of the Butherfurd Photographs. 285 



ments in the paper in reading over the proofs. Finally I desire 

 to express my thanks to Professor Rees, Director of the Ob- 

 servatoiy, for the interest he has shown in rny work, and for 

 securing its publication. 



Note on Refraction Formulas for Photographic Plates. 



Formulas for correcting the measured rectangular coordinates 

 of a star upon a photographic plate for refraction, may be easily 

 derived from the well known general formulas of Bessel. On page 

 166, Yol. 1. of his "Astronomische TJntersuchungen ' he gives the 

 following corrections to the differences of right ascension and 

 declination : 



A (a / — a)=s- k [tan 2 £ cos {p — q) sin q — tan C sin q tan S Q cos p 

 + sin p~\ sec ^ 



A ( (J ' — d)=s- k [ tan 2 C cos (p — q) cos q -\- tan f sin q tan <f sin p 

 + cos p] 



Substituting 



X= s sin p 

 Y—s cosp 

 G = tan '(, sin q 

 H= tan C cos q 



we obtain 



A (a' —a) =kX sec rf (J-f fl" 2 ) + k Y (G — tan J ) H sec fl 

 A (d'_ J) =fcX(G-h tan (5) fl" +&F(J+C? 2 ) 



These formulas become identical with those of Professor 

 Jacoby when we change h into /3 in order to allow for the in- 

 creased refrangibility of photographic rays. 



One point in the above deduction deserves mention ; the quanti- 

 ties <$ , etc., were intended by Bessel to be the means of corre- 

 sponding quantities for the two stars whose distance along the 

 arc of the great circle joining them has been measured. We have 

 treated them as though they referred to one end of that arc ; 

 however, this merely amounts to neglecting terms in the second 

 and higher powers of s, which may be done for most photographic 

 plates. 



If we omit the middle term in each bracket in Bessel 's formulas 

 we obtain the formulas given by Professor Turner; the omis- 

 sion of these terms, as has been repeatedly pointed out, corre- 



Annals N. Y. Acad. Sci., X., June, 1898—18. 



