PARTICULARLY APPLICABLE TO SOME GEODETICAL PROBLEMS. 137 



Hence compounding these ratios, 



. / . A A 



Sin bx . sm eg . s in ap 



T~7 '. A ?r = i (1). 



sm o^j.sm ex. sm aq 



In like manner, in the triangles ACD, ABD, ABC- 



• -^ , . A A 

 smpx _h sm eg x sm «6 c 



• 7^^ x' .A c' /T ~ A' 

 sinA/. smjo; '^ sin ac * 



therefore 5Hli^^iiiiL££i«^ _ , ,„, 



. A . A A = 1 (2)- 



sm 6;^ . sm 5^0.- . sin ac 



The triangles ACB, ABC, DBC give these equal ratios, 



■ it" ■ ^ A 



sm_bx^p sm ftc i sin «</ a 



smpx sm be " sin 4 ^ 



therefore !HL _^^-sin «c-sin /y _ 



. A . A A - ^ (3). 



smjtja;.sm 6c.sin ag- 



And again the triangles ABD, BBC, ABC give 



. ^ A A 



^"^ g-^' ^ 9 sm ab _ c sin p q a 



■ ^ c' A ~ « ' A" = - '■> 



smqx sm be sin pa * 



therefore «i" ^•^•«i" «&-sin y^'^y _ 



. A . A A - ^ (4)- 



smyx.sm be .smj)a 



From the formulae (1), (2), (3), (4), we form this table, from which 

 it appears that the problem may be resolved in four different ways 

 by the formula of Art. 28, which determine two angles whose sum or 

 difference is given, and in addition, the ratio of their series. 

 Vol. VI. Part I. S 



