(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 116 9' 36" in a circle whose radius is 50 yards. (4.) Prove the following formula sin (A - B) sin (A + B) cos 2 A . cos 2B } (1 + tan 2 tan (45 - A) 2 sin 2A 1 adapt the formula (3) to radius (r). А А (5.) Express sin in terms of the sides of a triangle, 2 2 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 . (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 48o. Determine the breadth of the river. and cos (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.) Shew 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, Meridian, 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. Shew 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 On Om Os according to each. What is Equinoctial Time ? Given the length of the mean tropical year equal to 365d 5h 48m 51.68 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 w + sin w tan o 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 1o. (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. Given log. tan. 9° 9.19971 9° - 9.19433 log. sin. 51° 30' 9.89354 log. cos. 51° 30' = 9.79415 log. sin. 7° 7' = 9.09304 log. sin. 14° 33' = 9.40006 log. sin. SECOND Class. PROBLEMS. Afternoon Paper. (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 li feet deep, to be filled with water to his of the top: what sized cube whose specific gravity is } heavier than water, 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 ellipse is 2 a’y2 + 6222 x2 y? с + + 1=0. aty2 + 64x2 THIRD CLASS. DYNAMICS. Afternoon Paper. (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 = 1 ft? 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 range on the inclined plane; find the greatest height which the projectile attains above the plane. (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 k v2 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 described in one oscilla 2f1 tion 9 (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 ε 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 a similar equal particle falling downwards, with a velocity equal to 2 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 shew which is best when one side is very large compared with the other two. 1 (2.) Explain the apparent absurdity of assuming x + 2 cos sin o) m Assuming (cosmo - N 1 sin mo)=(cos 0 1 express tan ma in terms of tan 6, and its powers, and shew clearly how you determine the sign of the last term in numerator and denominator. tan 70. ex. |