NEWSON I ONE-DIMENSIONAL TRANSFORMATIONS. 119 



which shows that every point on the line is transformed into the fixed 

 point — • The inverse of the transformation T is written 



. _ -dx, +b 

 ex, — a 



The determinant of this is also ad — be, which equated to zero also 

 gives d = — • Substituting this value of d in the last equation, we 

 have 



_Mcx-a_) = _.b. (7) 



a(cx — a) a x ' 



which shows that every point on the line is transformed by (7) into 

 the fixed point — — • 



The invariant points of a pseudo-transformation are also given by 

 equation (5). Patting d = ^ in this equation, it breaks up into 



(*-f) (* + t)=°; (8) 



thus showing that — and — ^ are the invariant points of the pseudo- 

 transformation. 



Theorem 3. A pseudo-transformation transforms every point on 

 the line into one or the other of its invariant points. 



Three conditions determine a projective transformation. — The 

 equation of a projective transformation T contains three independent 

 constants, viz., a : b : c : d. We infer, therefore, that three conditions 

 determine such a transformation. In particular, three points and 

 their corresponding points determine uniquely and completely a pro- 

 jective transformation. 



Let x', x", x'" be any three points on a line, and xi', xi", xi'" their 

 corresponding points, respectively. Substituting successively in 

 (1) the coordinates of each pair of corresponding points, we have 

 three equations, viz. : 



cx'x'i + dx'i — ax' — b = o, 

 cx"x"i + dx"i — ax"— b = o, (9) 



cx'"x'"i + dx'"i — ax'"— b = o. 



These equations are linear and homogeneous in a, b, c, d, and de- 

 termine the ratios of these quantities uniquely and completely, pro- 

 vided no two of these equations are identical or have their coefficients 

 proportional. 



Theorem 4. There is one and only one projective transformation 

 that transforms three given points on a line into three other given 

 points. 



