276 On the Surfaces of the Second Order. 



pose G and K to vanish, the equation of the cylinder becomes 



s 2 + %Hy = 0, (27) 



and we have 



H=y z - y l9 x l = x-t, 



(28) 



x? + y?. - x? - y? = ; 

 whence 



(29) 



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

 of #, the former passing through the focus, the latter meeting 

 the directrix of the parabolic section made by the plane of i/z. 

 The plane of ocy 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 modulus, 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 law of the modular focals ; 

 the law being, that the focus of the parabola shall be the focus 

 of the principal section in the plane of which the parabola lies, 

 and its vertex the focus of the principal section in the per- 

 pendicular plane. The parabola so described will have its 

 concavity opposed to that of the surface ; it will- cut the sur- 

 face in the umbilics, and will be its umbilicar focal, the only 

 such focal to be found among the surfaces of the second 

 class. We shall of course suppose further, that this focal has 

 a dirigent parabola connected with it by the same law as in 

 the other cases, the vertices of the focal and dirigent being equi- 

 distant from that of the surface and at opposite sides of it, while 

 the parameter of the dirigent is a third proportional to the para- 

 meters of the focal and of the principal section in the plane of 

 which the curve lies. The two focals of a paraboloid are so re- 

 lated, that the focus of the one is the vertex of the other. The 

 cylinders have no other focals than those which occur above. 



