lines may be drawn which shall lie entirely in the surface. 

 The plane of these right lines is of course the tangent plane 

 at that point, and therefore every tangent plane intersects 

 the surface in two right Hnes. This is otherwise evident 

 from considering that the sections parallel to a given tan- 

 gent plane are similar hyperbolas, whose centres are ranged 

 on a diameter passing through the point of contact, and 

 whose asymptotes, having always the same directions, are 

 parallel to two fixed right lines which we may suppose to be 

 drawn through that point. For as the distance between the 

 plane of section and the tangent plane diminishes, the axes 

 of the hyperbola diminish ; and they vanish when that dis- 

 stance vanishes, the hyperbola being then reduced to its 

 asymptotes. The tangent plane therefore intersects the 

 surface in the two fixed riglit lines aforesaid. The same 

 reasoning, it is manifest, will apply to any other surface of 

 the second order, which has hyperbolic sections parallel to its 

 tangent planes ; and therefore the hyperboloid of one sheet, 

 which is the only other such surface,* is also intersected in 

 two 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 sec- 

 tions (all its sections, except those made by planes parallel 

 to tlie axis, being hyperbolas), are parallel to the directive 

 planes. The generatrices of the hyperboloid of one sheet, 

 and the asymptotes of its hyperbolic sections, are parallel to 

 the sides of the asymptotic cone ; because any section of the 



* The double generation of these two surfaces by the motion of a right line 

 has been long known. It appears to have been discovered and fully discussed 

 by some of the first pupils of the Polytechnic School of Paris. This mode of 

 generation had, however, been remarked by Wren, with regard to the hypre- 

 boloid of revolution. It does not seem to have been observed, that the existence 

 of rectilinear generatrices is included in the idea of hyperbolic sections parallel 

 to a tangent plane. 



