The congruence T, which is made of all the generators of the first kind on the osculating hyperboloids of a ruled surface, has a great many interesting properties. Some of them have been considered in a previous paper.:}: We shall continue the consideration of this congruence and of configurations associated with it, completing in this way some of our previous investigations very essentially. We shall also study the osculating linear complex, and the pointto-plane correspondence to which it

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... s rise, enabling us to generalize some well-known theorems of Cremona and Lie. For the most part we shall, in this paper, confine our attention to the general case, when the flecnode curve intersects every generator in two distinct points. The case of coincidence will be left for a future occasion, as it requires the use of a different normal form for the equations than that here adopted. The notations are the same as in previous papers. To save space they are not again explained. § 1. The derivative cubic curve. If ax and an are arbitrary, ax yk + a2 zk will represent the coordinates of an arbitrary point on the generator g of the ruled surface, where (yk, zk) for ¿ = 1,2,3,4 are four simultaneous systems of solutions of our system of differential equations, whose determinant does not vanish. We shall usually write axy + a2z, suppressing the index k, as in previous papers. Of course this is essentially a form of vector analysis, which enables us to make one equation do the work of four. The point axp + a2cr of the corresponding generator g' of S', will then be such that the line joining it to axy + a2z is a generator of the sec-