166 ON THE SQUARE BAR MICROMETER. 



I 



A (a— a) = j5-^g (± f =F §J (J tan 2 £ sin 2^ + tan £ sin q tan 8) 



2 



15^ {d'-d) tan £ sin? tan 8. 



(18) 



To reduce the declination correction we note that the first term of (11) is 

 equivalent to one half the difference between the changes in refraction in right 

 ascension for the comet and star in the interval between entry and exit, reduced 

 to arc of a great circle; while the second term is the difference of the refrac- 

 tions in declination, at the middle points of their chords. So that, in a similar 

 way to that before used for (10), we transform the first two terms of (11) into 



15cosS k ( T ^±^)^-k(S' 



t 2 — t^ d fx. /cs/ ^ x dv 



dd' 



With regard to the third term of (11), an examination shows that its nu- 

 merical value must be very small, and that it may be expressed very nearly by 





15 sin S sin 1" [± %^ =? ^^1 A 8, 



where A 8 is equal to the refraction in declination for a star at the centre of 

 the field, and is expressed by k tan £ cos q ; for which we may substitute without 



sensible error, k cosec 1" tan £ cos q, as is manifest from the way k is derived 



from k. Therefore the third term of (11) may be written 



lScosS^i*^ 1 * V^]" tanS - 



Adding this to the above value of the first two terms of (11), and remem 

 bering that, from (5), 



15cosS[±^ T ^] = ±iVi-(S'-S), 



we obtain 



A(S'-S)=[ ± f' T f- ( S'-S)] K (,tanS-^)- K (^-S)^ (19) 



Differentiate the first and third of (15) with respect to t, and the second with 



respect to 



<7(cos £) = — sin £ d £ = — cos <j> cos 8 sin t dt 

 </(sin£sin^) = cos <j> cos idt\ 

 <?(sin £ cos q) = — cos £ d 8. 



