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The place which the configuration composed of a pair of rectilinear congruences with generators in one-to-one correspondence occupies in differential geometry is of some prominence. When corresponding generators do not intersect, a general theory for the projective study of the configuration may be constructed from a system of linear partial homogeneous differential equations of the first order in four dependent and two independent variables. The theory seems to be well adapted for the study of certain correspondences of Fubini, and a complete basis is furnished for their further investigation. The relation of Fubini's work to the earlier work of Darboux is also fully established. In connection with these correspondences a generalization to surfaces is secured for the concept of intersector curve which Lane introduced in his study of pairs of ruled surfaces. The consideration of corresponding nets of curves on the intersector surfaces suggests itself, and when the nets of curves are conjugate nets on the surfaces, two distinct possibilities occur. The first of these has been studied by Fubini, and the properties which he discovered are readily obtained by the methods of the present theory. In addition, one or two new theorems are discovered. The second case arises when a family of intersector surfaces exists in association with each of the two congruences in correspondence.

Pairs of rectilinear congruences with generators in one-to-one correspondence