278 Mr DAVIES on the Equations of Loci 



cosScosS 



C|. ti f -cosScosSA ) ( 



A J IcosS BcosS C-J 



Inserting these values in the first of equations (2), and performing a simi- 

 lar process with respect to cot a y , we shall obtain 



cos S cos S A + cos S B cos S C 

 -o 2V_ cog g cos s_ A cos S B cos S C 



.(4.) 



cosS cosS B + cos S AcosS C 

 cot a . = a i - 



*v_ cos g cos s- A cos S B cos S C 



In the numerators of these expressions, put for A, B, C, S their values, 

 and put 2 R instead of the denominator. Then we shall get 



cos S B cos S C = cos | (TT , (e,, ft, ft)} cos {TT e,+(e /; ft, ft)} 



= {COST e, + cos (e,, ft, ft 



cos S cos S A = cos {-TT e / +(e /y +ft / ft) cosi{ T *, + (f/,+ft, ft)} 



= | {cos -TT e, + cos (e,;+ ft, ft)} ; 



Theseaivecota - i cos7r ,+|cos(6,, ft ; PJ+ICOSTT ,+^(,,+ (3,, ft) 

 8 ' " yll 



_ COS 7T C, + i {COS (6 ;/ ft, ft) + COS ( 8// + ft, ft)} 



_ COS 7T , + COS COS ft, ft 



_ COS , + COS 6,, COS ft ft 



Similarly cot a/ = cos 6,, -cose cos ft, -ft 



^ It 



For cot a x and cot a //5 put their values just found in the equations (5, 6, 

 7, 8,) of Art. IV. Then we shall get 



(cos'e, cose,, cos ft, ft) cos ft + (cos e,, cos 6,008 ft, ft) cos ft, 



tflTl / ~~ ~ ^^ . - .... i i ... i . ,,,.-. i. __ ,,I | . I 



(cos e, cos 6,, cos ft ft) sin ft + k (cos e,, cos e, cos ft, ft) sin ft, 



_ (cos f, cos c,, cos ft, ft) cos ft + (cos 6,, cos 6,, cos ft, ft) cos ft, 



-f-^ . > =~ . 



sin ft x /3 V cos 2 e, 2 cos e, cos 6,, cos ft, ft -f cos 2 e, x 



