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two vertices. Then it is easy to see that (ab) (cd) — (ad) (bc) is LM - R2 multiplied by the factor (a — c) (b − d), and hence that if we compare, as in Art. 268, the forms (ab) (cd) — (ad) (bc), (af) (de) — (ad) (ef) we get the equation of the Pascal abcdef in

the form

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The same equation might also have been obtained in the forms, which can easily be verified as being equivalent,

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The three other Pascals which pass through (bc) (ef) are

(a−c) (b − d) (eƒ) = (a − ƒ) (e — d) (bc),

(a - b) (c — d) (eƒ) = (a − e) (ƒ − d) (be),

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these being respectively the Pascals abcdfe, acbdef, acbdfe. Consider the three Pascals

-

(a−c) (b − d) (ef) = (a − e) (ƒ − d) (bc) = (b −ƒ) (c − e) (ad); these evidently intersect in a point, viz. a Steiner g-point; but the three (a − c) (b − d) (eƒ ) = (a − e) (ƒ − d) (bc) = (b − e) (c − ƒ) (ad)

intersect in a Kirkman h-point.

Mr. Cathcart has otherwise obtained the equation of the Pascal line in a deter minant form. It was shewn (Art. 331) that the relation between corresponding points of two homographic systems is of the form

Aaa' + Ba + Ca' + D = 0.

Hence, eliminating A, B, C, D, we see that the relation between four points and other four of two homographic systems is

aa', a, a', 1

BB', B, p', 1 γγ', γ', γ', 1

ôô, ô, d', 1 | = 0,*

and the double points of the system are got by putting &'d, and solving the quadratic for d. But we saw Art. 289, Ex. 10, that the Pascal line LMN passes through K, K' the double points of the two homographic systems determined by ACE, DFB the alternate vertices of the hexagon. And since, if d be the parameter of the point K, we have M, R, L respectively proportional to ô2, ô, 1, it follows that the equation of the Pascal abcdef is

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SYSTEMS OF TANGENTIAL COORDINATES, Art. 311.

Through this volume we have ordinarily understood by the tangential coordinates of a line la + mẞ+ny, the constants l, m, n in the equation of the line (Art. 70); and by the tangential equation of a curve the relation necessary between these constants in order that the line should touch the curve. We have preferred this method because it is the most closely connected with the main subject of this volume, and because all other systems of tangential coordinates may be reduced to it. We

* On this determinant see Cayley, Phil. Trans., 1858, p. 436.

wish now to notice one or two points in this theory which we have omitted to mention, and then briefly to explain some other systems of tangential coordinates. We have given (Ex. 6, Art. 132) the tangential equation of a circle whose centre is a'B'y' and radius r, viz.

(la' + mẞ' + ny′)2 = p2 (l2 + m2 + n2 2mn cos A-2nl cos B-2lm cos C'); let us examine what the right-hand side of this equation, if equated to nothing, would represent. It may easily be seen that it satisfies the condition of resolvability into factors, and therefore represents two points. And what these points are may be seen by recollecting that this quantity was obtained (Art. 61) by writing at full length la+mẞ+ny, and taking the sum of the squares of the coefficients of a and y, cos a + m cos ẞ + n cos y, l sin a + m sin ẞ + n sin y. Now if a2 + b2 = 0, the line ax + by + c is parallel to one or other of the lines xy √(−1) = 0, the two points therefore are the two imaginary points at infinity on any circle. And this appears also from the tangential equation of a circle which we have just given: for if we call the two factors w, w', and the centre a, that equation is of the form a2 = r2ww', showing that w, w' are the points of contact of tangents from a. In like manner if we form the tangential equation of a conic whose foci are given, by expressing the condition that the product of the perpendiculars from these points on any tangent is constant, we obtain the equation in the form

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(la' + mß' + ny') (la" + mß" + ny") = b2ww',

conic is touched by the lines joining the two foci to the points

It appears from Art. 61 that the result of substituting the tangential coordinates of any line in the equation of a point is proportional to the perpendicular from that point on the line; hence the tangential equations aß = kyd, ay = kẞ2 when interpreted give the theorems proved by reciprocation Art. 311. If we substitute the coordinates of any line in the equation of a circle given above, the result is easily seen to be proportional to the square of the chord intercepted on the line by the circle. Hence if E, E' represent two circles, we learn by interpreting the equation E' that the envelope of a line on which two given circles intercept chords having to each other a constant ratio is a conic touching the tangents common to the two circles.

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Lastly, it is to be remarked that a system of two points cannot be adequately represented by a trilinear, nor a system of two lines by a tangential equation. If we are given a tangential equation denoting two points, and form, as in Art. 285, the corresponding trilinear equation, it will be found that we get the square of the equation of the line joining the points, but all trace of the points themselves has disappeared. Similarly if we have the equation of a pair of lines intersecting in a point a'ß'y', the corresponding tangential equation will be found to be (la' + mẞ' + ny')2=0. In fact, a line analytically fulfils the conditions of a tangent if it meets a curve in two coincident points; and when a conic reduces to a pair of lines, any line through their intersection must be regarded as a tangent to the system.

The method of tangential coordinates may be presented in a form which does not presuppose any acquaintance with the trilinear or Cartesian systems. Just as in trilinear coordinates the position of a point is determined by the mutual ratios of the perpendiculars let fall from it on three fixed lines, so (Art. 311) the position of a line may be determined by the mutual ratios of the perpendiculars let fall on it from three fixed points. If the perpendiculars let fall on a line from two points A, B be λ,, then it is proved, as in Art. 7, that the perpendicular on it from the + m point which cuts the line AB in the ratio of m: lis 7+ m and consequently that if the line pass through that point we have + mμ-0, which therefore may be

regarded as the equation of that point. Thus A+μ= 0 is the equation of the middle point of AB, λ = 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:l; or by cutting AC::1: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 0 in the form

BOC.X+COA. μ + AOB. v = 0.

Or, again, substituting for each triangle BOC its value p'p" sin 0 (see Art. 311)

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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 + mμ + nv will be at infinity when l+m+ n = 0; and generally a curve will be touched by the 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 = 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 18 (Art. 61)

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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 perpen. diculars A, μ, 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 A, μ, V. It is evident from the actual values that A, μ, are connected by the relation

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

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

λ

μ

sin COE' ein 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.

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intercepts made by a variable line on two fixed parallel lines, then any equation aλ + bμ + c = 0, denotes a point; and this equation may be considered as the form assumed by the homogeneous equation aλ + bu+ cv = 0 when the point = 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 . 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

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point, and an equation of the nth degree denotes a curve or the na class.

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 1, 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.

α = αλ? + 20λμ + όμ", y = αλ3 + 20 ́λμ + δ'με, ε = αλ2 + 20"λμ + 6μ. Then, solving these equations for λ2, 2λμ, μ2, we have (Higher Algebra, Art. 29) ▲λ2 = Ax + A'y + A′′z, 2^\μ = Hx + H'y + H"z, ▲μ2 = 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′′z) X2 + 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′′z) = (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

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

Aa = Al2 + Hlm + Bm2, Aẞ = A'l2 + II'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, u is

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Expanding this and regarding it as the equation of a variable line containing the parameter Aμ, 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 XXX' + Y (\μ' + λ'μ) + Zμμ' = 0, and we can otherwise exhibit it in this form, for the coordinates of either point satisfy the equations x=aλ2+2h\μ+bμ2, &c., and we have also μ'μ”X2 — λμ (X'μ” +λ′′μ')+X'λ′′μ3 = 0; hence, eliminating X2, λμ, μ3, we have

μ'μ", - ('μ" + λ"μ'), λλ"

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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, Ce 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

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