358 WILSON AND MOORE. 



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

 plane uzi + VZ2 + wzs = tangent to the surface in the standard form 

 (109'); the intersection is, 



u[h{x' + if) + c{x' - f-)] + vfix' - y') + iv[A{xr- if) + 2Bxy] = 0. 



There are <»- such hyperplanes. The discriminant of the quadratic 

 form 



[u{h + e) + vf-\- w.4]a-2 + [u{h - e) - vf - wA]f + 2wBxy] = (127) 



is 



A = w^B^ + {ue -\- vf + wA)"^ — n^P. 



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

 CX3 1 normal directions uw.io (forming a quadric cone) such that the tangent 

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

 directions. 



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

 (114), 



(/i2 - e'')u' -Pi)~ - (^2 -|_ B^)w'~ - 2fAvw - 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 ^ohich are perpendicular to the elements of Cone II 

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

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

 planes tvhich 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 indicatrix 

 cut the surface in coincident directions. 



Particular interest attaches to the hyperplane (si = 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 

 2iS h < e, h = e, ov h > e. This locus will be called the (generalized) 

 Dupin indicatrix. 



The condition a* (i = and [X,'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 li*|A = 0, however, 

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



