A geometric picture can be obtained as follows : let a be the axis, and 7r any plane perpendicular to a. If the plane be moved parallel to itself, and rotated into itself about a, the translation and the tangent of the angle of rotation being in proportion, then every point P of 7r will describe a right circular helix. The lines of the complex are the tangents to these 00 helices.
The lines which belong to two linear complexes A= o, B = o belong to every complex of the system X A--I--pB=o for every value of X and A. Within this system are two special complexes A'=o, B' = o, and the linear congruence may be defined as the 00' lines which meet two given lines, called the directrices of the congruence. They may be distinct, coincident or imaginary. When they are coincident, the congruence consists of cc' pencils of lines with vertices P on the directrix d, and lying in planes 7r through d, such that P, 7r are projectively related.
Two complexes A = o, B = o are in involution when on every line p of the complex, the poles of an arbitrary plane through p in A = o, B = 0 are harmonic as to the intersections of p with the axes of the two special complexes A' =o, B'=o in the system.
The lines which belong to three linear complexes A = o, C=o constitute the cc' lines of a regulus, i.e., one system of generators of a quadric surface; if they are real, they form a hyperboloid of one sheet, or a hyperbolic paraboloid. These lines also belong to every linear complex of the form XA-kuBd-PC---o. The other system of generators belongs to three independent linear complexes, each of which is in involution with B=o, C=o.
The lines belonging to four linear complexes are the two transversals of the axes of the four independent special linear complexes contained in the system. They may be distinct, co incident or imaginary.
These ideas had been recognized in part by various writers. but they were first put into systematic form in terms of co ordinates by Arthur Cayley in 186o. The results attracted the attention of several mathematicians, in particular of Julius Plucker, who soon contributed a number of notes leading to their further development. Eight years later appeared the first part of PlUcker's book on line geometry.
By means of a simple transformation of the line co-ordinates the quadratic identity o can be reduced to the sum of six squares, These xi are called Klein co-ordinates; they furnish one of the most striking examples in mathematics of the power and the ease of obtaining results in consequence of a proper notation.
A linear equation in sphere co-ordinates (linear complex of spheres) defines the spheres which cut a fixed sphere at a constant angle.
The line-sphere transformation together with its projective generalizations, is a powerful weapon in the study of certain curves and ruled surfaces. Thus, all the sextic ruled surfaces (120 types) and those of order seven having a straight line directrix (about 30o types) have been determined by this method.
Given a line l not belonging to a non-special linear complex A = o; as a point P describes 1, its polar plane 7r as to A =o describes a pencil whose axis 1' does not belong to A= o nor inter sect 1. Dually, every plane 7r' through 1 has a pole P'. As 7r' turns about 1, P' describes 1'. Two such lines 1, l' are called conjugate polars as to A= o. Any line of A =o which meets 1 will also intersect l' and every line which meets both l and 1' is a line of A= o. By this transformation a linear complex B=o is transformed into a complex B'=o. If A and B are in involu tion, B'=B. The six co-ordinate complexes xi = o in the Klein system are mutually in involution.