150 



AVlien C 6.2 we have from (23) Kj positive, negative or zero according 



as (11= — cd) = 0. But O <d 2 gives Od d= 2, so that u^lCd dV2 is 



the condition for the positive curvature. The curvature is zero or negative 

 vphen u^ = cd d^ 2 (u^d^ — 20d) -t- Od^ ^ giving the imaginary values 



for u). This shovrs that the tangent line to the tractrix which gives the 

 parabolic circle has a different slope than in the case where d =: 2C, since 

 in this case u/d<(2)» 2/2 , i. e. sin ; (2)' ^ 2. 



When d ' 20 we might consider three cases viz., d 20 , C = d 

 or 0>d. It will only be noted here that when O =: d the surface S' is the 

 same as the surface S and K^ is therefore — 1. 



In any case u^' — Od =z gives the valves of u for which the tangent 

 line to the curve O is parallel to the u — axis. 



Geodesic Lines on S^ 



Using the method of the calculus of variations as outlined in Ohapter I 

 we get for the geodesic lines on the syntractrix of revolution, 



Fv'=: (u2dv)/{Ejdu2 + Gidv-)' 2 — r 

 Here two cases may be considered according as 



r = or r dr 



( 1 ) When r = , then either u = or dv = . But u ziz , hence dv = 

 and therefore v := constant. Tliat is the meridians are geodesic lines, 



(2) When r ± we have 



dv = ((r/u2)(u2(d2 — 20d) + O^d^)' = ((d^ — u^jCr^ —u^)\ 2)dv 

 (The above equation is number 24.) 

 To reduce this expression on tlie riglit liand side to a convenient form sub- 

 stitute, 



u2(d2 — 2Cd) + C^d^ = (C^d^t^) (t= — 1) (25) 



Tliis may be written u^k -|- k, = (k'f^) (t- — 1) for convenience tlieu, 

 dv= (— k3 2r t^ dt)/((kr2 -f k,)\ = . (kd^ + kJ' - ((at-— 1) . 



(bt= — 1))' 2) (26) 



Where a.= (kd2)/(kd2 + k,) and b = (kr^) (kr= + k,) 



When r =b we may consider two cases 

 Wlien r = d and r :b d 



When r = d equation (26) becomes, 



dv= (— k-' M,dt) (kd- 4-k,(at- — 1)) (27) 



