462 



the equation of the cylinder, which is hyperboUc, being 



x^ tan^ - s2 _ _ jj.^ (26) 



The focal and dirigent are each a pair of right lines pa- 

 rallel to the axis of the cylinder ; the corresponding lines 

 passing through a focus and the adjacent directrix of any 

 section perpendicular to the axis. The directive planes are 

 parallel to the asymptotic planes of the cylinder. 



In this case, if k z: 0, the surface is reduced to two di- 

 rective planes, and the focal and dirigent to the intersection 

 of these planes. 



III. When TW zz 1, and 0=0, both a and b vanish, and 

 the surface is the parabolic cylinder. If, as is allowable, 

 we suppose g and k to vanish, the equation of the cylinder 

 becomes 



s2-f-2Hy=0, (27) 



and we have 



H = 2/2 - yu a?i = ^2> .„j,. 



xi + yi-x,'-y,^ = Q; ^"^^^ 



whence 



yi = — |h, 2^2 = 5H. (29) 



The focal and dirigent are each a right line parallel to the 

 axis of X, the former passing through the focus, the latter 

 meeting the directrix of the parabolic section made by the 

 plane oiy%. The plane oi xy is the directive plane. 



§ 7. We learn from this discussion, that, among the surfaces 

 of the second class, the hyperbolic paraboloid is the only 

 one which admits a twofold modular generation ; the modu- 

 lus, however, being the same for both its focals. In the 

 elliptic paraboloid the modular focal is restricted to the plane 

 of that principal section which has the greater parameter; 

 we shall therefore suppose a parabola to be described in 

 the plane of the other principal section, according to the 



