Involiitoric Transformation of the Straight 



Line. 



V,\ ARNOLD EMCH. 



A projective transformation is said to be iii7'i>lut(>rii if every point 

 of the original sj'steni J can be interchanged with tlie corresponding 

 point of the transformed S}'stem J,. This relation exists, for 

 instance, between the projective points (Xj), (x) of a straight line 

 whicli are related by the e(]nation: 



ax + b 



1 



, (I) 



ex — a 



where a, b, and c are real numbers. That equation (i) expresses 

 involution, is seen by the value of x which follows from equation 

 (i), i. e., 



ax^-j-b , . 



X = LJ , (2) 



ex J — a 



Like every general projective transformation of the straight line, 

 the involution has two double points, i. e. , places where the values 

 for X and Xj coincide. These values are foimd from e(]uation (i ) 

 by putting x^=x: 



ex- — 2ax— b = o. 

 Whence 



a-|-T'a- + bc 

 m= — L_l \ , 



c 



a — 1 a'+bc 



n= 1 1 , 



c 



m and n designating the values of the double points. According as 



^=a2+bc<o, 



A=-a2+bc^o, 



A=^a-+bc>o, 

 the involution is elliptic, parabolic, or hyperbolic, i. o. has imagin- 

 ary, coinciding and real, or two real doul)le points. 



Having stated these general facts we will now investigate the 

 involutoric transformation with respect to its s^roiip pi-(>p('i-tit-s. 



(Ill) K.K.N. I'NIV. (;T'.\I!.. VOli. IV. NO. ;.'. (XTOItKW. ISl)."). 



