CTCLOGEAPHIC TKANSFORMATION OF ORDINARY SPACE 37 



which shows that all circles of the congruence intersect the line s 

 under a constant angle, which is real or imaginary according, as 



a^ + P % c^, respectively. 



(See geometrical proof under 2). 



4. Linear Congruence of Circles. 



Let X"^ + J^^ = i?^ be the equation of a fixed circle, and 



{X-xy-\-{r-yy=7^ 



the equation of any circle orthogonal to the first. The condition for 



orthogonality is 



ic^ -\- y^ = JR^ -\- r^. 



In the cyclographic interpretation of space we have to put 

 7' = 2, SO that 



x^ -\- y^ — ^^ = -^^• 



This is the surface, i^(a', y, s) =0, and represents an orthogonal 

 hyperboloid of rotation of one nappe, having the s-axis as an axis of 

 rotation. 



A system of circles representing such a surface is called a linear 

 congruence of circles. If we now introduce another condition, 



G (a?, y, s) = 0, 



the circles of the congruence limited by this condition will be singly 

 infinite in number. Their centers are situated on a certain curve 

 (projection of intersection of x^ ^ y^ — jR^=0, and 6^ (a", y, s) = 0), 

 and they may envelope a curve. If {r = is a plane, the locus of the 

 centers is a conic and the envelope 2^ hiclrcular qiiartic. 



In tetracyclic co-ordinates^ the equation of the linear congruence 

 may be written 



«! m + a?2 ^2 + ^3 ^3 + a?^ m^ = 0, 



' See M. BOcher: Uber die Reihenentwickelungen der Potentialtheorie, Chapts. 1 and 2. 

 G. Darboux: Th6orie Q6n6rale des Surfaces, Vol. I. Chap. VI. 



