CONTENTS.
CHAPTER I.
SECTION I. - THE POINT.
RULE OF SIGNS,
1
Geometric sum or resultant,
2
Projections,
3
SECTION II. - CARTESIAN CO-ORDINATES.
Définitions,
4, 5
Distance between two points,
6
Condition that three points should be collinear,
8
Area of triangle or polygon in terms of the co-ordinates of vertices,
10
Co-ordinates of a point dividing in a given ratio the join of two given points,
12
Co-ordinates of harmonic and isotomic conjugates,
13
Theory of the mean centre,
14, 15
SECTION III.
Polar co-ordinates,
17
Biradial and biangular co-ordinates,
18
Transformation of co-ordinates,
19-22
SECTION IV. - COMPLEX VARIABLES.
Definition of, and mode of representation,
24
Sum or difference of two complex variables,
25
Product and quotient of ditto,
25
Examples on complex variables,
26
Miscellaneous Exercises,
27
CHAPTER II.
THE RIGHT LINE.
SECTION I. - CARTESIAN CO-ORDINATES.
To represent a right line by an equation,
30, 31
Standard form of equation,
32
Line parallel to one of the axes,
33
Equation of line defined,
33
Comparison of different forms of equation,
35
To find the angle between two lines,
36, 37
Power of point with respect to line or curve,
37
Length of perpendicular from a given point on a given line,
37
Equation of line through the intersection of two given lines,
39
Equation of line passing through two given points,
40
Co-ordinates of point of intersection of two given lines,
42
Equation of line through a given point and making a given angle with a given line,
44
Equation of line dividing the angle between two given lines into parts whose sines have a given ratio,
46
Condition that three lines should be concurrent,
48
Condition that general equation of the second degree should be the product of the equations of two lines,
51
If the general equation of the second degree represents two lines, to find the co-ordinates of their point of intersection,
52
Equation of bisectors of angles made by a given line-pair,
53, 54
Theory of anharmonic ratio,
55, 56
Harmonic system of points, harmonic conjugates,
56
Anharmonic ratio of four rays of a pencil,
57
Anharmonic ratio of four lines whose equations are given,
59
Harmonic pencil
59
SECTION II. - SYSTEMS OF THREE CO-ORDINATES.
Trilinear or Normal co-ordinates defined,
61
Isogonal conjugates; symmedian point and lines,
63
Brocard angle and Brocard points,
64
Barycentric or areal co-ordinates defined,
64
Relation between the normal and barycentric co-ordinates of a point,
65
Isotomic conjugates,
65
Trilinear equation of the join of two given points,
66
Trilinear poles and polars: Cotes' Theorem,
68
Complete quadrilateral or quadrangle,
69
Standard quadrilateral or quadrangle,
70
Relation (identical) between the equation of four lines, no three of which are concurrent,
70
Harmonic properties of complete quadrilateral,
70
Perspective; triangles in, axis and centre of,
72
Length of perpendicular from a given point on a given line,
73
Angle between two given lines,
73, 74
Line at infinity,
74
Cyclic points-Isotropic lines,
75
Conditions for parallelism and perpendicularity when general equation in trilinear co-ordinates represents two lines,
77
Condition that la+ mb+ ng= 0 may be antiparellel to g = 0,
77
Distance between two points,
78
Area of triangle the co-ordinates of whose vertices are given,
79
Area of triangle formed by three given lines,
80, 81
Complementary and anti-complementary points and figures,
81
Supplementary points,
82
Triangles in multiple perspective,
82
Isobaryc group of points,
85
Comparison of points and line co-ordinates, equation of a point,
86
The absolute co-ordinates of a line,
88
Equation of the cyclic points,
88
Miscellaneous Exercises,
90
CHAPTER II.
THE CIRCLE.
SECTION I. - CARTESIAN CO-ORDINATES.
To find the equation of a cirle,
96
Geometrical representation of power of point with respect to circle,
98
Equation of circle whose diameter is the intercept which a given circle makes on a given line,
100
Equation of tangent to a circle,
101
Equation of pair of tangents from given point to circle,
103
Equations of chords of contact of common tangents to two circles,
103
Pole and polar with respect to a circle,
105
Inverse points with respect to a circle,
105
Angle of intersection of two circles or curves
107
Mutual power of two circles,
107
Circle cutting three given circles at given angles,
108
Circle cutting three given circles orthogonally,
109
Circle touching three given circles,
109
Circle through three given points,
110, 111
Frobenius's Theorem,
111
Condition that four circles should cut a fifth orthogonally or be tangential to it
112
Coaxal system circles,
114
Radical axis and limiting points of coaxal system,
115
Radical centre,
117
Centre of similitude,
118
Circle of similitude,
119
SECTION II. - A SYSTEM OF TANGENTIAL CIRCLES.
Equations of circles in pairs touching three circles,
120
To investigate the condition that any number of circles should have one common tangential circle,
122
Tangent to nine-points circle at point of contact with incircle,
126
SECTION III. - TRILINEAR CO-ORDINATES.
Circumconic of triangle of reference,
126
Coates' Theorem,
127
Circumcircle of triangle of reference,
127, 128
Circumcircle of polygon of any number of sides,
129
Tangents to circumconic at angular points,
129
Chord of, and tangent to, circumcircle,
130
Incircle of triangle of reference,
131, 132
Sir Andrew Hart's method of finding equation of incircle,
132
Join of two points on incircle,
133
Tangent at any point on incircle,
134
Conditions that general equation should represent a circle,
135 - 137
Equation of circle thourgh three given points,
136
Equation of pedal circle of a given point,
136
Simson's line,
136
SECTION IV. - TANGENTIAL EQUATIONS.
Tangential equation of circumcircle of triangle,
138
Tangential equation of circumcircle of polygon,
139
Tangential equation of incircle of triangle,
Tangential equation of incircle n-sided polygon,
141
Tangential equation of circle whose centre and radius are given,
143
Nine-points circle, equation of,
144
Pascal's Theorem,
145
Miscellaneous Exercices on the circle,
141
CHAPTER IV.
THE GENERAL EQUATION OF THE SECOND DEGREE.
CARTESIAN CO-ORDINATES.
Contracted form of the equation,
151
Change of origin,
152
Intersection of line and conic, discussion of equation,
153
Co-ordinates of centre of conic,
154
Distinction between hyperbola, parabola and ellipse,
154, 165
Reduction of equation to centre as origin,
155
Line of centres,
155
Locus of middle points of system of parallel chords, diameters,
155
Conjugate diameters of central conics,
157
Axes; reduction of general equation to normal form,
158
Invariants of, and equation giving squares of semiaxes of conic represented by, the general equation,
159
Equation of new axes when referred to old,
159
Reduction of general equation to parabola,
160
Tangent to, and tangential equation of, general conic,
161
Ratio in which join of two given points in cut by conic,
162
Harmonic properties of polars,
162
Polar of a given point; tangent,
163
Condition that join of two points may be cut in given anharmonic ratio by the conic,
163
Equation of pair of tangents from an external point,
163
Orthoptic circle of conic,
164
Classification of conics,
165
Asymptotes defined and their equation found,
166
Asymptotes angle between,
166
Asymptotes hyperbola referred to as axes,
167
Newton's Theorem,
Locus of centre of conic through four points,
171
Exercices on general equation,
170
CHAPTER V.
THE PARABOLA.
The parabola; its axis, directrix, focus, vertex,
173
Latus Rectum,
160, 174
Intrinsic angle of point on curve,
175, 177
Equation of tangent to parabola,
176, 189
Subtangent bisected at vertex,
177
Pedal of, with respect to focus,
177
Locus of middle points of system of parallel chords,
179
Diameter through intersection of two tangents bisects chord of contact,
180
Equation of, referred to any diameter and tangent at its extremity
182
Isoptic curve of parabola,
184
Normal defined and equation found,
184, 190
Subnormal constant in parabola,
184
Joachimsthal's circle for,
185
Circle of curvature, radius and centre of,
185, 186
Circle of curvature, equation of,
186
Locus of centre of curvature, evolute,
187
Poral equation of, the focus being pole,
189
Length of line drawn from a given point in a given direction to meet the parabola,
191
Relation between the perpendiculars from the angular points of a circumtriangle on any tangent,
193
Exercices on the parabola,
195
CHAPTER VI.
THE ELLIPSE.
Focus; directrix; eccentricity,
201
Standard from of equation; centre; latus-rectum,
203
Method of generating ellipse. Pohlke, Boscovich, Hamilton,
205
Eccentric angle; co-ordinates of a point on curve in terms of,
206
The auxiliary circle,
206
Ellipse, the orthogonal projection of a circle,
206
Locus of middle points of a system of parallel chords,
208
Equation of tangent,
208, 237
Conjugate diameters; formulae of Chasles; theorem of Apollonius,
209, 210
Given any two conjugate diameters, to find the axis; construction of Mannheim,
210
Equation of, referred to a pair of conjugate diameters,
211
Supplemental chords,
213
Schooten's method of describing ellipse,
213
Equation of normal,
214, 237
Normals through a given point; Apollonian hyperbola,
216, 217
Evolute of ellipse
216
Radius of curvature at any point of,
216
Equation of the four normals from given point,
217
Joachimsthal's circle for,
218
Laguerre's Theorem,
219
Purser's Parabola,
220
Length of perpendicular from focus on tangent,
220
Pedal of, with respect to either focus,
221
New method of drawing tangents to curve,
221
Chords passing through a focus,
222
Angle between two tangents,
224
Locus of intersection of rectangular tangents:
224
Angle between two tangents expressed in terms of focal vectors to point of intersection,
225
Property of three confocal ellipses,
227
Fregier's Theorem,
227
Locus of pole of tangent with respect to circle whose centre is one of the foci,
228
Reciprocal polar of a curve defined,
228
Reciprocal polar of a circle with respect to a circle,
228
Reciprocal polars of a system of confocal ellipses or coaxal circles,
229
Rectangle contained by segments of any chord passing through a fixed point is to square of parallel semidiameter in a constant ratio,
230
Any two tangents are proportional to parallel semidiameters,
231
To find the major axis of an ellipse confocal with a given one, and passing through a given point,
232
Elliptic co-ordinates defined,
233
Polar equation of, a focus being pole,
236
Sum of reciprocals of segments of focal chord,
237
Exercises on the ellipse,
239
CHAPTER VII.
THE HYPERBOLA.
Focus; directrix; eccentricity,
250
Standard form of equation; centre, transverse and conjugate axis,
251
Latus-rectum; equilateral hyperbola,
252
Co-ordinates of a point on, expressed in terms of a single variable,
254
Locus of middle points of a system of parallel chords,
255
Equation of tangent,
256, 261, 273
Conjugate hyperbola,
257
Conjugate hyperbola equation,
258
Equation of, referred to a pair of conjugate diameters,
259
Equation of normal,
262, 273
Joachimsthal's circle for,
264
Lengths of perpendiculars from foci on tangent,
265
Pedal of, with respect to focus,
266
Reciprocal of, with respect to focus,
266
Polar equation of, the centre being pole,
267
Eccentricity given by secant of half angle between asymptotes,
268
Equation of, referred to asymptotes,
268
Co-ordinates of centre and radius of curvature at any point on curve,
271
Polar equation of, the focus being pole,
272
Area of equilateral hyperbola between an asymptote and two ordinates,
273
Hyperbolic functions defined,
275, 276
Co-ordinates of a point on, expressed in terms of a single variable,
276
Locus of pole of chord subtending a right angle at a fixed point,
277
Exercices on hyperbola,
279
CHAPTER VIII.
MISCELLANEOUS INVESTIGATIONS.
SECTION I. - FIGURES INVERSELY SIMILAR.
Double point or centre of similitude; double lines,
285
Triangles inversely similar are orthologique,
288
SECTION II. - PENCILS INVERSELY EQUAL.
Double directions of two pencils inversely equal,
288
Generation of the equilateral hyperbola,
289
Locus of centre of circumequilateral hyperbola of a triangle,
290
Orthocentre lies on circumequilateral hyperbola of a triangle,
290
SECTION III. - TWIN POINTS.
Twin points defined,
292
To construct the twin of a given point,
292
Isogonal conjugates of inverse points with respect to circumcircle are twin points.
293
Twin points are at the extremities of a diameter of circumequilateral hyperbola,
293
Locus of middle point of join of twin points,
294
Barycentric co-ordinates of twin points,
294
SECTION IV. - TRIANGLES DERIVED FROM SAME TRIANGLE.
Pedal triangle of a point: sides of same,
296
Pedal triangle area of,
297
Antipedal triangles,
297
Harmonic transformation of a triangle,
298
Ditto, sides and area of,
299
Locus of points whose pedal triangles have a constant Brocard angle,
300
SECTION V. - TRIPOLAR CO-ORDINATES.
Tripolar co-ordinates defined; tripolar pair,
301
To construct a point, being given the mutual ratios of its tripolar co-ordinates,
301
Pedal triangles of a tripolar pair are inversely similar,
302
Isodynamic points (Neuberg),
303
Relation between the tripolar and normal co-ordinates of a point,
303
Lucas's Theorem,
304
Equation of join of two points; of circle through three points; area of triangle formed by three points; &c., &c., in tripolar co-ordinates,
305, 306
CHAPTER IX.
SPECIAL RELATIONS OF CONIC SECTIONS.
S - kS' = O represents a curve passing through all the points of intersection of the curves S = 0 and S' = 0,
307
Special cases-
1°. General equation of a conic passing through four fixed points on a given conic,
307
2°. General equation of conics having double contact,
307
3°. General equation of conics touching two lines,
308
All circles pass through the cyclic points,
308
Every parabola touches the line at infinity,
308
Contact of different orders,
309, 310
Osculating circle,
309, 310
Parabola having contact of third order,
311
Focus defined as an infinitely small circle having imaginary double contact,
311
All confocal conics are inscribed in the same imaginary quadrilateral,
311
Antifoci,
311
Tangential equation of all conics confocal with a given one,
312
Construction of circle of curvature at any point of central conic,
312
Construction of chord of osculation at any point,
313
Four chords of osculation can be drawn through any point,
314
Through any point on a conic can be described three circles to osculate the conic elsewhere,
315
Steiner's Circle,
315
Six osculating circles of a given conic can be described to cut a given circle orthogonally, or to pass through a given point,
316
Malet's Theorem, viz. the centres of the six osculating circles which pass through a given point lie on a conic,
317
Four-pointic contact or hyperosculation,
318
Equation of a conic having double contact with two given conics,
318
Propeties of common chords of two conics having double contact with a third, or of three with a fourth,
319
Diagonals of quadrilaterals, inscribed in and circumscribed to a conic, form a harmonic pencil,
319
Brianchon's Theorem,
319
Twelve points of intersection of three conics, each of which has double contact with a fourth, lie six-by-six on four conics,
320
Through every point of a conic can be described a parabola to hyperosculate it at that point,
321
Extension of this theorem,
321
Equation representing foci of general conic,
322
Graves' Theorem,
322
Proprerties of arcs and tangents of confocal conics,
323
M Cullagh's, Chasles', and Fagnani's Theorem,
324
If a polygon circumscribe a conic, and if all the summits but one move on confocal conics, the locus of that summit will be a confocal conic,
325
Similar conics,
326
Homothetic figures defined,
326
Conditions that two conics should be homothetic,
326
Conditions of being similar, but not homothetic,
327
Properties of homothetic, and of similar conics,
327
Pascal's Theorem and line,
328
Steiner's Theorem,
329
M'Cay's Extension of Feuerbach's Theorem,
329
Equation giving squares of semiaxes of general conic,
330
Area of ellipse given by general equation,
331
Miscellaneous Exercises,
330
CHAPTER X.
THE GENERAL EQUATION - TRILINEAR CO-ORDINATES.
Aronhold's notation,
333
Some equations expressed in,
333, 334
Condition that ax2 = 0 should represent a line-pair,
334
Co-ordinates of double points,
334
Co-ordinates of pole of line with respect to conic,
335
Tangential equation of conic,
335
Condition that pole of lx = 0 should lie on mx = 0,
336
Equation of point-pair in which a line intersects a conic,
336
Geometrical signification of the vanishing of a coefficient in the general trilinear equation,
337
Equation of conic having triangle of reference as autopolar triangle,
338
Equation of conic referred to focus and directrix,
338
Conditions that general trilinear equation should represent an ellipse, parabola, or hyperbola,
340
Orthoptic circle of a conic referred to its autopolar triangle,
341
Discussion of the equation ab = g2, and properties of curve it represents,
342
Anharmonic properties of four points on a conic,
343
Tangential equation of ab = g2,
344
Equation of circle or curvature at any point on ab = k2g 2,
344
Theory of envelopes,
346
Envelope of a system of confocal conics
347
Condition that join of points in which a given line cuts a given conic, subtends a right angle at origin,
348
CHAPTER XI.
THEORY OF PROJECTION.
Base line; infinite line; projection of a point,
349
Projection of lines,
350
Anharmonic ratio of pencil unaltered by projection; projective properties,
351
Parallel are projected into concurrent lines,
351
Curve of any degree projected into curve of same degree,
351
Projection of concentric circles,
351
Projection of coaxal circles,
352
Any conic can be projected into a circle whose centre is the projection of any point in plane of conic,
352
The pencil formed by the legs of a given angle, and the isotropic lines of its vertex has a given anharmonic ratio,
353
Projection of focal properties,
356
Projection of locus described by vertex of constant angle,
357
Orthogonal projection; axis and modulus of,
358
Figures orthogonally related,
358
Conic of any species projected into conic of same species,
359
Lhuilier's problem, viz., to project a given triangle into another similar to a third,
359
Two triangles orthogonally related are orthologique,
362
Steiner ellipse of triangle,
362
SECTIONS OF A CONE.
Cone of second degree; base edge, vertex, axis,
363
Cone right and oblique,
363
Sections made by parallel planes,
363
Antiparallel section,
363
Sections which are parabolas, ellipses, hyperbolas,
364
Foci of plane section of right cone,
365
Directrices of section,
366
CHAPTER XII.
THEORY OF HOMOGRAPHIC DIVISION.
Condition that four points form a harmonic system,
368
Condition that three point pairs have a common pair of harmonic conjugates,
369
Condition that two line pairs form a harmonic pencil,
369
Locus of point whence tangents to two given conics form a harmonic pencil,
370
Point and line harmonic conics of two given conics,
371
Projective rows defined,
371
1 to 1 correspondence defined,
372
Projective rows have a 1 to 1 correspondence and are homographic,
372
Points which correspond to infinity,
373
Similar rows,
373
Projective pencils defined,
374
In any two projective pencils the anharmonic ratio of any four rays is the same as that of their homologous rays,
375
Maclaurin's and Newton's methods of generating conics,
376
Double points defined,
376
Double points and homologous point pairs have constant anharmonic ratio,
376
Double points found geometrically,
377
Concentric or superposed projective pencils,
378
Involution defined; double points and central point of,
379
Involution hyperbolic and elliptic,
379
Involution symmetric, isogonal, orthogonal,
380
A system of conics through four fixed points cuts any transversal in involution,
381
CHAPTER XIII.
THEORY OF DUALITY AND RECIPROCAL POLARS.
Principle of duality,
382
Reciprocation defined,
384
Substitutions to be made in any theorem in order to get the reciprocal theorem,
384
Reciprocal of Pascal's theorem,
385
Properties proved by reciprocation,
385
Special results when reciprocating conic is a circle,
386
Equation of the reciprocal of any conic,
387
To find the centre of reciprocation so that the polar reciprocal of a given triangle may be in species,
388
Lionnet's triangle,
389
Tangential equation of conic given a focus and circumtriangle,
390
Equation of the Brocard ellipse,
391
Metapolar quadrangles and their metapoles,
392
CHAPTER XIV.
RECENT GEOMETRY.
SECTION I. - ON A SYSTEM OF THREE FIGURES DIRECTLY SIMILAR.
Homothetic figures; double point or centre of similitude of,
393
To find the double point of two polygons directly similar,
394
Three directly similar figures; corresponding lengths, angles of rotation, double points, triangle and circle of similitude of
395
In ditto, triangle formed by three homologous lines is in perspective with triangle of similitude and locus of centre of perspective is the circle of similitude,
396
Invariable points and invariable triangle,
397
Invariable points form a system of three corresponding points,
397
Invariable triangle and triangle of similitude are in perspective,
397
Modular quadrangle,
398
Director point; adjoint points; annex triangles,
399
Annex triangles are directly similar to modular triangles,
400
Annex circles: Lionnet's circle and triangle,
401
Triangle formed by any three homologous points is orthologique with Lionnet's triangle,
401
Triangle formed by any three corresponding points is similar to pedal triangle of any of these points with respect to the corresponding annex triangle,
402
Some theorems deduced by last proposition,
403, 404
Exercises by Neuberg,
404
SECTION II. - THEORY OF HARMONIC CHORDS.
Symmedian point defined,
407
Centres of inversion; Brocard ellipse and circle,
408
Brocard points and angle,
409
Harmonic polygons defined,
411
Polar of Symmedian point is the same with respect to circumcircle and harmonic polygon,
413
Tucker's circle of harmonic polygon,
415
If figures directly similar be described on the sides of a harmonic polygon, every system of homologous points lies on the first pedal of a conic,
416
Exercises,
417
SECTION III. - THE TRIANGLE.
Parallels to sides of a triangle, through its symmedian point, meet sides in six concyclic points,
418
Lemoine circle and hexagon; Tucker's circles,
419, 421
Trilinear equation of the Brocard ellipse,
420
Equation and envelope of Tucker's circles,
421
First and second triangle of Brocard,
422
The invariable triangle is triply in perspective with original triangle,
422
Barycentric co-ordinates of centres of perspective,
423
SECTION IV. - VARIOUS CIRCLES AND CONICS.
Neuberg's circles,
423
Neuberg's circles equation of, in barycentric co-ordinates,
424
Neuberg's circles properties of,
425
First and second angle of Steiner,
426
Neuberg's circles-different modes of derivation,
426
M'Cay s circles,
427
M'Cay s circles, are special cases of annex circles,
427
Isogonal transformation, isogonal conjugates,
428
Isogonal transformation of any diameter of circumcircle is an equilateral hyperbola,
428
Isogonal transformation of a few lines,
429
Neuberg's hyperbolae,
Poles, with respect to triangle of reference, of Neuberg's hyperbolae, Kiepert's hyperbola, and circumcircle are collinear,
431
Fuhrmann's circles,
431
Point of contact of nine-points circle with incircle,
434
Three points on the common tangent of incircle, and nine-points circle,
435
Orthocentroidal circle defined and its equation found,
436
Orthocentroidal circle meets perpendiculars and medians in six points forming a harmonic hexagon,
437
The Brocard parabolae,
439
Artzt's parabolae (second group),
441
Kiepert's hyperbola,
431, 442
Kiepert's triangle,
443
Kiepert's hyperbola, any two points on, whose parametric angles differ by a right angle are collinear with centre of nine-points circle,
444
Kiepert's hyperbola; properties of in connexion with Neuberg's circles,
443, 444
Kiepert's hyperbola; parametric angles of some special points on,
446
If two Kiepert's triangles have their parametric angles complements, the orthologique centre of either and ABC is the centre of perspective of ABC and the other,
447
Jerabek's hyperbola,
448
Most general equation of circumequilateral hyperbola,
449
Steiner's ellipse, properties of,
451
Tarry's point is the centre of perspective of original triangle and that formed by centres of Neuberg's circles,
452
Steiner's axes are parallel to asymptotes of Kiepert's hyperbola,
453
The parametric angles of the first Brocard triangle and that formed by Neuberg's centres are complementary,
454
The foci of Steiner,
455
The foci of Steiner theorems concerning,
456, 457
Envelope of sides of Kiepert's triangle,
458
Co-ordinates of focus of Kiepert's parabola,
458
Equation satisfied by the Brocard angle,
459
Equation whose roots are Steiner's angles,
459
Relation between the Brocard and Steiner angles,
459
Equation of the seventeen-point cubic and the points (special) it passes through,
460, 461
CHAPTER XV.
INVARIANT THEORY OF CONICS.
Determinant of transformation,
462
Invariants, covariants, contravariants, mixed concomitants defined,
462, 463
Pencil and net of conics, trilinear and tangential,
463
Polar reciprocal of one conic with respect to a second,
463, 486
Three conics of the pencil S1 - kS2 = 0 represent line-pairs,
464
Equation of the three line-pairs,
465
Equation of tangent-pair to conic at points of intersection by a line,
465
Equation of asymptotes of conic given by general equation,
465
Lamé's equation,
465
Equation of bisectors of angles of a given line-pair in oblique co-ordinates,
466
Calculation of invariants for some particular conics,
467
Condition that a triangle may be inscribed in one conic, and circumscribed to another,
468
Three special relations which a triangle can have with respect to a conic,
468
Tact-Invariant of two conics,
469
Parallel to ellipse; equation of,
470
Condition that two conics should osculate,
471
Centres of six circles, which can be described through any point to osculate a conic, lie on a conic,
471
Invariant angles of two conics,
471
Tact-Invariant expressed as product of six anharmonic ratios,
472
Envelope of line cut harmonically by two conics,
473
Locus of point whence tangents to two conics form a harmonic pencil,
473, 477
Anharmonic ratio of pencil from any point of S1 - kS2 = 0 to common points of S1 and S2,
473
Locus of centres of all conics of a given pencil,
473
Anharmonic ratio of four conics of a pencil defined,
474
Conics harmonically inscribed or circumscribed,
475
Orthoptic circle of conic given by general equation,
476
Locus of centre of conic harmonically inscribed in four conics,
478
Other properties of harmonic conics,
479
Harmonic envelope of two conics for which Q2 vanishes,
480
Conics for which Q1 and Q2 vanish,
482
Harmonic system of conics and their harmonic invariant,
482
Examples of conics which are harmonic,
482, 484
Poncelet's Theorem,
484
Locus of third summit of a triangle circumscribed in a conic, two of whose summits move on another conic,
486
Fourteen-point conic of a quadrilateral,
486, 487
Equation of four common tangents to two conics,
488
Fourteen-point conic of two given conics,
489
Tangential equation of four points common to two conics,
489
Fourteen-line conic of two given conics,
Envelope of the eight common tangents of two conics at their points of intersection,
490
Autopolar triangle,
491
Autopolar triangle squares of sides of are covariants,
492
Mutual power of two conics,
493
Tact-Invariant of S - L12 = 0, and S - L22 = 0,
494
Orthogonal invariant, or the condition that two conics should cut orthogonally,
495
Frobenius's Theorem concerning two systems of five conics inscribed in the same conic,
495
Condition that four conics should cut a fifth orthogonally, or be tangential to it,
497
Equation of conic inscribed in a given conic and touching three given conics also inscribed in the same conic,
498
Orthogonal conics,
499
Equation of conic cutting orthogonally three given conics inscribed in the same conic,
499, 500
Locus of double points of a given trilinear net of conics,
501
Locus of point whose polars with respect to three conics are concurrent,
502
Jacobian of three conics defined,
502, 503
Ditto is the locus of the double points on lines cut in involution by the conics,
503
Ditto is the locus of the double point of all conics of the net l1S1 + l2S2 + l3S3 = 0,
503
Ditto various theorems concerning,
Envelope of line cutting three conics in involution,
505
Hermite envelope of net of conics,
506
Locus of point whence tangents to three conics form a pencil in involution,
507
Contravariants,
508
Conditions that general equation should represent an equilateral hyperbola or a parabola,
508, 509
The covariant F of the cyclic points and any conic gives the orthoptic circle of that conic,
509
Orthoptic circle of different conics,
510
Foci, equation of; antifoci,
511
General equation of conic confocal with a given one,
511
Co-ordinates of foci,
512
The covariant F of two conics having double contact passes through their points of intersection,
513
Identical relations,
513, 514
Fourteen-point conic of a quadrilateral expressed in terms of the equations of its four sides,
516
Any three conics are conjugate with respect to one infinite number of quadrilaterals,
516
Condition that three given conics should have a common point,
517
Number of independent invariants, &c., &c., of two conics,
517, 518
The six summits of two triangles, autopolar with respect to a conic, lie on a conic,
519
Condition that two given lines should intersect on a given conic,
520
Identical relation connecting coefficients in the equations of six conics harmonically circumscribed to the same conic,
520
Miscellaneous Exercises,
525