6. Two clocks are together at 12; when the first again comes to 12 it has lost a minute, and when the second comes to 12 it has gained a minute. How long must elapse before they are a quarter of an hour asunder?

7. Between 1801 and 1811 the population of Edinburgh increased by 24 per cent., and in the latter year it was 102,987; what was it in the former?

What would it have become in 1821 had the yearly rate of increase from 1811 been 2 per cent.?

8. At an examination, the respective marks attainable are as follows:-For classics, 1500; for English, 1200; for mathematics, 1400; for modern languages, 1000; for geography, 750. The relative proficiency of two candidates in these five departments is indicated by the following numbers:A's, 3, 4, 2, 5, 1; B's, 4, 3, 5, 0, 2; which should stand first in the examination?

2. If I buy 15 dozen of wine, 6 bottles containing a gallon, for £35, 10s., at what rate must I sell it per quart so as to gain £15 per cent.?

10. Of what is the logarithm? Find approximately the logarithm of 6.

ALGEBRA.

1. Multiply x-7xy-8y2 by 8y2-x2-7xy, and 1+2x+3x2+ &c.+nx-1 by 1-2x+x2.

2. Divide x3+y3+z3-3xyz by x+y+z; and admitting that 4b2c-(b+c-a) is divisible by a+b+c; resolve it into simple factors.

3. Extract the square root of

(ab)-2(a+b3)(a−b)2+2(a*-+b1),

and the cube root of

27x-27xy-45x1y2+35x3y3+30x3y+-12xy -8y".

4. Add together the fractions

(3.)

ca

+

(b−c)(c—a)' (a—c)(ba) † (a−b)(b—c)'

+

5. Prove that

x2+ y2+22xy+xz+yz;

and that the hypotenuse of a right-angled triangle, together with the perpendicular on it from the right angle, are greater than the two sides of the triangle.

6. Solve the following equations:

2x-1 3x-2 5x-4 7x+6

(1.)

(2.) '(5a+x)+√(5a—x)=√(10a+2b).
(3.) x2+y2=7+xy, x3+y3=6xy-1.

7. Sum the arithmetical series

1+3+5+etc. to n terms;

and the geometric series

1+3+etc. to 20 terms.

A sum of money rather more than doubles itself in 15 years at 5 per cent. compound interest. Suppose the interest to be such that the sum is exactly doubled in 15 years; it is required to determine how much a penny would amount to from the date of the death of Wallace, 1305, to 1860.

GEOMETRY.

1. Define a square; and a parallelogram. Prove that a square is a parallelogram. In what respect is Euclid's definition of a square faulty? What consequence can be deduced from it, if it be admitted?

2. If the square described upon one side of a triangle be equal to the sum of the squares described upon the other two sides of it, the angle contained by these two sides is a right angle.

3. If a straight line be divided into two equal and also into two unequal parts; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section.

If the point which, in this proposition, divides the line into two unequal parts, be taken in the line produced, what change ought, from the nature of the case, to be made in the enunciation ?

2 A

4. The angle at the centre of a circle is double the angle at the

circumference.

Assuming that there is no limitation as to the magnitude of an angle, deduce from this proposition, in immediate corollaries, the propositions that "the angles in the same segment of a circle are equal," and that "the opposite angles of a quadrilateral described in a circle are together equal to two right angles."

5. To inscribe an equilateral and equiangular pentagon in a given circle.

6. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular.

7. Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another which the polygons have; and that ratio is the duplicate ratio of homologous sides.

TRIGONOMETRY, ETC. .

1. Prove the relations which exist between radius and (1) sine and cosine, (2) tangent and cotangent, (3) secant and cosecant.

2. The three sides of a triangle are 148, 195, and 329; find its

area.

3. The three sides of a triangle are 2, √√6, and 1+√3; find the least angle.

4. Prove that R sin. (A + B) = sin. A cos. B+ cos. A sin. B. Find the numerical value of sin. 75°.

5. Determine the volume and convex surface of a cone, the area of whose base is 16 feet, and whose slant height is 3 feet.

PHYSICS.

1. If a straight lever support weights in equilibrium, they are to one another inversely as the lengths of the arms which support them.

2. Two forces, represented by 1 and 2, act at an angle of 45°; find the magnitude and direction of their resultant.

3. Water weighs 1000 oz. avoirdupois per cubic foot; find the pressure on a square yard 100 feet below the surface of the water.

A man just floats in water, what is his specific gravity? If he weighs 14 stone, what is his cubic measure?

4. Explain the action of the siphon. How does this action account for intermitting springs?

5. Enunciate the laws of reflexion and refraction of light. Account for total internal reflexion, and mention some of its applications. Find the positions of the successive images of objects placed between two mirrors inclined to one another by the angle of 30°.

6. State the physical explanations of sound and light. Give the

relative values of the notes in the diatonic scale. Are the values of the different colours in the solar spectrum similarly related?

APPENDIX II.

RESULTS OF EXAMINATIONS OF SCHOOLMASTERS

AND SUBSTITUTES.

TABLE NO. I.-SCHOOLMASTERS.

THE classification of Teachers, with respect to their examinations, during the ten years from 1854 to 1863 inclusive, has been made under four divisions, as shown from the Table, viz. :—

I. Teachers who have appeared for examination for the first

II. Teachers who did not pass the examination in whole at their
first appearance, but were examined a second time,
III. Teachers who failed to pass their first examination, and
appeared for a second examination,

49

6

IV. Teachers who have never been examined at all, although

two or more examinations have taken place since their
appointment,

12

These classes are further subdivided, thus, Class I., containtaining 61:

:

Teachers who have passed the whole of the examination at their first appearance, and received a special extra allowance as a mark of the Trustees' approbation for distinguished proficiency,

Teachers who have passed the whole of the examination at their first appearance, and received the usual allowance for scholarship, but without special commendation,

Teachers who have passed only part of the examination at their first appearance, but completed their trials at next appear

Teachers who have failed on a second appearance, but passed at a second examination,

Teachers who have died or resigned before time for second
appearance,
Teachers who have made a first appearance, but have not ap-
peared to complete examination,

Teachers who have failed at second appearance, and have not appeared for a second examination,

37

లు

3

Teacher who has made a first appearance, but has not returned to complete examination, being in ill health,

Class II., containing 49:

61

Teachers who, having passed only part of the examination at their first appearance, have returned and completed their trials at a second appearance,

41

Teachers who have failed on second appearance, but passed at a second examination,

3

[Two of these appeared twice at second examination, and one
passed at first appearance.]

Teachers who have failed on second appearance, and have not
returned for a second examination,

Class III., containing 6:

Passed second examination at first appearance,

5

49

1

Do.

at second appearance,

Failed to pass at second appearance,