489 



given central surface, describe a cone having its vertex at 

 the centre, the extremities of the diameter will lie in a sphe- 

 rical 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 bi- 

 sect 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 sec- 

 tion 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, r' the semiaxes of the section, con- 

 ceive two cones to be described by the revolution of two 

 constant semidiameters equal to r and r' 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 



- + ^ + -=1, (2) 



P Q R ^ 



and the directive axis to be that of t/, the axis of x will be 

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

 the axis of z will be the internal axis of the other cone. 

 Let K be the angle made by the two sides of the first cone 

 which lie in the plane xz, and k' the angle made by the two 

 sides of the second cone which lie in the same plane ; the 

 former angle being taken so as to contain the axis of a; within 

 it, and the latter so as to contain within it the axis of «. 

 2t2 



