NON-EUCLIDEAN GEOMETRY 27 



at a finite angle is represented in the r-plane by a curve cutting 

 the fixed circle orthogonally, and any curve in the r-plane 

 which cuts the fixed circle at a finite angle other than a right 

 angle corresponds in the r'-plane to a curve touching the fixed 

 circle. 



Let the equation of a curve in the r'-plane be f(r', 0')= 0. 



Then 



But 



Therefore 



dr' dr'ldd'' 



d_df dr = (f*-t) df , , dl_dj 



dr'~dr ' dr' 2p(r*+ k) ' dr' ' dd'~d6 ' 



dd 



dr'~ 2p(r z +k)'dr' 

 de r dd 



Hence when r z +k=0, r -,= oo unless -j-=Q, which proves the 



dr dr 



results. 



23. This transformation receives its simplest expression 

 through the medium of the sphere. 



Let a point Q be projected stereographically into P and 

 centrally upon the same plane or a parallel plane into P' 

 (Fig. 2). 



FIG. 2 



Then 

 and r=OP= 



