Group of the Pleiades. 



253 



III. 



CORRECTION FOR REFRACTION, 



The calculation of the refraction corrections, in itself somewhat 

 laborious, can be much facilitated by the use of special formulae 

 and tables. The usual Besselian refraction formulae are not very 

 well adapted to the reduction of photographic measures. For the 

 photographic plate gives us the position angle at the central star, 

 or at the centre of the plate, while the Besselian formulae require 

 the use of the position angle at the point midway between the two 

 stars. A special correction is needed, then, in photographic work, 

 to obtain this latter position angle, which is itself of no further 

 interest. Moreover, several quantities which would be different for 

 every star when using the Besselian formulae, 

 can be made constant for the whole plate by 

 means of the formulae now to be deduced. 



Let us consider the spherical triangle formed 

 by the zenith (Z), the central star (#), and the 

 star under observation (#'). Let z and z r be 

 the zenith distances of the two stars, and let 

 I and V be the angles which S S' makes with 

 the vertical circles at S and S'. All the quan- 

 tities in the figure are apparent; i. e., their 

 values are supposed to be uncorrected for re- 

 fraction. Thte effect of refraction is to vary 

 z and z' while the angle at Z remains unchanged. We are to deter- 

 mine the consequent variations of s and I. Now we can expand 

 these variations in terms of the variations of z and z' by means of 

 the general formula : 



d$ d 2 <p d 2 <p d 2 <p , x 



d<p 



a<j> u, y) = — &x 



dx 



