Lecture Delivered before the International Con- Gress of Mathematicians at Paris in 1900

node curve is fundamental in this theory. A number of Cremona's theorems on ruled surfaces with straight line directrices are generalized to apply to all ruled surfaces. Br. Wilczynski gives both analytic and synthetic proofs of these theorems. The third covariant furnishes another congruence associated with a given surface, and in particular a third ruled surface associated with the original one and the one already mentioned. A few brief remarks are made, showing how these covariant surfaces may serve to simplify the integration of the original system of differential equations. This paper will be combined with the previous paper on covariants for publication in the Transactions.