70. A circle can be inscribed in a quadrilateral, three of whose sides, taken in order, are 5, 4, 7; and the quadrilateral itself is inscribed in a circle. Show that the sine of the angle between the diagonals

is

8√70

67

This being very near unity, can it be safely inferred that the angle is very nearly a right angle? Give reasons for your answer. 71. Show that the area of a small segment of a circle is very nearly base x height, and that the error is very nearly 2÷80 of the area, where is the angle subtended by the arc at the centre. Calculate the numerical value of this fraction, when the sexagesimal measure of the arc is 5o.

72. In any plane triangle, if

(a—b) (s—c) = (b−c) (s—a)

show that the radii of the three circles, which touch one side and the other two produced, are in arithmetical progression.

73. Let A be the least of the three angles of the triangle A B C ; show how to determine points P and Q on the sides A C and A B respectively, such that the circle passing through the points A, P, Q shall equal that which passes through the points B, C, P, Q. 74. Show that the squares of the distances of the angular points of a triangle from the centre of the inscribed circle, in terms of the

75. The plane of a small circle of given spherical radius is inclined at a given angle to that of a great circle; find the condition that the two may intersect, and show how to calculate the arcs of the great circle included between the points of intersection.

Perform the calculation when the angle and the radius are each 60°.

76. Show that all places, whose latitude and longitude are 0 and 4, these being connected by the equation.

are equidistant from the point whose latitude and longitude are a and B.

What is meant by m?

77. The sides and angles of a spherical triangle being supposed all known, give formulæ for finding the angles of the chordal triangle. What relation must hold among the elements of the spherical triangle, in order that the chordal triangle may be right angled ?

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. The hypothenuse and a side of a right-angled triangle have the same ratio to one another as the hypothenuse and a side of a second right-angled triangle. Prove that the triangles are similar; and state the more general theorem of which the above is a particular case.

2. Construct a fourth proportional to three given straight lines.

If A, B, C, be three given straight lines, construct two others, D, E, such that C: D is the duplicate of A: B, and that A: B is the duplicate of C: E.

3. Show that one, and only one, circle can be described through any three given points, not in one straight line. If circles be circumscribed about any two triangles which have a common angle, prove that the diameters of the circles are in the same ratio as the sides of the triangles which are opposite to the common angle.

4. Prove that any plane which contains one, but not both, of two parallel lines, is parallel to the other. Hence, or otherwise, prove that two planes will be parallel if any two intersecting lines in the one are respectively parallel to two lines in the other.

5. Prove that the sum of the plane angles which contain any convex solid angle is less than four right angles.

Hence show that no closed convex solid can be contained by plane faces which all have six or more sides.

6. Define a right circular cylinder, and explain what is meant by a tangent plane to such a cylinder. Find the locus of the poles of the circular sections of a sphere made by tangent planes to a cylinder, the sphere being wholly outside the cylinder.

7. In what cases are the projections of a straight line upon two coordinate planes insufficient to determine the line?

Determine the intersection of two planes, whose traces are given, (1) when both planes are parallel to the axis, (2) when both planes cut the axis at the same point.

8. Find the projections, and the absolute length, of the line which joins the feet of the perpendiculars let fall from two given points on a given plane.

9. Find the angle between two lines, which meet at a point on the axis, and of which one is in the horizontal and the other in the vertical plane.

Prove that this angle cannot be a right angle unless one of the lines is at right angles to the axis.

10. Prove that the perpendicular from the focus of a parabola upon any tangent is a mean proportional between the focal distances of the point of contact and the vertex.

Having given the directrix of a parabola which touches two given straight lines, give a construction for finding the points of

contact.

II. If tangents be drawn to an ellipse from an external point, prove that they are equally inclined to the lines which join that point with the foci.

Hence find the locus of a point which moves so that the bisector of the angle between the two tangents drawn from it to an ellipse meets the major axis in a fixed point.

12. If a parallelogram be formed by drawing tangents at the extremites of a pair of conjugate diameters of an hyperbola, prove that the diagonals are the asymptotes; and that the area of the parallelogram is equal to the rectangle under the axes.

Fifth Stage. Subjects: Descriptive Geometry, Spherical Trigonometry, Co-ordinate Geometry of two and of three dimensions.

INSTRUCTIONS.

Read the General Instructions at the head of the Fourth Stage paper.
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.

21. Construct the projections of a straight line which intersects two given straight lines, and is perpendicular to a given plane; giving, in addition to the general case, the construction when the given plane is perpendicular to the axis.

In what cases is the problem impossible?

22. Having given the projection upon the horizontal and vertical planes of the centre of a sphere, and the radius of the sphere, construct the traces of the two planes which touch the sphere and are perpendicular to a given straight line.

23. The base of a cylinder being a given circle in the horizontal plane, and the generating lines being parallel to a line, whose projecttions are given, construct the points in which a given line meets the cylinder; and also the traces of the tangent plane to the cylinder at one of these points.

24. Prove that in any spherical triangle

cot b sin c =

cot B sin A + cos c cos A

Hence show that if the sum of the two sides b, c be a quadrant,

25. Having given two sides a, b, and the angle C, included by them, show how to solve the triangle by considering it as the sum or difference of two right-angled triangles.

26. Find the number of degrees, &c., in each of the angles of an equilateral spherical triangle whose area is equal to the square of the radius of the sphere.

If a, b be the sides including the right angle in a right-angled spherical triangle, and E the spherical excess of the triangle, prove that

cos acos b I + cos a cos b'

27. Find the equation to the straight line which joins two points whose co-ordinates are (— 4, 6) and (8, 3). Also find the equation to a straight line drawn through the point ( — 4, 3) at right angles to the line 4r 3y = 2. Prove that, these two straight lines are parallel to one another, and find the perpendicular distance

between them.

28. What is the locus represented by the equation

x2 + y2 + a2 = 2a ( x + y)?

Prove that the axes are tangents to the curve, and find the polar equation to the locus of the middle points of the chords which (when produced) pass through the origin.

29. If d', d" be the semi-diameters of an ellipse, respectively conjugate and perpendicular to the semi-diameter d, prove that