STARS IN COMA BERENICES. 455 



It now remains only to make use of the constants obtained 

 by the methods described above. This is a simple matter. 

 For we have to apply to the measured coordinates the correc- 

 tions 



-\- p ■ X %tz (Tg -(- r sec 6^- Y A^ k sec Jq to A^sec Jq 

 + p ■ Y —r-X + c to Y 



due to errors in the assumed constants, and 



-\- Afx ■ X sec (Jq + A^x y to X sec (\ 

 -\-MyX sec Jq + ^y y to Y- 



due to refraction. If then we add X sec o^ and Y, corrected by 

 the process explained above to a^ and o^ respectively, where a^ 

 and Og are the assumed coordinates of the center, we will obtain 

 for each star certain quantities, o.^ and o^, which are defined by 

 the equations 



« = ai-j- Ta 



where a and o are the right ascension and the declination re- 

 spectively of the given star, and T^ and T^ are the corresponding 

 transformation corrections, a^ and d^ may be called the " pro- 

 jected" coordinates of the star. Collecting all these operations 

 together, it is evident that we can write the following formulae : 



«i= (i +p -\- Vl/x)A'sec £^0 -}- (A4-}-rsec c^q) F-f (og-f- i sec fSg) 

 S^^{l+P + Ny)Y +{My^rcosi\)XsscS,.\-{6, + c), 



and 



a^^a^-[- Ta, (5 ^ (5j -[- Ts 



and when taken in connection with the preceding discussion, 

 it is evident that these equations express in mathematical lan- 

 guage all the steps necessary to transform the measured rec- 

 tangular coordinates on the plates, x and y, into the correspond- 

 ing right ascensions and declinations on the celestial sphere. 



(115) 



