Condition that,
that two lines should intersect on a
conic, 391.

a triangle self-conjugate to one may
be inscribed or circumscribed to
another, 340.

three conics have double contact with
same conic, 359.

have a common point, 365.

may include a perfect square in their
Syzygy, 366.

lines joining to vertices of triangle
points where conic meets sides
should form two sets of three, 351.
Cone, sections of, 326.
Confocal conics, 186.

cut at right angles, 181, 291, 322.
may be considered as inscribed in
same quadrilateral, 239.
most general equation of, 353.
tangents from point on (1) to (2)
equally inclined to tangent of (1),
182.

pole with regard to (2) of tangent to
(1) lies on a normal of (1), 209.
used in finding axes of reciprocal
curve, 291.

in finding centre of curvature, 376.
properties proved by reciprocation, 291.
length of arc intercepted between
tangent from, 377.
Conjugate diameters, 146.

their lengths, how related, 159, 168.
triangle included by, has constant
area, 159, 169.

form harmonic pencil with asymp-
totes, 296.

at given angle, how constructed, 171.
construction for 218.

Conjugate hyperbolas, 165.

Conjugate lines, conditions for, 267.
Conjugate triangles, homologous, 91, 92.
Continuity, principle of, 325.
Covariants, 347.

Criterion, whether three equations repre-
sent lines meeting in a point, 34.
whether a point be within or without
a conic, 261.

whether two conics meet in two real
and two imaginary points, 337.
Curvature, radius of, expressions for its
length, and construction for, 228,375.
circle of, equation of, 234.
centre of, coordinates of, 230.

Directrix of parabola is locus of rectangular
tangents, 205, 269, 352.

passes through intersection of per-
pendiculars of circumscribing tri-
angle, 212, 247, 275, 290, 342.
Discriminant defined, 266.

method of forming, 72, 149, 153, 155.
Distance between two points, 3, 10, 133.
Distance of two points from centre of
circle proportional to distance of
each from polar of other, 93.
when a rational function of coordi-
nates, 179.

of four points in a plane, how con-
nected, 134.

Double contact, 228, 234, 346.

equation of conic having d. c. with
two others, 262.

tangent to one cut harmonically by

other, and chord of contact, 312, 319.
properties of two conics having d. c.
with a third, 242, 282.

of three having d. c. with a fourth,
243, 263, 281.

tangential equation of, 355.
condition two should touch, 356.
problem to describe one such conic
touching three others, 356, 358.

Duality, principle of, 276.

Eccentric angle, 217, &c., 243.

in terms of corresponding focal angle,
220.

of four points on a circle, how con-
nected, 229.

Eccentricity, of conic given by general
equation, 164.

depends on angle between asymp
totes, 164.

Ellipse, origin of name, 186, 328.
mechanical description of, 178, 218.
area of, 372.

Envelope of

line whose equation involves indeter-
minates in second degree, 257, &c.
line on which sum of perpendiculars
from several fixed points is con-
stant, 95.

given product or sum or difference of
squares of perpendiculars from two
fixed points, 259.

base of triangle given vertical angle
and sum of sides, 260.

whose sides pass through fixed points
and vertices move on fixed lines,
259.

and inscribed in given conic, 250, 280,
319.

which subtends constant angle at fixed
point, two sides being given in
position, 284.

polar of fixed point with regard to a
Iconic of which four conditions are
given, 271, 280.

polar of centre of circle touching two
given, 291.

chord of conic subtending constant
angle at fixed point, 255.

Gaultier of Tours, 99.

Gergonne, on circle touching three others,
110.

Gordan, on number of concomitants, 363.
Graves, theorems by, 333, 377.

Infinity, line at, equation of, 64.

touches parabola, 235, 290, 329.
centre, pole of, 155, 296.
Inscription in conic of triangle or polygon
whose sides pass through fixed
points, 250, 273, 281, 307.
Intercept on chord between curve and
asymptotes equal, 191, 312.

on asymptotes constant by lines join-
ing two variable points to one fixed,
192, 294, 298.

on axis of parabola by two lines, equal
to projection of distance between
their poles, 201, 294.

Intercept on parallel tangents by variable
tangent, 172, 287, 299, 385.

313.

Harmonic, section, 56.

what when one point at infinity, 295.
properties of quadrilateral, 57, 317.
property of poles and polars, 85, 148,
295, 297, 318.

pencil formed by two tangents and
two co-polar lines, 148, 296.
by asymptotes and two conjugate
diameters, 296.

by diagonals of inscribed and circum-
scribing quadrilateral, 242.
by chords of contact and common
chords of two conics having double
contact with a third, 242.
properties derived from projection of
right angles, 321.

condition for harmonic pencil, 305.
condition that line should be cut har-
monically by two conics, 306.
locus of points whence tangents to two
conics form a harmonic pencil, 306.
Hart, theorems and proofs by, 124, 126,
127, 263, 378.

Harvey, theorem on four circles, 132.
Hearne, mode of finding locus of centre,
given four conditions, 267.
Hermes, on equation of conic circum-
scribing a triangle, 120.

Hesse, 381.

Hexagon (see Brianchon and Pascal),

property of angles of circumscribing,
270, 289.

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

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

criterion for, and method of forming,
304.

locus of intersection of corresponding
lines, 271.

envelope of line joining corresponding
points, 302, 303.

Homologous triangles, 59.

Hyperbola, origin of name, 186, 328.
area of, 373.

Imaginary, lines and points, 69, 77.

circular points at infinity, tangential
equation of, 352.

every line through either perpen-
dicular to itself, 351.

Inversion of curves, 114.
Involution, 307.

Jacobian of three conics, 360, &c.
Joachimsthal,

relation between eccentric angles of
four points on a circle, 229.
method of finding points where line
meets curve, 264.

vertex of triangle given base and a
relation between lengths of sides,
39, 47, 178.

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

and intercept by sides on fixed line, 300.
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, 208.
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
linea, 41, 42, 248, 280, 299.
generalizations of the last problem, 300.
of vertex of triangle which circum-
scribes a given conic and whose
base angles move on fixed lines,
250, 319, 349.

generalizations of this problem, 350.
common vertex of several triangles

given bases and sum of areas, 40.
vertex of right cone, out of which
given conic can be cut, 331.

point cutting in given ratio parallel
chords of a circle, 162.

intercept between two fixed lines, on
various conditions, 39, 40, 47.
variable tangent to conic between
two fixed tangents, 277, 323.
point whence tangents to two circles
have given ratio or sum, 99, 263.
taken according to different laws on
radii vectores through fixed point, 52.

Locus of,

such that Emr2 = constant, 88.
whence square of tangent to circle is
as product of distances from two
fixed lines, 240.

cutting in given anharmonic ratio,
chords of conic through fixed point,
320.

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.
point on line of given direction meeting
sides of triangle, so that oc2=oa.ob,
298.

on line cut in given anharmonic ratio,

of which other three describe right
lines, and line itself touches a conic,
324.

chords through which subtend right
angle at point on conic, 270.
whence tangents to two conics form
harmonic pencil, 306.

whose polars with respect to three
conics meet in a point, 360.
middle point of rectangles inscribed in
triangle, 43.

of parallel chords of conic, 143.

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, 209.

focal radius vector with corresponding
eccentric vector, 220.

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, 342.

eccentric vector with corresponding
normal, 220.
corresponding lines of two homogra-
phic pencils, 271.

polars with respect to fixed conics of
points which move on right lines,
271.
intersection of tangents to a conic
which cut at right angles, 166, 171,
269, 352.

to a parabola which cut at given
angle, 213, 256, 285.

at extremities of conjugate dia-
meters, 209.

whose chord subtends constant angle
at focus, 284.

from two points, which cut a given
line harmonically, 322.

each or both on one of four given
tangents, 302, 320.

Locus of,

at two fixed points on a conic satisfy-
ing two other conditions, 220, 320.
various other conditions, 215.
intersection of normals at extremity
of focal chord, 211.

or chord through fixed point, 214, 335.
foot of perpendicular from focus on
tangent, 182, 204, 351.

on normal of parabola, 213.
on chord of circle subtending right
angle at given point, 91.
extremity of focal subtangent, 184.
centre of circle making given inter-
cepts on given lines, 208.

centre of inscribed circle given base
and sum of sides, 208.

of circle cutting three at equal angles,
108.

of circumscribing circle given vertical
angle, 89.

of circle touching two given circles,
291, 320.

centre of conic (or pole of fixed line)

given four points, 153, 254, 268,
271, 281, 302, 320.

given four tangents, 216, 254, 267,
277, 281, 321, 339.

given three tangents and sum of
squares of axes, 216.

four conditions, 267, 389.

pole of fixed line with regard to sys-
tem of confocals, 209, 322.

pole with respect to one conic of tan-
gent to another, 209, 278.

focus of parabola given three tan-
gents, 207, 214, 274, 285, 320.
focus given four tangents, 275, 277.
given four points, 217, 288, 392.
given three tangents and a point, 288.
given four conditions, 389.
vertices of self-conjugate triangle,com-

mon to fixed conic, and variable of
which four conditions are given,
389.

Mac Cullagh, theorems by, 210, 220, 333,
374, 377.

Mac Laurin's mode of generating conics,
247, 248, 251, 299.
Mechanical construction of conics, 178,
Malfatti's problem, 263.
Middle points of diagonals of quadrilate-
194, 203, 218.
ral in one line, 26, 62.

Miquel, on circles circumscribing triangles
Möbius, 217, 278, 295.
formed by five lines, 247.
Moore, deduction of Steiner's theorem from

Brianchon's, 247.

Mulcahy, on angles subtended at focus, 331.
Newton's method of generating conics, 300.
Normal, 173, &c. 335.

Number of terms in general equation, 74.

of conditions to determine a conic, 136.
of intersections of two curves, 225.
of solutions of problem to describe
a conic touching five others, 390.

conics, 363.

O'Brien, 217.

Orthogonal systems of circles, 102, 131,
348, 361.

Osculating circle, 227, 234.

Pappus, 186, 295, 328.

Parabola (see Contents, pp. 195–207,

212-214).

origin of name, 180, 328.

has tangent at infinity, 235, 290, 329.
coordinates of focus, 239, 274, 354.
equation of directrix, 269, 352.
touching four lines, 274.

Parallel to conic, equation of, 337.
Parameter, 185, 197, 202.

same for reciprocals of equal circles,
286.

Pascal's hexagon, 245, 280, 301, 319, 380.
expression of coordinates by single,
217, 248, 386.

Perpendicular, equation and length, 26, 60.
condition for, 59.

extension of relation, 321, 354.

from centre and foci on tangent, 169,
179, 204.

Plücker, 278, 380.

Polar coordinates and equations, 9, 36,

87, 95, 160, 162, 184, 207.

poles and polars, properties of, 92, 148.
polar, equation of, 82, 147, 265.
pole of given line, coordinates of, 266.
polar reciprocals, 276, &c.

point and polar equivalent to two
conditions, 388.
Poncelet, 101, 278, 301, 314.
Frojection, 314, 332.

Quadrilateral,

middle points of diagonals lie on
a right line, 26, 62, 216.
circles having diagonals for diameters
have common radical axis, 277.
harmonic properties of, 57, 317.
inscribed in conics, 148, 319.
sides and diagonals of inscribed quad-
rilateral cut transversal in involu-
tion, 312.

diagonals of inscribed and circum-

scribed form harmonic pencil, 242.

Radical axis and centre, 99, 122, 224, 282.
Radius of circle circumscribing triangle
inscribed in conic, 213, 220, 333.
Radius of curvature, 227.

Reciprocals, method of, 66, 276, 294, 356.

Sadleir, theorems by, 184.

Self-conjugate triangles, 91.

Similar conics, 222.

condition for 224.

have points common at infinity, 236.
tangent to one cuts constant area
from other, 373.

Steiner,

theorem on triangle circumscribing
parabola, 212, 247, 275, 290, 342.
on points whose osculating circle
passes through given point, 229.
theorems on Pascal's hexagon, 246, 380.
solution of Malfatti's problem, 263.
Subnormal of parabola constant, 202.
Supplemental chords, 172.

Systems of circles having common radical
axis, 100.

of conics through four points cut a
transversal in involution, 312.

Tangent, general definition of, 78.
to circle, length of, 84.

to conic constructed geometrically,151.
determination of points of contact,
five tangents given, 247.
variable, makes what intercepts on
two parallel tangents, 172, 181.
or on two conjugate diameters, 172.
of parabola, how divides three fixed
tangents, 299.

Tangential equations, 65, 276, &c., 383,
&c.

of inscribed and circumscribing circles,
121, 125, 288.

of circle in general, 128, 384.
of conic in general, 152, 260.
of imaginary circular points, 352.
of confocal conics, 353, 384.

of points common to four conics, 344.
interpretation of, 384.

Townsend, theorems and proofs by, 252,
301, 375.

Transformation of coordinates, 6, 9, 157,
335.

Transversal, how cuts sides of triangle, 35.
Carnot's theorem of, 289, 318, 388.
met by system of conics in involu-
tion, 312.

Triangle, circumscribing, vertices or two
lie on a conic, 320.

Triangles made by four lines, properties
of, 217, 246.

Trilinear coordinates, 57, 60, 264.

Veronese, 382.