156 



Where k, = (d— G)- and k- =: (d- — 2Cd) (d— 0) = 



Here, 



R(t r= k^t^ — (k^ + l)t- + 1 



R^(t) = 4k2t3 — 2(k2 + l)t 



R'"(t) = 12k-'t- — 2(k2 + 1) 



g, = (1;12)(1 + 14k2 + k^) 



g,= (d + k'2j/(216))(l-34k= + k') 

 (44) may be reduced by the substitution, 



t = l + (ik-— 1) 2)(pu — pv) (46) 



Where pv = ( 1 12 )i 5k -—1 ) 



Then k-'t-' := k^ + (k-,k- — li) (pu — pv) + ((k^ 4)(k- — 1) -) (pu — pv: - 

 and since. dt/du = (R,t))V ^ we get by using (31), (32) and (83) + tan 

 X • V — (20— d) ( (k,) ' 2 (k=) ) ( (1/6) (k^ + l)u + ('5^ '5) (u + v) + ((5^ 'J) 



(U— V) ) + <y' (46) 



We have then the loxodromic lines on tlie surface S j given in terms of t 

 by the equations, 



uMd- — 2Cd) + O^d^ = (CM2t2)/(t = — 1) 



V = ^(t) + (V 

 where ?>(t) is given in (46) andu = p- '( (2 (k- — 1) (t — 1) ) + pv) v = p— ' 

 ((5k2-l)/(12)) 



Since kj = (d — c)^ is always positive it is to be noted that ^(t) is 

 always real. 



In particular when d =; 20 the equation, the general equation for the 

 loxodromic lines reduces to, 



((d^ 2) (u2(cl2 — u^)' 2) du = + tanoc- dv (47) 



and therefore 



(— ^d^ — u^)' V2u) = + tanoc- V + iV (47a) 



and these by the substitution ((d- — u-) ' - 2u) ^ y, v ^ x are given in the 

 x-y plane by the straight lines, 



y = 4- tan ax + y' .... (48) 



But this is the system of lines into which the loxodromic lines of the 

 pseudosphere may be transformed. Hence the loxodromic lines on S and 

 S, (when d =^ 20) may be represented by the same set of straight lines in 

 the plane. 



Suppose d = 20 = 1 and <^^' = and the tan cr. = I. Tlien 47ft becomes 



( — (d^ _u-)"'-) (2u) = + V. 

 This gives a line on the surface from the point u,, v,) = (1, 0) making an 

 angle of 46° with the lines v = constant. The line winds about the surface 

 as shown in figure IV. 



