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28. Give a statement of all the methods of payments of contractors for public works that have been laid before you, with the reasons for their adoption in different cases.

29. Give a description of the methods of setting the key course in the Grosvenor bridge at Chester, and in the Dora Ripaira; and also a sketch and description of the spandril filling in these two structures.

30. At the crossing of the Endrick valley on the line of the Glasgow waterworks, a cast iron pipe is employed as an inverted syphon, the internal diameter being 4 ft. at the deepest point, the vertical pressure being 235 ft.; calculate the thickness of the metal required, supposing that it is only strained to one-tenth of the ultimate strength, taking the approximation that the strain at every point from the inner to the outer surface is the same.

31. Calculate the discharge over a weir 900 ft. long on the crest, the depth of water flowing over, reckoned from the still water surface, being 0.44 ft.

32. A wrought iron beam, 100 ft. in clear span, and 10 ft. in depth, has the top and bottom flanges connected by a system of lattice bars forming single triangles, not intersecting, and loaded with a uniform weight of

ton per foot. Calculate the strain on a bar joining the points 20 ft. from the abutment on the lower flange, and 30 ft. from the same abutment on the upper; and prove the formula by which you work. In the case of a passing load of uniform weight per foot run, show when any particular diagonal sustains the greatest strain. State the advantages which arise from the lattice bars intersecting each other, and compare the system of boiler-plate sides with lattice work.

33. If, with the dimensions given in No. 32, we had instead of a uniform load a weight of thirty tons applied at the centre; calculate the strain on the diagonal, and prove the formula.


1. A shaft is sunk on a lode a certain distance, when the lode is found to be separated and thrown up by a slide; the lode is afterwards recovered by rising on the slide, and again worked; find the total depth of the working, and the horizontal distance at which a downright shaft should be sunk to reach the end

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2. It is required to sink a perpendicular shaft on the end of a level, whose bearings and drafts measured as follows:

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3. In a proposed mineral tramroad, you are supposed to know the weights P and Q of the loaded and unloaded waggons, and the coefficient of friction is supposed to be also known; what should be the inclination of the tramroad, in order that the work on the horses shall be the same in going up and down?

E. of S.

W. of S.

W. of S.
E. of S.

4. Explain the mode of finding the dip and strike of a coalbed without visible outcrop, from borings made at three given points on the surface of the ground.

5. Two lodes intersect each other, their backs making an angle of 30°, and their underlays are 18 inches and 12 inches in the fathom, respectively; find by calculation, or construct on the scale of 6 inches to the fathom, their line of intersection, along which there is supposed to run a good vein of ore, which we wish to extract by means of a diagonal shaft.



1. Write the formula of the variety of alum which upon ignition yields pure alumina; and state the grounds on which the earth is viewed as a sesquioxide.

2. Give the ordinary process for making chloride of iron. Explain also how this salt may be converted into the sesquichloride, and how the latter may be reduced to its former state.

Silicic acid,
Protoxide of iron,

Peroxide of iron,

3. A limestone was found to consist of the carbonates of calcium, magnesium, manganese, and iron, intermixed with a little siliceous sand; how would you make an accurate analysis of it?

Deduce from these its rational formula.

4. Phosphoric and arsenic acids may be estimated by analogous processes; what are they?

5. Write the formula of borax, and state how pure boracic acid may be procured from it. Mention also the volumetric process by which the degree of purity of borax may be determined.

6. The mineral called pinguite, on analysis, yielded the following results:






7. How would you analyze shot, so as to determine the exact amount of its lead, and of its arsenic ?

8. Give the details of the process for analyzing a mixed solution of the chlorides of iron, nickel, cobalt, zinc, and manganese.

9. How would you analyze a mortar so as to ascertain the amount of its free lime, of the lime in combination with carbonic, and of that united to silicic acid?

10. What are the proximate constituents of gunpowder?-and what should their proportions be so that the gaseous products of its combustion shall be exclusively nitrogen and carbonic acids? Calculate also the volume at 60°, and pressure of 30°, of the gases evolved by the explosion of an ounce of such powder.

11. Give the formulæ and crystalline systems of iron pyrites, magnetic pyrites, and mispickel; and describe a process for effecting the analysis of the latter mineral.

12. Assuming the composition of kaolin to be represented by the formula Al2O3, 2SiO3 + 2HO, and that it has proceeded from the decomposition of orthoclase; what change must the latter have experienced?

13. Enumerate the leading ores of iron, mentioning the formula of each, and the crystalline system in which it occurs.

14. If a water, including no earthy salts but sulphate of calcium, should be found to have 11° of hardness, how much gypsum does it include?

15. A solution of sesquichloride of iron, after reduction to protochloride required, in Marguerite's process, 173 measures of a volumetric solution, including 15.8 grains of permanganate of potash; how much of the sesquichloride existed in the original solution?

16. Mention the proper solvents for iron, copper, and gold; the forms in which these metals are usually estimated; and the methods by which each is brought into the state in which it should be weighed.





1. A pillar of uniform thickness, 20 ft. long, 2 tons weight, and inclined 67° to the horizon, rests against the top of a wall 15 ft. high; determine the moment of its pressure to overturn the wall.

2. A door, 10 ft. high, 5 ft. broad, and 5 cwt. weight, is supported on two hinges 8 ft. apart; determine the horizontal pressure on each hinge.

3. A wall of brickwork, 50 ft. long, 20 ft. high, and 1 ft. thick, supports the pressure of a roof inclined 55° to the horizon, and applied uni

formly over its top; required the greatest weight of the roof it could sustain without falling.

4. The end wall of a reservoir of water, 25 ft. deep, is a vertical trapezium 60 ft. broad at the top, and 40 ft. broad at the base; determine the pressure it sustains.

5. The span and fall of a suspension bridge are 100 and 25 ft., respectively, and the entire weight of the roadway, supposed uniformly distributed, is 75 tons; calculate the entire horizontal tension at the lowest point.

6. A horizontal beam, 10 ft. long, 5 in. deep, and 24 in. broad, is fixed at one end, and free at the other; calculate its deflection at the latter by the vertical force which, applied horizontally, would extend or compress it .005 inches.

7. A train, weighing 45 tons, is impelled along a horizontal road by a constant pressure of 600 lbs. ; if the friction be 7 lbs. per ton, calculate its velocity after moving from rest for 12 minutes, and the space it will describe in that time.

8. At the last athletic games in the College Park, a cricket ball, weight 5 oz., was thrown to the maximum distance of 91 yds.; neglecting the resistance of the atmosphere, calculate in foot pounds the entire work accumulated in it at the instant of projection.


1. Prove the rule for determining in magnitude and position the resultant of two parallel, unequal, and opposite forces, acting at different points.

2. Show that two pairs of parallel, equal, and opposite forces in the same plane, whose moments are equal and opposite, equilibrate each other.

3. In an ordinary door supported on two hinges, given the weight, dimensions, and interval between the hinges, determine the horizontal pressure on each hinge.

4. A body resting on a smooth inclined plane is sustained by a cord making an angle with the plane; given the angle and inclination, determine the tension on the cord and the pressure on the plane.

5. A body resting on a rough inclined plane is acted on by a force parallel to the ground; given the angles of inclination and friction, determine the extreme values of the force consistent with its equilibrium in opposite directions.

6. Determine the ratio in which the centre of gravity of a trapezium divides the line connecting the middle points of the parallel sides.

7. If the trapezium be immersed in water, having one of the parallel sides at the surface, determine the ratio in which the centre of pressure divides the same line.

8. For a surface of any shape immersed in water, show that the entire pressure is the same as for a horizontal surface of equal area whose plane passes through its centre of gravity.

9. A statical force F, acting freely for a time Tupon a mass M, gets up in it a velocity V; if W be the weight of M, required the ratio of F: W.

10. A mass M revolves uniformly in a time T in a circle whose radius = R; if F be the statical measure of its centrifugal force, and W its weight, required the ratio of F: W.

11. If a body be projected vertically upwards in vacuo with a velocity v, determine the extreme height to which it will ascend, with the times of ascent and subsequent descent.

12. If a body be projected obliquely in vacuo with a velocity v, and at an angle e, determine its range and time of flight on a horizontal plane passing through the point of projection.

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6. Trace the curve

; tan-1

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I + x

; log



when x=a.

sin x sin a

3. What is the result of substituting x + h for x in

2x3- 21x2 + 35x – 20?


I + x

4. What are the values of x which make the preceding function a maximum or minimum ?-and what are the maximum and minimum values of the function ?

5. Integrate with respect to x,


42 + √x+ + x (4+5)3.


y=3x+4+√/12 — 1 1x − x2,

7. Construct the traces of a plane which passes through a given point and a given line.


8. Construct the line of intersection of two planes, one of which is allel to the ground line, and the other perpendicular to the horizontal plane.

9. Exhibit the angle which two intersecting lines, whose projections are given, make with each other.

10. Draw a tangent plane to a cone from a point without it.

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