140 PROCEEDINGS OP THE AMERICAN ACADEMY. 



secutive points of a ^-spread intersect in a point at least. Tangent 

 planes at consecutive points of a surface in w-fold space intersect at least 

 in points. These tangent planes do not generally intersect in lines 

 unless the surface lies in a space of three ways. Let us take p to repre- 

 sent the tangent plane at any point P of the surface and take p', p", 

 p'", ... to represent the tangent planes at the points P', P", P'", . . . 

 consecutive points of an infinitesimal closed curve about P. If p and p' 

 intersect in a line they determine a three-flat. If the consecutive tan- 

 gent planes intersect in lines, then p" has a line in common with both p 

 and p' and so p" lies in this three-flat. In a similar manner it can be 

 shown that p', p", p'" . . . , all the tangent planes consecutive, to p lie in 

 the same three-flat with it, i. e. a unicpue three-flat is determined at each 

 point of the surface that contains the tangent plane at the point and all 

 the tangent planes consecutive to it. Since however this three-flat is 

 determined by any two of these tangent planes, the three-flats corre- 

 sponding to P and P' any two consecutive points are the same. Take 

 now any curve through P that lies on the surface. Since the three-flats 

 corresponding to any two consecutive points of the curve are the same, it 

 follows that the three-flats corresponding to all the points of this curve 

 are the same. If we take a different curve through P the same thing is 

 true of the points of it. The three-flats corresponding to all the points 

 of these two curves are the same since they are all the same as the 

 three-flat corresponding to P. From this it follows that the whole sur- 

 face and all of its tangent planes lie in the same three-flat. Hence if in 

 general all the tangent planes consecutive to any tangent plane of a 

 surface lie in the same three-flat with it, then the whole surface lies in 

 this three-flat. 



In the same way it may be shown that if in general all the tangent 

 planes consecutive to the tangent plane at any point of a surface lie 

 in the same four-flat with it that the whole surface lies in this four-flat. 

 Hence in w-fold space not only do the consecutive tangent planes of 

 a surface not intersect in lines, but all the tangent planes consecutive 

 to any tangent plane do not lie in the same four-flat with it. 



c. The locus of the intersections of the tangent plane at any point 

 of a surface with the consecutive tangent planes. 



In a four-fold space let the surface be given by 



ry=o, 

 v=o, 



