118 KANSAS UNIVERSITY SCIENCE BULLETIN. 



transforms the point x into xi. This equation may be solved for x, 



giving us 



_ dXi+b > 



cxj — a v ' 



The transformation expressed by this equation is called the inverse 

 of T, and is symbolized by T" 1 . T" 1 transforms a point xi into x. The 

 resultant of T and T" 1 is geometrically evident ; it leaves every point 

 on the line unchanged, and is therefore called an identical transforma- 

 tion. The resultant of T and T' 1 reduces to x-2=x, which shows that 

 every point on the line is transformed into itself. 



Invariant points. — When the points of a line are shifted into new 

 positions by a projective transformation T, does it ever happen that 

 one or more of the points are unaltered in position ? To answer this 

 question, we reason as follows : The coordinate of a point x, which 

 remains at rest or unaltered in position, i. e., which is transformed into 

 itself, must satisfy the equation 



__ ai + b | 

 ex + d 



Clearing of fraction, we see that the coordinates of all such points 

 satisfy the quadratic equation 



ex 2 + (d — a)x — b = o; 



whence we conclude that a projective transformation T leaves un- 

 altered two points on the line, and their coordinates are given by the 

 roots of equation (5). These two points are generally distinct, but 

 for special values of a, b, c, d, they may coincide. They are called 

 the invariant points of the transformation. Two transformations will 

 not generally have the same invariant points, but, as we shall learn, 

 an unlimited number of transformations may have one or both in- 

 variant points in common. 



Theorem 2. Every projective transformation of the points on a 

 line, except the identical transformation, leaves invariant two points 

 which are distinct or coincident. 



Pseudo-transformations. — If the determinant of T vanishes, the 

 transformation is called a pseudo-transformation. In denning the 

 transformation, it was expressly stated that the determinant must not 

 be zero. This condition excludes just these transformations called 

 pseudo-transformations. The equation of the transformation is written 



ax + b , 

 Xl ex + d ' 



if the determinant ad — be = o, then d = ^- Substituting this value 

 of d in the equation, we have 



a(ax+b) _a. //?\ 



A1 c(ax + b) — c' V U / 



