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.
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.
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.
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.
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.
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.
of mean position of given points, 50. of homology, 59.
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.
their eight points of contact lie on a conic, 332.
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,
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.
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.
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.
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.
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.
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,
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.
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