ON THE SQUARE BAR MICROMETKR. 



1C1 



which is the uncorrected dine 



This general expr 

 indicated below. 



of declination of the comet 



1 the star. 



assumes the following forms for the special cases 



Comet and star both north 

 Comet and star both south 

 Comet north, star south 

 Comet south, star north 



8 



S 





+9 



9 + ¥ [(' 



L ) cos 8' 

 ) cos 8' 

 .) cos 8' + 

 ) cos 8' + 



f x ) cos 8] 

 /, ) cos 8] 

 /^cosS] 

 fi) cos 8] 



(4) 



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

 write equation (3) with sufficient practical accuracy 



8' 



t 



8 



d' 



d 



15 



2 =F2"" Y cosS o [±(rf' a — *\) *(fe— *0]j 



(5) 



and the special forms (4) 



pondingly 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 

 of the square. 



the north and south angles 

 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 + x) = cos -V 6 - ft - h) cot 8. 



Developing this in the known way we find 



x 



tan 2 V (* 



2 



«.) - 



sin 1" 



an 28 + * -•-- '» — sin 4 8 + . . . 



2 



sin 1 



which 



ubtractive from the uncorrected value of d in (2). For the comet 



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



hi a 



<_5 



her powers, and employing the mean declination 8 



have 



•ly, puttin 





sines for tansrents 



8' 



8 



(<*' - d) 



2 sin 2 V- (X* 



t\) 



2 sin 2 -V- (h 



h) 



sin 1" 



cos 8n sin 8 



t) 



0) 



VOL. XI. 



21 



