SURFACES IN HYPERSPACE. 359 



(109') is used. We may say therefore that: The principal directions 

 (third generalization) at a point coincide ivith the principal directions of 

 the three diinensional surface obtained by projecting the surface on the 

 three space determined by the tangent plane and the mean curvature, or, 

 are along the axes of the {degenerate) conic in which a tangent Sn-\ 

 normal to the mean curvature cuts the surface. (The conic obtained as 

 the intersection of any S„_i which is normal to any Hne in the plane of 

 h. and the perpendicular $ • h on the plane of jJi-xS from has the same 

 axes.) 



In the special case /? = the conic (122) always has A > 0, and is 

 an hyperbola. There is no real hyperplane which cuts the surface in 

 a double direction. Cone II becomes a cylinder with elements per- 

 pendicular to the plane p.x5 of the indicatrix. From (109), p-x8 = 

 i?(ek3xki — ./Icoxks). The hyperplanes perpendicular to any direction 

 in the plane of the indicatrix have w = and cut the surface in the 

 same locus x- — y~ = 0, — except the particular one for which u :v = 

 f:—e which causes (121) to vanish identically and contains all direc- 

 tions on the surface. We may therefore define, if we choose, the 

 directions of the x and y axes as principal directions and the orthogonal 

 directions x'^ — y^ = as asymptotic lines on the surface at the point 

 where h = 0. 



A reference to (111) shows that h*ct = means h -\- ecos20 = 0. 

 On comparison with (122) we see that the directions 9 for which 

 h -\- e cos2^ = are the asymptotic directions of the intersection of 

 the surface with the tangent iS„_i perpendicular to h. Hence: The 

 asymptotic directions on a surface are the directions in which the surface 

 is cut by a tangent hyperplane perpendicular to the mean curvature vector. 

 This gives added corroboration of the generalization of the Dupin 

 indicatrix to the intersection of the surface and this particular 6'„-i. 



47. A second standard form for a surface. In the three dimen- 

 sional theory the condition 6' = is unchanged by the general linear 

 transformation. This is no longer the case in higher dimensions. 

 To discuss briefly projective properties of a surface we may proceed 

 as follows. The general surface has the property that the tangent 

 spaces Sn-\ which cut the surface in a double direction envelope a 

 nondegenerate cone. This statement is projective and the analytic 

 statement is hx[JLx8 ^ 0. The condition (hx|Xx8)'- = is therefore 

 invariant under projection. (The condition hx|xx5 = is the condi- 

 tion for the existence of Segre's characteristics, and as Segre was dis- 

 cussing projective properties, the result stated is but a corollary to 



