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the characteristics of systems of conics touching two fixed conics, and also satisfying the conditions two points, a point and a tangent, two tangents; viz. (36, 56), (56, 56), (56, 36). In like manner we have the number of conics of these respective systems which will touch a third fixed conic, viz. 184, 224, 184. The characteristics then of the systems three conics and a point, three conics and a line are (184, 224), (224, 184). And the numbers of these to touch a fourth fixed conic, are in each case 816, so that finally we ascertain that the number of conics which can be described to touch five fixed conics is 3264. For further details, I refer to the memoirs already cited, and only mention in conclusion that 2 - μ conics of any system reduce to a pair of lines, and 2μ - to a pair of points.

INDEX.

Angle,

between two lines whose Cartesian
equations are given, 21, 22.
ditto, for trilinear equations, 60.
between two lines given by a single
equation, 69.

between two tangents to a conic, 161,
201, 258.

between two conjugate diameters, 164.
between asymptotes, in terms of ec-
centricity, 159.

between focal radius vector and tan-
gent, 174.

subtended at focus by tangent from
any point, 177, 195.

subtended at limit points of system of
circles, 279.

theorems respecting angles subtended
at focus proved by reciprocation, 272.
by spherical geometry, 319.
theorems concerning angles how pro-
jected, 309, 311.
Anharmonic ratio,

fundamental theorem proved, 55.
what, when one point at infinity, 283.
of four lines whose equations are
given, 56, 293.

property of four points on a conic,
229, 240, 276, 306.

of four tangents, 241, 276.

of three tangents to a parabola, 287.
these properties developed, 284, &c.
properties derived from projection of
angles, 311, 312.

of four points on a conic when equal
to that of four others on same
conic, 241, 242.

on a different conic, 241, 291.

of four points equal that of their
polars, 260.

of four diameters equal that of their
conjugates, 290.

of segments of tangent to one of three
conics having double contact, by
other two, 307.

Apollonius, 316.

[blocks in formation]

Area,

of triangle inscribed in, or circum-
scribing a conic, 201, 208.

of triangle formed by three normals,
209.
constant, of triangle formed by join-
ing ends of conjugate diameters,
154, 164.

constant, between any tangent and
asymptotes, 182.

of polar triangles of middle points of
sides of fixed triangle with regard
to inscribed conic, 337.

of triangles equal, formed by drawing
from end of each of two diameters
a parallel to the other, 168.
found by infinitesimals, 352.
constant, cut from a conic by tangent
to similar conic, 354.

line cutting off from a curve constant
area bisected by its envelope, 355.

Asymptotes,

defined as tangents through centre
whose points of contact are at in-
finity, 150.

are self-conjugate, 162.

are diagonals of a parallelogram whose
sides are conjugate diameters, 180.
general equation of, 260, 328.

and pair of conjugate diameters form
harmonic pencil, 284.

portion of tangent between, bisected
by curve, 180.

equal intercepts on any chord between
curve and, 181, 300.

constant length intercepted on by

chords joining two fixed points to
variable, 182, 282, 286.

parallel to, how cut by same chords,
286.

by two tangents and their chord, 286.
bisected between any point and its
polar, 283.

parallels to, through any point on
curve include constant area, 182,
282, 286.

how divide any semi-diameter, 286.

Axes,

of conic, equation of, 151.

lengths, how found, 153.
constructed geometrically, 156.

how found when two conjugate dia-
meters are given, 168.

of reciprocal curve, 279.
axis of parabola, 185.

Axes,

of similitude, 108, 212, 270.
radical, 99, 126.

Bisectors of angles between lines given by
a single équation, 71.

of sides or angles of a triangle meet
in a point, 5, 34, 54.

Bobillier on equations of conic inscribed in,
or circumscribing a triangle, 120.
Boole on invariant functions of coefficients
of a conic, 154,

Brianchon's theorem, 233, 268, 362.
Burnside, theorems or proofs by, 80, 209,
210, 231, 235, 245, 261, 329.

Carnot, theorem of transversals, 277, 306,
367.

Cartesian, equations, a case of trilinear, 64.
Casey, theorems by, 113, 126, 130, 344, 366.
Cayley, theorems and proofs by, 129, 330,
337, 344, 360, 362, 368.

Centre,

of mean position of given points, 50.
of homology, 59.

radical, 99, 270,

of similitude, 105, 213, 270.
chords joining ends of radii through
c. s. meet on radical axis,107,212, 238.
of conic, coordinates of, 138, 148.
pole of line at infinity, 150, 284.
how found, given five points, 236.
of system in involution, 296.
of curvature, 219, 357.

Chasles, theorems by, 283, 288, 292, 358, 368.
Chord of conic, perpendicular to line join-

ing focus to its pole, 177, 309.
which touches confocal conic, propor-
tional to square of parallel semi-
diameter, 201, 210.

Chords of intersection of two conics, equa-
tion of, 322.

Circle, equation of, 14, 75, 87.

tangential equation of, 120, 124, 128,363.
passes through two fixed imaginary
points at infinity, 227, 313.
circumscribing a triangle, its centre
and equation, 4, 86, 118, 128, 276.
inscribed in a triangle, 122, 276.
having triangle of reference for self-
conjugate triangle, 243.

through middle points of sides (see
Feuerbach), 86, 122.

which cuts two at constant angles,
touches two fixed circles, 103.
touching three others, 110, 114, 130,
279.

cutting three at right angles, 102, 128,
347.

circumscribing triangle formed by

three tangents to a parabola, passes
through focus, 196, 203, 263, 273, 308.
circumscribing triangle formed by two
tangents and chord, 231.
circumscribing triangle inscribed in a
conic, 209, 321.

circumscribing, or inscribed, in a self-
conjugate triangle, 329.

Circles circumscribing triangles formed by
four lines, meet in a point, 235.
when five lines are given, the five
such points lie on a circle, 235.
tangents, area, and arc found by in-
finitesimals, 351.

Circumscribing triangles, six vertices of
two lie on a conic, 308, 362.
Class of a curve, 142.

Common tangents to two circles, 104, 106,
252.

to two conics, 332.

their eight points of contact lie on a
conic, 332.

Condition that,

three points should be on a right
line, 24.

three lines meet in a point, 32, 34.
four convergent lines should form
harmonic pencil, 56.

two lines should be perpendicular,
21, 59, 341.

a right line should pass through a
fixed point, 50.

equation of second degree should re-
present right lines, 72, 144, 148,
150, 255.

a circle. 75, 121, 339.

a parabola, 136, 263, 338.

an equilateral hyperbola, 164, 338.
equation of any degree represent right
lines, 74.

two circles should be concentric, 77.
four points should lie on a circle, 86.
intercept by circle on a line should

subtend a right angle at a given
point, 90.

two circles should cut at right angles,
102, 335.

a line should touch a conic, 81, 147,
255, 328.

two conics should be similar, 213.
two conics should touch, 324, 343.
a point should be inside a conic, 250.
two lines should be conjugate with
respect to a conic, 256.

two pairs of points should be harmonic
conjugates, 293.

four points on a conic should lie on a
circle, 218.

a line be cut harmonically by two
conics, 294.

in involution by three conics, 347.
three pairs of lines touch same conic,

258.

three pairs of points form system in
involution, 298.

a triangle may be inscribed in one
conic and circumscribed to another,
330.

a triangle self-conjugate to one may
be inscribed or circumscribed to
another, 328.

three conics have double contact with
same conic, 345.

have a common point, 348.

may include a perfect square in their
syzygy, 319.

Condition that,

lines joining to vertices of triangle
points where conic meets sides
should form two sets of three, 337.

Cone, sections of, 314.

Confocal conics,

cut at right angles, 175, 310.

Distance of two points from centre of
circle proportional to distance of
each from polar of other, 93.
when a rational function of co-ordi-
nates,. 173.

of four points in a plane, how con-
nected, 129.

may be considered as inscribed in Double contact, 215, 223.
same quadrilateral, 228.
most general equation of, 340.
tangents from point on (1) to (2)
equally inclined to tangent of (1),

equation of conic having d. c. with
two others, 251.

176.

pole with regard to (2) of tangent to
(1) lies on a normal of (1), 198.
used in finding axes of reciprocal
curve, 279.

in finding centre of curvature, 357.
properties proved by reciprocation,
279.

length of arc intercepted between
tangent from, 357.

Conjugate diameters, 141.

their lengths, how related, 154, 163.
triangle included by, has constant
area, 154, 164.

form harmonic pencil with asymp-
totes, 284.

at given angle, how constructed, 166.
construction for, 207.

Conjugate hyperbolas, 159.
Conjugate lines, conditions for, 256.
Conjugate triangles, homologous, 91, 92.
Continuity, principle of, 313.
Covariants, 333.

Criterion, whether three equations repre-
sent lines meeting in a point, 34.
whether a point be within or without
a conic, 250.

whether two conics meet in two real
and two imaginary points, 325.
Curvature, radius of, expressions for its
length, and construction for, 217,
357.

circle of, equation of, 223.
centre of, co-ordinates of, 219.

[blocks in formation]

tangent to one cut harmonically by
other, and chord of contact, 300, 307.
properties of two conics having d. c.
with a third, 231, 269.

of three having d. c. with a fourth,
232, 252, 270.

tangential equation of, 342.
condition two should touch, 343.
problem to describe one such conic
touching three others, 343, 345, 366.
Duality, principle of, 266.

Eccentric angle, 206, &c., 232.

in terms of corresponding focal angle,
209.

of four points on a circle, how con-
nected, 218.

Eccentricity, of conic given by general
equation, 159.

depends on angle between asymp-
totes, 159.

Ellipse, origin of name, 180, 316.
mechanical description of, 172, 207.
area of, 353.

Envelope of

line whose equation involves indeter-
minates in second degree, 246, &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, 248.

base of triangle given vertical angle
and sum of sides, 249.

whose sides pass through fixed points.
and vertices move on fixed lines, 248.
and inscribed in given conic, 239, 269,
307.

which subtends constant angle at fixed

point, two sides being given in
position, 273.

polar of fixed point with regard to a

Iconic of which four conditions are
given, 260, 269.

polar of centre of circle touching two
given, 279.

chord of conic subtending constant
angle at fixed point, 244, 272, 273.
perpendicular at extremity of radius
vector to circle, 194.

asymptote of hyperbolas having same
focus and directrix, 273.

given three points and other asymp-
tote, 261.

line joining corresponding points of
two homographic systems

on different lines, 290.

Envelope of

on a conic, 242, 291.

free side of inscribed polygon, all the
rest passing through fixed points,
239, 289.

base of triangle inscribed in one conic,
two of whose sides touch another,
336.

leg of given anharmonic pencil under
different conditions, 312.
ellipse given two conjugate diameters.
and sum of their squares, 249.
Equation, its meaning when co-ordinates
of a given point are substituted in
it; for a right line, circle, or conic,
29, 84, 127, 230.

ditto for tangential equation, 363.
pair of bisectors of angles between
two lines, 71.

of radical axis of two circles, 98, 127.
common tangents to two circles, 104,
106, 252.

circle through three points, 86, 128.
cutting three circles orthogonally,
102, 128.

touching three circles, 114, 130, 366.
inscribed in or circumscribing a tri-
angle, 118, 125, 276.

having triangle of reference self-
conjugate, 243.

tangential of circle, 128, 363.

tangent to circle or conic, 80, 141, 253.
polar to circle or conic, 82, 142, 254.
pair of tangents to conic from any
point, 85, 144, 257.

where conic meets given line, 260.
asymptotes to a conic, 260, 328.
chords of intersection of two conics,322.
circle osculating conic, 223.
conic through five points, 222.
touching five lines, 262.

having double contact with two given
ones, 251.

having double contact with a given one

and touching three others, 345, 366.
through three points, or touching three

lines, and having given centre, 256.
and having given focus, 276.
reciprocal of a given one, 281, 335, 342.
directrix or director circle, 258, 339.
lines joining point to intersection of
two curves, 259, 295.

four tangents to one conic where it
meets another, 336.
curve parallel to a conic, 325.
evolute to a conic, 220, 326.
Jacobian of three conics, 346.
Equilateral hyperbola, 163.

general condition for, 338.

given three points, a fourth is given,
204, 278, 329.

circle circumscribing self-conjugate
triangle passes through centre, 204,

329.

Euler, expression for distance between
centres of inscribed and circum-
scribing circles, 331.
Evolutes of conics, 220, 326.

Fagnani's theorem on arcs of conics, 358.
Faure, theorems by, 329, 337.
Feuerbach, relation connecting four points
on a circle, 87, 206.

theorem on circles touching four lines,
126, 128, 301, 345.
Fixed point, the following lines pass
through a

coefficients in whose equation are con-

nected by relation of first degree, 50.
base of triangle, given vertical angle
and sum of reciprocals of sides, 48.
whose sides pass through fixed
points, and vertices move on three
converging lines, 48,

line sum of whose distances from fixed
points is constant, 49.

polar of fixed point with respect to
circle, two points given, 100.

with respect to conic, four points
given, 148, 259, 269.

chord of intersection with fixed centre
of circle through two points, 100.
of two fixed lines with conic through
four points, one lying on each line,
290.

chord of contact given two points and
two lines, 251,

chord subtending right angle at fixed
point on conic, 170, 259.

when product is constant of tangents of
parts into which normal divides
subtended angle, 170.

given bisector of angle it subtends at
fixed point on curve, 310.
perpendicular on its polar, from point

on fixed perpendicular to axis, 178.
Focus, see Contents, pp. 171-179, 198-200.
infinitely small circle having double

contact with conic, 230.

intersection of tangents from two fixed
imaginary points at infinity, 228.
equivalent to two conditions, 367.
co-ordinates of, given three tangents,
263.

when conic is given by general equa-
tion, 228, 340.

focus and directrix, 173, 229,
theorems concerning angles subtended
at, 272, 319.

focal properties investigated by pro-
jection, 308.

focal radii vectores from any point
have equal difference of reciprocals,
201.

line joining intersections of focal nor-
mals and tangents passes through
other focus, 200.

locus of, given three tangents to a
parabola, 196, 203, 263, 273, 308.
given four tangents, 263, 265.
given four points, 206, 276.
given three tangents and a point, sce
Ex. 3, p. 276.

of section of right cone, how found, 319.
of systems in involution, 297.

Gaultier of Tours, 99.

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