ABSTRACT
To prove this theorem, we interpret the variables « i , « 2 and u3 as the components of a vector ti in a rectangular coordinate system and study the envelope Z of the family of planes:
XiUj + z 2 u2 + x3u3 = Z ( u i , u 2, u 3). (1)
Since Z is a positively homogeneous function of degree one, we can restrict ourselves to unit vectors u , . . . , e, so that y ju \ + tij + u§ = 1. Accordingly, we can consider our function on the surface of the unit ball E (a point on E is denoted by the same symbol as for the corresponding unit vector). Clearly, family ( 1 ) depends on two parameters and can therefore be expressed in vector form as
fix = Z(h). (2)
The coordinates of a point on the surface of Z are
At a point where d2 Z does not vanish, the surface Z has a definite tangent plane. The eigenvalues of d2Z are the principal radii of curvature of the surface Z. Therefore at a point where tPZ is an indefinite form, the surface Z intersects the tangent plane [l].
