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.
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.
perpendicular at extremity of radius vector to circle, 205.
asymptote of hyperbolas having same fccus and directrix, 285. given three points and other asymp- tote, 272.
line joining corresponding points of two homographic systems
on different lines, 302. on a conic, 253, 303.
free side of inscribed polygon, all the rest passing through fixed points, 250, 301.
base of triangle inscribed in one conic, two of whose sides touch another, 349.
leg of given anharmonic pencil under different conditions, 324. ellipse given two conjugate diameters and sum of their squares, 260. Equation, its meaning when coordinates of a given point are substituted in it; for a right line, circle, or conic, 29, 84, 128, 241.
ditto for tangential equation 384. pair of bisectors of angles between two lines, 71.
of radical axis of two circles, 98, 128. common tangents to two circles, 104, 106, 263.
circle through three points, 86, 130. cutting three circles orthogonally, 102, 130.
touching three circles, 114, 135, 359. inscribed in or circumscribing a tri- angle, 118, 126, 288.
having triangle of reference self- conjugate, 254.
tangential of circle, 129, 384. tangent to circle or conic, 80, 147, 264. polar to circle or conic, 82, 147, 265. pair of tangents to conic from any point, 85, 149, 269.
where conic meets given line, 272. asymptotes to a conic, 272, 340. chords of intersection of two conics, 334.
circle osculating conic, 234.
conic through five points, 233. touching five lines, 274.
having double contact with two given ones, 262.
having double contact with a given one and touching three others, 356. through three points, or touching three lines, and having given centre, 267. and having given focus, 288. reciprocal of a given conic, 292, 348, 356.
directrix or director circle, 269, 352. lines joining point to intersection of two curves, 270, 307.
four tangents to one conic where it meets another, 349. curve parallel to a conic, 337. evolute to a conic, 231, 338. Jacobian of three conics, 360.
Equilateral hyperbola, 168.
general condition for, 352. given three points, a fourth is given, 215, 290, 321.
circle circumscribing self-conjugate triangle passes through centre 215, 342.
Euler, expression for distance between centres of inscribed and circum- scribing circles, 343.
Evolutes of conics, 231, 338. Fagnani's theorem on arcs of conics, 378. Faure, theorems by, 341, 351, 392. Feuerbach, relation connecting four points on a circle, 87, 217.
theorem on circles touching four lines, 127, 313, 359.
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, 153, 271, 281.
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, 302.
chord of contact given two points and two lines, 262.
chord subtending right angle at fixed point on conic, 175, 270. when product is constant of tangents of parts into which normal divides subtended angle, 175.
given bisector of angle it subtends at fixed point on curve, 323. perpendicular on its polar, from point
on fixed perpendicular to axis, 184. Focus, see Contents, pp. 177-190, 209-212. infinitely small circle having double
contact with conic, 241.
intersection of tangents from two fixed
imaginary points at infinity, 239. equivalent to two conditions, 386. coordinates of, given three tangents, 274.
when conic is given by general equa- tion, 239, 353.
focus and directrix, 179, 241. theorems concerning angles subtended at, 284, 331.
focal properties investigated by pro- jection, 320.
focal radii vectores from any point have
equal difference of reciprocals, 212. line joining intersections of focal nor- mals and tangents passes through other focus, 211.
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.
Hamilton, proof of Feuerbach's theorem, Invariants, 159, 335.
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.
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.
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.
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
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.
Number of concomitants to system of | Self-conjugate triangle
Orthogonal systems of circles, 102, 131, 348, 361.
Osculating circle, 227, 234.
vertices of two lie on a conic, 322, 341. equation of conic referred to, 238, 253. common to two conics, 257, 362. determination of, 349, 361.
Serret on locus of centre given four tangents, 216.
three pass through given point on Similitude, centre of, 105, 223, 282. curve, 229.
Pappus, 186, 295, 328.
Parabola (see Contents, pp. 195–207,
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.
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.
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.
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.
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