ON THE SQUARE BAR MICROMETER. 163 



7. Correction for refraction when the square is adjusted to the true diurnal motion. — 

 Let it be assumed that the square is oriented with reference to the true diurnal 

 motion. Let 6 be the sidereal time when the star is actually on the north and 

 south diagonal, and Aia, Aga, the refraction in right ascension, and AjS, A28, the 

 refraction in decHnation, at entry and exit, respectively. Let the angular distance, 

 expressed in time, of the middle points of the true and apparent paths from the 

 diagonal be denoted by / and H, respectively. Then we have 



^ll'-g = jsec8 = ^-° + ^"^ -//sec8. 

 It will be seen by consideration of the geometrical conditions that 



■^ — * 15 2 ' 



the upper or under sign corresponding to positions north or south of the centre. 

 Hence, since a similar equation obtains for the comet, with accents, 



—2 £/ — 2 ^ 15 2 ^^^ ^> 



— 2 — 2 ^ 15 2 S^C ^ • 



The difference of the first members gives 



The first term of this being the true difierence of right ascension, and the 

 second term the apparent difierence (see equation (1) ), we have manifestly from 

 the difference of the second members of the above equations, the correction for 

 differential refraction in right ascension, 



A (a'- a) = I [(A^a + A,a) - {A,a'+ A^a')] + }l± (A^S'- A^S') T ( A^S - A,S)] j^- (10) 



To get the effect of refraction on the difference of declinations, we observe 

 that the length of the apparent path described by the star is 



15 cos (8) [(4 - ^1) - (A2 a - Ai a)], 

 where (8) is the apparent^ declination; and that, although not perpendicular to 



1 In this formula as originally written I had inserted the true declination. The manuscript having been transmitted 

 to Professor Young, and carefully examined by his assistant, Mr. Malcolm McNeill, they kindly pointed out that, rigor- 

 ously, the apparent declination should here be used. The effect of the difference is, of course, very small ; in fact, 

 inappreciable except for high declinations, near the horizon. 



