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Название: Principles of Geometry (vol. IV)
Автор: Baker H.F.
Аннотация:
The present volume, the first written and the most revised, of the book, for which indeed, mostly, the earlier volumes were undertaken, still bears many marks of the difficulty of compressing the matter into brief compass. But the writer hopes that it may seem to the reader as remarkable as it does to him, that it should be possible to comprehend under one point of view, and that so simple, the introduction to nearly all the surfaces ordinarily studied in the geometry of three dimensions, as well as the usual line geometry. Chapters v, vi, vii seek to make clear that this is so. To these the earlier chapters are auxiliary. But Chapters ii and iv have been introduced as much for their own interest as for their illustrative value; the results obtained in these two chapters are not required in the subsequent pages. It is hoped that the Table of Contents, and the Index, may make it easy to use the volume. It will of course be understood that the volume is throughout intended to be introductory and illustrative; hardly anywhere is it complete.
Anchor Ring, or Tore, and Dupin’s Cyclide193199 Angle between two circles, in projection from a quadric7 Angle between two circles, unaltered by inversion14 Apolar relation for point of contact of Hart section76 Apolar, or conjugate, linear complexes42 Apollonius’s problem for circles66 Associated lines, all met by planes120 Associated lines, associated system of five planes118 Associated lines, five, in fourfold space114 Associated lines, planes meeting these through any point122 Bauer, on a quartic surface20 Bennett, Hart section, equations for86 Bennett, theorem on circles through threes of four points17 Bertini, on Veronese’s surface54 Bitangents of Kummer’s surface, from lines of Segre’s cubic locus156 Bobillier, theorem for pedal circles22 Burkhardt, the group of the lines of a cubic surface105 Burnside a figure in fourfold space105 Burnside, invariants of a linear group210 Burnside, the group of the lines of a cubic surface105 Caporali, tetrahedral complexes containing a quadratic congruence242 Cardioid99 cartesian99 Castelnuovo and Kronecker, surface with double infinity of reducible plane sections55 Caustic, or focal, surface of a quadratic congruence225 Cayley, algebraic solution of Malfatti’s problem68 Cayley, on Veronese’s surface54 Cayley, representation of a line by a point of fivefold space40 Centre of a circle, in projection from a quadric7 Chords common to two curves51 Circle and two inverse points changed into same by inversion14 Circle of similitude, obtained by projection15 Circle represented by a line of fourfold space39 Circle, Hart and Feuerbach circlesChapter II 65 Circle, obtained by projection of plane section of a quadric1 Circle, three circles with three concurrent common tangents67 Circles in a plane, chain of theorems for29 Circles meeting given circles at equal, or at given angles67 Circles of similitude of the pairs of four circles15 Clifford, a theorem for circles13164105 COMPLEXsee also "Focal System" Complex, linear, determined by planes in fourfold space34 Complex, linear, point and polar plane represented in fivefold space48 Complex, linear, represented in fivefold space42 Complex, linear, six conjugate, constructed from six points139 Complex, linear, six conjugate, from lines of Segre’s cubic locus155 Complex, linear, six conjugate, poles of a plane lie on a conic204 Complex, lines common to three, or to four linear complexes4344 Complex, quadratic, condition for singular point214 Complex, quadratic, rationality of234 Complex, quadratic, represented in fivefold space213 Complex, six linear complexes mutually conjugate42 Complex, tetrahedral, determined by planes in fourfold space32 Complex, tetrahedral, in fivefold space239 Confocal Cyclides, cutting at right angles181 Confocal Cyclides, equation for183 Congruence of lines, for chords of a curve50 Congruence of lines, its order and class49 Congruence of lines, number of lines common to two51 Congruence of lines, quadratic, and Moebius tetrads241 Congruence of lines, quadratic, caustic or focal system of225 Congruence of lines, quadratic, in tangent solid of Segre’s quartic locus137 Congruence of lines, quadratic, lies in forty tetrahedral complexes242 Congruence of lines, quadratic, singular points and planes240 Congruence of lines, six congruences associated with a Kummer surface219223 Congruence of lines, six quadratic, equations for common singular points226 Conic, conics touching fives of six arbitrary lines1 Conic, in an arbitrary plane, determined by the joins of five points11 Conic, triangularly circumscribed to six conics35 Conic, two conics in fivefold space which are mutually conjugate43 Conics lying on quartic surface in fourfold space171 Conics through a point of Segre’s quartic locus127 Conjugate, or apolar, linear complexes42 Cremona, triangularly circumscribed conic and line coordinates6 Cubic curve, chords of, represented in fivefold space52 Cubic curve, conic touching seven lines associated with4 Cubic curve, five arbitrary points on11 Cubic curve, in plane, through two points, inverse of quartic with two double points97 Cubic curve, tetrads of points determined by two points29 Cubic curve, three tetrads of points on28 Cubic surface, by projection of quartic surface in fourfold space173 Cubic surface, cubic envelope, by projection of Segre’s figure149 Cubic surface, group of lines of105 Cubic surface, with four double points, theorem for28 Curvature, differential equation for190 Curvature, Dupin’s theorem188 Curvature, general theorems for189 Curvature, lines of, upon a Cyclide186 Curve, in n-fold space, intersection of (n-1) quadrics185 Cyclide, a particular surface, Kummer’s equation for197 Cyclide, angle between enveloping spheres of different systems179 Cyclide, arising by projection from fourfold space174 Cyclide, as locus of limiting points of fixed sphere and variable tangent plane of a quadric176 Cyclide, by inversion of a quadric192 Cyclide, Dupin’s Cyclide190193199 Cyclide, Dupin’s, points expressed by two parameters202 Cyclide, Dupin’s, treated tangentially201 Cyclide, enveloped by spheres175 Cyclide, equations for confocal cyclides183 Cyclide, four focal conics of principal quadric surfaces178 Cyclide, four points of double curve where tangent planes coincide180 Cyclide, general, irrational form of equation194 Cyclide, is inverted into itself from five centres179 Cyclide, lines of curvature correspond to inflexional curves on Kummer’s surface231 Cyclide, lines of curvature upon186 Cyclide, normal of, by projection from fourfold space184 Cyclide, spheres having stationary contact with187 Cyclide, tangent planes of, at a point of double conic177 Cyclide, three confocal through given point181 Cyclide, with cuspidal conic192 Cyclide, with four double points, equations for196 Cyclide, with planes of symmetry193 Cyclide, with two double points, equations for195 Cyclide, with two double points, inversion of194 Darboux, a generalisation of Dupin’s Cyclide198 Darboux, congruences of lines225 Darboux, generalisation of lines of curvature and of geodesic lines190 Darboux, inversion of Dupin’s Cyclide200 Darboux, on plane quartic curves, and Cyclides172 Darboux, on the existence of space of four dimensions200 Darboux, surface containing a doubly infinite system of conics54 Del Pezzo, on Veronese’s surface5455 Dickson, the group of the lines of a cubic surface105 Director conics, for plane quartic curve with two double points, are confocals91 Double four, of lines on quartic surface in fourfold space, twenty such170 Double four, of planes238 Double tangents of Kummer’s surface225 Double-six theorem, proved in fivefold space63 Dupinsee "Cyclide" Dupin, theorem on lines of curvature138 Dupin’s Cyclide190193199201202 Elte, on a six-dimensional poly tope105 Feuerbach’s theorem65 Feuerbach’s theorem, particular case of Hart’s theorem78 Figure, arising from four arbitrary planes117 Figure, of fifteen lines and points in fourfold space113 Figure, of twentyseven points in fourfold space104 Focal conics of a confocal system of quadrics92 Focal conics, descriptive form of theorem of intervals103 Focal systemsee "Complexlinear" Focal system, arising in tangent solid of Segre’s quartic locus135 Focal system, point and polar plane represented in fivefold space48 Focal system, six conjugate, scheme for133 Foci of plane quartic curve with two double points92 Focus, and focal line, of a singular plane in fivefold space219 Fontene, theorem for pedal circles22 Fresnel’s wave surface217 Gergonne, solution of Apollonius’s problem66 Goepel, equation of Kummers surface216 Grace, generalisation of a theorem for circles8
Grace, on systems of spheres105112 Grace, theorem for pedal circles23 Grace, theorem for six lines with a common transversal60 Grace, theorem for spheres touching planes58 Grace, Wallace’s theorem for space of even dimensions59 Group, linear, in four homogeneous variables208 Harmonic inversion, with fundamental line and plane119 Harmonic inversion, with fundamental point and solid117 Harmonic inversion, with two fundamental planes118 Harmonically conjugate quadric72 Hart circle for spherical triangle77 Hart circle of three circles, defined by angles of intersection88 Hart circle of three circles, orthogonal to their orthogonal circle71 Hart section of a quadric, sufficient condition is necessary82 Hart, apolar relation for point of contact76 Hart, Feuerbach circle as particular case78 Hart, general case7275 Hart, proof of Steiner’s solution of Malfatti’s problem68 Hart, radius of Hart section of a quadric81 Hart, theorem for circles65 Haskell, generalisation of a theorem for circles8 Hudson, on Kummer’s quartic surface208227233 Humbert, a curve on the Weddle surface234 Inflexional lines of a surface186 Inflexional lines of a surface, correspond to lines of curvature on Cyclide231 Inflexional lines of a surface, differential equation for190 Inflexional lines of a surface, of Kummer’s surface222230 Invariants, representative, for an involution in threefold space209 inversionsee also "Harmonic inversion" Inversion in a plane, deduced from projection in space12 Inversion of Cyclide179 Inversion of Cyclide with two double points194 Inversion of Dupin’s Cyclide, to quadric cone of revolution201 Inversion of plane quartic curve with two double points96 Involution of sets of sixteen points in threefold space208 Jessop, theorem for angles of intersection of tangent circles of a quartic curve99 Jordan, the group of the lines of a cubic surface105 Klein, inflexional lines of Kummer’s surface233 Klein, Lie’s correspondence between sphere and lines55 Klein, rationality of quadratic complex235 Klein, representation of a line by a point of fivefold space40 Klein, six conjugate linear complexes202208 Kronecker and Castelnuovo, surface with double infinity of reducible plane sections55 Kuehne, generalisation of Wallace’s theorem59 Kummer, equation for a particular Cyclide197 Kummer’s quartic surface, and Weddle’s surface, from four quadrics in threefold space160 Kummer’s quartic surface, arising by section of Segre’s quartic locus with tangent solid139 Kummer’s quartic surface, as locus of singular points of a quadratic complex212 Kummer’s quartic surface, bitangents of224 Kummer’s quartic surface, bitangents of, from lines of Segre’s cubic locus156 Kummer’s quartic surface, dual of217 Kummer’s quartic surface, equation expressed as sum of five squares215 Kummer’s quartic surface, equation shewing double points216 Kummer’s quartic surface, from confocal complexes227 Kummer’s quartic surface, from profile of Segre’s cubic locus156 Kummer’s quartic surface, identical with envelope of singular planes222 Kummer’s quartic surface, inflexional curves correspond to lines of curvature on Cyclide231 Kummer’s quartic surface, inflexional lines of222230 Kummer’s quartic surface, intersections with a line of fundamental complex229 Kummer’s quartic surface, obtained by transformation from fourfold space211 Kummer’s quartic surface, range of six points associated with215 Kummer’s quartic surface, related range of points and pencil of planes228 Kummer’s quartic surface, represented in fivefold space218 Kummer’s quartic surface, six congruences associated with219223 Kummer’s quartic surface, tangent lines at a point221 Lachlan, Hart circle defined by angles of intersection88 Larmor, A., circles arising from the intersection of three circles82 Leathem, theorem for four concyclic points of a sphere101 Lexell, theorem for four concyclic points of a sphere101 Lie, applied to Kummer’s surface and Cyclide233 Lie, correspondence of a sphere with two lines55 Limacon99 Line, in threefold space, represented by a point of fivefold space41 Line-cone, quadric, and point-cone, in fourfold space121 Malfatti’s problem68 Mannheim, inversion of Dupin’s Cyclide to Anchor Ring200 Mantel, symmetry of five points, for pedal property25 Maxwell, Dupin’s Cyclide193198 Meridian surface, Pluecker’s142 Miquel, a theorem for circles1 Miquel, generalisation of8 Moebius tetrads and quadratic congruence241 Moebius’s figure of two tetrads and Wallace’s theorem18 Morley, theorem for Feuerbach circle79 Noether, rationality of quadratic complex235 Normal of Cyclide, by projection from fourfold space184 Orr, theorem for four concyclic points of a sphere101 Orthocentric, Absolute conic for which five points are11 Pedal circles orthogonal to a fixed circle23 Pedal property of circumcircle generalised to rational curve26 Pedal, properties in threefold space242527 Picard, surface whose plane sections are rational55 Pluecker’s Meridian surface142 Point-cone, quadric, and line-cone, in fourfold space121 Polar line of a line, in regard to a focal system, or linear complex42 Polar lines, in a focal system, corresponding to the same sphere55 Polar of a line in regard to a quadric, represented in fivefold space49 Polar quadric of point of Segre’s cubic locus156 Polar system, by plane section of joins of five points10 Poncelet, on circles touching three circles66 Quadric in fivefold space, generation of237 Quadric surface, generators of, represented by a conic in fivefold space43 Quadric surface, generators touching curve of intersection with another quadric surface94170220223224 Quadric surface, harmonically conjugate72 Quadric surface, number of common points of two curves on51 Quadric surface, radius of section of, with respect to a point80 Quadric surface, sections of, touching three sections65 Quadric surfaces through five points give rise to Segre’s cubic locus152 Quadric surfaces with ring contact, theorem of intervals103 Quadric surfaces, ten, associated with six conjugate linear complexes206 Quadric, angles between four sections100 Quadric, condition four sections of quadric surface be touched by four planes72 Quadric, each its own polar plane47 Quadric, equation of73 Quadric, equations of69 Quadric, four sections and four tangent sections, mutual relations86 Quadric, in fivefold space, planes lying on45 Quadric, in fivefold space, representing lines41 Quadric, in fourfold space, in regard to which Segre’s figure is self-polar148149 Quadric, in fourfold space, lines of, through a point37 Quadric, sets of four with two common generators207 Quadric, system derived from four general quadrics160 Quadrics in n-fold space, through given points61 Quartic curve, angles of two tangent circles with variable tangent circle99 Quartic curve, as envelope of circles90 Quartic curve, foci of92 Quartic curve, general form of theorem for angles between tangent circles102 Quartic curve, in a plane, with two double points89 Quartic curve, in fourfold space120147 Quartic curve, inversion into itself, or another such curve96 Quartic curve, number of tangents from a double point93 Quartic curve, relation between distances to three concyclic foci94 Quartic curve, with four foci in a line98 Quartic surface, in fourfold space, algebraic treatment162 Quartic surface, in fourfold space, conics upon, five pairs through each point of surface171 Quartic surface, in fourfold space, descriptive theory of lines on167169 Quartic surface, in fourfold space, double-fours of lines on170 Quartic surface, in fourfold space, intersection of two quadrics161 Quartic surface, in fourfold space, lines of, by intersection with a quartic locus189 Quartic surface, in fourfold space, notation for lines170 Quartic surface, in fourfold space, other equations for lines185 Quartic surface, in fourfold space, representation on a plane165 Quartic surface, in fourfold space, self-polar pentad for168 Quartic surface, in fourfold space, sixteen lines of164 Quartic surface, with five double points, from quartic surface with fifteen double points160 Rao, co-residuation on quartic surface in fourfold space172 Related planes, three, variable plane containing corresponding points145 Reye, triply infinite system of quadric surfaces160 Richmond, figure arising from six points in fourfold space116 Richmond, pedal property for general rational curve25
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