288 On the Surfaces of the Second Order. 



right lines by any of its tangent planes. These right lines are 

 usually called the generatrices of the surface. 



From what has been said, it appears that the generatrices of 

 the hyperbolic paraboloid, and the asymptotes of its sections (all 

 its sections, except those made by planes parallel to the axis, 

 being hyperbolas), are parallel to the directive planes. The 

 generatrices of the hyperboloid of one sheet, and the asymp- 

 totes of its hyperbolic sections, are parallel to the sides of the 

 asymptotic cone; because any section of the hyperboloid is 

 similar to a parallel section of the asymptotic cone ; and when 

 the latter section is a hyperbola its asymptotes are parallel to 

 two sides of the cone. 



PART II. PROPERTIES OF SURFACES OF THE SECOND 

 ORDER. 



1. In the preceding part of this Paper it has been neces- 

 sary to enter into details for the purpose of communicating fun- 

 damental notions clearly. In the following part, which will 

 contain certain properties of surfaces of the second order, we 

 shall be as brief as possible ; giving demonstrations of the more 

 elementary theorems, but confining ourselves to a short state- 

 ment of the rest. 



Many consequences follow from the principles already laid 

 down. 



Through any directrix of a surface of the second order let a 

 fixed plane be drawn cutting the surface, and let S be any point 

 of the section. If the directrix and its focus F be modular, and 

 if a plane always parallel to the same directive plane be con- 

 ceived to pass through S and to cut the directrix in D, the direc- 

 tive distance SD will be always parallel to a given right line, 

 and will therefore be in a constant ratio to the perpendicular dis- 

 tance of S from the directrix. This perpendicular distance will 



however, been remarked by Wren, with regard to the hyperboloid of revolution. 

 It does not seem to have been observed, that the existence of rectilinear genera- 

 trices is included in the idea of hyperbolic sections parallel to a tangent plane. 



