141 



Lines on the Pset^dosphere and the Syntractrix of Rev- 

 olution. 



E. L. Hancock. 



INTRODUCTION. 

 Consider two surfaces of revolution S and Si generated by the revolu- 

 tion of the curves C and Ci about the Z axis. Oi is formed by taking on 

 the tangents to O distances equal to the constant-k' times the length of the 

 tangents. The length in each case is measured from the z-intercept toward 

 the point of tangency. Let O = O be given by z — f(u), then Oi — O will 

 be given by, 



zi = (L — l)uif'(Lui) + f(Lui) 

 where L ^ 1 k' and the equations of transformation from S to Si are, 



u = Lu] 



(1) 



V = Vl 



When the length of the tangent to the curve C is constant, as in the 

 tractrix, the curve Ci is the syntractrix (see Note), and the surfaces S and Si 

 are therefore the pseudosphere and the syntractrix of revolution. 



What follows is the study of lines on these surfaces. The geodesic 

 lines on the pseudosphere have been studied by means of lines in the plane. 

 This surface being one of constant negative curvature ( — 1) may, accord- 

 ing to Beltrami (see Note 2), be represented geodesically by a system of 

 straight lines in the plane. 



Much of the work outlined here for geodesies] on the pseudosphere may be 

 found in Darboux, Theorie des Surfaces, Vol. Ill, and is given liere only in 

 the way of review and for completeness. 



The claim made for the originality in this part of the work is in (1) 

 the classification of the geodesic lines and the study of certain systems of 

 geodesic lines and their corresponding lines in the plane; (2) the transforma- 

 tions of the system of circles into straight lines by making use of the sphere. 



Note 1.— The syntractrix is defined as the curve generated by taliingr a constant dis- 

 tance on the tangents to the tractrix. Peacoclc, p. 175. 



Note 2.— Beltrami, Annali di Matematica. Vol. 7, p. 185 



Bianchi, Lulsat, Differential— Geometrie, p. 436. 



