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For the purpose of finding simple metric characterizations of the invariants I, J, we choose an orthogonal Cartesian coordinate system in such a way that the point 0 is the origin, the line 00* is the X„_i-axis, and the tangent hyperplane f„_i is the coordinate hyperplane Xx = 0. n-lXi(Xn-l i,*-i »-i + /un_i,„_i(X„_i — hy + • • • , ~ h) where h is the distance between the points 0, 0*, and M = cot w, w being the angle of the tangent hyperplanes tn-i, tfLu In order to express the two invariants I, J in terms of the coefficients of expansions (36), (37) we have first as in §3 to consider the homogeneous coordinates y0, y\, • • • , yn of a point defined by formulas (15) and the most general projective transformation of coordinates, which shall leave the point 0 and the tangent hyperplane tn-\ invariant and carry the point 0* and the tangent hyperplane ^"Li into the vertex (0, • • • , 0, 1, 0) and the coordinate hyperplane 19451 INVARIANTS OF CERTAIN PAIRS OF HYPERSURFACES 581 yi = 0 of the new coordinate system respectively.
A projective characterization of the invariant I. Let T, I \ be the asymptotic curves of the surface Si at the point 0 whose tangents are t, h respectively. Among the four-point quadrics of the asymptotic curve r at the point 0 we can determine a two-parameter family such that every one of the family has t% for a generator and has at the point 0 contact of the second order with the surface 5 2 . /Z)*2 - mxy + Eyz + Fz2 = 0, where E, F are arbitrary. The quadric (20) is cut by the plane r 2 in the line t% and a line with equations (21) 3px - Imy = 0, z = 0.
_i sin*< -Deo' /^y-3 /ffn-l#2*\ ViU U„liJ BIBLIOGRAPHY 1. E. Bompiani, Invarianti proiettivi di una particolare coppia di elementi super ficiali del 2° ordine, Bollettino della Unione Matematica Italiana vol. 14 (1935) pp. 237-243. 2. C. L. Bouton, Some examples of differential invariants, Bull. Amer. Math. Soc. vol. 4 (1898) pp. 313-322. 3. P. Buzano, Invariante proiettivo di una particolare coppia di elementi di superficie, Bollettino della Unione Matematica Italiana vol. 14 (1935) pp. 93-98.