50. A square is described with two angular points on the side B C of a triangle A B C, and the other angular points on the sides CA, A B respectively if a denote the side B C, p the perpendicular drawn from A to BC, and x the side of the square; show that

:

51. AP and A B are the opposite sides of a valley-the points A,B,P being in the same vertical plane; the inclination of AB to the horizon is 10°, and A B is 2 miles long. An observer finds that the vertical elevation of P at A is 43°; on going to B he finds that he is in the same horizontal line as P. Calculate the vertical height of P above A, and its horizontal distances from A and B. (40.) 52. A C is a line 7 miles long, and B a point in it 4 miles from A; at a certain point P it is found that the angles A P B and B PC are 30° each; find the lengths of A P, B P, and C P, without logarithms. (45.)

Section C.-Spherical Trigonometry.

53. Find the distance in geographical miles (or minutes of a great circle) between the place where the meridian of Greenwich cuts the equator and the point whose longitude is 58° 17′ 51′′ W and latitude 71° 34′ 16′′ N. (25.) 54. What arc of a great circle, making an angle of 60° with the equator is cut off by the parallel of 45°? Answer to minutes. (30.) 55. Two sides of a spherical triangle are 62° 15′ 24′′ and 103° 18′ 47′′. The angle opposite the shorter side is 53° 42′ 38′′. angle opposite the longer side to seconds.

Find the

(35.)

56. Two regular hexagons, A B C ...., A B C . . . . in planes at right angles to one another, have a common side A B. What is the plane angle between the next sides, namely CB C1? Answer to minutes. (40.)

57. A and B are two places in the same latitude, namely 57° 36′ 3′′, and the longitude of B is 66° 49′ 17′′ west of A. In what direction should a ship start from A in order to take the nearest course to B? (40.) 58 Two sides and the included angle of a spherical triangle being given, the third side is found, with the help of an auxiliary angle,

Read the General Instructions at the head of the Elementary paper.

You may not answer more than ten questions.

The value attached to each question is (50).

Three hours are allowed for this paper.

71. Show that, if a = b,

(a2x2 + b2 y2 + c2 22) (x2 + y2 + z3) + a2 b2 c2

− b2 c2 (y2 + ≈2) — c2 α2 (≈2 + x2) — a2 b2 (x2 + y2). is divisible by x2 + y2 + z2 — a2; and find the quotient.

show (without solving the equation) that a +ß = a and aß = b. Find the equation whose roots shall be the cubes of the roots of the equation x2 − 4 x + 5 = 0.

73. Express as a surd the value of the continued fraction

show that either x+y+z=0, and

ax+by+cz=0, or else a=b=c.

76. A, B, C, D, are the angular points of a square, taken in order; show how to draw through C à line cutting A B and D A produced in P and Q, so that A P and A Q together may equal half the perimeter of the square.

77. Find all the values of x which satisfy the equation sin 2x + sin x

=0.

78. A B is a diameter of any circle; with centre B and radius B A describe a second circle; from B draw any straight line cutting the larger circle in P and the smaller in Q: join A Q. Show that

the area contained by P Q, arc A P, and chord A Q, is bisected by the arc A Q.

79. A, B, and C are three fixed points in a right line which are joined to any fourth point O: prove that the ratios

OA cosec В OC: O B cosec COA: O C cosec A OB are independent of the position of the point O.

80. In a right angled triangle AB C, A D is the perpendicular on the hypotenuse. Prove that the areas of the circles inscribed in the triangles A B D and A C D are proportional to the triangles, and that the areas of the circumscribed circles are together equal to that of the circle described on B C as diameter.

81. Let P and Q be the areas of two regular polygons inscribed and circumscribed respectively to the same circle, and let P' Q' be the areas of the inscribed and circumscribed polygons, in the same circle, with double the number of sides. Prove that

Hence find an expression for the area of the octagon circumscribed

to a circle whose radius is r.

82. Given the radii of the four circles which are touched by the three sides of a triangle, find the sides.

May arbitrary values be assigned to all four of these radii?

83. Prove that in a right angled spherical triangle the number of sides which exceed a quadrant must either be two or none, and cannot be one or three.

84. On a sphere, whose radius is, describe a small circle with spherical radius 0, and also a great circle one of whose poles is on the circumference of the small circle: show that the length of the common chord of the two circles is √(— cos 2 0), and ex

2 r

sin

plain the negative sign in this expression.

85. The side c of a spherical triangle is a quadrant: the arc of a great circle drawn through C at right angles to c is ß, and it divides the angle C into two parts C', C": show that

cotan C', cotan C′′ = (sin ẞ)2

86. Show that the sum of the nth roots of unity, and the sum of their reciprocals, are each equal to zero.

87. Having given xy-5 y3, expand y in a series by ascending powers of x, by means of indeterminate co-efficients, as far as x5.

SUBJECT V. PURE MATHEMATICS, STAGES 4 & 5.

EXAMINER, THOMAS SAVAGE, ESQ., M.A.

GENERAL INSTRUCTIONS.

If the rules are not attended to, the paper will be cancelled.

You are permitted to answer questions from the Fourth Stage, or from the Fifth Stage, or from the Honours paper, but you must confine yourself to one of them.

Put the number of the question before your answer.

The figures in descriptive geometry should not only be constructed with ruler and compasses, but the construction should in all cases be explained and its accuracy demonstrated.

You are to confine your answers strictly to the questions proposed. Your name is not given to the Examiner, and you are forbidden to write to him about your answers.

The examination in this subject lasts for three hours.

Fourth Stage. Subjects: Plane, Solid, and Descriptive Geometry and Geometrical Conics.

INSTRUCTIONS.

You are not permitted to attempt more than eight questions, and of these not more than two should be on the same subject.

The values attached to the questions differ little from one another.

1. Define similar figures, and prove that, if two triangles have the sides about each of their angles proportional, the triangles will be similar.

In the quadrilateral, A B C D, the diagonal A C is a mean proportional between A B and A D, and is also a fourth proportional to C B, A B, C D. Show that the triangles A B C, A B D, are similar; and state which are the equal angles in the two triangles.

2. Prove that the areas of two rectangles have to one another the ratio which is compounded of the ratios of their sides; and thence show that, if a perpendicular be drawn from one of the angles of a rectangle upon the diagonal, which does not pass through that angle, the area of the rectangle is a mean proportional beween the square on the diagonal and the rectangle under the segments of the diagonal.

3. If a perpendicular be drawn from one angle of a triangle to the opposite side, prove that the diameter of the circle circumscribed about the triangle is a fourth proportional to this perpendicular and the two sides which contain that angle.

From a point O, in the circumference of a circle, the two chords O A, O B are drawn so that the rectangle under O A and O B is equal to a given square. Prove that A B touches a fixed circle,

and determine its radius.