GEOM Th.e tangents drazen. from a point P to a guadric .su:_'/"ace form (2 4-one of the second order, for the polar plane of l’ cuts it in a conic. (5. If the pole describes a line a, its polar plane will turn about another line a’, as follows from 2. These lines a and a’ are said to be conjugate with regard to the surface. § 100. '.l‘lie pole of the line at infinity is called the centre of the surface. If it lies at the infinity, the plane at iiilinity is a tangent; plane, and the surface is called a paraboloid. Th.e polar plane to any point at infinity passes through the centre, and is called a dianwtrical plane. A line. through the centre is called a diameter. It is bisccted at the centre. The line conjugate to it lies at infinity. If a point moves along a diameter -its polar plane turns about the umjugntc line at infinity; that is, it mores parallel to itself, its centre nun-ing on the first line. Thv: middle points of parallel chords lie in a plane, viz., in the polar plane of the point at infinity through which the chords are drawn. The centres ofparallcl sections lie in. a diameter zrhich is a line conjugate to the line at infinity in 'u‘hich the planes -meet. 'rw1s1‘i-:D CUi3iCs.] TWISTEI) Criiics. § 101. If two pencils with ccntrcs S , and S2 are made projective, thcn to a ra_v in one corresponds a ray in the other, to a plane a plane, to a llat or axial pcncil a piojective flat or axial pcncil, and so on. 'l‘hci'c is a double infinite number of liiics in a pencil. Ve shall see that a single iufiiiite nniiibcr of lines in one pencil meets its a-arresponding ray, and that the points of intersection form a curve in space. of the double infinite iiunibcr of planes in the pencils cach will meet its corresponding plane. This gives a system of a. double infinite number of lines in space. Ve know (§ 5) that there is a «pi-aclruple infinite number of lines in space. From among these we may select those which satisfy one or more given eoii- ditioiis. The systems of lines thus obtained was first systeiiiati- cally investigated and classified by Pliickcr, in his Geometric dcs Itaunics. lle uses the following iiaiiics:— A trchlzz in_/initc iiumher of lines, thatis, all lines vliicli satisfy oiie condition, are said to form a complex of lines; c.g., all lines cutting a. given line, or all lines touching a suiface. A double in_/inite number of lines, that is, all lines which satisfy two conditions, or which are common to two coniplcxcs, are said to lhriii a congruence of lines; c.g., all lines in a plane, or all lines cutting two curves, or all lines eutting a given curve twice. A single in_/inite number of lines, that is, all lines which satisfy three conzlitions, or which belong to three complexes, form a ruled .s-mjfaec ; e.g., one set of lines on a ruled quadric siirfacc, or develop- ulile sni-faces which are formed by the tangents to a curve. lt follows that all lines in which coii‘e.spondiiig planes in two projective pencils mcct l'oriu a coiigruciicc. “'0 shall see this con- grueii.-e consists of all lines which cut a twisted cubic twice, or of all .m-ants to a twisted cubic. § 102. Let l, he the line S,S._, as a line in the pencil S1. To it mi'i'cspoii«_ls a line l._, in S1. At each of the centres tzco corrcspono’ing lines meet. The two axial pencils with I, and l2 as axes are pro- jt-('li'(', and, as their axes meet at S2, the intersections of corre- sponding planes form a cone of the second order ('§ 58), with S2 as (L‘lltTC. If 1r, and We be corresponding planes, then their intersec- tion will be a line ])g which passes through S2. Correspoiiding to it in S, will be a line p, which lies in the plane 1r,, and which therefore meets p2 at some point P. (.'oiivci.<-ely, if p_, be any line in S._, which meets its corresponding line p, at a point 1’, then to the plane l2 p._, will correspond the plane l,p,, that is, the plane S,S2l’. 'l‘ln-sc planes intersect in p._,, so that p._, is a line on the quadric eoiic gr-uci'.itcd by the axial pencils 1, and l,. Hence 2-- Alt lines in one pencil zI:h1'ch meet their corresponding lines in the o/hcrforin a cone of the second order which has it ccntrc at the centre of the firs! pencil, and passes tln‘ough. thc centre of the second. I-‘roni this follows that the points in which corresponding rays meet lie on two cones of the second order which have the ray joining their centres in common, and form tlicreforc, l(ig;£tll(‘I with the line Sp‘, or Z,, the iiitcrscction of thcse coin-s. Any plane cuts cach of the cones in a conic. These two conics have necessarily that point in common in which it cuts the line l,, and therefore l.-csidcs either one or three other points. It follows that the curve is of the third order as a plane may cut it in three, but not in more than three, points. lIcnce:——- The locus of points in ichich corresponding lines on t zro projectirc pencils meet is a curve of the third 0?‘(lr..:' or a “t-u'i.stc¢l cubic" k, u-hit.-h passes through the centres or" the pencils, and irhich. appears as the intersection of two con.es of the second order, ’lt'/I-it'll hare one line in common. .-1 l inc l:rlo‘ngi7Ig to the congruence (lctcrniined by the pencils is a arrnnt of the cubic; it has two, or one, or no points in common u'ith this cubic, and is called accordingly (1. sccant proper, a tangent. or a ETRY 407 sccant improper of the cubic. A seeant improper may be considered. to use the language of coordinate geometry, as a sccant with imaginary points of intersection. § 103. If a, and an be any two coirespondiiig lines in the two pencils, then corresponding planes in the axial pencils having (1, and a2 as axes generate a ruled quadric surface. If I’ be any point on the cubic iv, and if 11,, _2)2 be the corresponding rays lll S, and S2 which meet at P, tlicu to the plane (4,1), in S, corre- sponds (L2 12., in S2. These thcrcfore meet in a line through I’. This may be stated thus :— Those secants of the cubic zchich cut a ray a,, clrazcn through the centre S, of one pencil, form a ruled guadric surface which passes through both centres, and zchieh contains the twisted cubic k. such surfaces an infinite number exists. Ercry my through S, or S2 which is not a sccant determines one of them. If, however, the rays a, and a2 are secants meeting at A, then the ruled quadric surface becomes a cone of the second order, having .1 as centre. Or all lines of the congruence echich pass through a point on the twisted cubic k form a cone of the second order. In othcr words, the projection of a twisted cubic from any point in the curve on to any plane is a conic. _ If a, is not a sccant, but made to pass through any point Q in space, the rulcd quadric surface determined by a, will pass through Q. There will therefore be one line cy’ the congruence passing through Q, and only one. For if two such lines pass through Q. then the lines S,Q and S2Q will be corresponding lines; hence Q will be a point on the cubic la, and an infinite number of secants will pass through it. Hcnce :— Through every point in space not on the twisted cubic one and only onc sccant to the cubic can be drawn. § 104. The fact that all the secants through a point on the cubic form a quadric cone shows that the centres of the projective pencils generating the cubic are not distinguished from any other points on the cubic. If we take any two points S, S’ on the cubic, and draw the secants through cach of them, we obtain two quadrie cones, which have the line SS’ in common, and which intersect besides- along the cubic. If we make thcse two pencils liaviiig S and S’ as centres projective by taking four rays on the one cone as corro- sponding to the four rays on the other which meet the first on the ciibic, the coircspoiidciice is determined. These two pencils will generate a ciihic, and the two cones of secants having S and S’ as centres will be identical with the above cones, for each has fivc rays in common with one of the first, viz., the line SS’ and the four liiics determined for thc corrcspoiidciicc ; therefore these two cones intersect in the original cubic. This gives the theorem :— On a twisted cubic any two points may be taken as centres of pro- jcctirc pencils which generate the cubic, coircspoiuling planes being those zchirh meet on the same sccant. Of the two projective pencils at S and S’ we may keep the first fixed, and move the centre of the other along the curve. The pencils will licrehy remain projective, and a plane a in S will he eut by its corresponding plane u’ always in the same sccant (1. Whilst S’ moves along the curve the plane a’ will turn about in. describing an axial pencil. In this article we have given a purely geometrical tlicoi y of coiiics, cones of the second order, quadric surfaces, &c._ In doing so we have followed. to a great cxtent, llcye s Geometric (lcr Lagc, and to this cxccllciit work those readers are rcfcrrcd who wish for a more exhaustive treatment of the subject. lt will have been observed that scarcely any use has been made of algebra, and it would have been even possible to avoid this little, as is done by Ileye. There me, however, other svstenis ol gcoincti-_v which start more or less from theorems known to the Greeks, and using more or less algebia. We cannot do more hcre than eiiuinci-ate a few of the more no- I mincnt works on the siibjcct, which, however, are almost all Con- tinental. These aie the lollowiiig, :- Monac, Geometric Dcsci-iplirc: ('air.ol. Géometrze dc Position (ISO3). contain- ing a thcoiy of tinnsvcrsals. l’oIi(clct's git.-at woik. Truile drs I’roprz'etes I’:o- Jeclzres des I"1'gure: (1822): Mohins. lmryceiilrisclier Culcnl (I826): Steiner. Abluingigkezt Geomelrischer Geslalten (IS-'32). containing the first full discussion of the projective iclutioiis between IUWS. rcncils, il'c.: Von Slnudt, (:'eomelru- (ler Luge (1847) mid Ileilriige zur Geometric dcr Luge (lS.3C--60). in which :1 systcni of txcoinetry is built up fioni the bcgniniiig without any icl'crciice to iiuinhci. so that iihimatcly a number itsclf acts a gcoiiictricnl definition. and in Villltll iiiia,r_,riniii_v elements are s_'stcinaticnll_v intiodiiccd into pure gcomc-tr_v; Cli:islc:-. _-lpercu Ilzstorique (1837). in which the author gixes a brilliant account. of the piogiess of modern [IL'0l]lCll'll.'{ll nicthods. pointiiig out the advantages of the diticicnt purely gcoiiictricul methods as coniparcd with the aiialytical ones. hut w'ithout. taking as much account of the G('l‘n1:lll as of the I-‘iench authors; l:l.. Rapport am‘ Ies Progrés de la Ge'omr5trie (1870). a continuation of the Apr-rcu: ld.. Traité de Géoniélrie Supe'riem'e (1852); Ciemoiia. Inlroduzione ml mm Tcoria Gcometrica della Cm-re Prime (1362) and its coniiiiiiatioii l'rclimimu'I. «It una Tcoria Geomelrica delle .S'uper;/icie. which at present. are most easily pio- cui able in their German tinnslaiioiis by Cuitre. As more elementary books. we mention Steiner. l'orIe.mrigen ill-er S_unIlietisclic Geometric. edited by Cciscr and Schi'o'dci (1567); Ciciiiona, l;‘Iemenls (Ir Geome'trie I’ro_1ect are (1.875). iraiislat(-d from the llilllflll by Dcwulf . Towiisciid. Jloclern Geometry of the l’oir.!. Linc, amt Circle (l8G=5)_ which contains a variel_v of niodcrii methods, but. nnfortiiiiatcly. is confiiied to (‘lI'('l-JR, without ciitcriiu: into conics. A grcut niaiiy of the pro- positions nic. liowcvcr, easily extended to conics. (0 “-,'
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