450 KRETZ. 



For, the photographs behig taken at three different dates, namely 

 1870.3 (Plates I-III), 1875.4 (Plates IV-VIII), and 1876.4 

 (Plates IX-XIV), the relative positions of the standards will not 

 be the same for them all, due to the cause mentioned. As the 

 epoch of reduction is to be 1875, we must not apply any cor- 

 rection for precession or nutation, otherwise what has been said 

 above regarding this matter would not apply. We can, then, 

 construct Table VIII. This table gives the Right Ascensions 

 and Proper Motions in Right Ascension, and the Declinations 

 and Proper Motions in Declination, for 1875. Then follow six 

 columns showing the seconds of the Right Ascensions and 

 Declinations with the proper motions applied to reduce them to 

 the three dates mentioned ; and in the last two columns will be 

 found the Weights in Right Ascension and in Declination, respec- 

 tively, of each star for 1870 and 1875. The same weights were 

 used for 1875 and 1876. They were calculated by reversing 

 the process for obtaining the probable errors explained in Part I, 

 Sect. II, " FoRMUL/E FOR Adjustment." From these quantities 

 we can then obtain Ja and Jo with their weights for each, 

 "standard" on every plate. 



Now let us compute for each standard on each plate the 

 quantities n_^ and w as follows : From Table V we obtain x and 

 J for each star. Form the products X sec 0^ = x sec d^ . 5 2". 87 

 and V=j/ . 5 2''. 87. Correct these for refraction by means of 

 Table VII and the corresponding formulae, and apply the trans- 

 formation corrections from Table VI. Subtract from the sums 

 thus obtained the corresponding J« or Jd : the differences will be 

 n sec 3. and n . It is then evident that ;/ and n should be 



X y X y 



zero, if it were not for errors of observation, and for errors in 

 the assumed constants. We are to find the values of the latter. 

 Let us introduce the notation : 



/ = the correction to the assumed scale-value, so that the 



true value is 5 2". 8 7 (i + p) ] 

 r = the orientation correction, or small angle through which 

 the axes must be rotated in the direction of decreasing 

 position angles ; 



(110) 



