238 Prsesepe Group; Measurement and Reduction 



and Ad, eliminating the errors of observation as far as possible 

 by means of a least-square solution. 



I. Transformation Corrections. 



An astronomical photograph may be regarded as a central pro- 

 jection of a portion of the celestial sphere upon a tangent plane. 

 The point on the plate which "corresponds to the point of tan- 

 gency is the foot of the perpendicular let fall from the optical centre 

 of the object glass upon the plane of the plate. The rigorous re- 

 lations between the rectilinear coordinates referred to this point 

 as origin, and the right ascension and declination of a star, were 

 given in simple form by Professor Turner in Vol. XVI, of " Ob- 

 servatorj^," page 374. Previous to this, however, Ball and Ram- 

 baut gave these relations in the form of series in their paper " On 

 the Relative Positions of 223 stars in % Persei," Transactions of 

 the Royal Irish Academy, Vol. XXX, page 241. In our notation 

 these formulas would be 



Aa — Xsec d = (Xsecrf ) Ftan^ — \ (Xsecc? ) 3 + (Xsec^ ) Ftan 2 ^ 



AS— Y = — K^sec(J ) 2 sin2rf — JF 3 — H^secJ ) 2 F 



I 

 The elegance of these formulas lies in the fact that the coefficients 



of the powers and products of X and Y, are functions of 3 Q only, 

 and are therefore constant for a plate, or indeed for an entire 

 zone. For most plates these series are sufficiently accurate, but 

 when the declination or the measured coordinates are large they 

 fail ; in such cases we do not need to resort to the rigorous for- 

 mulas but we have [merely to extend the series to higher terms, 

 as was done by Professor Jacoby in a review of a paper by Pro- 

 fessor Donner, in the Vierteljahrschrift for 1895, page 114. In 

 the same place, formulas are also given in which Jaand Ad appear 

 in the second members, instead of X and Fas above. Omitting 

 terms of higher degree than the third, which is permissible for the 

 Prsesepe plates, these formulas may be written 



Aa — Xsec^ = Aa ■ Ad- tan<5 — \ Acfi ( i — f sin 2 6 ) 

 A6 — Y= — \ Aa* ■ sin 2 S — \ Aa 2 . AS • cos 2cS — J A6* 



The use of these formulas presupposes a knowledge of the ap- 

 proximate values of Aa and A8 for each star. They possess two 

 points of advantage over the inverse forms : first, there is one 

 term less in the expression for Aa — X sec d Q • and second, they 

 give slightly more accurate values for the corrections as they do 



