670 SURFACE there are on the surface two systems of lines each of which is a regulus. A cubic surface may be a regulus (see No. 11 infra). Surfaces of tJie Orders 2, o, and 4- 10. A surface of the second order or a quadric surface is a surface such that every line meets it in two points, or what comes to the same thing such that every plane section thereof is a conic or quadric curve. Such surfaces have been studied from every point of view. The only singular forms are when there is (1) a conical point (cnic- node), when the surface is a cone of the second order or quadricone ; (2) a conic of contact (cnictrope), when the surface is this conic ; from a different point of view it is a surface aplatie or flattened surface. Excluding these de- generate forms, the surface is of the order, rank, and class each = 2, and it has no singularities. Distinguishing the forms according to reality, we have the ellipsoid, the hyperboloid of two sheets, the hyperboloid of one sheet, the elliptic paraboloid, and the hyperbolic paraboloid (see GEOMETRY, ANALYTICAL). A particular case of the ellip- soid is the sphere ; in abstract geometry this is a quadric surface passing through a given quadric curve, the circle at infinity. The tangent plane of a quadric surface meets it in a quadric curve having a node, that is, in a pair of lines ; hence there are on the surface two singly infinite sets of lines. Two lines of the same set do not meet, but each line of the one set meets each line of the other set; the surface is thus a regulus in a twofold manner. The lines are real for the hyperboloid of one sheet and for the hyperbolic paraboloid ; for the other forms of surface they are imaginary. 11. We have next the surface of the third order or cubic surface, which has also been very completely studied. Such a surface may have isolated point singularities (cnic- nodes or points of higher singularity), or it may have a nodal line; we have thus 21 + 2, = 23 cases. In the general case of a surface without any singularities, the order, rank, and class are = 3, 6, 12 respectively. The surface has upon it 27 lines, lying by threes in 45 planes, which are triple tangent planes. Observe that the tangent plane is a plane meeting the surface in a curve having a node. For a surface of any given order n there will be a certain number of planes each meeting the surface in a curve with 3 nodes, that is, triple tangent planes ; and, in the particular case where n = 3, the cubic curve with 3 nodes is of course a set of 3 lines ; it is found that the number of triple tangent planes is, as just mentioned, = 45. This would give 135 lines, but through each line we have 5 such planes, and the number of lines is thus = 27. The theory of the 27 lines is an extensive and interesting one ; in particular, it may be noticed that we can, in thirty-six ways, select a system of 6 x 6 lines, or " double sixer," such that no 2 lines of the same set intersect each other, but that each line of the *ne set intersects each line of the other set. A cubic surface having a nodal line is a ruled surface or regulus ; in fact any plane through the nodal line meets the surface in this line counting twice and in a residual line, and there is thus on the surface a singly infinite set of lines. There are two forms ; but the distinction between them need not be referred to here. 12. As regards quartic surfaces, only particular forms have been much studied. A quartic surface can have at most 16 conical points (cnicnodes) ; an instance of such a surface is Fresnel's wave surface, which has 4 real cnicnodes in one of the principal planes, 4x2 imaginary ones in the other two principal planes, and 4 imaginary ones at infinity, in all 16 cnicnodes ; the same surface has also 4 real +12 imaginary planes each touching the surface aiong a circle (cnictropes), in all 16 cnictropes. It was easy by a mere homographic transformation to pass to the more general surface called the tetrahedroid ; but this was itself only a particular form of the general surface with 16 cnicnodes and 16 cnictropes first studied by Kummer. Quartic surfaces with a smaller number of cnicnodes have also been considered. Another very important form is the quartic surface having a nodal conic ; the nodal conic may be the circle at infinity, and we have then the so-called anallagmatic surface, otherwise the cyclide (which includes the particu- lar form called Dupin's cyclide). These correspond to the bicircular quartic curve of plane geometry. Other forms of quartic surface might be referred to. Congruences and Complexes. 13. A congruence is a doubly infinite system of lines. A line depends on four parameters and can therefore be determined so as to satisfy four conditions ; if only two conditions are imposed on the line we have a doubly infinite system of lines or a congruence. For instance, the lines meeting each of two given lines form a congruence. It is hardly necessary to remark that, imposing on the line one more condition, we have a ruled surface or regulus ; thus we can in an infinity of ways separate the congruence into a singly infinite system of reguli or of torses (see infra, No. 16). Considering in connexion with the congruence two- arbitrary lines, there will be in the congruence a deter- minate number of lines which meet each of these two' lines; and the number of lines thus meeting the two lines is said to be the order-class of the congruence. If the two arbitrary lines are taken to intersect each other, the con- gruence lines which meet each of the two lines separate themselves into two sets, those which lie in the plane of the two lines and those which pass through their intersec- tion. There will be in the former set a determinate number of congruence lines which is the order of the congruence, and in the latter set a determinate number of congruence lines which is the class of the congruence. In other words, the order of the congruence is equal to the number of congruence lines lying in an arbitrary plane, and its class to the number of congruence lines passing through an arbitrary point. The following systems of lines form each of them a congruence : (A) lines meeting each of two given curves ;. (B) lines meeting a given curve twice; (C) lines meeting a given curve and touching a given surface ; (D) lines touch- ing each of two given surfaces ; (E) lines touching a given surface twice, or, say, the bitangents of a given surface. The last case is the most general one ; and conversely for a given congruence there will be in general a surface having the congruence lines for bitangents. This surface is said to be the focal surface of the congruence ; the general surface with 16 cnicnodes first presented itself in this manner as the focal surface of a congruence. But the focal surface may degenerate into the forms belonging; to the other cases A, B, C, D. 14. A complex is a triply infinite system of lines, for instance, the tangent lines of a surface. Considering an arbitrary point in connexion with the complex, the com- plex lines which pass through the point form a cone; considering a plane in connexion with it, the complex: lines which lie in the plane envelope a curve. It is easy to see that the class of the curve is equal to the order of the cone ; in fact each of these numbers is equal to the number of complex lines which lie in an arbitrary plane and pass through an arbitrary point of that plane ; and we then say order of complex = order of curve ; rank of complex = class of curve = order of cone ; class of complex = class of cone. It is to be observed that, while
