CHAPTER XV. THE PRINCIPLE OF DUALITY; AND THE METHOD OF 298. THE methods of abridged notation, explained in the last chapter, apply equally to tangential equations. Thus, if the constants A, u, v in the equation of a line be connected by the relation ν (aλ+bμ+ cv) (a'λ+b′μ + c ́v) = (a′′λ+b′′μ+c′′v) (a′′′′λ+b′′”μ+c′′v}, the line (Art. 285) touches a conic. Now it is evident that one line which satisfies the given relation is that whose λ, μ, v are determined by the equations μ aλ+bμ+cv=0, a′′λ+b′′μ+ c′′v=0. That is to say, the line joining the points which these last equations represent (Art. 70), touches the conic in question. If then a, ß, y, & represent equations of points, (that is to say, functions of the first degree in A, μ, v) ay = kßd is the tangential equation of a conic touched by the four lines aß, By, yd, da. More generally, if S and S' in tangential coordinates represent any two curves, S-kS represents a curve touched by every tangent common to S and S. For, whatever values of λ, μ, v make both S=0 and S′ = 0, must also make S-kS'=0. Thus, then, if S represent a conic, S-kaß represents a conic having common with S the pairs of tangents drawn from the points a, B. Again, the equation ay=k8” represents a conic such that the two tangents which can be drawn from the point a coincide with the line aß; and those which can be drawn from y coincide with the line yẞ. The points a, y are therefore on this conic, and B is the pole of the line joining them. In like manner, S-a2 represents a conic having double contact with S, and the tangents at the points of contact meet in a; or, in other words, a is the pole of the chord of contact. 2 2 So again, the equation ay = k3ß" may be treated in the same manner as at Art. 270, and any point on the curve may be 2 represented by μ2a + 2μkß+y, while the tangent at that point joins the points μa + kß, μkß + y.* Ex. 1. To find the locus of the centre of conics touching four given lines. Let Σ= 0, Σ'0 be the tangential equations of any two conics touching the four lines; then, by Art. 298, the tangential equation of any other is E+E' = 0. And (see G+kG' F + kF" C+kC'' C + kC' Art. 151) the coordinates of the centre are the form of which shows (Art. 7) that the centre of the variable conic is on the line joining the centres GFG' F' of the two assumed conics, whose coordinates are; and that it divides the distance between them in the ratio C: kC'. Ex. 2. To find the locus of the foci of conics touching four given lines. We have only in the equations (Ex. Art. 258a) which determine the foci to substitute A+ kA' for A, &c., and then eliminate k between them, when we get the result in the form {C (x2- y2)+2Fy - 2Gx + A- B} {C'xy - F'x - G'y + H'} = {C' (x2 — y2) + 2F'y — 2G′x + A′ – B'} {Cxy — Fx — Gy + H}. This represents a curve of the third degree (see Ex. 15, p. 275), the terms of higher order mutually destroying. If, however, E and E' be parabolas, Σ + k' denotes a system of parabolas having three tangents common. We have then C and C" both = 0, and the locus of foci reduces to a circle. Again, if the conics be concentric, taking the centre as origin, we have F, F', G, G′ all = 0. In this case Σ + kƐ' represents a system of conics touching the four sides of a parallelogram and the locus of foci is an equilateral hyperbola.† Ex. 3. The director circles of conics touching four fixed lines have a common radical axis. This is apparent from what was proved, p. 270, that the equation of the director circle is a linear function of the coefficients A, B, &c., and that therefore when we substitute A + kA' for A, &c. it will be of the form S+kS' = 0. This theorem includes as a particular case, "The circles having for diameters the three diagonals of a complete quadrilateral have a common radical axis.” 299. Thus we see (as in Art. 70) that each of the equations used in the last chapter is capable of a double interpretation, according as it is considered as an equation in trilinear or in tangential coordinates. And the equations used in the last chapter, to establish any theorem, will, if interpreted as equations * In other words, if in any system x'y'z', x"y"z", be the coordinates of any two points on a conic, and x""y"""" those of the pole of the line joining them, the coordinates of any point on the curve may be written μ”x' + 2μkx" + x", μ2y' + 2μky""'+y", μ2z' + 2μkz""' + z", while the tangent at that point divides the two fixed tangents in the ratios μ: k, μk: 1. When k = 1, the curve is a parabola. Want of space prevents us from giving illustrations of the great use of this principle in solving examples. The reader may try the question :-To find the locus of the point where a tangent meeting two fixed tangents is cut in a given ratio. + It is proved in like manner that the locus of foci of conics passing through four fixed points, which is in general of the sixth degree, reduces to the fourth when the points form a parallelogram. in tangential coordinates, yield another theorem, the reciprocal of the former. Thus (Art. 266) we proved that if three conics (S, S+ LM, S+ LN) have two points (S, L) common to all, the chords in each case joining the remaining common points. (M, N, M-N), will meet in a point. Consider these as tangential equations, and the pair of tangents drawn from L is common to the three conics, while M, N, M- N denote in each case the point of intersection of the other two common tangents. We thus get the theorem, "If three conics have two tangents common to all, the intersections in each case of the remaining pair of common tangents, lie in a right line." Every theorem of position (that is to say, one not involving the magnitudes of lines or angles) is thus twofold. From each theorem another can be derived by suitably interchanging the words "point" and "line"; and the same equations differently interpreted will establish either theorem. We shall in this chapter give an account of the geometrical method by which the attention of mathematicians was first called to this "principle of duality."* 300. Being given a fixed conic section (U) and any curve (S), we can generate another curve (s) as follows: draw any tangent to S, and take its pole with regard to U; the locus of this pole will be a curve s, which is called the polar curve of S The conic U, with regard to which the pole is taken, is called the auxiliary conic. with regard to U. We have already met with a particular example of polar curves (Ex. 12, Art. 225), where we proved that the polar curve of one conic section with regard to another is always a curve of the second degree. We shall for brevity say that a point corresponds to a line when we mean that the point is the pole of that line with regard to U. Thus, since it appears from our definition that every point of s is the pole with regard to U of some tangent to S, we shall * The method of reciprocal polars was introduced by M. Poncelet, whose account of it will be found in Crelle's Journal, vol. IV. M. Plücker, in his "System der Analytischen Geometrie," 1835, presented the principle of duality in the purely analytical point of view, from which the subject is treated at the beginning of this chapter. But it was Möbius who, in his "Barycentrische Calcul," 1827, had made the important step of introducing a system of coordinates in which the position of a right line was indicated by coordinates and that of a point by an equation. briefly express this relation by saying that every point of s corresponds to some tangent of S. 301. The point of intersection of two tangents to S will correspond to the line joining the corresponding points of s. This follows from the property of the conic U, that the point of intersection of any two lines is the pole of the line joining the poles of these two lines (Art. 146). Let us suppose that in this theorem the two tangents to S are indefinitely near, then the two corresponding points of s will also be indefinitely near, and the line joining them will be a tangent to s; and since any tangent to S intersects the consecutive tangent at its point of contact, the last theorem becomes for this case: If any tangent to S correspond to a point on s, the point of contact of that tangent to S will correspond to the tangent through the point on s. Hence we see that the relation between the curves is reciprocal, that is to say, that the curve S might be generated from s in precisely the same manner that s was generated from S. Hence the name "reciprocal polars." 302. We are now able, being given any theorem of position concerning any curve S, to deduce another concerning the curve s. Thus, for example, if we know that a number of points connected with the figure S lie on one right line, we learn that the corresponding lines connected with the figure s meet in a point (Art. 146), and vice versâ; if a number of points connected with the figure S lie on a conic section, the corresponding lines connected with s will touch the polar of that conic with regard to U; or, in general, if the locus of any point connected with S be any curve S', the envelope of the corresponding line connected with s is s', the reciprocal polar of S'. 303. The degree of the polar reciprocal of any curve is equal to the class of the curve (see note, Art. 145), that is, to the number of tangents which can be drawn from any point to that curve. For the degree of s is the same as the number of points in which any line cuts s; and to a number of points on s, lying on a right line, correspond the same number of tangents to S passing through the point corresponding to that line. Thus, if S be a conic section, two, and only two, tangents, real or imaginary, can be drawn to it from any point (Art. 145); therefore, any line meets s in two, and only two points, real or imaginary; we may thus infer, independently of Ex. 12, Art. 225, that the reci procal of any conic section is a curve of the second degree. 304. We shall exemplify, in the case where S and s are conic sections, the mode of obtaining one theorem from another by this method. We know (Art. 267) that "if a hexagon be inscribed in S, whose sides are A, B, C, D, E, F, then the points of intersection, AD, BE, CF, are in one right line." Hence we infer, that "if a hexagon be circumscribed about s, whose vertices are a, b, c, d, e, f, then the lines, ad, be, cf, will meet in a point (Art. 265). Thus we see that Pascal's theorem and Brianchon's are reciprocal to each other, and it was thus, in fact, that the latter was first obtained. In order to give the student an opportunity of rendering himself expert in the application of this method, we shall write in parallel columns some theorems, together with their reciprocals. The beginner ought carefully to examine the force of the argument by which the one is inferred from the other, and he ought to attempt to form for himself the reciprocal of each theorem before looking at the reciprocal we have given. He will soon find that the operation of forming the reciprocal theorem will reduce itself to a mere mechanical process of interchanging the words "point" and "line," "inscribed " and "circumscribed," "locus" and "envelope," &c. If two vertices of a triangle move along fixed right lines, while the sides pass each through a fixed point, the locus of the third vertex is a conic section. (Art. 269). If, however, the points through which the sides pass lie in one right line, the locus will be a right line. (Ex. 2. p. 41). In what other case will the locus be a right line? (Ex. 3, p. 42). If two sides of a triangle pass through fixed points, while the vertices move on fixed right lines, the envelope of the third side is a conic section. If the lines on which the vertices move meet in a point, the third side will pass through a fixed point. In what other case will the third side pass through a fixed point? (p. 49). If two conics touch, their reciprocals will also touch; for the first pair have a point common, and also the tangent at that point common, therefore the second pair will have a tangent common and its point of contact also common. So likewise if two conics have double contact their reciprocals will have double contact. |