A Theorem on Conjugate Nets in Protective Hyperspace

A theorem proved by C. C. Hsiung in a recent paper [l]1 may be stated as follows : In a linear space Sn of n ( ̂ 3) dimensions let Nx be a conjugate net and w be a fixed hyperplane ; then the points M, M of intersection of the fixed hyperplane w and the two tangents at a point x of the net Nx describe two conjugate nets Nm, Nm in the hyperplane ir, respectively, and one of the two nets Nm, N¡¡ is a Laplace transformed net of the other. The purpose of this note is to prove, in an elementary manner, a general theorem of which the above stated theorem of Hsiung is a specialization. The statement of the theorem follows: In a linear space Sn of n (^3) dimensions let Nx be a conjugate (parametric) net. Let M, M be points on the u-, v-tangents at x of the net Nx, respectively, which describe two nets Nm, Nm having the property that the tangent plane of Nm (Nm) at M (M) passes through M (M). The nets Nm, N¡¡ are conjugate nets and each one of them is a Laplace transformed net of the other one. For the proof let us observe first that since Nx is a conjugate net, the points M, M, dM/du, and dM/dv lie in the tangent plane to Nx at x; this plane is therefore determined by the points M, dM/du, dM/dv. The conditions that the tangent planes to Nm and Nm at M and M, respectively, pass through the points M and M are equivalent to the conditions that the matrices