162 ON" THE SQUARE BAR MICROMETER 



where the last term is the required correction for curvature. If we take 

 values m' and m from Table V., Chauvenet's Astronomy, Vol. II., using the side 

 arguments \ (f 2 — t\) and J (^ — - ^), we can write this equation very simply, 



8' — 8 = (d r — d) — (m f — m) cos 8 sin S ; (6) 



where id' — d) is the uncorrected value given by (3). For a high declination 



we can, without sensible error, write cos S for cos 8 sin 8 



o 



For very high declinations, say above 85°, we should, for greater accuracy in 

 computing (d\— d), substitute the sines for the arcs described by the star and 



comet; so that instead of (6) we have the more rigorous expression 



v a — 4- g ' i 9 ± sin ¥ ( *' 2 ~ *'*> cos 8 ' * sin ^ & ~ ^ cos 8 (™> ™w«, a /*t\ 

 S — 8 — ±2^5 ^jr, (m— w) cos So. (7) 



It will, however, seldom be necessary to employ this equation, since in practice, 

 for stars so near the pole as to require it, the use of the square or of any 

 micrometer dependent on transits, would be abandoned as inconvenient. 



6. Correction for proper motion. — Let A a be the increase in time-seconds of 

 the comet's right ascension, and AS' the increase in arc-seconds of its declina- 

 tion (positive for northerly motion), each during one sidereal second. It is man- 

 ifest that the time of the comet's arrival at both preceding and following bars 

 is accelerated or retarded by the amount necessary for it to describe an arc 

 equal to the comet's motion in declination during half the duration of the tran- 

 sit. Consequently, to the uncorrected difference of right ascension given by (1) 

 is to be added the correction 



t\— t\ A 8' 



2 15 cos 8' ' 



(8) 



where the upper or under sign is to be used, according as the comet passes north 

 or south of the centre. 



Another effect of the proper motion is to alter the length of the chord 



described in the interval {f 2 — t\) by an amount equal to the comet's motion 



in right ascension during that interval. The corrected length of the chord is 

 2 ~" ^i) (1 - A a ) cos 8', whence the uncorrected difference of declination found 



by (3) requires the correction 



± ~^~2 — l A a. 15 cos 8' ; 



(9) 



the double sign having the same significance as before 



