ON THE SQUARE BAB MICROMETER. 



161 



which is the uncorrected difference of declination of the comet and the star. 

 This general expression assumes the following forms for the special cases 

 indicated below. 



Comet and star both north 

 Comet and star both south 

 Comet north, star south 

 Comet south, star north 



U'-s= < 



- -¥- [(^'2 - ^'1) cos 8' - (4 - ^1) cos 8] 



+ ¥- [(^2 - i'l) cos 8' - (4 - ^1) cos 8] 



+ff- -y- [{i'o - 1\) cos 8' + (4 - ^1 ) cos 8] 



-9 + -r [(^'2 - ^'1) cos 8' + (4 - 4) cos 8] 



(4) 



For moderate declinations, say under 40°, we may take 80 = J (8' +8) and 

 write equation (3) with sufficient practical accuracy 



h'-h = il'-d^±(^l- YCOsSo [±(i^2-A)T(4-4)]; (5) 



and the special forms (4) in a correspondingly simplified manner. 



5. Correction for curvature. — When the declination of the objects observed is 

 considerable, the difference of declination found by (3) requires a correction for 

 the curvature of their paths, which may be obtained in the following manner : — 



The value of d given by (2) corresponds to the distance from the centre of 

 the square of the point of intersection of a great circle joining the places of 

 the star's entry and exit, with the diagonal joining the north and south angles 

 of the square. The actual path of the star, being a small circle, cuts the diag- 

 onal at a small distance, which we will call x, from the great circle. From the 

 triangle formed by the pole, the star when on the bar, and the intersection of 

 the great circle and diagonal, we have 



cot (8 + cc) = cos "Y- (4 — 4) cot 8. 



Developing this in the known way we find 



^ ^^V< sm 2 8 + 1 ^^^, sm4 8 + 



which is subtractive from the uncorrected value of d in (2). For the comet we 

 have a precisely similar equation with accents. Hence, neglecting the 4th and 

 higher powers, and employing the mean declination 8^, we have, nearly, putting 

 sines for tangents, 



2 sin^ V- {t'l — «'i) — 2 sin' y {U_ — t^) 



S'-8 = {d'- d) - 



VOL. XI. 



sin 1" 



cos So sin 8, 



'o> 



21 



