Connex

Clebsch connex

A connection between the points and lines of the plane expressible by the equation

(1)

where and are homogeneous coordinates of points and lines, respectively. For example, the equation

(2)

defines the so-called principal connex, expressing the incidence of the point and the line . What two connexes have in common is called a coincidence. The notion of a connex was introduced by A. Clebsch in 1871 for a uniform formulation of differential equations.

Thus, the equation

(3)

is defined by the coincidence of connexes (1) and (2), and the problem of integrating equation (3) consists in composing curves from the points and lines thus defined, such that and are, respectively, the points of the integral curve and the tangents to it. The introduction of this projective point of view (the coordinates and have equal status) also provides a principle of classifying differential equations.

Similar constructions can be carried out for partial differential equations, not necessarily of the first order.

References

[1]	F. Klein, "Vorlesungen über höhere Geometrie" , Springer (1926)