Introduction. Let Xi and u i} (i=l, 2, 3), be the projective coordinates in plane xu of a point and line, respectively, and yi and Vi the coordinates of a point and line in plane yv. The planes may be superimposed. Let 5 be a transformation in mixed coordinates (1) pyi = i(x 9 u), avi = fc(x, u), i = 1, 2, 3, where the <£» and the \f/i are polynomials homogeneous in the Xi of degrees a and c, respectively, and are also polynomials in the Ui of degrees b and d, respectively. This transformation is said to be birational if from (1) it is possible to obtain the inverse s-1 given by (2) px % = 4>l (y, v), a'ut = ^/ (y, v), i = 1, 2, 3, where the <£/ and ^/ are polynomials homogeneous in the yi of degrees a' and c\ and in the Vi of degrees ft' and d', respectively, and p"yi = *iW,V), c7^ = ^(^,^). This transformation is a line element transformation f if each of the equations implies the other. An element transformation is a birational contact transformation if, in addition to the above requirements, each of the two systems of equations X Xidui = X Uidxi = 0 and J2 Ji^i = X) My» = 0 implies the other. J Simple examples of birational contact transformations in the plane * Presented to the Society, February 26, 1938. f A line element transformation is not a contact transformation unless it preserves unions. % The representation of line elements by means of six coordinates Xi, m is due to Clebsch who also gave the necessary and sufficient conditions that (1) be a contact transformation.
