NON-EUCLIDEAN GEOMETRY 29 



where <j> is the angle which the tangent at P to the curve 

 f(r, 0)=0 makes with the initial line. 



Draw the tangents P'T 19 P'T Z from P' to the circle (Fig. 3) 



P' 



FIG. 3 



and let LOP'T^OP'T^a. Also draw P'X' parallel to the 

 a>axis. Then 



~-k r . tan <!>' 



Therefore 



2i ^_sin (a-<ft')_sin X / P / T l si 



'sin (a+^psin X'P'T Z ' sin OP f T t 

 =P'(X'0, T^T 2 }. 

 Thus the true angle <f> is given by 



Hence the angle between two lines P'X', P'Y' through P' 

 is given by 



Jiog(OF', r^-i log (ox'.rjr^ log (XT, T^J. 



Next to determine the distance function ; let P, Q be- 

 come P', Q' (Fig. 4). The orthogonal circle PQXY becomes 

 a straight line P'Q'X'Y', and OPP', OQQ', etc., are collinear 

 since angles at O are unaltered. 



