AZIMUTH IN TRIGONOMETRICAL OPERATIONS. 105 



/. S (T~ tan PS. (l—cos^ P). cos" PS = sin 2 PS. sin" \^P (y) 



2nd Tan S'a — sin P a. tan hP — sin [P S—S ff). tan d P 

 :. S' ff = (sin PS — cos PS. Sff). tanb P 

 Or S' ff = sin PS. tan h P.{\~cot PS. S a) 



= sin PS. tan a P. (1-2. cos- PS. sinr i ^ P) (s) 



Combining now the equations ((3) and (e) we obtain 



ffff' = sin PS. tan d P. (1-2. cos" PS. sin" ^^^P) cos SZ. 8 Z 



= tan PS. sin X. tan d P. (1-2. cos" PS. sin' \lP).l Z (0 



cos PZ sin 1. \ 



because cos ZS — _ i 



\ cos PS cos PiS / 



and this value of ffa' answers for both the upper and lower positions, 

 being subtractive in the former, and additive in the latter, with respect 

 to the mean distance Sa. 



We have, therefore, generally 

 Sff z= Sc/ ^ ffff' in which, by substituting the values given in Equa- 

 tions (y) (a) iO we get 



Sin 2 PS. sin^ h^^ = (^i^ P- cos X + tan PS. sin \. tan d P. (1-2. cos' PS. sin^ ^hP)).tiZ 



sin2PS.sin^ oP 



r tanPS.tan-k.tan^ P.{l-2.cos' PS.sin-^d P)^ 



sin P.cosl..< ITjl V 



i sin P ) 



sin 2 PS. sin" ^ 8 P 

 ^ sin P. cos X. (1 + cot P. tan 3 P. (1-2. cos" PS. sin" J- d P)) 



^ . ^ sin 2 PS. sin" ^ P . , , , r> . ^ n\ „ i 



Ov Z z=: — J, — : — Tz — . ( 1 + cot p. tan o P) nearly. 



S'ln V. sin P. cos ? 



AVhence Log S Z = Log sin 2 PS -\- 2. Log. sin i 3 P -f- Log. coscc P 



-I- Log. sec X -I- A. C. Log. sin V ± 31. cot P. tan a P Where 31 



denotes the number 0.4;i42})4 1819, ivcc. whose Log is }).();17784.3, the upper 

 or lower sign being used according as the star is above or below the 

 maximum. 



2 ]) 



