74 



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



dii 



(^) ^ = -¥/^^^" + '^' = 2;i 





Noting that in the present case 



and remembering tliat 



-Ai^-i— = p(u + v) -p)u -V) — 2 p V ^- 



(4) becomes 



^ = 2aS^ — '^' + P(ii + V) — p(u — V) — 2pv]du + '5' 



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

 geodesic lines given by the equations 



v = f(t)+r5' 

 1 



The constant iV being additive lias 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. Tlie curves may be 

 completely discussed when iV^^o. 



Since the parameter lines of the surface consist of geodesic lines 

 tlirough a point and tlieir ortliogoual trajectories E may be taken equal to 

 unity.* E du^ ^ du'^ 



Hence - A log ['^ + /d-^-uO ^^ ^^, ^^. ^^^^^ ^^^.j^ 2u: 

 2 ^ I, u J ' d 



Because of tlie relations of the 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 Fliichen, p. 49. 

 t Bianchi, Differential Geometrie, p. 419, 



