6a 



KANSAS UNIVERSITY QUARTERLY. 



X, yi I 

 A B I 



1/I-I-Pl^_l/I+Pl' 



X, yi I 



A B I 



X y I 

 A B I 

 I p o 



+ a. 



l/I+p2 l/l+p2 l/l+P' l/l+P* 



These forms reduce to 



A B I 



o 



Xi yi I 



A B I 



I p o 



X, y 

 I p 

 I p 



X, y, I 



A B I 



I p o 



X y 

 I P 

 I Pi 



X y I 

 A B I 



IPC 



But p, is purely arbitrary and may be so chosen that p, 



(27 r 



-p=r=i. 



29. Explicit Normal Form of Type V. The above implicit 

 normal form of a transformation S' of type V may be solved for x, 

 and yj giving us the explicit normal form for S'. Thus 



and y^: 



r p 

 I Pi 



(.28). 



Here a is . of the implicit normal form. 



cos^ 



In homogeneous coordinates the last equations become 



px,= 



X y z o 

 A B C A 

 I p o I 



^y 



p, O Aa + I| 



jx y z o 



:a B C B 



I p o p 



il p, O Ba + p, 



pz, 



X y z o 



A B C C 



I p o o 



I P] O Ca 



(29) 



In the last form we have the projective transformation of type V 

 expressed in terms of the essential parameters of the transformation. 



