On the surfaces of the third degree (From Mr. J. Steiner.) (Presented at the Gesammtsitzung the Berlin Academy on the 31st January 1856.) The higher algebraic surfaces are investigated with respect to their characteristic geometric properties slightly. For the long-term investigations chief this subject is communicated to a portion of those results, which relate to the surfaces of the third degree. It can be seen from them, that these areas are now treated almost as easily and einläfslich than previously, the surfaces of the second degree. Of the fine qualities of the former, like here, in brevity, are given below. First, several different types of production areas are shown the third degree, from which the essential qualities of those areas stand out immediately, and the following of which are noteworthy. I. By the 9 lines, g, in which the cut surfaces of two arbitrary given binoculars and through each other any given point, P is a surface-degree drill, f * is determined. Namely, each layer laid through the point intersects the 9 lines in 9 points, which, together with that any curve of degree specify the place and all these curves, the aforementioned area. Among the 9 g --- there are six, three such lines which intersect each other, and which therefore determine a hyperboloid, each of these 6 hyperboloids intersects the surface f * still in three new lines, so that also contains the same 27 lines. With regard to the two bands line that contains each hyperboloid are the three defining lines on one side and the three new lines to the other party, cut those three words those, but not each other. (134) II Become a given surface tufts of the second degree, B (f *%) and a given sheaf of B (E) ^ projectives relative to each other, they create an area of some third degree, f * which) by the basic curve, RA * the first, just as through the axis, g, and the other tuft is, that is, all conics, C 7, which are cut in each plot of the second degree, f 1, the corresponding levels, E,, **) are an area of the third degree. Here there are five levels affect E, f which the corresponding surfaces 2, so that the corresponding conic C in two straight lines, g, falls, etc. III. If a bunch of second-degree surface, B (/*), given, then the Pampolare of any pole, P, in respect of the same size drill some degree f 3, which is always through the base curve of the sheaf R 4, and also by the pole about. That is, the pole P from each surface, f 2, of the given sheaf circumscribed cone, they touch along a conic C 2 and all these conics lying on a surface of degree / * ', the planes of the conics in the polar planes of the pole respect to the respective surfaces of the pencil to go collectively by a particular line, g, which is f * is also in the area. The fan given area contains four cones in particular, as Poncelet was first shown for each of them called the conic C 7 is divided into two straight lines, ^ i, which intersect at the vertex of the cone, forming a triangle g with that line, also in the comparison area of the pencil, which passes through the pole P, and touches of its polar plane is there, the conic C 2 is divided into two straight lines, y 2, which intersect at the pole, and form g equally with that line a triangle, which together are already 11 p lying in the plane straight. Through each of the last two lines g 2 can be four such planes specify which affect the general curve of the sheaf R ^ 4, and each of these planes intersects the surface f ^ 3 in two new lines, which (in the plane with the Berübrungspunct curve) cross, which, together with those of f 27 in the area * lies precisely constitutes. --- *) That is to say, the 4th-degree curve in space, f the common intersection of all surfaces of the B (^ 3). **) The projective relationship of the given fan is done, among other things, that we in any point P on the curve f ^ R ^ 4 to all members of the tuft area B (2) tangent planes E_1 sets, which form a sheaf B (E_1) then those with the given sheaf B (E), shall, by voluntary adoption of three pair of corresponding planes, E and E_1, Projective covers in place and then E_1 every level, that surface f ^ 2, which it touches. (135) IV cut surfaces are given any three second degree, so the three polar planes of each pole P in relation to the same, in general, each in a different points P t, P is the pole moves in any given plane, it describes the punct P_1 any surface third degree. Or thinking: you look at all surfaces of the second degree, which pass through any given seven points, so that any given plane in relation to the same appropriate poles sümmtlich third in a surface level. The many interesting circumstances, by far, which must take place here yet, here are ignored. For these types of development --- --- and continue through the help of some polarity phrases arise following remarkable properties of the main surfaces of the third degree. , "A general surface / third degree * contains 27 straight lines g (real or imaginary), and each of them will be cut by 10 of rest, from five pairs to intersect itself, so that they make with those five triangles. All 27 g of two straight lines intersect each other, consequently, in 135 J and the parties are on the whole, 46 triangles A. The five pairs of intersection, d, in each line, g, belong to an involution - Punctensystem; is the same hyperbolic, it contains two Asymptotenpuncte (Doppelpuncte) \ pi. The sides of each triangle \ Delta contains all three hyperbolic either 1 ° or 2 ° only one hyperbolic and two elliptic Puncte - system. Or wider: There are 27 different systems of such planes, E, which is f ^ 3 in the plane conics, C ^ 2, cut, and although there are the same from 27 Ebenenbüscheln, B (E), by the 27 straight axes respectively to g, and conversely, any plane which intersects f ^ 3 tu the surface of a cone cuts, cuts the same nor necessarily in one of the 27 lines and is one of the sheaf. The body of conics, C 2, f belong to the planes of the same sheaf, cut its axis, g, in that Puncte - system, each level is regarded as one of the area * double-touching, and their conic sections with the axis than Berührungspuncte; among the conics there are two particular, C ^ 2_0, which affect the axis, in the above asymptotes-s point; addition there are five conics which decompose into two g straight, so that the corresponding level p touches the surface in three discoveries, namely in the corners of triangles lying in her dee A. The levels of the 45 triangles are touching the only ones who f * surface in three punct. (136) It also gives cut 45 such systems from surfaces of the second degree, f 2, f which the area third degree ^ 3 in three conics C 2; each triangle corresponds to such a system, namely, any of three levels, which relate to go through its three pages containing three such conics C 2, by which an area is always some second degree, and vice versa: If a surface of second degree f 2 with the area third degree f * any three conics in common, so go the same levels each time by the three sides cut one of its 45 triangles, or is a surface f 2 by two f * lying in the plane conics, so both still in some areas always a third conic, and the levels of the three conics pass through the ropes of the same triangle A . The sides of each triangle A are pi affected by the aforementioned special conics CÂ £-points in their asymptotes \; the three pairs or six-Puncte are asymptotes to three and three in the straight line l, and the three corresponding conics C 2 for a cone of the second degree, f ^ 2_0, how long the plane of the triangle of the corresponding line I touched, and the apex of all four cones lie in a straight line. Besides containing the corresponding area of the triangle system, second degree, f 2, infinitely many cone, its apex are collectively in an area of the fourth degree. The three conics C 2, through which one each surface of the second degree is f 2, g may exist especially in three straight couple, and then the surface a simple hyperboloid, h 2,. If we take the 27 grams of any three straight lines, which do not intersect each other, so they determine such a hyperboloid, for the same intersects the surface f 3 g in three different yet always straight lines which meet those three, but not each other, "There is in entire 360 such hyperboloids h 2, each of the 45 systems surfaces of the second degree includes 48 of them, and each hyperboloid is found in 8 different systems. " (137) Selects one of the 45 triangles A two such that no straight line g have in common whose planes intersect then in another line, say k, which is called its edge, so make the triangles in pairs on both sides of this edge, in three puncte \ delta. Is called (by the triangles A and B, their sides are straight line g) or by a, a_1, a_2 and b_2 cut, so for example, the pairs a and b, a_1 and b_1, a_2 and b_2 k on the edge, and then these pairs are from the other three sides are triangles \ Delta: abc, a_1b_1c_1, a_2 b_2 c_2 or A_1, B_1, C_1, the third side c, c_1, c_2, by itself, the pages of a sixth triangle \ delta or C. The planes of the triangles A, B, C form a binoculars, T, whose three edges intersect at their sites k in pairs, and so are the planes of the triangles A ^ B u C x is a binoculars, 1 \, on the edges of their pages meet each other, these triangles, as these have the same 9 lines g, L, or a 7 aa bb cc lb 2 lc 2 on sides, and cut surfaces of both binoculars each other in the same (as above I). Two such are called conjugate binoculars binoculars. "The levels of the 45 triangles \ delta formed in this way the whole 240 binoculars, or 120 pairs of binoculars conjugate T and T_1." * These pairs are arranged on three and three in 40 groups, each of which groups all 27 lines g contains. , Every triangle A is found in 16 different tried, to fall so that binoculars or 16-crown to his level, the 16 head are always in a curve of fourth degree, which has the ropes of the triangle to double tangents, and that these in their asymptotes -points \ pi prejudice " The 240 binoculars have a combined 720 k distinct edges, so are the 186 d of intersection of the 27 lines g k to three and three in 720 straight lines which meet at three and three in 240 new Puncte T (vertices of binoculars). Through each of intersection \ delta ever go Just 16 k, each of which still by two others of intersection, such as dx and d give 2 (instead of \ delta), one takes in each of them a fourth point, \ lambda, so that \ delta \ delta_1 \ lambda \ delta_2 are harmonious, then the 16 Puncte k are twice to four and four in four lines, and these 8 lines to the two lines together with g, the cut of that first point 3, located in a hyperboloid. (138) If by some in the space ^ f 3 lying conic C 2 an arbitrary surface of the second degree, f 2, placed so that it intersects surface, in general, nor in a spatial fourth-degree curve, R ^ 4, which always countless other Go surfaces of the second degree, or a bunch of second degree surface is, in these areas, there are 5 such that the given surface f * touching each in one point, and the contact layers in these five Puncte together with the level of that conic C 2 and go through a same line g; also includes each of the two other levels of contact and line g, which contains cross in Berührungspunct, so that, therefore, each one a triangle. --- If one by any two mutually non-intersecting line g an arbitrary hyperboloid, then cut the same f * besides the surface yet in such a space curve of the fourth degree, u /, which passes through no other surface of the second degree, which curve is therefore essential different from the previous R ^ 4, which is regarded as the intersection of any two surfaces of the second degree, and which has hitherto been believed to be the only one fourth-degree curve in space. The two curves differ notably more in the following characteristics. "The surface tangent to the curve R? (ie the area in which all its tangents lie] & tn is of degree and of the 6 ^ (class)) ien while the tangent space of the curve R * is the 8 " 'grade and the W s class." Furthermore, : Â »Straight From the two bands, which lie in the one passing through the curve Rf hyperboloid intersects every line of the one body of the curve and every line in three of the other party only in one point, while each hyperboloid, which by the Curve RA's, meets every line from the one or the other party in the same two points. " , Thus there are two essentially different kinds of space-curves of the fourth degree, R 4 and R ^ ^ 4_1. " If the given surface third degree, f, from any one point or pole P defined a cone is the same as of 6 th degree, and touches the surface along a curve in space 6 th of the degree by which each time any one surface of the second degree, f 1, Give that the first polar of the pole P with respect to the given area / * is called. There are infinitely many such special pole, the first ever a polar cone of the second degree, f_0 ^ 2, and there is the law that, that if P x is the vertex of this cone, then its first polar also a cone is, and that the peak of it in that first pole P lies. "These two points P and P_1 torrid reciprocal poles in relation to the area / *. Mentioned, the common locus of inverse pole is a certain area of the fourth degree, P \ f "which the core area of the third degree * is given surface. (139) "The core area is P ^ 4 including through the apex of the binoculars above 240, namely the apex of each of the 120 pairs of binoculars are conjugate T and 1 \ a reciprocal pair of poles." It still takes place closer to the fact that the polar cone of the vertex T f * T t is the conjugate binoculars circumscribed, that is, by its three edges is k, and likewise vice versa. , Furthermore, the two Asymptotenpuncte \ pi are in each of the 27 lines g P and a pair of reciprocal poles P_1, namely the line affected in the same area of the core P ^ 4, '--- "It gives the whole 10 such special poles P, or P_0, whose polar conic f ^ 2_0 in two planes, F and F_1, falls, (so that even from the pole of f * of the area circumscribed cone in the third grade, two cones and also the curve of contact of two plane curves of third degree decays) and he is then the reciprocal of Pol, P k, not absolutely determined, but it lies along the cut line or edge, pt, the two planes everywhere, so that for each of these edge lying punct P_1 the first polar f ^ 2_0 is a cone, and that the apex of cone Disser in that pole P are united ", The 10 Poland P_0 correspond Thus, 10 degrees p_1 reciprocal." , The 10 poles are nodal points of the core area P ^ 4 and 10 lines all lie in it. "The relative positions of these poles and lines is such that in each edge p_1 are three of the 10 poles, and that through each Pol P_0 go three of the 10 edges. Or more precisely, the 10 poles P_0 and the 10 straight p_1 are the rough edges of a complete pentahedron, ie there are 5 provides that certain levels e_0 which pairs in the 10 lines and Cut three each in the 10 Poles, and where the line of intersection of any two planes of intersection of each particular and the other three are reciprocally. " 9 The core area P ^ 4 will thereafter be cut from each of the 5 levels e_0 in four straight p_1. The p_1 passing through each edge, above two levels F and F_1 are assigned to the corresponding two levels e_0 harmonious. The ten pairs of levels F and F t also have interesting reciprocal relationships among themselves. (140) Now there are also still those poles P, the polar cone f ^ 2_0 particular cylinder. 'The location of these poles is an area lying at the core of the 6th degree curve in space, B?, Which through the 10 nodal points P is the same "since (the polars, F and F x, regarded as a cylinder) are." The Ax a, each R 6 cylinder intersects the curve in three points, and through every point of the curve ever go three axes "The common place of all cylinders - axes, a is a (straight) Area 8th degree, aP, which the curve R? has a threefold line, and in which, especially the 10 edges of the aforementioned px pentahedron are "several curious properties of this surface can not be developed here. The core area P * f intersects the given surface along a curve in space * 12th level, R ^ (12), which is very characteristic for the latter area. First, this curve passes through the 54 Asymptotenpuncte of the 27 lines g n and touched them in the same, so that every line she has the double tangent. , Then dividing the curve R n on the surface f * those regions from each other, where the curvature is positive and where the same is negative) along the curve itself is the same zero Furthermore, the curve R ^ (12) f the place of all those on the surface Puncte ^ 3, in which the corresponding tangent plane, the area with return-punct cuts, cuts of such a curve that is 4n third degree C *, for which the punct Rückkehrpunct has, therefore, the return so that tangent, t, the curve C ^ f * 3, the area in the same Puncte osculates dreipunctig or touched. , The locus of this cusp-t is a developable surface the 30th level, t ^ (30), f ^ 3 which the area along the curve osculates R ^ (12), and g has 27 lines to double lines, so that ie, the intersection curve both surfaces, t ^ (30) and f ^ 3, by the 90th grade must be triple the curve R ^ (12) and from the double-g is to be counted 27 lines, 'etc. (141) Any level, e, f intersects the given surface in a curve * third degree, to the area along this curve is defined * developable surface is between the 12th and the 6th grade class, and her return line (arrête de rebroussement) is from the 18th degree. The second polar of a pole of some f P with respect to the given surface is a plane * such e., the pole moves in that firm-level P e E, then the envelope of its polar surface level, a third degree s ^ 3 only fourth-class *), which four nodal points, Q_0 has, and is enrolled that developable land \ Psi, e ^ 3, the area always the core area P ^ 4 are enrolled and touches the same curve in space along a 6th degree R ^ 6, which especially also the 4 nodal points Q are, therefore, so that the latter always in the core area P ^ 4 are. " 9 The area e 3 is called the second polar E level in terms of the given surface f \ has the same (in other areas of the same degree, the remarkable special characteristic: that the (for some Puncte her lying in her circumscribed cones) of other puncte of 6th degree) in two cone of the second degree and the associated tangent plane) decays latter touches both cones, and these two always go through the four nodal points Q. But if their plane E to infinity, it is their second polar a 3, the envelope Average diameter of all levels of the given surface f *, the same shall retain all specified properties, it is inscribed in the land, and P *