On a Property of W-Congruences
Guido Fubini · Annals of Mathematics · 1940
In a recent paper (Math. Ann. vol. 114, p. 237), Jonas has given a very interesting new property of W-congruences. After some general remarks, this paper will give here new, easy, and purely projective demonstrations of these theorems. On the last page we state a new problem which seems to be important for the theory of W-congruences. We shall say that two surfaces, whose points are in a given one-to-one reciprocal correspondence, are congruence-transforms of each other if they are the focal surfaces of the congruence generated by the straight lines joining two corresponding points of the two surfaces. We shall denote by y, *. the points whose homogeneous projective coordinates are xi, yi, .** (i = 1, 2, 3, 4), by ax + by + * ** the point whose coordinates are axi + byi + * i i . If (p is a function of two parameters u, v, we denote by so, so,,, #uu, Or.,, ... the derivatives 8co/cu, 8cp/8v, l2V/8u2, 8 2V/8U8V, ... .When xi are functions of u, v, we denote by x, X, * the points whose coordinates are 8xi/8u, 8xi/8v, * . By (x, y, z, t) we denote the determinant