On the Extra- 31 eridian Determination of Time. 213 



For a second series of approximate values, m^, ??.,, -^ and -/ which 

 satisfy the condition A, = a — T — -, ^ a — T' — r.,', we have 



sin ?n.2 = — U:, tan (p, ( LI ) 



sin (r., — Wo) = ??., tan f5, (12) 



sin (-./ — m.j) = w,, tan '5'; (l*^) 



. • . - — r, = {n — n.,) (tan r) — tan (p) -\- c sec 5 ^ A, — t (14) 



==t' — -,' = (?? — ??„) (tan 5' — tan tp) -\- c sec ')'; (15) 



-, — r, = (rto — w,) (tan (5 — tan cc), (1^) 



-,' — 7i'=(«., — ??j) (tan 'J' — tan cp). (1'^) 



Adding (6) successively to (7) and (8) we have approximately. 



r, = ll■^ (tan 8 — tan ^), (IS) 



7,'^Wi (tan 5' — tan <p). (19) 



.-. i,=iL_-::r:^>'_J!!z:!^_.. (20) 



Also, if 

 and 

 we have 



whence 



«> = «— T— r,, (21) 



t^ = a'.-r~r;, (22) 



^. = ^i-(^-^,) = ^/-(-/-- ') 



= ^ — r,v = A/— r/ .. (23) 



V = _!__>. (24) 

 '1 '~~'\ 

 In the same manner, for two stars observed in the reversed position 

 of the horizontal axis, 



tr = ^/'— - " v" = t;"—T-' v", (25) 



when ce. 



•^" = ^'^->,- (26) 



Equations (14) and (15) give, 



cos o' — cos 5 ,^_, 



w — ?i., = c . {Ti\ 



sin (-J'— '5) ^ ' 



... /;=^.,_csec^''°iili^llfl, (28) 



^cosi(^'4-3) ^ ^ 



