jngate. to it, and the METRICAL PROPE RTIES.:| The point M there- are harmonic with regard to the asymptotes. It follows fore bisects EF. But by construction M bisects CD. that I)F=- ICC, and l5D=Cl" ; or / I Fig. 25. . 'l‘ii iaoi-.i~:.i. —On any set-ant of an hyper-bola the segments between the curve and the as_i/mptotes are equal. If the chord is clniiigcd into a tangent, this gives The se_r/mrnt between the asymptotes on any tangent to an h_i/pcrbola is bz‘sr'clert by the point of contact, The first part allows a simple solution of the problem to find any nnniber of points on an hyperbola, of which the asyniptotes and one point are given. This is eqiiivalent to three points and the tan- gents at two of them. This construction re- quires lllC:'lSl11‘(‘li1CIlt-. § 74. For the para- bola, too, follow some metrical pa-operties. A dian1et<-1' PM (fig. 26) bisects every chord con- pole 1’ of such a chord EU lies on thedianietcr. But a d iaineter cuts the parabola once at iii- linity. Hence- T/zr'nrem.—'l‘lie seg- ment l’.I which joins the middle point M of a chord ofa parabola to the pole 1’ of the chord is bisected by the parabola at A. 3‘ 75. Two asyniptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the liyperbola. But in such a quadrilateral the intersections of the Fig ‘.26. Fig. '27. diagonals and the points of contact of opposite sides lie in a line (,3 54). If therefore. DEFG (fig. 27) is such a quadrilateral, then the diagonals DF and GE will meet on the line which joins the points of contact of the asyniptotes, that is. on the line at iiifinil y; hence GEOMETRY 401 they are parallel. From this the following theorem is a simple deduction :— Theorem.—.-ll triangles formed by a tangent and the asymptotos of an liypcrbola are equal in area. If we draw at a point P (fig. 27) on an hyperbola a tangent, the part IIK between the asymptotes is bisected at 1’. The parallelo- gram PQUQ’ formed by the asyniptotes and lines parallel to them through P will be half the triangle OHK, and will therefore be constant. If we now take the asymptotes OX and OY as oblique axes of coordinates, the lines 06) and Q1’ will be the coordinates of I’, and will satisfy the equation mg/=—const. =a2. Th-e0rem.—For the asyinptotes as axes of coordinates the eq1ia- tion of the hyperbola is a'y=const. It is not diflicult to get the equations to the ellipse aIltlll_‘1)€1'l.J0lR referred to their axes as axes of coordinates. We are satisfied to have shown in one case that the curves of the second order as generated by projective pencils are the same as those which are in coordinate geometry defined by equations of the second degree. lI0L1.'TlO.'. § 76. If we have two projective rows, ABC on u and A’B’C’ on a’, and place their bases on the same line, then each point in this line counts twice, once as a point in the row a and once as a point in the row a’. In fig. 28 we denote the points as points in the one row by letters above the line A, B, C . . ., and as points A B in the second row by A’, B’, C’ 3, i, F . . . . below the line. Let now A B A and B’ be the same point, then to A will correspond a point A’ in the second, and to B’ a point B in the first row. In general these points A’ ainl B will be difl'ereiit. It may, however, happen that they coincide. Then the correspondence is a peculiar one, as the followiiig theorem shows :— Tiii-10111-:.I.—If two gtrojeetiee rows lie on the same base, and if it happens that to one point in the base the same point eu1're.s',2)omls, ushcthcr we consider the point as belonging to the first or to the second row, then. the same will ]t((11])G7L for every poi-at in the basc—— that is to say, to crery point in the line C07"I‘€8]}071(lS the same point in the first as in the second row. Proof. In order to determine the correspondence, we may assume three pairs of corresponding points in two projective rows. Let then A’, B’, C’, in fig. 29, corre- spond to A, B, C, so that A and B’, ' D I3 C and also B and A’, denote the _i, mi‘, 5, i same point. Let us further de- 3 C A 1) note the point C’ when considered Fig_ 29, as a point in the first row by D ; then it is to be proved that the point D’, which corresponds to D, is the same point as C. Ve know that the cross-ratio of four points is equal to that of the coi'responding row. Hence (A BCD) = (A’B'C’D’) but replacing the dashed letters by those undashed ones which denote the same points, the second cross-ratio equals (BADD’), which, according to§ 15, iv., equals (ABD’D) ; so that the equation becomes Fig. 28. (ABCD)=(Al3D’D). This requires that C and D’ coincide. _ § 77. Two projective rows on the same base, which have the above property, that to every point, whether it be considered as a point in the one or in the other row, corresponds the same point, are said to be in involution, or to form an -involution of points on the line. We mention, but without proving it, that any two projective rows may be placed so as to form an involution. _ An involution may be said to consist of a row of pairs of points, to every point A corresponding a point _A’, and to A’ again the point A. These points are said to be conjugate. _ From the definition, according to which an involution may be considered as made up of two projective rows, follow at once the following important properties :— (1.) The cross-ratio of four points equals that of the. four con- jugate points.
- 2.) If we call a point which coincides with its conjugate point a
“ focus” of the involution we may say : An involution has either two foci, or one, or none,’and is called respectively a hyperbolic, paZ'§b)olIic, 0I}'1elll1){)icl.lI1}’0ll1fl(;i1 (§ 34).t _ _ t h . n a yper o ic invo u ion any wo conjugate pom s are ar- inonic conjugates with regard to the two foci, For if A, A’ be two conjugate points, F1, F, the two foci, then to the points F,, F._., A, A’ in the one row correspond the points F1, F2, AI’, A in the other, each focus corresponding to itself. Hence (F,F._,AA_) = (F,F.,A’A)—tliat is, we may interchange the two points AA’ with- out altering the value of the cross-ratio, which is the charac- teristic property of harmonic conjugates (§ 18). v
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