2. Prove the rule for dividing one fractional expression by another.

2. What number is that, the double of which exceeds its half by 6? The difference of two numbers, and a quarter part of their sum, are each equal to 2; find the numbers.

3. If A does a piece of work in 10 days, which A and B can do together in 7 days, how long would B take to do it alone?

Find the amount of P£ at compound interest for n years, the interest being paid yearly.

HIGHER BRANCHES OF MATHEMATICS.

Section 1.

1. Find x from the equation 12x=

35.

A draper bought a piece of silk for £16 48., and the number of shillings which he paid per yard was the number of yards. How much did he buy?

2. Find x and y from the equations x • y = 4, x2 + y2

=40.

There is a rectangular field whose length exceeds its breadth by 16 yards, and it contains 960 yards: find its dimensions.

3. A fast passenger train starts at 20 minutes past 12; it overtakes a luggage train which travels 15 miles an hour, and after having gone 15 miles further, overtakes a slow passenger train which travels 20 miles an hour. Another fast train, which travels at the same rate as the first, starts from the same station at 2 o'clock of that day, overtakes the luggage train, and, after having gone 65 miles farther, overtakes the slow train also, and finds that it has then travelled 120 miles. Required the rate at which the fast trains travel.

Section 2.

1. Prove the formula for finding the sum of an arithmetical series. The first term of an arithmetical progression is 1; the common difference 1; the sum of the series 36. Required the number of terms. 2. When are quantities said to be proportional? And when is one quantity said to vary as another?

Prove that, if a b c d; then a abc: cod.

3. Find the number of different combinations that may be made of n different things, taking r of them together.

Section 3.

1. In a circle the angle in a semicircle is a right angle, but the angle in a segment greater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle. 2. Describe a circle in a given triangle.

3. Similar triangles are to each other in the duplicate ratio of their homologous (or corresponding) sides. Prove this, and indicate the steps of the proof of the corresponding proposition for all similar figures.

Section 4.

1. Define the sine and the tangent of an angle. What are the values of sin. 90°, tan. 180°, and tan. 45°? Prove that cos. 2 A= 2 cos. 2 A112 sin. A.

2. Having given one side of a right-angled triangle, and the angle adjacent to that side, show by what calculations the other parts of the triangle may be obtained.

3. Write down the expression for the cosine of an angle of a triangle in terms of the sides, and prove the expression for the sine of an angle, and for the area of the triangle, in terms of the sides.

4. What is the logarithm of a number? What are the properties of logarithms on which the utility of logarithmic tables depends?

Section 5.

1. Prove the expressions for the circumference and area of a circle in terms of its radius.

2. Show that the solid content of any cone, or pyramid, is found by multiplying the area of its base by one-third its height.

MECHANICS.

[Note. The following Paper on Mechanics contains Questions taken exclusively from the Rules given in Tate's Elements of Mechanics,' which, for the present, has been recommended as a convenient Manual. Success in answering these Questions will be received as evidence of merit in any Candidate for a Certificate. Though the Committee of Council are desirous to promote the general introduction of these studies, from their obviously great practical utility, they do not require that every Candidate should succeed in answering this Paper in the present year.]

Section 1.

1. Define the unit of work, and show that if a pressure of m pounds be exerted over a space of n feet, the number of units of work done is represented by m + n.

2. A locomotive engine working at 40 H.P. ascends an incline of 1 in 250 steadily at the rate of 25 miles per hour; what is the weight of the train? 3. The traction of a waggon upon a level road is

Leb

m

th of the gross load

W; there is an ascent on this road of 1 in n; show that if the friction on the ascent be supposed the same as that on the level the traction P up it is represented by the formula P = W (

Section 2.

n

1. There is a fall upon a stream of 11 feet, down which 22,400 lbs. of water descend per minute, and on which there is erected a water-wheel whose modulus is 6; what is its horse power?

2. A well, 100 feet deep and 5 feet in diameter, is to be deepened 30 feet. Two men are employed in the work. The material is such that four times as much time is employed in the use of the pick as of the shovel. Supposing that each man could, when using the shovel alone, throw out 400 cubic feet into the vessel which conveys it to the surface in one day, how long would they be in completing the work?

3. Three men undertake to pump out the water from a shaft a feet in

depth, working in succession: how must they divide the work that each may do an equal share of it?

Section 3.

1. What must be the length of the stroke of the piston of an engine whose area is 1000 square inches, that, making 20 single strokes per minute under a mean effective pressure of 15 lbs. per square inch, the engine may yield 10 H.P. after one-fifth of the work done on the piston has been lost by friction?

2. State concisely the method by which the number of units of work done per stroke upon each square inch of the piston of an engine may be determined when the steam is worked expansively.

3. Investigate an expression for the work accumulated in a body of a given weight moving with a given velocity.

4. A train which weighs 400 tons is travelling at the rate of 20 miles an hour; what friction must be put upon it by the breaks, in addition to the friction of the rail, that it may be brought to rest within the space of 200 yards, the steam being thrown off?

Section 4.

1. State concisely the statical principle of the equality of moments, and describe a method of proving it by experiment.

2. Investigate an expression for the velocity acquired by a body falling by gravity freely through a given space.

3. Show how it may be determined whether a pillar will stand or fall, when any given pressure is applied obliquely to its summit.

4. Show generally how the traction of a body up an inclined plane, subject to friction, may be determined, and investigate the direction of least

traction.

MENSURATION.
Section 1.

1. Prove the rule of cross multiplication.

2. Prove a rule for determining the number of standard rods of brickwork in a wall.

3. Prove a rule for determining the area of a trapezoid.

Section 2.

1. How many cubical feet of timber are there in the flooring of a room in. thick, and 17 ft. 6 in. in length by 15 ft. 3 in. in breadth?

2. In a wall, 10 ft. high, 15 ft. long, and 2 bricks thick, there is an arched doorway 4 ft. wide and 6 ft. high to the springing of the arch, which is semi-circular; how many standard rods of brickwork are there in the wall?

3. What is the weight of a circular iron ring whose inner diameter is 18 in., and whose section is 2 in. in diameter, the weight of a cubic foot of the iron being 450 lbs.?

Section 3.

1. After measuring a piece of cloth to contain 90 yards, I find that the yard measure that I have used is too short by 1-30th part; what is the true measure of the cloth?

2. How many square inches of tin plate are required to make an open cylindrical vessel to contain a gallon whose height is equal to one-half its diameter?

N.B. An imperial gallon contains 277,274 cubical inches.

3. Show that less tin will be used in making a vessel of the dimensions

given in the last example than in making one of any other dimensions, but of the same cylindrical form and capacity.

4. Investigate Thomas Simpson's rule for determining the area of a plane surface bounded by an irregular line.

5. Investigate a formula for determining the quantity of earth to be taken out of a cutting for a road, the width of the road and the slope of the banks being given.

1. Describe Gunter's chain.

Section 4.

2. Construct a field-book for a three-sided field, of which one side has an irregular form, and the other two are straight lines, assuming any dimensions whatever.

3. Describe and explain the vernier.

4. Describe the spirit level and its adjustments.

GEOMETRY.
Section 1.

1. Define a plane superficies, and a circle.

2. Draw a straight line perpendicular to a given straight line from a given point within it.

3. The angles at the base of an isosceles triangle are equal to each other, and if the equal sides be produced the angles on the other side of the base shall be equal.

Section 2.

1. Prove that the sum of the three angles of a triangle equals two right angles. Given the value of two angles of a triangle, how is the value of the third ascertained?

2. Prove that parallelograms on the same base and between the same parallels are equal to one another.

3. In any right-angled triangle, the square which is described upon the side subtending the right angle is equal to the square described upon the sides containing the right angle.

Section 3.

1. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts together with the square of the line between the points of section is equal to the squares of half the line.

2. In any triangle the square of the side subtending any acute angle is less than the squares of the sides containing that angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle and the acute angle; prove only the first case of this proposition.

Section 4.

1. Construct a triangle whose area shall be equal to that of a given trapezium.

2. Show how to make a square double a given square.

3. Show that the diagonals of a parallelogram bisect each other.

POPULAR ASTRONOMY.

Section 1.

Give one reason, and that the simplest, for believing

1. That the earth is isolated in space.

2. That the form of the earth is nearly that of a sphere.

3. That the dimensions of the earth are those assigned to it in books on astronomy.

Section 2.

.1. Explain the phases of the moon, and illustrate your explanations by a diagram.

2. Under what circumstances is an eclipse of the sun total, or partial, or annular?

3. How often would similar eclipses return if there were no regression of the moon's nodes?

Section 3.

1. Describe the apparent path of the sun in a summer's day in the Arctic Circle.

2. Show that more of the sun's rays fall on a given portion of the earth's surface when they are incident vertically, than when obliquely.

3. The extreme summer heat of Moscow is equal to that of Nantes, that of Tobolsk to that of Cherbourg, and that of Astrachan to that of Bordeaux. Account for these and similar facts.

Section 4.

1. Account for the apparently retrograde motion of the planets.

2. On what causes do the variations of brightness in the planet Venus depend?

3. What is meant by parallax? To what uses is the consideration of parallax applied, and what conclusions have been drawn from it?

PHYSICAL SCIENCE.
Section 1.

1. Mention proofs of the extreme divisibility of matter.

2. What do you understand by density? How are the densities of different bodies compared ?

3. Explain the use of the terms heat and cold. What is latent heat? What specific heat?

4. How does a thermometer enable us to compare the temperatures of bodies?

Section 2.

1. Draw a diagram of Bramah's hydrostatic press, and show how to determine the pressure which may be produced by means of it.

2. There is a barge whose section is an equilateral triangle, each side being a feet in length, and whose length is b feet, how deep will it sink in the water, its weight and that of its lading being together w lbs. ?

3. Define the centre of pressure of a fluid, and show that the centre of pressure of a rectangular flood-gate is situated at two-thirds the depth.

Section 3.

1. How does the intensity of the light of a candle vary with the distance of the illuminated surface from the flame? Illustrate your meaning by a numerical comparison in some supposed distance.

2. If a ray of light passes from air into water, what change takes place in its direction, and according to what law? Under what circumstances is it impossible for a ray to pass out of water into air?