338 



WILSON AND MOORE. 



The expansion of the products gives expressions like 



(mxn) • (m^nxp) = 



m n p 



m- m-n iii'p 

 in«n n- n«p 



such an expression represents the component of p perpendicular to 

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



[(mxn)«(mxnxp)].[(mxn)«(mxnxr)] = (mxn)-(inxnxp).(mxnxr). 



Hence 



<i>s = (mxnxp) • (mxnxr) — (mxnxq)^ = G a, 



Ga = 



in«n n- nT 

 m-p ii'p pT 



m- m • n m • q 

 ni'ii n^ n-q 

 m • q n • q q^ 



(106) 

 (106') 



If the surface is a ruled surface the form 



y = i(u) + vg(u) 

 is a possible parametric form. Then 



m = f + rg', n = g, q = g', r = 0, 



Ga = — (inxnxq)2 = — {fxgxg'y. 



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 

 f'>^g>^g' = or g' = bi' + eg, where b and c are functions of u alone. 

 Then, 



jyj^ [ f' + v{br + eg)] X g = (1 + bv)i' X g ^ f'Xg 



(i + Hlf'xg| If xg| 



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 nded 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 (m) orthogonal 

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



