SURFACES IN HYPERSPACE. 357 



conclude that neither the principal directions nor the asymptotic 

 lines as defined by Kommerell are the best generalizations of corre- 

 sponding lines on ordinary surfaces. That we have found other and 

 better definitions may be attributed in part to the broader view point 

 that we get by working in higher than four dimensional space, but 

 must be credited in large measure to the suggestiveness of the method 

 of attack developed by Ricci in his Lezioni. 



Kommerell's type of asymptotic lines will exist only when ax|i, = 0, 

 that is, only when the indicatrix lies in a plane through the surface 



FiGUKE 3. 



point and the surface becomes four dimensional at the point. This 

 condition has been discussed in §43. It will be seen that Kommerell's 

 asymptotic lines are identical with Segre's characteristics; they are 

 therefore important lines for those surfaces on which they exist. 

 (Levi has asymptotic lines only in case the two characteristics coin- 

 cide, their common direction being then called asymptotic.) 



In the four dimensional case the important lines on the surface are 

 as follows: Two principal directions corresponding to the points Pi 

 and P2 for which h-\i- = 0; two asymptotic directions (in our sense) 

 Ai and A2 for which h«a = 0; two characteristic directions Ci and C2 

 for which axp, = 0. The principal directions are orthogonal and 

 bisect the asymptotic directions, but need not bisect the character- 

 istic directions; the asymptotic and characteristic directions divide 

 each other harmonically. If lies on the indicatrix, Ai, Ci, C2 coin- 

 cide. If the indicatrix reduces to a linear segment Pi and Ci, P2 and 

 C2 coincide. 



46. The Dupin indicatrix. Another way of getting at the prop- 

 erties of a surface in ordinary space is by the Dupin indicatrix, which is 

 the intersection of the surface by a tangent plane (or a plane parallel 

 thereto). In five dimensions we must take a hyperplane (a four 



