356 WILSON AND MOORE. 



coincide with one of the 'principal directions) is that h shall he normal to 

 a tangent plane to Cone I, hence an element of Cone II, that is, 



h.$.h = = (h-h)2 - (h-K')2 - (h-8)2. 



In ordinary surface theory this reduces to h^G = 0; but here h, |JL, 

 and 8 are not colHnear and the condition is not satisfied by G = 0. 

 Indeed, 



h-*-h - h-$s = h^ji' - {h'\i.)- + h252 - (h-8)2 



= (hx|Ji)2 + (hx8)2 = h-$-h - h'-6'o. 



We see therefore that surfaces for which G = make h'$*h positive 

 unless h, [t-, 8 are coUinear, that is unless the surface is three dimensional 

 at the point in question. The condition h*$'h > means, however, 

 that the plane tt perpendicular to h through does not cut the indi- 

 catrix, that is that the asymptotic directions on the surface are imag- 

 inar\'. Hence : For all developahles except the twisted curve surfaces the 

 asymptotic directions are imacjinary. 



The special cases which arise when the surface is four dimensional, 

 with the indicatrix either an ellipse or a linear segment are not espe- 

 cially different from the general case. 



The scalar form h • ^ which for -the definition of asymptotic direc- 

 tions (in our sense) has taken the place of the scalar form \p (second 

 fundamental form) in three dimensions may be used to define a con- 

 jugate system of curves upon the surface as in the ordinary three 

 dimensional case. The asymptotic lines h*^ = are then the double 

 elements in the involution. It is easy to see that the lines of curva- 

 ture (in our sense) are the pair of orthogonal elements in this involu- 

 tion. For if we use the lines of curvature (in our sense) as parametric 

 curves the forms h*^ and <p = ds~ are simultaneously reduced to a 

 sum of squares since ai2 = and h • yi2 = in this case. 



Kommerell's generalization of asymptotic directions in case n = 4 

 was to those directions which correspond to the infinite points of his 

 characteristic (our Conic II, inverse pedal to the indicatrix), i. e., to 

 directions which make the normal curvature a tangent to the indi- 

 catrix, namely cix|JL = 0. On a surface in Sa there are two such direc- 

 tions; but the generalization breaks down for w > 4 because the con- 

 dition ax|jL = cannot be satisfied.*^ We are therefore forced to 



46 Kommerell's second fundamental form, the vanishing of which determines 

 his asymptotic directions, has therefore no relation at all to fjencral surface 

 theorj^, because his asymptotic directions do not exist in general. The second 

 fundamental forms which we develop are vital to the theory. 



