155 



</> 



r,G. - 4 



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

 u = It 



v = i'(t) + <y 



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



LoxoDROMio Lines on S^ 



The general equations for the loxodromic lines on a surface 

 ((E)V2 (G)' 2), du= + tan oc dv becomes in the case of Sj ((u2(d=—2Cd) 

 + CM-)\''-) (u = (d2— u2)i 2) )^xiz=i + tan ex- dv which by the substitu- 

 tion u2(d2 — 20d) + 0-d= = (C^d^t^) (t^ — 1) reduces to the form, 

 ((20 — d)(t2dt)) ((t= — l)((d=— 2Cd)t-— (d— C)=)> = = + tan x • dv. 



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

 ((20 — d) (k,)i =)■ ((t=Ot) ((k^t*) — (k2 + l)t2 + D' - = + tanGc. dv. 



(Tiie above is equation 44. ) 



