348 WILSON AND MOORE. 



Further: the rate of change of the tangent plane squared is, from 



(73), (80), (82), 



— - X -T^= (axil + IxjA) X (Ctx-q + gxjjL) = 2|x'nx|ixa. (117) 

 ds as 



As |xT| and H^xa are completely perpendicular, the only possibility for 

 the product to vanish is that H-xa = 0. This will vanish when |Ji. = 

 and hence: If the indicatrix is a linear segment there are two directions 

 for which the rate of change d^l/ds is a simple plane vector. In this 

 case M and dM. intersect in a line. There are then only tivo directions 

 in ivhich consecutive tangent planes intersect in a line. 



If the indicatrix does not reduce to a linear segment the only way 

 that H'-^o, can vanish is to have 1^ and a parallel or a vanish. Now 

 the latter alternative will happen when and only when the indicatrix 

 (now an actual ellipse) passes through the surface point and in this 

 case there is only a single direction in which c/M is a simple vector. 

 If hx|Xx6 = 0, that is, if the surface at the point is four dimensional 

 (i. e., planar), there are two directions on the s^irface for which P'Xa = 0, 

 namely, those which make a tangent to the indicatrix, for these directions, 

 and only for these, c?M is a simple plane and consecutive tangent planes 

 intersect in a line. These two directions cannot be perpendicular 

 and may be imaginary, they are coincident in the case where ci may 

 vanish. // the surface at the point is five dimensional, f/M is never a 

 simple plane. 



It appears therefore that in no case above three dimensions can 

 conjugate directions be defined by considering intersections of consecu- 

 tive tangent planes. 



We may express upon the y's the condition of degeneracy. First 

 if the surface is four dimensional at the point, then at that point yn, 

 yi2, y22 must be coplanar and a linear relation 



^yn + 5yi2 + Cy22 = (118) 



must subsist between these. Next if the ellipse collapses into a seg- 

 ment, the condition (102) for |Ax8 = may be used to show that the 

 normals, yn/an, Yn/aio, Yii/ciii are termino-collinear and the relation 



Aan + Bai2 + Ca^ = 

 subsists between the coefficients A, B, C in (118). Finally if the 



