394 PROCEEDINGS OF THE AMERICAN ACADEMY 



whose spherical pole is x'y' . Hence, to a point and its polar with refer- 

 ence to a conic, there correspond an arc and its conic polo with refer- 

 ence to the supplementary conic. 



35. From Art. 30 can be deduced : If, from four fixed points on 

 a spherical conic, arcs be drawn to a fifth point of the conic, their 

 anharmonic ratio is constant. Reciprocally : If four fixed tangents 

 be drawn to a conic, thoy will cut a fifth tangent in four points wliose 

 anharmonic ratio is constant. 



36. If the spherical conic be projected upon a plane tangent to the 

 sphere at the pole of a cyclic arc, the conic becomes a pkme circle, 

 and the cyclic arc a line at infinity. The plane passing through the 

 centre of the sphere parallel to the plane of projection is the cyclic 

 plane ; and, if two planes be drawn through the centre of the sphere 

 and throiigli any two lines in the plane of projection, these two planes 

 will intersect the parallel cyclic plane in two radii making the same 

 angle as the lines. Since the centre of the circle is the pole of the 

 line at infinity, its projection on the sphere will be the conic pole of 

 the cyclic arc. By means of tliis method, many properties of the cir- 

 cle can be extended, with suitable mollifications, to splierical conies. 

 The propositions of Arts. 21 and 22 can be proved in this way, though 

 in the inverse order. 



37. If two tangents to a conic intercept upon a cyclic arc a seg- 

 ment of constant length, the locus of the point of intersection of these 

 tangents is a second conic, and the arc joining the points of contact 

 of the tangents will envelope a third conic. The cyclic arc will be a 

 cyclic arc of the new conies, and will have the same conic pole for all 

 three conies. For, if two tangents to a circle make a constant angle, 

 the locus of the intersection is a circle, and the chord of contact envel- 

 opes a third circle, and these three circles are all concentric. Recip- 

 rocally: If a constant angle has its vertex at either focus of a 

 spherical conic, the arc joining the points in which the sides of the 

 angle cut the curve will envelope a second conic, and the tangents to 

 the given conic at the points of cutting will intersect on a third conic, 

 then (Art. 34) the focus at which the vertex of the constant angle is 

 placed will be a focus for the three conies, and the directrix will also 

 be the same for all three. 



This example is sufficient to illustrate the method of applying the 

 principle. It is plain that all graphic properties of the circle can be 

 extended to spherical conies. 



