OF ARTS AND SCIENCES. 895 



38. If now we suppose the radius of the sphere to become infinite, 

 the spherical conies become plane. As the properties ah-eady proved 

 still hold good, we can deduce the well-known properties of plane 

 conies from corresponding ones of spherical conies. A remarkable 

 analogy has been shown to exist between the foci and cyclic arcs ; and, 

 as the properties of the foci hold good in the plane conies, the question 

 naturally arises, What becomes of the reciprocal properties of the 

 cyclic arcs ? In the case of the ellipse, the cyclic arcs become lines 

 at infinity ; but, in the hyperbola, the cyclic arcs become the plane 

 asymptotes. 



The Calculus can be applied to the equations of the spherical conies, 

 and expressions can thus be found for the area and arc. As these, 

 however, involve elliptic integrals, I have omitted them. 



