149 



CHAPTER II. 



Lines on the Syntraotrix of Revolution. 



Taking the equation of the syntractrix in the form, 



x=:(d2 — y2)i 2 + C cosh-' (d/y) (19) 



the surface S is given by, 



X = u cos V ~| 



y = u sin v l (20) 



z = — (d^ — u^)' 2 + Ocosh-Md u) J 



or we may transform the equation of the tractrix by 



y=(C/d)y, \ on 



x = x, + ((d-C) d)(d=-yf)V/2 I ^^^^ 



Giving as the relation between the surfaces S and S ^ , 



u ^ (c d)Uj 



V = Vi 

 In this work O represents the length of the tangents to the tractrix and 

 d the constant distance taken on tliese tangents to get the syntractrix. 

 Hence d = constant . C 



We get for the fundamental qualities : 

 E, = (u2 — Gd)V(u='(d2 — u2)) + 1, F^ = 0, G, =u= and 

 D, — (u-(d2 — 20d) + Cd3) (u(d2 — u^)^ 2) 



D'j = ,D"i = (u(u2 — Cd)) (d2 — u2)i,2 (22) 



K, = ((u- — od)(u2(d2 — 20dj4 Cd^;) ((d- — u2)(u2(d= 2cd) + O-'d-) 

 (Above equation is number 23 and is the equation of the Gaussian cur- 

 vature. ) 



When C = d , (23) becomes —1 or the curvature of the pseudosphere. 

 When C = d 2,Ki becomes (2u^ — d^) (d- — u^) 



Since for the surface d =i u the denominator is always positive and the 

 numerator is positive or negative according as 



2u2 — d2 



That is, according as u >(d/(2)\ ^) and u . — (d)/((2) ' -') or — (d ((2)',-) 

 < u < d/((2)' '') . For u = + d ((2}'r^) , Kj = 0. This means tliat for 



the particular surface S ^ defined by d = 20 the Gaussian curvature is zero 

 for the circles u = constant, given by taking the distance d on tlie tangent 

 whose inclination to the z — axis is - 4 or (3-) 4. Tangents to the tractrix 

 whose inclination to the z axis is something between -4 and ySn-) 4 give 

 the cun^es u = constant along which the surface have a negative curvature. 



