William Kingdon Clifford. # Elements of dynamic; an introduction to the study of motion and rest in solid and fluid bodies (Volume 2) online

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two given curves on the fixed plane.

Investigate analogous determinations of the motion in space of a

solid body which has one, two, or three degrees of freedom.

4. What relations must hold of the relative positions of n axes in

order that small rotations about them may be equivalent to rest? (n=3,

4, 5, 6.) In the case n=5, show that, if one axis touch the hyperboloid

containing three others, each axis touches the hyperboloid containing any

three others.

CHAPTER H.

1. When a body has a given spin of magnitude w, find the locus of

those points in it which have a velocity of given magnitude v.

2. When a body has a twist about a certain screw, find the locus of

those points whose velocity is in a given direction.

3. Two spins of equal magnitude about non-concurrent axes are

compounded into a twist. Prove that its axis is equidistant from and

equally inclined to the axes of spin.

4. Spins about the sides of a triangle and represented by their

lengths taken in order, compound into a translation represented by twice

the area of the triangle.

5. Extend this proposition to a plane or skew polygon.

6. If a line A revolve about a line B with angular velocity w, prove

that the lengthwise velocity of A is kw sin 6, where k is the shortest dis-

tance and 6 the angle between A and B.

CHAPTER HI.

A circle rolls uniformly inside another circle of double the radius;

show that every point rigidly connected with it moves in an elliptic

harmonic motion, and that the sum or difference of the axes of all the

ellipses is constant.

BOOK HI. CHAPTER I.

1. In a flat board subjected to uniform stress in its plane, if an

ellipse can be found such that the pressure across each axis is repre-

sented in magnitude and direction by the other, prove that the same

property belongs to any two conjugate diameters.

112 DYNAMIC.

2. A regular hexagon undergoes a pure shear by sliding parallel to

one of its sides. Find the amount of the shear that the deformed

hexagon may have two right angles, and determine its other angles.

3. Any system of forces is equivalent to a wrench about a certain

screw.

Prove that a rotating body will do no work against this wrench if

where a is the pitch of the screw, k the shortest distance between the

axis of the screw and the axis of rotation, and <f> the angle between these

axes.

4. In the case of a plane homogenous stress, show that a conic may

be drawn such that the stress across any diameter is represented in

magnitude and direction by the conjugate diameter.

5. Prove that a solid body can be moved from any one position to

any other position by a twist about a certain screw. What becomes of

the screw in the cases of pure translation and pure rotation ?

A body revolves uniformly about an axis so as to make one revolution

during the time in which it moves uniformly 2 feet along the axis. Find

the locus of those points the direction of whose motion makes an angle

of 60 with the axis.

6. Define a simple shear, the amount of a shear, the ratio of a shear.

If the shear be small, show that the amount is twice the elongation or

contraction.

7. Prove that the deformation at any point of a strained body is

compounded of a dilatation or compression and two simple shears.

BOOK IV. CHAPTEE I.

1. On the latus rectum of a parabola, as diagonal, a square is

described; find the centre of gravity of that portion of the square which

is inside the parabola.

2. For every plane 0,rea there exists an ellipse, such that if the area

be subject to pressure at every point of it proportional to the distance of

the point from a certain line in its plane, the resultant pressure acts at

a point E, such that if G is the centre of gravity of the area, and P the

pole of the line in regard to the ellipse, BG GP, and RGP is a straight

line.

EXERCISES. 113

CHAPTER H.

Find swing-radii in the following cases :

1. A right circular cylinder about its axis.

2. A right circular cylinder about a line cutting axis at right angles.

3. A right circular cylinder about a tangent to the circular rim.

4. A sphere about any tangent.

5. A regular tetrahedron about any edge.

6. A regular tetrahedron about a line bisecting two opposite edges.

7. A paraboloid of revolution in regard to a plane touching it at the

vertex.

8. A paraboloid of revolution in regard to its base.

9. Determine the swing-conic of three particles of equal mass ; and

show that so far as second moments are concerned any area may be

replaced by three such particles.

1. Define a homogeneous strain, a pure strain, a simple shear. Show

how to represent the pure strain of a plane area by means of a certain

ellipse, such that the new position of any diameter of the ellipse is per-

pendicular to the tangents at its extremities.

What kind of strain would take place if the same construction were

made with a hyperbola ?

2. Define the centre of parallel forces, and find it (1) when the

forces are distributed uniformly over the surface of a quadrilateral: (2)

when they are distributed over the surface of a triangle, proportionally to

the distance from one side of it.

3. A regular hexagon is made of equal and uniform rods jointed

together, and the ends of two opposite sides are joined by two parallel

strings ; the whole framework is then allowed to hang from one of these

sides, which is held horizontal. Suppose (1) that the strings are

inextensible, and find their tension ; (2) that either of them would be

stretched to twice its natural length by the weight of the six rods, and

find the position of equilibrium.

4. Find the momentum and the kinetic energy of a rigid body

rotating about a fixed axis.

When a pendulum performs small oscillations under the influence of

gravity, prove that the centres of oscillation and suspension are con-

vertible.

5. In a fluid at rest under the action of any forces, prove that the

level surfaces are also surfaces of equal pressure and density.

If two tubes were constructed so as to lie along a meridian, one always

C. 8

114 EXERCISES.

half-a-mile above the level of the sea, and the other always half-a-mile

below, in what direction would water flow in these two tubes respectively?

6. A man stands with his feet on the upper bars of the backs of two

chairs whose seats are turned away from him. His legs are straddled at

an angle of 60. Find the conditions of equilibrium : (1) regarding the

man as a rigid body, (2) assuming that his legs are jointed freely at the

hip for lateral motion.

7. Explain the operation of "getting up swing." Is this possible if

you are placed perfectly at rest in a seat freely suspended by two equal

ropes from points in the same horizontal plane? A trapezist who is

swinging with a range of 30 on either side of the vertical, can increase

the distance of his centre of inertia from the axis of oscillation by one

eighth ; show how by graphical construction to determine the number of

swings in which ha can increase his range to 60.

8. In a plane lamina under homogeneous stress in its plane, prove

that a conic may always be found such that the stress across any diameter

is represented in magnitude and direction by the conjugate diameter.

9. If a rigid body be in motion, show that at any instant the

straight lines which, if rigidly connected with it, would have no

longitudinal motion, form a linear complex.

Investigate the geometrical relations between a system of forces and

the small motion by which no work is done on the forces.

10. State the Law of Energy for a single particle under the action of

gravity. Describe Attwood's Machine ; and show how the definition of

mass derived from it enables us to extend the law of energy to two or

more bodies. What motion can be communicated to a ton of material

by the fall of 5 Ib. through 160,000 ft.?

11. Define Best and Equilibrium. Can a body have either without the

other? "When a system of bodies is passing through a position of

equilibrium, there is no change of kinetic energy, and therefore no work

is being done." Explain this statement, and show that it amounts to the

principle of virtual velocities. Four uniform rods, AB, BC, CD, DA, are

jointed together at A, B, C, D, and placed in a vertical plane with AB

resting on the ground. AB=BC=DA% CD. A load X times the

weight of CD is placed upon it ; find what positions of the load exert the

greatest and least compression upon AB.

12. Give a construction for finding the resultant of a number of

forces given in magnitude and direction by lines drawn on a piece of

paper. When the forces are all parallel, in what way does this construc-

tion determine their moments about any point in the plane?

NOTES.

p. 51, 1. 20, "its velocity," i.e. the velocity of the mass-centre.

p. 57, 11. 2 and 4 from bottom, "immediately before" and "imme-

diately after "

p. 60, 1. 1. This statement seems questionable mass-acceleration

=m , rate of change of momentum = (mv), and the two are not always

equal.

p. 61, 1. 6 up. I have received the following remarks on this para-

graph: "Cut a perfectly homogeneous spherical shell of indian-rubber, so

that there is no strain, and turn it inside out. This can be done if a

small piece be removed. Surely the body would have surface-strain.

Does not Clifford mean normal surface stress?"

p. 61, 1. 11 up, "compression of disc" strikes one as being obscurely

worded though the meaning is clear.

p. 62, 1. 8, " stress across section," i.e. per unit area.

p. 66, no definition given of strength.

1. 21, for " battery " should we not read "cell" ?

p. 71, introduce O at lower angle of figure.

p. 74, 1. 16, rather "look back along it," i.e. in opposite sense to spin-

rotor.

p. 76, 1. 6 from bottom. Supply "Vorlesungen iiber Math. Physik."

p. 78, 1. 10 up, 'T-t-T'," the reference is, of course, to a well-known

text-book.

INDEX.

VOL. I.

Abscissa, 29

Absolute measure, 49

,, unit, 49

Adams, J. C., 108

Amplitude, 18, 21

Angular velocity, 63

Anomaly, 107

Archimedes, 15

Area, projection of, 27

equal, 31

Argument, 19

Asymptotes, 90

Asymptotic cone, 178, 184

Attraction, 3

Auxiliary circle, 31

Axes of coordinates, 13

of ellipse, 29

,, of hyperbola, 90

Axis of cylinder, 22

,, of parabola, 40

Axodes, 137

Ball, R. S., 126, 131

Braun, 36

Cardioid, 154

Cayley, 149

Central projection, 25

Centre of ellipse, 28

Centrode, 136

Circular motion, 12

Circulation, 194

Clock, 24

Complex number, 86

Complex of lines, 133

Component, 5

Composition of steps, 4

,, of rectilinear motion,

17

of circular motion, 19

, , of harmonic motion, 23

,, of velocity, 59

Compound harmonic motion, 33

Compression, 158

Cone, oblique, 179

Congruent, 119

Conic sections, 9

Conjugate axis, 31, 90

,, diameters, 28

Coordinates, 13

Cross-ratio, 42

Crossed-rhomboid, 146

Curvature, 75

Curve of positions, 15

,, harmonic, 23

,, Lissajous', 37

,, of velocities, 69

Cusp, 75

Cycloidal curves, 152 7

Cylinder, 22

,, motion on, 33

Cylindroid, 128

Derived functions, 64, 66

Description, 11

Diameter, 28

,, of parabola, 39, 40

Diametral plane, 172

Diaphragms, 207

Dimensions of velocity, 50

Direction of motion, 41

Displacement conic, 162

INDEX.

117

Displacement function, 163

,, quadric, 183

Dynamic, 1, 2

Eccentric Anomaly, 107

follower, 107

Eight, figure of, 35

Elastic, 2

EUipse, 22, 27

,, eccentricity, directrix, 102

,, latus rectum, 106

Elliptic compasses, 154

,, harmonic motion, 31

,, velocity in, 61

Elongation, 158

Epoch, 18, 21

Equation to motion, 17, 19, 21, 32

Equipotential surface, 203

Euclid, 16, 41

E volute, 144

Exponential series, 83

Falling body, 38

Figure of eight curve, 35

Flow, steady, line of, tube of, 199

Fluent, 63

Fluids, 3

Fluxions, 62

,, of sum, product, 64

,, of quotient, 65

Force, 2

law of, 2

Fourier's theorem, 37

Functions, 64, (vector), 162

Galileo, 38

Generator of cylinder, 22

Graphic mode of description, 12

Great circle, 119

Hamilton, 105

Harmonic motion, 20

,, ,, elliptic, 31

,, compound, 33

curve, 23, 43

range, 42

Hodograph, 67

Hyperbola, 89, 103

Hyperboloid, 178

Ideal motions, 191

Inflexion, point of, 44, 74

Instantaneous axis, 141

centre, 136

,, screw, 137

Integral, 73

Inverse curves, 100

Involute, 144

Kepler, 107

Kinematic, 2

Kinetic, 2

Kite, 150

Lambert, 108

Linear-function, 185

Line-integral, 195

Lissajous, 36, 37

Logarithm, 80

Logarithmic rate, 80

spiral, 86

Major axis, 31

Matrix, 163

Maximum distance, 30

Maxwell, 209

Mean point, 11

,, velocity, 53

Measure, 49

Mid-centre, 11

Minimum distance, 30

Minor axis, 31

Modulus, 86

Moment, 92

Motion, 1

representation of, 14

uniform, 15

circular, 18

harmonic, 20

elliptic harmonic, 21

compound harmonic, 33

parabolic, 38

elliptic, 107

clockwise and counter-

clockwise, 19

irrotational, 203

logarithmic, 78

steady, 199

Motor, 125

Newton, 45, 70, 98, 107

Oblique projection, 25

Ordinate, 29

Origin, 13

Orthogonal projection, 24

Osculating plane, 76

Parabola, 34

Parabolic motion, 38, 103

118

INDEX.

Particle, 2

Pedal curves, 101

Period, 18, 21

Perversion, 119

Phase, 18, 21

Pitch, 125

Pitch-conic, 129

Planes, coordinate, 14

Plucker, 73

Position, vector, 14

Positions, curve of, 15

Projection, 24

of velocity, 60

Quantity, vector, 13

fluent, 62

scalar, 62

Quasi-harmonic motion, 89

Bate, 48

Eeciprocal curves, 101

Eectilinear motion, 16

Eepulsion, 3

Eesultant, 5

Eigid, 1

Eoberts, S., 148, 150

Eotation, 1

velocity, 123

Eotor, 123

Eoulette, 140

Scalar, quantity, 62

,, product, 95

Scrolls, 137

Shear, 167

Shell, double, 218

Sines, curve of, 22

Sink, 214

Solids, 3

Sources, 214

Spheroid, 175

Spin, 123, 193

,, -system, 200

tube of, 200

Squirt, 214

Static, 2

Steps of translation, 3

,, product of, 94

Steps, displacement of, 161

Strain, 2, 158160, 172

,, -function, 163

principal axes and elonga-

tion, 176

pure, 176

-flux, homogeneous, 193

Stress, 3

Surface-integral, 201

Sylvester, 107, 150

ff, convergence, divergence of, 209

Tangent to ellipse, 41

to parabola, 43

to harmonic curve, 43

,, Newton's criterion, 45

definition, 47

Thomson and Tait, 23, 35

Tidal clock, 24

Three-bar motion, 146

Translation, 1

Transverse axis, 31

Twist, 2

,, magnitude of, 125, 193

Uniform motion, 15, 48

Vector, 13

product, 95

Velocity, 48, 50

variable, 51

mean, 53

to find, 51, 53

definition, 58

composition of, 59

projection of, 60

moment of, 98

lengthwise, 132

-potential, 203

,, maximum, 210

minimum, 210

Vortex, 217

-filament, 217

Wallis, 72

Wave, 23

Whirls, 214

Wry (shear, strain), 170

INDEX.

VOL. II.

Acceleration by place, 74

Archimedes, 14

Aristotle, 73

Attraction, 63, 64, 66

Battery, 66

Cavendish, 63

Charge-accelerations, 65

Charges, like, unlike, 64

Composition, law of, 70

Conductor, 65

Confocal, 39

Conjugate lines, 18, 25

,, points, 25

,, planes, 31

motions, 80, 83

Core, 19, 44

Cross-centre, 7

Culmann, 18 (note)

Density, 1, 59

Ellipsoid of gyration, 39

momental, 54, 75, 82

Fluid friction, 68

Focal ellipse, 40

hyperbola, 40

Force, 67, 69, 70, 73

internal, external, 71

Fourier's theorem, 88

Friction, 68

Gram, 59

Gravity, 63

Insulators, 65

Kirchhoff, 76

Loaded area, 15, 16

body, 31

Magnet, 65

Mass, 2, 58, 70

,, resultant, 2

,, vector, 2

Mass-centre, 3, 11 14

triangle and tetrahe-

dron, 5

,, trapezium, 6

,, ,, solids, 8

Mid-centre, 7

Middle quarter, 20, 21, 44

fifth, 44

Moment, 3, 77

,, circular arc, 10

first, 15

,, second, 15, 31

mixed, 18, 31

Momentum, 48, 51, 60

moment of, 49, 53

Motion, laws of, 70

Neutral axis, 15

Newton, 63

120

Null-conic, 24

-quadric, 36

Opposite points, 27

Poinsot, 75

Polar, 25, 28

planes, 35

Pole, 15, 16,, 25, 28, 31, 35

(north on the magnet), 65

Principal axes, 29

screws, 56

Eeciprocal, 44

Reciprocity, law of, 63, 70, 72

Eepulsion, 64, 66

Eesistance, 68

INDEX.

Botor, 48

Self-conjugate, 29

quadric, 34

tetrahedron, 36

Stress, 59

across section, 62

Swing-radius, 17

,, -conic, 24

-ellipsoid, 34

Sylvester, 10

Thomson and Tait, 73

Vector, 48, 49

Velocity, instantaneous, 92

,, arrival, departure, 93

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Investigate analogous determinations of the motion in space of a

solid body which has one, two, or three degrees of freedom.

4. What relations must hold of the relative positions of n axes in

order that small rotations about them may be equivalent to rest? (n=3,

4, 5, 6.) In the case n=5, show that, if one axis touch the hyperboloid

containing three others, each axis touches the hyperboloid containing any

three others.

CHAPTER H.

1. When a body has a given spin of magnitude w, find the locus of

those points in it which have a velocity of given magnitude v.

2. When a body has a twist about a certain screw, find the locus of

those points whose velocity is in a given direction.

3. Two spins of equal magnitude about non-concurrent axes are

compounded into a twist. Prove that its axis is equidistant from and

equally inclined to the axes of spin.

4. Spins about the sides of a triangle and represented by their

lengths taken in order, compound into a translation represented by twice

the area of the triangle.

5. Extend this proposition to a plane or skew polygon.

6. If a line A revolve about a line B with angular velocity w, prove

that the lengthwise velocity of A is kw sin 6, where k is the shortest dis-

tance and 6 the angle between A and B.

CHAPTER HI.

A circle rolls uniformly inside another circle of double the radius;

show that every point rigidly connected with it moves in an elliptic

harmonic motion, and that the sum or difference of the axes of all the

ellipses is constant.

BOOK HI. CHAPTER I.

1. In a flat board subjected to uniform stress in its plane, if an

ellipse can be found such that the pressure across each axis is repre-

sented in magnitude and direction by the other, prove that the same

property belongs to any two conjugate diameters.

112 DYNAMIC.

2. A regular hexagon undergoes a pure shear by sliding parallel to

one of its sides. Find the amount of the shear that the deformed

hexagon may have two right angles, and determine its other angles.

3. Any system of forces is equivalent to a wrench about a certain

screw.

Prove that a rotating body will do no work against this wrench if

where a is the pitch of the screw, k the shortest distance between the

axis of the screw and the axis of rotation, and <f> the angle between these

axes.

4. In the case of a plane homogenous stress, show that a conic may

be drawn such that the stress across any diameter is represented in

magnitude and direction by the conjugate diameter.

5. Prove that a solid body can be moved from any one position to

any other position by a twist about a certain screw. What becomes of

the screw in the cases of pure translation and pure rotation ?

A body revolves uniformly about an axis so as to make one revolution

during the time in which it moves uniformly 2 feet along the axis. Find

the locus of those points the direction of whose motion makes an angle

of 60 with the axis.

6. Define a simple shear, the amount of a shear, the ratio of a shear.

If the shear be small, show that the amount is twice the elongation or

contraction.

7. Prove that the deformation at any point of a strained body is

compounded of a dilatation or compression and two simple shears.

BOOK IV. CHAPTEE I.

1. On the latus rectum of a parabola, as diagonal, a square is

described; find the centre of gravity of that portion of the square which

is inside the parabola.

2. For every plane 0,rea there exists an ellipse, such that if the area

be subject to pressure at every point of it proportional to the distance of

the point from a certain line in its plane, the resultant pressure acts at

a point E, such that if G is the centre of gravity of the area, and P the

pole of the line in regard to the ellipse, BG GP, and RGP is a straight

line.

EXERCISES. 113

CHAPTER H.

Find swing-radii in the following cases :

1. A right circular cylinder about its axis.

2. A right circular cylinder about a line cutting axis at right angles.

3. A right circular cylinder about a tangent to the circular rim.

4. A sphere about any tangent.

5. A regular tetrahedron about any edge.

6. A regular tetrahedron about a line bisecting two opposite edges.

7. A paraboloid of revolution in regard to a plane touching it at the

vertex.

8. A paraboloid of revolution in regard to its base.

9. Determine the swing-conic of three particles of equal mass ; and

show that so far as second moments are concerned any area may be

replaced by three such particles.

1. Define a homogeneous strain, a pure strain, a simple shear. Show

how to represent the pure strain of a plane area by means of a certain

ellipse, such that the new position of any diameter of the ellipse is per-

pendicular to the tangents at its extremities.

What kind of strain would take place if the same construction were

made with a hyperbola ?

2. Define the centre of parallel forces, and find it (1) when the

forces are distributed uniformly over the surface of a quadrilateral: (2)

when they are distributed over the surface of a triangle, proportionally to

the distance from one side of it.

3. A regular hexagon is made of equal and uniform rods jointed

together, and the ends of two opposite sides are joined by two parallel

strings ; the whole framework is then allowed to hang from one of these

sides, which is held horizontal. Suppose (1) that the strings are

inextensible, and find their tension ; (2) that either of them would be

stretched to twice its natural length by the weight of the six rods, and

find the position of equilibrium.

4. Find the momentum and the kinetic energy of a rigid body

rotating about a fixed axis.

When a pendulum performs small oscillations under the influence of

gravity, prove that the centres of oscillation and suspension are con-

vertible.

5. In a fluid at rest under the action of any forces, prove that the

level surfaces are also surfaces of equal pressure and density.

If two tubes were constructed so as to lie along a meridian, one always

C. 8

114 EXERCISES.

half-a-mile above the level of the sea, and the other always half-a-mile

below, in what direction would water flow in these two tubes respectively?

6. A man stands with his feet on the upper bars of the backs of two

chairs whose seats are turned away from him. His legs are straddled at

an angle of 60. Find the conditions of equilibrium : (1) regarding the

man as a rigid body, (2) assuming that his legs are jointed freely at the

hip for lateral motion.

7. Explain the operation of "getting up swing." Is this possible if

you are placed perfectly at rest in a seat freely suspended by two equal

ropes from points in the same horizontal plane? A trapezist who is

swinging with a range of 30 on either side of the vertical, can increase

the distance of his centre of inertia from the axis of oscillation by one

eighth ; show how by graphical construction to determine the number of

swings in which ha can increase his range to 60.

8. In a plane lamina under homogeneous stress in its plane, prove

that a conic may always be found such that the stress across any diameter

is represented in magnitude and direction by the conjugate diameter.

9. If a rigid body be in motion, show that at any instant the

straight lines which, if rigidly connected with it, would have no

longitudinal motion, form a linear complex.

Investigate the geometrical relations between a system of forces and

the small motion by which no work is done on the forces.

10. State the Law of Energy for a single particle under the action of

gravity. Describe Attwood's Machine ; and show how the definition of

mass derived from it enables us to extend the law of energy to two or

more bodies. What motion can be communicated to a ton of material

by the fall of 5 Ib. through 160,000 ft.?

11. Define Best and Equilibrium. Can a body have either without the

other? "When a system of bodies is passing through a position of

equilibrium, there is no change of kinetic energy, and therefore no work

is being done." Explain this statement, and show that it amounts to the

principle of virtual velocities. Four uniform rods, AB, BC, CD, DA, are

jointed together at A, B, C, D, and placed in a vertical plane with AB

resting on the ground. AB=BC=DA% CD. A load X times the

weight of CD is placed upon it ; find what positions of the load exert the

greatest and least compression upon AB.

12. Give a construction for finding the resultant of a number of

forces given in magnitude and direction by lines drawn on a piece of

paper. When the forces are all parallel, in what way does this construc-

tion determine their moments about any point in the plane?

NOTES.

p. 51, 1. 20, "its velocity," i.e. the velocity of the mass-centre.

p. 57, 11. 2 and 4 from bottom, "immediately before" and "imme-

diately after "

p. 60, 1. 1. This statement seems questionable mass-acceleration

=m , rate of change of momentum = (mv), and the two are not always

equal.

p. 61, 1. 6 up. I have received the following remarks on this para-

graph: "Cut a perfectly homogeneous spherical shell of indian-rubber, so

that there is no strain, and turn it inside out. This can be done if a

small piece be removed. Surely the body would have surface-strain.

Does not Clifford mean normal surface stress?"

p. 61, 1. 11 up, "compression of disc" strikes one as being obscurely

worded though the meaning is clear.

p. 62, 1. 8, " stress across section," i.e. per unit area.

p. 66, no definition given of strength.

1. 21, for " battery " should we not read "cell" ?

p. 71, introduce O at lower angle of figure.

p. 74, 1. 16, rather "look back along it," i.e. in opposite sense to spin-

rotor.

p. 76, 1. 6 from bottom. Supply "Vorlesungen iiber Math. Physik."

p. 78, 1. 10 up, 'T-t-T'," the reference is, of course, to a well-known

text-book.

INDEX.

VOL. I.

Abscissa, 29

Absolute measure, 49

,, unit, 49

Adams, J. C., 108

Amplitude, 18, 21

Angular velocity, 63

Anomaly, 107

Archimedes, 15

Area, projection of, 27

equal, 31

Argument, 19

Asymptotes, 90

Asymptotic cone, 178, 184

Attraction, 3

Auxiliary circle, 31

Axes of coordinates, 13

of ellipse, 29

,, of hyperbola, 90

Axis of cylinder, 22

,, of parabola, 40

Axodes, 137

Ball, R. S., 126, 131

Braun, 36

Cardioid, 154

Cayley, 149

Central projection, 25

Centre of ellipse, 28

Centrode, 136

Circular motion, 12

Circulation, 194

Clock, 24

Complex number, 86

Complex of lines, 133

Component, 5

Composition of steps, 4

,, of rectilinear motion,

17

of circular motion, 19

, , of harmonic motion, 23

,, of velocity, 59

Compound harmonic motion, 33

Compression, 158

Cone, oblique, 179

Congruent, 119

Conic sections, 9

Conjugate axis, 31, 90

,, diameters, 28

Coordinates, 13

Cross-ratio, 42

Crossed-rhomboid, 146

Curvature, 75

Curve of positions, 15

,, harmonic, 23

,, Lissajous', 37

,, of velocities, 69

Cusp, 75

Cycloidal curves, 152 7

Cylinder, 22

,, motion on, 33

Cylindroid, 128

Derived functions, 64, 66

Description, 11

Diameter, 28

,, of parabola, 39, 40

Diametral plane, 172

Diaphragms, 207

Dimensions of velocity, 50

Direction of motion, 41

Displacement conic, 162

INDEX.

117

Displacement function, 163

,, quadric, 183

Dynamic, 1, 2

Eccentric Anomaly, 107

follower, 107

Eight, figure of, 35

Elastic, 2

EUipse, 22, 27

,, eccentricity, directrix, 102

,, latus rectum, 106

Elliptic compasses, 154

,, harmonic motion, 31

,, velocity in, 61

Elongation, 158

Epoch, 18, 21

Equation to motion, 17, 19, 21, 32

Equipotential surface, 203

Euclid, 16, 41

E volute, 144

Exponential series, 83

Falling body, 38

Figure of eight curve, 35

Flow, steady, line of, tube of, 199

Fluent, 63

Fluids, 3

Fluxions, 62

,, of sum, product, 64

,, of quotient, 65

Force, 2

law of, 2

Fourier's theorem, 37

Functions, 64, (vector), 162

Galileo, 38

Generator of cylinder, 22

Graphic mode of description, 12

Great circle, 119

Hamilton, 105

Harmonic motion, 20

,, ,, elliptic, 31

,, compound, 33

curve, 23, 43

range, 42

Hodograph, 67

Hyperbola, 89, 103

Hyperboloid, 178

Ideal motions, 191

Inflexion, point of, 44, 74

Instantaneous axis, 141

centre, 136

,, screw, 137

Integral, 73

Inverse curves, 100

Involute, 144

Kepler, 107

Kinematic, 2

Kinetic, 2

Kite, 150

Lambert, 108

Linear-function, 185

Line-integral, 195

Lissajous, 36, 37

Logarithm, 80

Logarithmic rate, 80

spiral, 86

Major axis, 31

Matrix, 163

Maximum distance, 30

Maxwell, 209

Mean point, 11

,, velocity, 53

Measure, 49

Mid-centre, 11

Minimum distance, 30

Minor axis, 31

Modulus, 86

Moment, 92

Motion, 1

representation of, 14

uniform, 15

circular, 18

harmonic, 20

elliptic harmonic, 21

compound harmonic, 33

parabolic, 38

elliptic, 107

clockwise and counter-

clockwise, 19

irrotational, 203

logarithmic, 78

steady, 199

Motor, 125

Newton, 45, 70, 98, 107

Oblique projection, 25

Ordinate, 29

Origin, 13

Orthogonal projection, 24

Osculating plane, 76

Parabola, 34

Parabolic motion, 38, 103

118

INDEX.

Particle, 2

Pedal curves, 101

Period, 18, 21

Perversion, 119

Phase, 18, 21

Pitch, 125

Pitch-conic, 129

Planes, coordinate, 14

Plucker, 73

Position, vector, 14

Positions, curve of, 15

Projection, 24

of velocity, 60

Quantity, vector, 13

fluent, 62

scalar, 62

Quasi-harmonic motion, 89

Bate, 48

Eeciprocal curves, 101

Eectilinear motion, 16

Eepulsion, 3

Eesultant, 5

Eigid, 1

Eoberts, S., 148, 150

Eotation, 1

velocity, 123

Eotor, 123

Eoulette, 140

Scalar, quantity, 62

,, product, 95

Scrolls, 137

Shear, 167

Shell, double, 218

Sines, curve of, 22

Sink, 214

Solids, 3

Sources, 214

Spheroid, 175

Spin, 123, 193

,, -system, 200

tube of, 200

Squirt, 214

Static, 2

Steps of translation, 3

,, product of, 94

Steps, displacement of, 161

Strain, 2, 158160, 172

,, -function, 163

principal axes and elonga-

tion, 176

pure, 176

-flux, homogeneous, 193

Stress, 3

Surface-integral, 201

Sylvester, 107, 150

ff, convergence, divergence of, 209

Tangent to ellipse, 41

to parabola, 43

to harmonic curve, 43

,, Newton's criterion, 45

definition, 47

Thomson and Tait, 23, 35

Tidal clock, 24

Three-bar motion, 146

Translation, 1

Transverse axis, 31

Twist, 2

,, magnitude of, 125, 193

Uniform motion, 15, 48

Vector, 13

product, 95

Velocity, 48, 50

variable, 51

mean, 53

to find, 51, 53

definition, 58

composition of, 59

projection of, 60

moment of, 98

lengthwise, 132

-potential, 203

,, maximum, 210

minimum, 210

Vortex, 217

-filament, 217

Wallis, 72

Wave, 23

Whirls, 214

Wry (shear, strain), 170

INDEX.

VOL. II.

Acceleration by place, 74

Archimedes, 14

Aristotle, 73

Attraction, 63, 64, 66

Battery, 66

Cavendish, 63

Charge-accelerations, 65

Charges, like, unlike, 64

Composition, law of, 70

Conductor, 65

Confocal, 39

Conjugate lines, 18, 25

,, points, 25

,, planes, 31

motions, 80, 83

Core, 19, 44

Cross-centre, 7

Culmann, 18 (note)

Density, 1, 59

Ellipsoid of gyration, 39

momental, 54, 75, 82

Fluid friction, 68

Focal ellipse, 40

hyperbola, 40

Force, 67, 69, 70, 73

internal, external, 71

Fourier's theorem, 88

Friction, 68

Gram, 59

Gravity, 63

Insulators, 65

Kirchhoff, 76

Loaded area, 15, 16

body, 31

Magnet, 65

Mass, 2, 58, 70

,, resultant, 2

,, vector, 2

Mass-centre, 3, 11 14

triangle and tetrahe-

dron, 5

,, trapezium, 6

,, ,, solids, 8

Mid-centre, 7

Middle quarter, 20, 21, 44

fifth, 44

Moment, 3, 77

,, circular arc, 10

first, 15

,, second, 15, 31

mixed, 18, 31

Momentum, 48, 51, 60

moment of, 49, 53

Motion, laws of, 70

Neutral axis, 15

Newton, 63

120

Null-conic, 24

-quadric, 36

Opposite points, 27

Poinsot, 75

Polar, 25, 28

planes, 35

Pole, 15, 16,, 25, 28, 31, 35

(north on the magnet), 65

Principal axes, 29

screws, 56

Eeciprocal, 44

Reciprocity, law of, 63, 70, 72

Eepulsion, 64, 66

Eesistance, 68

INDEX.

Botor, 48

Self-conjugate, 29

quadric, 34

tetrahedron, 36

Stress, 59

across section, 62

Swing-radius, 17

,, -conic, 24

-ellipsoid, 34

Sylvester, 10

Thomson and Tait, 73

Vector, 48, 49

Velocity, instantaneous, 92

,, arrival, departure, 93

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