Percival Frost.

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/ and Bij^ -^-Cz^^Ax . . 133

CoMCOiD defined ...... 134



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CHAPTER XI.

ON generation by straight links.

Geometrical account of generation of hyperboloida of one ahoet by straight lines 137



XVI CONTENTS.

Page

Two systems of generating lines include all straight lines which lie entirely

on an hyperbcloid ...... 139

Surface generated by a line meeting thi-ee non-intersecting lines . . 139
Conditions that a line drawn through a given point of a conicoid may lie entirely

on the surface ...... 140

Points on an hyperboloid for which the generating lines are perpendicular . 141

Equations of the generating lines of an hyperboloid of one sheet . . 142
Projections of generating lines on the principal planes are tangents to the traces

on those planes . . . . . . .143

Two generating lines of the same system do not intersect . . 143

Generating lines of opposite sj'stems intersect .... 144

Equations of the generating lines of an hyperbolic paraboloid . . 146
Lines of the same system on an hyperbolic paraboloid do not and those of

opposite systems do intersect ..... 147
Projection of generating lines on the principal planes are tangents to the traces

on those planes ...... 147



CHAPTER XII.
SIMILAR SURFACES. PLANE SECTIONS OF CONIOOIDS. CYCLIC SECTIONS.

Similar surfaces defined . . . . . .153

Surfaces similarly situated ..... 153

In what sense hyperboloids of one and two sheets may be similar . . 154

Sections of the same conicoid by parallel planes and sections of similar and
similarly situated conicoids by the same plane are simUar and similarly
situated conies . . . . . .154

Similar and similarly situated conicoids intersect the plane at infinity in the

same conic ....... 156

Nature of a plane section of a conicoid examined by projections . 155

Locus of centres of sections of a central conicoid by parallel planes . .156

Position of the cutting plane when the section is a point-ellipse or line-
hyperbola ...... 157

Locus of centres of sections of a paraboloid by parallel planes . . 157

Position of the plane for a point-ellipse or line-hyperbola . . 168

Magnitude and direction of axes of a plane central section of a conicoid . 168

Direction of the plane section whose axes arc of given magnitude . 159

Nature of a central plane section of a central conicoid . . . 160

Nature of any plane section of a central conicoid . . . 161

Angle between the real or imaginary asymptotes of a plane section . . 161

Area of an elliptic non-central section of a conicoid . . . 1G2

Volume of asymptotic cone cut o£E by a plane touching an hyperboloid of two

sheets is constant . . . . . .163

Magnitude and direction of axes of any plane section of a paraboloid . 163

Nature of any plane section of a paraboloid .... 164

Cyclic sections of a conicoid ..... 165

Generation of a conicoid by the motion of a variable circle . . .166

Umbilics of a conicoid defined ..... 166

Any two cyclic sections of opposite systems lie on one spliere . . 167

Geometrical investigation of the direction of cyclic sections . . 168



CONTENTS. xvii

CHAPTER XIII.

TANGENTS. CONICAL AND CYLINDRICAL ENVELOPES. NORMALS.
CONJUGATE DIAMETERS.

Pa(?c

Tangent LINE to a given central conicoid at a given point . .171

Tangent plane . . 172

Equation of a tangent plane to a conicoid drawn in a given direction . 172

Equation of a tangent plane of a cone .... 173

Equations of an asymptote to a central conicoid . . . ,173

Nature of the intersection of a central conicoid with a tangent plane : 173

Axes of the section by a plane through the centre of a conicoid . .174

Locus of points of contact of tangent planes passing through a given point 174

Polar plane of a point and pole of a plane with respect to a given conicoid

defined •••.... 174

Fundamental property of poles and polar planes . . . 175

Conical envelope of a central conicoid . . . .175

Cylindrical envelope of a central conicoid . , .176

Corresponding results for a non-central conicoid . . . .177

Equations of the normal to a central conicoid at a given point . 177

Six normals to a central conicoid through a given point . , . 177

Locus of a point from which three normals to a conicoid have their feet in a

given plane section . . . . , .178

Diametral planes of a conicoid defined . . . .180

Diametral plane of a central conicoid for a given system of parallel chords . 180

Conjcgate diameters and conjugate planes . . .181

Relations between the coordinates of the extremities of a system of conjugate

diameters ....... 182

The sum of the squares of the projections of three conjugate diameters on any

line or any plane is constant ..... 182

Relations between the lengths of three conjugate diameters and the angles

between them . . . . . .182

Diametral plane of a paraboloid for given parallel chords , . . 185

Harmonic definition of a polar plane .... 186



CHAPTER XIV.

CONFOCAL CONICOIDS. FOCAL CONICS. BIFOCAL CHORDS. CORRESPONDING POINTS.

Confocal CONICOIDS defined . . . . .192

Two modes of representing a system of confocal conicoids . . 192

Through every point pass three conicoids confocal with a given conicoid . 193

If two parallel planes touch two confocals, the difference of the squares of the

perpendiculars on the planes from the common centre will be constant 194

Locus of poles of a given plane with respect to a system of confocala . .194

Elliptic COORDINATES explained ..... 195

When three confocals pass through a point, each of the normals at that point

is' perpendicular to the Other two .■ . , , .196

Axes of a central section of a conicoid expressed by means of two confocals

to the conicoid . . . . . .197

If P be a point in the intersection of two confocala A, B, the diameter of A

paraUel to the normal at P to jB is constant . . . .198

Lengths of perpendiculars on tangent planes at a point common to three confocala 198

C



XVlll CONTENTS.

Page

pd constant for every point in the intersection of two confocals . . 199

Principal axes of a cone enveloping a given central conicoid , . 199
Equation of the enveloping cone referred to normals to the three confocals

through the vertex as axea ..... 200

Equation of the enveloped conicoid referred to three normals , . 201

Properties of a line touching two confocals to a given conicoid . . 201
Two conicoids can be constructed which are confocal with a given conicoid and

which touch a given straight Une ..... 202

Length of a chord of a conicoid touching two conicoids confocal with it . 202

Two confocals appear to cut at right angles .... 203

Gilbert's method of soh^ng problems in confocals . . . 203

Focal conics are particular confocals ..... 205

Locus of vertices of right cones envelopmg a conicoid . . . 207
Bifocal chords defined . . . . . .208

Properties of bifocal chords ..... 208

Construction for the four bifocal chords through a point in an ellipsoid . 209

Corresponding POINTS of two ellipsoids defined . . . 211
Relation between two points on one eUipsoid and the corresponding points

on another . . . . . . .211

Correspondence of extremities of conjugate diameters of two ellipsoids . 212
Correspondence of curves of intersection of a series of confocal ellipsoids with

a fixed confocal hyperboloid ..... 212

Umbihcs of confocal ellipsoids correspond .... 213

Plane^cui-ve con-esponding to the curve of intersection of confocal conicoids . 213



CHAPTER XV.

MODULAE AND UMBILICAL GENERATION OP CONICOIDS.
PROPERTI^ OF CONES AND SPHER0-C0NIC3.

Mac Ccllagh and Salmon's modular and umbihcal generation of conicoids
Locus of a point moving according to the modular method

umbilical

Chaslbs' definition of confocal conicoids
Focal and dirigent conics for central conicoids
Focal and dirigent conics reciprocals of each other



Online LibraryPercival FrostSolid geometry → online text (page 1 of 29)