16. Shew the method of comparing the mean addititious force of the sun upon the moon, with the force of gravity at the earth's surface.

17. Let AB and DC be two diam. of a given circle, drawn at right angles to each other; AEB a circular arc described with rad. DB or DA: prove that the area of the lune AEBC area of triangle ADB.

C

E

B

18. Suppose a comet, in its descent towards the sun, to impel the earth from a circular orbit, in a direction making any acute angle with the earth's dist. from the sun; and the vel'. after impact: vel'. before ::

32: find what change would be produced

in the length of the year.

19. A globe and its circumscribed cylinder revolve uniformly round a common axis in the same time: compare the motion of the cylinder with that of the globe.

20. Shew that the fluent of

x

generated

22

whilst x from O becomes 1, is

0

where 2 is the

3

quad. arc of a circle whose rad. is 1.

21. A body descends in a parabola in a resisting medium, by a force tending to the focus, which

1 dist.

compare the resist. with the centrip. force:

and find the law of the velocity.

22. The relation between the abscissa and ordinate of an algebraic curve is expressed by the equation y"—a+bx.yn-1+c+dx+ex2. yn—2—, &c. = 0: prove that the sum of the ordinates divided by the respective subtangents, is a constant quantity.

23. A cylinder of a given weight and dimensions,

is put in motion round an axis parallel to the horizon, by a given weight (P) suspended by a small string wound round the surf. of the cylinder: find the actual time in which P would descend from the surf. of the earth to the center.

24. Let A and B be two particles of matter, connected by an inflexible line AB; and let a force be impressed perpen. at any point D which is not the center of gravity: find the point of initial spontaneous rotation; and then determine the path of that point in one revol. of A and B.

Morning Problems.—Mr. Walker..

FIRST AND SECOND CLASSES.

1. An equiangular prism is placed upon an inclined plane with its axis parallel to the horizon, and is just supported: - find the plane's inclination.

2. A cylinder is just immersed in a fluid with its axis perpendicular to the surface; find at what point it must be cut by a plane perpendicular to its axis,

that the pressures upon the convex surfaces may be equal.

3. Prove that a small rectilinear object and image subtend equal angles at the vertex of a spherical reflector.

4. The latitude being given, find at what time on the longest day, the variation of the sun's altitude is greatest.

5. Given the moon's horizontal parallax and periodic time, with the sun's apparent diameter and length of a sidereal year; shew the method of comparing the densities of the earth and sun.

6. aż=ÿ√ a2+4y, where z is the arc of a certain curve, and y the ordinate; determine the relation between the ordinate and abscissa.

7. Prove that the solid generated by the revolution of an ellipse round the minor axis, is a mean proportional between the solid generated round the major axis and its circumscribed sphere.

8. Find the distance of the point of suspension from the center of gravity of a given system of bodies, that the time of an oscillation may be the least possible.

9. Investigate and construct Cotes's fifth spiral.

10. The moon performs a revolution round the earth in a certain period, whilst they revolve round their common center of gravity; in what ratio must the mean distance of the moon be diminished that it may revolve round the earth at rest, in the same time?

11. Describe the curve which is the locus of the equation x-a3x2+a2y=0; and find its whole area.

13. Prove the differential method of summing' series, by the method of increments.

14. Find the attraction of a corpuscle to a sphere,

when the attractive force to each particle

1

dist.

Third and Fourth Classes.-Mr. Walker.

1. Find the interest of £555. for 2 years, at 44 per cent.

2. A piece of ground, containing 4970.25 square yards, is to be laid out in the form of a square: find the length of a side.

4. Prove that the ratio 1+x: 1 is nearly equal

5. Shew the method of ascertaining the height of a mountain above the level of an horizontal plane contiguous to a sloping side.

6. Prove that the minor axis of an ellipse is a mean proportional between the major axis, and the latus rectum.

7. In a right angled spherical triangle, sine of hypoth. rad. sine of a side sine of the angle opposite that side. Required a demonstration.

8. Given the two apparent weights in each end of a pair of false scales, find the true weight: and shew whether it is greater or less than their sum.

9. Find the height to which a body will rise, if projected perpendicularly from the horizon, with a velocity of 144 feet in 1" and find how far it will ascend in 2′′.

10. Compare the time of an oscillation in a cycloid, with the time in which a body would fall through a space-length of the pendulum.

11. A cubical vessel, whose altitude is 32 inches, stands upon an horizontal plane and is filled with water: find where a small orifice must be made in a side, that the fluid may spout to a distance equal to the height of the vessel.

-

12. Find the height of an homogeneous atmosphere: and shew that it is about 51⁄2 miles.

13. Find the principal focus of rays refracted through a sphere, denser than the ambient medium : and, supposing the focus to be in the surface of emergence, determine the ratio between the sines of incidence and refraction..

14. Determine the place of a double convex lens, between the eye of a spectator and an object at a given distance, that the apparent magnitude may be a maximum.

15. Construct two angles geometrically, whose sines are in the ratio a: b, and tangents in the ratio m: 1; and prove that rad. cos. of greater ang. :: √ m2—1.b2 : √ aa—b2.