Connex

Clebsch connex

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

$$\tag{1 } f ( x ^ {1} , x ^ {2} , x ^ {3} , u _ {1} ,\ u _ {2} , u _ {3} ) = 0 ,$$

where $x ^ {i}$ and $u _ {i}$ are homogeneous coordinates of points and lines, respectively. For example, the equation

$$\tag{2 } u _ {1} x ^ {1} + u _ {2} x ^ {2} + u _ {3} x ^ {3} = 0$$

defines the so-called principal connex, expressing the incidence of the point $x$ and the line $u$. 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

$$\tag{3 } F \left ( x , y , \frac{dy}{dx} \right ) = 0$$

is defined by the coincidence of connexes (1) and (2), and the problem of integrating equation (3) consists in composing curves from the points $x$ and lines $u$ thus defined, such that $x$ and $u$ are, respectively, the points of the integral curve and the tangents to it. The introduction of this projective point of view (the coordinates $x$ and $u$ 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)