CONTINUOUS GROUPS OF PROJKCTIVK TKANSFORlMA TIONS. 249 



Case IV. When the two eonies have a eoiitaet of the first order on 1, 

 tlie invariant fi^i^i/re eonsists of ait points on t/ie line 1. a si ni;;/e point A 

 not on 1, a /id the pencil of lines through A. 



Case V. Jf'hen the eonies have contact of the second or third order 

 on 1, ///(• invariant figure eonsists of all points on the line 1 and all lines 

 through a point A on 1. 



We need a convenient and expressive notation for these five 

 kinds of projective transformations in the plane. We shall desig- 

 nate a transformation of the first kind by the single letter T. When 

 we wish to designate the conies which determine the transforma- 

 tion, we shall write it T (KK^ ). WHien we wish to call attention 

 to the triangle which remains invariant we shall use the notation 

 T(ABC). Transformations belonging to cases II and III will be 

 designated by T^ and T ' ' respectiveh'. Transformations belong- 

 ing to cases IV and V will be designated by S and S' respectively. 



(To be continued.) 



