A Method of investigating the Geometry of Families of Curves, with Examples

Extract

Consider the equation f(ξ η, x, y) = 0. If definite values of ξ and η be taken, and x and y be current co-ordinates, f(ξ η, x, y) = 0 is represented by a curve in the plane of xy. If all the possible values of ξ and η be taken in turn, f(ξ η, x, y) = 0 will be represented by a doubly infinite family of curves in the xy plane. Similarly if all possible definite values of x and y be taken in turn and ξ and η be regarded as current co-ordinates, f(ξ η, x, y) = 0 will be represented by a doubly infinite family of curves in the ξη plane. Since the first and the second doubly infinite families of curves represent the same equation, it may be expected that some geometrical correspondence exists between them. It is the object of the present paper to investigate this correspondence.