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embankment between it and the sea. Give the details of the construction of the self-acting sluices, noting the advantages derived therefrom.
28. Describe the Fen district of the east of England, and the character of the rivers flowing through it; and give in detail an account of the remedial works of the Nene outfall and Eau brink cut.
29. State fully, and prove the truth of Mr. Galbraith's rule for the number of gallons per 24 hours delivered from a rain basin of a given number of square miles' area, and having a known annual rain-fall.
30. Explain by a sketch what is meant by the “Average Lead” (Day on the construction of Railways); and show also the most and the least advantageous form of longitudinal section of excavation and embankment with reference to this point.
31. Calculate the inclination of a coal tramway so that the work of the horses may be equal in ascending and descending; each waggon carrying i ton of coal in descending, and 2 cwt. of stores, &c., in ascending, and weighing unloaded 6 cwt.; the friction being tłoth part of the load.
CHEMISTRY AND MINERALOGY.
1. How would you make the analysis of a gaseous mixture, including oxygen, nitrogen, carbonic acid, and muriatic acid ?
2. Give the theory of the production of chlorine from pyrolusite, and from bichromate of potash.
3. Write the formula of ferrocyanide of potassium; and enumerate and explain the products obtained when it is distilled with dilute sulphuric acid.
4. Enumerate the leading minerals which include phosphoric acid. Write the formula of the native phosphate of lime, and explain the mode of making its analysis.
5. The sulphide of hydrogen in a mineral water may be estimated either volumetrically, or by precipitation. Explain each of these pro
6. Write the formula of the Prussian gunpowder, and enumerate the products of its combustion.
7. Common iron pyrites, like mispickel, contains sulphur, iron, and arsenic; how, then, are these minerals distinguished from each other?
8. Enumerate the metallic ores which include carbonic acid, and write the formula and crystalline system of each.
9. Bromide of potassium sometimes contains iodide of same metal; how is the latter best detected ?
10. A mixture of copper pyrites with an ore of iron gave, upon analysis, the following results :
29.31 31.75 Iron,
28 How was the analysis made ?-what is the ore of iron ?-and what the atomic proportion in which the two minerals are mixed ?
11. How would you make a complete analysis of a limestone, including, besides carbonate of lime, the carbonates of magnesia, iron, and manganese; and argil, the alumina of which is partly soluble?
12. Explain how, by chemical means, you would estimate the value of a hydraulic lime.
13. Give Rose's notation of a closed and an open form in the second, third, fourth, fifth, and sixth crystalline systems.
14. How would you deprive the oil of vitriol of commerce of its arsenious acid, its nitrous acid, and its saline matter?
15. Enumerate the bases and acids usually present in ordinary well water occurring in a limestone district; specify the mode of estimating each; and explain what circumstance connected with the analysis would lead you to suspect the presence of nitrates. Lastly, give the direct test usually employed for nitric acid occurring in water.
SUPPLEMENTAL EXAMINATION.-PRACTICAL MECHANICS.
1. A ton weight rests on a rough inclined plane, whose height and length are 21 and 100 feet, respectively; required the least coefficient of friction that will be just sufficient to prevent motion.
2. In the same case, the friction being just sufficient prevent motion, what constant force should be applied parallel to the plane so as to make the weight fall through its entire length in one minute ?
3. Each rafter of an ordinary isosceles roof sustains a weight of 256 lbs.; if the pitch of the roof be 36°, required the horizontal thrust of each rafter against the sustaining wall.
4. In the same case, if the wall be of brickwork of 112 lbs. weight per cubic foot, and of any height; required its limiting thickness to sustain the pressure, the rafters being supposed to be 2 ft. apart.
5. A coffer-dam sustaining a pressure of 26 ft. of water is supported by props 20 ft. long, 20 ft. apart, and applied along the line of pressure of the water; required the thrust on each prop.
6. A raft 30 yds. long, 20 yds. broad, and 16 in. deep, is made of wood whose specific gravity = 0.6; required the greatest load it will just sustain without sinking below the surface of water.
7. The extremities of the supporting chains of a suspension bridge are inclined 20° to the horizon; if the tension at the middle point of each be 100 tons, required the tensions at its extremities.
8. Show that the strength of a rectangular beam to resist fracture at a given point, under a given moment, varies as the product of its breadth into the square of its depth at the point.
LLOYD EXHIBITION EXAMINATION.
1. A heavy particle is allowed to descend along a rectilinear groove laid on a smooth inclined plane; what is the pressure on the side of the groove?
2. If at any point an opening be cut in the side of the groove, determine completely the motion of the particle after it reaches this point.
3. A particle resting on a rough inclined plane is attached by a fine string to a fixed point in the plane; if the particle be disturbed from its position of equilibrium, determine the angle through which it must be moved in order that it may just return to its original position.
[N. B.— The original position of equilibrium is supposed to be that which it would be if the plane were smooth.]
4. A rectangular block is laid on the less steep of two smooth inclined planes which slope in the same direction; one of the edges of the block being coincident with the line of intersection of the planes. The block is sustained by a cord passing over a given pulley, and supporting a weight. Find the condition of equilibrium.
5. A right cone is placed on its base on a rough inclined plane, the inclination of which is gradually increased. Find the condition that motions of rolling and sliding may take place simultaneously.
6. An ellipse is placed with its major axis vertical; to find the radius vector by which a particle will descend in the shortest time from the upper focus to the curve.
MR. M. ROBERTS.
1. If a, b, y are the direction angles of a right line in reference to three rectangular axes, find the value of the determinant
1, - 1, tany 2. Find the condition that the equations
ax3 + 36x2 + 3cx + d=0,
(ac – 62) x2 + (ad — bc) x + bd – c2 = 0, should have a common root. 3. Prove that the sum of the pth powers of the roots of the equation
2+ pixn-1 + P2an-2 +....=0 is the coefficient of mp in the expansion by powers of m of
n+ (n − 1) pim + (n − 2) p2m2 +....
1 + pim + P2m2 + . + Pnmn 4. If a +b+c=0, express ab + ac + bc and abc by means of the quantities a -6, a
x2m-4 dx x2m-2 dx
R find the values of A, B, C.
7. Find the condition that the ellipse
and the circle
(x-a) +(y-B)a= ka should touch each other.
8. A perpendicular is dropped from the centre of an ellipse on any tangent to the curve; show that two circles may be found such that the rectangle under the tangents drawn to them from the foot of the perpendicular is constant.
ASTRONOMY AND OPTICS.
1. Explain the method of determining the latitude of a place from an observed zenith distance of a known heavenly body taken very near the meridian of the place.
2. Explain the method of determining the longitude of a place, supposed approximately known, from the observed occultation of a known star by the Moon.
3. When the latitude and time are determined from two altitudes of the Sun and the interval between, required the corrections to be applied for given small errors in the altitudes.
4. For a given day and place, express the difference between the lengths of morning and evening in terms of the Sun's change in declination from his rising to his setting.
5. Investigate the formulæ for the effects of aberration on the right ascension and declination of a known star; and show from them the positions of those stars for which the effects vanish.
6. The Earth being supposed very nearly spherical, express the lengths of its equatoreal and polar radii in terms of the lengths of any two measured arcs of meridians whose terminal latitudes are known.
7. Investigate the differential equation of the course of a ray of light through a medium composed of spherical strata varying according to any
law; the incidence being supposed to take place in a plane passing through their common centre.
8. Explain the most accurate method with which you are acquainted of determining the refractive index and dispersive power of a given transparent substance.
9. Investigate the formula for the angular semidiameter of a halo on the hypothesis of its formation by refraction through prisms of ice placed in their positions of minimum deviation.
10. The eye of an observer being situated in a plane passing through a luminous point, and the common axis of a system of thin lenses; exhibit by a figure the course of the pencil by which the point is seen by the observer.
M CULLAGH PRIZE EXAMINATION.
CALCULUS OF VARIATIONS.
dan and if, moreover, the limiting values of
do-ly X, Y,
dx2 be given, show that 82 U may be put under the form *1 dey
U=S Vdxdy, where
V= A (rt – 52) + Br + Cs + Dt + E, A, B, &c., being functions of X, Y, Z, P, q, show that 82 U may be put under the form
82 U=SS Odady, where is a homogeneous quadratic function of dz, dp, 89.
N.B.—The limiting values of x, y, z, and p or 9, are supposed to be given.
3. In determining, by the method of the Calculus of Variations, the surface of given superficial area which makes
R R' a maximum or minimum, show that we find as the equation of the surface
R+ R = const., R, R' being the principal radii of curvature.