Line Geometry

Every generator of a non-developable ruled surface of order n meets n-2 other generators.

The Quadratic Complex.

If d = o is the equation of the complex of order 2, then the lines belonging to it satisfy the equation 4)+XP=o for every value of X. When the discriminant of this form has six distinct roots Xi; the equation can be re duced to 4) = when P = =o. The surface of singu larities is in this case the most general Kummer surface of order and class 4, with 16 double points and 16 singular planes, i.e., planes which touch the surface at every point of a conic. Each conic passes through six double points, and each double point lies on six such conics. The same surface is also a complete focal sur face for six congruences of order and class 2, namely those defined by 4)=o, xi =o (i= I, 2, . . . , 6).

The surface is invariant under the operation of interchanging the points of contact of the lines of each congruence; these generate a finite group of order 32, of which 16 are harmonic homologies, and 16 are involutorial correlations. For particular forms of 4), corresponding particularities exist in the Kummer surface. Thus, if the complex consists of lines which intersect two given quadrics harmonically. The surface of singularities is now the tetrahedroid. The six nodes on each

conic are in involution. It is a projective generalization of the wave surface of Fresnel. This complex is known as Battaglini's complex. Another particularization is that in which the X-dis criminant is a square and each Xi makes every first minor vanish. The complex consists of the lines which meet the faces of a fixed tetrahedron in projective tetrads. The surface of singularities consists of the faces of the tetrahedron. This complex is called the tetrahedral complex. The first systematic study of the quad ratic complex was made by Battaglini, who assumed, erroneously, that the most general one could be expressed as a linear function of the squares of the pik.

The correction of this error led to the complete classification according to the form of the X-discriminant, thus affording one of the early applications to the theory of elementary divisors. The Kummer surface also furnished one of the first illustrations of a surface which remains invariant under an infinite number of distinct birational transformations, but this property is shared by the focal surface of many other congruences. The geometry of the straight line has shown a close relation to exist among many apparently divergent fields of mathematics.