SURFACES IN HYPERSPACE. 355 



Indeed equation (78), namely, dM'd'M. = — Gds^ -^ 2h«^ contains 

 an important property of asymptotic lines on ordinary surfaces; 

 the asymptotic lines are those for which the rate of turning of the 

 normal, in this case the torsion, is ■^J { — G). Along the asymptotic 

 directions in the general case of ?i dimensions, dM-dM = — Gds^, 

 tiuit is, the rate of turning of the tangent plane is V (— G), if by anal- 

 ogy (dM/dsy may be called the rate of turning when successive 

 planes do not intersect in a line.*^ 



In the ordinary case a = for the asymptotic lines. In the general 

 case tt'h = 0. To prove this consider |see (67)] 



= h-^/ = h-i:„[a'XA* + K-(XrX, + XAs) + pX,X.]r/.iv/a-, . 



As d.Vr-Avs = X'"' -.X^') for the curves defined by X, these curves will be 

 along the asymptotic dir?ctions when and dhly when h»a = 0, as the 

 other terms vanish in the summation. 



The condition h*a = means that the curvature of the asymptotic 

 line is perpendicular to h and consequently the osculating plane of the 

 curve is perpendicular to h. C'onversely if the osculating plane is 

 perpendicular to h then a must be perpendicular to h. Hence: The 

 asymptotic line is characterized by the property that its osculating plane 

 is perpendicidar to the mean curvature vector as in the three dimensional 

 case. 



As in the case of principal directions (Definition 3), the asymptotic 

 lines we have defined become illusory for minimal surfaces. For 

 surfaces, not minimal, the asymptotic lines cannot he orthogonal, as may 

 be seen from the configuration of the indicatrix. 



That the condition h-a = may be satisfied for a real direction on 

 the surface, the plane tt through the surface point perpendicular to h 

 must cut the indicatrix in real points. Now: The asymptotic lines 

 here defined for any surface are bisected by the principal directions 

 (Definition 3). For the plane tt is parallel to K- if h-K- =^ and cuts 

 the plane of the indicatrix in a line parallel to |x. The two vectors 

 a go to points of the indicatrix which represent equal amounts of the 

 surface angle d, above and below the directions for which h*JJ' = 0. 

 If this plane tt is tangent to the indicatrix the asymptotic lines fall 

 together. The condition that the asymptotic directions fall together {and 



45 If we define the angle between two plane vectors, whether or not these 

 be simple planes, by the formula cos e = M'N/CM^N^)! we have a real angle 

 whenever the planes or complexes are real. If M is a unit plane, N a nearby 

 unit plane M + AM, then 2M.AM + (AMJ= = and by a familiar trans- 

 formation we find {de/dsY = {dM/dsy. 



