SURFACES IN HYPERSPACE. 347 



is negative, zero, or positive. ^'^ The conic breaks up into two lines 

 if Br = 0, that is if the indicatrix is a Hnear 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 Zi = 1/h, S3 = — e/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/h, — e/Ah) in this special case; the 

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

 the normal plane in the lines Si (h + ^) + Azs = 1 and Zi{h -\- e) — 

 Az^ = 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 Avhich 

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

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

 axial in Le\'i'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- 

 ti\e 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 where 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 xvhich H'' = Oj (i-nd these are orthogonal directions, and for them 

 the normal curvature is a maximum 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, 



(p = OnfArr + a-ndx'^, "^ = Ynd'i-T + Y^^dxo^. 



41 Kommerell distinguishes these cases bj' saying that the surface point is 

 elliptic, hyperboUc, or parabohc, 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. 



