143 



In considering these curves two cases may arise, (1 ) when oc ^ 0, 

 (2) X • 0. Case (1) when at — 0, either u = or v^ = 0. But u ± 0. 

 hence v' = and so v = constant. That is the meridians are geodesies. 



Case (2 1 when x - 0, (3) becomes 



V = (O x u) (u^ — X 2(1/2 _!_,;; (4) 



This may, liowever, be put in a more convenient form, since in the present 

 case the geodesic lines v = constant all meet in a point and the curves 

 u = constant form a system of geodesic circles — the orthogonal trajec- 

 tories of the meridians. Under such conditions E may be equated to unity 



(see Note 1). The new ua is then given by the relation U2 =^ | (E)\/2du. 



Hence u = e"^ c. Replacing in (4) u by its value just found the equation 

 of the geodesic lines becomes 



v= (C X )(I — X 2e-2" c)i 2 -f fS (see Note 2) (5) 



This equation may be used to determine the allowable values of a 

 and 5. The constant ,3 being additive has no effect except to turn the sur- 

 face about the z axis. Thus a geodesic line given by one value of /? may 

 be made to coincide with one given by another value of 3 by revolution 

 about the z axis, x remaining constant. ,3 may vary from — X)' to + x . 



From (5) it is seen that the lines are real or imaginary according as 



Qc 2e-2"/c = 1, 

 <^ 



(1) Let x'2e-2» c>l, then ! x | >e"/ c. 



But for the pseudosphere U2<0 log C so that the geodesies will be imagi- 



< < 



nary when I x | O. (2 & 3). Let x ^e-su/c — 1, then | x | = e"/c. 



Hence | x 1 =: O gives real geodesies. 



Equations (5) may be transformed into 



X 2(v2 -(- C^e— 2" c) — 2 ,3 x'^v -(- ( ,?2.x ^ — C^) =0 which when 

 v2 zt O^e— 2"/C ^= y 



v = x (6) 



may be represented in the plane by the straight lines, 



y = 2;3x - (^92 — OVx 2) (7) 



(6) may be broken up into two transformations 



(a) V = X 



Ce-"/c 



..(8) 



Note 1.— Knoblauch, Theorie derKnimmen Flachen, p. 133. 

 Note 2.— Bianchi, p. 419. 



