149 



CHAPTER II. 



Lines on the Syntractrix of Revolution. 



Taking the equation of the syntractrix in the form, 



x=(d" — y-)' - + O cosh-Md/y) (19) 



the surface S is given by, 



X = u cos V 



y = u sin V }■ (20) 



z = — (d-' — U-) ' 2 -f O cosh-i (d u) 



) 



or we may transform the equation of the tractrix by 



y=(Cd)y, ■» 2n 



x = x, + ((d-(J) d)(d^-yf)^^ [ ^^^^ 



Giving as the relation between the surfaces S and Sj, 



u =: (c d)Ui 



V = Vi 



In this work O represents the lengtli of the tangents to tlie tractrix and 

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

 Hence d = constant . C 



We get for the fundamental qualities: 

 E, = (u= — Od)2 (u2(d2 — u2)) + 1, Fj = 0, G, =u- and 

 D, — (u-(d= — 20d) + Cd^) (u(d- — u^)^ 2) 



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



K, = ((u= — od)(u2(d2 — 20d)+ Od^j) ((d^ — u2)(u2(d2 — 2cd) + C-d^j 

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

 vature. ) 



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

 When C = d 2,Ki becomes (2u2 — d-) (d^ — u^) 



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

 numerator is positive or negative according as 



2u2 — d2 



That is, according as u >(d/(2)"2) and u<— (d)/((2) ' -; or — (d ((2) '/ -) 

 u r d/((2) ' -) . For u = 4 d, ((2) '^-) , K, = 0. This means that for 



tlie particular surface S i defined by d = 2C the Gaussian curvature is zero 

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

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

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

 the curves u = constant along which the surface have a negative curvature. 



