construct a dynamogram of areas which shall replace the several lines in the representation of the preceding question.

3. The top of a rectangular wall, of any material and dimensions, being supposed to sustain the pressure of an ordinary roof, of any weight and inclination, distributed uniformly over its entire area; determine the form and position of the line of pressure in any transverse section of the wall.

4. Either face of a trapezoidal wall, of any material and dimensions, being supposed to sustain the pressure of an ordinary embankment, of any slope and plasticity, distributed uniformly over every horizontal course of its area; determine the magnitude and point of application of the entire resultant pressure against the face.

5. A heavy linked chain, consisting of four uniform and equal rectilinear bars suspended from two fixed terminals in a common horizontal plane, being supposed to hang in free equilibrium under the action of gravity; determine its form of equilibrium, with the equal strains on the terminals arising from its weight,

6. The several supporting chains of an ordinary suspension bridge, of uniform weight per unit of length, being supposed to spring from terminal abutments of inconsiderable elevations above the roadway compared with the span of the bridge; determine approximately the common depression of their lowest points arising from a common small extension of their equal lengths.

7. An elastic rod of any original form being supposed subjected to the deforming action of any system of straining forces; show that its directions of greatest and least stiffness, to resist change of form by bending under their action, are at right angles to each other throughout its entire length.

8. An elastic beam of any transverse section being supposed slightly deflected from its original rectilinear form by a system of straining forces acting only at right angles to its unstrained length; show that the neutral line of its flexure coincides all through with the bent axis of its original form.

9. The bending strain at any point of an elastic beam of any form, supported in any manner, and bent by its own weight, being supposed equal to the nth part of the breaking strain of the material at the point; show that a similar and similarly supported beam of n times its dimensions, in the same material, would break at the corresponding point under the bending action of its own weight.

10. The two extremities and the middle point of a uniform elastic beam of any transverse section being supposed held at a common horizontal level by three rigid supports; determine the distribution of its weight between them in its state of strained equilibrium under the action of gravity.

11. The flange of an ordinary Warren girder which rests on the two abutments being supposed to sustain a permanent load uniformly distributed over its entire length; determine, for the two different cases of an odd or even number of pairs of braces, the law of variation of the resulting strain in the several segments of the other flange.

12. Determine also, for the two different cases, the law of variation of the resulting strain in the several pairs of anticlinal braces equidistant from the centre of the girder; distinguishing, at the same time, by positive and negative signs the pairs that are in tension and compression in the two cases, respectively.

APPLIED MECHANICS-PRACTICAL.

1. A weight of 112 lbs. rests on a rough plane, inclined 15° to the horizon; the coefficient of friction being supposed to be 0.75; calculate, in lbs., the limiting forces which, applied horizontally, must be exceeded in order to give it upward, downward, and horizontal movement, respectively, on the plane.

2. The plane, weight, and friction being again as aforesaid; calculate, also in lbs., the limiting forces which, applied in the direction of greatest efficiency in each case, will suffice just not to disturb the equilibrium of the weight on the plane in the three aforesaid directions, respectively.

3. A rough isosceles wedge, whose angles of friction and cleavage are 45° and 5°, respectively, being supposed to have been forced into a fissure by a pressure of 112 lbs. applied at its back; calculate, in lbs., the limiting force which must be exceeded in order to extract it again directly from the position attained.

4. The top of a rectangular wall, sp. gr. = 2.5, 100 ft. long, and 2 ft. thick, being intended to sustain the uniformly distributed pressure of a roof, 20 tons weight, inclined 47° to the horizon; calculate, in feet, the limiting height to which the wall may, under the circumstances, be raised.

5. The internal face of a rectangular wall, of the same thickness and material as in the preceding question, being intended to sustain the pressure of earth, sp. gr. 1.5, and natural slope = 45°, receding backwards from its upper edge at the latter inclination; calculate, again in feet, the limiting height to which the wall may be raised.

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6. Calculate, in inches, the height to which water will rise in a pump at the first stroke of the piston; the length of the stroke being supposed 2 ft., the mean height of the valve above the reservoir 20 ft., and the pressure of the atmosphere 15 lbs. to the square inch.

7. The span and dip of a suspension bridge are 250 and 25 feet, respectively, and the weight of the roadway, supposed uniformly distributed along its entire length, is 100 tons; calculate, in tons, the entire central and terminal tensions on the supporting chains.

8. The several supporting chains, in the preceding, being supposed to undergo a common expansion or contraction of 1 foot; calculate, in inches, the central depression or elevation of the roadway, and, in tons, the changes in the aforesaid tensions, consequent on their change of length.

9. A locomotive, 20 tons weight, being supposed to acquire a velocity of 20 miles an hour, while running through 1 mile of distance, under

the action of a constant difference of moving and resisting forces; calculate, in lbs., the requisite difference of forces.

10. The locomotive of the preceding question, moving with the aforesaid velocity round a curve of a mile radius, being supposed to exert no horizontal pressure against the rails of the roadway, supposed 5 ft. 6 in. apart; calculate, in inches, the requisite superelevation of the outer over the inner rail.

II. The rim of a fly-wheel, sp. gr. 7,75, whose inner and outer radii are 4 and 5 feet, respectively, and whose thickness is I foot, being supposed to revolve uniformly 20 times per minute round its axis; calculate, in foot lbs., the entire amount of work accumulated in it.

12. The fly-wheel of the preceding question, revolving with the aforesaid velocity, being supposed suddenly brought to rest by the collision against a fixed obstacle of a knob projecting from the outer circumference of its rim; calculate, in feet, the height through which an equal mass should fall vertically, from rest, so as to strike directly against the obstacle with the same force of percussion.

MR. LESLIE.

1. Show how to calculate the temperature produced by the combustion of fuel.

2. How is the evaporating power of fuel found?

3. There is an economy of power in using steam at a high, rather than at a low pressure?

4. There would not be any economy in using liquids such as alcohol or ether, which boil at a lower temperature than water?

5. What is meant by the mechanical equivalent of heat, and how has its value been ascertained?

6. Calculate the mechanical effect due to the evaporation of water.

7. Give the theory of a double-acting engine.

8. Show how to find the maximum useful effect in a double-acting engine.

9. Form the equation on which the theory of a locomotive depends. 10. Deduce the relation connecting the speed of a locomotive with the pressure in the cylinders.

CHEMISTRY AND MINERALOGY.

DR. APJOHN.

1. Explain the methods of preparing nitrate of silver, chloride of Barium, sulphide of hydrogen, and sulphide of ammonium.

2. Given a soluble metallic salt, what are the successive experiments which would enable you to determine in which of the groups (used in qualitative analysis) the metal is found?

3. Write the formula of potash alum, and the system in which it crystallizes. Mention also its different Isomers, and the action which Sulphide of Ammonium exerts upon it.

4. What is the cause of the hardness of a water, and how is degree of hardness determined?

5. Write a list of the mono-basic, bibasic, and tribasic acids with which you are acquainted, and mention the number of distinct salts which each is capable of forming with a monad metal.

6. What is the composition of salt of phosphorus, and of ammoniacomagnesian phosphate, and, when each of these salts has been exposed to a red heat, what is the nature of the residua left by them?

7. Explain how you would make the analysis of a clay consisting of silex, alumina, ferric oxide, and the carbonates of lime and magnesia.

8. If a mixture of potash and soda, in the form of chlorides weighs a grain, and in the form of sulphates, 6 grains, what is the amount of each alkali estimated as a chloride?

9. Write the composition of what is called Nessler's Test, and explain how it is applied by Wanklyn in estimating the free ammonia of a water, and the ammonia derivable from its organic matter.

10. There are two arsenical acids. How are they named, and what is the composition of each? State also how they may be distinguished from each other.

II. There are three native sulphides of iron, and one arsenio-sulphide of same metal. Give the formula, the name, and the crystalline system of each. State also the readiest means of distinguishing them from each other.

12. Give the composition and the crystalline system of the minerals generally known under the names of heavy spar, celestine, anhydrite anglesite, leadhillite, lanarhite.

13. Write the formula of the two principal acids of which Si O2 is the anhydride; and mention the native silicates which have a composition in accordance with one or other of these acids.

14. Write the formulæ of leucite, and specify the mineral of the felspathic group which has the same oxygen ratio. State also the points

of agreement and disagreement between leucite and analicme.

15. A cube of fluor may be modified in various ways. Its edges may be truncated. Its edges may be bevilled. Its angles may be truncated. Its angles may be acuminated by three planes resting symmetrically on its faces; also by three planes set symmetrically on its edges. Name the simple forms corresponding to these modifications, and give the notation of each.

16. What are the differences as respects composition between common pig-iron, speigeleisen, bar-iron, and steel; and how would you estimate the amount of carbon present in any of them?

17. A mineral analysed by Rammelsberg gave the following results:

What is its empirical formula, and oxygen ratio?

18. Write lists of the uniaxal and of the biaxal native carbonates.

SCHOOL OF ENGINEERING.

MIDDLE CLASS.

MECHANICS AND HYDROSTATICS.

MR. GALBRAITH.

1. If P and Q be two weights attached, the one to the wheel, the other to the axle of a wheel and axle whose weight is W; investigate the angular motion of the system, and the velocities of P descending and Q ascending at any instant in terms of P, Q, W, a and b, a being the radius of the wheel, and b that of the axle.

2. Let any set of forces act on a body retained by a fixed axis, say of z, give the general expressions for the strains on the axis arising from the so-called "lost forces."

3. A body retained by a fixed axis is struck with a given momentum ; find with what angular velocity it will start.

4. Find also the shock upon the axis, and investigate the circumstances of the case in which the axis shall receive no shock.

5. A vertical retaining wall, whose height is h, is pressed by a bank of earth, the inclination of which to the horizon is a. If o be the natural slope of the earth when free, prove that the pressure of any portion of the bank whose slope is i, for each unit of length, is

6. From this expression deduce Dr. Hart's construction for the angle corresponding to the maximum pressure, show also that it may be found by the equation

7. How may this equation be solved by means of a subsidiary angle? Apply the method to the example a = 30° and p = 45°.