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TUESDAY, January 6, 1857. 13...4. 1. The imperial gallon contains 277.27 cubic inches, and a cubic foot of water at its maximum density weighs 62:42lbs.; find the weight of a pint of water correctly to two places of decimals.

2. Supposing the cost of digging a trench to vary as the depth to which it is sunk and the quantity of earth taken out, and that the cost of digging a trench 3 feet broad by 8 feet deep is 9 pence per yard, what should be the cost of digging a trench 120 yards long, 5 feet broad, and 10 feet deep ?

3. Define a fraction ; and from your definition prove a rule for adding together two fractions with different denominators. Add together the fractions, a2 - bc

C2 - ab (a + b)(a + c)' (b + c) (6 + a)' (c + a)(c + b) 4. Prove a rule for extracting the square root of a compound algebraical quantity. Shew that, if

24 + ax: + bx? + cx + d be a complete square, the coefficients satisfy the equation

0% - a?d=0. Is it necessary that the coefficients satisfy any other equation ?

5. Solve the equations,

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X + y P 6. Find the number of permutations of n things taken r together.

If the number of permutations of n things taken r together be denoted by the symbol

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shew that the number of such permutations, in which p particular things occur, will be

'Pp.n-PP-p. 7. Define a logarithm, and find logs 3125.

Prove that loga N = logab.logo N; and, having given log102=•301030 and log10 7= •845098, find log1998, and log1000

343

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5

4

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8. Define the sine of an angle, and prove from your definition that for all values of 0 numerically less than , sin (- 0)=sin 0. Trace the variation in sign of the expression cos (= sin ). cos ( T cos ),

TT as 0 varies from 0 to

2. 9. Find an expression for all the angles which have the same sine. Hence, if sin 38 be given, find the number of values of tan 0 which will be generally obtained ; and illustrate the result geometrically,

10. Prove the formula

cos (A - B)=cos A.cos B + sin A .sin B, A being greater than B, and each angle less than 90°.

Also shew that cos a + cos B+ cos y +cos(u+B+3)=4 cos (B+y).cos 3 (y+a).cos } (a +B), and

sin a +2 sin 3 a + sin 5 a sin 3a sin 3a + 2 sin 5 a + sin 7 a sin 5a

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11. Determine the expression for the cosine of an angle of a triangle in terms of the sides, and deduce the expression for the sine.

If 0 and $ be the greatest and least angles of a triangle, the sides of which are in arithmetic progression, prove that

4(1-сos 0) (1 - cos 0) = cos 0 + cos . 12. A quadrilateral can be inscribed in a circle; find the tangent of half of one of its angles in terms of its sides. If a circle can be inscribed in the quadrilateral, shew the fourth root of the product its sides is a mean proportional between its semi-perimeter and the radius of the inscribed circle.

.

WEDNESDAY, January 7, 1857. 9...12. 1. ASSUMING that the resultant of two forces, acting at a point, is represented in direction by the diagonal of a parallelogram, the sides of which represent the forces in direction and magnitude; shew that the diagonal will also represent the resultant in magnitude.

Shew that within a quadrilateral, no two sides of which are parallel, there is but one point, at which forces, acting towards the corners and proportional to the distances of the point from them, can be in equilibrium.

2. Shew that if three forces acting in one plane hold a body in equilibrium, they either pass through a point or are parallel to each other.

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A heavy equilateral triangle, hung up on a smooth peg by a string the ends of which are attached to two of its angular points, rests with one of its sides vertical; shew that the length of the string is double the altitude of the triangle.

3. Find the relation of the Power to the Weight in the single moveable pully, when the strings are not parallel.

An endless string hangs at rest over two pegs in the same horizontal plane, with a heavy pully in each festoon of the string ; if the weight of one pully be double that of the other, shew that the angle between the portions of the upper festoon must be greater than 1200.

4. Find the ratio of the Power to the Weight in the Wheel and Axle, in order that there may be equilibrium.

Explain the meaning of the terms 'mechanical advantage' and effici. ency,'as applied to the machines; and shew that, in the Wheel and Axle, what is gained in power is lost in velocity.

5. Define the centre of gravity of a heavy body; and determine the position of the centre of gravity of a pyramid on a triangular base.

Find the centre of gravity of the solid included between two right cones on the same base, the vertex of one cone being within the other; and determine its limiting position if the vertices approach to coincidence.

6. State the laws of friction; and explain what is meant by the term coefficient of friction.'

A uniform rod is held at a given inclination to a rough horizontal table by a string attached to one of its ends, the other end resting on the table; find the greatest angle at which the string can be inclined to the vertical without causing the end of the rod to slide along the table.

7. Define uniform motion and uniformly accelerated motion, and explain how they are measured.

If f be the measure of a uniform acceleration, when t minutes and a feet are taken as the units of time and space, and f' the measure of the same acceleration, when a' feet are taken as the unit of space, find the number of minutes in the unit of time.

8. State the second law of motion; and apply it to prove that a force, of uniform intensity and direction, acting on a given particle originally at rest, produces a uniform acceleration of its motion.

State the convention with respect to units which is necessary, in order that the equation P=Mf may represent the relation between the numerical measures of force, mass and acceleration; and supposing the unit of force to be 5lbs, and the unit of acceleration, referred to a foot and a second as units, to be 3, find the unit of mass.

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9. An elastic ball A, moving with a given velocity on a smooth hori. zontal plane, impinges directly on a ball B of the same radius at rest ; determine the velocity of each after the impact, indicating at what points of your reasoning any law of motion or other result of experiment is assumed.

The balls being equal in mass as well as in volume, shew that, if B afterwards impinge perpendicularly on a smooth wall, the original distance of which from the nearest point of B is given, the time, which elapses be. tween the first and second impact of the balls, will be independent of their radius.

10. Shew that a particle, projected in any direction not vertical, and acted upon by gravity only, will describe a parabola.

An inclined plane passes through the point of projection ; find the condition that the particle may impinge perpendicularly on the plane; and, in that case, shew that its range on the plane is equal to

2v2
g

1 + 3 sino a where v is the velocity of projection, and a the inclination of the plane to the horizon.

11. Two given weights are connected by an inextensible string, which passes over a smooth pully; determine the motion of each weight and the tension of the string.

The system being initially at rest, find the weight which, let fall at the beginning of the motion from a point vertically above the ascending weight, so as to impinge upon it, will instantaneously reduce the system to rest. Will the system afterwards remain at rest ?

12. A seconds pendulum is carried to the top of a mountain 3000 feet high; assuming that the force of gravity varies inversely as the square of the distance from the Earth's centre, and that the Earth's radius is 4000 miles, find the number of oscillations lost in a day.

Also determine how much the pendulum must be shortened in order that it may oscillate seconds on the mountain.

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WEDNESDAY, January 7, 1857. 11...4. 1. Give the meanings of the several symbols which are employed in the formula p=gpx.

If one second be the unit of time, what must be the unit of length, in order that the above formula may give the pressure in pounds, supposing the unit of volume of the standard substance to weigh 16 lbs ?

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2. Prove that the pressure of a fluid on any surface is equal to the weight of a column of the Auid, the base of which is equal to the area of the surface, and altitude equal to the depth of the centre of gravity of the surface below the surface of the fluid.

The inclinations of the axis of a submerged solid cylinder to the vertical in two different positions are complementary to each er; P is the difference between the pressures on the two ends in the one, and P' in the other position: prove that the weight of the displaced Auid is equal to

(P + P 2). 3. Describe an experiment to shew that the pressure of a given mass of air at a given temperature varies as its density. How is this ratio to be modified when the temperature, as well as the density, varies ?

A volume of air of any magnitude, free from the action of force, and of variable temperature, is at rest : if the temperatures at a series of points within it be in arithmetical progression, prove that the densities at these points are in harmonical progression. 4. A body of given volume is immersed totally in a given fluid ; find

l the magnitude and direction of the resultant Auid pressure.

A body is floating in a fluid ; a hollow vessel is inverted over it and depressed: what effect will be produced in the position of the body,(1) with reference to the surface of the fluid within the vessel, (2) with reference to the surface of the fluid outside ?

5. Describe the Diving Bell, and find the volume of the air in the bell at any depth below the surface.

If P be the weight of the bell, P' of a mass of water the bulk of which is equal to that of the material of the bell, and W of a mass of water the bulk of which is equal to that of the interior of the bell, prove that, supposing the bell to be too light to sink without force, it will be in a position of unstable equilibrium, if pushed down until the pressure of the enclosed air is to that of the atmosphere as W to P-P'.

6. Explain the principle of the common Barometer. Given the pres. sure of the air at a given time on a square inch, shew how to find the height in inches of the barometric column.

Why is the rising or falling of a barometer generally an indication of coming fair or foul weather ? Why is a sudden fall a sign of a coming gale?

7. Find the geometrical focus (1) of a pencil of rays incident directly upon a plane refracting surface, and (2) of a pencil of incident directly upon a refracting plate.

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