246 



Let (x, y, z) be any point P on the warped surface, and E' E" W^^ 

 the rectilinear element containing it. 



Let O M = x' , O N = X , O R = q, O Q = p. 



Then by similarities and projections the following equations exist : 



x_' _ E^^^E^i _ W"2 E^2 _ p , _ px 



X ~ E'^'i Pi ~ W2 P2 ~' p— z ' ^ ~ p— z 



q y 



Similarly, v = 



q— z 



Substituting these values ot x' y' in f (x' y') = o, there results the cor- 

 responding functional equation, 



ffJ^, '^] = o, 

 Ip — z q — zj 



whii-h is the equation in Cartesian co-ordinates X, Y axes general, Z 

 axis perpendicular to X and Y of the warped surfaces as defined above 

 and includes every warped surface with two distinct rectilinear directrices. 

 For its application it requires that a section of the surface should be 

 known parallel to the riglit-line directrices and not including either of 

 them. This general surface is referred directly to the orthogonal pro- 

 jections of two warped lines in space upon a plane parallel to both, 

 and to their common perpendicular. The angle at which the lines 

 intersect is inq)licitly contained in the equation of the surface. The 

 form of the equation of the surface does not change, therefore, when 

 the surface itself is deformed by changing the angle in space of the 

 right line directi-ices, provided the form of tlie equation of the plane 

 curve directrix remains unchanged. 



It is also at once evident that the method derives immediately the 

 Cartesian equation of the warped surface determined by the fact that 

 an element cuts a curved directrix, a linear directrix and is parallel to a 

 given plane. This is equivalent to saying that one of our parameters 

 p, q. remains tinite while the other becomes indetinitely great. 



For simplicity suppose the three axes always at right angles to each 

 other unless otherwise specified. 



The Hyperbolic Paraboloid. 



(a) Let f (x^ Y) — x' — y^ = O. 



Tiien f ~ — , -- — = — i^^— =: o. 



I p — z q — z I p — z q — z 



Let p = 1, q = — 1 



Then x -|- xz — y -}- yz = o. 



