944 



SCIENCE 



[N. S. Vol. XXV. No. 650 



non is more extensively and clearly exhibited 

 than in Viola. Ezra Beainerd 



MiDDLEBimr, Vermont 



FORMULAS FOR THE COMPARISON OF ASTRONOM- 

 ICAL PHOTOGRAPHS 



The present paper contains formulas suit- 

 able for the direct comparison of rectangular 

 coordinates measured on different astronom- 

 ical negatives. The problem here involved 

 supplements what may be called the funda- 

 mental transformations in the reduction of 

 celestial photographs; viz., the calculation of 

 right-ascensions and declinations from rect- 

 angular coordinates, and rectangular coordi- 

 nates from right-ascensions and declinations. 

 The writer has published formulas for all 

 these transformations in * Tables for the Re- 

 duction of Astronomical Photographs,' Con- 

 trib. Obs. Col. Univ., No. 23. In these 

 formulas the problem is solved by expansion 

 into series, taking advantage of the fact that 

 the photographs under consideration cover but 

 a very small part of the sky, so that meas- 

 ured coordinates may be regarded as small 

 quantities. 



It is sometimes desirable to compare rect- 

 angular coordinates of the same stars meas- 

 ured on two different overlapping photo- 

 graphs without computing right ascensions 

 and declinations. For instance, Conner used 

 this method for strengthening his determina- 

 tion of plate-constants in his reduction of the 

 astrophotographic catalogue plates (' Sur le 

 Eattachement des cliches astrophotograph- 

 iques,' Acta Soc. Sci. Fenn., Tom XXI., 'So. 

 8). Another important application will 

 doubtless occur in the calculation of the solar 

 parallax from Eros observations by the diur- 

 nal method. 



For these reasons, the writer has thought 

 it desirable to expand directly the x and y of 

 a star on one plate in terms of its x and y on 

 a second plate. The resulting series, though 

 clumsy in appearance, are rapidly convergent, 

 and in most practical cases, convenient in use. 

 As here given, all terms to the fifth order, 

 inclusive, have been retained; but a table is 

 attached to the formulas showing the declina- 

 tion at which any term may be omitted in 



actual applications of the method. When this 

 declination is greater than Y5°, the table con- 

 tains the number 75+. Inasmuch as we 

 require a precision of 0".01 up to 75° declina- 

 tion, the table has been arranged so as to 

 exclude only terms less than 0".005. 



To obtain the desired expansions, we let: 

 ^1^ Hi, be the coordinates of a star on a cor- 

 rectly oriented plate whose center 

 corresponds to the right-ascension a^ 

 and declination 8j on the sky. 

 x^, 2/2, be the coordinates of the same star on 

 a second correctly oriented plate 

 whose center corresponds to the 

 right-ascension a^ and declination 8, 

 on the sky. 

 M^, M^ . . . N^, N^, ... be certain auxiliary 

 quantities, constant for aU stars on 

 a given pair of plates. 

 If we now put: 



da = ai — 02, dS = 5i — Sj, Sr=J(5i + 8j), 



we can express x^ 

 follows : 



2/j, in terms of z^, y^, as 



(I) 



iTj ^ a?! -j- Ml -f M^i -f- Af ,2/1 -\- MiX^ -f- M^^i 



+ U,y^ -\- MtX^ + MsX^y^ + M^jh', 



yj = yi + ^1 -f- N2X1 + N^yi + Ntx^' + NiW^i 



+ ^ny' + y7a!i'j/i + NsXiy^' + NtVi'. 



Expressions for the M'a and N'a, with the 

 table mentioned above, are given at the end of 

 the present paper. The writer is under spe- 

 cial obligations to Mr. G. W. HartweU, as- 

 sistant in mathematics, Columbia University, 

 for help in this part of the work. Demonstra- 

 tions are omitted here, because the formulas 

 can be verified satisfactorily by means of a 

 numerical example, such as the following 

 particularly unfavorable one. Let us assume 

 two plates and an imaginary star such that : 



ai = 0* 0' 0".00, 

 81 = 74° 0' O".0O, 

 <Fi = -1-3600", 

 y^ = + 3600", 



a, = 2* 0' 0".00, 

 8, = 75* 0' 0".00, 

 8 =74" 30' O".0O. 



The right ascension and declination of the 

 imaginary star, which we will call A and D, 

 can then be computed readily from x^, y^, 

 Oi, 8„ by means of our former series published 

 in Contrib. Obs. Col. Univ., No. 23. 



