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I. GEOMETRY.

1. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes, &c.

2. The sides about the equal angles of equiangular triangles are proportionals; and those which are opposite to the equal angles are, &c.

3. If two straight lines be cut by parallel planes, they shall be cut in the same ratio.

4. The triangles formed by joining the corresponding extremities of three equal and parallel straight lines, not all in the same plane, are equal, and their planes are parallel.

5. The traces of two planes being given, find the projections of their intersection, when the traces of the given planes intersect in two points.

6. Draw a plane that shall pass through a given point and through a given straight line.

CO-ORDINATE GEOMETRY. :

7. Determine the equation to a straight line which is drawn through a point whose co-ordinates are x = 5, y'-2, perpendicular to the straight line whose equation is

y=2x+5,

and construct the line by scale.

8. The co-ordinates to the centre of a circle referred to rectangular axes are a and ẞ: find the equation to the circle. Also determine the position and magnitude of the circle which is the locus of the equation

9. Give the definition of 66 а conic section," and state when the conic section is a "parabola," an "ellipse," or an "hyperbola."

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II. ARITHMETIC AND ALGEBRA,

1. What sum of money lent at £3. 48. per cent. per annum, simple interest, will amount to £10000 in 7 years?

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2. A father left of his property to his eldest son, of the

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remainder to his second, and the rest to his third: the difference between the shares of the second and third was £1800: what was the share of each ?

3. Given the sum of three numbers equal to 13; the sum of the products of every two equal to 47; and the squares of the numbers in arithmetic progression; find the numbers.

4. When the price, per cwt., of tin to that of copper was as 3:2, 11 cwt. of gun-metal cost £64. 8s., but when the prices of tin and copper were reversed, 11 cwt. of gun-metal cost £89. 128. : what was the price of copper in the first case, and what the weight of copper and tin in the metal?

5. Solve the following equations:

6. The equation x3- 11x2+7x+147 = 0 has two equal roots: find them by means of the derived equation, and the third root by depressing the equation.

7. Decompose the fraction

x +5+1 x3- 2x2-x+2 3

into three fractions

having denominators of the first degree.

8. Investigate an expression for the number of shot in a square pile of n courses, that is, an expression for the sum of n terms of the series 12+ 2 + 32...... n2.

9. By means of a table of logarithms find the value of the

and

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III.

TRIGONOMETRY AND MENSURATION.

1. A and B being any two angles, show that

3. Show that, in any triangle, the sides are to each other as the sines of their opposite angles; and apply this to finding the remaining side and angles of a triangle of which two sides are 315-753, 238.825, and the angle opposite the greater of these two sides is 97° 13'.

4. From the top of a hill I observed two church spires exactly in the same direction on the horizontal plane below the hill: the angle of depression of the base of the nearer spire was observed 15° 23′, and of the further spire 4o 12': the top of the hill being known to be 315 feet above the horizontal plane on which the churches stand, it is required to find, from these observations, the distance between the two spires.

5. Two sides of a court-yard in the form of a parallelogram are 40 feet and 70 feet, and its shorter diagonal is 50 feet; what will it cost paving at 38. 9d. a square yard?

6. The side of an equilateral triangle, of a square and of a regular hexagon are in arithmetic progression; the sum of this progression is 36 feet; and the perimeters of the three figures are equal; find their areas.

7. The diameter of the horizontal bottom of a circular reservoir is 200 feet; the sides of the reservoir being inclined all round at an angle of 45o, find the number of cubic feet of water contained in it when the depth of the water is 10 feet.

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IV. STATICS.

1. Define the terms (1) "force," (2) "gravity," (3) "weight," (4) "resultant," and (5) state what is meant by the "parallelogram of forces."

2. Assuming the parallelogram of forces, show that if ƒ and f represent two forces acting at a point,

the angle which their

directions make with each other, and r their resultant,

find the resultant of two pressures acting at a point in directions making an angle of 60° with each other, and find also the angle which the resultant makes with the direction of the pressure 7 lbs.

3. The extremities of a string 14 inches long are fixed to two points 10 inches apart, in the same horizontal line, and a weight of 8 lbs. is suspended,

1st, from a knot in the string 6 inches from one extremity; 2nd, from a smooth ring which slides freely along the

find in both cases the tensions of each of the parts into which the string is divided, and state clearly the mechanical principles involved in each step of the investigation.

4. Define the centre of gravity of a body, or system of bodies; and show that if m, m,, m,, m, are bodies in a straight line, and a, a, a,, a are their distances from a fixed point in that line, the distance of their centre of gravity from the same point is

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am + am, + am + am 3

m + m21 + m2 + m2

5. If a and b be the two parallel sides of a trapezoid, and h the line which bisects these sides, show that the centre of gravity of the trapezoid will be in this line, and its distance from a along

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V. DYNAMICS.

A body projected vertically upwards from the bottom of a tower with a velocity of 60 feet per second reaches the top in 2 seconds: what is the height of the tower? and how much above the top does the body rise?

2. A weight of 3 oz. draws a weight of 12 oz. down a plane inclined 30° to the horizon, by means of a string passing over a pulley at the bottom of the plane: find the vertical descent of each of the weights in 3 seconds, the friction and inertia of the pulley being 1 oz., and g = 32 feet.

3. Investigate the expression for the velocity of a shot which, being fired at an angle of elevation e, shall strike a mark at the distance r, on a plane passing through the point of projection and making an angle i with the horizon.

4. Show how the value of e is determined from the equation

in terms of r, v, i; and prove, that the range on a given plane is a maximum when the direction of projection bisects the angle between the plane and the vertical.

5. At what elevation must a shot be fired with a velocity of 400 feet that it may range 2500 yards on a plane which descends at an angle of 30o ?

6. The length of the seconds' pendulum being 39·139 inches, find (1) the value of g the force of gravity, (2) the length of the pendulum which vibrates 80 times in a minute.

7. A clock intended to beat seconds gains 7.4 seconds a day: how must its pendulum be altered that it may go right?