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Higher Mathematics.



1. A string 9 feet long has one end attached to the end of a smooth uniform heavy rod 2 feet long, and at the other end carries a ring which slides upon the rod. The rod is hung by means of the string from a smooth peg. If o be the inclination of the rod to the horizon, show that tand = } (31 – 3).

2. What is the Funicular Polygon ? Examine the conditions of equilibrium when the weights are all equal.

3. Prove the principle of virtual velocities for a system of forces in one plane acting on a rigid body.

4. If the unit of velocity be a velocity of a feet per t seconds, the weight of 1 lb. being the unit of weight, and 1 lb. the unit of mass, find the units of length and time.

5. A body moving down a smooth inclined plane is observed to fall through equal spaces a in consecutive intervals of time to tz; show that the inclination of the plane to the horizon is


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gun recoils.

6. A shot of 700 lbs. is fired with a velocity of 1,600 feet per second from a thirty-five ton gun; find the velocity with which the

7. What is the synodic period of a planet ? Given that of Venus (584 sidereal days), find its sidereal period.

8. At a place on the Arctic circle where will the sun rise at the summer solstice ?

9. The sun rose one morning at 8h. 7m., and set the same evening at 4h. 5m. ; find the value of the equation of time that day.

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3. If 0 and $ be the greatest and least angles of a triangle, the sides of which are in arithmetical progression, prove that 4 (1 – cos 0) (1 – cos)= cos 0 + cos $.

4. Obtain by the use of De Moivre's theorem the six values of (-1).

5. If QVQ be a double ordinate of a diameter PV of a parabola, show that Q V2= 4 SP. PV.

6. In the ellipse or hyperbola the rectangles contained by the segments of any two chords which intersect are in the ratio of the squares on the parallel diameters.

7. Define a rectangular hyperbola, and state and prove some of its special properties.

8. Find the locus of the vertex of a triangle, given base and difference of base angles.


9. The ordinate of a point P on an ellipse, produced, meets the auxiliary circle in Q ; show by using the eccentric angle that the locus of the intersection of CQ with the normal at P is a concentric circle.

Natural Philosophy.

Junior and Senior.



Junior Students will only be examined in three of the subjects (a), (b), (c), (d),

(g), (h). Senior Students will only be examined in three of the subjects (a), (b), (c), (d), (e), (f), (g), (k).

NOTE.—(6) cannot be taken with (a), nor (9), nor (k), withorit ($).

(a) 1. How would you prepare chlorine from hydrochloric acid, and what weight of it would you obtain from 292 grammes of the acid ?

2. Mention the most characteristic properties of chlorine, and explain as far as you can its bleaching power.

3. What is chloride " and chlorate ” ? How may each be known ?

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(6) 4. How is bromine prepared ? How much can you get from 170 grammes of potassium bromide ?

5. How are HF and H S prepared ? Illustrate your answer by diagrams of the apparatus used.

6. Show how by means of HS we can divide metals into

various groups.

(c) 7. What is the path of a projectile? How does the velocity vary at different points, and where is it greatest and least ? What is meant by the range, and how must the body be projected to obtain the greatest range ?

8. A stone thrown vertically upwards reaches a certain height. Show that if the weight of the stone be doubled, other circumstances remaining the same, the height reached will be one-fourth of its former value.

9. Define the centre of gravity of a body. Find the centre of gravity of weights of three, four, and five pounds respectively placed at three of the angular points of a square whose sides each measure one foot.

10. Explain how you would discover experimentally the centre of gravity of a chair.

(d) 11. State the laws connecting the pressure, density, and temperature of a gas.

What is meant by absolute temperature ? Give the absolute temperature corresponding to 60° C.

12. A gas occupies 98 cubic inches at 185° C. Find what it will occupy at 10° C. under the same pressure.

13. What is meant by the boiling point of water ? Under what circumstances does it vary? Account for the bubbles observed in water as it gradually approaches the state of boiling.

14. How are quantities of heat measured ? Is a greater or a less quantity of heat required to raise 1 lb. of water than to raise 1 lb. of mercury from 0° C. to 100° C.?

(e) 15. Two metal spheres, A and B, are suspended by silk threads near one another. Originally A is charged and B ancharged with electricity. State the electric condition produced on B (i) when A and B are connected by means of a metal rod similarly suspended ; (ii) when there is no connection except through the surrounding air ; (iii) when an insulated conductor is held between them.

16. Explain carefully how quantities of electricity are measured.

17. Describe the gold leaf electroscope. Show how by means of it to determine the character of the electricity of any electrified body.

18. What are condensers of electricity ? Give examples of condensers, and describe the condensing electroscope and the principle it illustrates.

(f) 19. Describe minutely the structure and reproduction of Pencillium and Mucor, comparing them with each other.

20. What facts, as to the necessary food of these moulds, do you learn from the substances on which they are found growing ? Compare them in this respect with Torula.

(g) 21. Write out in a tabular form a classification of the Crustacea.

22. Describe the habits and structures of a Lobster, a Crab, and a King-Crab.

23. Draw a diagram of all the bones of either the Cat, or Man, or any other vertebrate you know : affix the name to each bone.

24. Give a short account of the habits, structure, and geographical distribution of the Felidæ.

(h) 25. Give the characters which distinguish Monocotyledons from Dicotyledons, and the three sub-classes into which Dicotyledons are divided, with examples of each.

26. Trace the various steps by which you would decide the place of a Wallflower in the Vegetable Kingdom.

27. Give the meaning of the terms hypogynous, symmetrical, perfect, and hermaphrodite as applied to a Wall-flower.

28. Describe from observation, and as much as you can in botanical language, the flowers of a Wallflower and a Primrose, and compare them with each other.

(k) 29. Distinguish between continental and oceanic islands. In what

ways may oceanic islands be formed ? 30. What is the cause of currents in the ocean ? Describe or draw a chart showing the system of currents in the Atlantic and Pacific Oceans, giving their direction and names.

31. How do tide waves differ from wind waves ? Account for spring and neap tides, and explain why high tides occur at different hours each day.

32. Give the position and depth of the deepest ocean basins. What is known of the floor of these deep basins, and of the general sea bottom ?


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