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)