SURFACES IN HYPERSPACE. 351 



Definition 1. Kommerell calls the directions, which give a maxi- 

 mum or minimum magnitude to the normal curvature, principal 

 directions and points out that for the principal directions fin the four 

 dimensional case) the point of intersection of consecutive normal 

 planes lies in the osculating plane of the normal section through that 

 direction. This is clear from the configuration of the indicatrix when 

 a»|JL = 0. For here the tangent PT a,t the point P of the indicatrix 

 is perpendicular to OP the vector from the surface point to P, 

 and consequently P' the inverse of the pedal of P, lies on OP. [He 

 calls the conic which is the inverse of the pedal of the indicatrix (our 

 Conic I) the characteristic (our Conic II) and shows that the princi- 

 pal directions correspond to the lines OP' which may be drawn from 

 perpendicular to the characteristic. These lines are the same as 

 those perpendicular to the indicatrix.] In this way he generalizes 

 one property of the principal directions in three dimensions. The 

 generalization is far from perfect. For in the case of three dimensions 

 consecutive normals do not intersect in general, but do intersect for 

 principal directions, and the intersection lies in the osculating plane 

 of the normal section through a principal direction. In the four 

 dimensional case the normal planes in general intersect, but except 

 for the principal directions the point of intersection does not lie in the 

 normal ci, and is not in the osculating plane of the normal section 

 through the direction. 



The result may be generalized to the general case of a surface in 

 five (or more) dimensions. For if be the surface point, F the foot 

 of the perpendicular upon the plane of the indicatrix, and P a point 

 such that the tangent PT is perpendicular to OP, then as PT is per- 

 pendicular to OF, PT is perpendicular to the plane OF P. Hence FP 

 is perpendicular to PT. Thus the points P of the indicatrix which 

 correspond to Kommerell's principal directions on the surface are those 

 for which the radius FP from the foot of tlie perpendicular OF is 

 perpendicular to the indicatrix. Moreover, if F' be the inverse of F 

 all the normal three spaces pass through F\ Consecutive normal 

 spaces intersect in a line through F' perpendicular to the plane of 

 0. = OP and [it = PT. Hence the intersection F'P' of consecutive 

 normal spaces cuts the line OP in some point P'. Thus: one of the 

 <x> ^ points of intersection of consecutive normal spaces lies in the osculating 

 plane of the normal section in case that section corresponds to a direction 

 of maximum or viinimum normal curvature. 



As Kommerell points out, in the special case of a surface which at 

 is of the three dimensional type, the condition a'|JL = 0, breaks down 



