( 309 ) 



6. If Z7r and Vy represent tlie differential quotients of the qua- 

 dratic function U with regard to x and y, the centres of tlie conies 

 of the net 



f/ + A T' + ^ T-F = 

 are indicated by the reUitions 



So in general any point (.c, ?/) is the centre of one conic. On the 

 other hand any of the four points determined bv 



U^: U^= V, : T'^ = W^ : Wy 



is the common centre of an infinite number of conies. For each of 

 these points the above mentioned linear equations are dependent of 

 each other, so that A and // are connected by a linear relation ; so 

 in this way a pencil of conies is characterized. 



Conseqitently the surface Sr, contains four right lines perjjendi- 

 cular to V. 



7. Let us now consider the orthpptical circles belonging to the 

 conies with four common tangents. Any right line through the point 

 P determining a conic of the tangential pencil, the tangents through 

 F form an involution. This contains only one pair of rectangular 

 rays; so P lies on one orthoptical circle. 



The wellknown property accoi'ding to which the centres of the 

 tangential pencil lie in a right line, also proves that the orthoptical 

 circles of a tangential pencil form a pencil. 



According to the circles of that pencil passing through two fixed 

 points or intersecting a fixed circle orthogonal or touching each 

 other in a fixed point, the system (co) will be represented by a rec- 

 tangular hyperbola with its real axis perpendicular to V or situated 

 in Fj or by two right lines cutting V at angles of 45°. 



So the system («) contains no more than two point-circles or, in 

 other words, the tangential pencil can contain only two rectangular 

 hyperbolae. 



A conic of the tangential pencil degenerating into two points, the 

 line joining these points is the diameter of the corresponding m. In 

 connection with the statement above the wellknown property results 

 from this: 



The circles described on the diagonals of a complete quadrilateral 

 as diameters belong to a pencil. 



