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equations to determine its equilibrium, so as to give 3s equations between
e unknown quantities, if s be the number of points and e the number of
connexions. There are, however, six equations of equilibrium of the system
which must be fulfilled necessarily by the forces, on account of the equality
of action and reaction in each piece. Hence if

e = 3s-6,

the effect of any external force will be definite in producing tensions or pressures
in the different pieces; but if e>35 — 6, these forces will be indeterminate.
This indeterminateness is got rid of by the introduction of a system of e equa-
tions of elasticity connecting the force in each piece with the change in its
length. In order, however, to know the changes of length, we require to assume
3s displacements of the s points ; 6 of these displacements, however, are equiva-
lent to the motion of a rigid body so that we have 3s — 6 displacements of
points, e extensions and e forces to determine from 3s — 6 equations of forces, e

* [Owing to an oversight this paper is out of its proper place ; it should have been immediately
before the memoir on "The Electro-magnetic Field." (No. XXV.)]



ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 599

equations of extensions, and e equations of elasticity ; so that the solution is
always determinate.

The following method enables us to avoid unnecessary complexity by treating
separately all pieces which are additional to those required for making the frame
stiff, and by proving the identity in form between the equations of forces and
those of extensions by means of the principle of work.



On the Stiffness of Frames.

Geometrical dejinition of a Fram^i. A frame is a system of lines connecting
a number of points.

A stiff frame is one in which the distance between any two points cannot
be altered without altering the length of one or more of the connecting lines
of the frame.

A frame of s points in space requires in general 35 — 6 connecting lines to
render it stiff. In those cases in which stiffness can be produced with a smaller
number of lines, certain conditions must be fulfilled, rendering the case one of
a maximum or minimum value of one or more of its lines. The stiffness of
such frames is of an inferior order,, as a small disturbing force may produce
a displacement infinite in comparison with itself.

A frame of s points in a plane requires in general 26- — 3 connecting lines to
render it stiff.

A frame of s points in a line requires s — 1 connecting lines.

A frame may be either simply stiff, or it may be self-strained by the intro-
duction of additional connecting lines having tensions or pressures along them.

In a frame which is simply stiff, the forces in each connecting line arising
from the application of a force of pressure or tension between any two points
of the frame may be calculated either by equations of forces, or by drawing
diagrams of forces according to known methods.

In general, the lines of connexion in one part of the frame may be affected
by the action of this force, while those in other parts of the frame may not
be so affected.

Elasticity and Extensibility of a connecting piece.

Let e be the extension produced in a piece by tension-unity acting in it,
i-hen e may be called its extensibility. Its elasticity, that is, the force required



GOO ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES.

to produce extension-unity, will be - . We shall suppose that the efiect of

pressure in producing compression of the piece is equal to that of tension in
producing extension, and we shall use e indifferently for extensibility and com-
pressibility.

Wcyrk done against Elasticity.
Since the extension is proportional to the force, the whole work done will
be the product of the extension and the mean value of the force ; or if x is
the extension and F the force,

x = eF,

work = iFx = ^eF' = ^-af.

When the piece is inextensible, or e = 0, then all the work applied at one end
is transmitted to the other, and the frame may be regarded as a machine whose
efficiency is perfect. Hence the following

Theorem. If p be the tension of the piece A due to a tension-unity
between the points B and C, then an extension-unity taking place in A will
bring B and C nearer by a distance p.

For let X be the tension and x the extension of ^4, F the tension and
// the extension of the line BC; then supposing all the other pieces inextensible,
no work will be done except in stretching A, or

iXx + iYy = 0.
But X=pY, therefore y= —px, which was to be proved.

Problem I. A tension F is applied between the points B and C of a
frame which is simply stiff; to find the extension of the line joining D ai d F,
all the pieces except A being inextensible, the extensibility of A being e.

Determine the tension in each piece due to unit tension between B and C,
and let p be the tension in A due to this cause.

Determine also the tension in each piece due to unit tension between D
and F, and let y be the tension in the piece A due to this cause.

Then the actual tension of ^ is Fp, and its extension is eFp, and the
extension of the line DE due to this cause is -Fepq by the last theorem.



ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 601

Cor. If the other pieces of the frame are extensible, the complete value
of the extension in DE due to a tension F in BC m

-F%{epq),
where 'Z{epq) means the sura of the products of epq, which are to be found
for each piece in the same way as they were found for A.

The extension of the line BC due to a tension F in BC itself will be

-Ft{ep%
t{ep') may therefore be called the resultant extensibility along BC.

Problem IL A tension F is applied between B and C; to find the
extension between D and E when the frame is not simply stiff, but has
additional pieces R, S, T, &c. whose elasticities are known.

Let p and q, as before, be the tensions in the piece A due to unit
tensions in BC and DE, and let r, s, t, &c. be the tensions in A due to
unit tension in R, S, T, &c. ; also let R, S, T be the tensions of R, S, T,
and p, (T, T their extensibilities. Then the tension A

= Fp + Rr + Ss+Tt + &c.;
the extension of A

= e{Fp + Rr + Ss + Tt + &c.);
the extension of R

= -Ft (epr) - RXer" - Sters - TXert + &c. = Rp ;
extension of S

= - Ft{eps) - Rt{ers) - Stes' - Tt{est) = Sa ;
extension of T

= - FX(ept) - Rt{ert) - SX{est) - Tt(ef) = TV ;
also extension of DE

= - FX{epq) - Rt(eqr) - S%(eqs) - Tt(eqt) = x,
the extension required. Here we have as many equations to determine R, S, T,
&c. as there are of these unknown quantities, and by the last equation we
determine x the extension of DE from F the tension in BC.
Thus, if there is only one additional connexion R, we find



R=-F



t(epr)



X(er^) + p'
and



t(^^)-'-%^S?}



VOL. I. 76



602 ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES.

If there are two additional connexions R and S, with elasticities p and <r,
x=-F

r % (epr) t (ers) t (eqs) + % (eps) t (eqr) % (ers) + 1 {epq) %e (r= + p) le {s' + a)^
\-t (epr) t {eqr) %e {5^ + a-)-t {eps) S {eqs) te{r' + p)-t {epq) (2 {ers)Y J *
The expressions for the extensibility, when there are many additional pieces,
are of course very complicated.

It will be observed, however, that p and q always enter into the equations
in the same way, so that we may estabhsh the following general

Theorem. The extension in BC, due to unity of tension along DE, is
always equal to the tension in DE due to unity of tension in BC. Hence we
have the following method of determining the displacement produced at any
joint of a frame due to forces applied at other joints.

1st. Select as many pieces of the frame as are sufficient to render all its
points stiff. Call the remaining pieces R> S, T, &c.

2nd. Find the tension on each piece due to unit of tension in the
direction of the force proposed to be applied. Call this the value of _p for each
piece.

3rd. Find the tension on each piece due to unit of tension in the
direction of the displacement to be determined. Call this the value of q for
each piece.

4th. Find the tension on each piece due to unit of tension along R, S, T,
&c., the additional pieces of the frame. Call these the values of r, s, t, &c.
for each piece.

5th. Find the extensibility of each piece and call it e, those of the
additional pieces being p, <t, t, &c.

6th. R, S, T, &c. are to be determined from the equations
Rp + Rt (er") + S{ers) + Tt {ert) + Ft {epr) = 0,
So- + Rt{ers) + S{es') + T% {est) + Ft {eps) = 0,
Tt + Rt{ert) + S{est) + Tt{ef) + Ft{ept) = 0,
as many equations as there are quantities to be found.

7th. X, the extension required, is then found from the equation
x= -Ft{epq)-Rt{erq)-St{eqs)-Tt{eqt).



ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 603

In structures acted on by weights in which we wish to determine the
deflection at any point, we may regard the points of support as the extremities
of pieces connecting the structure with the centre of the earth ; and if the
supports are capable of resisting a horizontal thrust, we must suppose them
connected by a piece of equivalent elasticity. The deflection is then the
shortening of a piece extending from the given point to the centre of the
earth.

Example. Thus in a triangular or Warren girder of length l, depth d,
with a load W placed at a distance a from one end, ; to find the deflection
at a point distant h from the same end, due to the yielding of a piece of
the boom whose extensibility is e, distant x from the same end.

The pressure of the support at 0=W -j~ ; and if x is less than a, the



W

force at x will be -jr x{l-a), or



If X is greater than a,
Similarly, if a: is less than 6,
but if ic is greater than b,



x(l-a)
P~ dl



a{l-x)
^~ dl



^" dl



^" dl



The deflection due to x is therefore Wepq, where the proper values of p
and q must be taken according to the relative position of «, h, and x.

If a, b, I, X represent the number of the respective pieces, reckoning from
the beginning and calling the first joint 0, the second joint and the piece
opposite 1, &c., and if L be the length of each piece, and the extensibility of
each piece =e, then the deflection of b due to W at a will be, by summation
of series,

= \ WeD . ^^^^ {2b{l -a)- (b<- af + 1}.

76—2



604 ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES.

This is the deflection due to the yielding of all the horizontal pieces.
The greater the number of pieces, the less is the importance of the last term.

Let the inclination of the pieces of the web be a, then the force on a

. „^ I — a
piece between and a v& W j-^'^, or

/ l — <^ 1-
r/ = :r—. — when x<a,
^ t sm a

and

p' = i— — when x>a.
^ f sin a

Also

when x<o,



I sin a

b
I sin a



when a; > 6.



If e be the extensibility of a piece of the web, we have to sum Wte'pq
to get the deflection due to the yielding of the web,



INDEX TO VOL. I.



Absorption, Electric, 573
Ampere, 193

Beam : bent into a circular ring, 55 — 57, 65
Bending, Lines of, 87, 97, 105, 106, 107, 108
Brewster, Sir David, 43, 63, 68, 263, 413

Cauchy, 32, 40, 71

Challis, 453, 503, 505

Clapeyron, 30, 32, 70, 72

Clausius, 377, 386, 387, 405

Coil, Coefficient of induction of, 592—597

Colour-blindness, 119, 137, 441

Colour box, 420 ; Method of observation, 426

Coloured beams of light, Mixtures of, 143 — 147

Coloured discs, 128, 264, 265

Coloured powders, 142

Colour equations, 128, 129, 138—141, 148, 268—270,
427, 428, 442

Colour: Experiments on, 126, 263; Law of Per-
ception of, 130; Theory of the Perception of,
135

Colours : History of, 411 ; Mathematical theory of
Newton's diagram of, 416 ; Method of repre-
senting them by straight lines, 418 ; Theory of
Compound, 149, 243, 410; Three primary.
Theory of, 445

Colour sensations, Relations of to the pure rays of
the spectrum, 150

Colour top, 127, 147

Colour triangle, 121, 131— 135, 268, 416

Compression, Equations of, 36

Condenser, Capacity of, 572 ; Theory of, 572 — 576

Contact, Conic of, 93, 94, 102

Cotes, 271



Coulomb, 33, 71

Current Electricity, Conduction of, 180

Currents: Action of closed, 183; Intrinsic energy
of, 541 ; Mutual action between, 537; Produced
by induction, 185; Quantity and Intensity as
properties of, 189

Curvature, entire and specific, 89, 92

Cylinder : acted on by centrifugal force, 60 ; hollow,
exposed to pressure, 45 — 50 ; hollow, dilated
by heat, 62; of parallel wires twisted, 59;
twisted, 42, 43, 44, 66

D'Alembert, 248

Diagrams, Conditions of indeterminateness in draw-
ing reciprocal diagrams, 516; of Force, 514
Dielectrics, Theory of, 177
Dynamometer, Weber's, 546

Earth's motion, 259

Elastic bodies. Collisions of, 405

Elasticity : Ajcioms, 31 ; Equations of, 38 ; Co-
efficients of, 41

Elastic solids : Equilibrium of, 30 ; Pressures in the
interior of, determined by the action on polar-
ized light, 68

Electric absorption, 573

Electrical images, 209

Electromagnet, Effect of the core, 222

Electromagnetic disturbances. Propagation of, 578,
583

Electromagnetic Field : Dynamical theory of, 526 ;
General equations of, 534, 551—552, 554—564;
Mechanical actions in, 565 — 570

Electromagnetic Induction, 536 ; Calculation of
coefficients, 589



606



INDEX TO VOL. I.



Electromagnetic momentum, 538 ; Dynamical illus-
tration of, 537
Electromagnetism : Ampere's laws, 193
Electrotonic state, 188, 205, 538
Equipotential surfaces, Magnetic, 553
Euler, 29, 32, 248, 271

Faraday, 155, 188, 205, 241, 504, 529, 531, 535,

542, 573, 585
Pelici, 538
Figures : General relation between the numbers of

points lines and polygons in, 515; Reciprocal,

514
Fizeau, 500, 580
Fluid : Application to Lines of Force, 175; Theory

of motion, 160; through a resisting medium, 163
Foramen Centrale, Sensibility to light, 242
Forbes, Professor, 1, 124, 142, 145, 146,243
Force : Diagrams of, 514 ; Lines of, 155, 158, 241 ;

Magnetic Lines of, 551 ; Physical lines of, 451
Forces : Absolute values of, concerned in propagation

of light, 588; Electromotive, 181
Foucault, 229, 248, 580
Fourier, 361
Frames, Equilibrium and stiffness of, 598

Gases: Conductivity of, 403—405; Diffusion of,
392, 403; Dynamical Theory of, 377; Fric-
tion, 390 ; Ratio of specific heats of, 409

Gauss, 81, 88—90, 271

Graham, 403

Grassman, 125, 414, 419

Gravitation, arising from the action of surrounding
medium, 571

Green, 196

Gregory, Dr, 126

Gutta percha, optical properties of, 43

Heavy body, descent of, in a resisting medium, 115
Helmholtz, 125, 141, 144, 145, 146, 152, 204, 243,

271, 414, 415, 488, 533
Herapath, 377
Herschel, Sir J. F. W., 142

Image, perfect defined, 273
Images, Electrical, Theory of, 209



Induction: by motion of conductor, 540; Coefficients
of, for two circuits, 539 ; Determination of
coefficients of, 547 ; Electromagnetic, 536 ;
Magnecrystallic, Theory of, 180; of one cur-
rent by another, 540 ; Paramagnetic and dia-
magnetic. Theory of, 178

Instrument, perfect optical, 274

Isinglass, optical properties of, 43, 67

Jenkin, F, 532, 573
Joule, 377

Kohlrausch, 492, 499, 535, 569, 579

Lame, 30, 32, 70, 72

Laplace, 292, 293, 294, 369

Leslie, Sir John, 16

Light : distinction between optical and chromatic
properties of, 411 ; Electromagnetic theory of,
502, 577 ; propagation of, forces called into
play, 587 ; propagation of, in a crystallized
medium, 583

Liquids, Compressibility of, 50

Listing, 271

Magnetic field of variable intensity, 214

Magnetic lines of force, 552

Magnetism, Quantity and Intensity as indicated by

lines of force, 192
Magnets, Permanent, Theory of, 178
Momentum, Electromagnetic, 538

Nasmyth, J., 57
Navier, 30, 31, 32, 72
Neumann, 208, 512, 527

Newton, 3, 124, 135, 142, 143, 144, 145, 146, 149,
151, 249, 267, 410, 411, 412

CErsted, 30, 33, 50

Optical Instruments, General laws of, 27 1 ; Mathe-
matical treatment of, 281 — 285
Oval Curves, 1

Plate, bent by pressures, 57
Plateau, 243, 295
Platometer, New form of, 230



INDEX TO VOL.



607



Plucker, 585
Poinsot, 248, 250

Poinsot's Theory of Rotation, Instrument to illus-
trate, 246
Poisson, 30, 32, 72

Polyhedron, inscribed in a surface, 94, 98, 99
Pressures, Equations of, 37
Problems, Solutions of, 74

Ray. Reduced path of, 280

Reciprocal Figures, 514; Application to Statics,
522; Possibility of drawing, 516; Relation
between the number of points lines and polygons
in, 515

Refraction, index of, how related to specific in-
ductive capacity, 583

Regnault, 33, 71

Resistance, Electric, how related to transparency,
586

Resisting medium, descent of a heavy body in, 115

Rigid body, Stability of the steady motion about a
fixed centre of force, 374 ; motion of about a
sphere, 296

Ring : motion of, when rigid, about a sphere, 296 —
310 ; motion of, when the parts are not rigidly
connected, 310; of equal satellites, 360

Rings, Effect of long continued disturbances on a
system of, 352; mutual perturbations of two,
345 ; Fluid, Loss of energy due to friction, 354

Rolling Curves, 4 ; Examples of, 22—29

Rolling of Curves on themselves, 19

Rotation, Theory of, 249

Saturn's Rings, Stability of, 286, 288—376

Sources and sinks defined, 163

Spectrum, Relation of the Colours of, to Compound

Colours, 410
Sphere, hollow, dilated by heat, 64 ; exposed to

pressures, 51 — 55
Sphere : magnecrystallic, 217 ; magnetic in uniform

field of magnetic force, 212



Spherical shell: Electromagnetic, 220, 222; Per-
manent magnetism in, 220 ; Revolving in
magnetic field, 226

Spherical electromagnetic coil machine, 224

Spheres perfectly elastic : Motions and Collisions of,
378 ; Boyle's law, 389; Mean distance between
collisions, 386 ; Mean Velocity, 381 ; Mean Velo-
city-square, 381 ; Two systems in same vessel,
383

Spheres: Two, between poles of a magnet, 216;
Two, in uniform magnetic field, 215

Stokes, 32, 33, 71, 72, 143, 209, 391, 410

Struve, 292

Surfaces: Applicability of, 95; Conjugate systems
of curves on, 95, 96 ; Transformation of-, by
bending, 80

Telescopes, perfect, 275 — 279

Thomson, Sir W., 157, 196, 199, 209, 212, 374,
453, 503, 505, 528, 529, 533, 588

Top, Dynamical, 248 ; Instantaneous axis, 255 ;
Invariable axis, 252—255 ; Method of observ-
ing the motion, 257

Tractory of a curve, 13

Tractory of circle, 15, 17

Tubes, unit, 161

Unit, Electrostatic and Electromagnetic, 569

Verdet, 504, 506, 507, 513, 529

Vortices, molecular : Applied to Electric Currents,

467 ; Applied to Magnetic phenomena, 451 ;

Applied to Statical Electricity, 489

Wave length, Method of determining, 423

Weber, 208, 492, 499, 507, 527, 535, 545, 569,

579
Wheatstone, 434
Wilson, Dr George, 137, 415

Young, 32, 60, 124, 136, 137, 412, 419, 447



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