142 



as indicated; (3) the study of the asymptotic lines and the loxodromic lines 

 on the pseudosphere and their representations in the plane. 



In the second part of the work the lines on the syntractrix of revolu- 

 tion are studied. This work so far as I know has never been done before 

 In it I have worked out the equations of the geodesic, asymptotic and loxo- 

 dromic lines. These have been studied in particular by classifying the 



surfaces Si according as d = 20, where C is the length of the tangent to 



the tractrix and d the constant distance taken on that tangent. When 



d = 20 it happens that the geodesic lines on Si are all real and that the 

 geodesic lines for d 20 are real or imaginary according as r^ ki k . 



The loxodromic lines are represented in the plane by the same system 

 of straight lines as the loxodromic lines of the pseudosphere. The draw- 

 ings are given for the sake of clearness. 



CHAPTER I. 

 Geodesic Lines on the Pseudosphere. 

 Taking the equation of the tractrix in the form, 



X = C cosh.-^c/y — (O^ — y^)^^ we get for the given surface, 



X =: u cos V (2) 



y := u sin v 



z -1^ O cosh.-ic u — (02 — u2)i/2 

 and the fundamental quantities of the Gaussian (see Note 1) notation are, 

 E = C2/ii2, F = 0. G = u2, D = (02) (u(02 - nH^ •'), D' = 0, 



D" = — U(02 — U2)l, 2^ K =r — 1. 



Using the method of calculus of variations as developed by Weier- 

 struss (see Note 2) to obtain the equations of the geodesic lines, we have 

 to minimize the integral, 



1= il (Edu2 4 2Fdudv + Gdv^) i ^dt 



. ' to 



— n ((0-u'2),;(u-'') + u-x'-y ■■'dt = Jfo^'^'dt 

 Legendre's condition for a minimum is Fv — (d /dv)Fv' = where 

 ( Fv) = ((^F)/(tW) and Fv' = (c5F)/(.V). 



Here Fv — 0, so that we get as the equations of the geodesies 



Fy' = (•a''v')/((C^n' w'') ^ n'-v' - }\ "" = <x (3) 



Where oc is the constant of integration. 



Note 1.— Bianchi, Differential-Geometrie, pp. 61 and 87. 

 Note 2.— Kneser, VariationsrecLniung-. 



Ossrood, Annals of Mathematics, V^ol. 2, p. 105. 



