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where the joins cut it, these tangents forming the second triangle. From A,B, C, a third triangle A,B,C2 is derived in the same fashion, and so on. Express the area of A,B,C, in terms of the angles of the triangle ABC and the radius of its incircle.
Prove that, as n increases indefinitely, the triangle A„B„C„ tends to become equilateral.
9. Find the coordinates of the orthocentre of the triangle whose vertices are
(2, 3), (− 6, − 1), (− 3, − 2). Find also the equation to its circumcircle.
10. If perpendiculars be dropped from the foci of an ellipse on any tangent, prove that the rectangle contained by them is equal to the square on half the minor axis of the ellipse.
11. Three fixed tangents to a parabola meet a variable tangent in A, B, C respectively prove that the ratio AB: BC is constant.
12. A variable tangent to a given equilateral hyperbola meets the asymptotes in P and P', and an equilateral triangle is described on PP' as base and having its third vertex on the side of PP' away from the centre of the hyperbola. Find the locus of the third vertex.
1. Obtain an expression for the rate of flow of a liquid through a hole in the side of a vessel at a given depth below the surface of the liquid.
2. What are the laws of vibration of a stretched string? Draw sketches showing a vibrating string, giving its second overtone.
3. Explain the term 'beats.' Show clearly that the number of beats per second of two notes nearly in unison is equal to the difference of their vibration numbers.
4. Describe fully how you would determine the index of refraction of water,
5. Explain the construction of a compound microscope, and give a method for determining experimentally its magnifying power.
6. Show how to determine the specific heat of a liquid by observations on the rate of cooling of a vessel containing the liquid.
7. What do you know of the spheroidal state of a liquid?
8. How would you magnetize iron so as not to produce any magnetic poles? How can such magnetization be detected?
9. Describe the causes which render a simple copperzinc cell useless after being joined up for some time. What explanation would you give of the fact that the failure of the cell is principally due to the state of the copper plate?
10. A plate of metal connected with the Earth is brought near a second plate connected with an electrical machine in action. Explain, as fully as you can, the effect of the presence of the first plate on the charge carried by the second.
1. Show graphically how the current given by the simple. cell falls off in time.
2. Determine the water equivalent of the given calorimeter.
3. Measure the refractive index of the given prism.
DRAWING AND DESCRIPTIVE ARCHITECTURE.
[Lithograph Sketch, fig. 3a, accompanies this Paper.]
1. In an isometric drawing, show how to draw a scale for measuring lines drawn in any direction on the coordinate planes.
2. Find the vanishing point of lines inclined 30° to the horizontal plane, and lying in planes inclined 35° to the horizontal, whose horizontal traces are perpendicular to the ground line, given distance and height of eye.
3. Three isometric axes being OX, OY, OZ, radiating from 0, on OX take a point A, and OY a point B; join AB, and find the projection of the perpendicular from O on AB.
4. Give sketches illustrating in plan and section (a) a Greek theatre, (b) a Roman circus.
5. Sketch capitals suitable for shafts (a) of the Early English Pointed, (b) English Later Decorated styles.
6. In a trihedral angle the facial angles are denoted by a, b, c, and the dihedral angles by A, B, C: if a = 45°, b=56°, and A=52°: find c.
7. Construct a cycloid, the diameter of the rolling circle being 2 inches, and show how to draw a tangent to the curve parallel to a given line.
8. The directrix of an oblique cylinder is a circle of 2 inches diameter lying in the horizontal plane, the horizontal projection of its axis is parallel to the ground line, and its vertical projection makes an angle of 40° to the right with the ground line, the horizontal trace of a plane is perpendicular to the ground line, and its vertical trace makes an angle of 35° with the ground line to the left construct the horizontal projection of the intersection of the plane with the cylinder, and find its real form.
9. In the accompanying diagram, fig. 3a, of an oblique pyramid, abed is the base of the pyramid lying in the horizontal plane, and ve' the projections of its vertex, AB, A'B', are the horizontal and vertical projections of a line: construct the projections of the points where the line cuts the pyramid.
10. The base of an oblique cone is a circle lying in the horizontal plane; its diameter is 2 inches, and its centre is 1 inches from the ground line; the horizontal projection of the vertex of the cone is 2 inches from the ground line and 2 inches from the centre of the base, and its height 24 inches: construct the traces of a plane touching the cone, and making an angle of 75° with the horizontal plane.
All Chemical changes must be expressed both in words and by equations. Candidates who neglect this instruction will not receive full credit for their answers.]
1. How would you prepare the sesquioxide and the heptoxide of sulphur?
Of what acids are these substances the anhydrides?
2. Give commercial methods for the preparation of— oxygen, iodine, hydrochloric acid, ammonia, and caustic soda.
3. Give the formula of the substances produced when excess of aqueous ammonia acts on
(a) silver chloride,
(b) mercurous chloride,
(d) copper sulphate.
4. Give a detailed account of Pattinson's process for the extraction of silver from argentiferous lead.
5. How was fluorine isolated?
Describe fully how you would prepare pure anhydrous hydrofluoric acid.
6. How would you isolate carbon from limestone, and silicon from orthoclase?
7. Give the mode of preparation, properties, and uses of the oxides of lead. How would you proceed to test the purity of a sample of white lead"?
8. What are the chemical changes involved in the 'setting' of 'mortar,' 'cement,' and plaster of Paris'?
9. Describe two methods of testing for arsenic.
10. Explain and give examples of dissociation,' 'electrolysis,' and hydrolysis.' Give a technical application of 'electrolysis.'