3. What sum will amount to £57 17s. 74d. in 3 years, at 5 per cent. compound interest? SECTION III-1. If 60 yards of Holland cost £18, how must I sell it per yard to clear 8 per cent?

2. A debt of £1000 is to be discharged by monthly payments of £50. If it were proposed to pay the whole of this sum at once, when should the payment be made?

3. A bookseller purchases the copyright of a book for £40, and pays for the printing, paper, binding, &c., of 1000 copies, £45. If he should sell 100 copies a year, what would he gain per cent., allowing compound interest?

SECTION IV.-Solve the following Equations

SECTION V.-1. Investigate an expression for the sum of a series in arithmetical progression.

2. Extract the square root of 2 + √ 3.

3. Given the first term, the last term, and the number of terms of a series in geometrical progression. Investigate an expression for the sum of the

terms.

SECTION VI.-1. I take a number, multiply it by 34, take 60 from the product, multiply the remainder by 24, and after this subtract 30 when nothing remains. What is the number?

2. There is a number consisting of three digits which are in arithmetical progression. If the number be divided by the sum of its digits the quotient is 48, and if 198 be subtracted from it a number will be obtained consisting of the same digits, but in an inverted order. What is the number? 3. A person buys 124 head of cattle-pigs, goats, and sheep, for £400. Each pig cost £4 10s., each goat £3 3s. 4d., and each sheep £1 5s. How many were there of each kind?

SECTION VII.-1. Solve the Equation

23 + ах + b = 0

and investigate the conditions under which the roots are possible. 2. Approximate by the method of continued fractions to the value of

587

1943

and show generally, that, in such a series of fractions, the approximating values are alternately greater and less than the true one.

3, A sum of £a having been borrowed, £b are re-paid annually, partly as interest at £r per cent. upon the remaining debt, and partly in liquidation of it. In how many years will the debt be paid off?

GEOMETRY AND TRIGONOMETRY.

SECTION 1.-1. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

2. Equal triangles, upon equal bases in the same straight line, and towards the same parts, are between the same parallels.

3. 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.

SECTION II.-1. To divide a straight line, so that the rectangle contained by the whole and one of the parts shall be equal to the square of the other part. 2. In every triangle the square of the side subtending either of the acute

angles is less than the squares of the sides containing that angle by twice the rectangle contained by either of those sides, aud the straight line intercepted between the acute angle and the perpendicular let fall upon it from the opposite angle.

N.B. The first case only of this proposition is to be proved.

3. The sides about the equal angles of equiangular triangles are proportionals. SECTION III.-1. From one extremity of the base of an isosceles triangle a straight line is drawn perpendicular to the opposite side; show that it makes with the base an angle equal to one-half the vertical angle of the triangle.

2. Show that, if from the point of bisection of either of the sides of a triangle a straight line be drawn parallel to the base, the area of the triangle cut off by it will be equal to one-third the area of the whole triangle.

3. A rectangular field of a given area is to be enclosed from the waste, on the banks of a stream which supplies the place of a fence for one of its sides: show that the least length of fencing will be required for the other three sides, when that which is parallel to the stream is double the length of either of the others.

2. In any spherical triangle

c)

Cos a Cos b Cos c + Cos A Sin b Sin c

3. In any spherical triangle

Sin

Sin

(A B)
(A+B)

Tanc

SECTION VI.-1. At what rate is a place in 450 N. latitude carried per hour by

the diurnal revolution of the earth?

2. The height of the Peak of Teneriffe is 12,172 feet; at what distance at sea would it appear under an angular elevation of 13° 17'?

3. By what means can the distance from one another of two remote objects C and D be determined, by observations taken from stations A and B, whose distance from one another is known, and whence the objects C and D are visible?

SECTION VII.-1. Trace the line whose equation is

y= - ax + b

2. Investigate the polar equation to a parabola.

3. Investigate the equation to an ellipse referred to any conjugate diameter. SECTION VIII-1. The distances from one another of three objects in the same

horizontal plane but not in the same straight line, being known; I observe the angles they make with one another from a distant point in the same plane. How can the position of this point, in respect to them, be determined, 1st, by construction, and 2ndly, by calculation.

y = n Sin x--- n2Sin 2 x + n Sin 3 x - &c.

3. Investigate the general equation to the surface of an oblique cone on a circular base and the equation to the intersection with it, of a plane passing through the centre of the base; showing under what circumstances the section is a circle, a parabola, an ellipse, or a hyperbola.

MECHANICS AND HYDROSTATICS.

SECTION I.-1. A rope, each foot of which weighs 6lbs. and which has, suspended from its extremity, a weight of half a ton, is to be raised from a depth of 60 fathoms in 4 minutes. How many horses' power must be employed in raising it?

2. How many bushels of coals must be expended in a day of 24 hours, in raising 150 cubic feet of water per minute, from a depth of 100 fathoms -the duty of the engine being 60 millions?

3. The piston of an engine is 3 feet in diameter: the length of the stroke is 6 feet, and 6 strokes are made per minute; under what effective pressure must the engine work, that it may yield 75 horse power upon the piston? SECTION II.-1. A rod 16 feet long, is of uniform thickness, and weighs 13lbs., a weight of 25lbs. is suspended from one extremity, and one of 9lbs. from the other extremity. On what point will it balance?

2. A stone is let fall from the top of a tower, and one second afterwards another stone is let fall from a window distant 100 feet below the top. When and where will the one stone overtake the other?

3. A train weighing 50 tons is impelled along a horizontal line by a constant pressure of 410lbs.; what space will it have described in the first 10 minutes of its motion?

SECTION III-1. There is a water wheel which is worked by a stream whose section is 2 feet by 3 and its mean velocity 2 feet per second. The fall is

15 feet, and the modulus of the wheel .6; it is used to raise water from the upper level of the stream to a height of 40 feet above it. How many cubic feet will it raise per minute?

2. Steam is admitted into the cylinder of an engine whose stroke is 10 feet, at a pressure of 34lbs. per square inch, and cut off at one-fourth the stroke. How many units of work will it do per stroke on each square inch of the piston?

3. A weight of 1lb. is attached to the extremity of a string 3 feet long, which is made to perform 25 revolutions horizontally in 1 minute. What is the tension upon the string?

SECTION IV.-1. Show that the work accumulated in a moving body is equal to one-half its "vis viva."

2. A block of granite 50 feet long, 24 feet wide, and 1 foot thick, each cubic foot of which weighs 164lbs., is supported in an inclined position, resting on its end, by means of a rope 60 feet long fastened to a point distant 3 feet from its top, and fixed to the ground at a distance of 25 feet from the point on which it rests. What is the tension on the rope? 3. A rectangular beam of given dimensions is allowed to fall over on its extremity from a vertifical into a horizontal position. Investigate a formula determining the angular velocity it will acquire.

SECTION V.-1. What is the pressure upon the plug of a water-main 2 inches in diameter, situated 100 feet beneath the surface of the reservoir which supplies the main?

2. What is the total pressure upon a flood-gate 36 feet high, and 12 feet wide, when the water reaches to its surface, and what is the pressure upon the lower half of the gate?

3. A body is to be drawn along a horizontal plane; investigate generally the best direction of traction, friction being taken into account.

SECTION VI.-1. There is a sluice 3 feet high, and 1 foot wide, situated at 20 feet from the top of a flood-gate. What is the pressure upon this sluice when the water reaches to the top of the flood-gate, and where may a single horizontal pivot be placed, so as to sustain it?

2. What are the general conditions of the equilibrium of a floating body?
3. A cylinder is allowed to roll down an inclined plane. Investigate a
general expression for the velocity it will acquire.

SECTION VII-1. Find the centre of gravity of a triangle.

2. Find the centre of gravity of any portion of a circular ring.

3. Investigate the moment of inertia of a sphere about an axis touching its

surface, and find the time of oscillation of such a sphere when suspended from a thread of inappreciable thickness.

GEOGRAPHY AND ASTRONOMY.

SECTION I.-By what familiar illustrations would you explain to a class in your school?

1. The isolation of the Earth in space.

2. The heat of tropical regions of the Earth.

3. The apparent rotation of the Heavens.

SECTION II.-1. Why do the time of high water, and its height, depend upon the age of the moon; and why does it return twice in every twenty-four hours? What are co-tidal lines?

2. By what provision are the lengths of the days and nights of different regions varied throughout the year; and in what respects are the wisdom and goodness of God apparent in this provision?

3. What is meant by mean temperature, and what by isothermal lines? How are these related to vegetation?

SECTION III.-In what parts of the world is the difference of the extreme temperatures of winter and summer most remarkable ? What remarkable contrast is there between the distribution of mean temperature and that of extreme summer heat, and in what respects are the wisdom and goodness of God apparent in it?

2. How does the proximity of a sea-board, and the indentation of the land by gulphs and arms of the sea, affect the temperature; and for what reasons? 3. Why, beyond the tropics, do the western sea-boards of the great continents enjoy a milder climate than their eastern sea-boards?

SECTION IV.-1. What winds prevail in tropical regions, and what in higher latitudes? How would these winds influence a navigator in shaping his course from Liverpool to Sidney and back? On what causes are they dependant?

2. What is meant by the dew-point? Why does more rain fall in mountainous, than in level districts?

3. What is the nature, and what are the objects of Mercator's projection of the sphere?

SECTION V.-1. Give some account of the central mountain system of Asia, and its dependant rivers?

2. What is the area of Great Britain? and what proportion of it is under cultivation?

3. What is the population of Great Britain, what proportion of it is employed in agriculture, and what in manufactures?

SECTION VI-1. Where are the forests of Arden, Dean, and Sherwood?

2. Where are the vales of Blackmore, Clwyd, and Avon?

3. Give some account of the agriculture of Essex, Cheshire, and Hertfordshire.

SECTION VII.-1. What is that range of mountains known as the Pennine range? 2. What are the principal coal fields of Great Britain, and what manufacturing districts are situated upon them respectively?

3. Enumerate the British West India Islands and the British possessions in America.

SECTION VIII.-1. Describe the apparent motions of the planet Venus, an account for the periodical variations in her brightness.

2. Why do not eclipses occur every month? and why do nearly similar eclipses return every nineteen years?

3. How has the distance of the sun been measured? and how has it been discovered that the earth's orbit is an ellipse?

SECTION IX.-1. Explain fully what is meant by parallax, and show that the region of the fixed stars is vastly distant as compared with the diameter of the earth's orbit.

2. What is that property of light which is called refraction? To what law is it subjected, and what phenomena of the visible heavens result from it?

3. What is the phenomenon known as the aberration of light, by what is it caused, and in what way does it serve to prove the annual revolution of the earth ?

SECTION X. 1. What is the difference between mean solar and sidereal time? 2. What is the difference bet een mean time and apparent time? Knowing the hour angle of the sun, how can the mean time be calculated? 3. How may the sidereal time be calculated from the observed altitude of a star whose declination and right ascension are given?

Intelligence.

CAMBRIDGE CLASSICAL TRIPOS,

MARCH 23rd, 1847.

EXAMINERS.

Francis France, M.A., St. John's College.

Francis Whaley Harper, M.A., St. John's College.

Benjamin Wrigglesworth Beatson, M.A., Pembroke College.
John Alexander Frere, M.A., Trinity College.