ON THE SQUAKE BAR MICKOMETER. 165 



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 f sin q 



H-= cos 5 ' I' = tan ^ cos 2-. (13) 



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

 general principle that, when ?« is a function of x, the change of tt, corresponding 

 to a small finite change oo — x, is ^^ (^ — x) ), 



A(a'-a) = -,V'c(S'-S)^-Ksec8„[±^:F^],^. (14) 



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

 relations, 



cos ^ = sin <^ sin 8 + cos ^ cos 8 cos t ; 

 sin 4 cos 5' = sin (^ cos 8 — cos <^ sin 8 cos ^; (15) 



sin ^ sin (^ = cos ^ sin t. 



Differentiating these with respect to 8, 



d (cos ^) = — sin ^ cZ ^ = sin ^ cos qdZ; 

 d (sin ^ cos §') = — cos tdh; 

 d (sin ^ sin 5-) = ; 



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



^ = — [1^ tan^ ^ sin 2 5' — tan t, sin q tan 8] sec S. (16) 



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



d (cos ^) = — sin ^ (? ^ = — sin ^ sin g- cos 8 (/ ^ ; 

 c? (sin ^ cos <^) = sin ^ sin §' sin 8 J ^ ; 



and substituting in ^ from (13), we get 



^ = i tan^ ^ sin 2 5" cos 8 + tan I, sin q 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, 



