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, 166, 189, 212, 213, 269, 391.
between two conjugate diameters, 169. between asymptotes, 164, 392.
between focal radius vector and tan- gent, 180.
subtended at focus by tangent from any point, 183, 206. subtended at limit points of system of circles, 291.
theorems respecting angles subtended at focus proved by reciprocation, 284, by spherical geometry, 331. theorems concerning angles how pro- jected, 321, 323. Anharmonic ratio, 295.
fundamental theorem proved, 55. what, when one point at infinity, 295. of four lines whose equations are given, 56, 305.
property of four points on a conic, 240, 252, 288, 318.
of four tangents, 252, 288.
of three tangents to a parabola, 299. these properties developed, 297. properties derived from projection of angles, 321, 323.
of four points on a conic when equal to that of four others on same conic, 252.
on a different conic, 252, 303. of four points equal that of their polars, 271.
of four diameters equal that of their conjugates, 302.
of segments of tangent to one of three conics having double contact, by other two, 319.
of triangle formed by three normals, 220.
constant, of triangle formed by join- ing ends of conjugate diameters, 159, 169.
constant, between any tangent and asymptotes, 192.
of polar triangles of middle points of sides of fixed triangle with regard to inscribed conic, 351, 392.
of triangles equal, formed by drawing from end of each of two diameters a parallel to the other, 173. found by infinitesimals, 371. constant, cut from a conic by tangent to similar conic, 373.
line cutting off from a curve constant area bisected by its envelope, 374. of common conjugate triangle of two conics, 362.
defined as tangents through centre whose points of contact are at in- finity, 155.
are self-conjugate, 167.
are diagonals of a parallelogram whose sides are conjugate diameters, 190. general equation of, 272, 340. and pair of conjugate diameters form harmonic pencil, 296.
portion of tangent between, bisected by curve, 191.
equal intercepts on any chord between curve and, 191, 312.
constant length intercepted on by
chords joining two fixed points to variable, 192, 294, 298.
parallel to, how cut by same chords, 298.
by two tangents and their chord, 298.
bisected between any point and its polar, 295.
parallels to, through any point on curve include constant area, 192, 294, 298.
how divide any semi-diameter, 298.
of similitude, 105, 224, 282. chords joining ends of radii through c.s. meet on radical axis, 107,224, 250. of conic, coordinates of, 143, 153. pole of line at infinity, 155, 296. how found, given five points, 247. of system in involution, 308. of curvature, 230, 376.
Chasles, theorems by, 295, 300, 304, 377,389. Chord of conic, perpendicular to line join-
ing focus to its pole, 183, 321. which touches confocal conic, propor- tional to square of parallel semi- diameter, 212, 221, 391. Chords of intersection of two conics, equa- tion of, 334.
Circle, equation of, 14, 75, 87.
tangential equation of, 120, 124, 128, 288, 385.
trilinear equation of, 128.
passes through two fixed imaginary points at infinity, 238, 325. circumscribing a triangle, its centre and equation, 4, 86, 118, 130, 288. inscribed in a triangle, 122, 288. having triangle of reference for self- conjugate triangle, 254. 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, 135, 291.
cutting three at right angles, 102, 130, 361.
or at a constant angle, 132.
cutting three at same angle have common radical axis, 109, 132. circumscribing triangle formed by three tangents to a parabola, passes through focus, 207, 214, 274, 285, 320. |
Circle circumscribing triangle formed by two tangents and chord, 241, 376. circumscribing triangle inscribed in a conic, 220, 333.
circumscribing, or inscribed, in a self- conjugate triangle, 341. circumscribing triangles formed by four lines meet in a point, 246. when five lines are given, the five such points lie on a circle, 247. tangents, area, and arc found by in- finitesimals, 370.
Circumscribing triangles, six vertices of two lie on a conic, 320, 381. Class of a curve, 147. Common tangents to two circles, 104, 106, 263.
their eight points of contact lie on a conic, 345. 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, 354.
a right line should pass through a fixed point, 50.
equation of second degree should re- present right lines, 72, 149, 153, 155, 266.
a circle, 75, 121, 352.
a parabola, 141, 274, 352. an equilateral hyperbola, 169, 352. 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, 348.
that four circles should have common
orthogonal circle, 131.
a line should touch a conic, 81, 152, 267, 340.
two conics should be similar, 224. two conics should touch, 336, 356. a point should be inside a conic, 261. two lines should be conjugate with respect to a conic, 267.
two pairs of points should be harmonic conjugates, 305.
four points on a conic should lie on a circle, 229.
a line be cut harmonically by two conics, 306.
in involution by three conics, 363. three pairs of lines touch same conic, 270.
three pairs of points form system in involution, 310.
a triangle may be inscribed in one conic and circumscribed to another, 342.
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, 31. 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,
De Jonquières 388.
Determinant notation, 129. Diagonals of quadrilateral,
middle points lie in a line, 26, 62, 216. circles described on, as diameters, have common radical axis, 277. Diameter, polar of point at infinity on its conjugate, 296.
Director circle, 269, 352.
when four tangents are given, have common radical axis, 277.
of parabola, equation of, 269, 352.
Directrix of parabola is locus of rectangular tangents, 205, 269, 352.
passes through intersection of per- pendiculars of circumscribing tri- angle, 212, 247, 275, 230, 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,
which subtends constant angle at fixed
point, two sides being given in position, 284.
polar of fixed point with regard to a conic 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, 856.
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,
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.
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.
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.
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