Henry Moseley. # Syllabus of a course of experimental lectures on the Theory of Equilibrium, to be delivered at the King's College, London, in the October term of the year 1831 online

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SYLLABUS

OF A

COURSE OF EXPERIMENTAL LECTURES

â‚¬fftots of iBquiXihvium,

TO BE DELIVERED AT

THE KING'S COLLEGE, LONDON,

OCTOBER TERM OF THE YEAR 183L

THE REV. H. J^OSELEY,

PKOl'ESSOn OF NATURAL AND EXPERIMENTAL PHILOSOPHY

LONDON:

B. FELLOWES, BOOKSELLER & PUBLISHER TO THE COLLEGE,

39, LUDGATE-STREET.

1881.

LOAN STACK

TJiese Lectures require no introductory course of

mathematical reading; the method of demonstration

being exclusively experimental.

LONDON :

R. CLAY, PRINTER, BREAD-STREF.T-HILL, CHEAPSIDE.

'?339J)

SYLLABUS,

&c. &c.

Time Space, â€” Matter Force,

On the nature of a Property or Quality.

The Properties of Matter, â€” Impenetrability â€”

uselessness of the term, as simply implying the

distinction of matter and space. â€” Divisibility.

Molecules, â€” Quantity, â€” Motion, â€” Force. â€”

Quantity of motion, â€” Velocity, â€” Direction Resist-

ance, â€” Pressure, â€” Equilibrium,

All the mutual relations of Time â€” Space â€”

Matter â€” and Force, belong properly to the science

of Mechanics.

a2

" 090

According to the usual acceptation of the term.

Mechanics, that science is, however, confined to the

investigation of the conditioms of the equilibrium

and the rnotiori of masses, or aggregates of matter,

acted upon hy known and appreciable forces.

To the theory of Equilibriwn belong â€” the science

of Statics, or the Equilibriwn of Solids â€” and the

science of Hydrostatics, or the Equilibrium of

Fluids,

To the tJieory of Motion belong â€” the science of

Dynamics, or the Motion of Solids â€” and of Hy-

dronamics, or the Motion of Fluids,

On the Abstract or Mathematical Method in

Physics.

On the Experimental Method â€” Nature and limits

of the proof by experiment.

ON STATICS,

OR THE EQUILIBRIUM OF SOLIDS.

Substitution of the term pressure for force in

Statics, as implying force held in equilibrium.

The equality of pressures.

The unit of pressure.

The quantity of pressure-

The direction of pressure.

The representation of pressures, in quantity and

direction, by lines.

Forces, to sustain one another upon a flexible line,

must be equal ; act from one another, and in the

direction of the line, at the points where they are

applied to it. These conditions being satisfied, the

form of the line is immaterial, and the pressures

may be any where appHed in it.

Application of pressure by means of the cord and

pulley.

If any number of pressures, acting upon the

parts of a rigid system, are in equilibrium among

themselves, and if to the same system there be

applied other pressures, such as do not disturb the

equilibrium thus existing ; these last are in equili-

brium amongst themselves.

Difficulty of ascertaining the conditions of the

equilibrium of given pressures applied to a rigid

system, arising from the impossibility of obtaining

any such system not already acted upon by the

pressure of gravity, or weight.

This difficulty obviated by placing the system in

equilibrium with respect to that pressure.

On the equilibrium of two j^i'^'s^^^fc's acting upon

6

a rigid system, â€” They must be equal â€” they must

act in opposite directions, but in the same straight

line. These conditions being satisfied, the form of

the system and the points of application of the

pressures may be any whatever, and they may act

either to or from one another.

Pressure is propagated through rigid bodies in

right lines.

On the equilibrium of three pressures in the

same plane acting upon a point.

The parallelogram of forces.

The equilibrium of a weight suspended by two

cords. '

The resistance of a surface is in a direction per-

pendicular to that surface.

The equilibrium of three pressures, one of which

is supplied by the resistance of an inclined plane,

or a curved surface.

The equilibrium of a weight upon an inclined

plane â€” when the power acts (1st) parallel to the

surface of the plane â€” (2d) parallel to its base â€”

(3d) at any angle above the plane â€” (4th) at any

angle below it.

The force required to retain the plane at rest,

when acted on by given forces.

The wedge when it is in equilibrium. â€” The

screw.

The equilibrium of weights sustaining one another

on two inclined planes, by means of a string passing

over a pulley.

The equilibrium of weights sustaining one another,

as above, on two curved surfaces.

On curves of equilibration.

The equilibrium of any number of pressures

acting, in the same plane, upon a point.

The polygon of forces.

The equilibrium of three pressures in different

planes, acting upon a point.

The equilibrium of any number of pressures in

different planes, acting upon a point.

The equilibrium of three pressures acting uj)on

a rigid system in the same plane.

Case in which the three pressures are parallel.

Case in which one of the pressures is supplied by

the resistance of a fixed axis, about which the sys-

tem is moveable.

Pressure upon the axis.

Conditions of equilibrium in the case in which the

forces are parallel â€” deduced from the above â€” proved

independently.

8

The straight lever (1st) when the forces are ob-

lique, â€” (2d) when they are parallel. The balance. â€”

The false balance â€” means of weighing correctly with

it. â€” The steelyard. â€” The handspike. â€” The crooked

lever. â€” The crooked balance. â€” The hammer lever. â€”

The crank. â€” Combinations of levers. â€” The weighing

machine. â€” The Russel press lever. â€” The Stanhope

press lever. â€” The genou.

The wheel and axle. â€” The capstan. â€” The fusee.

The cog-wheel.â€” Pinion. â€” Trundle. The forms of

the cogs, leaves, and staves of these, necessary to an

uniform action of the machinery. Precautions as

to form, &c. may be in a great measure neglected,

where the cogs are small. â€” System of toothed

wheels. â€” The crane.

The screwâ€” the relation of the power and weight

independent of the diameter of the screw. â€” The

endless screw. â€” Hunter's screw. â€” Combinations of

the screw and lever. â€” The camb.

The pulley. â€” The single moveable pulley. â€”

Smeaton's pulley. â€” White's pulley. â€” The Spanish

burton. â€” The system of pulleys in which the last

supports the weight. â€” The system of pulleys in

which each string is attached to the weight. â€”

The American burton.

9

Different mechanical contrivances for varying the

quantity and direction of pressure and motion. â€”

The crank. â€” Crown, spur and bevelled wheels. â€”

Hook's joint. â€” Parallel motions. â€” The sun and

planet wheels^ &c. â€” The eccentric.

The equilibrium of any number of forces in tJie

same plane applied to a rigid system.

Lemma, â€” I-f there be a parallelogram in a given

plane, and a point be anywhere taken in the same

plane ; the difference of the areas of the triangles

formed by drawing lines from the given point to the

extremities of two adjacent sides of the parallelo-

gram, shall be equal to the area of the triangle

formed by lines drawn from the given point to the

extremities of the diagonal.

Theory of areas.

Theory of moments.

Case in which the system revolves upon an axis

perpendicular to the plane of the forces.

Pressure upon the axis.

Equilibrium of any number of parallel forces

acting upon a rigid system in the same plane.

Equilibrium of any number of parallel forces

acting any where upon a rigid system.

The centre of gravity.

10

Examples, â€” the centre of gravity of a parallelo-

gram â€” of a triangle â€” of a pyramid â€” of a prism â€” of

a circle â€” of a semicircle â€” of a parabola â€” of a cycloid.

Conditions of the equilibrium of a heavy body

sustained on a horizontal plane â€” on an inclined

plane â€” and on a curved surface. â€” (1) When the

base of the body is a plane. â€” (2) When the base is

a curved surface.

The hanging tov/ers of Pisa and Bologna.

Comparative stability of structures of different

forms.

Theory of the carriage v^^heel. â€” The drawbridge.

Stability of loaded vehicles.

Guldinus's properties.

The centre of gravity of a body or system of

bodies, in equilibrium, is at its highest or lowest

possible point. This proposition proved generally.

â€” Exemplified in the case of the inclined plane. â€”

The crooked lever, &c.

The equilibrium of a beam supported upon a

roller and a vertical plane^ â€” of a rectangle upon

two rollers â€” of a beam upon two surfaces.

The equilibrium of a beam supported by a string

fastened at its extremities â€” supported by two

strings passing over pulleys, and carrying weights.

it

The position of the centre of gravity in animals.

The attitudes of animals dependent upon the

position of the centre of gravity.

TJie conditions of stable â€” unstable â€” aud mixed

equilibrium.

Examples. â€” A hemisphere and a parabolid upon a

plane surface. â€” The same upon a spherical surface.

The common balanceâ€” its sensibility â€” the rapi-

dity of its vibrations â€” its adjustments. These

require to be different for every different loading,

in order that the sensibility may be greatest of

which the balance is capable.

The equilibrium of a body, or system of bodies,

stable or unstable, according as the centre of

gravity is at its low^est or highest point.

Conditions of the equilibrium of any number of

forces acting, in any given number of directions,

upon a system of invariable form.

Case in which any number of forces act in dif-

ferent directions perpendicular to a rigid line.

Case in which a system acted upon by any

number of forces is moveable about a fixed axis.

Pressure upon the axis.

Conditions of the equilibrium of any number of

forces acting upon a system of variable form.

12

The jointed polygon of r^ocls.

The equilibrium of a frame-work of two or more

jointed polygons of rods connected together.

The conditions of the equilibrium of a jointed

frame-work loaded with weights, and placed in an

upright position, is the same as though the frame

were suspended and the same loading applied.

Stability of its equilibrium when suspended.

Instability in the opposite position.

Easy practical method of determining the proper

form of a roof, bridge, or other jointed frame under

a given loading ; and the pressure, on its different

parts and its abutments.

Equilibrium of the arch with polished voussoirs.

Instability of its equilibrium.

On tlie equilibrium of the funicidar polygon. â€”

Case in which the pressures are applied to rings,

moving freely on the thread. â€” The elastic polygon.

The equilibrium of the funicular curve. â€” The

common catenary. â€” The tension on the catenary

at the lowest point varies as the radius of curvature

at that point. â€” Easy method of ascertaining the

tension at any other point.

On the position of equilibrium of a string of a

given length, suspended over two given points.

13

On the relation between the length of the string

and the tension on its parts when suspended.

On the positions of equilibrium of a string whose

extremities hang freely over two pulleys, in the

same horizontal line.

Of all the curves, of given length, which can be

drawn so as to terminate in two given points, in

the same horizontal line, the catenary is that whose

centre of gravity is most distant from that line.

On the catenary loaded with weights.â€” A catenary

may be so loaded as to assume any required form.

Variation in the tension of the catenary, produced

by an irregularity in its loading, and consequent

variation in its form.

The catenary approximates at its vertex very

nearly to a parabola.

ON FRICTION.

The statical laws of friction,

Rennie's apparatus and experiments.

The friction of hard metals under pressures of

less than 32lbs. 8oz. the square inch, nearly one-

sixth of the pressure.

With higher pressures this ratio increases.

The friction of woods.

14

The friction of stones.

The diminution of friction by unguents, varies as

the insistant weights and the nature of the un-

guents; the lighter the weight, the finer and more

fluent should be the unguent, and vice versa.

On the modifications introduced hy friction in

the conditions of the equilibrium of tJie different

mechanical 'powers.

On the two states bordering upon motion in the

inclined plane.

The wheel and axle.

Methods of diminishing friction by means of fric-

tion wheels. â€” Friction of the carriage wheel. â€” The

screw. â€” The system of toothed wheels.

ON THE RIGIDITY OF CORDS.

Coulomb's experiments.

States bordering on motion in the different sys-

tems of pulleys. â€” The proper ratios of the wheels,

axles, and cords of the different pulleys of each

system.

ON THE STRENGTH OF MATERIALS.

Absolute resistance. â€” Rennie's experiments.â€” The

resistance of different masses of metal, wood, and

15

stone to the compression of their parts in given

directions. â€” The resistance of different masses of

metal and wood to the separation of their parts in

given directions. â€” Anomalous results.

On the strain w^hich produces permanent altera-

tion of structure.â€” There is reason to believe that

all bodies are perfectly elastic, as to any pressure

less than that which produces permanent alteration

of structure.

Galileo's hypothesis of the rigidity of fibres. â€”

Leibnitz's hypothesis of the extensibility of fibres. â€”

Theory which admits the compressibility as well as

the extensibility of fibres.

Relative resistance. â€” The neutral line â€” its pro-

perties.

On the strength of a horizontal bar fixed im-

moveably at one end, and carrying a weight at the

other.

On the strength of a bar fixed immoveably at

both ends, and carrying a weight between.

On the strength of a bar supported in the middle^

and carrying weights at the ends.

On the strength of a bar carrying weights

variously distributed over its surface.

On the strongest forms of beams.

16

On the construction of open beams.

On the deflexion of beams by their own weight,

when supported horizontally at their extremities â€”

when inclined to the horizon.

On the deflexion of columns sustaining weights.

On the proper forms of columns sustaining

weights.

The elasticity of flexure.

On the deflexion of elastic laminae.

The equilibrium of springs.

The elasticity of torsion.

On the proper forms , and tlie strength of solid

arches of wood and iron.

On the loading of solid arches.

On the strength of open arches.

THEORY OF ROOFS.

The Shed Roof

The angle of its elevation dependent on the height

and the strength of the walls or pillars on which it

abuts. â€” The strength of its timbers. â€” Ingenious

method of getting rid of the horizontal thrust, by

supporting the timbers beneath their centres of

gravity.

TJie commonTruss Roof â€” The theory of this roof.

17

â€” The horizontal thrust on its abutments. â€” The

different pressures upon its parts, and the consequent

variation in the strength of its timbers.

The deflexions in the tie beam and principal

rafters.

The variations in the lengths of the timbers.

The strength of the joints.

On the different forms of the trussed roof

Examples, â€” The roof of the Bazilica of St. Paul's

at Rome. â€” The roof of the Theatre Argentina at

Rome. â€” The roof of the Birmingham Theatre.

Methods of giving support to roofs â€” by means of

additional frame-work abutting in the wall beneath

the tie beam â€” by means of buttresses, &c.

On truncated roofs.

The roof of Drury Lane Theatre.

On the loading of roofs, hy the suspension of

ceilings, S^c,

The roof and ceiling of the Teatro Alia Scala.

On various methods of dispensing with the tie

beam at the foot of the rafters.

The collar beam â€” great objections to its use

unless supported by pillars, or otherwise, at its

extremities.

Roof of the Church St. Genevieve at Paris.

B

18

Roof of the Theatre Odeon.

The roof of a church in Wiltshire.

On g'othic roofs.

The roof of Westminster School.

The roof of the Middle Temple Hall.

The roof of Westminster Hall.

On the polygonal roof.

Roofs formed with four principal rafters.

Conditions of he equilibrium of four such raf-

ters.

The roof of the Theatre at Bordeaux.

Roofs of sheds in the Arsenal at Cherbourg.

Roofs of sheds in the Dockyard at Plymouth.

The roof of the shed for containing Mahogany in

the West India Docks.

Method of supporting roofs hymeans of polygonal

frames or arches of short rafters.

Great advantages of this method of support.

The proper form and strength of the polygon.

The roof of the Riding House at Moscow^

On the arch of curved timber.

On the use of iron in the framing of roofs.

The roof of the Brunswick Theatre

Roof at Mr. Maudeslay's manufactory.

On trussed floors.

19

Example.â€” The trussed floor in the Teatro Alla-

scala at Milan.

ON THE THEORY OF WOODEN BRIDGES.

The wooden bridge in which the timbers are

straight, and rest immediately upon the piers.

The bridge of Caesar over the Rhine.

The bridge of Cayuga in America.

Method of constructing a wooden bridge over a

rapid torrent.

The bridge across the rapids of Niagara.

On the straight wooden bridge where the timbers

of the roadway are trussed from above, and there

is no horizontal pressure upon the abutments.

The bridge of Palladio over the Cismone.

The great bridge formerly at SchafFhausen.

Wooden bridge near Baltimore, N. America.

The wooden bridge, in which the roadway is

principally supported yrow? beneath by timbers, which

rest obliquely upon the abutments.

The bridge over the Kendal near Berne.

On wooden bridges supported by polygonal

ribs.

The wooden bridge at Lyons.

On wooden bridges with curved ribs.

B 2

20

The bridge of Trajan over the Danube.

The bridge of Freysingen in Bavaria.

The bridge of Bamburgh on the Regnitz in

Germany.

The great wooden arch at Scuykill, in N. America.

The proper variation in the strength of the parts

of the curve of a wooden arch.

ON THE THEORY OF IRON ARCHES.

Strength requisite in the different portions of the

arch. â€” Colebrook-Dale bridge. â€” Buildwas bridge. â€”

Sunderland bridge. â€” Bonar bridge. â€” The bridge

of the Louvre. â€” Vauxhall bridge. â€” Southwark

bridge. â€” Telford's proposed bridge over the

Thames,

ON SUSPENSION BRIDGES.

Application of the theory of the loaded cate-

nary. â€” The method of constructing the chain. â€”

Easy practical method of determining the tension. â€”

It is less at the vertex as the curvature is greater.

On the method of suspending the roadway. â€” On

the piers of the bridge, and the attachment of the

chain. â€” The wire bridge over the Tees near

Durham. â€” The wire bridge over the Tweed near

21

Peebles. â€” The Kelso suspension bridge. â€” The sus-

pension pier at Leith. â€” Brunell's suspension bridge

erected in the Island of Bourbon. â€” The suspension

bridges over the Cataracts of Sckuylkill, and at

Merimac, in North America. â€” The Hammersmith

suspension bridge. â€” The Menai suspension bridge.

Any jointed polygon, or arch, placed in an upright

position so as to be sustained by the pressure of its

parts upon one another and upon its abutments, is

in a position of unstable equilibrium, the centre of

gravity being at its highest possible point. To the

stability of such a polygon, or arch, it is therefore

necessary that its joints should be rendered rigid, in

the directions in v^^hich their position is liable to

disturbance, â€” by additional framing or otherwise.

In the suspended polygon, or curve, the equili-

brium is stable, the centre of gravity being at its

lowest possible point. There is therefore no neces-

sity for rendering the joints rigid ; and the material,

requisite in the other case for producing this rigi-

dity, may be here dispensed with.

The advantage of the upright over the suspended

arch of the same materials, lies in this, that in

the former case the arch is sustained by the resis-

tance of its parts to compression, and in the latter

22

by their resistance to separation, and that mate-

rials are torn asunder more readily than they are

crushed.

ON THE THEORY OF THE STONE ARCH.

True theory of the arch, allowing for the friction

of the voussoirs.

General conditions of the stability of the arch.

The two states bordering upon motion.

Method of describing the line of the least loading

necessary to the equilibrium of an arch whose key-

stone is given.

Line of the greatest loading which such an arch

will bear.

On the comparative strength of different por-

tions of the arch. Generally the strength of an

arch is greater as its curvature is less.

The curvature being given, the stability of an

arch properly constructed increases with the loading.

On the line of pressure.

When the voussoirs are exceedingly narrow, and

the loading considerable, the curve of equilibrium is

the catenary.

Different steps in the fall of an arch.

Example, â€” Pont y Prydd.

23

The circular arch.

Easy method of describing the lines of greatest

and least practicable loading.

The great strength of the circular arch, and

variety of loading under which conditions of its

equilibrium obtain.

The segment of a semi-circular arch.

Its advantages over the whole semicircular arch.

No semicircular arch can be safely constructed

with equal voussoirs â€” A segment may.

A straight wall of any height may be built over

a segment of a circular arch.

Exarnjdes of circular arches.

The bridge of Rimini.

The aqueduct bridge at Nismes.

The bridge of Avignon.

The bridge of Briande.

The bridge of Ulm, &c. &c.

The elliptical arch.

Method of describing the lines of loading in the

two states bordering upon motion, and for given

dimensions of the key-stone.

Weights which an elliptical arch of given dimen-

sions is capable of sustaining on its crown.

24

Comparative weakness of the semi - elliptical

arch.

Cases of semi-elliptical arches, in which, under

the irregular pressures to which they are subjected,

their stability must be dependent upon other causes

than the friction of their parts.

Examples of elliptical arches.

Bridge of the Rialto at Venice.

Bridge over the Arno at Florence.

Bridge of Neuilly.

Waterloo bridge.

London bridge.

On segments of elliptical arches.

Advantages in the use of segments raised upon

high vertical piers, where a clear water way is

required.

A segment of an elliptical arch may be built with

equal voussoirs.

The loading on certain points about the haunches

of a semi-elliptical arch may be any whatever.

Examples of flat arches.

Bridge over the Oise.

Pont de la Concorde.

25

The arch between the western towers of Lincoln

Cathedral.

On the pointed gothic arch.

Equilibrium of the pointed arch.

Method of determining the states bordering upon

motion.

The loading of the key-stone.

Comparative strength of the varieties of the

pointed arch.

Conditions of the equilibrium of an arch, taking

into account the tenacity of the cement.

On the piers of arches.

On the centering of arches.

On the equilihrium of the dome.

The weight on the haunches may be increased

without limit, and their convexity diminished, but

not the contrary.

A dome may be built w^ithout centering. â€” Ex-

ample, â€” The dome of the cathedral of Florence.

On the equilibrium of a dome loaded on the

crown â€” The domes of the cathedral at Florence, St.

Peter's of Rome, and St. Paul's in London. â€” On

^he equilibrium of a dome in which the crown is

wanting.

26

On the theory of the groin.

On the conoidal groin,

ON THE PRINCIPLE OF VIRTUAL VELOCITIES.

Lagrange's proof.

The principle of virtual velocities shewn to obtain

in the following cases of equilibrium.

The equilibrium of any number of forces acting

upon a point. â€” Of weights on the straight lever. â€” Of

forces acting obliquely on the crooked lever. â€” In

the case of the single pulley where the strings are

inclined. â€” Of a system of compound levers. â€” Of the

inclined plane. â€” Of equilibrium on a curve â€” on two

curves â€” of the screw â€” of the wheel and axle â€” of

toothed wheels, &c. &c. â€” In the case of motion about

the centre of gravity of any system of bodies.

The centre of gravity of any system of weights

is at its highest or lowest points, when those weights

are in equilibriumâ€” proof deduced from the prin-

ciple of virtual velocities.

On the quantity of motion.

The quantity of motion, a measure of the moving

force.

Demonstration of the principle of virtual velocities

founded on this consideration.

27

ON THE USEFUL EFFECT OF MACHINES.

OF A

COURSE OF EXPERIMENTAL LECTURES

â‚¬fftots of iBquiXihvium,

TO BE DELIVERED AT

THE KING'S COLLEGE, LONDON,

OCTOBER TERM OF THE YEAR 183L

THE REV. H. J^OSELEY,

PKOl'ESSOn OF NATURAL AND EXPERIMENTAL PHILOSOPHY

LONDON:

B. FELLOWES, BOOKSELLER & PUBLISHER TO THE COLLEGE,

39, LUDGATE-STREET.

1881.

LOAN STACK

TJiese Lectures require no introductory course of

mathematical reading; the method of demonstration

being exclusively experimental.

LONDON :

R. CLAY, PRINTER, BREAD-STREF.T-HILL, CHEAPSIDE.

'?339J)

SYLLABUS,

&c. &c.

Time Space, â€” Matter Force,

On the nature of a Property or Quality.

The Properties of Matter, â€” Impenetrability â€”

uselessness of the term, as simply implying the

distinction of matter and space. â€” Divisibility.

Molecules, â€” Quantity, â€” Motion, â€” Force. â€”

Quantity of motion, â€” Velocity, â€” Direction Resist-

ance, â€” Pressure, â€” Equilibrium,

All the mutual relations of Time â€” Space â€”

Matter â€” and Force, belong properly to the science

of Mechanics.

a2

" 090

According to the usual acceptation of the term.

Mechanics, that science is, however, confined to the

investigation of the conditioms of the equilibrium

and the rnotiori of masses, or aggregates of matter,

acted upon hy known and appreciable forces.

To the theory of Equilibriwn belong â€” the science

of Statics, or the Equilibriwn of Solids â€” and the

science of Hydrostatics, or the Equilibrium of

Fluids,

To the tJieory of Motion belong â€” the science of

Dynamics, or the Motion of Solids â€” and of Hy-

dronamics, or the Motion of Fluids,

On the Abstract or Mathematical Method in

Physics.

On the Experimental Method â€” Nature and limits

of the proof by experiment.

ON STATICS,

OR THE EQUILIBRIUM OF SOLIDS.

Substitution of the term pressure for force in

Statics, as implying force held in equilibrium.

The equality of pressures.

The unit of pressure.

The quantity of pressure-

The direction of pressure.

The representation of pressures, in quantity and

direction, by lines.

Forces, to sustain one another upon a flexible line,

must be equal ; act from one another, and in the

direction of the line, at the points where they are

applied to it. These conditions being satisfied, the

form of the line is immaterial, and the pressures

may be any where appHed in it.

Application of pressure by means of the cord and

pulley.

If any number of pressures, acting upon the

parts of a rigid system, are in equilibrium among

themselves, and if to the same system there be

applied other pressures, such as do not disturb the

equilibrium thus existing ; these last are in equili-

brium amongst themselves.

Difficulty of ascertaining the conditions of the

equilibrium of given pressures applied to a rigid

system, arising from the impossibility of obtaining

any such system not already acted upon by the

pressure of gravity, or weight.

This difficulty obviated by placing the system in

equilibrium with respect to that pressure.

On the equilibrium of two j^i'^'s^^^fc's acting upon

6

a rigid system, â€” They must be equal â€” they must

act in opposite directions, but in the same straight

line. These conditions being satisfied, the form of

the system and the points of application of the

pressures may be any whatever, and they may act

either to or from one another.

Pressure is propagated through rigid bodies in

right lines.

On the equilibrium of three pressures in the

same plane acting upon a point.

The parallelogram of forces.

The equilibrium of a weight suspended by two

cords. '

The resistance of a surface is in a direction per-

pendicular to that surface.

The equilibrium of three pressures, one of which

is supplied by the resistance of an inclined plane,

or a curved surface.

The equilibrium of a weight upon an inclined

plane â€” when the power acts (1st) parallel to the

surface of the plane â€” (2d) parallel to its base â€”

(3d) at any angle above the plane â€” (4th) at any

angle below it.

The force required to retain the plane at rest,

when acted on by given forces.

The wedge when it is in equilibrium. â€” The

screw.

The equilibrium of weights sustaining one another

on two inclined planes, by means of a string passing

over a pulley.

The equilibrium of weights sustaining one another,

as above, on two curved surfaces.

On curves of equilibration.

The equilibrium of any number of pressures

acting, in the same plane, upon a point.

The polygon of forces.

The equilibrium of three pressures in different

planes, acting upon a point.

The equilibrium of any number of pressures in

different planes, acting upon a point.

The equilibrium of three pressures acting uj)on

a rigid system in the same plane.

Case in which the three pressures are parallel.

Case in which one of the pressures is supplied by

the resistance of a fixed axis, about which the sys-

tem is moveable.

Pressure upon the axis.

Conditions of equilibrium in the case in which the

forces are parallel â€” deduced from the above â€” proved

independently.

8

The straight lever (1st) when the forces are ob-

lique, â€” (2d) when they are parallel. The balance. â€”

The false balance â€” means of weighing correctly with

it. â€” The steelyard. â€” The handspike. â€” The crooked

lever. â€” The crooked balance. â€” The hammer lever. â€”

The crank. â€” Combinations of levers. â€” The weighing

machine. â€” The Russel press lever. â€” The Stanhope

press lever. â€” The genou.

The wheel and axle. â€” The capstan. â€” The fusee.

The cog-wheel.â€” Pinion. â€” Trundle. The forms of

the cogs, leaves, and staves of these, necessary to an

uniform action of the machinery. Precautions as

to form, &c. may be in a great measure neglected,

where the cogs are small. â€” System of toothed

wheels. â€” The crane.

The screwâ€” the relation of the power and weight

independent of the diameter of the screw. â€” The

endless screw. â€” Hunter's screw. â€” Combinations of

the screw and lever. â€” The camb.

The pulley. â€” The single moveable pulley. â€”

Smeaton's pulley. â€” White's pulley. â€” The Spanish

burton. â€” The system of pulleys in which the last

supports the weight. â€” The system of pulleys in

which each string is attached to the weight. â€”

The American burton.

9

Different mechanical contrivances for varying the

quantity and direction of pressure and motion. â€”

The crank. â€” Crown, spur and bevelled wheels. â€”

Hook's joint. â€” Parallel motions. â€” The sun and

planet wheels^ &c. â€” The eccentric.

The equilibrium of any number of forces in tJie

same plane applied to a rigid system.

Lemma, â€” I-f there be a parallelogram in a given

plane, and a point be anywhere taken in the same

plane ; the difference of the areas of the triangles

formed by drawing lines from the given point to the

extremities of two adjacent sides of the parallelo-

gram, shall be equal to the area of the triangle

formed by lines drawn from the given point to the

extremities of the diagonal.

Theory of areas.

Theory of moments.

Case in which the system revolves upon an axis

perpendicular to the plane of the forces.

Pressure upon the axis.

Equilibrium of any number of parallel forces

acting upon a rigid system in the same plane.

Equilibrium of any number of parallel forces

acting any where upon a rigid system.

The centre of gravity.

10

Examples, â€” the centre of gravity of a parallelo-

gram â€” of a triangle â€” of a pyramid â€” of a prism â€” of

a circle â€” of a semicircle â€” of a parabola â€” of a cycloid.

Conditions of the equilibrium of a heavy body

sustained on a horizontal plane â€” on an inclined

plane â€” and on a curved surface. â€” (1) When the

base of the body is a plane. â€” (2) When the base is

a curved surface.

The hanging tov/ers of Pisa and Bologna.

Comparative stability of structures of different

forms.

Theory of the carriage v^^heel. â€” The drawbridge.

Stability of loaded vehicles.

Guldinus's properties.

The centre of gravity of a body or system of

bodies, in equilibrium, is at its highest or lowest

possible point. This proposition proved generally.

â€” Exemplified in the case of the inclined plane. â€”

The crooked lever, &c.

The equilibrium of a beam supported upon a

roller and a vertical plane^ â€” of a rectangle upon

two rollers â€” of a beam upon two surfaces.

The equilibrium of a beam supported by a string

fastened at its extremities â€” supported by two

strings passing over pulleys, and carrying weights.

it

The position of the centre of gravity in animals.

The attitudes of animals dependent upon the

position of the centre of gravity.

TJie conditions of stable â€” unstable â€” aud mixed

equilibrium.

Examples. â€” A hemisphere and a parabolid upon a

plane surface. â€” The same upon a spherical surface.

The common balanceâ€” its sensibility â€” the rapi-

dity of its vibrations â€” its adjustments. These

require to be different for every different loading,

in order that the sensibility may be greatest of

which the balance is capable.

The equilibrium of a body, or system of bodies,

stable or unstable, according as the centre of

gravity is at its low^est or highest point.

Conditions of the equilibrium of any number of

forces acting, in any given number of directions,

upon a system of invariable form.

Case in which any number of forces act in dif-

ferent directions perpendicular to a rigid line.

Case in which a system acted upon by any

number of forces is moveable about a fixed axis.

Pressure upon the axis.

Conditions of the equilibrium of any number of

forces acting upon a system of variable form.

12

The jointed polygon of r^ocls.

The equilibrium of a frame-work of two or more

jointed polygons of rods connected together.

The conditions of the equilibrium of a jointed

frame-work loaded with weights, and placed in an

upright position, is the same as though the frame

were suspended and the same loading applied.

Stability of its equilibrium when suspended.

Instability in the opposite position.

Easy practical method of determining the proper

form of a roof, bridge, or other jointed frame under

a given loading ; and the pressure, on its different

parts and its abutments.

Equilibrium of the arch with polished voussoirs.

Instability of its equilibrium.

On tlie equilibrium of the funicidar polygon. â€”

Case in which the pressures are applied to rings,

moving freely on the thread. â€” The elastic polygon.

The equilibrium of the funicular curve. â€” The

common catenary. â€” The tension on the catenary

at the lowest point varies as the radius of curvature

at that point. â€” Easy method of ascertaining the

tension at any other point.

On the position of equilibrium of a string of a

given length, suspended over two given points.

13

On the relation between the length of the string

and the tension on its parts when suspended.

On the positions of equilibrium of a string whose

extremities hang freely over two pulleys, in the

same horizontal line.

Of all the curves, of given length, which can be

drawn so as to terminate in two given points, in

the same horizontal line, the catenary is that whose

centre of gravity is most distant from that line.

On the catenary loaded with weights.â€” A catenary

may be so loaded as to assume any required form.

Variation in the tension of the catenary, produced

by an irregularity in its loading, and consequent

variation in its form.

The catenary approximates at its vertex very

nearly to a parabola.

ON FRICTION.

The statical laws of friction,

Rennie's apparatus and experiments.

The friction of hard metals under pressures of

less than 32lbs. 8oz. the square inch, nearly one-

sixth of the pressure.

With higher pressures this ratio increases.

The friction of woods.

14

The friction of stones.

The diminution of friction by unguents, varies as

the insistant weights and the nature of the un-

guents; the lighter the weight, the finer and more

fluent should be the unguent, and vice versa.

On the modifications introduced hy friction in

the conditions of the equilibrium of tJie different

mechanical 'powers.

On the two states bordering upon motion in the

inclined plane.

The wheel and axle.

Methods of diminishing friction by means of fric-

tion wheels. â€” Friction of the carriage wheel. â€” The

screw. â€” The system of toothed wheels.

ON THE RIGIDITY OF CORDS.

Coulomb's experiments.

States bordering on motion in the different sys-

tems of pulleys. â€” The proper ratios of the wheels,

axles, and cords of the different pulleys of each

system.

ON THE STRENGTH OF MATERIALS.

Absolute resistance. â€” Rennie's experiments.â€” The

resistance of different masses of metal, wood, and

15

stone to the compression of their parts in given

directions. â€” The resistance of different masses of

metal and wood to the separation of their parts in

given directions. â€” Anomalous results.

On the strain w^hich produces permanent altera-

tion of structure.â€” There is reason to believe that

all bodies are perfectly elastic, as to any pressure

less than that which produces permanent alteration

of structure.

Galileo's hypothesis of the rigidity of fibres. â€”

Leibnitz's hypothesis of the extensibility of fibres. â€”

Theory which admits the compressibility as well as

the extensibility of fibres.

Relative resistance. â€” The neutral line â€” its pro-

perties.

On the strength of a horizontal bar fixed im-

moveably at one end, and carrying a weight at the

other.

On the strength of a bar fixed immoveably at

both ends, and carrying a weight between.

On the strength of a bar supported in the middle^

and carrying weights at the ends.

On the strength of a bar carrying weights

variously distributed over its surface.

On the strongest forms of beams.

16

On the construction of open beams.

On the deflexion of beams by their own weight,

when supported horizontally at their extremities â€”

when inclined to the horizon.

On the deflexion of columns sustaining weights.

On the proper forms of columns sustaining

weights.

The elasticity of flexure.

On the deflexion of elastic laminae.

The equilibrium of springs.

The elasticity of torsion.

On the proper forms , and tlie strength of solid

arches of wood and iron.

On the loading of solid arches.

On the strength of open arches.

THEORY OF ROOFS.

The Shed Roof

The angle of its elevation dependent on the height

and the strength of the walls or pillars on which it

abuts. â€” The strength of its timbers. â€” Ingenious

method of getting rid of the horizontal thrust, by

supporting the timbers beneath their centres of

gravity.

TJie commonTruss Roof â€” The theory of this roof.

17

â€” The horizontal thrust on its abutments. â€” The

different pressures upon its parts, and the consequent

variation in the strength of its timbers.

The deflexions in the tie beam and principal

rafters.

The variations in the lengths of the timbers.

The strength of the joints.

On the different forms of the trussed roof

Examples, â€” The roof of the Bazilica of St. Paul's

at Rome. â€” The roof of the Theatre Argentina at

Rome. â€” The roof of the Birmingham Theatre.

Methods of giving support to roofs â€” by means of

additional frame-work abutting in the wall beneath

the tie beam â€” by means of buttresses, &c.

On truncated roofs.

The roof of Drury Lane Theatre.

On the loading of roofs, hy the suspension of

ceilings, S^c,

The roof and ceiling of the Teatro Alia Scala.

On various methods of dispensing with the tie

beam at the foot of the rafters.

The collar beam â€” great objections to its use

unless supported by pillars, or otherwise, at its

extremities.

Roof of the Church St. Genevieve at Paris.

B

18

Roof of the Theatre Odeon.

The roof of a church in Wiltshire.

On g'othic roofs.

The roof of Westminster School.

The roof of the Middle Temple Hall.

The roof of Westminster Hall.

On the polygonal roof.

Roofs formed with four principal rafters.

Conditions of he equilibrium of four such raf-

ters.

The roof of the Theatre at Bordeaux.

Roofs of sheds in the Arsenal at Cherbourg.

Roofs of sheds in the Dockyard at Plymouth.

The roof of the shed for containing Mahogany in

the West India Docks.

Method of supporting roofs hymeans of polygonal

frames or arches of short rafters.

Great advantages of this method of support.

The proper form and strength of the polygon.

The roof of the Riding House at Moscow^

On the arch of curved timber.

On the use of iron in the framing of roofs.

The roof of the Brunswick Theatre

Roof at Mr. Maudeslay's manufactory.

On trussed floors.

19

Example.â€” The trussed floor in the Teatro Alla-

scala at Milan.

ON THE THEORY OF WOODEN BRIDGES.

The wooden bridge in which the timbers are

straight, and rest immediately upon the piers.

The bridge of Caesar over the Rhine.

The bridge of Cayuga in America.

Method of constructing a wooden bridge over a

rapid torrent.

The bridge across the rapids of Niagara.

On the straight wooden bridge where the timbers

of the roadway are trussed from above, and there

is no horizontal pressure upon the abutments.

The bridge of Palladio over the Cismone.

The great bridge formerly at SchafFhausen.

Wooden bridge near Baltimore, N. America.

The wooden bridge, in which the roadway is

principally supported yrow? beneath by timbers, which

rest obliquely upon the abutments.

The bridge over the Kendal near Berne.

On wooden bridges supported by polygonal

ribs.

The wooden bridge at Lyons.

On wooden bridges with curved ribs.

B 2

20

The bridge of Trajan over the Danube.

The bridge of Freysingen in Bavaria.

The bridge of Bamburgh on the Regnitz in

Germany.

The great wooden arch at Scuykill, in N. America.

The proper variation in the strength of the parts

of the curve of a wooden arch.

ON THE THEORY OF IRON ARCHES.

Strength requisite in the different portions of the

arch. â€” Colebrook-Dale bridge. â€” Buildwas bridge. â€”

Sunderland bridge. â€” Bonar bridge. â€” The bridge

of the Louvre. â€” Vauxhall bridge. â€” Southwark

bridge. â€” Telford's proposed bridge over the

Thames,

ON SUSPENSION BRIDGES.

Application of the theory of the loaded cate-

nary. â€” The method of constructing the chain. â€”

Easy practical method of determining the tension. â€”

It is less at the vertex as the curvature is greater.

On the method of suspending the roadway. â€” On

the piers of the bridge, and the attachment of the

chain. â€” The wire bridge over the Tees near

Durham. â€” The wire bridge over the Tweed near

21

Peebles. â€” The Kelso suspension bridge. â€” The sus-

pension pier at Leith. â€” Brunell's suspension bridge

erected in the Island of Bourbon. â€” The suspension

bridges over the Cataracts of Sckuylkill, and at

Merimac, in North America. â€” The Hammersmith

suspension bridge. â€” The Menai suspension bridge.

Any jointed polygon, or arch, placed in an upright

position so as to be sustained by the pressure of its

parts upon one another and upon its abutments, is

in a position of unstable equilibrium, the centre of

gravity being at its highest possible point. To the

stability of such a polygon, or arch, it is therefore

necessary that its joints should be rendered rigid, in

the directions in v^^hich their position is liable to

disturbance, â€” by additional framing or otherwise.

In the suspended polygon, or curve, the equili-

brium is stable, the centre of gravity being at its

lowest possible point. There is therefore no neces-

sity for rendering the joints rigid ; and the material,

requisite in the other case for producing this rigi-

dity, may be here dispensed with.

The advantage of the upright over the suspended

arch of the same materials, lies in this, that in

the former case the arch is sustained by the resis-

tance of its parts to compression, and in the latter

22

by their resistance to separation, and that mate-

rials are torn asunder more readily than they are

crushed.

ON THE THEORY OF THE STONE ARCH.

True theory of the arch, allowing for the friction

of the voussoirs.

General conditions of the stability of the arch.

The two states bordering upon motion.

Method of describing the line of the least loading

necessary to the equilibrium of an arch whose key-

stone is given.

Line of the greatest loading which such an arch

will bear.

On the comparative strength of different por-

tions of the arch. Generally the strength of an

arch is greater as its curvature is less.

The curvature being given, the stability of an

arch properly constructed increases with the loading.

On the line of pressure.

When the voussoirs are exceedingly narrow, and

the loading considerable, the curve of equilibrium is

the catenary.

Different steps in the fall of an arch.

Example, â€” Pont y Prydd.

23

The circular arch.

Easy method of describing the lines of greatest

and least practicable loading.

The great strength of the circular arch, and

variety of loading under which conditions of its

equilibrium obtain.

The segment of a semi-circular arch.

Its advantages over the whole semicircular arch.

No semicircular arch can be safely constructed

with equal voussoirs â€” A segment may.

A straight wall of any height may be built over

a segment of a circular arch.

Exarnjdes of circular arches.

The bridge of Rimini.

The aqueduct bridge at Nismes.

The bridge of Avignon.

The bridge of Briande.

The bridge of Ulm, &c. &c.

The elliptical arch.

Method of describing the lines of loading in the

two states bordering upon motion, and for given

dimensions of the key-stone.

Weights which an elliptical arch of given dimen-

sions is capable of sustaining on its crown.

24

Comparative weakness of the semi - elliptical

arch.

Cases of semi-elliptical arches, in which, under

the irregular pressures to which they are subjected,

their stability must be dependent upon other causes

than the friction of their parts.

Examples of elliptical arches.

Bridge of the Rialto at Venice.

Bridge over the Arno at Florence.

Bridge of Neuilly.

Waterloo bridge.

London bridge.

On segments of elliptical arches.

Advantages in the use of segments raised upon

high vertical piers, where a clear water way is

required.

A segment of an elliptical arch may be built with

equal voussoirs.

The loading on certain points about the haunches

of a semi-elliptical arch may be any whatever.

Examples of flat arches.

Bridge over the Oise.

Pont de la Concorde.

25

The arch between the western towers of Lincoln

Cathedral.

On the pointed gothic arch.

Equilibrium of the pointed arch.

Method of determining the states bordering upon

motion.

The loading of the key-stone.

Comparative strength of the varieties of the

pointed arch.

Conditions of the equilibrium of an arch, taking

into account the tenacity of the cement.

On the piers of arches.

On the centering of arches.

On the equilihrium of the dome.

The weight on the haunches may be increased

without limit, and their convexity diminished, but

not the contrary.

A dome may be built w^ithout centering. â€” Ex-

ample, â€” The dome of the cathedral of Florence.

On the equilibrium of a dome loaded on the

crown â€” The domes of the cathedral at Florence, St.

Peter's of Rome, and St. Paul's in London. â€” On

^he equilibrium of a dome in which the crown is

wanting.

26

On the theory of the groin.

On the conoidal groin,

ON THE PRINCIPLE OF VIRTUAL VELOCITIES.

Lagrange's proof.

The principle of virtual velocities shewn to obtain

in the following cases of equilibrium.

The equilibrium of any number of forces acting

upon a point. â€” Of weights on the straight lever. â€” Of

forces acting obliquely on the crooked lever. â€” In

the case of the single pulley where the strings are

inclined. â€” Of a system of compound levers. â€” Of the

inclined plane. â€” Of equilibrium on a curve â€” on two

curves â€” of the screw â€” of the wheel and axle â€” of

toothed wheels, &c. &c. â€” In the case of motion about

the centre of gravity of any system of bodies.

The centre of gravity of any system of weights

is at its highest or lowest points, when those weights

are in equilibriumâ€” proof deduced from the prin-

ciple of virtual velocities.

On the quantity of motion.

The quantity of motion, a measure of the moving

force.

Demonstration of the principle of virtual velocities

founded on this consideration.

27

ON THE USEFUL EFFECT OF MACHINES.

1 2