NON-EUCLIDEAN GEOMETRY 29 



where <f> 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 lt P'T Z from P' to the circle (Fig. 3) 



Fio. 3 



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

 or-axis. Then 



, I~-V~ 



tana=v-7o r/ * 

 v r* 



Therefore 



_ _ 



sin (a+^')~sin X'P'T Z ' sin 

 =P'(X'0, TiTj. 

 Thus the true angle <j> is given by 



<^=| log (OX', TM. 



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

 is given by 



log (OY', TW-'log (OX', T.T^log (XT, TJTj. ' 



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., ar collinear 

 since angles at are unaltered. 



