OF ARTS AND SCIENCES. 



381 



If the cutting arc becomes a tangent, the parts intercepted between 

 the point of contact and the cyclic arcs are equal. 



12. From the two properties announced, we can deduce another. 

 The area of the spherical triangle formed by a tangent and the cyclic 

 arcs is constant. 



sin ON = sin oil sin m, 



sin OH = sin OK(sin okh == sin okc'). 



, • , sin^ OK sin M sin ic =: constant. 



But 



cos c' = — COS (m -(- k) — 2 sin^ ok sin M sin K. 



Hence, since c' is constant for a given conic, m -j- K is constant, and 

 therefore the area of the triangle is constant, for it equals c' -f- ^^^ ~h 

 K — 180°. 



"We may then define a spherical conic as the envelope of the' base of 

 a spherical triangle, of which the vertex, the vertical angle, and the 

 area are given. The arcs forming the vertical angle are the cyclic arcs 

 of the conic. 



13. This property is also true of the supplementary conic; and 

 therefore, remembering the relation existing between arcs and their 

 poles, we may define a spherical conic as the locus of the vertex of a 

 spherical triangle, of which the base is given in length and position, 

 and of which the sum of the sides is al§o given. The extremities of 

 the base are the foci of the conic ; and we now wish to determine 

 their position. It is evident that they are the poles of the supple- 

 mentary cyclic arcs. Since, by Art. 11 (^'), a conic is symmetrical with 

 respect to its cyclic arcs, their poles must lie on an axis at equal dis- 

 tances from the centre ; and, since the axes of conies and those of sup- 

 plementary conies are parts of the same circles, the foci of a conic 

 must lie on an axis of that conic, at equal distances from the centre. 

 It can be shown that this axis is the major axis. Let f and f' be the 

 foci ; then, as the sum of the sides of a spherical triangle is greater 

 than the base, the foci fall inside 

 the conic. When p is at A, 



f'a -f- fa = constant =: 2 CA ; 



and, when at b, 



f'b -|- fb = 2 fb = 2cA ; 



,•, CA = FB : 



