5. (1). If upon the sides BA, CA, of any triangle, any two parallelograms be drawn, and their sides produced to meet in K and KA be joined, then the parallelogram constructed upon the base BC, with one side equal and parallel to KA, will be equal to the sum of the other two.

(2). When the triangle becomes right-angled at A and the parallelograms on the sides become squares, show that the parallelogram on the base is also a square. (This is = Prop. 47. Eucl. B. I.)

6. The square described upon the side of a regular polygon inscribed in any circle is equal to the sum of the squares described upon the sides of a regular hexagon and decagon inscribed in the same circle.

7. If a straight line be drawn from C, the point of bisection of a given arc ACB, cutting the chord AB or the chord produced in any point E and the circumference of the circle in D, prove that in each case the rectangle contained by CD and CE is equal to the square described on CB.

8. The greatest of all straight lines passing through either of the points of intersection of two given circles which cut each other, and terminated

both

ways by the two circumferences, is that which is parallel to the line joining the centres of the two circles.

9. If the sides of a regular polygon of n sides be produced to meet, the sum of the angles made by the lines thus produced at the points of intersection is equal to 2n-8 right angles.

10. Represent the arithmetic, geometric, and harmonic means, between two given lines geometrically.

11. The centre of the circle circumscribed about any triangle, the point of intersection of the perpendiculars let fall from the angular points of the same triangle to the opposite sides, and the point of intersection of the lines joining the angular points with the middle of the opposite sides, all lie in the same right line.

12. If four circles touch each either internally or externally, three sides of any quadrilateral figure, the centres of these circles will lie in the circumference of the same circle.

13. Describe a circle passing through a given point which shall touch both a given circle and a given straight line.

14. If from the centre and angular points of a regular hexagon perpendiculars be drawn to any

given right line, six times the perpendicular from the centre is equal to the sum of the perpendiculars from the angular points.

TRINITY COLLEGE.

PLANE TRIGONOMETRY.

1. TRACE the signs of the Sine, Cosine, Tangent, and Secant, through the circle.

2. Transform the formula,

(cos. A)" + a. (cos. A)". (sin. B)" + b. (cos. A). (sin. C)+&c. where the radius = 1, to an equivalent formula, where the radius = r; and prove the rule.

3. Given the sines and cosines of two arcs A and B; it is required to find sin. (A+B) and sin. (A−B).

4. Prove that, sin. (A+B). sin. (A-B)= (sin. A)2 — (sin. B)2, and cos. (A+B). cos. (A−B) =(cos. A)-(sin. B).

1

5. If x+ -=2 cos. A, prove that a3+

2 cos. 3A.

6. Tan. (45° +A) Tan. (45°- A) +

2 Tan. 2A.

=

Prove this, and explain what is meant

by a Formula of Verification.

7. Tan. A + cotan. A=2 cosec. 2A.

Tan. A cotan. A=2 cotan. 2A.

8. In any triangle, the sum of any two sides : their difference: the tangent of half the sum of the angles subtended by those sides: the tangent

of half their difference.

9. Given the sine of 1'; show how the sines of all arcs from 1 to 90° may be found, rad. = 1.

10. Given two sides of a triangle, and an angle opposite to one of them, solve the triangle; and show the ambiguity in this case.

11. Given two sides and the included angle; solve the triangle.

12. Explain the method of finding the distance between two visible but inaccessible objects on an horizontal plane; and show how the requisite triangles are to be solved.

13. Two sides of a triangle and the angle included being given, find the area of the triangle.

14. The perimeter and the three angles of a triangle being given, find each of the sides.

At three being again called together, the Sophs were to display their astronomical acquirements on the paper below.

TRINITY COLLEGE.

ASTRONOMY.

1. By what arguments is it inferred that the Earth revolves about its axis, and about the Sun ?

2. Find the right ascension and declination of a comet, when its distance from two known stars is given.

3. Required the time of the Sun's transit over the vertical wires of a telescope, on a given day at a given place.

4. The apparent meridian altitude of the Sun's lower limb 53° 40', his apparent semi-diameter =15° 50′, his mean refraction=29", the parallax =4′′.5, collimation=34".5, and declination=16° 13'. Find the latitude of the place of observation. 5. The altitude of the Sun was observed to be half of his declination at 6 o'clock. Prove that