Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/417

Rh GEOM At the saine time the following problem has been solved :— Prublcin. ——'l‘o dcterniine the centre and also the point correspond- ing to any given point in an elliptical involution of which two pairs of conjugate points are given. § 81. iy the aid of § 53, the points on a conic may be made to correspond to those on a line, so that the row of points on the conic is projective to a row of points on a line. ’e may also have two projective i'ows on the same conic, ang these will be in involution as soon as one point on the conic has the same point corresponding to it all the same to whatever row it belongs. An involution of points on a conic will have the property (as follows from its deﬁni- tion, and from § 53) that the lines which join conjugate points of the involution to any point on the conic are conjugate lines of an involution in a pencil, and that a ﬁxed tangent is cut by the tangents at conjugate points on the conic in points which are again eoiiju- gate points of an involution on the ﬁxed tangent. For such invol- ution on a conic the following theoreni holds :— ’I‘iiI;onI5.i. — The lines iehich join corresponding points in an in- rnlution on a conic all pass through. a_ﬁ.1:cd point; and reciprocally, the points of intersection of conjugate lines in an involution among l:In_4[I’ll/S to a conic lie on a line. We prove the first part only. The involution is determined by two pairs of conjugate points, say by A, A’ and B, ll’ (ﬁg. 32). Let AA’ and BB’ meet in 1’. If we join the points in involution to any point on the conic, and the conjugate points to another point on the conic, INVOLUTIO.'.] we obtain two projective pencils. 'e take A and A’ as centres of these pencils, so that the pencils A(A’Bl5') and A'(AB’l§) are pro- jective, and in perspective position, because AA’ corresponds to A'A. Hence corresponding rays meet in a line, of which two points ai'e found by joining AD’ to A’B and A ll to A’l3’. It follows that the a;cis of peixspectivc is the polar of the point P, where AA’ and BB’ ineet. lf we now wish to construct to any other point C on the conic the corresponding point C’, we join C to A’ and the point where this line cuts p to A. The latter line cuts the conic again in C’. But we know from the theory of pole and polar that the line CC’ passes through 1’. l_''OI.1'TIO.' DE'l'El‘.‘.II.'ED BY A Coxic o.' A LI.'E.—FOCI. _ ,$ 82. The polars, with regard to a conic, of points in a row p torni a pencil 1’ projective to the row (§ 66). This pencil cuts the lus-- of the row p in a projective row. It A is a point in the given row, A’ the point where the polar of A cuts 11, then A and A’ will be corresponding points. lf we take A’ a point in the lirst row, then the polar of A’ will pass through A, so.that A corresponds to A’——in other words, the rows are in in- volution. The conjugate points in this involution are conjugate points with regard to the conic. Conjugate points coincide oiil_v if thc_polar of a point A passes through A—tliat is, if A lies on the come. Hence- . Till.-‘.Ol:E_I'.——.f conic determines on every line in its plane an involu- tiop, in 11'/uch those points are conjugate which are also conjugate 1/‘ll/t rcgarrl to the conic. the line cuts the conic the involution is hyperbolic, the points Q] intersection being thcfoci. If the line touches the conic the involution is parabolic, thc tzro _/l1"l coinciding at the point of contact. . ll the line does not cut the conic the i-ncolution is elliptic, having no Incl. ll’, 011 the other hand, we take a point P in the plain of a conic, we get to each line a through P one conjugate line which joins 1’ to the pole of (I. These pairs of conjugate lines through P form an iiivoliition in the pencil at P. The focal rays of this involution_ nr-"tne tangents drawn from P to the conic. This gives the theorem reciprocal to the last, viz. 2-- E T It Y -103 THEOP.E.I.-—/f conic determines in every pencil in its plane an involution, corresponding lines bcing conjugate lines with regard to the conic. If the point is without the conic the involution is hyperbolic, the tangents from the points being the focal rays. I f the point l ics on the conic the in col at ion is parabolic, the tangent at the point counting for coincident focal rays. If the point is within the conic the involution is elliptic, hazing no focal rays. It will further be seen that the involution determined by a conic on any line p is a section of the involution, which is determined by the conic at the pole 1’ of p. § 83. Definition. ——'l‘lie centre of a pencil in which the conic deter- mines a circular involution is called a “ focus" of the conic. In other words- /! focus is such a point that crery line through it is perpendicular to its conjugate line. The polar to a focus is called a dircctria: of the conic. From the deﬁnition it follows that :— Every focus lies on an axis, for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular; and Every line joining two foci is an axis, for the pcrpendiculars to this line through the foci are conjugate to it. These conjugate lines pass through the pole of the line, the pole lies therefore at in- liiiity, and the line is a diameter, hence by the last property an axis. It follows that all foci lie on one axis, for no line joining a point in one axis to a point in the other can be an axis. As the conic dctcrinines in the pencil which has its centre at a focus a circular involution, no tangents can be drawn from the focus to the conie. }Icnce each focus lies ieithin a conic; and a (ll7‘t'Cl7‘l.1: does not cut the conic. Further properties are found by the following considerations :— § 84. Through a point P one line p can be drawn, which is with rcgai'd to a given conie conjugate to a given line q, viz., that line which joins the point P to the pole of the line 9. If the line q is made to describe a pencil about a point Q, then the line p will describe a pencil about 1’. These two pencils will be projective, for the line p passes through the pole of g, and whilst (1 describes the pencil Q, its pole describes a projective row, and this row is perspec- tive to the pencil l’. Ve now take the point P on an axis of the conic, draw any line p tlii'ougli it, and from the pole of p draw a perpendicular q to 1». Let g cut the axis in Q. Then, in the pencils of conjugate lines, which have their centres at P and Q, the lines p and q are conjugate lines at right angles to one another. Besides, to the axis as a ray in either pencil will correspond in the other the perpcii- dicular to the axis (§ 72). The conic generated by the iiitersec- tioii of corresponding lines in the two pencils is therefore the circle on PQ as diaineter, so that crcry line in P is perpendicular to its corresponding line in Q. 'l‘o every point P on an axis of a conic corresponds thus a point Q, such that conjugate lines through P and Q are perpendicular. Ve shall show that these point-pairs P, Q form an involution. To do this let us move I’ along the axis, and with it the line p, keeping the lattei- parallel to itself. Then P describes a row, p a perspective pencil (of parallels), and the pole of p a projective row. At the same time the line g describes a pencil of parallels perpen- dicular to p, and perspective to the row formed by the pole of p. The point Q, tliereforc, where g cuts the axis, describes a row pro- jective to the row of points P. The two points P and Q describe thus two projective rows on the axis ; and not only does P as a point in the ﬁrst now correspond to Q, but also Q as a point in the first corresponds to P. The two rows therefore form an involution. The centre of this involution, it is easily seen, is the centre of the come. A focus of this inrolation has the property that any two co-n- jugate lines through -it are perpendicular; hence, it is a focus to the come. Such involution exists on each axis. But only one of these can have foei, because all foci lie on the same axis. The involution on one of the axis is elliptic, and appears (§ 80) therefore as the section of two circular involutions in two pencils whose centres lie in the other axis. These centres are foci, hence the one axis contains two foci, the other axis none ; or entry central conic has two foci which lie on one aatis equidistant from the centre. The axis which contains the foci is called the principal axis ; in case of an hyperbola it is the axis which cuts the curve, because the foci lie within the conic. In case of the parabola there is but one axis. The involution on this axis has its centre at inﬁnity. One focus is therefore at inﬁnity, the one focus only is ﬁnite. A parabola has only one focus. §85. If through an_v point 1’ (ﬁg. 33) on a conic the tangent PT and the normal PN (i. e., the perpendicular to the tangent through the point of contact) be drawn, these will be conjugate lines with regard to the conic, and at right angles to each other.