NEWSON: FKOJECTIVE TRANSFORMATIONS. 6l 



t^'pe II, when a^iro in type II. We shall now show that type V is 

 also a special case of type II. In type II when k, the character- 

 istic cross-ratio of the hyperbolic transformation along AB and 

 through A, is unity, these two one-dimensional transformations are 

 both identical transformations and hence every ray through A is an 

 invariant ray; therefore for k=i, the transformation T'(ka) of type 

 II degenerates into S' of type V. 



27. Type V also a Special Case of Type III. A transformation 

 of type V ma}^ be also regarded as a special case of type III. In 

 type III the characteristic constants of the parabolic transforma- 

 tions along 1 and through A satisfy the relation a'^aa. When a 

 is zero a is always zero and the transformation along 1 is identical 

 having all points in 1 invariant. But a is the radius of curvature 

 of the invariant conic K touching 1 at A. The equation of the 

 invariant conic K is 



ax^ — 2hxy + by2=2y. 



When ar=o this breaks up into two lines 



y=o and by — 2hx=2, 

 which are invariant lines. The line by — 2hx — 2^0 is a second in- 

 variant line through A and hence all lines through A are invariant 

 lines. Our transformation of type III thus degenerates into one of 

 t\'pe V. 



28. Invariant Relations and Imollcit Normal Form. Let P and 



P, be any two corresponding poiuts in an elation S' and let PI and 

 Pjl, be perpendiculars let fall from P and P^ on the line 1 of 

 invariant points; let Pd and P,d, be perpendiculars let fall from 

 the same points on any other line through O as OD; evidently from 

 the figure we have the relation 



P,d, Pd 



P,I, PI • 

 We also have 



III I usin^ 



P,l, PI OP^sin^ OPsin^ sin< 



(25) 



(26) 



where 6 is the angle which the line OPPj makes with 1. Let the 

 coordinates of P, P,, and O be (x,y) (Xj,yi) and (A,B) respectively. 

 Let p and Pj be the tangents of the angles which the lines Ol and 

 OD make with the axis of x. 



We can now replace the perpendiculars in the above invariant 

 relations by their expressions in terms of the coordinates of the 

 invariant points and lines. Thus we get 



