ON THE SQUARE BAR MICROMETER. ]()£> 



Take a common value for £ and q corresponding to the centre of the square, 

 and Bessel's coefficient of differential refraction k for this point. Put also for 

 brevity, 



tan £ sin q 



l x ~~ ^sT~> * = tan£cos£ 



(13) 



Then the addition of (12) gives the value of (10), which may be written (on the 

 general principle that, when u is a function of x, the change of v, corresponding 



du 



to a small finite change x' — x, is ^- {x' — x) ), 



i 



A(a'-a)=- A K(S'-S)^-KSecS l) [±^ T ^-'']^. (14) 



The differential coefficients in (14) may be got from the known fund 



relations, 



cos £ = sin <f> sin 8 + cos <j> cos 8 cos / ; 

 sin £ cos q = sin <f> cos 8 — cos <j> sin 8 cos t ; (15) 



sin £ sin q = cos </> sin t. 



Differentiating these with respect to 



d (cos £) = — sin £ J £ = sin £ cos q d 8 ; 

 e? (sin £ cos <?) = — cos £ e? 8 ; 



d (sin £ sin <?) 

 and substituting in the value of -j^ obtained from (13) we get 



-ns == — [J tan 2 £ sin 2 ^ — tan £ sin $- tan 8] sec 8. (16) 



Differentiating the first and second of (15) with reference to t, 



* 



^(cos£) — — sin £^£ = — sin £ sin q cos Sdl; 

 d (sin £ cos q) = sin £ sin qsiwh dt; 



and substituting in ^ from (13), we get 



d 



dt a 



I tan 2 £ sin 2 ? cos 8 + tan £ sin ? sin 8. (17) 



Substituting (16) and (17) in (14), remembering that da = — dt and also the 

 relation (5), we derive, finally, the refraction correction in right ascension, 



