214 ^ Cincinnati Society of Natural Hi-storij. 



where z=^(p — 8 and z' ^S' — ^, assuming tJ' ^ ^ ^ o, and for the 

 stars observed in the opposite position of the horizontal axis, 



CCt'S, J- ( y'" y" \ 



t = ^.,"+c sec -^ ^ :i_i, ( 29 ) 



cos ^ (z"'-\-z") 



where 



z" = <p — 15", z'" = ')'" — <p and rf" ^^ ^>o". 



Assuming that the stars observed are not far from the zenith, and 

 neglecting small quantities, we have 



c sec ^=1 (^,— ^,"), (30) 



cos \ (z' — z\ 1 — 2 sin- -^ (z' — z) 



cos ^ (z'-\-z) 1 — 2 sin- ^ (z'-\-z) 



= 1-2 (sin^ i (-_^)_sinH (^'+.^) ) 

 = 1+2 sin 4 z sin ^ z\ (31) 



cos -1 (z"'—z") ^ 2 sin i z" sin ^ z'". (32) 



cos 4- {z"'-[-z") ^ - - , ^ ' 



. • . taking the mean of (28) and (29) 

 t = ^ (t.^-\-t,")-^ ^ (t,—t,") (sin ^ z" sin | 2'"— sin -k z sin 4- 2'); (33) 

 or, if 



Z' =2 



sin 1° 



sin 4- i' 

 sin 1° 



7"_9 SI" ¥ 

 sin 1° ' 



~"^ sin 1° ' 

 t = \{ t.r^f.;' )-|-0-000038 ( t~t.:' ) ( Z" Z'"— Z Z' ), (34) 



in which Z, Z', Z", Z'" may be assumed to be tlie zenith distances ex- 

 pressed in degrees. 



After making the preliminary reductions and correcting for level 

 error, we may assume for the first approximation, 



^ _^ sin(a--T'-(«-T)) sin(a--T--(a--T-)) 



' tan 5'— tan 8 ^^ tan rf"— tan <5" ' ^' ' 



Witli this value of /i, compute 



sin hi^ = ■ — n, tan (f, (30) 



