1057 



If we take i]^2, the curve C{i),q) is a parabola of which 

 and Y^ are ordinary points and we have as a special case the 

 well-known proposition : 



Let a quadrangle be inscribed in a conic, the tangents in the 

 vertices and a pair of opposite sides touch a conic. 



^ 6. The tangent in the point P{.v^fj^) to tiie curve Cp{p,q) 

 has the equation : 



■'g— ^•i _ y-yi 



The points of contact of tangents from a point L {)•, (i), (o tlie 

 curves of the pencil C{p, q) are lying on a conic ki- 



k — .^• fi — // 

 p,v qy ' 



or: 



{q—p)xy + pnx — qXy = 0. 

 The conic k/^ passes through the three vertices of the triangle of 

 co-ordinates and through the point L. And as 'v is a root of tlie 

 equation (2) ki, passes also through the point L^, of which the 

 co-ordinates are: 



X (x 



The curve CL{p,q) touches ki in L and intersects ki in L^. 

 This point L^ is the point of the curve Cl ip, q) that has the point 

 L as satellite point. 



If the point L is chosen on one of the sides of the triangle of 

 co-ordinates, ki degenerates into two straight lines. 0?ie of these 

 straight lines is the side of the triangle of co-ordinates on which L 

 lies, the second straight line passes through the opposite vertex. Of 

 the tangents to a curve C{p,q) that touch in ordinary points, 

 only two or one are real, if k^ degenerates. 



§ 7. Let (p be an arbitrary curve of order n, each intersection 

 Q of (f with kj, not being a vertex of the triangle of co-ordinates 

 possesses then the property that the tangent in Q to C(^. (/;,</) passes 

 through L. If therefore in any point of q the tangent is drawn to 

 the curve C{p,q) passing through this point, these tangents envelop 

 a curve of class 2 n — k, if k is the number of intersections of (f 

 with ki that coincide with the vertices of the triangle of coordinates. 



If (p is in particular a straight line the tangents to the curves 

 C{p,q) in the points of q envelop a conic. If the straight line 



