On the Surfaces of the Second Order. 287 



approaches to 90, and m indefinitely diminishes, this quantity 

 will approach indefinitely to n* (x - # 2 ) 2 , which will be its limit- 

 ing value when <j> = 90. But x - x z is the distance of the point 

 S from a fixed plane intersecting the axis of x perpendicularly 

 at the distance x from the origin of co-ordinates ; and there- 

 fore, in the limit, the equation expresses that the distances of 

 any point S of the surface, from the focus F and from this fixed 

 plane, are to each other as n to unity, that is, in a constant 

 ratio, which is a common property of the three surfaces in 

 question. This property also belongs to the right cone, but the 

 right cone does not rank among the excepted surfaces. 



12. We have seen that, when the modulus is unity, any 

 plane parallel to either of the directive planes intersects the 

 surface in a right line; whence it follows, that through any 

 point on the surface of a hyperbolic paraboloid two right 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 lines. This is otherwise evident from considering 

 that the sections parallel to a given tangent 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 distance vanishes, the hyperbola being then 

 reduced to its asymptotes. The tangent plane therefore inter- 

 sects the surface in the two fixed right 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 



* 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, 



