On the Surfaces of the Second Order. 299 



its vertex at the common centre, is of the second order. The 

 cylinder also, which passes through the same curve and has its 

 side parallel to any of the axes of the given surface, is of the 

 second order ; and the cone, the cylinder, and the given sur- 

 face are condirective, that is, the directive planes of one of them 

 are also the directive planes of each of the other two. This may 

 be seen from the equations of the different surfaces ; for, in ge- 

 neral, two surfaces whose principal planes are parallel will he 

 condirective, if, when their equations are expressed by co-ordi- 

 nates perpendicular to these planes, the differences of the coeffi- 

 cients of the squares of the variables in the equation of the one 

 be proportional to the corresponding differences in the equation 

 of the other. 



If any given surface of the second order be intersected by a 

 sphere whose centre is any point in one of the principal planes, 

 the cylinder passing through the curve of intersection, and hav- 

 ing its side perpendicular to that principal plane, will be of the 

 second order, and will be condirective with the given surface. 

 This cylinder, when its side is parallel to the directive axis, is 

 hyperbolic ; otherwise it is elliptic. If a paraboloid be cut by 

 any plane, the cylinder which passes through the curve of sec- 

 tion, and has its side parallel to the axis of the paraboloid, will 

 be condirective with that surface ; and it will be elliptic or hy- 

 perbolic, according as the paraboloid is elliptic or hyperbolic.* 



If two concentric surfaces of the second order be reciprocal 

 polars with respect to a concentric sphere, the directive axis of 

 the one surface will coincide with the mean axis of the other, 

 and the directive planes of the one will be perpendicular to the 

 asymptotes of the focal hyperbola of the other. When one of 

 the surfaces is a hyperboloid, the other is a hyperboloid of the 

 same kind ; the asymptotes of the focal hyperbola of each sur- 



* I have introduced the terms directive and condirective, as more general than 

 the terms cyclic and biconcyclic employed by M. Chasles. The latter terms suggest 

 the idea of circular sections, and therefore could not properly he used with refe- 

 rence to the hyperbolic paraboloid, or to the hyperbolic or parabolic cylinder, in each 

 of which surfaces a directive section is a right line. 



