a conic of the system with one of the tangents drawn from B. Let us examine in how many points the locus can meet the line AB; and we see at once that if a point of the locus be on AB, this line must be a tangent to the conic. Consider then any conic touching AB in a point T, then the tangent AT meets the tangent BT in the point T, which is therefore on the locus; and likewise the tangent AT meets the second tangent from B in the point B, and the tangent BT meets the second tangent from A in the point A. Hence every conic which touches AB gives three points of the locus on AB. The order of the locus is therefore 3v, and A and B are each multiple points of the order v. Thus the locus of foci of conics touching four lines is a cubic passing through the two circular points at infinity. If one of the conditions be that all the conics shonld touch the line AB, then it will be seen that any transversal through A is met by the locus in v points distinct from A, and that A itself also counts for v; hence the locus is in this case only of the order 2v; which is therefore the order of the locus of foci of parabolas satisfying three conditions. An important principle in these investigations is that if two points A, A' on a right line so correspond that to any position of the point A correspond m positions of A', and to any position of A' correspond n positions of A, then in m + n cases A and A' will coincide. This is proved as in Arts. 336, 340. Let the line on which A, A' lie be taken for axis of x; then the abscissæ x, x' of these two points are connected by a certain relation, which by hypothesis is of the mth degree in x' and the nth in x, and will become therefore an equation of the (m+n)th degree if we make x = x'. To illustrate the application of this principle, let us examine the order of the locus of points whose polar with respect to a fixed conic is the same as that with respect to some conic of the system; and let us enquire how many points of the locus can lie on a given line. Consider two points A, A' on the line, such that the polar of A with respect to the fixed conic coincides with the polar of A' with respect to a conic of the system, and the problem is to know in how many cases A and A' can coincide. Now first if A be fixed, its polar with respect to the fixed conic is fixed; the locus of poles of this last line with respect to conics of the system, is, by the first theorem, of the order v, and therefore determines by its intersections with the given line positions of A'. Secondly, examine how many positions of A correspond to any fixed position of A'. By the reciprocal of the first theorem, the polars of A' with respect to conics of the system, envelope a curve whose class is μ, to which therefore μ tangents can be drawn through the pole of the given line AA' with respect to the fixed conic. It follows then, that μ positions of A correspond to any position of A'. Hence, in μ + v cases the two coincide, and this will be the order of the required locus. Hence we can at once determine how many conics of the system can touch a fixed conic : for the point of contact is one which has the same polar with respect to the fixed conic and to a conic of the system; it is therefore one of the intersections of the fixed conic with the locus last found; and there may evidently be 2 (u + v) such intersections. We have thus the number of conics which touch a fixed conic, and satisfy any of the systems of conditions, four points, three points and a tangent, two points and two tangents, &c., the numbers being respectively 6, 12, 16, 12, 6. From these numbers again we find the characteristics of the system of conics which touch a fixed conic and also satisfy three other conditions, three points, two points and a tangent, &c.; these characteristics being respectively (6, 12), (12, 16), (16, 12), (12, 6). We find hence in the same manner the number of conics of the respective systems which will touch a second fixed conic, to be 36, 56, 56, 36. And thus again we have 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 2v - μ conics of any system reduce to a pair of lines, and 2μ — v to a pair of points. MISCELLANEOUS NOTES. (1). Art. 293, p. 267. In connection with the determinant form here given it may be stated that the condition that the intersection of two lines Ax + μy + v2, λ'x + μ'y + v'z should lie on the conic, is the vanishing of the determinant (2) Art. 228, Ex. 10, p. 217. Add, Either factor combined with lp+mp'+np"+pp""=0 gives a result of the form λp + up' + vp′′ = 0, where λ + μ + v = 0, which represents a curve of the third degree. (3). Art. 372, p. 337. The discrimination of the cases of four real and four imaginary points has been made by Kemmer (Giessen, 1878). His result is that if D= 2'2+18AA’®®' – 27A2▲22 – 4A®'3 — 4A′®3, L = 2 (Θ' - 3ΔΘ) Σ - (ΘΘ' - 9ΔΔ') Φ + 2 (Θ2 – 3ΔΘ') Σ', M = ‡ {L2 — (Þ2 — 4££') D}, N = D{A'23 - A′O'ĢE2 + (®22 — 2A′O) Σ2Σ' + A'®Eğ2 + (®2 — 2A®') ΣΣ'2 – ▲A'μ3 + ΔΘ'ΦΩΣ' - ΔΕΦΣ" + Δ2Σ" - (ΘΘ' - 3ΔΔ) ΣΣ'Φ}, we must have D and M positive, L and N negative, in order to have four real points of intersection. I add a selection from some miscellaneous notes which had been sent me at various times by Messrs. Burnside, Walker, and Cathcart, to be used when a new edition was called for, but which I did not remember to insert in their proper places. (4) B. Art. 231, Ex. 10. If the normals at four points meet in a point, their eccentric angles are connected by the relation a+ẞ+y+8 = (2m + 1) π. Hence (see Art. 244, Ex. 1) the circle through the feet of three of the normals from any point passes through the point on the conic opposite to the fourth point. (5) B. If 1, 2, 3, 4 be the feet of four normals from a point, and r12 denote the semi-diameter parallel to the chord 12, then r212 + r234 = a2 + b2. where F 2 √(- S'A) = " F (6) B. Art. 169, Ex. 3. To any rectangular axes, tan has the same meaning as in Art. 383. If the coordinates be trilinear, the right-hand side is multiplied by M, which is the value of x sin A + y sin B + z sin C. 211 √(-2) Θ'Σ - ΔΩ' (7) B. If the tangents be drawn from the pole of ax+By+yz, tan p= where Σ, O, O' have the same meaning as in Art. 382, is the quantity representing tangentially the circular points at infinity, viz. a2 + ß2 + y2 - 2ẞy cos A - 2ya cos B - 2aß cos C'; and II=0 is the condition that ax + By + y2, and the line at infinity should b conjugate, or II = Aa sin 4 + BB sin B + Cy sin C + F 3 sin C + y sin B) + G (y sin A + a sin C) + H (a sin B + ẞ sin 4). As a particular case, the angle between the asymptotes, for which 2 = 0, Σ = 0 = IIe 2 (−→) is tan = (8) B. The length of the chord intercepted on any line is given by the two, following equations, p being the parallel semi-diameter: (9) B. If II = Aaa' + Bßß' + Cyy' + F (By' + B′z) +G (ya' + y'a) +H (aß' + a′ß), the Jacobian of II, Σ, Q is a parabola touching aʼx + f'y + y'z = 0, the normals where this line meets the conic, and the two axes. (10) B. The area of a triangle circumscribing a conic is (11). The squares of the semi-axes of the conic are given by the quadratic RO3+RM2A00' + M1A2 = 0. (12). The equation of a conic circumscribing a triangle, of which a, b, c are the sides and b', b', b'" the semi-diameters parallel to them, is (13) W. The area of the triangle formed by the polars with respect to an ellipse of a2b2 (PQR)2 points P, Q, K, is 4 (QOR) (ROP) (POR)' formed by P, Q, and the centre. where (QOR) is the area of the triangle (14) W. If P, Q, R be the middle points of the sides of a circumscribing triangle, and a, ẞ, y the eccentric angles of the point of contact, (QOR) = ab tan (B − y). From this expression can easily be deduced Faure's theorem (Art. 381, Ex. 12). (15) C. The relation (Art. 388a) is a particular case of the following connecting the covariants of three conics: 4AA'A′′UVW + F1F1⁄2F ̧ − ▲UF¡¦2 – ▲'VF2 – A′′WF3 = 12, where I=0 denotes the locus of the point whence tangents to the three conics are in involution (see Art. 388c). (16) C. Art. 383, p. 352. The expression in the trilinear equation of the director circle there given, may be written where O'S − {L2 + M2 + N2 – 2MN cos A - 2NL cos B – 2LM cos C' }, L = ax + hy + gz, M = hx + by +fz, N=gx+fy + cz. INDEX. Angle, between two lines whose Cartesian between two tangents to a conic, 166, between two conjugate diameters, 169. between focal radius vector and tan- subtended at focus by tangent from subtended at limit points of system of theorems respecting angles subtended at focus proved by reciprocation, 284, fundamental theorem proved, 55. property of four points on a conic, of four tangents, 252, 288. of three tangents to a parabola, 299. of four points on a conic when equal on a different conic, 252, 303. of four diameters equal that of their of segments of tangent to one of three Apollonius, 328. Area, of triangle formed by three normals, constant, of triangle formed by join- constant, between any tangent and of polar triangles of middle points of of triangles equal, formed by drawing line cutting off from a curve constant Asymptotes, Axes, defined as tangents through centre are self-conjugate, 167. are diagonals of a parallelogram whose portion of tangent between, bisected equal intercepts on any chord between constant length intercepted on by chords joining two fixed points to parallel to, how cut by same chords, by two tangents and their chord, bisected between any point and its parallels to, through any point on how divide any semi-diameter, 298. of conic, equation of, 156. lengths, how found, 158, 392. how found when two conjugate dia- of reciprocal curve, 291. EEE. of mean position of given points, 50. radical, 99, 282. of similitude, 105, 224, 282. chords joining ends of radii through c.s. meet on radical axis, 107,224, 250. Chasles, theorems by, 295, 300, 304, 377, 389. ing focus to its pole, 183, 321. Circle, equation of, 14, 75, 87. tangential equation of, 120, 124, 128, trilinear equation of, 128. passes through two fixed imaginary through middle points of sides (see which cuts two at constant angles, cutting three at right angles, 102, 130, or at a constant angle, 132. cutting three at same angle have Circle circumscribing triangle formed by circumscribing, or inscribed, in a self- Circumscribing triangles, six vertices of to two conics, 344. their eight points of contact lie on a three points should be on a right three lines meet in a point, 32, 34. two lines should be perpendicular, a right line should pass through a equation of second degree should re- a circle, 75, 121, 352. a parabola, 141, 274, 352. two circles should be concentric, 77. subtend a right angle at a given two circles should cut at right angles, that four circles should have common a line should touch a conic, 81, 152, two conics should be similar, 224. two pairs of points should be harmonic four points on a conic should lie on a a line be cut harmonically by two in involution by three conics, 363. three pairs of points form system in a triangle may be inscribed in one |