## Birational space transformations associated with congruences of lines |

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0's are homaloidal American Mathematical Society associated F base curve g bundle of lines cjk.iXi CjkX class 84 coefficients common tangent surface conic curve of order denoted by g DePaolis direct transformation directrices fundamental curves fundamental surfaces gives the homaloidal hence the homaloidal homaloidal net H homaloidal surfaces gives i-O i-O image of g image of P(y Images of fundamental Invariant and homaloidal jacobian locus meets the associated multiple point n-ic non-composite NON-INVOLUTORIAL SPACE TRANSFORMATION non-involutorial transformation associated order 2m parameter parasitic lines passes a single pencil of transversals pencils of surfaces plane curve point P'(x point P(y points of intersection residual base curve residual pencils residual point Q(x respectively ruled surface Similarly single F space transformations associated surface G Table of characteristics take the equations tangent lines tetrahedron unique line unique transversal Vogt West Virginia University write the following