446 KRETZ. 



A very simple way of verifying the above formulae is the fol- 

 lowing ^: Bessel ^ gives corrections for clearing apparent differ- 

 ences in right ascension and declination, obtained by micromet- 

 ric observations, from the effects of refraction in the form : 



A («^ — a) = SK [tan^ C^ cos [p — c/) sin (/ 



— tan i^Q sin ^ tan 6^ cos p -}" ^in /] sec Jg 



A (J^ — J) = jK [tan2 ^g cos (/ — ^) cos g 

 -\- tan Cg sin ^ tan Jg sin J> -[- cos /] 



where s and / are the measured distance and position angle, 

 ^Q and (J are the true zenith distance and parallactic angle at the 

 middle pomt between .the two stars, whose coordinates are 

 («, o) and (a', o'), and o,, is the declination of that point. Now 

 (Chauvenet, Astronomy, Vol. II, p. 453) 



K tan^ Cq^=z b — a 



where, r being the refraction, 



sin Co 1 I 



b = 



sin (Cg — r) I — r cot Co i — ^^ 



rt'Co — ^ dr dr dk' 



l—~ijr \ — k' sec2 Co — -T^ tan Co 

 "So "So 



placing cos r ^ i, sin r = r and remembering that r =: /^' tan (Tg 

 (Chauvenet, Vol. I, p. 171), where r and k' are expressed in 

 parts of the radius. Expanding the expressions for a and b by 

 division, we easily obtain 



dk' 

 b - — a ^= K tan^ Cg = k^ tan ^ Cn ~t~ '~t?~ tan Cg -|- ' "> 



«So 



dk' 

 the succeeding terms being higher powers in k' and which 



can be neglected. For zenith distances less than 70° the term 



dk' 

 in may also be neglected. For inside that limit we have 



1 Cf. Schlesinger's " Praesepe," Note, p. 285, where the above method was first 

 pointed out 



2 Astronomische Untersuchungen, Vol. I, p. i66; or Chauvenet, Astronomy, 

 Vol. TI,.p. 458. 



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