CYCLOGRAPHIC TRANSFORMATION OF ORDINARY SPACE 41 



The trace of the singular surface of the cyclographic congru- 

 ence hased upon the curve of intersection of an orthogonal hyper- 

 holoid of rotation of one nappe and a right cylinder of the n"* 

 order or m*^ class is a curve of the 27n*^ order. 



6. General Representation of Curves. 



A curve in space may always be considered as the intersection 

 of two surfaces 



F {x, y, z)=o and G (x, y, z) = o. 



The system of circles in the plane of reference, representing the 

 curve, is therefore common to two linear congruences and may have 

 an envelope. Representing the curve by 



^=f{t\y = g {t\z = h it), 



where ^ is a parameter, the equation of the system of circles 



becomes 



\x-f{t)\^^\y-g{t)X'={h{f)J. 



The envelope is obtained by eliminating t from this equation and 



In general this elimination is diflScult, so that other methods must be 

 applied. 



7. Representation of Conics. 



The circles representing a plane curve, all intersect a certain 

 straight line of the plane of reference under a constant angle. Thus, 

 the system of circles intersecting a given straight line under a con- 

 stant angle (real or imaginary) and having their centers on a conic, 

 represent a conic in space. The determination of the envelope of the 

 system of circles representing a conic in space requires the considera- 

 tion of the congruence of rays which pass through two given conics 

 in space. Through every point in space (not on one of the conics) 

 four rays of the congruence may be passed and in every plane lie six 

 rays. Hence, the congruence is of the 4th order and 6th class. The 



