SURFACES IN HYPERSPACE. 349 



linear segment is collinear with h, the normals are all collinear and must 

 satisfy (118) and an additional relation 



^'yii + 5'yi2 + C'yo2 = 0. (118') 



Segre *^ showed that if the coordinates of a surface satisfy the rela- 

 tion (lis) at each point, there is traced on the surface a double system 

 of curves, called characteristics, having the property that tangent 

 planes to the surface in two infinitely near points in the direction of 

 one of these characteristics will intersect in a line tangent to the other. 

 Also an Sn-i passing through the tangent plane will cut the surface in a 

 curve ha\nng a node at the point of contact and such that the tangents 

 at the node are separated harmonically by the tangents to the char- 

 acteristics. The direction of the nodal tangents correspond to the 

 points in which a line drawn through the surface point cuts the indi- 

 catrix. The considerations above given show that these surfaces 

 are those for which hxfiixS = (see §39). 



If the equation (118) is of the parabolic type, that is if B^ — AAC = 0, 

 the two characteristics will coincide. If this happens Moore *^ showed 

 that the characteristics have the property that their tangents have 

 three point contact with the surface. For this type the indicatrix 

 passes through the surface point and consequently one of the nodal 

 tangents always coincides with the tangent to the (double) character- 

 istic. 



Segre also showed that a surface whose coordinates satisfy two equa- 

 tions (118), (118') either lies in a three space or else consist of the 

 tangents to a twisted curve. These are the surfaces for which our 

 invariant (|ix8)2 _ $2s/«^ of §39 vanishes, and the statement there 

 made is thus substantiated. 



If the indicatrix degenerates into a linear segment not passing 

 through the surface point, the two characteristics are perpendicular, 

 and this is the only case in which the characteristics are at right 

 angles. If the linear segment passes through the surface point two 

 cases arise. 1° If one end of the segment lies on the surface then at 

 that point the surface has the character of a twisted curve surface. 

 If the condition is satisfied identically the surface is a twisted curve 

 surface. 2° If the segment does not have an end in the surface then 



42 Segre, Su una classa di superficie degl'iperspazi, ecc, AUi di Torino, 1907. 



43 C. L. E. Moore, Surfaces in hyperspace which have a tangent line with 

 three point contact passing through each point, Bull. Amer. Math. Soc, 18, 

 1912. 



