( 307 ) 



two pairs of rays in common with the involution of the lines, 

 intersecting in P at right angles ; so through F pass two circles (o. 

 Two planes intersecting the plane V in the right line I at angles 

 of 45° have four pairs of points in common with the curve 124, 

 This shows that I is touched by four orthoptical circles. Their 

 points of contact are the coincidences of the (2,'2)-correspondence, 

 determined on I by the system («). 



a. If we represent the form 



^" +r + «•'■ + ^'j + <• 



by C, the system (w) is represented by the equation 



Ci + 2 A C3 + A2 Cg = 0. 



The power of the point (.r, </) with regard to the circle indicated 

 by a definite value of A is then represented by 



Ci + 2 A C2 4- A2 Q 



1 + 2 A + A2 



This expression becomes independent of A if we assume the radi- 

 cal centre of C], Cj,,, C3 for (j-, y). So all the circles (o have a com- 

 mon radical centre or, in other words, all orthoptical circles cut a 

 fixed circle 1 at right angles. 



As ^f must bear the point circles of (w) we may conclude to the 

 following theorem: 



The circle throayh the diagonal points of a complete quadrancjle 

 contains the centre of the orthogonal hyperbola determined by the ver- 

 tices of the quadrangle. Through its centre pass the directrices of 

 the parabolae ivhich can be drawn through those vertices. 



Considering the obtained results as a property of the rectanguhir 

 hyperbola it can be expressed in the following terms : 



The diagonal points of each complete quadrangle inscribed in a 

 rectangular hyperbola can be connected by a circle with the centre 

 of that curve. 



4. All circles « being orthogonal to the fixed circle yj their 

 images lie on the orthogonal hyperboloid of revolution with one 

 sheet cutting V in yl at right angles. 



So the image £2^ is the section of this hyperboloid with the cylin- 

 der, which is orthogonally cut by V in the conic ;'. 



21* 



