156 



Where k, — (d— C)^ and k^ = (d^ — 20d) (d— 0) = 



Here, 



R(t == k^t^ — (k^ + l)t2 + 1 



R'(t) = 4k2t3 — 2(k2 + l)t 



R"(t) ^ 12k2t2 — 2(k2 4- 1) 

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

 g,= ((1 + k = )/(216))(l-34k2 + k*) 

 (44) may be reduced by the substitution, 



t ^ 1 + (|k2— l),2)(pu — pv) (46) 



Where pv = ( 1 ' 12 ) 1 5k ^—\ ) 



Then k^t^ = k^ + (k^k^ — In (pu — pv) + ((k^ 4)(k = — 1) 2); (pu — pvi ^ 

 and since dt/du ^ (R t))' - we get by using (31), (32) and (33) + ^an 

 oc- V— (20— d) ( (k,) V 2 (k2) ) ( (1/6) (k2 + l)u + (rf' rS) (u + v) + (S'/6) 



(u— V) ) + 6'' ~ (46) 



We have then the loxodromic lines on the surface Sj given in terms of t 

 by the equations, 



u2(d2 — 20d) + 0^2 = (02d2t2)/(t2 — 1) 



V = 0(t) + ^5'^ 

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

 ((5k2 — 1),(12)) 



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

 always real. 



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

 loxodromic lines reduces to, 



((d2 2) (u2(d2 — u2)' 2) du = + tan (X • dv ...... (47) 



and therefore 



(— id^ — u2)V 2/2u) = + tana- v + rf^' (47a) 



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

 x-y plane by the straight lines, 



y = + tan ax + 'V' .... (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 , 1 when d =^ 20 ) may be represented by the same set of straight lines in 

 the plane. 



Suppose d = 20 = 1 and (^^^ = and the tanoc = 1. Then 47a becomes 



( — (d2 — u2)i'2) (2u) = + V. 

 This gives a line on the surface from the point Uj, vj =; (1, 0) making an 

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

 as shown in figure IV. 



