358 WILSON AND MOORE. 



dimensional linear spread) to cut the surface. If we consider a hyper- 

 plane uzi -\- VZ2 + tvzs = tangent to the surface in the standard form 

 (109'); the intersection is, 



u[h(x^ + y^) + c(.r2 - zy2)] + vfix"" - if) + iv[A(x'' - if) + 2Bxy] = 0. 



There are oo^ such hyperplanes. The discriminant of the quadratic 

 form 



[u{h -\-e)-\-vf+ wAjx"^ + [ii(h - e) - vf - loAly"- + 2wBxy] = (127) 



is 



A = ^2^2 + (we + vf + ivAy - uW. 



The equation A = determines a quadric cone. Hence: There are 

 00 1 normal directions u:vuo (forming a quadric cone) such that the tangent 

 hyperplanes normal to any of these directions cut the surface in coincident 

 directions. 



If u:v:w be the directions of an element of Cone II we have, from 

 (114), 



{h2 _ g2)^2 _p^2 _ (^2 ^_ 52)^,2-_ ofAviv - 2Aeuw - 2feuv = 0, 



as the equation of Cone II (with its vertex transferred to 0). This is 

 identical with A = except for sign. We see therefore that: The 

 tangent hyperplanes which are perpendicular to the elements of Cone II 

 cut the surface in coincident lines; these hyperplanes are also the tangent 

 hyperjjlanes to Cone I. Hence we may state: The tangent hyper- 

 planes which cut the indicatrix in real points cut the surface in 

 real directions; those which cut the indicatrix in imaginary points cut 

 the surface in imaginary directions; and those tangent to the iyidicatrix 

 cut the surface in coincident directions. 



Particular interest attaches to the hyperplane (zi = 0) perpendicular 

 to h. This cuts the surface in the directions {h + e)x'^ -\- {h — e)y'^ 

 = 0. These directions are real, coincident, or imaginary according 

 as h < e, h = e, or h > e. This locus will be called the (generalized) 

 Dupin indicatrix. ■ ■ . 



The condition a • |X = and |Ji« 8 = which give the first two general- 

 izations of principal directions may be calculated from (111), (112), 

 but exhibit no special properties relative to the axes used in standard- 

 izing the equation of the surface. The condition h'jJL = 0, however, 

 is satisfied by the x and y axes in the tangent plane when the form 



