CYCLOGRAPHIC TRANSFORMATION OF ORDINARY SPACE 35 



tan a=X. 



A plane P in a general 

 position intersects the 

 plane of reference E in a 

 straight line s (Fig 1). 

 For any point A of the 

 plane P and the respect- 

 ive circle, with 2V as a 

 center and lYA"^ as a 

 radius, the relation exists 

 that the ratio of the ra- 

 dius AA^ and the distance 

 A'B of the center J.' 

 from s is constant. Hence, 

 if <A B A'=i, Fig. 1. 



AA' 

 BA' 



-tan i, 



A'B 

 and since A''Z^=^AA\ -j^ = const. = cos ^, the cosine of the angle 



which the tangent at L makes with s. Conversely, it can easily be 

 shown that any circle intersecting s at a constant angle ^, represents 

 a point of the plane F. Hence the theorem: 



The sy stern of all circles intersecting a fixed line in their plane 

 under a constant angle, represents a plane in space. 



This theorem is general and applies to all planes in space if 

 imaginary angles of intersection are included. In any real or im- 



TT 



maginary case of <^, tan i. cos <^^1. For tan t<^l, i. e., ^<C^ 

 cos </>>!, so that </> becomes imaginary. 



3. Complexes and Congruences of Circles in a Plane. 



The fact that ordinary space contains a triply infinite number of 

 points, and a plane a triply infinite number of circles, makes it pos- 

 sible to establish a uniform correspondence between the points on one 



