Lessons 23–25.Conical Intrados—Intrados of Revolution.
Skewtrompein the angle. Suggestions on the general problem of conical skew vaulted roofs.
Spherical domes, &c.
Lessons 26–27.Intrados, a Ruled Surface.
Winding staircases, &c., &c.
Lesson 28.Helicodial Intrados.
Staircase on the Saint-Giles screw.
Lessons 29–31.Composite Vaulted Roofs.
Various descriptions of vaults.
Suggestions on vaulted roofs with polygonal edges and with ogival edges.
Lesson 32.Revision.
Spirit and method of stereotomy.
Degree of exactness necessary. Approximate solutions. Case where it is proper to employ calculation in aid of graphical constructions.
Review and comparison of differentappareils.
MECHANICS AND MACHINES.
GENERAL ARRANGEMENTS.
The pupils execute during the two years of study:—
1. Various drawings or plans of models in relief representing the essential and internal organs of machines, such as articulations of connecting rods, winch-handles and fly-wheels, grease-boxes, eccentrics worked by cams or circles giving motion to rods; the play of slides, &c.; cylinders of steam-engines, condenser, pistons, and various suckers; Archimedes’ screw, and other parts of machines.
The sketches of the plan drawings are traced by hand and figured. The drawings in their finished state are washed and colored according to the table of conventional tints; they all carry a scale suitably divided.
2. A drawing of wheel-work by the method of development, and tracing the curves of teeth by arcs of circles from which they are developed. This drawing represents, of the natural size, or on any other scale of size considered suitable to show the nature of the partial actions only, a small number of teeth either in development or projection; the entire wheel-work is represented by the usual method of projection, where in drawings on a small scale the teeth are replaced by truncated pyramids with a trapezoidal base.
3. Finally, numerical exercises concerning the loss of work due to the proejudicial resistances in various machines, the gauging of holes, orifices, &c.
Models in relief or drawings on a large scale, of the machines or elements of the machines mentioned in the course, assist in explaining the lessons. They are brought back, as often as found necessary, under the eyes of the students. When possible, lithographic sketches of the machines, or the elements of the machines, which ought to enter into the course, are distributed among the pupils.
The pupils, divided into s, pay their first visit to the engine factories towards the end of their first year of study; they make one or more additional visits at the end of the second year.
FIRST YEAR.
PART I. KINEMATICS.—PRELIMINARY ELEMENTARY MOVEMENTS OF INVARIABLE POINTS AND SYSTEMS.
Lessons 1–2.
Object of kinematics, under the geometrical and experimental point of view. Its principal divisions.
Re-statement of the notions relative to the motion of a point, its geometrical representation, and more especially the determination of its velocity.
Simultaneous Velocities of a Point and the Increments of its Velocities.
Ratio of the elementary displacement and the velocity of a point to the displacement, and velocity of its projection upon a straight line or plane. Use of infinitesimals to determine these ratios. Example:—Oscillatory motion of the projection upon a fixed axis of a point moving uniformly upon the circumference of a circle.
Analogous considerations for polar co-ordinates. Relations of the velocity of a point, of its velocity of revolution and its angular velocity about a fixed pole; of its velocity in the direction of the radius vector; of the velocity of increase of the area which this radius describes.
Simple Motions of Solids, or Rigid Systems.
1. Motion of rectilinear or curvilinear translation; simultaneous displacements, and velocities of its different points.
2. Motion of rotation about a fixed axis; relation of the velocities of different points to the angular velocity.
Geometrical notions and theorems relative to theinstantaneous centerof rotation of a body of invariable figure and movable in one plane, or to theinstantaneous axisof rotation of a rigid system situated in space, and movable parallel to a fixed plane. Relation of the velocities of different points to their common angular velocity. Use of the instantaneous center of rotation for tracing tangents; examples—and amongst others—that of the plane curve described by a point in a straight line of given length, whose extremities slide upon two fixed lines. Rolling of a curve upon another fixed curve in a plane. Descartes’ theorems upon the inter of the normals at the successive points of contact: cycloids, epicycloids, involutes, and evolutes. Extension of the preceding motions to the instantaneous axis of rotation of a rigid system movable about a fixed point.
COMPOSITION OF MOTIONS.
Lessons3–6.Composition of the Velocities of a Point.
Polygon of velocities. Example of movements observed relatively to the earth. Particular cases; composition of velocities taken along three axes; composition of the velocity of a point round a fixed pole, and its velocity along the radius vector. Method of Roberval for tracing tangents.
Composition of the Simple Motions of a Solid System.
Composition of any number of translatory displacements of a solid. Composition of two rotations about two intersecting axes. Composition of any number of rotations about axes cutting one another at the same point; parallelopiped and polygon of rotations. Composition of two simultaneous rotations about parallel axes; case where the rotations are equal and of opposite kinds. Decomposition of a rotation about an axis into an equal rotation about any axis whatever parallel to the first, and a translation perpendicular to the direction of this axis. Direct and geometrical decomposition of the most general motions of a body into a rotation about, and a translation along, an axis called theinstantaneous axis. Composition of any two motions whatever. Every movement of an invariable system is at each instant of time decomposable into three movements of rotation, and three movements oftranslationwith respect to three axes, which are neither parallel nor lying in the same plane, but otherwise arbitrarily chosen.
Relative or Apparent Motions.
Relative motion of two points whose absolute motions are given graphicallyà priori. Trajectory of the relative motions, relative velocities, and displacements upon curves or upon the direction of the mutual distance of the two points; use of the parallelogram to determine its amount. Relative motion of a point in motion in respect of a body turning about a fixed axis; relative motion of two bodies which turn about parallel or converging axes, and in general of two rigid bodies or systems impelled by any motions whatever. How this problem is immediately reduced to that of the composition of given motions.
The most general continued motion of an invariable figure in a plane is anepicycloidalmotion, in which the instantaneous center describes a curve fixed in relation to absolute space, and traces relatively to the proposed figure a movable curve, which is rigidly connected with that figure and draws it along with it in its motion of rolling upon the other fixed curve. Case of space or spherical figures.
ON THE ACCELERATED MOTION OF A POINT.
Lessons7–9.Accelerated Rectilinear Motion.
Re-statement of the motions acquired relatively to the acceleration in the variable rectilinear motion of a point. Brief indication of the solution of six problems arising out of the investigation of the laws of the motion in terms of the space, time, velocity, and accelerating force. For the most part these solutions may be brought to depend on exact or approximate quadratures. Numerical exercises.
Accelerated Curvilinear Motions.
Re-statement of the notions acquired relative to the composition of accelerating forces; the resulting acceleration, the normal and tangential acceleration animating a point in motion on a curve. The total acceleration of a point upon an axis or plane is the projection upon this axis or plane of the acceleration of the moving body in space. In uniform curvilinear motion the total or resultant acceleration becomes normal to the curve. Particular case of the circle; value of the normal acceleration in terms of the velocity of revolution or the angularvelocity of the radius vector. Case of any curve whatever; geometrical expression of the total or resultant acceleration.
Accelerated Compound and Relative Motions.
Geometrical investigation of the simple and compound accelerations arising out of the hypothesis in which the motion of any system of points whatever is referred to another system of invariable form, but also inmotion.Geometrical and elementary explanations of the results obtained by means of the transformation of co-ordinates.
Examples or Exercises chosen from among the following Questions:—
Projection of circular and uniform motion upon a fixed straight line or plane; motion of a circle which rolls uniformly on a straight line; comparison of the motions of the planets relatively to each other, treating them as circular and uniform: comparison of the accelerating force on the moon with that of bodies which fall to the earth.
GEOMETRICAL THEORY AND APPLICATION OF MECHANISMS OR CONTRIVANCES FOR THE TRANSFORMATION OF MOTION.
Lessons 10–19.
Succinct notions on the classification of elementary motions and organs for transmission of motion in machines after Monge and Hachette, Lanz and Bétancourt.
The most essential details upon this subject are set forth in the following order, and made clear by outline drawings previously distributed among the pupils.
Organs fitted to regulate the direction of the circular or rectilinear motion of certain pieces.
Axle; trunnions, gudgeons; pivots and bearings; couplings of axes; adjustment of wheels and of their arms. Joints with hinges, &c.; sheaves and pulleys; chains, ropes, and straps; means of securing them to the necks. Grooves and tongue-pieces. Eyelet-holes sliding along rectilinear or curvilinear rods. Advantages and disadvantages of these different systems of guides under the point of view of accuracy.
Rapid indication of some of their applications to drawbridges and to the movable frames or wagons of saw-works and railways.
Transmission at a Distance of Rectilinear Motion in a determinate Direction and Ratio.
Inclined plane or wedge guiding a vertical rod. Wedge applied to presses. Rods, winch-handles, &c. Disposition of drums or pulleys in the same plane or in different planes; geometrical problem on this subject. Fixed and movable pulleys. Blocks to pulleys. Simple and differential wheel and axle moved by cords. Transmission through a liquid. Ratios of velocities in these different organs.
Direct Transformation of circular progressive motion into progressive and intermittent rectilinear motion.
Rod conducted between guides: 1o, by the simple contact of a wheel; 2o, by cross-straps or chains; 3o, by a projecting cam; 4o, by means of a helicoidalgroove set upon the cylindrical axis of the wheel. To-and-fro movement, and heart-shaped or continuous cam, waves, and eccentrics. Simple screw and nut. Left and right handed screws; differential screw of Prony, called the micrometric screw. Ratio of the velocities in these different organs.
The example of the cam and pile-driver will be particularly insisted upon; 1o, in the case where this cam and the extremity of the rod have any continuous form given by a simple geometrical drawing; 2o, in the case where this form is defined geometrically by the condition, that the velocity is to be transmitted in an invariable ratio, as takes place for cams in the form of epicycloids or involutes of circles.
Transformation of a circular progressive motion into another similar to the first.
1o, by contact of cylinders or cones, the two axes being situated in the same plane; 2o, by straps, cords, or endless chains, the axes being in the same situation; 3o, by cams, teeth, and grooves, at very slight intervals; 4o, by the Dutch or universal joint. Case, where the axes are not situated in the same plane; use of an intermediate axis with beveled wheels or a train of pulleys; idea of White or Hooke’s joint in its improved form. Endless screw specially employed in the case of two axes at right angles to one another. Combinations or groupings of wheels. Idea of differential wheels. Relations of velocities in the most important of these systems of transmission.
Transformation of circular progressive Motion into rectilinear or alternating circular motion.
Ordinary circular eccentric. Eccentrics with closed waves or cams. Examples and graphical exercises in the class-rooms relative to the alternate action of the traveling frames of saw-mills, of the slides or entrance valves of steam-engines. Cams for working hammers and bellows.
Transformation of alternating circular motion into alternating rectilinear motion, or into intermittent and progressive circular motion.
Pump rods with or without circular sectors, &c. Examples taken from large exhausting pumps, fire-engines, and common pumps. Suggestions as to the best arrangement of the parts. Lagarousse’s lever, &c. Application of the principle relative to the instantaneous center of rotation to give the relations of the velocities in certain simple cases.
Transformation of alternating circular or rectilinear motions into progressive circular motion.
The knife-grinder’s treadle. System of great machines worked with connecting rods, fly-wheel, &c. Watt’s parallelogram, and the simplest modifications of it for steamboats, for instance. The most favorable proportions for avoiding the deviation of piston-rods. Simplification of parts in the modern steam-engines of Maudsley, Cavé, &c. Variable ratios of the velocities.
Of organs for effecting a sudden change of motion.
Suspendorsor moderators, &c. Dead wheels and pulleys, &c. Mechanisms for stretching cords or straps, and make them change pulleys during the motion. Brakes to windmills, carriages, &c., &c. Case where the axes are renderedmovable. Means for changing the directions and velocity of the motions. Coupled and alternate pulleys; alternate cones; castors moving by friction and rotation upon a plate or turning-cone; eccentric and orrery wheels. Means of changing the motion suddenly and by intervals; wheels with a detent pile-drivers; Dobo’s escapement for diminishing the shock, &c.
Geometrical Drawing of Wheel-work.
General condition which the teeth of toothed wheels must satisfy. Consequence resulting from this for the determination of the form of the teeth of one of two wheels, when the form of the teeth of the other wheel is given.
Cylindrical action of toothed wheelsor toothed wheels with parallel axes. External engagement of the teeth; internal engagement. Particular systems of toothed wheels; lantern wheels, flange wheels, involutes of circles. Reciprocity of action; case where the action can not be rendered reciprocal. Pothook action. Details as to the form and dimensions given in practice to the teeth and the spaces which separate them.
Conical action of toothed wheels, or toothed wheels with converging axes. Practical approximate method of reducing the construction of a conical to that of a cylindrical engagement of toothed wheels.
Means of observation and apparatus proper for discovering experimentally the law of any given movement.
Simple methods practiced by Galileo and Coulomb in their experiments relative to the inclined plane and the motion of bodies sliding down it. Various means of observing and discovering the law of the translatory and rotatory motion of a body according as the motion is slow or rapid. Determination of the angular velocity, &c. The counter in machines. Apparatus of Mattei and Grobert for assigning the initial velocity of projectiles (musket balls.) Colonel Beaufoy’s pendulum apparatus. Chronometrical apparatus for continuous indications by means of a pencil. Eytelwein’s apparatus with bands, and its simplest modifications. Apparatus with cylinders or revolving disks. Use of the tuning-fork for measuring with precision very small fractions of time.
(The principal sorts of the apparatus above described are made to act under the eyes of the pupils.)
PART II.—EQUILIBRIUM OF FORCES APPLIED TO MATERIAL SYSTEMS.
Lesson 21.
Résumé of the notions acquired upon the subject of forces, and their effects on material points.
Principle of inertia, notion of force, of its direction, of its intensity. Principle of the equality of action and reaction. What is meant by the force of inertia? Principle of the independence and composition of the effects of forces. Forces proportional to the acceleration which they produce on the same body. Composition of forces. Relation between the accelerating force, the pressure, and the mass. Definition of the work done by a force. The work done by the resultant is equal to the sum of the works done by the components. Moment of a force in relation to an axis deduced from the consideration of the work of the force applied to a point turning about a fixed line. The moment of theresultant of several forces applied to a point is equal to the sum of the moments of the components. Corresponding propositions of geometry.
Lessons 22–25.
Succinct Notions upon the Constitution of Solid Bodies.
Every body or system of bodies may be regarded as a combination of material points isolated or at a distance, subject to equal and opposite mutual actions. Interior and exterior forces. Example of two molecules subject to their reciprocal actions alternately, attractive and repulsive, when the forces applied draw them out of their position of natural equilibrium. Different degrees of natural solidity, stability, or elasticity; they can only be appreciated by experience.
Equilibrium of any Systems whatever of Material Points.
General theorem of the virtual work of forces applied to any system whatever of material points. It is applicable to every finite portion of the system, provided regard be had to the actions exercised by the molecules exterior to the part under consideration. Determination of the sum of the virtual works of the equal and reciprocal actions of two material points. Demonstration of the six general equations of equilibrium of any system whatever. They comprise implicitlyeveryequation deduced from a virtual movement compatible with the pre-supposed solidification of the system.
Theorem on the virtual work in the case of systems where one supposes ideal connections, such as the invariability of the distance of certain points of the system from one another, and the condition that certain of them are to remain upon curves either fixed or moving without friction.
Equilibrium of Solid Bodies.
The six general equations of equilibrium are sufficient as conditions of the equilibrium of a solid body. Theory of moments and couples.
APPLICATIONS.
Lessons 26–29.Equilibrium of Heavy Systems.
Recapitulation of some indispensable notions for the experimental determination of the center of gravity of solids when the law of their densities is unknown. Re-statement of the theorem relative to the work done by gravity upon a system of bodies connected or otherwise. In machines supposed without friction submitted, with the exception of their supports, to the action of gravity alone, the positions of stable or unstable equilibrium correspond to the highest or lowest points of the curve which would be described by the center of gravity of the system when made to move. Influence of defect of centering in its wheels, upon the equilibrium of a machine. Case where the center of gravity always remaining at the same height the equilibrium is neutral. Examples relative to the most simple drawbridges, &c.
Equilibrium of Jointed Systems.
Equilibrium of the funicular polygon deduced from direct geometrical considerations: Varignon’s theorem giving the law of the tensions by anotherpolygon whose sides are parallel and proportional to the forces acting upon the vertices of the funicular polygon. Case of suspension bridges; investigation of the curve which defines the boundary of the suspension chain; tensions at the extremities.
Equilibrium of systems of jointed rigid bodies without friction. Determination of the pressure upon the supports and the mutual actions at the joints.
Equilibrium and stability of solid bodies submitted to the action of stretching or compressing forces.
Permanent resistance and limiting resistance of prisms to longitudinal extension and compression. Equilibrium and stability of a heavy solid placed upon a horizontal plane and submitted to the action of forces which tend to overset it. Resultant pressure and mean pressure; hypothetical distribution of the elements of the pressure on the base of support. Conditions of stability, regard being had to the limit of resistance of solid materials, co-efficient of stability deduced from it.
PART III.—ON THE WORK DONE BY FORCES IN MACHINES.
Lessons 30–39.General Notions.
Principle of work in the motion of a material point. Extension of this principle to the case of any material system whatever in motion. Considerations relative to mechanical work in various operations, such as the lifting of weights, sawing, planing, &c. It is the true measure of the productive activity of forces in industrial works. It may always be calculated either rigorously or approximately when the mathematical or experimental law which connects the force with the spaces described is given. Uniform work, periodical work, mean work, for the unit of time. Horse-power unit. Examples and various exercises, such as the calculation of the work corresponding to the elasticity of gases on the hypothesis of Mariotte’s law, the elongation of a metallic prism, &c.
Dynamometrical Apparatus.
Dynamometer of traction by a band or rotating disc or register. Dynamometer of rotation with simple spring, with band or register. Dynamometer of rotation with multiple springs and with register for the axles of powerful machines. Improved indicator of Watt.
(These pieces of apparatus are made to act under the eyes of the pupils.)
Work of Animal Prime Movers upon Machines.
Results of experience as to the values of the daily work which animal motors can supply under different circumstances without exceeding the fatigue which sleep and nourishment are capable of repairing.
Theory of the Transmission of Work in Machines.
Principal resistance. Secondary resistances. Two manners in which bodies perform the duty of motors. Ratio of work done to work expended always inferior to unity. Different parts of machines; receiver; organs of transmission; tools as machines.
Calculation of the Work due to the passive resistances in machines.
Résuméof the notions previously acquired on friction. Application to the inclined plane, to the printing-press, to guides or grooves, to the screw with a square thread; different cases of uniform motion being impossible under the action of forces of given directions. Friction of trunnions, pivots, eccentrics, and insertions of winch-handles. Prony’s dynamometrical brake; conditions of its application. Resistance to rolling; its laws according to experiment. Use of rollers and friction-wheels; their practical inconveniences.
Mixed friction of toothed wheels; the Dobo escapement: friction of the teeth in the endless screw.
Stiffness and friction of cords. Results of experience. Friction of cords and straps running round drums. Different applications; brakes; transmission by cords, endless straps, or chains.
Examples and exercises; effects of passive resistances in the capstan, the crane, pulleys, &c.
Lesson 40.Revision.
SECOND YEAR.
PART I.—DYNAMICS.— DYNAMICS OF A MATERIAL POINT.
Lessons 1–2.Completion of the Notions acquired on this Subject.
Differential equations of the motion of a material point submitted to the continued action of one or more forces. The acceleration of the projection of a point upon any axis or plane is due to the projection of the forces on this axis or plane. The acceleration along the trajectory is due to the tangential force. Relation of the curvature to the centripetal force. Introduction of the force of inertia into the preceding enunciations.
The increase of the quantity of motion projected upon an axis or taken along the trajectory is equal to the impulsion of the projected resultant, or to that of the tangential force. The total impulsion of a force is got by methods of calculation and of experiment analogous to those which relate towork. The increase of the moment of the quantity of motion in relation to any axis is equal to the total moment of the impulsions of the forces during the same interval of time; direct geometrical demonstration of this theorem. In decomposing the velocity of the moving body into a velocity in the plane passing through the axis of the moments, and a velocity of revolution perpendicular to this plane, we may replace the moment of the quantity of motion in space by the quantity of motion of revolution. Particular case known under the name of the principle of areas.
Extension of the preceding theorems to the case of relative motions. Apparent forces which must be combined with the real ones that the relative motion of a point may be assimilated to an absolute motion. Particular case of relative equilibrium. Influence of the motion of the earth upon the accelerating force of gravity.
DYNAMICS OF ANY MATERIAL SYSTEMS.
Lessons 3–8.
Principle or general rulewhich reduces questions in dynamics to questions in equilibrium by the addition of the forces of inertia to the forces which reallyact on the system. Equation of virtual work which expresses this equilibrium; it comprises in general the external and internal forces.
General Theorems.
These theorems, four in number, are founded upon the principle of the equality of action and reaction applied to internal forces. They may be deduced from the preceding rule, but the three last are obtained more simply by extending to a system of material points analogous theorems established for isolated material points.
General theorem of the motion of the center of gravity of a system. Particular case calledprinciple of the conservation of the motion of the center of gravity.
General theorem on the quantities of motion and impulsions of exterior forces projected on any axis.
General theorems of moments of quantities of motion and impulsions of exterior forces, projected on any axis whatever.
General theorems of the moments of quantities of motion and impulsions of exterior forces about any axis. Analogy of these two theorems with the equations of the equilibrium of a solid, in which the forces are replaced by impulsions and quantities of motion.
Composition of impulsions, of quantities of motion, or the areas which represent them. All the equations which can be obtained by the application of the two theorems relative to quantities of motion and impulsions, reduce themselves to six distinct equations. Particular case calledprinciple of the conservation of areas. Fixed plane of the resulting moment of the quantities of motion calledplane of maximum areas.
General theorem of work andvis viva. Part which appertains to the interior forces in this theorem. Particular case called principle of the conservation ofvires vivæ, where the sum of the elements of work done by the exterior and interior forces is the differential of a function of the co-ordinates of different points of the system. Application of the theorem of work to the stability of the equilibrium of heavy systems.
Extension of the preceding theorems to the case of relative motions. Particular case of relative equilibrium. Motion of any material system relative to axes always passing through the center of gravity, and moving parallel to themselves. Invariable plane of Laplace. Relation between the absolutevis vivaof a material system, and that which would be due to its motion, referred to the system of movable axes above indicated.
Examples and Applications.
The following examples, amongst others, to be taken as applications or subjects of exercises relative to the general principles which precede.
Walking. Recoil of guns. Eolypile. Flight of rockets.
Pressure of fluid veins, resistance of mediums, &c. Direct collision of bodies more or less hard, elastic, or penetrable. Exchange of quantities of motion. Loss ofvis vivaunder different hypotheses. Influence of vibrations and permanent molecular displacements.
Pile driving; advantage of large rammers. Comparison of effects of theshocks and of simple pressures due to the weight of the construction. Oblique collision, and ricochet. Data furnished by experiment.
Oscillations of a vertical elastic prism suspended to a fixed point, and loaded with a weight, neglecting the inertia, and the weight of the material parts of this prism. Case of a sudden blow. What is meant by the “resistance vive” of a prism to rupture? Results of experiments.
Work developed by powder upon projectiles, estimated according to thevis vivawhich it impresses on them, as well as upon the gun and the gases upon hypothesis of a mean velocity.
SPECIAL DYNAMICS OF SOLID BODIES.
Lessons 9–12.Simple Rotation of an invariable Solid about its Axis.
In applying to this case the first general rule of dynamics, the theorem of the moments of the quantities of motion, and the theorem of work, we are led to the notion of the moment of inertia; explanation of the origin of this name. The angular acceleration is equal to the sum of the moments of the exterior forces divided by the moment of inertia about the axis of rotation. Sum of the moments of the quantities of motion relative to this axis.Vis vivaof a solid simply turning about an axis. What is meant byradius of gyration?
Remindof the geometrical properties of moments of inertia, of the ellipsoid which represents them, of the principal axes at any point, of those which are referred to the center of gravity.
Pressure which a rotating body exercises on its supports. Reduction of the centrifugal and tangential forces of inertia to a force which is the force of inertia of the entire mass accumulated at the center of gravity, and a couple.
Particular case where the forces of inertia have a single resultant; different examples. Center of percussion. Compound pendulum; length of the corresponding simple pendulum. Center of oscillation; reciprocity of the centers or axes of suspension and oscillation. Pressure upon the axis. Influence of the medium; experience proves that the resistance, varying with the velocity, changes the extent of the oscillations, but does not sensibly affect the time. Experimental determination of the center of oscillation and the moment of inertia about an axis.
Motion of an invariable Solid subject to certain Forces.
General notions on this subject. Motion of the center of gravity; motion of rotation about this point.
Lessons 13–19.Various Applications.
Motion of a homogeneous sphere or cylinder rolling upon an inclined plane, taking friction into account.
Motion of a pulley with its axis horizontal, solicited by two weights suspended vertically to a thread or fine string passing round the neck of the pulley, the axle of which rests upon movable wheels. Atwood’s machine serving to demonstrate the laws of the communication of motion.
Motion of a horizontal wheel and axle acted on by a weight suspended vertically to a cord rolled round the axle, or upon a drum with the same axis, and presenting an eccentric mass. To take account of the variable friction of thebearings, and the stiffness of the cord, with recourse, if necessary, to approximation by quadratures. Oscillations of the torsion balance.
Balistic pendulum. Condition that there may be no shock on the axis. Experimental determination of the direction in which the percussion should take place.
Theory of Huyghen’s conical pendulum considered as a regulator of machinery. How to take account of the inertia and friction of the jointed rods, as well as of the force necessary to move the regulating lever, &c.; appreciation of the degree of sensibility of the ball apparatus with a given uniform velocity.
Windlass with fly-wheel. Dynamical properties of the fly-wheel. Reduced formulæ for a crank with single or double action. Advantages and disadvantages of eccentric masses. Tendency of the tangential forces of inertia to break the arms. Numerical examples and computations.
Mutual action of rotating bodies connected by straps or toothed wheels in varying motion.
The wedge and punching-press. Stamping screw or lever used in coining, cams, lifting a pile or a hammer. To take account of the friction during the blow, and afterwards to estimate the loss ofvis vivain cases which admit of it.
PART II.—SPECIAL MECHANICS OF FLUIDS.—HYDROSTATICS.
Lessons 20–22.
Principle of the equality of pressure in all directions. Propagation of the pressures from the surface to the interior of a fluid, and upon the sides of the vessel. Equations of equilibrium for any set of forces. Pressure exerted in the containing orifices. Measure of the pressure upon a plain portion of surface inclined or vertical (sluice-gate, embankments, &c.) Center of push or pressure. Pressure against the surfaces of a cylindrical tube. Effect, and resistance to oppose to the pressure. Manometer and piezometer. Equilibrium of a body plunged in a heavy fluid or floating at its surface. Stability of floating bodies. Metacenter. Laws of the pressure in the different atmospheric strata.
HYDRAULICS.
Lessons 23–27.Flow of Fluids through small Orifices.
Study of the phenomena which accompany this flow in the case of a thin envelop and a liquid kept at a constant level. Conditions of this constancy in the level, and the permanence of the motion in general. Motion of the lines of fluid; form; contraction; reversal and discontinuity of liquid veins. Fundamental formulæ for liquids and gases based upon the principle ofvis viva, and Bernoulli’s hypothesis of parallel s or Borda’s of contiguous threads. Torricelli’s theorem relative to small orifices. What is called the theoretical expenditure, effective expenditure, and co-efficient of geometrical contraction. Co-efficient deduced from the effective expenditure. Its variations with the volume of the fluid contents, and the form of the inner surfaces of the reservoir. Results of the experiments of Michelotti, Borda, Bossut, &c. Phenomenon of adjutages. Venturi’s experiments; influence of atmospheric pressure; loss ofvis viva; reduction of the velocity and augmentation of the expenditure. Results of experience relative to the co-efficient of expenditure, the form and range of the parabolic jets, showing the initialvis viva, and the loss ofvis viva.
Large orifices.—Sluice holes and floodgates; reservoirs or open orifices; expenditure; practical formulæ and results of experiment. Influence of the proximity of the sides and the walls. Arrangement to avoid the effects of contraction or the losses ofvis viva.
Flow through conducting Pipes and Open Canals.
Practical formulæ relative to the case of uniform s of great length. Measure of the pressures at different points of a conduit-pipe. Expression for the losses of effect due to corners and obstructions. Flow of gases. Principal methods of measuring the volume consumed adopted in practice. Floats. Pitot’s tube. Woltman’s mill. Register mill in air or gas. Waste in such instruments. Modulus and scale for water-supply.
PART III.—DIFFERENT MACHINES CONSIDERED IN THE STATE OF MOTION.
Lesson 28.General Considerations. Résumé of the Notions acquired on this Subject.
Equation ofvis viva, and transmission of work in machines, account being taken of the different causes of power and resistance. Physical constitution of machines;receiver,communicators, andoperator. Influence of the weights, of frictions, of shocks, and any changes in thevis viva. Parts with continuous or uniform motion, with alternating or oscillating motion. Laws of the motion on starting from rest, and when the stationary condition is established. The positions to which the maximum and minimum of thevis vivacorrespond are those in which there is equilibrium between all the forces, exclusive of the forces of inertia. Advantage of uniform or periodic motion. General methods for regulating the motion; symmetrical distribution of the masses and strains; flys and various regulators. Brakes and moderators; their inconveniences. Object and real advantages of machines.
Lessons 27–35.Hydraulic Wheels.
Vertical wheels with float-boards, with curved ladles, and with spouts. Figure of the surface of the fluid in these latter. Horizontal wheels working by float-boards, buckets, and reaction. Turbines. Description, play, and useful effects compared according to the results of experiment. Vertical wheels of windmills and steamboats. Screw propeller.
Windmills.
Description. Result of Coulomb’s observations.
On the principal kinds of Pumps.
Special organs of pumps. Valves and pistons, force pump, sucking pump; limit to the rise of the water. Sucking and force pump. Dynamical effects. Indication as to the losses ofvis vivaand the waste in different pumps. Explanation of the hydraulic ram. Air vessel. Fire pumps. Double action pumps.
Various Hydraulic Machines.
Hydraulic press. Water engine. Exhausting machines;norias; under and overshot wheels; Archimedes’ screw, construction and experimental data.
Lessons36–39.Steam Engines.
Succinct description of the principal kinds of steam-engine with or without detent. Effects and advantages of the detent. Condenser. Air Pump. Furnace and feeding-pump.
Variable detent. Formulæ and experimental results.
Lessons40–42.Revision.
Reflections on the totality of the subjects of the course.
IV. PHYSICS.—FIRST YEAR.
GENERAL PROPERTIES OF BODIES.—HYDROSTATICS.—HYDRODYNAMICS.
Lessons1–5.Preliminary Notions.
Definitions of physics. Phenomena. Physical laws. Experiments are designed to make them spring out of the phenomena. Method of induction. Physical theories; different character of the experimental and mathematical methods.
General Properties of Bodies.
Extension. Measure of lengths. Vernier. Cathetometer. Micrometer screw. Spherometer. Dividing engine.
Divisibility. Porosity. Ideas generally received on the molecular constitution of bodies. These conceptions, which are purely hypothetical, must not be confounded with physical laws. Elasticity. Mobility. Inertia. Forces; their equilibrium, their effects, their numerical estimation.
Weight or Gravity.
Direction of gravity. Plumb-line. Relation between the direction of gravity and the surface of still water.
Weight. Center of gravity.
Experimental study of the motion produced by weight. In vacuum, all bodies fall with the same velocity. Disturbing influence of the air. Inclined plane of Galileo. Atwood’s machine. To prove by experiment; 1othe law of the spaces described; 2othe law of velocities. Morin’s self-registering apparatus with revolving cylinder.
Law of the independence of the effect produced by a force upon a body, and the motion anteriorily acquired by this body. Law of the independence of the effects of forces which act simultaneously upon the same body. Experimental demonstration and generalization of these laws. Law of the equality of action and reaction.
Mass. Acceleration. For equal masses the forces are as the accelerations which they produce. Relation between the force, mass, and acceleration. Collision.
General laws of uniformly accelerated motion. Formulæ.
Pendulum. Law of the isochronism of small oscillations and law of the lengths deduced from observation.
Method of coincidences or beats. Use of the pendulum as the measure of time. Simple pendulum; formulæ. Compound pendulum: the laws of the oscillations of a compound pendulum are the same as the laws of the oscillations of a simple pendulum whose length may be calculated.