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sphere after describing a space equal to eight diameters, the resistance to the motion arising only from the inertia of the medium.

Third and Fourth Classes.-Mr. Hornbuckle.

1. Required the price of paving a floor, whose length is 10yds. 2ft. and breath 5yds. lft. at 2s. per square foot.

2. Reduce 17s. 94d. to the decimal of a pound. 3. Prove the rule for finding the greatest common

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measure of two quantities, and reduce to its lowest terms.

4. If an angle of a triangle be bisected by a line which also cuts the base, the rectangle under the sides of the triangle is equal to the rectangle under the segments of the base, augmented by the square of the bisecting line.

5. Given the three angles of a triangle and the radius of the circumscribing circle, to find the sides.

6. The sum of the tangents of two arcs: diff. of tangents the sine of the sum of those arcs: sine of the difference.

7. Prove that a body cannot easily be balanced on a point under the center of gravity.

8. Given the length of a pendulum (1) that oscillates seconds, find by means of it how far a body will fall freely by the force of gravity in ť".

9. Given the velocity and direction of projection, find the range on a plane of given elevation, and the greatest altitude.

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11. Inscribe the greatest cone in a given sphere. 12. Find the chords of curvature perpendicular and parallel to the axis at a given point of the common parabola.

13. The roots of the cubic x3—px2+qx—r=0 are in musical progression, shew how they may be found.

14. Find generally the sums of the powers of the roots of an equation.

15. Explain the principle of the hydrometer, and shew that if it be made to sink to the same depth in different fluids, the specific gravities of these fluids are as the weight of the instrument in the several

cases.

16. Find the center of a Meniscus, and prove it to be a fixed point.

17. Place an object before a double convex lens, so that the image may be double of the object, and

erect.

18. Given the right ascension and declination of a star. Required its latitude and longitude.

19. Supposing the earth's orbit a parabola, find the apparent path of aberration of a fixed star parallel to the ecliptic.

20. The velocity at any point of a conic section: velocity in a circle at the same distance :: √1⁄2L× SP : SY; Prove this, and shew it to be the same ratio as that of HP: AC, AC being the semi-axis major.

21. How far must a body fall internally to acquire the velocity in a parabola, the force varying inversely as the square of the distance?

22. Compare, by means of a circle, the times of describing freely different spaces from the same distance towards the same center of force, the law of the force being the inverse square of the distance.

23. Explain the principle of Cassegrain's telescope, and find its magnifying power.

24. Sum the following series 1.2.4+2.3.5+ 3.4.6+ &c. to n terms.

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Fifth and Sixth Classes.-Mr. Hornbuckle.

1. 17 ells of cloth, each containing 5qrs. cost £6. 17. 104; how much will 18 yards cost at the

same rate.

2. Required the interest due on £115. for 5 years at 4 per cent. per ann. simple interest.

3. Solve the following equations

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4. In any plane oblique angled triangle, given two sides and the included angle, to solve the triangle.

5. A body falls from rest by the force of gravity down a given inclined plane; compare the times of describing the first and last halves of it.

6. Prove the velocity in any point of a parabola equal to that acquired in falling down & the parameter.

7. Given a rectilinear object, and its inclination to a known refracting surface; required the magnitude and inclination of the image.

8. Prove that the image of a strait line before a spherical reflector is the arc of a conic section.

9. Find the time of emptying a cylinder of given base and altitude through a small orifice in its base and investigate the fluxional expression for the time of emptying vessels in general.

10. If all the coefficients of an equation be whole numbers, shew that it cannot have a fractional root.

11. Draw a meridian line on a horizontal plane.

12. Compare the velocity in a curve with that in a circle, at the same distance, in general, and in the conic sections; the center of force being in the focus.

Evening Problems.-Mr. Walker.

1. Find the exact value of . 1666, &c. of a £. 2. Find the length of a pendulum that would oscillate seconds at the dist. of the earth's rad. above its surface; and then determine the point below the surf. where it would oscillate in the same time.

3. A body, projected from the top of a tower at an angle of 45° above the horizontal direction, fell in 5" at a dist. from the bottom of the tower equal to its altitude: find that alt. in feet.

4. Compare the resist. upon the curve of a semicircle, moving in a fluid in the direction of its axis, with the resist. upon its base.

5. Two equal hollow cubes are immersed in water, having a side parallel to the surface, and their depths are in the ratio of 4: 1; compare the times of filling through equal orifices in the bottoms.

6. What must be the form of a glass lens placed in water, that all rays, incident parallel to its axis, may converge accurately to one point within the water?

7. The reflecting curve is a semi-circle, and rays fall parallel to its axis: construct the caustic; and compare the density of the rays at different points.

8. Given the sun's alt. at 6 o'clock, and the alt. when due east; find the lat. of the place.

9. Given the mean horary motion of a planet in its orbit; shew the method of finding the horary motion in longitude.

10. The hyp. logarithm being given, find the corresponding number.

11. Sum the series

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2 1 3 1 4 1
+ X
X
+
1.3 3 3.5 32 5.7 33

+

And 1.2.4+3.4.6+5.6.8+7.8.10+ &c. to n terms, by increments.

12. Shew the method of finding the content of a pyramid, whatever be the figure of its base; the area of the base, and alt. of the solid, being given.

13. Investigate the ratio of the centripetal and centrifugal forces in any curve; and apply it to the hyperbolic spiral.

14. Find the actual vel'. and per. time, of a body revolving in a circle, at the dist. of the earth's rad. above its surface.

15. The equation to a certain curve is y =

4x2

√ a2x2

where x is the absc. and y the ord. : find the whole area of the curve, and shew that it area of a circle whose rad. is a.

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