fit / 



Now, since ^= i/R(t) we get from (2) 



ft A^ ] 



Noting that in tlie present case 



p'^ Y=— 62(^(^2 _,S2) 



and remembering that 



(P'V)2 



p(u + v)— p)ii— V) — 2 p V 



pil — pv)2 P Tl — pv 



(4) becomes 



v = ~J[-rS2 + p(ia + v)_p(u-v)-2pv]du + rf' 



The functions -^ may be expressed in power series. We have then the 

 geodesic lines given by the equations 



V = f(t)+rK 

 1 



Tlie constant <y being additive has no effect upon the nature of the 

 geodesies. It determines their position. All lines given by rf' may be 

 made to coincide by a revolution about the z-axis. The curA'es may be 

 completely discussed when fr=ro. 



Since the parameter lines of the surface consist of geodesic lines 

 tlu-ough a point and tlieir orthogonal trajectories E may be taken equal to 

 unity.* Edu2 = du'2 



„ d 1 rd + v/d= — uO , T , 2u^ 



Hence — — log t I ^::=u , or u=:d sech — 



2^1 u J • d 



Because of the relations of tlie surface to the pseudo-sphere it may be 

 represented upon the upper part of the Cartesian planet. The relation be- 

 tween the surfaces is given by the equations 



v=v' 



c / 

 u ^ -^ u 

 d 



■■' Knoblauch, Krummen FlUchen, p. 49. 

 t Bianohi, Differential Geometrie, p. 419. 



