CHAPTER II.-DYNAMICAL LAWS AND PRINCIPLES.

Ideas of Matter and Force introduced-Matter-Force-Mass

-Density-Measurement of Mass-Momentum-Change

of Momentum-Rate of change of Momentum-Kinetic

Energy-Particle and Point-Inertia.

Force-Specification of a Force-Place of Application-Direc-
tion-Magnitude-Accelerative Effect-Measure of Force
Standards of Weight are Masses, and not primarily intended
for Measurement of Force-Clairaut's Formula for the
Amount of Gravity-Gauss's absolute Unit of Force-
Maxwell's two suggestions for Absolute Unit of Time-
Third suggestion for Absolute Unit of Time-British
Absolute Unit-Comparison with Gravity

SECTIONS

205-216

217-220

221-226

Resolution of Forces-Effective Component of a Force

227, 228

Geometrical Theorem preliminary to Definition of Centre of

Inertia Centre of Inertia.

229, 230

Moment-Moment of a Force about a Point-Moment of a

233

234

Digression on Projection of Areas

Couple-its Moment, Arm, and Axis

Moment of Velocity-Moment of Momentum-Moment of a

Rectilineal Displacement-For two Forces, Motions, Ve-

locities, or Momentums, in one Plane, the Sum of their

Moments proved equal to the Moment of their Resultant

round any point in that Plane-Any number of Moments

in one Plane compounded by addition-Moment round

an Axis-Moment of a whole Motion round an Axis-Re-

sultant Axis

Virtual Velocity-Virtual Moment

Work-Practical Unit-Scientific Unit-Work of a Force-

Work of a Couple-Transformation of Work-Potential

Energy

-

235, 236

237

Newton's Laws of Motion-First Law-Rest-Time-Ex-

amples of the Law-Directional Fixedness - The "In-

variable Plane" of the Solar System Second Law—

Composition of Forces-Measurement of Force and Mass

-Translations from the Kinematics of a Point-Third

Law-D'Alembert's Principle-Mutual Forces between

Particles of a Rigid Body-Motion of Centre of Inertia

of a Rigid Body-Moment of Momentum of a Rigid

Body-Conservation of Momentum, and of Moment of

Momentum-The "Invariable Plane" is a Plane through

the Centre of Inertia, perpendicular to the Resultant Axis

-Terrestrial Application-Rate of doing work-Horse-

power-Energy in Abstract Dynamics

238-241

242-270

SECTIONS

Conservative System-Foundation of the Theory of Energy-
Physical axiom that "the Perpetual Motion is impossible"
introduced-Potential Energy of Conservative System 271-274
Inevitable loss of Energy of Visible Motions-Effect of Tidal
Friction-Ultimate tendency of the Solar System

Conservation of Energy

.

275-277

Kinetic Energy of a System-Moment of Inertia-Moment of
Momentum of a Rotating Rigid Body-Radius of Gyration
-Fly-wheel-Moment of Inertia about any Axis

Momental Ellipsoid-Equilibration of Centrifugal Forces—

Definition of Principal Axes of Inertia-Principal Axes—

Binet's Theorem-Central Ellipsoid-Kinetic Symmetry

round a Point; round an Axis

Energy in Abstract Dynamics

280, 281

282-285

286-288

Equilibrium-Principle of Virtual Velocities-Neutral Equi-

librium-Stable Equilibrium-Unstable Equilibrium—

Test of the nature of Equilibrium

Deduction of the Equations of Motion of any System-Inde-

terminate Equation of Motion of any System-of Conser-

vative System-Equation of Energy-Constraint intro-

duced into the Indeterminate Equation-Determinate

Equations of Motion deduced-Gauss's Principle of Least

Constraint

Impact-Time-integral-Ballistic Pendulum-Direct Impact
of Spheres-Effect of Elasticity-Newton's Experiments-
Distribution of Energy after Impact-Newton's experi-
mental Law consistent with perfect Elasticity

Moment of an Impact about an Axis-Ballistic Pendulum-

Work done by Impact-Equations of Impulsive Motion

Theorem of Euler, extended by Lagrange-Liquid set in Motion
impulsively-Impulsive Motion referred to Generalized
Co-ordinates-Generalized Expression for Kinetic Energy
-Generalized Components of Force-of Impulse-Im-
pulsive Generation of Motion referred to Generalized
Co-ordinates-Momentums in terms of Velocities-Kinetic
Energy in terms of Momentums and Velocities - Velo-
cities in terms of Momentums-Reciprocal relation be.
tween Momentums and Velocities in two Motions-Ap-
plication of Generalized Co-ordinates to Theorems of
§ 311-Problems whose data involve Impulses and Velo-
cities-General Problem (compare § 312)-Kinetic Energy
a minimum in this case-Examples

Lagrange's Equations of Motion in terms of Generalized Co-

ordinates deduced direct by transformation from the

Equations of Motion in terms of Cartesian Co-ordinates

-Equation of Energy-Hamilton's Form-"Canonical

289-292

293

294-306

307-310

311-317

form" of Hamilton's general Equations of Motion of a

Conservative System-Examples of the use of Lagrange's

Generalized Equations of Motion-Gyroscopes and Gy

rostats Gyroscopic Pendulum-Ignoration of Co-ordi-

Kinetics of a perfect fluid-Effect of a Rigid Plane on the

Motion of a Ball through a Liquid-Seeming Attraction

between two ships moving side by side in the same

direction-Quadrantal Pendulum defined-Motion of a

Solid of Revolution with its axis always in one plane through

a Liquid—Observed phenomena—Applications to Nautical

Dynamics and Gunnery - Action Time Average of

Energy-Space Average of Momentums-Least Action-

Principle of Least Action applied to find Lagrange's

Generalized Equations of Motion--Why called "Station-

ary Action" by Hamilton-Varying Action - Action

expressed as a Function of Initial and Final Co-ordinates

and the Energy; its differential Coefficients equal re-
spectively to Initial and Final Momentums, and to the
time from beginning to end-Same Propositions for Ge-
neralized Co-ordinates-Hamilton's "Characteristic Equa-
tion" of Motion in Cartesian Co-ordinates-Hamilton's
Characteristic Equation of Motion in Generalized Co-or-
dinates-Proof that the Characteristic Equation defines
the Motion, for free particles-Same Proposition for a
Connected System, and Generalized Co-ordinates-Ha-
miltonian form of Lagrange's Generalized Equations de-
duced from Characteristic Equation

Characteristic Function-Characteristic Equation of Motion-

Complete Integral of Characteristic Equation-General

Solution derived from complete Integral-Practical In-

terpretation of the complete Solution of the Characteristic

Equation-Properties of Surfaces of Equal Action-

Examples of Varying Action-Application to common

Optics or Kinetics of a Single Particle-Application to

System of free mutually influencing Particles--and to
Generalized System

Slightly disturbed Equilibrium-Simultaneous Transformation

of two Quadratic Functions to Sums of Squares-Gene-

ralized Orthogonal Transformation of Co-ordinates-Sim-

plified expressions for the Kinetic and Potential Energies

--Integrated Equations of Motion, expressing the fun-

damental modes of Vibration; or of falling away from

Configuration of Unstable Equilibrium-Infinitely small

Disturbance from Unstable Equilibrium-Potential and

Kinetic Energies expressed as Functions of Time-

Example of Fundamental Modes

SECTIONS

318, 319

320-330

331-336

337

General Theorem of Fundamental Modes of infinitely small
Motion about a Configuration of Equilibrium-Normal
Displacements from Equilibrium-Theorem of Kinetic
Energy-Of Potential Energy-Infinitesimal Motions in
neighbourhood of Configuration of Unstable Equilibrium

Case of Equality among Periods-Graphic Representation-

Dissipative Systems-Views of Stokes on Resistance to a

Solid moving through a Liquid-Friction of Solids—

Resistances varying as Velocities-Effect of Resistance

varying as Velocity in a simple Motion

SECTIONS

338

339-341

.342-343 p

Infinitely small Motion of a Dissipative System—Cycloidal

System defined-Positional and Motional Forces-Differ-

ential Equations of Complex Cycloidal Motion-Their

Solution-Algebra of Linear Equations-Minors of a De-

terminant-Relations among the Minors of an Evanescent

Determinant-Case of Equal Roots-Case of Equal Roots

and Evanescent Minors-Routh's Theorem-Case of no

Motional Forces-Conservative Positional, and no Mo-

tional, Forces-Equation of Energy in Realized General

Solution

Artificial or Ideal Accumulative System-Criterion of Sta-

bility-Cycloidal System with Conservative Positional

Forces and Unrestricted Motional Forces-Dissipativity

defined-Lord Rayleigh's Theorem of Dissipativity-In-

tegral Equation of Energy-Real part of every Root of

Determinantal Equation proved negative when Potential

Energy is positive for all real Co-ordinates; positive for

some Roots when Potential Energy has negative values;

but always negative for some Roots-Non-oscillatory sub-

sidence to Stable Equilibrium, or falling away from Un-

stable-Oscillatory subsidence to Stable Equilibrium, or

falling away from Unstable-Falling away from wholly

Unstable Equilibrium is essentially non-oscillatory if

Motional Forces wholly viscous-Stability of Dissipative

System-Various origins of Gyroscopic Terms-Equation

of Energy-Gyrostatic Conservative System-simplifica-

tion of its Equations-Determinant of Gyrostatic Conser-

vative System-Square Roots of Skew Symmetric De-

terminants-Gyrostatic System with Two Freedoms-Gy-

rostatic Influence dominant-Gyrostatic Stability--Ordi-

nary Gyrostats-Gyrostats, on gimbals; on universal

flexure-joint in place of gimbals; on stilts; bifilarly slung in

four ways-Gyrostatic System with Three Freedoms-Re-

duced to a mere rotating System-Quadruply free Gyro-

static System without force-Excepted case of failing gy-

rostatic predominance-Quadruply free Cycloidal System,

gyrostatically dominated-Four Irrotational Stabilities

confirmed, four Irrotational Instabilities rendered stable,

by Gyrostatic Links-Combined Dynamic and Gyrostatic

Stability gyrostatically counteracted-Realization of Com-

pleted Solution-Resultant Motion reduced to Motion of

a Conservative System with four fundamental periods

equal two and two-Orthogonalities proved between

two components of one fundamental oscillation; and

equality of their Energies-Orthogonalities proved be-

tween different fundamental oscillations-Case of Equal

Periods Completed Solution for case of Equal Periods

Two higher, and two lower, of the Four Funda-

mental Oscillations, similarly dealt with by Solution

of two similar Quadratics, provided that gyrostatic in-

fluence be fully dominant-Limits of smallest and

second smallest of the four periods-Limits of the next

greatest and greatest of the four periods Quadruply

free Cycloidal System with non-dominant gyrostatic in-

fluences-Gyrostatic System with any number of freedoms

-Case of Equal Roots with stability-Application of

Routh's Theorem-Equal Roots with instability in tran-

sitional cases between Stability and Instability-Condi

tions of gyrostatic domination-Gyrostatic Links ex-

plained-Gyrostatically dominated System: its adynamic

oscillations (very rapid); and precessional oscillations

(very slow)-Comparison between Adynamic Frequencies,

Rotational Frequencies of the Fly-wheels, Precessional

Frequencies of the System, and Frequencies or Rapidities

of the System with Fly-wheels deprived of Rotation-

Proof of reality of Adynamic and of Precessional Periods

when system's Irrotational Periods are either all real or

all imaginary-Algebraic Theorem

SECTIONS

344-345xxvili

Kinetic Stability-Conservative disturbance of motion-Ki-

netic Stability and Instability discriminated-Examples

-Circular Simple Pendulum-Kinetic Stability in Cir-

cular Orbit-Kinetic Stability of a Particle moving on a

Smooth Surface-Incommensurable Oscillations-Oscil-

latory Kinetic Stability-Limited Kinetic Stability-

Kinetic Stability of a Projectile-General criterion-Ex-

amples-Motion of a Particle on an anticlastic Surface,

unstable;-on a synclastic Surface, stable-Differential

Equation of Disturbed Path

General investigation of Disturbed Path-Differential Equa-

tion of Disturbed Path of Single Particle in a Plane-Ki-

netic Foci-Theorem of Minimum Action-Action never a

Minimum in a course including Kinetic Foci-Two or more

Courses of Minimum Action possible-Case of two mini-

mum, and one not minimum, Geodetic Lines between two

Points-Difference between two sides and the third of a

Kinetic Triangle-Actions on different courses infinitely

346-355