OF ARTS AND SCIENCES. 



391 



If arcs be drawn from any point of a spherical conic to the foci, 

 the product of the tangents of the halves of the angles made by these 

 arcs with the major axis is constant. For the intersection of tlie sup- 

 plementary cyclic arcs is the pole of the major axis, and the poles of the 

 arcs drawn from the foci lie on these cyclic arcs, at the extremities of a 

 tangent to the supplementary conic. If the angles be measured in 

 opposite directions, the ratio of the tangents of the semi-angles is con- 

 stant ; and a similar modification may be made in the reciprocal 

 theorem. 



29. If a tangent be drawn to the inner of two concyclic conies, the 

 parts included between the point of contact and 



the outer curve are equal. This is an immediate 

 consequence of Arts. 11 and 12. Then, by the 

 method of infinitesimals used in plane conies 

 (Salmon's Conic Sections, § 396), the area in- 

 cluded between the tangent and the outer curve 

 is constant, as the point of contact moves along 

 the inner curve. 



The conies supplementary to concyclic conies 

 are confocal. The pole t' of pp' lies upon the outer conic, and, if from t' 

 tangents be drawn to the inner conic, these tangents measure the 

 angles which the tangents at p and p' make with pp'. Moreover, the 

 curve between the points of contact of tangents from successive posi- 

 tions of t' measures the infinitely small angle made by consecutive tan- 

 gents along the curve prp'. But the sum of these angles with those 

 at p and p' mentioned above is constant. Hence this theorem follows : 

 — If, from a point on the outer of two confocal conies, tangents be 

 drawn to the inner one, the sura of these tangents and of the concave 

 part of the curve included between them is constant. 



30. I shall now give some principles of conic poles and polars with 

 reference to spherical conies. The spher- 

 ical anharmonic ratio is 



sin AD sin bc 



sin AB sin CD 



To prove this ratio constant for any given 

 pencil. 



sin AD = 



sin ADD sin OA sin ADD sin OA sin CD 



