ox THE SQUARE BAR MICROMETER. 167 



Also multiply the first of (15) by cos S and the second by sinS, subtract and 

 reduce, and we get 



cos <!> cos t T.J, ,5 



cosS cos^ = 1 - tan ^ cos^ tanS. 



Employing these relations in the values of -^-^ and ^ from (13) we get, 



^ = — tan^ 4 sin^ q + tan (, cos q tan S — 1 ; 



-jl = — 1 — tan^ C cos^ q ; 

 which, introduced into (19) give, finally, for the refraction correction in declination, 



A (S' - 8) = ^ (± I' T ^) (tan^ ^ sin^ ^ + 1) + K ( J' - d) tan^ t, cos 2 q. (20) 



Equations (18) and (20) are, therefore, the expressions (general for any posi- 

 tions of the comet and star, inside or outside the square) by which the refrac- 

 tion corrections may be accurately computed, when the square has been oriented 

 with respect to the true diurnal motion. We can often substitute for these 

 the simpler approximate expressions, which involve the terms in tan^ ^ only, 



A (a' - a) = 3o478-„ (± I T f) tanH sin 2 ^ ; 



A (S' - 8) = k[± ■{ T {) tan^ C sin^ q + K{d'-d) tan^ ^ cos 2 ^ ; 



where, it will be noticed, the whole correction in right ascension, and the first 

 term of that in declination, disappear when the star and the comet are observed 

 in the same half of the square. 



The precaution should always be exercised, before using the approximate values 

 given by (21), to see, by a general inspection of the case in hand, whether the 

 smaller terms of which the more rigorous equations (18) and (20) take account, 

 are fairly negligible. 



The constant k can be computed from section C of Talkie II., in Vol. II. 

 of Chauvenet's Astronomy, if necessary for the particular state of the air, by 

 article 295, which also gives the formulae for t, and q. Except for extreme 

 atmospheric conditions, or near the horizon, however, it will be sufficiently accu- 

 rate to take log k for the mean state of the air, or equal to log a" of the table. 



