Congruences of curves

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where u and v denote the parameters determining the curves and t is the parameter which determines the points on the curve. By this method of definition the equations of a congruence of curves are given a form similar to that usually adopted in the discussions of rectilinear congruences, so that one proceeds naturally to a generalization of some of the properties of the latter. At the same time the curves, as above defined, are looked upon as the intersections of surfaces, namely, those defined by (1), when u and v respectively are constants. In ?? 1, 2 we show how the theorems of DARBOUX can be established when the congruence is defined in this manner. Several examples are given to illustrate the different theorems. In ? 3 we consider the problem of determiining a function 4 (u, v ) such that the tangents to the curves of the congruence at the points of intersection with the surface, defined by (1) after t has been replaced by 4,, shall form a normal congruence, or in the second place the ruled surfaces u = const., v = const. of this congruence of tangents shall be developable. It is shown that for every congruence a function exists which furnishes a solution to the first of these problems, and the condition is found which the functions/, f2, f3 must satisfy in order that 4 may be constant. However, there does not always exist a function such that the second condition is satisfied. But when the curves t = const.,