153 



In particular when d = 20 and r = d (29) becomes 



V = 4- (d/2a) + (1/4) log (d— u)/(d+u) + '^ 

 For the purpose of illustration let d = 1 then (35) becomes 



V = = (d/2u) ^ (1/4) log (l-u)/(l+Ti) — <^ 



And since <' is an added constant we may without loss of generality let 

 '5 + 0. 



This particular geodesic line has been drawn in figure 3. It is to be 

 noted that the line winds around the surface as it approaches smaller 

 values, and then again winds around approaching the circle u = 1. The 

 lines r = d = 1 are all similar to this one and may be obtained by giving 

 different values to '^ . 



When d -20 , k = (d- — 2Cd) is positive and ab is positive and since 

 k, = C^d^ is always positive and we have K always real so that the geo- 

 desic lines on the surface Si defined by d 20 are all real. 



When d < 20 , k = (d - — 2Cd) is negative and ab is positive or nega- 

 tive according as r^ . kj/k or ^ C-d^)/{d.- — 20d | So that on the 

 surface S j defined by d< 20 , K will be real or imaginary according as 

 r^^lkj/k) . Hence the geodesic lines on such surfaces become imaginary 

 lines when t^> | k,/k | , that is when r> | k^k | V2andr< - | k,/k | -/^ 



Asymptotic Lines on Sj. 



From the general equation of the asymptotic lines on a surface we get 

 for the asymptotic lines on S,, 



(u-(d= — 2Cd) + 0d3)VV(u((0d — u=)(d= — u=) 7=)) du = + dv 



(The above equation is number 37). 



The substitution of \i-{d^ + 20d) — Od^ =l/t2 reduces (37) to the form, 



(— kdt) ((1— k,t2)((at2 — i)(bt2 — l))i 2 1_ dv. 



Where k = d^ — 2Cd, k. = Od^ a =r Odk -f- k„ b = d=k + k,. 

 In the particular case when d= 20 (37) becomes 



((d = ) (u((d- — 2u-)(d2 — u2))i 2)) . du = + dv 

 Which when u ^ 1 t reduces to 



(— d^t . dt), ((d^t^ — 2)(d=t= — 1))' 2-^4^ dv (39) 



Here R(t) = dH* — 3d2t2 -f 2 



R'(t) = 4dn3 — ed^t 

 R"(t) = 12dn- — 6d = 

 g, = (ll 4)d^ 

 g,- (9 8)d'' 



