684 MOORE. 



circles of the same generation intersect in two points while those of 

 opposite generations intersect in one point. Equations (47) show 

 that the surface is also a translation surface. 



Wilson and Moore ^* discussed the locus of the end of the normal 

 curvature vector (the indicatrix) of curves passing through a given 

 point of a surface and found that in general it is a conic. But when 

 this indicatrLx becomes a linear segment the surface has some proper- 

 ties of surface in 3-space. On such a surface lines of curvature can 

 be defined as in 3-space and will be orthogonal. If we define lines of 

 curvature as lines of maximum or minimum normal curvature we find 

 in general there are four directions through each point but in 3-space 

 these four directions divide into two sets of two, one the asymptotic 

 lines and the other the lines of curvature. For surfaces whose indi- 

 catrix reduces to a linear segment not passing through the surface 

 point in question these four directions of maximum and minimum 

 radii of curvature again factor into two sets; one giving the curves 

 called by Segre characteristics and the other giving lines analogous 

 to lines of curvature in 3-dimensions. For the surface here consid- 

 ered all four sets of these curves are circles. 



We saw that the curvature segment or indicatrix cut the planes 

 Ml and M2 in the ends of the projection of r on these planes. Then 

 as r is rotated the curvature segment will cut the circles generated by 

 these projections. Hence the locus of the curvature segment will be 

 the congruence of lines cutting two given circle. Also the mean 

 curvature defined by the curvatures of two orthogonal directions. 



2h =Ci + C2 



is the vector from the surface point to the middle of the curvature 

 segment. Then the locus of the end of the mean curvature vector 

 will be the surface 



,.2 , ,.2_ ^«"- IV 

 .ll -T .12 — 



Xs^ -\- Xi- = 



a- 



}f- - IV 



62 



which is a surface like (43). 



We have here considered general positions of the vector r but an 

 interesting case arises when /• is so located that its projection on J/i 



14 Differential geometry of two-surfaces in hvperspace. These Proceedings, 

 62, 1916. 



