( 30G ) 

 is the circle represented by 



a^ + y^ = iA+B) C. 



This orthoptical circle (Monge's circle) is real for the ellipses 

 and the hyperbolae situated in the acute angles of their asymptotes, 

 and imaginary if the hyperbola lies in the obtuse asymptote-angles; 

 it degeneratos into a point for the rectangular hyperbola {A-\-B=0) 

 and for the system of two right lines {C=0). With the parabola 

 it is represen<:ed by the directrix. 



The orthoptical circle being concentric with the conic to which 

 it belongs, the centres of the orthoptical circles « of a pencil of 

 conies lie on the conic 7, containing the centres of the pencil. 



If now, following Mr. Fiedlek, we represent each circle by the 

 vertices of the two right cones of which it is the base-circle, the 

 system (0) is represented by a skew curve of the 4'^^ order £2i. 

 Indeed, each plane perpendicular to the plane V of the orthoptical 

 circles intersects the conic in two points and bears two pairs of 

 points representing orthoptical circles. 



The skew curve £2i being intersected in four points by a plane 

 parallel to T', there are in («) four circles with given radius. 



So the system («) contains four point- circles; it is evident that 

 we find these in the double points of the three degenerated conies 

 and in the centre of the orthogonal hyperbola belonging to the 

 pencil. 



2. The system («) contains two orthoptical right lines, viz. the 

 directrices of the two parabolae in the pencil. Each of these lines 

 is represented by two planes inclined to the plane V at angles of 

 45°, i.e. parallel to four generatrices of each rectangular cone //j 

 having its vertex in a point P of V and placed symmetrically with 

 respect to this plane. 



The cone 11^ bearing the images of all the cii'cles through P, and 

 the above named four points at infinity )'epresenting two right lines 

 not passing through P, the remaining four points of intersection 

 of Hi with £2i will represent two circles (w) intersecting in P. 

 Therefore : 



The orthoptical circles of a pencil of conies form a system with 

 index two. 



This can also be shown as follows. The tangents through P to 

 the conies of the pencil are arranged in a (2,2)-correspondeuce, 

 each ray through P being touched by two curves. Tnis system his 



