7. A tower stands on a horizontal plane, and it is observed that at a position on the plane, distant 237 feet from the foot of the tower, the angle of elevation of the top of the tower is 37° 15′ 15′′; find, to two places of decimals, the height of the tower.

9. Find expressions for the radii of the inscribed and escribed circles of a triangle ABC.

If r, ~1, ̃1⁄2, ̃1⁄2, represent these radii, prove that

IO. A spectator on one side of a straight road is looking in the direction perpendicular to the direction of the road, and observes that a telegraph post is exactly opposite and that a is the angular elevation of the top of this post. He then turns round and observes that ẞ is the angular elevation of the next post. Assuming that the height of each post is the same, and that c is the distance between the posts, prove that the height of each is equal to

II.

ctan a tan B÷√tan'a - tan2ß.

Two cylindrical vessels are filled with water; the radius of one vessel is six inches and its height one foot, and the radius of the other is eight inches and its height one foot and a half; find the radius of a cylindrical vessel eleven inches in height which will just contain the water in the two vessels.

12.

Having given that the length of each edge of a regular tetrahedron is four inches, determine, to three places of decimals, the number of square inches in the total surface of the tetrahedron.

Also find the number of cubic inches in the volume of the tetrahedron.

IV. STATICS AND DYNAMICS.

I. Enunciate and prove the proposition known as the "Polygon of Forces."

OA, OB, OC are three straight lines inclined at angles of 120° to one another; a force of 3P acts from A to O, a force of 4P from 0 to B and a force of 5P from O to C. Show by a carefully drawn figure how you would graphically determine the magnitude and direction of the resultant of the three forces.

2. B, C are two smooth rings fixed in space at a distance apart = 25 inches, B being 9 inches, and C 16 inches above the ground. A string ABCD passes through the rings and supports equal weights W, W at its extremities A, D. Find the resultant pressures of the string upon the

rings.

3. Two parallel forces 5P, 7P act at points A, B respectively. State clearly the position of the line of action and the magnitude of their resultant (i.) when the forces are like, and (ii.) when unlike.

A horizontal bar AB, 7 feet long, is supported at its extremities, and a man of 150 lbs. weight hangs from it by his hands, one being I foot from A, the other 3 feet from B. Find the pressures on the supports due to the weight of the man.

4.

Three forces (not parallel) acting in one plane upon a rigid body are in equilibrium. Show that if any triangle be formed by drawing straight lines perpendicular to the directions of the forces, its sides will be proportional to the forces.

A triangle ABC (whose weight may be neglected) rests in a vertical plane with the middle points of the sides AB, AC in contact with two smooth pegs, the line joining them being horizontal and parallel to the base BC. Determine graphically, or otherwise, the point in BC where a weight W may be placed without disturbing the equilibrium; and if W = 10 lbs., and AB, AC and BC be 4, 5 and 6 feet respectively, find the pressures on the pegs.

5.

Show that if a body be placed on a horizontal plane it will stand or fall according as the vertical line through its centre of gravity falls within or without the base. Explain what is meant by the "base."

Why does a bicycle rider when on the point of falling over to one side steer his machine to that side?

6. Distinguish between the three kinds of levers and give examples of each.

Two levers OA, OB of lengths 3 and 4 feet respectively can turn in a vertical plane about a common fulcrum 0, and their middle points are connected by a string whose length is 2 feet. Find the least force which applied at A will keep OB horizontal with a weight of 12 lbs. suspended from B. Find also the tension of the string.

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7. 'Suppose mud composed of coarser particles to fall at the rate of two feet per hour, and these to be discharged into that part of the Gulfstream which preserves a mean velocity of three miles an hour for a distance of two thousand miles; in twenty-eight days these particles will be carried x miles, and will have fallen only to a depth of y fathoms." (Lyell's Principles of Geology.)

Find the numbers given for x and y in this passage.

8. Assuming that the space described from rest in time t by a particle moving with uniform acceleration a is equal to hať“, deduce the corresponding expression for the case when the particle has an initial velocity u.

If the space described in the fifth and sixth seconds from rest be 25 feet, find the acceleration.

9. Distinguish between (i.) acceleration and accelerating force, and (ii.) the mass and weight of a body; and investigate an expression for the acceleration of a particle sliding down a smooth inclined plane under the action of gravity.

Two particles start simultaneously from rest, the one down an inclined plane AC of length 25 feet, the other down a plane BC of length 70 feet, the heights of A, B above the horizontal plane through C being 7 and 56 feet respectively. Find which particle will arrive at C first, and when at C how far it will be from the other particle.

IO.

Two bodies whose masses are P, Q are connected by a string which passes round a smooth pulley; find the acceleration.

Show that, if W be added to one of the weights in question (2), and W be taken from the other, the pressures on the rings will be unaltered.

II. A particle is projected at an angle of 45° to the horizon with the velocity it would acquire in falling freely for one second; show in a figure the position of the particle at the end of the first, second, and third seconds of the motion; and taking any unit of length to represent the initial velocity, mark against each straight line of your diagram the measure of its length.

FURTHER EXAMINATION.

V. PURE MATHEMATICS.

No

[Full marks may be obtained for about three-fifths of this paper. Candidate must attempt more than ten questions. Great importance will be attached to accuracy.]

I. A moveable circle with constant radius cuts the two fixed straight lines APR, AQS in P, Q, R, and S, prove that for all positions of the circle the sum of the arcs RPQ and PQS is constant.

If the roots of x2+ Px+q=o be a and ẞ, and the roots of x2+px + Q=0 bey and 8, find the roots of x2+px+q= 0 in terms of a, ß, y, and 8.

4. Find the number of combinations of n things r together.

There are 2n letters, of which 2 are a, 2 are b, and so on: find how many different algebraical products can be formed, in each of which the sum of the indices of the letters is 3.

5. Prove that the sum of the squares of the coefficients of (1+x)" is

2n nn

and find the sum of the products taken two and two together.

6. Expand loge(1+x) in ascending powers of x. Prove that the coefficient of x" in the expansion of