1

1.3 2.4 3.5

ad infinitum. And 12+22+32 +, &c. ... to n terms.

19. Construct an horizontal dial for a given latitude; and find the angle between the hour lines of 12 and 3.

20. Given the sun's declination and lat. of the place; find the time of its rising.

21. Shew that every equation, whose roots are possible, has as many changes of the signs as it has positive roots.

22. Prove that the periodic times in all ellipses round the same center are equal: and, round different

23. Investigate the ratio of the angular velocities of the distance and perpendicular upon the tangent, in any curve: and apply it to the log. spiral.

24. A body descends from a given altitude by a

force which ∞

1

dist. 12,

-

and, at the middle point of

its descent, is projected with the velocity acquired, in a direction making an acute angle with the distancé from the center: what orbit will the body describe? and what will be its periodic time, when compared with the periodic time in a circle whose rad. is the given alt.?

Evening Problems. Mr. Hornbuckle.

-

1. Investigate the rule for extracting the cube root, and apply it to find the cube root of 738,763264.

2. Required the discount on a given sum (£p) due 11⁄2 years hence, at 5 per cent. per ann.

3. Resolve the recurring series a+bx+cx2+ dr3+ &c. whose scale of relation is f+g, into two geometrical series.

4. Two equal weights are suspended by a string passing over three tacks, which form an isosceles triangle, the base being parallel to the horizon, and the vertical angle 120°. Compare the respective pressures on the tacks with each other, and with the weights.

5. Suppose a strait rod to be partly immersed in a vessel of water; determine the angle at which it must be inclined to the surface, that the apparent bending at the surface may be a maximum.

6. A cylinder full of water, whose length is equal to the diameter of the base, is supported with its sides parallel to the horizon: compare the time of discharging half the fluid through a small orifice in the lower side in this situation, with that of discharging the same quantity through an equal orifice in the base, when the sides are perpendicular to the horizon.

7. Apply Napier's rule to find the declination of a star, which, in a given latitude, rises in the north east point.

8. Investigate the nature of the curve in which a body descends from one given point to another in the least time possible; the velocity at each point

being supposed to vary as the corresponding ordinate of the curve.

9. A cylindrical rod suspended at one end, whose weight is (W) and length (1) inches, oscillates seconds; on what part of the rod must a given weight (w) be suspended that it may oscillate twice in a second.

10. Approximate to the roots of the equation x2+xy=5, 2xy-y2=2, and shew on what the accuracy of an approximation depends.

11. Find the area of the curve, whose equation

a — √ a2x2

is y=ax hyp. log.

a + √ a2x2

contained between

the values (a) and (b) of the abscissa; x being the abscissa and y the ordinate.

12. ABC is a semi-cycloid, DC its base, AD its axis, AFG a curve traced out by assuming the ordinate EF always equal to the cycloidal arc AB: investigate the equation to the curve AFG, and compare its area with the area of the cycloid.

p

root is

.1

14. The roots of the equation x” —px2¬1+qxn—2 &c. =0 are in arithmetical progression; the least

n-1.3p2-6ng, and the com

mon difference

n—1.3 p2 — 6ng. The investi

16. Prove that when the first point of Aries rises, the ecliptic makes the least angle with the horizon, and when it sets, the greatest; and thence explain the phænomenon of the harvest moon.

17. Suppose a body describing a logarithmic spiral in a resisting medium, the density varying inversely as the distance, and the centripetal force as the square of the density, to be deprived of its angular motion at a given distance from the center: compare the time of its descent to the center in a strait line, with the time of descent in the spiral.

18. Two bodies, whose weights are A and B, are projected together with the respective velocities, a and b, from the same point, in the same direction, and at a given angle of inclination to the horizon. Required the greatest altitude to which their common center of gravity will ascend, and the path described by it.

19. A cylindrical vessel, of given altitude and base, is situated on a horizontal plane; an eye is placed so as to see only the farther extremity of that diameter of the base which passes through the point,

in which a perpendicular drawn from the eye to the plane meets it: To what depth must it be filled with water, that the eye in the same situation may see the center of the base?

20. Construct a vertical south-east dial.

1

21. ABD is a semi-circle, AD its diameter, EF any chord parallel to AD, produced indefinitely, CB, CP, CR radii; the parts Bb, Pp, Rr of the radii, intercepted between the chord, or the chord produced, and the circumference, are bisected in x, y, z: find the nature of the curve passing through all the points of bisection, both within, and without the circle.

22. Investigate the number expressing the probability of throwing an ace at least (t) times in (n) trials with a die of (p) faces marked 1, 2, 3...p.

23. Shew that Newton has properly applied the principles of the golden rule in his investigation of the ratio between the equatorial and polar diameters of the earth; i. e. if this ratio be that of 1+n: 1, n being very small, and the figure of the earth an oblate spheroid, prove that the excess of weight supported at the equator is proportional to the difference (n) of the diameters.

per

24. Find at what angle a plane, which is pendicular to the plane of the meridian, must be

C