143 



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

 (2) X. it 0. Case (1) when oc = 0, either u = or v' = 0. But u± 

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



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



V = (O au) (u= — oc -(\ 2+ .:; (4) 



This may, however, 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 — tlie orthogonal trajec- 

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



(see Note 1). The new us is then given by the relation ua = I (E)\/Mu. 



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

 of the geodesic lines becomes 



v= (O, X )(1 — oc 2e_2u c)i 2-^/3 (see Note 2) (6) 



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

 and ;>. The constant /? 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 witli one given by another value of :3 by revolution 

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



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



X 2e-2u,'c = 1, 



(1) Let x'2e-2" c >I, then 1 x | >e» <-. 

 But for the pseudosphere u, C log C so that the geodesies will be imagi- 

 nary when I X I ■ O. (2 & 3). Let x -e— =" c — l, then | x | = e"/c. 



Hence i x 1 == O gives real geodesies. 



Equations (5) may be transformed into 



X 2(v2 -|- O^e-^" c) — 2 ,3 x'-v -f- ( ,^-.x - — O^) = which when 

 v^ ± 02e-2"/c ;= y 



v = x (6) 



may be represented in the plane by the straight lines, 



y = 2,/3x — (.^2 — 0^ X 2) (7) 



(6) may be broken up into two transformations 



(a) V := X 



Oe-", c 



(8) 



Note 1.— Knoblauch, Theorie der Krummen Plachen. p. 133. 

 Note 2.— Bianehi, p. 419. 



