SURFACES IN HYPERSPACE. 347 



is negative, zero, or positive.*^ The conie breaks up into two lines 

 if Be = 0, that is if the indicatrix is a linear segment. 



The degenerate case 5 = requires a little more investigation to 

 find what happens to consecutive.normal spaces. If we observe what 

 happens as we pass from a true ellipse to a segment, we see that the 

 points of intersection of consecutive normal planes bunch themselves 

 more and more closely' about the point Si = 1/h, S3 = — ej Ah, which 

 is the inverse of the foot of the perpendicular to the segment from the 

 surface point. It therefore appears that the normal planes all pass 

 through a common point (0, 0, 1/A, — e/Ah) in this special case; the 

 two nearby planes in the direction of the axes .r, y may be said to cut 

 the normal plane in the lines 21 (A + e) + Azz — 1 and Zi{h + e) — 

 Az'i = 1 respectively. These lines are those into which (116) factors 

 and are perpendicular to the lines which join the surface point to the 

 extremities of the indicatrix. 



There is a special case under the case B = 0, namely that in which 

 the indicatrix, now a segment, is coUinear wath h. The surface is then 

 three dimensional in the neighborhood of the point, or the point is 

 axial in Levi's nomenclature. The common intersection of the con- 

 secutive normal planes has retreated to infinity and the locus reduces 

 to two parallel straight lines which are the intersection of the consecu- 

 tive normal planes in the x and y directions with the normal at the 

 given point — thus consecutive normal planes do not in general meet 

 that normal plane. . 



43. Segre's Characteristics. The points for which B = 0, that 

 is, those w^here the indicatrix reduces to a linear segment have one 

 property of importance in common with surfaces in three dimensions. 



For if the indicatrix reduces to a linear segment, there are two directions 

 on. the surface, namely those corresponding to the ends of the linear seg- 

 ment, for which [i- — 0, and these are orthogonal directions, and for them 

 the normal curvature is a inaximum or a minimum. If these lines he 

 taken as parametric curves the second fundamental (vector) form and the 

 first fundamental form reduce simultaneously to the sum of squares, 



41 Kommerell distinguishes these cases by saying that the surface point is 

 elliptic, hj'perbolic, or parabolic, but though this distinction may be useful 

 in the case of surfaces lying in a 4-space and possibly at planar points in general, 

 there is apparently no similar classification in general surface theory. 



