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 /-plane to a curve touching the fixed 

 circle. 



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



Then 



But 



Therefore 



dr' dr'jdd'' 



dj dj cLr \r K] CI/T j u/r CLJ 

 dr'~dr' dr' 2p(r 2 +k)'dr' ' d6' dB' 



dO' _ (r 2 k) 2 d6 

 dr 7 ~~2p(r 2 +k)'dr' 



Hence when r 2 +A;=0, ^-,=< unless ^-=0, which proves the 



dr o/r 



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 7 

 (Fig. 2). 



s 



Then 

 and r=OP= 



