IX. On multiple contact of surfaces

Abstract

In a paper “On the Contact of Quadrics with other Surfaces,” published in the Pro­ceedings of the London Mathematical Society (May 14,1874, p. 70), I have shown that it is not in general possible to draw a quadric surface V so as to touch a given surface U in more than two points, but that a condition must be fulfilled for every additional point. The equations expressing these conditions, being interpreted in one way, show that two points being taken arbitrarily the third point of contact, if such there be, must lie on a curve the equation whereof is there given. The same formulæ, interpreted in another way, serve to determine the conditions which the coefficients of the surface U must fulfil in order that the contact may be possible for three or more points taken arbitrarily upon it; and, in particular, the degrees of these conditions give the number of surfaces of different kinds which satisfy the problem. In another paper, “Sur les Surfaces osculatrices” (Comptes Rendus, 6 Juillet, 1874, p. 24), the corresponding conditions for the osculation of a quadric with a given surface are discussed.