Divide a++ a3b-3a2b25a b3-2b4 by a2+2 ab+b2.

6. Under what circumstances is am+xm divisible by a+x? is always divisible by a—x when (m) is a

Show that am

positive intiger.

1262

7. Find the greatest common measure of a2+ab· and a2-5 ab + 6 b2. And of a3-b3 and a3-3 a2b+ 3 ab2—b3.

8. Add together the fractions

10. A number consists of two

is greater by unity than the first.

digits, the second of which

If the number be divided

by 4 and multiplied by 7 the result is a number consisting of the same digits inverted. Find the number.

THURSDAY, DECEMBER 9, 1852.
GEOMETRY.

1. Define a right angle. Prove that all right angles are equal. What extension of the axiom regarding straight lines is requisite in order to prove this?

2. Shew how to besect a given angle.

3. Make a triangle of which the sides shall be equal to three given lines, any two of which are together greater than the third.

Shew distinctly how and why the construction will fail if this condition be not satisfied.

4. If in two right angled triangles the hypothenuse of the one be equal to that of the other and one side of the one to one side of the other, shew that the triangles are equal, without assuming Prop. 47, B. I.

5. If from any point in the base of an isosceles triangle perpendiculars be let fall upon the equal sides, shew that the sum of these perpendiculars is independent of the position of the point in the base.

duced.

Discuss the case when the point lies in the base pro

6. If the sum of the squares upon two sides of a triangle

be equal to the square upon the remaining side, shew that the angle opposite to this latter side is a right angle.

7. In obstuse angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of 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.

8. The angle at the centre of a circle is double of the angle at the circumference upon the same base. Prove this, and thence shew that the angle in a semi circle is a right angle.

9. Two circles intersect in A and B: at A, the tangents A C, A D are drawn to each circle and terminated by the circumference of the other. If B C, B D be joined, shew that A B or A B produced will besect the angle C B D.

10. Shew that angles in the same segment of a circle are equal.

11. Shew that the three perpendiculars from the angular points of a triangle upon the opposite sides, pass through one point. If the feet of the perpendiculars be joined, shew that the triangle so formed has its angles bisected by the perpendiculars.

12. Under what conditions may a quadrilateral figure be inscribed in a circle?

Under what conditions may a circle be inscribed in a quadrilateral figure?

13. Find the sum of all the angles of a gnomon, explaining the result.

' 14. Inscribe a circle in a given triangle.

15. Divide a right angle into 15 equal parts.

The Ban. Sir Charles Dichalsan's Prizes.

SUBJECTS FOR 1853.

I. LATIN HEXAMETER VERSE.

Australia.

II. GREEK IAMBIC. (Trim. Acat.)

SHAKSPEARE.-" AS YOU LIKE IT : Act II. Scene 3. Begining-" But do not so: I have five hundred crowns." To-" Not my master's debtor."

The Exercises for these prizes must be sent in to the REGIStrar of the UNIVERSITY, on Monday, July 11, 1853, in an envelope with a motto, and accompanied by a sealed letter, containing upon the outside, the motto chosen by the writer, and upon the inside his name.