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PROCEEDINGS OF THE AMERICAN ACADEMY 



or, since a and ^ are perpendicular to the cyclic planes, 



sin Q sin q' = k, 



where q and q' denote arcs from any point of the conic perpendicular 

 to the cyclic arcs. 



T S M 



(j3) If a great circle cut a spherical conic, the parts of it between 

 tlie points of intersection and the respective cyclic arcs are equal. For 



sin BO sin rp . sin rt 'sinxB 



^^" ^ ^^ sin BD ^^ sin rd' ^^" ^ ^^ sin rs ^^ sin b9 ' 



and therefore, by («), 



sin BS sin db = sin rs sin dr, 

 or sin (br -j- rs) sin db = sin rs sin (db -}- br) ; 



,*, sin DB cos RS = sin rs cos db, 



.', DB = RS. 



I shall also insert a quaternion proof, as given by Tait. If a conic 

 be cut by a plane whose ecjuation is Sj'p = 0, the intersections of this 

 with the cyclic planes are Nay and N^y. Then, since a point of the 

 curve can be reached by moving in the directions of these intersections, 



we may write 



Q = x\]Yay + yUV|3j', 



SaQ = ySaUV(3;', 



S^Q = xS^Way. 



, •, q"^ — SoqS^q = may be written 



x"^ -\- y"^ -\- Axy = 0, 



where -4 is a scalar function of «, ^, and y only. The form of the 

 equation shows that any two values of x and y can be interchanged. 

 This, then, establishes the theorem. 



