W . QUADRIC sURi«"m_:i5s.] Gr 1‘: O lll If we cut these pencils by t we get" (EFM X) = (F’E'lIN) or (F.Fil.') -= ( E’F'NM). But this is, according to§ 77 (7), the condition that M, I' are cor- responding points in the involution determined by the point pairs li, E’, 1'‘, F’ in which the line t cuts pairs of opposite sides of the four-point ABC 1). This involution is independent of the particular conic chosen. ,5 88. There follow several important consequences :— Tin:or.|~:.i.—Th7-ough four points two, one, 07' no conic may be rlniu-n -uvhieh touch a-n y given line, according as the involution deter- mined by the given four-point on the line has real, coincident, or inaryina-ry foci. ’l‘ni-:o1:E.i.—Two, one, or no conics may be drawn irhich touch four given lines and pass through a given point, according as the involution clcterntiiial by the given four-side at the point has real, coincident, or i-mayinary focal rays. For the conic through four points which toilclics a given line has its point of contact at a focus of the involution determined by the four-point on the line. As a special case we get, by taking the line at infinity :— Tn l£0l:[£.[. —Throz(_r/h four points of which none is at i'n_/inity either two or no parabola-s may be drawn. The problem of drawing a conic through four points and to11cli- in; a given line is solved by determining the points of contact on the line, that is, by determining the foci of the involution in which the line cuts the sides of the four point. The corresponding re- l'll.‘ll‘l{ holds for the problem of drawing the conics which toneh four lines and pass through a given point. ll['LED QL'.Dl'.lC SURFACES. § 89. Formerly we have considered projective rows which lie in the same plane. In that case, lines joining corresponding points envelope 2». conic. 'e shall now consider projective rows whose bases do not meet. In this case, corresponding points will be joined by lines which do not lie in a plane, but on some surface, which like every surface generated by lines is called a ruled surface. This surface clearly contains the bases of the two rows. If the points in either row be joined to the base of the other, we obtain two axial pencils which are also projective, those planes being corresponding which pass through corresponding points in the given rows. If .,A’be two eorresponding ioints, a,a' the planes in the axial pencils passing through them, tlien AA’ will be the line of intersection of the corresponding planes a,a', and also the line joining corresponding points in the rows. If we cut the whole figure by a plane this will cut the axial pencils in two projective flat pencils, and the curve of the second order generated by these will be the curve in which the plane cuts the surface. llence 'l‘iiI-:o1:F..I.—T/ze locus of lines joining com'esponding points in two pi-.y'ceti-'vc rows which do not lie in the same plane is a surface which euntains the bases of the rows, and wlticli can also be generated by the lines of in/vrsrctz'o,i 0 correspomlmg planes in two projective a.::ial pencils. This surface is cut by every plane in a curve of the second ordrr, hence either in a conic or in a l inc-pair. No line which does not lie altogether on the surface can have more than two points in common with the surface, n'h.ich is therefore said to be of the second order, or is called a, -ruled guarlric surface. 'l‘liat no line. which does not lie on the surface can cut the sur- face in more than two points is seen at once if a plane be drawn t.hrongh the line, for this will cut the surface in a conic. It follows also that .1 line 1/'h.i:_'h contains more than two points of the surface lies alto_r/ether on the suiface. § 90. Through any point in space one line can always be drawn cutting two given lines which do not themselves meet. If therefore three lines in space be given of which no two meet, then through every point in either one line may be drawn cutting the other two. T iiiaonl-:‘.I.—[/’ a line moves so that it always cuts three given lines of which -no two -meet. then it generates a ruled quadric surface. I’roQf.—Let a, b, c be the given lines, andp, g, r . . . . lines cutting them in the points A,.’’, A". . .; B, B',B". . . ; C, C’, C”. . . rcspcctivelv; then the planes through a containing 1), g, r, and the planes through tr containing the same lines, may be taken as coi'i-espoiidiiig planes in two axial pencils which are projective, because both peneils cut the line c in the same row C, C’, C”. . . ; the surface ean therefore be generated by projective axial pencils. ()f the lines 12, q, r . . . no two can meet, for otherwise the lines a_, b, c which cut them would also lie in their plane. There is a single infinite number of them, for one passes through each point of 0. T liese lines are said to form a set of lines on the surface. If now three of the lines p, q, r be taken, then every line (I cutting them will have three points in common with the surface, and will therefore lie altogether on it. This gives rise to a second set E T R Y 405 of lines on the surface. From what has been said the theorem follows :— TllEOl:E.[.—-A ruled guadric surface contains two sets of straight lines. Every line of one set cuts every line of the other, but no two lines of the same set meet. Any two lines of the same set may be taken as bases of two pro- jective rows, or of two projective pencils which generate the surface. They are cut by the lines of the other set in two projective rows. The plane at infinity like every other plane cuts the surface either in a conic proper or in a. line-pair. In the first ease the surface is called an Ilyperboloid qf one sheet, in the second an I[yper- bolic Paraboloid. The latter may be generated by aline cutting three lines of which one lies at infinity, that is, cutting two lines and remaining parallel to a given plane. QUADRIC SURFACES. §91. The eonics, the cones of the second order, and the ruled quadric surfaces complete the figures which can be generated by projective rows or flat and axial pencils, that is, by those aggregates of elements which are of one dimension (§§ 5, 6). VVe shall now consider the simpler figures which are generated by aggregates of two dimensions. The space at our disposal will not, however, allow us to do more than indicate a few of the results. § 9:3. We establish a correspondence between the lines and planes in pencils in space, or reciprocally between the points and lines in two or more planes, but consider principally pencils. In two pencils we may either make planes correspond to planes and lines to lines, or else planes to lines and lines to planes. If hereby the condition be satisfied that to a flat, or axial, pencil corresponds in the first case a projective flat, or axial, pencil, and in the second a projective axial, or flat, pencil, the pencils are said to be projective in the first case and reciprocal in the second. For instance, two pencils which join two points S1 and S2 to the different points and lines in a given plane 1r are projective (and in perspective position), if those lines and planes be taken as corre- sponding which meet the plane 1r in the same point or in the same line. In this case every plane through both centres S1 and S2 of the two pencils will correspond to itself. If these pcneils are brought into any other position they will be projective (but not perspective). _ The correspondence between two projective pencils is uniquely determined, if to four rays (or planes) in the one the corresponding rays (or planes) in the other are given, provided that no three rays qf either set lie in a plane. I’roof.——Let a., b, c, d be four rays in the one, a’, b’, c’, d’ the corre- sponding rays in the other pencil. 'c shall show that we can find for every ray e in the first a single corresponding ray e’ in the second. To the axial pencil a (b, c, d . . . ) formed by the planes which join a to b, c, d . . . , respectively corresponds the axial pencil a’ (b', e’, d’. . . ), and this c0i'respoiidciice is determined. Hence, the plane a'e' which corresponds to the plane ac is deteiniiiicd. Similarly the plane b'e’ may be found and both together determine the ray e’. Similarly the correspondence between two reciprocal pencils is determined if for four rays in the one the corresponding planes in the other are given. § 923. We may now coiiibine— 1. Two reciprocal pencils. Each ray euts its corresponding plane in a point, the locus of these points is a quadrie surface. 2. Two projective pencils. Each plane cuts its eorresponding plane in a line, but a ray as a rule does not cut its corresponding ray. The locus of points where a ray cuts its corresponding ray is a twisted cubic. The lines where a plane cuts its corresponding plane are secants 3. Three projective pencils. The locus of intersection of corresponding planes is a cubic surface. Of these we consider only the first two cases. § 94. If two pencils are reciprocal, then to a plane in either corresponds a line in the other, to a flat pencil an axial pencil, and so on. Every line cuts its cor1'csponding plane in a point. If S, and S2 be the centres of the two pencils, and P be a point where a line a, in the first cuts its corresponding plane a2, then the line lag in the pencil S2 ’lt'ltiCh passes through P will meet its corresponding plane B, in P. For b2 is a line in the plane ag. The corresponding plane B1 must therefore pass through the line a,. hence through_ P. The points in which the lines in S1 cut the pl-aiies corresponding to them in S2 are therefore the same as the points in which the lines in S._, cut. the planes corresponding to them in S1. The locus of these points is a surface which is cut by a plane in a. conic or -in a line-pair and by a li'ne in not more than two points nnlcss -it lies altogether on the surface. The surface itself is there.
fore called a quadric surface, or a sm;facc of the second order.