249 



d. In the most general form with elliptic sections: 

 Let p ^ 1, q = cc, c =: 1. 



Then x=* -f (1 — z)- j^ = {I — z)2, the equation of Wallis's Coneo- 

 Ctmeus, or the ship carpenter's wedge. 



e. Assume case a. Tlie central section at z = o is a circle. Deform 

 the surface by rotating one directrix about the Z axis any angle less than 

 7r/2. The section z = o will now be an ellipse referred to its equi conju- 

 gate diameters. The form of the equation of this section will not change; 

 also the form of the equation of the deformed surface will be invariant. 



Order of the Resulting Warped Surfaces. 



Let fn (x y) represent a homogenous algebraic expression involving x 

 and y and of the nth degree. 



In the fundamental demonstration, 



1. Let f (x' yO = f , (X y) — c = o. 



If X and y are both present, the corresponding warped surface is of 



the 2d order. 

 If X or y is absent, the resulting surface is a plane. 



2. Let f (x' yO = f2 (x y) + fi (x y ) — c = o. 

 x^ and y* both present, 4th order. 



x^ or y2 absent, other terms present, 3d order. 



x^ and y* both absent, xy present, x and y present or one or both 

 absent, 2d order. 



3. Let f (x' yO = fs (x y) + f2 (x y) + fi (x y) — c = o. 

 x^ and y3 both present, 6th order. 



x3 or y'^ absent, other terms present, 5tji order. 



x^ and terms involving x^ absent ; or, y'' and terms involving y^ ab- 

 sent, 4tli order, 



x^ and y^ both absent, other terms present, 4th order. 



x^, y'\ and xy^ and terms involving y^ absent, other terms present ; 

 or, x', y3, and x^y and terms involving x^ absent, other terms 

 present, 3d order. 



To deduce the general law of order of the resulting scrolls, construct 

 Fig. 2. Within the squares are present all the powers and combinations 

 that can occur in a complete equation in x, y, of the 5th degree. The 

 numbers at the intersections of the lines show the order of the resulting 

 scroll provided at least two terms remain in our original f (x', y') =o, one 



