120 KANSAS UNIVERSITY SCIENCE BULLETIN. 



The identical transformation. — Suppose that the transformation 

 (1) leaves three points of the line invariant. If we put x'i = x' 

 x"i = x" and x'"i = x'" in equations (9), these reduce to the following : 



ex" 2 -f (d — a)x' — b = o, 

 cx" 2 + (d — a)x"— b = o, (10) 



cx'"2+ (d — a)x'" — b = o. 



The determinant of these equations, 



= (x' — x'") (x" — x'") (x' — x"), (11) 



does not vanish so long as the three points are distinct ; consequently, 

 the coefficients of the above equations must vanish identically. 

 Thus, c = o, b = o, d = a. Putting these values in (1) we get xi=x, 

 which is the identical transformation. The identical transformation 

 we know transforms every point of the line into itself. 



Theorem 5. A projective transformation which leaves three points 

 of a line invariant is the identical transformation and leaves all points 

 of the line invariant. 



§2. Types and Normal Forms of Projective Transformations. 



Two types of projective transformations. — The invariant points of 

 a transformation T are given by the roots of the quadratic equation 

 (5). The roots of this equation are : 



{A,A') = a - d± V / (a+ 2 ^-4(ad-bc) | ^ g) 



These two roots are distinct or coincident, according as 



(a + d) 2 — (ad — be) 



is not or is equal to zero. Thus there are two distinct types of 

 transformation. The first type is characterized by the fact that it 

 has two invariant points, while the second type has only one. Every 

 transformation not identical belongs to one or the other of these types. 



Normal form of type I. — A transformation T of type I, whose in- 

 variant points are A and A', may be written in the form : 



It^T = k ^X' (13) 



where the constant k is expressible in the terms of the coefficients a, 

 b, c, d, as follows : 



i_ (a + d + y'(a + d)'-4(ad-bc)) 2 /-ijx 



K— 4(ad-bc) \ L *' 



In order to verify this, we must solve equation (13) for xi ; this 

 gives us 



(1 — k)x + (kA — A') 



v _ (A-kA')x-AA'(l-k) /-.--n 



A1 M — fcW -+- fWA — AM ' \ ±0 ) 



