428 PROCEEDINGS OF THE AMERICAN ACADEMY. 



lines and also (7)-lines; the latter set, the (7)-planes, contain only 

 (7) -lines. 



Any plane passed through a given (6)-line cuts the cone in two ele- 

 ments and is therefore a (5) -plane. The geometry of such a plane is 

 the non-Euclidean plane geometry above described, and the elements 

 of the cone are the fixed directions. The -perpendicular in this plane 

 to the given (5)-line is a (7)-line. The locus of the lines perpendicular 

 to the given (5)-line in all the planes through the line is a (7)-plane. 

 This (7)-plane will be called perpendicular to the (5)-line. Such a 

 plane possesses no elements of the cone, that is, no lines which are 

 fixed in rotation; hence the geometry of a (7)-plane is ordinary 

 Euclidean geometry. In the plane any line may be rotated into any 

 other line, and the locus of the extremity of a given segment issuing 

 from the center of rotation is a closed curve which is the circle in that 

 plane. Moreover, the idea of angle, and of perpendicularity between 

 lines in the (7)-plane, being the same as in ordinary Euclidean geome- 

 try, need not be further defined. 



A plane passed through a (7) -line may cut the cone in two elements 

 and be a (5)-plane, or may fail to cut the cone and will then be a (7)- 

 plane.^^ The perpendiculars to a (7)-line will therefore be in part 

 (5)-lines and in part (7)-lines, and the plane perpendicular to a (7)- 

 line will therefore be a (5) -plane. Thus a plane perpendicular to a 

 (5)-line is a (7)-plane, and a plane perpendicular to a (7)-line is a 

 (5)-plane. 



In any three dimensional rotation one line, the axis of rotation, 

 remains fixed, and points in any plane perpendicular to the axis remain 

 in that plane. If the axis is a (5)-line, the rotation is Euclidean; if 

 a (7)-line, non-Euclidean. 



When all possible rotations, Euclidean and non-Euclidean, about 

 axes through a given point are considered, the locus of the termini 

 of a (7)-vector of fixed interval, and a (5)-vector of equal interval, 

 issuing from the common center of the rotations, is a surface which 

 from a completely Euclidean point of view appears to be the two 

 conjugate hyperboloids of revolution asymptotic to the fixed cone, 

 but which from our non-Euclidean viewpoint is really analogous to 

 the sphere. The (5)-lines cuts the two-parted hyperboloid; the (7)- 

 lines, the one-parted. 



27. If we construct at a point three mutually perpendicular axes, 

 two will be (7)-lines, and one a (5)-hne. The unit vectors along these 



23 Planes tangent to the cone will be discussed later. 



