OF ARTS AND SCIEXCES. 



385 



at the focus, — the equation shows that the projection on tlie plane is 



a conic with the same focus and an eccentricity equal to . 



•' ^ sin 2a 



The importance of this principle is due to the fact that it enables us 

 to establish many properties of spherical conies with reference to a 

 single focus from known {properties of plane conies. 



21. The angle subtended at the focus by any chord is bisected by 

 the arc joining the focus to its conic pole. The spherical conic can be 

 projected into a plane conic having the same focus, of which this prop- 

 erty is true ; but the angles of the plane conic at the focus are the 

 measures of the angles of the spherical conic. 



From this can be established a reciprocal property by the aid of 

 the sujjplementary conic. 



Two tangent arcs to a conic and the arc joining their points of con- 

 tact cut the cyclic arc in three points, one of which bisects the distances 

 between the other two. 



Suppose A represents a conic 

 and B its supplementary conic, 

 and Zj^'fp = Z^tfp. The poles 

 of ft', fp, and ft all lie on a 

 cyclic arc of the supplementary 

 conic. Let these poles be t'^, p^, 

 and Tj respectively ; then t'jP^ = ' 

 PjTj. The poles of pt and pt 

 lie on the supplementary conic ; 

 denote them by r and s. Then 

 R, s, and p^ lie on the circle of 

 which p is the pole. Moreover, 

 Tj' and s lie on the circle of which 



TOL. XIII. (n. 8. V.) 



