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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 .

205–216

Force-Specification of a Force—Place of Application-Direc-

tion—Magnitude-Accelerative Effect—Measure of Force 217—220
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

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
Force about an Axis

231, 232

Digression on Projection of Areas

233

Couple-its Moment, Arm, and Axis

234

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

235, 236

Virtual Velocity-Virtual Moment

237

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

Work of a Couple-Transformation of Work-Potential

Energy

238-241

Newton's Laws of Motion-First Law-Rest-Time-Es.

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

242—270

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271–274

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275—277

278, 279

.

280, 281

282-285

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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 8 312)—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

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.

nates

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

320-330

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331-336

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339-341

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

. 312—343 p

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.
stableOscillatory 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

344-345ıxvill

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

346-355

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

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