164 ON THE SQUARE BAE MICROMETER. 



the diagonal, it will be equal in length, nearly, to a line drawn perpendicular 

 to the diagonal at its intersection with the apparent path. Hence the distance 

 of the middle point of the apparent path from the centre of the square will be 



q 15 

 ± 2 T Y ^'OS (S) [(^2— ^i) — (Aj a — Ai a)] ; 



and, since the distance between the apparent and true paths is ^^(AaS + AiS), we 

 have, putting D = the declination of the centre, 



S - Z> = ± I T T cos (8) [(4 — h) - ( Ao a — Ai a)] - 1 ( A, 8 + Aj 8). 

 Putting (S) = 8 + |^(A2 8 + AiS) and developing, neglecting squares of the refractions, 



8 — Z> = ± f-' T Y cos ^ [(^2 ~ 'i) ~ (^2a — Aia)] - +( A28 + Ai8)[1t ^^ sin 1" sin 8 (4-^1)] . 



For the comet a similar equation obtains, with accents. Taking the difference 

 and remembering the relation (3) we get (8'— 8) — (d' — d), or the correction 

 for refraction in declination, 



A (8' - 8) = ^^ cos 80 [± (A2a' - Aia') ^ {^^a - A^a)] - \ [( AjS' + A18') - ( AgS + AjS)] 



+ -V- sin 8 sin 1" [ ± ^^' ( A28' + A18') t ^^ (A2S + AjS)] . (11) 



It remains to reduce (10) and (11) to a form convenient for direct com- 

 putation. 



Assuming the refraction to vary at a uniform rate over the field of view, the 

 first term of (10) is manifestly the difference of the refractions in right ascen- 

 sion for the star and the comet, at the middle points of their chords; and the 

 second term is one half the difference between the changes in the refraction 

 in declination for the comet and star, during the interval between their respec- 

 tive entries and exits, reduced to time at the given declination. Hence, if I, and 

 ^' be the true zenith distances of the star and comet, and q and §-' be their 

 parallactic angles, we have (Chauvenet's Astronomy, Vol. II., equation 234), 



wa a- a ^ wa 'J-a '\—i.^^^k^^ tan r sing' 



^(Aaa + Aja) — l(A2a +Aia) — A xido^Z ~ '^ 15cosS'' 



i5^-*[±(A2S'-A,8')T(A2S-A,8)]= (12) 



2(.ogg • [q: {i'^ — i'i) ^'tan^' cosg'' ± (4~^i)^ tan^ cosj']. 



