SrfeWSON: PROJECTIVE TRANSFORMATIONS. 



45 



Let p and p^ be a 

 pair of corresponding 

 lines in the transfor- 

 mation T; let PP,, 

 NNj, and MM^, be 

 the pairs of points of 

 intersection of p and 

 Pj with the sides of 

 the invariant triangle. 

 Since k,. = (ABMM,), 



kjjr=(BCPP,),, and ky = (CANN,) (observe the order in which the 



points are taken), we have 



AM. BM, BP. CP, 



Fig. - 



kyk^.ky: 



AM^. BM BPj. CP 

 But by the theorem of Menelaus we have 



,^BP, 

 BM 



CN AN^ 

 CNT^AN" 



AM. BP. CN , AM 



— I and 



BM. CP. AN 



CP. 



Hence 



,k. 



(I) 



T/ieoreiii i. — Every projcitive {raiisfori)iatio)i of txpc I /// tlw plane 

 determines a cfiaracteristie erois-ralio along eaeli of the invariant lines 

 and through each of the invariant points. When these three cross- 

 ratios are reckoned in the same order around the triangle their product 

 is unity. 



3. Elliptic Sub-type eT. In the elliptic sub-type of type I 

 the invariant triangle has one real and two conjugate imaginary ver- 

 tices, Fig. 3. Along the side BC and 

 in the pencil through the vertex A the 

 characteristic cross-ratio is necessaril}' 

 of the form e and this one-(iimen- 

 sional transformation is therefore 

 elliptic. But along the sides AB and 

 AC the one-dimensional transforma- 

 tions are not elliptic, /. (■. the charac- 



|l\ \ teristic cross-ratio is not of the form 



e . But the above theorem evidently 

 holds whether the triangle is all real or whether it is part real and 

 part conjugate imaginary: so that k^^kyk^^^i for all transforma- 

 tions of type I. 



4. Cross-Ratio of Corresponding Areas. Let (ABC) be 

 the invariant triangle, Fig. 4, of a transformation T and let P 

 and P, be any pair of corresponding points in the plane, P 



