( 308 ) 



When the base points of' tlie circle form an orthocentric group i) 

 all the conies are rectangular hyperbolae ; so the system (a) consists 

 entirely of point-circles. 



The orthogonal circle y/, bearing them, passes through the diago- 

 nal points of the complete orthogonal quadrangle and coincides with 

 the circle of Feuerbach (nine-points circle), containing as is known 

 the centres of all the orthogonal li^perbolae through the vertices and 

 the orthocentre of a triangle. 



5. According to the method of Fiedler the orthoptic circles 

 of a net of conies will be represented by a surface, intersecting the 

 plane V of the net in the locus of the orthoptical point-circles. 



Any point P determines a pencil belonging to the net, one of 

 the base points of which is -P ; so through P passes an orthogonal 

 hyperbola. From this follows, that the orthogonal hyperbolae form 

 a pencil situated in the net. As was said above the orthoptical 

 point- circles of this pencil lie on the circle y/, passing through the 

 diagonal points and the middle points of the six sides of the qua- 

 drangle of the base points. 



The remaining point-circles « are centres of pairs of lines and 

 form the cubic curve, called the PIessian of the net. 



Three pairs of lines of the net belong to the pencil of orthogonal 

 hyperbolae and consist of orthogonal i-ight lines. The double points 

 must lie on the Hessian as well as on the circle yl; so these curves 

 must touch each other in three points. 



Both loci of point-circles forming together a curve of the 5* order, 

 the image of the system of orthoptical circles is a surface of the b^^ 

 order .S'j. 



Each right line of V is touched in each of its points by owe conic 

 of the net; each right line determines the direction of the axis of 

 a parabola belonging to the net. From this follows, that the point 

 at infinity of each right line cutting V at angles of 45° is to be 

 regarded as the image of an orthoptical right line. 



So the points of contact of the asymptotes of the rectangular 

 hyperbola representing all the circles through two points belong to 

 the ten points common to S^ and that hyperbola. The remaining 

 eight points of intersection representing four circles it is evident 

 from this that we can draw four orthoptical circles of t/ie system 

 through any two points. 



') That is to say: four points, each of which is the orthocsntre of the triangle 

 havino' the others for vertices. 



