Appendix: C.

it in equilibrium, the sum of the moments of the forces tending to turn it in one direction is equal to the sum of the moments of those tending to turn it in the other.

How does the moment of a force measure its effect to turn it round a fixed point?

(3.) Assuming the parallelogram of forces, determine the resultant of any number of forces in the same plane acting on a point.

At any point in the circumference of a circle, two equal forces act in directions passing through two fixed points on the circumference. Shew that the resultant of these forces passes through a fixed point.

(4.) Find the ratio of the power and weight in that system of pullies where each hangs by a separate string (1) when the strings are parallel (2) when they are inclined to the horizontal bar at angles 01, 02, 03, &c., respectively.

Suppose the number of parallel strings to be 8, and 1, 2, 3, &c. inches, their respective distances from each other, find where the weight must be attached to the cross bar in order that it may be horizontal: the weights of the pullies being taken into consideration.

(5.) Explain the term virtual velocity; and apply it to find the condition of equilibrium on the screw. Would it be applicable if there were no friction between the outer and inner screw?

(6.) All couples tending to turn a system in the same direction, are statically equivalent whose planes are parallel and moments equal.

How are couples estimated numerically, and why?

(7.) Find the distance of the centre of gravity of the frustum of a cone from the base; a and b being the radii of the two ends, and c the altitude of the frustum.

(8.) ABGC, DEF, are two horizontal levers without weight, B and F their fulcrums,

the end D of one lever rests upon the F end of C of the other, Hk is a rod without weight suspended by two equal parallel strings, from the points E and G. Prove that a weight P at A will balance a weight W placed anywhere on Hk.

(9.) A uniform rod rests on a smooth fulcrum with one end on a rough horizontal plane, show that the extreme position in which it will rest is given by the equation

2a being the length of the rod h, the height of the fulcrum above the plane and p = tan a.

FOURTH CLASS.

PLANE TRIGONOMETRY.

Morning Paper.

(1.) Explain the principle by which the signs of the Trigonometrical lines in the different quadrants are determined; and from this give the proper signs to the tangent, secant, and versed sine in the third quadrant.

(2.) Expand cos (A-B) when A is 180, and < 270°, and B of the form (180-C), where C is 45°. Construct the figure for the quadrant in which the angle (A-B) may

be situated.

(3.) Find the number of degrees both French and English in an arc, which is equal to the length of the radius.

Find the length of an arc subtending an angle of 11° 9' 36" in a circle whose radius is 50 yards.

(4.) Pro

and explain the meaning of the double sign in both results.

(6.) Prove Demoivre's theorem when the index is fractional, and shew that it has as many values as units in the denominator of the index.

(7.) Express the length of an arc in terms of its tangent, and apply the formula to obtain a rapidly converging series for calculating π.

(8.) A person standing at the edge of a river observes that the top of a tower on the edge of the opposite side subtends an angle of 55° with a line drawn from his eye parallel to the horizon; receding backwards 30 feet, he then finds it to subtend an angle of 48". Determine

the breadth of the river.

(9.) Having given the logarithm of two consecutive numbers to find the logarithm of a number next superior.

Construct a table of proportional parts by which the logarithms of all numbers between 3.75450 and 3.75460 may be computed, and prove the process.

(10.) Show fully how to construct a table of natural sines. What is the use of formulæ of verification? Prove one.

FIRST CLASS.

ASTRONOMY.

Afternoon Paper.

(1.) Define the terms Pole of the heavens, Meridan, Zenith, Equator. What two causes principally prevent the line joining the centre of the earth with a point on its surface from being, in general, the vertical line at that point? At what point on the Earth's surface is it vertical.

(2.) Explain the cause of the change of the seasons. In different years are they of different lengths?

(3.) Describe the transit instrument and the errors of adjustment to which it is liable. Find the azimuthal deviation from the meridian of a transit instrument, from the observed superior and inferior transits of the same circumpolar star.

(4.) Enumerate the different methods of finding the latitude of a place on the Earth's surface.

Show how to find the latitude and hour angle, from two altitudes of the sun and the time between.

(5.) What different kinds of time are employed in Astronomy?

When is it 0h 0m 0s according to each. What is Equinoctial Time?

Given the length of the mean tropical year equal to 365d 5h 48m 51.6s, find the length of the sidereal day.

(6.) Explain the physical causes of the Precession of the Equinoxes. And shew that the precession of a star in right ascension in t years

=t50". 2 (cos + sin o tan & sin a.)

(7.) Explain the cause of Astronomical refraction, and the effect produced by it on the apparent positions of the heavenly bodies.

Determine the coefficient of refraction from observations of circumpolar stars.

(8.) What is parallax? Express the parallax of a heavenly body in terms of its distance. from the earth, its observed zenith distance, and the radius of the earth.

If p be the moon's parallax, find approximately the greatest proportional error which would arise in putting sin p = p, cos. p. = I supposing the greatest horizontal parallax of the moon to be 1°.

Appendix C.

Appendix C.

(9.) Explain the cause of aberration. By whom was it discovered and, in what manner? By what observations had the velocity of light been previously determined?

Shew how to find the aberration of a given star in latitude and in right ascension. (10.) What is the equation of time? Explain the cause to which it is principally due. Shew that whatever be the position of the perihelion of the earth's orbit, it must vanish four times a year.

(11.) What is the reason that in tropical climates the twilight generally is very short compared with its duration in higher latitudes?

Find at what times of the year the twilight is shortest, and its duration then in London, the latitude being 51° 30', assuming that near the equinoxes (March 21, September 22,) the sun moves with a motion in declination of 23 daily.

(1.) It is found that on mixing 63 pints of sulphuric acid, whose specific gravity is 1·82, with 24 pints of water, one pint is lost by their mutual penetration; find the specific gravity of the compound.

(2.) Suppose a vessel one foot long, nine inches wide, and 13 feet deep, to be filled with water to 15 of the top: what sized cube whose specific gravity is heavier than water, 18 should be placed in it to make the water reach the brim.

(3.) A cylinder floats in water, its base being 4 inches below the surface, when an ounce weight is placed upon it it sinks another inch; shew that its weight is 4 ounces.

(4.) A person employs three sets of men to pump the water from a well which is 20 feet deep and 6 feet in diameter; the pressure of the atmosphere being equal to a column of water 32 feet in height-and the pump discharges 1017.8784 cubic inches of water at every stroke. How must they divide the work so that each may do an equal share of it, supposing the well to be quite full at the commencement, and that the first set of men finish their work previous to the commencement of the second, and the second before the third.

(5.) Two conjugate diameters are produced to intersect the same directrix of an ellipse, and from the point of intersection of each one a perpendicular is drawn on the other, prove that these perpendiculars will cut one another in the nearer focus.

(6.) Find the locus of a point such that if from it a pair of tangents be drawn to an ellipse, the product of the perpendiculars dropped from the foci upon the line joining the points of contact shall be constant.

(7.) Shew that the equation to the locus of the middle points of all chords of the same length (QC) of an an elipse is

(1.) State the third law of motion, and explain the several terms in it; apply it directly to the following question. Two bodies, whose masses are given, are placed on a horizontal table, at the extremities of a fine elastic string, which is stretched; determine the motion. If the bodies are inelastic, and impinge on each other with the velocity acquired, what will be the motion after impact?

(2.) Two smooth bodies of given masses moving with given velocities strike directly against each other. It is required to find the velocity of each, after impact.

(3.) Prove the formulæ v=ft, s = I fi2

Divide the length of an inclined plane into two parts, so that the times of descent down them may be equal.

(4.) Shew that the curve described by a projectile is a parabola, and the velocity at any point is that acquired by falling from the directrix.

(5.) To find a point where a projectile will strike an inclined plane through the point of projection, and its distance, or ranged on the inclined plane; find the greatest height which the projectile attains above the plane.

(6.) What

(6.) What must be the inclination of a cannon to the horizon, and the velocity of a ball projected from it, that the latter may strike the ground at two miles distance, after having just passed over a hill 100 feet high at the distance of one mile, neglecting the resistance of the atmosphere?

(7.) If a body be thrown directly upwards with a given velocity, the resistance of the air being = kv2 where k is small, find the height to which it ascends, and the time of

ascent.

(8.) A body oscillates in a cycloidal arc, acted upon by gravity and by a small constant retarding force (f) in the direction of its motion at every point; shew that the time of oscillation is the same as if this force had not acted, and that the decrement of the arc 2fl g

described in one oscillation =

(9.) A perfectly elastic ball falls from a height h, on a plane inclined 30 degrees to the horizon; shew that it will strike the plane again after an interval equal to twice the time of its fall, and that its range on the plane will be 4 h.

(10.) A spherical particle of which is the elasticity, is projected with a velocity v at any angle of projection a, and at the instant of attaining the greatest altitude strikes

v

a similar equal particle falling downwards, with a velocity equal to at the point of collision; to find the distance of the particles at the end of t seconds after impact.

FOURTH CLASS.

PLANE TRIGONOMETRY.

Afternoon Paper.

(1.) Having given the three sides of a triangle, give the different methods of calculating the angles; and show which is best when one side is very large compared with the other two.

(2.) Explain the apparent absurdity of assuming x +

mỹ

Assuming (cos mo—√ — 1 sin m0)

=

(cos 0

1

= 2 cos 0

x

m

1 sin 0) express tan mo in terms of tan 0, and its powers, and shew clearly how you determine the sign of the last term iu numerator and denominator.

(3.) If a, b, c, A, B, C, be the sides and angles of a triangle, then the radius of a circle described about a triangle whose sides are a cos A, b cos B, and c cos C=that described about the original triangle.

(4.) Two equal circles intersect at right angles, and with the points of intersection, as centres two arcs are drawn touching the circles, so as to form an oval; shew that the space common to the two circles is equal to each of the spaces exterior to both.

(5.) If two observers A and B, at the distance of one mile from each other, see at the same moment a large bird, directly West and North-west of them respectively, A finds the angle of elevation made by the bird and a horizontal line to be 45°, and B finds it to be 30°; (88. APP.)

Appendix C.

Appendix C.

required the distance of the bird from each of the observers, and its perpendicular height above the plane.

(6.) If (r) be the radius of the circle inscribed between the base of a right angled triangle, and the other two sides produced and r' be the radius of the circle inscribed between the altitude of the same triangle and the other two sides produced; the area of the triangle shall be equal to the rectangle rr'

(7.) Expand cos 0 in a series ascending by powers of 0, and thence prove that

Deduce cos (0 + $) = cos 0 cos & sin sin

(8.) Having given the chord of an arc of a circle; deduce an approximate rule for finding the length of the arc.

A semicircular arch is made with stones three feet long, the span of the arc being 40 feet, and its height 16 feet, what is the area of the front of the arch?

ENGLISH ESSAY.

FOR ALL THE CLASSES.

On language as an instrument of civilization, with special reference to the effects which may be expected from the diffusion of knowledge through the medium of the English language in India.

VERNACULAR ESSAY.

Diligence, Industry and Honestry are the principal means of increasing national wealth.

যত্ন পরিশ্রম এবং সরলতা দেশীয় সম্পত্তি বৃদ্ধির প্রধান উপায় ।

LATIN ESSAY.

Quis inter Romanos summum Imperatoris laudem, quis boni sanctique viri præ cæteris meruerit?

JUNIOR SCHOLARSHIPS, 1851.

CROMBIE'S ETYMOLOGY AND SYNTAX, PART II.

Morning Paper.

(1.) Give rules for prefixing or rejecting the article in the phrases subjoined:

"A man, considered as a moral being, may be defined to be the responsible animal." "Whoever has power abuses it: every page of history proves the fact:-individual, body, the people,—it is all the same, power is abused."

"More I try, less I succeed."

(2.) In what cases does the verb precede its nominative case?

(3.) Certain nouns of the singular form require sometimes a plural, sometimes a singular, verb.

Why is this? Give an example.

(4.) Priestley contends for the expression, "He is greater than me," in preference to greater than I." Explain his reasons, and Crombie's answer to them.

(5.) Adverbs have sometimes an article (definite or indefinite) prefixed to them. State the reasons, and give examples of the above usage.

(6.) With what cases are interjections joined?

(7.) What is necessary to form a complete sentence?

In punctuation, how does the colon differ from the period?

(8.) Name the different members of the following sentence:

"

Though for no other cause, yet for this; that posterity may know we have not loosely, hrough silence, permitted things to pass away as in a dream, there shall be for men's information extant thus much concerning the present state of the church established amongst us, and their careful endeavour which would have upheld the same." (9.) The relative agrees with its antecedent in what particulars?

Point