Second Afternoon. Mr. Sowerby.

1. Investigate a rule for extracting the square root of a binomial, one or both of whose factors are quadratic surds; and apply it to determine the square root of 2 1.

2. The spaces described in any times by bodies uniformly accelerated, are proportional to the squares of the times, or to the squares of the last acquired velocities.

3. Explain the principle and construction of the air-pump; and compare the density of the air in the receiver at first, with its density after any number of turns.

4. Find the principal focus of a sphere, the radius of the sphere and the ratio of the sines of incidence and refraction being given.

5. Suppose the body to revolve in an ellipse, it is required to find the law of the force tending to the focus.

6. Divide a given angle into two other angles, so that their sines may be to each other in a given ratio; and shew how the value of these angles may be calculated.

Second Evening.-Mr. Hornbuckle.

1. In a given latitude, a pendulum will oscillate once in one second, supposing the earth not to revolve round its axis. Required the angular motion round the axis, that the pendulum may oscillate once in two seconds.

2. a. by.c is a minimum, and x+1.y +1. z+1 2 a constant 'quantity. Required the relation between x, y, z,

3. Resolve

1

1—ez+ƒz2 — gz3+&c.

into a series

of fractions whose denominators are binomials. 4. Determine the apparent magnitude of a straight rectilinear object, placed at a given depth, parallel to a surface of water; the eye being situated at any point in the plane passing through the object perpendicular to the surface.

5. Two perpendiculars of given lengths are situated at a given distance from each other in a horizontal plane. Determine geometrically that point on the plane between them, to which, if lines be drawn from the extremity of each perpendicular, the times of falling down the two inclined planes may be equal.

6. Find the center of gravity of a cylindrical portion of the atmosphere measured from the earth's surface, the force of gravity being constant.

7. If the ordinate (y) of a curve be composed of powers of the abscissar and constant quantities; having given the increment of x, find the contemporary increment of y.

8. A string of given length is suspended to two tacks any where situated, the length of the string being greater than the distance between them. It is required to find the position of a given weight (w), which slides freely on the string, when at rest.

9. Construct the spiral in which the areas are the measures of the ratios between the ordinates which terminate them.

10. The angle between the apsides in an orbit very nearly circular: 180°::/b+c.mb+nc. Prove that the law of the force hence deduced by NEWTON'S bAm +cAn method coincides with

11. Required the time in which a given cylindrical wheel will roll from the top of a given conical hill to the bottom.

12. Required the present value of £1. to be paid at the end of n years, if either of the individuals A or B, whose ages are given, be alive at that time.

13. It is required to find the altitude of the first point of Aries, at a given hour, day, and place; the angle also, and point in which the ecliptic cuts the horizon at that time.

14. Given the length and weight of an elastic string, and the force which stretches it, to find the number of vibrations in a second.

15. Construct the fluent

less than unity.

z + 2 az2 + 23.

, a being

16. A reservoir being supplied with water at a given rate, determine the height to which a sluice must be drawn, that the reservoir may be always kept just full; the dimensions of the sluice, and the depth of its base from the surface, being given.

17. Explain the reason why at spring-tides in summer, in north latitudes, the afternoon tide is greater than the morning tide.

18. A particle is attracted towards a straight line given in position and magnitude, the law of the force being the inverse square of the distance. Determine the direction in which the particle will begin to move towards the line.

19. Find the fluents of the following quantities xa. a2 i (a being a given

21. The roots of the quadratic equation 2-px +10 are a and b. Prove that a+b" is equal to

22. BRADLEY observed, that the apparent motion in declination of every star tended the same way when they passed the meridian at the same hour, each being farthest north when it passed at six o'clock in the evening, and farthest south at six in the morning. Explain the reason why this is nearly true in those stars whose declinations are not very great.

23. A ball is shot from a cannon with a given velocity, at a given angle of elevation, situated at a given distance from the foot of a hill, whose elevation is also known. Determine the point in which the ball will strike the hill.

24. Given the force of gravity at the surface of a primary planet, the mean distance and periodic time. of its secondary with the ratio of their respective diameters. It is required to compare their densities.

1806.

First Morning.-Mr. J. Brown.

MONDAY, JANUARY 13, 1806.

1. How many years' purchase is an estate worth, when the rate of interest is 4 per cent.?

2. An inclined plane is a tangent to a cycloid in the middle point between the highest and lowest points; what must be the length of the plane, that the time of falling down it may be equal to the time of half an oscillation?

3. Having given the specific gravities of water and iron, it is required to determine what proportion the thickness of an hollow iron globe must bear to its diameter, that it may just float in water.

4. In a given latitude, and at a given time of the year, how many hours will be shewn upon a vertical south dial? And at what time of the year will the greatest number be shewn?

5. Having given the altitudes of the sun, and of any particular colour in the primary rainbow, it is required to determine the sines of incidence and refraction.

6. The curve ANC is a cycloid, and the curve AMD is formed by taking PM equal to the arc