, (2) when n infinity. = 9. When does one quantity vary (1) directly, (2) inversely, as another? If x varies as y, prove that x2+y2 will vary as x2-y2. 10. Expand a3 (a3 —x3) } in a series of ascending powers of x by means of the Binomial theorem, writing down the first 4 terms and the rth term. 11. Express the numbers 957 and 23.125 in the septenary scale. xy ху 2. Add together 4 (22-2ry) and 3–7 (22-y2-3) 5 3. Divide 4x5 - x3+4x by 2x2+3x+2. 5 4. Investigate a rule for finding the square root of an algebraical quantity, and explain how the method of extracting the square root of a numerical quantity may be deduced. 5. Find the square root of (a+b)2 — c2+(a+c)2 − b2+(b+c)2 — a2. 6. Find the greatest common measure of 4x4 - 12x3 +5x2+14x-12 and 6x4-11x+9x-13x+6. 8. Investigate an expression for the sum of a geometrical series; and find the sum of 30 terms of the series 2++7+&c. 9. A number consists of 2 digits; if it be multiplied by 2, and the product diminished by 4, the digits are inverted; and if 10 be subtracted from it, the remainder is equal to 3 times the sum of the digits; find the number. MATHEMATICAL QUESTIONS. Set to Candidates for the Colonial Office who selected Mathematics as a subject of examination. A. PURE MATHEMATICS. 1. To a given straight line apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle. 2. A segment of a circle being given, describe the circle of which it is the segment. 3. Give Euclid's definition of proportion and of similar figures, and show that similar polygons may be divided into the same number of similar triangles having the same ratio to one another that the polygons have. 4. Investigate a rule for extracting the cube root of an algebraical expression. 5. Reduce to its lowest terms the expression 6. Solve the following equations 3x4+14x3+9x+2 2x4+9x+14x+3° In equation (2) investigate the conditions that the values of may be two positive integers. 7. Write down the (r+2)th term and the middle term of the expansion of (a-b)2". 9. Given two sides of a rimge and the angle opposite one of them: show how a sive me ringle, and point out when the case is amcigicus. If o=5. b=7, and A-sin is there ambiguity? 10. Given log18=49877044; £ fr 1=6), fnd the value of $1712926 w seven places of decimals 11. If the three sides of a triangle are — 1, 25+1, and 2—1, show that the greatest angle is 1207. 12. A right ccne is cur by a plane which meets the cone on both sides of the vertex; show that the section is a hyperbola. Under what condition is it possible to cut an equilateral hyperbola from a given cone. 13. In an ellipse prove that the line drawn through the focus at right angles to the focal distance intersects the tangent and the directrix in the same point B.-MIXED MATHEMATICS. 1. Assuming the truth of the parallelogram of forces as to direction, prove its truth as to magnitude 2. Given the sum of two forces and their resultant, and also the angle which one of them makes with the resultant. Determine the forces, and the angle at which they wor 3. State the result of any experiments do with reference to A body weighing 12,000 tons, placed on a plane whose li re |