482 



found to be equal, as it ought to be, to one of the angles 

 made by the two sides of the cone which are in the plane of 

 the focal lines, namely to the angle within which the inter- 

 nal axis lies. 



If we conceive the cone to have its vertex at the centre of 

 a sphere, and the points F, F', S to be on the surface of this 

 sphere, the arcs of great circles connecting the point S with 

 each of the 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' to the opposite extremity of the diame- 

 ter of the sphere, the difference of the arcs SF and SF' will 

 be constant, which shows that the spherical curve is analo- 

 gous 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, ac- 

 cording as the diameter along which the enlargement 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 hmiting 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 constant. This is identical with a property already de- 

 monstrated relative to the tangent planes of the cone. In- 

 deed 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 tangent 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 perpendicu- 



