155 



The asymptotic lines in tliis case are then given by the equations, 

 u = It 



where ^t(t) is given in (42) & u = p '((Sd^— td^) (4 — 4td))v— p '(dV4) 



LoxoDROMio Lines on S^ 



Tlie general equations for the loxodromic lines on a surface 

 ((E) V 2 (G)^ 2 ) ^ (ju — _(_ tan ex dv becomes in the case of Si ((u = (d2— 2Cd) 

 + CM = )'/2), (u-(d2 — u^)^ -) )du= + tan a-- dv which by the substitu- 

 tion u2(d2 — 20d) + C^d- = (C^d^t-) (t^ — 1) reduces to the form, 

 ((20 — d)(tMt)) ((t-' — l)((d- — 2Cd)t2— (d— 0)2)1, 2 trz ^ tanoc dv. 



(Tlie above equation is number 43.) This may be put in the form, 

 ((20 — d)/(k2)\ 2). ((t^Ot) ((k't^) — (kM- l)t' + 1)\'' = + tanoc- dv. 



(The above is equation 44.) 



