488 



portional 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 having 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 eUiptic. If 

 a paraboloid be cut by any plane, the cylinder which passes 

 through the curve of section and has its side parallel to the 

 axis of the paraboloid, will be condirective with that surface ; 

 and it will be elliptic or hyperbolic, according as the para- 

 boloid is elliptic or hyperbolic* 



If two concentric surfaces of the second order be recipro- 

 cal 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 perpen- 

 dicular 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 hy- 

 perbola of each surface are the focal lines of its asymptotic 

 cone; and the two asymptotic cones are reciprocal. 



When any number of central surfaces of the second order 

 are confocal, or, more generally, when their focal hyperbolas 

 have the same asymptotes, it is obvious that their reciprocal 

 surfaces, taken with respect to any sphere concentric with 

 them, are all condirective. 



§8. If a diameter of constant length, revolving within a 



* I have inti'oduced the terms directive and condirective, as more general 

 than the terms cyclic and hicoHcyclic employed by M. Chasles. The latter terms 

 suggest the idea of circular sections, and therefore could not properly be used 

 with reference to the hyperbolic paraboloid, or to the hyperbolic or parabolic 

 cylinder, in each of which surfaces a directive section is a right line. 



