148 



Asymptotic Lines on S. 



The asymptotic lines ou the surface are defined by the equation 



D du2 + .2D'du.dv + D"dv2 =r (See Note 1) (12) 



This becomes for the surface S, 



C -h (O^ — e2" c) 1 2 = e" ce(+v + j3) (13) 



and by (8) becomes in the x — y plane 



y = — (y'-l)' ' + e(+x + J) (14) 



LoxoDROMic Lines on S. 



The differential equation of the loxodromic lines of a surface are 

 given *by 



((E) 1 2(G) ^ 2) (du/dv) = tan a: (15) 



Where x is the constant angle which the curves make with the curves 

 V := constant. For S (15) becomes, 



(Cdu u^) = 4- tan ac .dv. 



Hence tan x.uv + kjU-j-O = 



This by the relation u = e"' c becomes, 



tan X .e" c v + k jC" c -(- C = (17) 



which by (8) gives, 



y =: — tan x.x — k, (18) 



This is a system of straight lines parallel to the line 



y = — tan x .x 

 and so a system of lines making a constant angle with the lines x ^= con- 

 stant. And this is as it should be since the geodesic lines v = constant go 

 over into the lines x = constant by the same transformation. 



By selecting lines from different systems of loxodromic lines we may 

 envelop any geodesic except the meridians. This may be seen by changing 

 (17) to tlie form, 



x sin X -j- y cos x -|- k j cos x ^^ 

 Where if k i and cos x cliange so that k , cos x = constant we get a sys- 

 tem of lines enveloping a circle with the centers at the origin. This cor- 

 responds to the loxodromic lines on the surface enveloping a geodesic. 



*Bianchi, p. 109. 



