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shall not think it worth while to treat separately the cases where any of these are at an infinite distance, for which the constructions for the general case only require to be suitably modified. Thus to be given a parallel to an asymptote is equivalent to one condition, for we are then given a point of the curve, namely, the point at infinity on the given parallel. If, for example, we were required to describe a conic, given four points and a parallel to an asymptote, the only change to be made in the construction (Art. 269) is to suppose the point E at infinity, and the lines DE, QE therefore drawn parallel to a given line.

To be given an asymptote is equivalent to two conditions, for we are then given a tangent and its point of contact, namely, the point at infinity on the given asymptote. To be given that the curve is a parabola is equivalent to one condition, for we are then given a tangent, namely, the line at infinity. To be given that the curve is a circle is equivalent to two conditions, for we are then given two points of the curve at infinity. To be given a focus is equivalent to two conditions, for we are then given two tangents to the curve (Art. 258a), or we may see otherwise that the focus and any three conditions will determine the curve; for by taking the focus as origin, and réciprocating, the problem becomes, to describe a circle, three conditions being given; and the solution of this, obtained by elementary geometry, may be again reciprocated for the conic. The reader is recommended to construct by this method the directrix of one of the four conics which can be described when the focus and three points are given. Again, to be given the pole, with regard to the conic, of any given right line, is equivalent to two conditions; for three more will determine the curve. For (see figure, Art. 146) if we know that P is the polar of R'R", and that T is a point on the curve, T', the fourth harmonic, must also be a point on the curve; or if OT be a tangent, OT' must also be a tangent; if then, in addition to a line and its pole, we are given three points or tangents, we can find three more, and thus determine the curve. Hence, to be given the centre (the pole of the line at infinity) is equivalent to two conditions. It may be seen likewise that to be given a point on the polar of a given point is equivalent to one condition. For example, when we are given that the curve is an equilateral hyperbola, this is the same as saying that the two points at infinity on any circle lie each on the polar of the other with respect to the curve. To be given a self-conjugate triangle is equivalent to three conditions; and when a self-conjugate triangle with regard to a parabola is given three tangents are given.

Given five points.-We have shown, Art. 269, how by the ruler alone we may deter mine as many other points of the curve as we please. We may also find the polar of any given point with regard to the curve; for by the help of the same Article we can perform the construction of Ex. 2, Art. 146. Hence too we can find the pole of any line, and therefore also the centre.

Five tangents.-We may either reciprocate the construction of Art, 269, or reduce this question to the last by Ex. 4, Art. 268.

Four points and a tangent.-We have already given one method of solving this question, Art. 345. As the problem admits of two solutions, of course we cannot expect a construction by the ruler only. We may therefore apply Carnot's theorem (Art. 313),

Ac. Ac'. Ba. Ba'. Cb. Cb' Ab. Ab'. Bc. Bc. Ca. Ca'.

Let the four points a, a', b, b' be given, and let AB be a tangent, the points c, c' will coincide, and the equation just given determines the ratio Ac2: Be2, everything else in the equation being known. This question may also be reduced, if we please, to those which follow; for given four points, there are (Art. 282) three points whose polars are given; having also then a tangent, we can find three other tangents immediately, and thus have four points and four tangents.

Four tangents and a point.-This is either reduced to the last by reciprocation, or

by the method just described; for given four tangents, there are three points whose polars are given (Art. 146).

Three points and two tangents.—It is a particular case of Art. 344, that the pair of points where any line meets a conic, and where it meets two of its tangents, belong to a system in involution of which the point where the line meets the chord of contact is one of the foci. If, therefore, the line joining two of the fixed points a, b, be cut by the two tangents in the points A, B, the chord of contact of those tangents passes through one or other of the fixed points F, F", the foci of the system (a, b, A, B), (see Ex. Art. 286). In like manner the chord of contact must pass through one or other of two fixed points G, G' on the line joining the given points a, c. The chord must therefore be one or other of the four lines, FG, FG', F'G, F'G' ; the problem, therefore, has four solutions.

Two points and three tangents.-The triangle formed by the three chords of contact has its vertices resting one on each of the three given tangents; and by the last case the sides pass each through a fixed point on the line joining the two given points; therefore this triangle can be constructed.

To be given two points or two tangents of a conic is a particular case of being given that the conic has double contact with a given conic. For the problem to describe a conic having double contact with a given one, and touching three lines, or else passing through three points, see Art. 328, Ex. 10. Having double contact with two, and passing through a given point, or touching a given line, see Art. 287. Having double contact with a given one, and touching three other such conics, see Art. 387, Ex. 1.

ON SYSTEMS OF CONICS SATISFYING FOUR CONDITIONS.

If we are only given four conditions, a system of different conics can be described satisfying them all. The properties of systems of curves, satisfying one condition less than is sufficient to determine the curve, have been studied by De Jonquières, Chasles, Zeuthen, and Cayley. References to the original memoirs will be found in Prof. Cayley's memoir (Phil. Trans., 1867, p. 75). Here it will be enough briefly to state a few results following from the application of M. Chasles' method of characteristics. Let u be the number of conics satisfying four conditions, which pass through a given point, and the number which touch a given line, then μ, are said to be the two characteristics of the system. Thus the characteristics of a system of conics passing through four points are 1, 2, since, if we are given an additional point, only one conic will satisfy the five conditions we shall then have; but if we are given an additional tangent two conics can be determined. In like manner for three points and a tangent, two points and two tangents, a point and three tangents, four tangents, the characteristics are respectively (2, 4), (4, 4), (4, 2), (2, 1). We can determine a priori the order and class of many loci connected with the system by the help of the principle that a curve will be of the nth order, if it meet an arbitrary line in n real or imaginary points, and will be of the nth class if through an arbitrary point there can be drawn to it n real or imaginary tangents. Thus the locus of the pole of a given line with respect to a system whose characteristics are u, v, will be a curve of the order v. For, examine in how many points the locus can meet the given line itself. When it does, the pole of the line is on the line, or the line is a tangent to a conic of the system. By hypothesis this can only happen in cases, therefore v is the degree of the locus. This result agrees with what has been already found in particular cases, as to the order of locus of centre of a conic through four points, touching four lines, &c. In like manner let us investigate the order of the locus of the foci of conics of the system. To do this let us generalize the question, by the help of the conception of foci explained Art. 258a, and we shall see that the problem is a particular case of the following: Given two points A, B to find the order of the locus of the intersection of either tangent drawn from A to

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 should 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 z' and the nth in z, 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 linev 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 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, 86). 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 u conics of any system reduce to a pair of lines, and 2μ- to a pair of points.

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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 λx + y + vz, X'x + p'y + v'z should lie on the conic, is the vanishing of the determinant

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(2) Art. 228, Ex. 10, p. 217. Add, Either factor combined with lp+mp'+np"+pp""=0 gives a result of the form Xp + μp' + 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'OO' – 27▲2A”2 – 4A®'3 — 4A′O3,

L = 2 (Θ" - 3Δ'Θ) Σ -- (ΘΘ' - 9ΔΔ') Φ + 2 (Θ – 3ΔΘ') Σ',

M = {L2 - ($22 — 4££') D},

N = D {▲22Σ3 — A'O'¢E2 + (O22 — 2A′O) Σ2E'

+ Δ'
ΘΣΦ? + (Θ' - 2ΔΘ') ΣΣ' - ΔΔ' Φο

+ ΔΘ'Φ'Σ' - 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 + d = (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 r2 12 + r234 = a2 + b2.

2 √(- S'A)

where F

9 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 a sin A + y sin B + 2 sin C.

211 √(-2) (7) B. If the tangents be drawn from the pole of ax+By+y%, tan p = μ'£ - A' where Σ, O, O' have the same meaning as in Art. 382, is the quantity representing tangentially the circular points at infinity, viz.

a2 + B2 + y2 — 2ẞy cos A - 2ya cos B-2aß cos C;

and II0 is the condition that az + By + yz, and the line at infinity should b conjugate, or

II = Aa sin 4 + BB sin B + Cy sin C + F (B sin C + y sin B) + G (y sin A + a sin C) + H (a sin B+ ẞ sin A). As a particular case, the angle between the asymptotes, for which 20, 20 = Пe is tan =

e'

=

(8) B. The length of the chord intercepted on any line is given by the two, following equations, p being the parallel semi-diameter:

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(9) B. If II = Aaa' + Bßß' + Cyy' + F (By' + B′y) + G (ya' + y'a) + H (aß' + a'ß), the Jacobian of II, 2, 2 is a parabola touching a'x + ẞ'y + y'z = 0, the normals where this line meets the conic, and the two axes.

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(11). The squares of the semi-axes of the conic are given by the quadratic

R13+ R2M2AOO' + 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

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(13) W. The area of the triangle formed by the polars with respect to an ellipse of a2b2 (PQR)2 points P, Q, R, 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) = fab tan § (ẞ − 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'▲"UVW + F1F2F ̧ − AUF,? — ▲'VF? — ▲′′WF32 = 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.

L = ax + hy + gz,

2MN cos A 2NL cos B-2LM cos C'},

M = hx + by +fz, N = gx +fy + cz.

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