338 



WILSON AND MOORE. 



The expansion of the products gives expressions like 



(mxn) • (mxnxp) = 



m 



n 



m'' m-n in«p 



in*n n- 



ii'p 



such an expression represents the component of p perpendicular to 

 m^n multiphed by the square of mxn. The product 



[(inxn)'(mxnxp)]«[(mxn)'(mxnxr)] = (mxn)-(mxnxp).(mxnxr). 



Hence 



If the surface is a ruled surface the form 



y = f(u) + n{u) 



is a possible parametric form. Then 



m = f + vg', n = g, q = g', r 

 Ga = — (mxnxq)2 = — (f'xgxg')2. 



0, 



Hence: The total curvature of any ruled surface with real rulings is 

 negative. If the surface is developable, i. e., if G = 0, we have 

 I'y^gy^g' = or g' = hi' + eg, where h and c are functions of u alone. 

 Then, 



_ [ f ^ + v{hi' + eg)] X g = (1 + hv)r X g ^ f X g 



(i + M|f'xgI If xgl 



is a function of the single variable u and remains constant as v changes, 

 the tangent plane is tangent along the whole generator, and the 

 surface is the tangent surface of a twisted curve. Hence: All 

 developable ruled surfaces are twisted curve surfaces. 



If the ruled surface is not developable we select as a simple canoni- 

 cal form that obtained by taking the directrix y = f (u) orthogonal 

 to the rulings and u as the arc along this curve. Then, 



