Attraction of Uniform Spherical Shell on Internal Point
-Digression on Cones and Elements of a Surface,

Attraction of Uniform Spherical Shell on an External

Point On an element of its Surface,

SECTIONS

462-470

.

471, 472

473-476

Density of Shell varies inversely as the cube of the

distance from a given point,

Direct Analytical Calculation of Attractions, Variation of
Force in crossing a Surface, Alteration of Latitude
by Hill, Cavity, or Crevasse, etc., .
Potential-Potential Energy-Exhaustion of Potential
Energy-Force in terms of the Potential-Equi-
potential Surface-Intensity of Force at different
points of such a Surface,

Line of Force. Variation of intensity along it,

Potential of a Point, of a Mass. Force within a Homo-
geneous Sphere-Laplace's Equation-Poisson's ex-
tension of it-Examples,

Integral of Normal Attraction over a Closed Surface-

Consequent Properties of the Potential,

Theorem of Gauss,

Green's Problem-Reducible cases-Electric Images-Ex-

amples,

Attraction of an Ellipsoid-Ellipsoidal Shells-Chasles'
Theorem-Attraction of Spheroid of Revolution-
Maclaurin's Theorem - Ivory's Theorem-Caven-
dish's Theorem,

477-481

482-488

489, 490

491

492-496

497, 498

499-518

519-525

Centre of Gravity. Centrobaric Bodies proved possible

by Green-Properties of Centrobaric Bodies,

526-535

Origin of Spherical Harmonic Analysis-Applications—

Potential of Distant Body-Attraction of Particle on

Distant Body-Approximation in the common idea

of Centre of Gravity-Examples,

Exhaustion of Potential Energy-Methods of Gauss and

Green,

536-546

547-550

CHAPTER VII.-STATICS OF SOLIDS AND FLUIDS.

Rigid Body-Conditions of Equilibrium for Translation

and Rotation-Particular case of Forces in one Plane

-Resultant of any set of Forces,

Couples-Composition and Resolution of Couples-Force

resolved into Force and Couple-Application to Rigid

Body-Examples-Central Axis-Reduction of any

number of Forces applied to a Rigid Body to two-

Symmetrical case,

Special case-Parallel Forces-Centres of Inertia and
Gravity-Couple on Magnetized Steel,

Conditions of Equilibrium of three Forces acting on a
Rigid Body-Physical Axiom,

Equilibrium under the action of gravity and constraints
alone-Rocking Stones-Equilibrium about an Axis,
or on a Surface,

Pappus' Theorem,

Examples-Balance, Rod under constraint with or with-
out friction, Block on Rough Plane, Gate supported
by rings passing round a post,

Flexible and Inextensible Cord-Catenaries - Kinetic
Analogies-Inverse Problem,

Flexible String on Surface-Rope coiled about Rough

Cylinder-Cord on Rough Surface,

Flexure and Torsion of Elastic Wire,

Composition and Resolution of Curvature in a Curve,

General Laws of Torsion and Flexure of a Wire, .

Warping of Normal Section, and Rotations, due to

Flexure and Torsion,

Potential Energy of Bent and Twisted Wire-Components

of Restituent Couple-Principal Torsion-flexure Axes

and Rigidities-Three Principal Spirals,

SECTIONS

553-560

561-563

564

565-568

569

570-573

573-583

584-587

588, 589

590

591, 592

Particular cases-Elastic Central Line a Normal Axis of

Torsion-Flexibility equal in all directions-Spiral

Elastic Curve transmitting Force and Couple-Kirchhoff's

Kinetic comparison-Common Pendulum and Plane
Elastic Curve-Equation of Plane Elastic Curve—
Bow slightly bent,

Wire of any shape disturbed by Forces and Couples
applied through its length-Longitudinal Tension-
Equations of Torsion-flexure-Terminal conditions, .

Straight Beam infinitely little bent-Case of Independent

Flexure in two Planes-Plank bent by its own

weight,

593

594-598

599-608

609-613

614, 615

616-620

Rotation of a Wire round its Elastic Central Line-Elastic

Universal Flexure Joint of § 189-Practical in-

equalities-Straight Wire made into a Hoop-Wire

circular when unstrained-Conical bending of de-

velopable Surface,

-

Flexure of a Plane Elastic Plate-Definitions-Geometri-

cal preliminaries-Limitations-Stretching of a Plane

by Synclastic or Anticlastic Flexure-Stretching of

curved Surface by Flexure not fulfilling Gauss's con-

dition,

Flexure of Plane Elastic Plate Limitations of Forces

and Flexures-General statement of results in ad-

vance-Laws of Flexure assumed in advance-Stress-

couple across Normal Section-Axes of Bending

Stress - Definitions of Synclastic and Anticlastic

Stress-Resolution and Composition of Stresses and

Curvatures, .

Work done in Bending Plate-Partial Differential Equa-

tions of Work done-Potential Energy of Bent Plate

-Case of equal flexibility in all directions, .

Plate bent by any forces-Equations of Equilibrium—

Equations between Stress and Curvature-Partial

Differential Equation of the Bent Surface-Boundary

conditions-Examples-Circular Strain-Symmetri-

cal Flexure of Flat Ring,

Reduction of general problem to that of no load over the

area-Rectangular Plate held and loaded by diagonal

pairs of corners-Transition to Finite Flexures,

Transmission of Force through Elastic Solid-Specifica-

tion of a Stress by six independent elements-

Tangential Tractions-Force across Surface in terms

of Stress-Stress-quadratic-Normal Planes and

Axes of a Stress-Composition of Stresses-Laws of

Stress and Strain compared-Elements of Strains

and Stresses,

Work done by Stress within a Varying Solid-Work

done on Surface-Differential Equation of Work

done by Stress,

Definition of Perfectly Elastic Body-Potential Energy

of Strained Solid-Green's twenty-one Coefficients—

Stress-components in terms of Strain-Strain-com-

ponents in terms of Stress-Average Stress through

any Strain,.

SECTIONS

621-626

627-631

632-639

640-642

643-653

654-657

658-669

670, 671

672-674

Definition of Homogeneousness-Isotropic and Aeolo-

tropic Bodies-Measures of Resistance to Distortion

and to Compression-Compressibility-Rigidity,

Relation between common reckonings of Shear and Dis-

torting Stress. Strain by a Single Longitudinal

Stress-Young's modulus. Ratio of lateral contrac-

tion to longitudinal expansion-Navier's Ideal Per-

fect Solid untenable,

Definition of Young's modulus-Different modes of reckon-

ing it. Specific modulus. In terms of absolute unit.

Different metrical denominations,

Stress required to maintain a given Strain, and vice versa.

Equation of energy for isotropic substance,

Fundamental problems of the mathematical theory. Equa-

tions of internal equilibrium. Proof that they agree

with those of equilibrium of a rigid body. Simplifi-

cation for an isotropic body,

Torsion applications by De St. Venant.

Preliminary

lemma. Circular cylinder. Traction on sides re-

quired for simple twist in any prism or rod, except

circular cylinder. Hydrokinetic analogue to St.

Venant's problem. St. Venant's solvable cases.

Elliptic cylinder. Equilateral triangular prism.

Curvilinear square, and star prisms. Reduction to

Green's problem. Extension to certain curvilinear

SECTIONS

675-680

681-685

686-691

692-695

696-698

rectangles. Plane isothermals. Fan-shaped sections, 699–707

Diagrams of contour lines in sections of warped prisms—

from St. Venant. Diagrams of twisted elliptic, square,

and flat rectangular bars,

False extension of Coulomb's law-Corrected values.

Places of greatest distortion in twisted Prisms,

Passage to broad

Problem of flexure of a bar of isotropic material. Geo-

metrical interpretation. Anticlastic and conical cur-

vatures in bent rectangular prism.

thin band-thence to flexure of a plate,

General problem of infinite solid: solved for isotropic sub-
stances,

Application of spherical harmonic analysis to distortion
of spheres and shells,

Plane strain-Solution by plane harmonics of plane strain
of cylinders, .

708

709, 710

711-729

.

730-734

735-737

738, 739

Small bodies stronger than larger ones in proportion to

their weights. Examples,

Transition to Hydrodynamics. Imperfect elasticity in

solids-Viscosity of solids: of fluids. Forbes' "Vis-

cous Theory" of Glaciers. Plasticity of solids,

Fluid pressure: proved

Application to statics of

Application of the

Definition of a perfect fluid.

equal in all directions.

solids. Centre of pressure.

principle of energy,

Effect of external forces-Surfaces of equal pressure: are

surfaces of equal density and potential when the

forces are conservative,

Particular case-gravity the only external force, .

Rate of increase of pressure. Calm atmosphere of uniform

temperature. Homogeneous atmosphere,

Condition of equilibrium when fluid fills a closed vessel.

Non-conservative forces. Imaginary example. Actual

case, .

SECTIONS

740

741

742-747

748-750

751

752, 753

Relation between density and potential,

Resultant pressure on plane area,

754-759

760

761

762

Loss of apparent weight by immersion in a fluid,

Equilibrium of a floating body- Lemma - Stability—

Work done in vertical displacement-Work done in

displacement about an axis in the plane of flotation.

Work required in general displacement. Hence the

conditions of stability. Metacentre-Conditions of

its existence,

Synthetical proof that an oblate ellipsoid of revolution is

a figure of equilibrium of a rotating mass of homo-

geneous liquid. For different liquids the squares of

the angular velocities giving the same ellipticity are

proportional to the densities. Table of eccentricities

and corresponding angular velocities and moments of

momentum for a liquid of the earth's mean density.

One definite spheroid shown to satisfy given data of

mass and moment of momentum,

Equilibrium of rotating ellipsoid of three unequal axes,
Digression on Spherical Harmonics. Harmonic Spheroid.

Harmonic Nodal Cone. Polar Harmonics. Zonal,
Sectorial, and Tesseral Harmonics. Murphy's ex-
pansions. Biaxal Harmonic. Treatment of problems

763-768

769-777

778