246 KANSAS UNIVF:KSrrV (^)UAkJ ERIA . 



llicorciii 2. The conic K dividi's ilw plane tt into the rci^^ions O (^out- 

 side) and I (/nsidr) ; ///(■ t-(i//ic K^ //rridrs tiic plane tt' into two correspond- 

 ing regions O^ and I^. T/ie projective transformation determined liy K 

 and K'^ transforms t/ie region O into O^ and tlie region I into /' ; // 

 transforms the boundary of O and I into the t>onndarr of O^ and /'. 



These two conies touching the line 1 have generally three other 

 common tangents which will be designated by x, y, z, (Fig. i). 

 We shall next examine the relation of these three lines to the trans- 

 formation determined by the two conies K and K^. Let us take a 

 point P on the line z and draw the two tangents from it to the conic 

 K; one of these tangents is the line z which cuts 1 at C,, the other 

 cuts 1 at some point as Q. The corresponding point to P is found 

 by drawing tangents to K^ from C^ and Q. One of these tangents 

 is again the line z; the tangent to Ri from Q intersects z in P^ the 

 corresponding point to p. In like manner every point on the line 

 z is transformed into a point on the line z; in other words the line 

 z is an invariant line of the transformation. In the same way it 

 may be shown that the lines x and y are also invariant lines of the 

 transformation. The fixed line 1 is not an invariant line, although 

 a common tangent to the two conies. 



The points A, B, C, which are the intersections of the invariant 

 lines X, y, z, are invariant points of the transformation. This is 

 evident from the fact that the two tangents from A to K are the lines 

 y and z; the two tangents from C^ and B^ to K^ are also y, and z, 

 which intersect at A the startling point. Thus A is a self corre- 

 sponding point, or an invariant point, of the transformation; the 

 same is true of B and C. 



It is easy to see from the construction that these three points 

 and these three lines are the only ones left invariant by the trans- 

 formation determined by K and K^. In particular cases where the 

 conies K and K' are specially related to one another, e. g. touch one 

 another, the invariant figure may be different. These special cases 

 will be determined later. When the conies K and K^ do not inter- 

 sect, the sides and vertices of the invariant triangle are all real; but 

 if the conies intersect in two real and two imaginary points, they 

 have only one real common tangent other than 1, the other two 

 being imaginary. In this case the invariant triangle has one real 

 and two imaginary vertices, one real and two imaginary sides. If 

 the two conies intersect in four real points, tlie invariant triangle is 

 again real in all its parts. We shall call this most general transfor- 

 mation, Case I . 



For special positions of the conies K and K^ the invariant figure 



