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.

conics, 363.

O'Brien, 217.

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

Osculating circle, 227, 234.

curve, 229.

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.

Pliicker, 278, 380.

Polar coordinates and equations, 9, 36,

87, 95, 180, 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
transverzal 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.

IN

ANNOUNCE

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