336 



WILSON AND MOORE. 



CHAPTER III. SPECIAL DEVELOPMENTS IN SURFACE 



THEORY. 



40. The twisted curve surfaces and ruled surfaces. As an 



illustration and application of the foregoing analysis, we may treat the 

 case of the surface formed by the tangents to a twisted curve in n 

 dimensions. Let y = i{ii) be the equation of the curve, u being the 

 arc. The surface is then, 



y = f(u) + rf(w), r-r ^ 1, I'-i" = 0, 



dy = (f + vi")du + i'dv, 

 els'- = dydy = (f + vty-dir + 2du dv + dv"" 

 = (1 + vyB?)dii- + 2du dv + dv", 



where R = (f"*f")~^ is the radius of curvature of the curve. 

 That the surface is developable follows from the familiar argument, 

 namely: ds does not depend upon the torsion of the curve and hence 

 the surface is applicable upon the tangent surface to all curves for 

 which R is the same function of u, and a plane curve can be found 

 satisfying this condition. 



To calculate ^ two methods are available based on (75) and (76). 

 The advantage of the first form (75) is that the expression dyxd'M. 

 may be replaced by {dy>^dU'M)/U, where U is any scalar function; — - 

 since dy lies in M and f/yxM = 0. Now 



M 



(/yxr/M 



(f +. vr')xi' f"xf 



= R{r + vr'yr'xf'du, 



= Rvi"xf"'xrdu = - Mx^. 



r = Rt'xr, 



Multiply by M us in the text and repeat the argument there given. 

 Then 



- ^ = R-iifxt) • {t"xr"xf')du' 

 = Rh[i"'r"f" - i'.r'r" + f"-f"f'-f'"f']c/u2, 

 ^ = v[f"' + i'/R' - f"'f"'r']du-, 



smce 



