3OO On the Surfaces of the Second Order. 



face 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 sur- 

 faces, taken with respect to any sphere concentric with them, are 

 all condirective. 



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

 given central surface, describe a cone having its vertex at the 

 centre, the extremities of the diameter will lie in a spherical 

 conic. And if ,!the cone be touched by any plane, the side of 

 contact will evidently be normal to the section which that plane 

 makes in the given surface, and will therefore be an axis of the 

 section. As the axes of a section always bisect the angles made 

 by the two right lines in which its plane intersects the directive 

 planes of the surface, and as the cone aforesaid has the same 

 directive planes with the given surface, it follows that the right 

 lines in which a tangent plane of a cone cuts its directive planes 

 are equally inclined to the side of contact a theorem which has 

 been already obtained in another way. 



If a section be made in a given central surface by any plane 

 passing through the centre, the cone described by a constant 

 semidiameter equal to either semiaxis of the section will touch 

 the plane of section ; for if it could cut that plane, a semiaxis 

 would be equal to another radius of the section. Denoting by 

 r, / the semiaxes of the section, conceive two cones to be de- 

 scribed by the revolution of two constant semidiameters equal 

 to r and / respectively. These cones are condirective with the 

 given surface, and have the plane of section for their common 

 tangent plane. Supposing that surface to be expressed by the 

 equation 



and the directive axis to be that of y, the axis of x will be the 

 internal axis of one cone, say of that described by r, and the axis 



