A Treatise on Infinitesimal Calculus: The dynamics of material systems. 1862

University Press, 1862

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Índice

THE MOTION OF A MATERIAL SYSTEM	1

Degenerate forms of surfaces of the second degree	15

The most general motion consists of a translation and of	22

On central radii vectores of an ellipsoid	26

CHAPTER II	32

rotations Definition and explanation of angular velocity	33

Relation of linear velocity to angular velocity	36

Linerepresentatives of angular velocities	38

The instantaneous rotationaxis the instantaneous pole and the couple of impulsion	248

Examples in illustration	250

The pressure at the fixed point	252

Rotation of a rigid body about a fixed point under the action of finite forces 148 The general equations of motion Eulers equations	253

The equations and their results when no forces act	255

Explanations of these results the invariable axis	256

Position determined by the three angles Qpty	258

The component of the instantaneous angular velocity along the invariable axis is constant	260

Composition of angular velocities whose rotationaxes meet	40

Particular forms when the rotationaxes are at right angles	41

Bohnenbergers machine	42

Composition of many angular velocities whose rotationaxes pass through one point	43

Geometrical explanation of Foucaults pendulum experiment	44

Composition of angular velocities whose rotationaxes are parallel	45

Couples of angular velocities	47

General case of the composition of angular velocities	48

Particular cases of the preceding Article	50

3537 The central axis	53

Determination of the linear velocity due to given angular ve locities	56

Motion of a body defined by two systems of reference	58

Relation between angular velocities and the differentials of directioncosines	60

Analytical proof that all motion consists of a translation and of a rotation	64

CHAPTER III	67

Explanation and statement of DAlemberts principle	69

Illustrations of its application to particular problems	73

47 The statement and proof given by DAlembert	77

The equations of motion of a material system the internal forces of which are in equilibrium	78

The equations of motion expressed in a single equation by means of the principle of virtual velocities	80

The number of equations necessary for the complete solution of a problem	83

Application to the motion of a flexible string	84

Independence of the motions of translation of centre of gra	86

Conservation of motion of centre of gravity of moments	92

If f g be any two integrals g is constant example	97

Conservation of areas	98

Conservation of vis viva critical value of vis viva when	105

Carnots theorem	114

CHAPTER IV	124

Transformation of equations instantaneous forces	127

Moment of inertia and radius of gyration	128

The equations deduced from first principles	130

The axial components of the resulting angular velocity	131

77 Transformation of equations finite forces	132

The resulting angular velocityincrement	133

Axial components of velocityincrement	134

Analysis of the equations	135

Centrifugal forces	137

Resulting equations of motion	138

Simplification of the equations principal axes	139

Proof of the existence of a system of principal axes and the position of it at a given point	141

87 Interpretation of the results by means of an ellipsoid	143

Particular forms of the ellipsoid of principal axes	145

Principal axes determined in a particular problem	146

One principal axis being given the determination of the two others	147

Examples in illustration	148

Reduced forms of equations of motion	150

Permanent axes	153

Foucaults gyroscope	154

General value of moment of inertia	156

9 Example in illustration	157

Singular values of principal moments	160

The equimomental cone	161

Moments of inertia relative to parallel axes	162

Moments of inertia in relation to the central ellipsoid	163

The position of principal axes at any point	166

The central ellipsoid of gyration	167

The symmetry of a body	169

The cone reciprocal to that of principal axes is equimomental	171

Professor Maccullaghs construction by Apsidals	173

109110 Particular forms of the equimomental surface	174

Conditions when a line is a principal axis	179

Examples of moments of inertia	185

CHAPTER V	199

Conditions when there is no pressure on the axis	206

The axis of percussion	212

Examples of the simple isochronous pendulum	220

Experimental determination of the radius of gyration	226

Motion of machines with fixed axes	233

CHAPTER VI	245

The equations of motion in their general and reduced forms	246

Poinsots interpretation of the preceding result	261

General and particular properties of the polhode	264

158159 Do of the herpolhode	266

The stability of the rotationaxis	269

Particular cases of the preceding theorems depending on particular initial circumstances	271

162163 Do depending on particular constitutions of the body	273

Discussion of the case when the oscillation of the rotation axis is small	277

The cone described by the rotationaxis in the body	278

Certain properties of the principal axes of the moving body	279

Rotation of a heavy body about a fixed point	281

Particular case when a b and the initial axis of rotation is the principal axis of unequal moment	288

170171 Do when the axis of unequal moment is inclined to the vertical at a constant angle	291

Bohnenbergers and Fessels machines	295

Precession and nutation of the earth	297

Determination of small quantities	300

16 Transformation of certain terms of the equations	304

177 General integral of the equations	306

Determination of the lunisolar precession and nutation	311

Poinsots determination of the preceding results from gene ral reasoning	315

all the integral equations by means of a single function	317

The pressure on the fixed point	318

CHAPTER VII	320

The equations of motion	322

The components of velocity resulting from a given force	323

The locus of points which move with the same velocity	324

The spontaneous axis	326

The vis viva of the system is a maximum when the rotation axis is the spontaneous axis	327

The motion of a body due to a blow parallel to a central principal axis and in a central principal plane	329

Properties of centre of percussion and spontaneous centre of rotation	332

The centres of greatest percussion	334

cules of given masses at the ends of an inflexible bar	335

The position of the point on which the moving body	338

The centres of greatest reflexion and greatest conversion	339

Motion of a body due to a blow parallel to a central princi pal axis	342

Theorems relating to the spontaneous axis and spontaneous centre corresponding to a given centre of impulsion	344

Other incidents of the motion	346

Centres of greatest percussion	349

Case in which a couple of impulsion initially acts	350

Case when the body strikes against a moveable mass	352

Points of greatest reflexion and greatest conversion	355

Points of perfect reflexion and perfect conversion	356

The initial motion of a billiard ball	363

The general case of rocking or titubation	379

Examples of small oscillations	385

Small oscillations of a body of which one point is fixed	391

The motion of a top on a smooth horizontal plane	402

CHAPTER VIII	413

The equations derived from fictitious forces	420

Motion of a heavy particle in a rotating tube	429

Adaptation of the equations with the omission of small	435

The investigation carried to a higher approximation	442

Relative motion of a particle on a smooth inclined plane	453

All combined into one equation by means of the principle	459

Relations between absolute and relative angular velocities	465

Particular cases of the preceding	474

26 Moving work and resisting work useful work and lost	487

On mechanical units	493

Two modes of forming the equations of motion of an elastic	499

The properties of a vibrating string deduced from the pre	506

Another mode of expressing the integrals	513

The general equations of motion of a molecule of an elastic	519

THEORETICAL DYNAMICS	524

Case in which the equation 8 A 8 A is significant	530

Illustration taken from the transformation to polar coordi	536

Considerations on the integrals of such a system	544

Definition of canonical elements The initial values	550

canonical solution is reducible to quadratures	562

The results of the gyroscope 480	566

Interpretation of the three new elements	570

Conclusion	578

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Términos y frases comunes

angle angular velocity axes fixed axial components axis passing becomes blow body rotates central ellipsoid centre of gravity centrifugal forces coefficients cone confocal constant coordinate axes corresponding couple curve D'Alembert's principle determined differential direction-cosines displacement distance earth elastic ellipse ellipsoid of gyration equa equal equations of motion evident expressed fixed in space fixed point focal conic Hence horizontal infinitesimal initial instantaneous axis integration intersection invariable axis invariable plane isochronous length Let the rotation-axis let us suppose line of action mass moment of inertia momental ellipsoid moments of inertia momentum momentum-increments moving system nutation origin oscillation parallel pendulum perpendicular plate polhode position pressure principal moments principal plane quantities radii radius of gyration radius vector respectively second degree shews sphere string surface system of particles theorem tion values velocity-increment vertex vertical vibrations vis viva viva

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Información bibliográfica

QR code for A Treatise on Infinitesimal Calculus: The dynamics of material systems. 1862

Título	A Treatise on Infinitesimal Calculus: The dynamics of material systems. 1862 Volumen 4 de A Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics, Bartholomew Price Volumen 4 de A treatise on infinitesimal calculus, Bartholomew Price
Edición	2
Editor	University Press, 1862
Procedencia del original	la Universidad de California
Digitalizado	5 Jun 2007

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