nevvson: projectivk transformaiions. 



65 



7. Obtain the explicit normal form of type IV by solving the 

 equations of Art. zi, for x, and yj and show that the determinant 

 of the explicit form is 



/x^k' 



A B I 3 

 A, B, I . 



=^ p in t3pe I. 



r p o 



S. Obtain the explicit normal form of type IV from that of t\'pe 

 I In' making k,=k and 



B,— B 

 A,— A 



9. Obtain the explicit normal form of type IV from that of 

 type II by making a--o in t}'pe II. 



10. Obtain the explicit normal form of type V from the implicit 

 form by solving equations (27) for x, and y^; show that l\, = i. for 

 type V. 



11. Show that the explicit normal form of type V results when 

 k -I in t3'pe II. 



12. Show that t)pe III degenerates into type V when a=o in 

 the explicit form. 



ADDENDA. 



1. The theor}^ of the normal forms of projective transformations 

 in three dimensional space is wholly analogous to the theory of 

 those in the plane set forth in this paper. The explicit normal 

 forms for the thirteen t\pes in space have been determined and 

 most of the forms generalized for n dimensions. These results 

 will be published later. 



2. In one-dimension there are but two types of the transforma- 

 ax-f b 



tion. X , = 



cx-f-d 

 X,— A 

 1 



, these are the well known forms: 



, u^— A, I I 



A -^^ x=A ' 



X, — A X — A X, — A X — A 



These explicit forms when solved for x^ give respectively 



Ix T O j 



A I A : 



1 1 o Att-!-i 



A I I 

 A, I k 



X I o 



lA I I 



I O a 



The analogy of these forms with the forms for the plane is evident. 



