12. Given the latitudes of two places together with their difference of longitudes, to find the declination of the sun, when it sets to the two places at the same time.

13. Required the equation to a curve, whose subtangent is equal to n times its abscissa.

14. If the force vary

1
Dn+1'

how far must a

body fall externally to acquire the velocity in any curve, whose chord of curvature at the point of projection is c; and apply the expression to the parabola and logarithmic spiral.

Afternoon Problems.-Mr. Palmer.

THIRD AND FOURTH CLASSES.

1. Find the value of £.123333 &c.

2. Determine geometrically a mean proportional between the sum and difference of two given straight lines.

3. What is the general form of parallelograms, whose diameters cut each other at right angles?

4. Investigate the area of a circle, whose diameter is unity and prove that the areas of different circles are in a duplicate ratio of their diameters..

5. Divide a given line into two parts, such that their product multiplied by their difference may be a maximum.

6. Prove that in any curve the velocity velocity in a circle at the same distance (SP) :: chord of curvature: 2SP.

7. A body projected from one extremity of the diameter of a circle, at an angle of 45°, strikes a Ꭰ .

mark placed in the center. Required the velocity of projection and greatest altitude.

8. Find the area of a curve whose equation is

as

upon

9. In how many years will the interest due £100. be equal to the principal, allowing compound interest?

10. Admitting the periods of the different planets to be in a sesquiplicate ratio of the principal axes of their orbits, shew that they are attracted towards the sun by forces reciprocally proportional to the squares of their several distances from it.

11. Prove that in the course of the year the sun is as long above the horizon of any place as he is below it.

12. Determine the limits within which an eclipse of the sun or moon may be expected and shew what is the greatest number of both which can happen in one year.

13. Prove that the time in which any regular vessel will freely empty itself time in which a body will freely fall down twice its height : : area of base : area of orifice.

15. Find the principal focus of a lens; and shew how an object may be placed before a double convex lens, that its image may be inverted and magnified so as to be twice as great as the object.

16. Prove that Cardan's rule fails unless two roots of the proposed cubic be impossible; and determine

whether that rule be applicable to the equation x3-237x-884-0.

17. Deduce Newton's general expression in sect. 9. for the force in the moveable orbit.

18. Define logarithms, and explain their use: also, prove that log. Ax Blog A+log. B.

19. Explain the different kinds of parallax; and shew from the want of parallax in the fixed stars, that their distance from the earth bears no finite ratio to that of the sun.

Afternoon Problems.-Mr. Palmer.

FIFTH AND SIXTH CLASSES.

1. How many yards of cloth, worth 3s. 74d. per yard, must be given in exchange for 935 yards, worth 18s. 1d. per yard?

2. Find the interest of £873. 15s. Od. for 2 years, at 44. per cent.

3. Prove that the diameters of a square bisect each other at right angles.

4. Prove the opposite angles of a quadrilateral figure inscribed in a circle equal to two right angles. 5. Prove that if Ax B when C is given, and Ax C when B is given, when neither B nor C is given, Ax BC.

6. Prove radius a mean proportional between tangent and cotangent; and that sine x cosine ∞ sine of twice the angle.

7. Given the sine of an angle, to find the sine of twice that angle.

8. Prove that in the parabola ordinate) abscissa > parameter.

9. Extract the square root of a3—x3.

10. Solve the equation 3x2-19x+16=0.

11. Prove that motion when estimated in a given direction is not increased by resolution.

12. Find the ratio of P: W when every string in a system of pullies is fastened to the weight.

13. Prove that time of oscillation a

✓ length
✔force

14. Prove that when a fluid passes through pipes

kept constantly full, velocity

1

area of section

15. Define the center of a lens; and find the center of a meniscus.

16. Find the fluxion of √a3 +x3—√ a2—x2. 17. Prove elevation of the æquator above the horizon co. latitude.

19. Prove that in the same orbit velocity <<

Evening Problems.—Mr. Palmer.

1

perp

FIRST, SECOND, THIRD, AND FOURTH CLASSES.

1. When £100. stock may be purchased in the per cents. for £59. at what rate may the same quantity of stock be purchased in the 5 per cents.

with equal advantage?

2. A ball of wood being balanced in air by the same weight of iron, how will the æquilibrium be affected when the bodies are weighed in vacuo? and by what weight of wood, properly disposed, may the equilibrium be restored?

1

3. Investigate the value of the circumference of a circle whose radius is unity.

4. Compare the areas of the parabolas described by two bodies projected together from the same point, and with the same velocity, towards a mark situated in an horizontal plane, the angles of elevation being to each other 2: 1.

5. Prove the rule for finding the quadratic divisors of any equation; and apply it to the equation x*—17x3 +88.x2-172x+112=0.

6. On what point of the compass does the sun rise to those who live under the æquinoctial, when he is in the northern tropic?

7. How many equal circles may be placed around another circle of the same diameter, touching each other and the interior circle?

8. Determine the resistance of the medium in which a body by an uniform gravity may describe a parabolic orbit?

9. Prove that a body moving in the reciprocal spiral, approaches or leaves the center uniformly.

10. Find the velocity and time of flight of a body projected from one extremity of the base of an equilateral triangle, and in the direction of the side adjacent to that extremity towards an object placed in the other extremity of the base.

11. Define similar curves; and prove that conterminous arcs of such curves have their chords of curvature at the point of contact in a given ratio.

12. Compare the time of a revolution about the center of a given ellipse, with that about its focus.

13. Find the attraction of a corpuscle placed in the axis of a cylindrical superficies, whose particles attract in an inverse duplicate ratio of the distance.