390 PROCEEDINGS OF .THE AMERICAN ACADEMY 



which is a spherical conic. ' 



27. Several additional properties of cyclic arcs can also be stated. 

 First, to find their equation. Their poles are at the same distance from 

 the centre on the axis of 7/ as the foci of the supplementary conic are 

 from its centre. Suppose a, §, c, and a', §', c', are the semi-diameters 

 and focal distances of the two conies, then 



cos a' sin /3 



cos c' =■ — 1^ = -. — , 

 COS 13' sm a' 



tan'^ c' 



Hence, the co-ordinates of the poles of the cyclic arcs are 



and ± 4 / — ; 



and the equation of the arcs can (Art. 2) be written 



Y a2 — ta 



Let 2d = the angle between the cyclic arcs; then 20 -\- 2c' = 9. 

 Hence, 



•a , sin /3 



sin o = cos c = — : — , 



sin a 



SO that the angle of the cyclic arcs is equal to the angle subtended by 

 the minor axis at either focus. 



28. Every tangent to a spherical conic cuts the cyclic arcs in two 

 points, such that the product of the trigonometric tangents of the 

 halves of the arcs lying between these points and the point of inter- 

 section of the cyclic arcs is constant. It is known by spherical trigo- 

 nometry that (see fig. B, Art. 21) 



tan i CD tan i ce sin c 



tan (area ced) =z r~rz — i : — ^ • 



^ '1-4- tan i CD tan ^ ce cos c 



,•, (by Art. 12) tan ^ CD tan ^ ce = constant. 



