160 ON THE SQUARE BAR MICROMETER. 



Call ^1 the time of the star's transit over the north following or the south follow- 

 ino- bar (i. e., the bars which intersect at the position angle 90°), whether the 

 transit occurs inside or outside the square; and 4 its time of transit over the 

 north preceding or the south preceding bars (i. e., those intersecting at 270°), 

 also irrespective of the place on the bar, inside or outside. Take for the comet 

 the same letters accented, so that i\ and t'^ are, respectively, the times of the 

 comet's transit over a following and a preceding bar, as just defined, whether 

 inside or outside. Thus 4 is numerically greater than i^, and 4 greater than t\, 

 when the transit occurs inside, and less when it occurs outside. This notation 

 preserves, algebraically, the relations which represent the geometric conditions 

 for all the positions which the objects can occupy in the field, with reference 

 to each other and to the square; and, if strictly adhered to, render all the 

 formulae hereafter given entirely general. 



4. Determination of the uncorrected appareni differences of right ascension and decima- 

 tion of two objects. — By the term uncorrected differences are meant those found by 

 assuming the micrometer to be perfectly adjusted, and the paths of the objects 

 to be rectilinear, and also by ignoring the effects of refraction, and of the mo- 

 tion of the comet relative to the star during the observation. 



Let a, a' be the apparent right ascensions, and S, 8' the apparent declinations, 

 of the star and comet, respectively; and let g be the value, in arc, of the 

 diagonal of the square. 



The uncorrected difference of right ascension will then be 



a' - a = J- (^ + 4) - * (^ + ^2)- (1) 



Calling d the difference of declination of the star and the centre of the square, 

 we have 



(? = ± I T -^-^' 15 cos 8 ; 



and, similarly for the comet, v^) 



j'=±|V-4^15cos8'; 



where g and / are of course numerically equal, but the upper or under sign is 

 to be used, according as the path is north or south of the centre of the square. 

 The difference of these equations gives 



§'_8 = J'-f;=±f=Ff-¥[±(^'2-^'i)cos8'=F(if,-/i>cosS], (3) 



