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Gergonne, on circle touching three others, | Intercept on parallel tangents by variable

110.

Graves, theorems by, 321, 357.

Harmonic, section, 56.

what when one point at infinity, 283.
properties of quadrilateral, 57, 305.
property of poles and polars, 85, 143,
283, 285, 306.

pencil formed by two tangents and
two co-polar lines, 143, 284.
by asymptotes and two conjugate
diameters, 284.

by diagonals of inscribed and circum-
scribing quadrilateral, 231.
by chords of contact and common
chords of two conics having double
contact with a third, 231.
properties derived from projection of
right angles, 309, 310.
condition for harmonic pencil, 293.
condition that line should be cut har-
monically by two conics, 294.

locus of points whence tangents to two
conics form a harmonic pencil, 294.
Hart, theorems and proofs by, 123, 125,
126, 252, 359.

Hearne, mode of finding locus of centre,
given four conditions, 256.

Hermes, on equation of conic circumscrib-
ing a triangle, 120.

Hesse, 362.

Hexagon (see Brianchon and Pascal),

property of angles of circumscribing,
258, 277.

Homogeneous, equations in two variables,
meaning of, 67.

trilinear equations, how made, 64.
Homographic systems, 57, 63.

criterion for, and method of forming,
292.

locus of intersection of corresponding
lines, 260.

envelope of line joining corresponding
points, 290, 291.
Homologous triangles, 59.
Hyperbola, origin of name, 180, 316.
area of, 354.

Imaginary, lines and points, 69, 77.
circular points at infinity, equation
of, 338.

every line through either perpen-
dicular to itself, 338.

Infinity, line at, equation of, 64.

touches parabola, 224, 278, 318.
centre, pole of, 150, 284.
Inscription in conic of triangle or polygon
whose sides pass through fixed
points, 239, 261, 269, 289.
Intercept on chord between curve and

asymptotes equal, 181, 300.

on asymptotes constant by lines join-
ing two variable points to one fixed,
182, 236.

on axis of parabola by two lines, equal
to projection of distance between
their poles, 190, 281.

tangent, 167, 275, 287, 365.

Invariants, 154, 323.

Inversion of curves, 114.
Involution, 295.

Jacobian of three conics, 345, &c.
Joachimsthal,

relation between eccentric angles of
four points on a circle, 218.
method of finding points where line
meets curve, 283.

Kirkman's theorems on hexagons, 360.

Latus rectum, 179.

Limit points of system of circles, 101, 279.
Locus of

vertex of triangle given base and a
relation between lengths of sides,
39, 47, 172.

and a relation between angles, 39, 47,
88, 107.

and intercept by sides on fixed line, 288.
and ratio of parts into which sides

divide a fixed parallel to base, 41.
vertex of given triangle, whose base
angle moves along fixed lines, 197.
vertex of triangle of which one base
angle is fixed and the other moves
along a given locus, 51, 96.

whose sides pass through fixed points
and base angles move along fixed
lines, 41, 42, 237, 268, 287.
generalizations of the last problem, 288.
of vertex of triangle which circum-
scribes a given conic and whose
base angles move on fixed lines,
239, 307, 336.

generalizations of this problem, 337.
common vertex of several triangles

given bases and sum of areas, 40.
vertex of right cone, out of which
given conic can be cut, 319.
point cutting in given ratio parallel
chords of a circle, 157.

intercept between two fixed lines, on
various conditions, 39, 40, 47.
variable tangent to conic between

two fixed tangents, 265, 311.
point whence tangents to two circles
have given ratio or sum, 99, 252.
taken according to different laws on
radii vectores through fixed point,
52.

such that Emr2 = constant, 88.
whence square of tangent to circle is
as product of distances from two
fixed lines, 229.

cutting in given anharmonic ratio,
chords of conic through fixed point,
308.

on perpendicular at height from base
equal a side, given base and sum of
sides, 59.

such that triangle formed by joining
feet of perpendiculars on sides of
triangle has constant area, 119.

Locus of,

point on line of given direction meeting
sides of triangle, so that oc2= oa.ob,
286.

on lines cut in given anharmonic ratio,
of which other three describe right
lines, and line itself touches a conic,
311.

chords through which subtend right
angle at point on conic, 259.
whence tangents to two conics form
harmonic pencil, 294.

whose polars with respect to three
conics meet in a point, 345.
middle point of rectangles inscribed in
triangle, 43.

of parallel chords of conic, 138.

of convergent chords of circle, 96.
intersection of bisector of vertical
angle with perpendicular to a side,
given base and sum of sides, 51.
of perpendicular on tangent from
centre, or focus, with focal or central
radius vector, 198.

focal radius vector with corresponding
eccentric vector, 209.

of perpendiculars to sides at extremity
of base, given vertical angle and
another relation, 47.

of perpendiculars of triangle given base
and vertical angle, 88.

of perpendiculars of triangle inscribed
in one conic and circumscribing
another, 329.

eccentric vector with corresponding
normal, 209.

corresponding lines of two homogra-
phic pencils, 260.

polars with respect to fixed conics of
points which move on right lines,

260.

intersection of tangents to a conic
which cut at right angles, 161, 166,
258, 339.

to a parabola which cut at given
angle, 202, 245, 273.

at extremities of conjugate dia-
meters, 198.

whose chord subtends constant angle
at focus, 272.

from two points, which cut a given
line harmonically, 340.

each or both on one of four given
tangents, 290, 308.

at two fixed points on a conic satisfy-
ing two other conditions, 209, 308.
various other conditions, 204.
intersection of normals at extremity
of focal chord, 200.

or chord through fixed point, 203, 323.
foot of perpendicular from focus on
tangent, 176, 193.

on normal of parabola, 203.
on chord of circle subtending right
angle at given point, 91.
extremity of focal subtangent, 178.
centre of circle making given inter-
cepts on given lines, 197.

Locus of

centre of inscribed circle given base
and sum of sides, 197.

of circle cutting three at equal angles,
108.

of circumscribing circle given vertical
angle, 89.

of circle touching two given circles,
279, 308.

centre of conic (or pole of fixed line)
given four points, 148, 243, 256,
260, 290, 308.

given four tangents, 243, 256, 265,
269, 309, 327.

given three tangents and sum of
squares of axes, 205.

four conditions, 256, 368.

pole of fixed line with regard to sys-
tem of confocals, 198, 310.
pole with respect to one conic of tan-
gent to another, 198, 266.

focus of parabola given three tan-
gents, 196, 203, 263, 273, 308.
focus given four tangents, 263, 265.
given four points, 206, 276.
given three tangents and a point, 276.
given four conditions, 369.
vertices of self-conjugate triangle, com-

mon to fixed conic, and variable of
which four conditions are given, 369.
Mac Cullagh, theorems by, 209, 319, 355,
357.

Mac Laurin's mode of generating conics,
236, 240, 287, 288.

Malfatti's problem, 252.
Mechanical construction of conics, 172,
183, 192, 207.

Middle points of diagonals of quadrilate-
ral in one line, 26, 62.
Miguel, on circles circumscribing triangles
formed by five lines, 238.
Möbius on tangential co-ordinates, 266.
on harmonic properties, 283.
Moore, deduction of Steiner's theorem from
Brianchon's, 236.

Mulcahy, on angles subtended at focus, 319.

[blocks in formation]

Parallel to conic, equation of, 325.

Parameter, 179, 186, 191.

same for reciprocals of equal circles,

274.

Similar conics, 211.

condition for, 213.

have points common at infinity, 225.
tangent to one cuts constant area
from other, 354.

Pascal's hexagon, 234, 268, 289, 307.
Perpendicular, equation and length, 26, 60. Steiner,
condition for, 59.

extension of relation, 342.

from centre and foci on tangent, 164,
174, 193.

Plücker, 266, 360.

Polar co-ordinates and equations, 9, 36,
87, 95, 156, 179, 196.

poles and polars, properties of, 92, 143.
polar, equation of, 82, 142, 254.
pole of given line, co-ordinates of, 255.
polar reciprocals, 266, &c.

point and polar equivalent to two
conditions, 367.
Poncelet, 101, 266, 289, 302.
Projection, 303, 321.

Quadrilateral,

middle points of diagonals lie on
a right line, 26, 62.
circles having diagonals for diameters
have common radical axis, 265.
harmonic properties of, 57, 305.
inscribed in conics, 143, 307.
sides and diagonals of inscribed quad-
rilateral cut transversal in involu-
tion, 300.

diagonals of inscribed and circum-
scribed form harmonic pencil, 231.

Radical axis and centre, 99, 122, 212, 270.
Radius of circle circumscribing triangle
inscribed in conic, 202, 321.
Radius of curvature, 216.
Reciprocals, method of, 66, 264-282, 342.

Sadleir, theorems by, 178.
Self-conjugate triangles, 91.

circle having triangle of reference for,
243.

of equilateral hyperbola, 204.

vertices of two lie on a conic, 310, 329.
equation of conic referred to, 227, 242.
common to two conics, 246.

determination of, 335, 347.

theorem on triangle circumscribing
parabola, 201, 236, 263, 278, 329.
on points whose osculating circle
passes through given point, 218.
theorems on Pascal's hexagon, 235,
360.

solution of Malfatti's problem, 252.
Subnormal of parabola constant, 191.
Supplemental chords, 166.

Systems of circles having common radical
axis, 100.

of conics through four points cut a
transversal in involution, 300.
Tangent, general definition of, 78.
to circle, length of, 84.

to conic constructed geometrically,

143.
determination of points of contact,
five tangents given, 236.
variable, makes what intercepts on
two parallel tangents, 167, 175.
or on two conjugate diameters, 167.
of parabola, how divides three fixed
tangents, 287.

Tangential equations, 65, 264, &c., 363,
&c.

of inscribed and circumscribing circles,
120, 124, 275.

of circle in general, 128, 363.
of conic in general, 147, 249.
of imaginary circular points, 337.
of confocal conics, 340, 363.

of points common to four conics, 331.
interpretation of, 363.

Townsend, theorems and proofs by, 241,
289, 355.

Transformation of co-ordinates, 6, 9, 151,
323.

Transversal, how cuts sides of triangle, 35.
Carnot's theorem of

met by system of conics in involu-
tion, 300.

Triangle circumscribing, vertices of two
lie on a conic, 308.

Serret on locus of centre given four Trilinear co-ordinates, 57, 60, 253.

tangents, 205.

Similitude, centre of, 105, 212, 270.

Zeuthen, 368.

THE END.

W. METCALFE, PRINTER, GREEN-STREET, CAMBRIDGE.

BY THE SAME AUTHOR.

A TREATISE ON THE GEOMETRY OF

THREE DIMENSIONS.

Second Edition.

Dublin; HODGES, SMITH, AND FOSTER.

LESSONS INTRODUCTORY TO THE MODERN HIGHER ALGEBRA.

Second Edition.

Dublin: HODGES, SMITH, AND FOSTER.

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