52 



KANSAS UNIVERSITY QUARTERLY, 



and y ^ = 



A B I I j 



'A, B, I k I 



I p O a 1 



In honiot^eneous form this result luav be written 



COSi// 



(12) 



px 



X V z o 

 A B C A 



A, B, C, kA, 



I p O Aa-)-I 



X 3' z o 

 _ A B C B 

 ^^'i— A, B, C, kB, 



1 1 p O Ba4 p 



PZ 



X y z o 

 A B C C 

 A, B, C, kC, 



1 p O Ca-j-O 



03) 



In the final form we have the projective transformation of the 

 second type, 



/3X,=ax-f by-(-cz, p}' , =a jX-j-b^y-f c,z, pz j^aoX-j-bgy-fCo 

 expressed in terms of its essential parameters. 



>?.l Type III; Its Properties and Normal Form. 



13. The Most Difficult Case. We coine now to the considera- 

 tion of projective transformations of type III. The geometric 

 properties of transformations of this type are more difficult of 

 determination than those of an}' other type and in consequence a 

 different method is employed in the discussion of this type. A 

 more direct method is desirable but is not at hand. 



14. The Relation a'^ /x a. The invariant figure of T" is a 

 lineal element Al; /. t\ an invariant point A and an invariant line 

 1 through the point A. It is evident that T" produces a one- 

 dimensional parabolic transformation of the points on 1 and also a 

 parabolic transformation of the pencil of lines through A. We 

 must first determine the relation of these two one-dimensional 

 transformations. 



Type III may be considered as a special case of type II in which 

 the invariant point B is brought to coincide with A and at the same 

 time the invariant line AB is brought to coincide with 1. We 



proved in Art. 9 that u'= ,^a, where </> is the angle BAl, d 



the distance AB, u and a the characteristic constants of the trans- 



