137 



Determination of all Surfaces for Which, When Lines 

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

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

 M, N, ARE Functions of One Variable Only. 



Wm. H. Bates. 



The following simplificatioas 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. 



(^x Jy 6z 



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



'^u (iu 6u 



1 rSx 1 <^J 1 fSz 



(2) X2 = -— — ; Y2 = ^— ; Z2 = — - — 



yG'^y j/Gfiv i/Grfv 



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

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



^^Xi 



(3) — = -LX 



<^Xi di/G 



(4) — = X2 



<W du 



(6) 



'IXo 



'iu 



