--

==

regarded as the equation of that point. Thus A+μ = 0 is the equation of the middle point of AB, X-μ= 0 that of a point at infinity on AB. In like manner (see Art. 7, Ex. 6) it is proved that lλ + mμ + nv = 0 is the equation of a point 0, which may be constructed (see fig. p. 61) either by cutting BC in the ratio n m and AD in the ratio m+n: 1; or by cutting AC:::n and BE::1+n: m, or by cutting AB: ml and CF:: 1+m: n. Since the ratio of the triangles AOB: AOC is the same as that of BD: BC, we may write the equation of the point in the form

BOC.λ+ COA. μ + AOB. v = 0.

Or, again, substituting for each triangle BOC its value p’p" sin ✪ (see Art. 311)

μ sin ' v sin e"

0.

ρ

+mu + ny will touched by the

Thus, for example, the coordinates of the line at infinity are λ=μv, since all finite points may be regarded as equidistant from it; the point be at infinity when l+m+ n = 0; and generally a curve will be line at infinity if the sum of the coefficients in its equation = 0. So again the equations of the intersections of bisectors of sides, of bisectors of angles, and

the perpendiculars, of the triangle of reference are respectively λ + μ + v = 0, λ sin A + μ sin B + v sin C = 0, λ tan A + μ tan B + v tan C = 0. It is unnecessary to give further illustrations of the application of these coordinates because they differ only by constant multipliers from those we nave used already. The length of the perpendicular from any point on la + mẞ + ny is (Art. 61)

the denominator being the same for every point. If then p, p', p" be the perpendiculars let fall from each vertex of the triangle on the opposite side, the perpendiculars λ, μ, v from these vertices on any line are respectively proportional to lp, mp', np"; and we see at once how to transform such tangential equations as were used in the preceding pages, viz. homogeneous equations in l, m, n, into equations expressed in terms of the perpendiculars λ, μ, v. It is evident from the actual values that λ, μ, v are connected by the relation

It was shown (Art. 311) how to deduce from the trilinear equation of any curve the tangential equation of its reciprocal.

When O is given every thing in this equation is constant except the two variables

λ

μ sin COE' sin COE'

but since sin COE = sin ODA, these two variables are respectively AD, BE.

In other words, if we take as coordinates AD, BE the
DDD.

=

intercepts made by a variable line on two fixed parallel lines, then any equation αλ + όμ + c = 0, denotes a point; and this equation may be considered as the form assumed by the homogeneous equation aλ + bμ + cv = 0 when the point v = 0 is at infinity. The following example illustrates the use of coordinates of this kind We know from the theory of conic sections that the general equation of the second degree can be reduced to the form aß k2, where a, ẞ are certain linear functions of the coordinates. This is an analytical fact wholly independent of the interpretation we give the equations. It follows then that the general equation of curves of the second class in this system can be reduced to the same form aß = k2, but this denotes a curve on which the points a, ẞ lie and which has for tangents at these points the parallel lines joining a, ẞ to the infinitely distant point k. We have then the well known theorem that any variable tangent to a conic intercepts on two fixed parallel tangents portions whose rectangle is constant.

Again, let two of the points of reference be at infinity, then, as in the last case the equation of a line becomes

point, and an equation of the nth degree denotes a curve of the n1à

class.

C

It is evident that tangential equations of this kind are identical with that form of the tangential equations used in the text where the coordinates are the coefficients l, m, in the Cartesian equation lx + my = 1, or the mutual ratios of the coefficients n the Cartesian equation lx + my + n = 0.

EXPRESSION OF THE COORDINATES OF A POINT ON A CONIC BY A SINGLE

PARAMETER.

We have seen (Art. 270) that the coordinates of a point on a conic can be expressed as quadratic functions of a parameter. We show now, conversely, that if the coordinates of a point can be so expressed, the point must lie on a conic. Let us write down the most general expressions of the kind, viz.

x = aλ2 + 2hλμ + bμ2, y = a'λ2 + 2h'λμ + b'μ2, z = a′′λ2 + 2h′′λμ + b′′μ2. Then, solving these equations for λ2, 2λμ, μ2, we have (Higher Algebra, Art. 29) AX2 = Ax+ A'y + A′′z, 2Aλμ = Hx + H'y + H"z, Aμ2 = Bx + B'y + B′′z, where A is the determinant formed with a, h, b, &c., and A, H, B, &c. are the minors of that determinant. The point then, evidently, lies on the locus

(Hx + H'y + H'z)2 = 4 (Ax + A'y + A′′z) (Bx + B'y + B'z).

If we look for the intersection with this conic of any line ax + By + yz, we have only to substitute in the equation of this line the parameter expressions for x, y, z, and we find that the parameters of the intersection are determined by the quadratic (aa + a'ß + a′′y) λ2 + 2 (ha + h'ß + h′′y) λμ + (ba + b′ß + b′′y) μ2 = 0.

The line will be a tangent if this equation be a perfect square, in which case we

must have

(aa + a'ß + a"y) (ba + b'ß + b′′y) = (ha + h'ß + h''y)2, which may be regarded as the equation of the reciprocal conic. If this condition is satisfied, we may assume

whence Δα

aa + a'ß + a′′y = l2, ha + h'ß + h" y = lm, ba + b′ß + b′′y = m2,

Al2 + Hlm + Bm2, Aß = A'l2 + H'lm + B'm2, Ay = A′′l2 + H"lm + B′′m2 ; that is to say, the reciprocal coordinates may be similarly expressed as quadratic functions of a parameter, the constants being the minors of the determinant formed with the original constants.

The equation of the conic might otherwise have been obtained thus: The equation of the line joining two points is (Art. 132a) got by equating to zero the determinant formed with x, y, z ; x', y', z'′; x", y", z". If the two points are on the curve, we may substitute for their coordinates their parameter expressions; and when the two points are consecutive, we see, by making an obvious reduction of the determinant, that the equation of the tangent corresponding to any point λ, μ is

Expanding this and regarding it as the equation of a variable line containing the parameter : μ, its envelope, by the ordinary method, gives the same equation as before.

The equation of the line joining two points will be found, when expanded, to be of the form Xλλ' + Y (λμ' + λ'μ) + Zμμ' = 0, and we can otherwise exhibit it in this form, for the coordinates of either point satisfy the equations x=aλ2+2hλμ+bμ2, &c., and we have also μ'μ”λ2 — λμ (\'μ" +λ′′μ')+λ'λ′′μ2 = 0; hence, eliminating λ2, λμ, μ2, we have

μ'μ", - (λ'μ" + λ"μ'), λλ"

If the parameters of any number of points on a conic be given by an algebraic equation, the invariants and covariants of that binary quantic will admit of geometric interpretation (see Burnside, Higher Algebra, Art. 190). A quadratic has no invariant but its discriminant, and when we consider two points there is no special case, except when the points coincide. In the case of two quadratics their harmonic invariant expresses the condition that the two corresponding lines should be conjugate and their Jacobian gives the points where the curve is met by the intersection of these lines. If we consider three points whose parameters are given by a binary cubic, the covariants of that cubic may be interpreted as follows: Let the three points be a, b, c, and let the triangle formed by the tangents at these points be ABC; these two triangles being homologous, then the Hessian of the binary cubic determines the parameters of the two points where the axis of homology of these triangles meets the conic; and the cubic covariant determines the parameters of the three points where the lines Aa, Bb, Cc meet the conic. In like manner, if there be four points the sextic covariant of the quartic determining their parameters, gives the parameters of the points where the conic is met by the sides of the triangle whose vertices are the points ab, cd; ac, bd; ad, bc.

ON THE PROBLEM TO DESCRIBE A CONIC UNDER FIVE CONDITIONS. We saw (Art. 133) that five conditions determine a conic; we can, therefore, in general describe a conic being given m points and n tangents where m + n = 5. We

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. 258ɑ), or we may see otherwise that the focus and any three conditions will determine the curve; for by taking the focus as origin, and reciprocating, 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 determine 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 μ be the number of conics satisfying four conditions, which pass through a given point, and the number which touch a given line, then μ, v 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