137 



Deteeminatiox of all Surfaces for Which, When^ Liistes 

 OF Curvature are Parameter Lines (ii^const., v= 

 const.), THE Six Fundamental Quantities, E, F, G, L, 

 M, N, are Functions of One Variable Only. 



Wm, H. Bates. 



The following simplifications come out of the data: 



(1) F = = M (Since lines of curvature are parameter lines). 



(2) The V — derivatives of E, G, L, N vanish. (Since the latter are 

 functions of u only. ) 



(3) We may substitute for u a function defined by the equation, 



Edu2 = du'^ 

 which makes E, G, L, and N functions of u' only. Also the system of 

 parameter curves is (as a whole) not changed, for when u = const, u^ = 

 const also. Now if we drop the prime from u', the substitution has exactly 

 the effect of making E = 1. 



Let (Xi, Yi, Zi) and (X2, Y2, Z2) be direction cosines of tangents to 

 the v-curve and u-curve at any point of the surface. These tangents, 

 together with the normal to the surface (direction cosines of which are X, 

 Y, and Z) at the point form a rectangular system of axes. 



r^X rfy dz 



(1) Xi = — ; Yi = — ; Zi = — (since E = 1). 



(iu f'u (5u 



1 '^x 1 f5y 1 H 



(2) X2 = -— -— ; Y2 = ^— ; Z2 = -— — 



i/Gdv ^/Gf5v |/G rfv 



Then the differential equations for the general surface (see Bianchi, 

 1902 Edition, p. 123) become after introducing the above simplifications, 



r^Xi 



(3) = LX 



<}u 



f^Xi d^/G 



(4) _ = X2 



•H- du 



(5) 



<*X2 



du 



