294 On the Surfaces of the Second Order. 



fixed points F, F' will have a constant sum. The curve formed 

 by the intersection of the sphere and the cone may therefore, 

 from analogy, be called a spherical ellipse, or, more generally, a 

 spherical conic, because, by removing one of its foci F, F 7 to the 

 opposite extremity of the diameter of the sphere, the difference 

 of the arcs SF and SF / will be constant, which shows that the 

 spherical curve is analogous to the hyperbola as well as to the 

 ellipse. Either of these plane curves may, in fact, be obtained 

 as a limit of the spherical curve when the sphere is indefinitely 

 enlarged, according as the diameter along which the enlarge- 

 ment takes place, and of which one extremity may be conceived 

 to be fixed while the other recedes indefinitely, coincides with 

 the internal or with the directive axis of the cone. The fixed 

 extremity becomes the centre of the limiting curve, which is an 

 ellipse in the first case, and a hyperbola in the second. 



The great circle touching a spherical conic at any point 

 makes equal angles with the two arcs of great circles which join 

 that point with the foci, because the sum of these arcs is con- 

 stant. This is identical with a property already demonstrated 

 relative to the tangent planes of the cone. Indeed it is obvious 

 that the properties of the cone may also be stated as properties 

 of the spherical conic, and this is frequently the more convenient 

 way of stating them. 



5. If the sides of one cone be perpendicular to the tan- 

 gent planes of another, the tangent planes of the former will be 

 perpendicular to the sides of the latter. For the plane of two 

 sides of the first cone is perpendicular to the intersection of the 

 two corresponding tangent planes of the second cone ; and as 

 these two sides approach indefinitely to each other, their plane 

 approaches to a tangent plane, while the intersection of the two 

 corresponding tangent planes of the second cone approaches in- 

 definitely to a side of the cone. Thus any given side of the one 

 cone corresponds to a certain side of the other ; and any side of 

 either cone is perpendicular to the plane which touches the other 

 along the corresponding side. This reasoning applies to cones 

 of any kind. 



