58 



KANSAS UNIVERSITY QUARTERLY. 



22. Type IV a Special Case of Type II. A transformation T' 

 of type II is characterized by a hyperbolic one-dimensional trans- 

 formation along its invariant line AB and a parabolic one-dimen- 

 sional transformation along its invariant line Al. If a the charac- 

 teristic constant of the parabolic transformation be equal to zero, 

 the parabolic transformation along AI is the identical transformation 

 and every point on Al is an invariant point. The invariant figure 

 is now the point B, all lines through B, and all points on 1. 



The iransforniation T' degenerates into S ivlien the eharacteristic 

 eonstajit u is zero. 



In T' it was proved that the characteristic cross-ratios along AB 

 and through A were equal. Hence it follows in the degenerate 

 form S that the characteristic cross-ratio along any invariant line 

 is equal to that in any invariant pencil. 



23. Implicit Normal Form. The cross-ratio of the pencil 



* A(01PP,), Fig. 8, is k. 



whence we have 



sinPAO sinP,AO 



written 



\K 



PI 



sinPAl ■ sinP^Al 



= p- : ^S^ where Po and 

 Fl P] i ] 



P^o are the perpendiculars 

 from P and Pj on the line 

 OA, and PI and P.l^ are 

 the perpendiculars from P 

 and P, on the line 1. In 

 another form this may be 



(19) 



PjO Po 



In like manner the cross-ratio of the pencil O(Al'PPj), where 

 or is parallel to Al is unity; for OP coincides with OPj. This is 



given by 



sinPOA sinP.OA 



Po PjO 



PI P,l, 



(20) 



sinAOr sinPjOr 

 where Po and P,o are the perpendiculars from P and Pj on 

 AO, and PI and P^l. are the perpendiculars from P and P, 

 on 01. Let the coordinates of P, P^, O and A be respectively 

 (x,y) (x,y,) (A^,BJ and (A,B) and let p be the tangent of 

 the angle which the line Al makes with the axis of x. The 

 perpendiculars in the above expression may now be replaced by 

 their analytic expressions in terms of the coordinates giving us 



