CYCLOGRAPHIC TRANSFORMATION OF ORDINARY SPACE 39 



The equation of the bieirciilar quartic is 



^j (^Z + ttj *'2^ + ^'3 ''^'3' + ^4 *'4^=0? 



and has the four orthogonal circles as co-ordinate circles. 



We shall illustrate these results by an example. Assume as 

 one of the cones of intersection an hyperbolic cone having its vertex 

 Vj on the sphere. In the adjoining plate a symmetrical arrange- 

 ment has been effected. The horizontal plane of projection has been 

 assumed through the center of the sphere, and the base of the cone 

 ( ^1 ) ^ppsars as a hyperbola in the hotizontal plane. The vertical 

 projection of the C\ is an arc of an ellipse. The same is true of the 

 profile-projections (not shown in the figure). The tangents at the 

 double-point of the horizontal -projection of O^ are the asymptotes of 

 the base-hyperbola. 



The four cones ( V^), ( V^), ( V.^), ( V^), consist, therefore, of the 

 original cone, which is double-counting ( V^), ( V^), and two elliptical 

 cylinders (V^) and (1^4), so that only three systems of circles are 

 obtained. The stereographic projection was determined by taking 

 the profile-plane through the center of the sphere as a plane of pro- 

 jection. In the plate the three systems of circles are distinguished 

 by black, red and green lines, respectively. 



The stereographic projection of this plate admits of an interest- 

 ing interpretation by the cyclographic method. Consider any of the 

 systems of circles — for instance, the red circles. As these are all 

 orthogonal to a fixed circle (red dot), they represent points of 

 an orthogonal hyperboloid of rotation of one nappe. As their 

 centers are on a conic (red dash), they also represent points on a 

 cylinder. This system of circles consequently represents a twisted 

 curve of the fourth order C'^, the intersection of the cylinder and the 

 hyperboloid. It follows, therefore, from my previous paper, that ihe 

 hi-cireular quartic in thejplane of projection is the trace of the sur- 

 face of singularities of the cyclographic congruence defined by the 

 twisted curve C^, in space. 



