5. Given the latitude of the place, and the sun's declination: find at what time of the day the azimuth of the sun increases the slowest. 6. Investigate, as Cotes has done, the variation of the density of the atmosphere, supposing the force of gravity to vary inversely as the square of the distance from the earth's centre. 7. One root of the equation x3 – 4.22 – 3x+12=0, is of the form vā, where a is not a square number. Solve the equation. 8. A given paraboloid is perforated by a cylinder whose axis coincides with that of the solid. Required the dimensions of the cylinder, so that the part taken away may be equal to that which remains. 9. In a combination of wheels and axles, in which the circumference of each axle is applied to the circumference of the next wheel, and in which the ratios of the radii of the wheels and axles are those of 2:1, 4:1, 8:1, &c. there is an equilibrium when the power : the weight ::1:p. Required the the number of wheels. 10. If the roots of the equation x" — pamol + gxn-> - &c. =0, be in arithmetic progression, the least 1-1.3/" – 6nq root will be and the na_1 2 1 - 1.3p2 -- 6ng. common difference Re n? - 1 quired the investigation. 11. Given the major and minor axes of an ellipse. Required the radius of a circle described round the focus as a centre, in which the periodic time is equal to the time of moving through the aphelion, from mean distance to mcan distance. PN-1 X Vn 12 n Х n 12. Construct Newton's telescope, and investigate its magnifying power. 13. Compare the quantity of water discharged by two equal parabolas in the side of a reservoir kept constantly full; one of the parabolas having its base, and the other its vertex downwards, and the summits of both coinciding with the surface of the fluid. 14. Suppose that a body falls from a given altitude to a centre of which the attractive force varies as the distance, and that the system moves in a direction perpendicular to the line of descent, with a velocity equal to the greatest velocity which the body could acquire in its fall; construct the curve traced out by the body, and then find its area. 15. If z be an integral increasing or decreasing 1 1 &+? _ ?+22+? - &c. unequally; then 2% 1 12 continued to m + 1 terms. 16. Find the fluents of the following quantities: viz. di x2.C ał + 22 xĆ, 2018 ? +22: (x – a) (x (x - a) (x – b) (x – c): when the fluent of aí +2m+" XC=A; -"y, where z is a circular arc, and y its sine to radius 1. 17. Sum the following series: 1 + &c. ad infin. 1.4.7 4.7. 10' 7.10.13 22 + + 82 + &c. to n terms, 1 +&c. ad infin. 9.10 4 1.3.3+3.5.92 + + &c. to n terms. 3.5.3 '5.7.33 + + 18. Materials are to be raised, through a given altitude, by a wheel and axle whose radii are known; the power, which is given, being applied to the circumference of the wheel. Find the quantity raised at each ascent when the greatest quantity in the whole is raised in a given time; the inertia of the machine being neglected. 19. Find the area on the plane of the horizon that is bounded by the shadow of a tower of given altitude, between the hours of 8 and 2, in a given latitude: the sun being in the equinoctial. 20. Find the relation of x to y in the equation s(a+bx+cy)=ýld+ex+fy). 21. Let a sphere of given diameter be projected in a fluid whose specific gravity is to that of the body as I to n: Having given the velocity of projection, it is required to find the velocity after describing any space and also the time of describing it. 22. Prove that the projection of the rhumb-line, on the plane of the equinoctial to an eye situated in the pole, is a logarithmic spiral; and hence determine the length of any arc of the meridian, on the planisphere. 23. Determine, as Newton has done, the path of a projectile in a medium in which the resistance varies as the velocity; the force of gravity being uniform and acting in parallel lines. 24. Determine the dimensions of a conic frustum, of given altitude, on which, when moving in a resisting medium, in the direction of its axis, with its less end foremost, the resistance will be equal to that on the base of a given cylinder, moving with the same velocity; and at the same time, less than the resistance on any other frustum of the same base and altitude. Morning Problems.-Mr. Turton. 1. Define similar curves when referred to their axes; and prove that similar and conterininous arcs have a common tangent at the common point of termination. 2. If (a) and (b) be two sides of a trapezium that are parallel to each other; prove that the centre of gravity of the figure will divide a perpendicular to those sides into two parts that are to each other as 2a+b to 2b ta. 3. If a body fall from a finite altitude towards a centre of force, and the time of falling vary as the nth power of the space fallen through; required the law of the variation of the force. 1 4. Resolve into trinomial fractions without the aid of fluxions; n being an even number. 5. If the refracting curve bethe logarithmic spiral, and rays issue from the centre, investigate the nature of the caustic, 6. Find the force of elasticity, so that, in the case of direct impact, the sum of the products of each body into the cube of its velocity may be the same before and after impact. 7. If a weight P be suspended by an inflexible line, whose length is (a), to what point must a given weight p be attached, so that the pendulum may oscillate in the least time possible? 8. There is a small aperture, whose area is (m) at a given distance (a) from the bottom of a vertical X cylinder filled with water. When full, the fluid falls on the horizontal plane, at the distance (6) from the base; and after (t) seconds at the distance (c) from the base. Required the content of the vessel. 9. Let y= A + Bxm + C#m + D.UP + &c. where A, B, C, &c. are constant quantities: then if x be Ý ӱ j , xx+ xr? + x 23+ &c. 1.2.22 1.2.3.23 Required the investigation. da pn tin-lì 10. Find the fluent of ; and also of a + bz" dzin-z va + bz* 11. Find the amount of £l. for the time (1), at compound interest at a given rate; interest being due every moment. 12. In a given latitude, at a given hour, and on a given day, the altitude and azimuth of a star are observed. Required its right ascension and declination. 13. Suppose the earth a perfect sphere, and that a penduluin whose length is (a) inches, vibrates seconds in latitude 60°. What will be the length of a pendulum that vibrates seconds at the equator? 14. Deduce Cotes's construction of his first spiral, by means of Newton's general proposition in the 8th section. Tuesday Afternoon.—Mr. Walter. THIRD AND FOURTH CLASSES. 1. How much ready money can I receive for a note of £75, due 15 months hence, at 5 per cent. discount. |