150 



When C d 2 we have from (23) K, positive, negative or zero according 



as (u- — cd) = 0. Bnt G <d 2 gives Cd d- 2, so that u=(Cd d=, 2 is 



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

 when u^ = cd d= 2 (u-id^ — 2Cd) -j- Od^ = giving the imaginary values 



foru). This shows that the tangent line to the tractrix which gives the 

 parabolic circle has a different slope than in the case where d = 20, since 

 in this case u;d<:(2) ' ^ 2 ^ i_ e. sin (2) ' - 2. 



When d < 20 we might consider three cases viz., O ; d 20 , O = 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 = gives the valves of u for which tlie 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'=z(u'dv) (Ejdu^ -f G,dv-)' - = r 

 Here two eases may be considered according as 



r = or r r^ 



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

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



(2) When r = we have 



dv=i ((r u2)(u2(d= — 2Cd) + 0=d = )' = ((d- — u^^r^ — u-)\ ')Av 



(The above equation is number 24.) 

 To reduce tliis expression on the riglit liand side to a convenient form sub- 

 stitute, 



u2(d2 — 2Cd) + O^d- = (C=d=t2) (t= — 1) (25) 



This may be written u^k + kj = (k^tv) (t- — 1) for convenience then, 

 dv= (— k^ -r t= dt) ((kr= + k J^ ■ . (kd- + kj' - ((at-— 1) . 



(bt= — 1))^ -') (26) 



Where a.= (kd^) (kd^ + kj) and b = (kr-) (kr- + ki) 



When r ± we may consider two cases 

 When r = d and r ± d 



When r ^ d equation (26) becomes, 



(iv= (— k^ ^djdt) (kd= + k,(at= — 1)) (27) 



