COMPARISON OF CATALOGUES OF STARS. 399 



Both of these methods of comparing different catalogues, are open to the objection 

 that the coordinates of any given star, when referred in this way to a normal system, 

 may differ by a small amount from the values of the coordinates derived by a direct 

 reduction of the instrumental constants from the fundamental stars of the system. I 

 estimate the uncertainty arising in this way to be .015 s in Aa and 0.20" in AS. In the 

 reduction of the zone stars observed at the Harvard College Observatory, the instru- 

 mental constants from 1871 to 1876 have been recomputed from the positions given 

 in Publication XIV. When, therefore, the final catalogue is completed, the data will 

 be at hand for a definite determination of the amount of this uncertainty. But since 

 this reduction is wholly impracticable in the case of catalogues already published 

 finally, some method of reducing different catalogues to a homogeneous system is a 

 necessity in the determination of proper motions. 



Admitting this necessity, that method is to be preferred which will reduce the 

 residuals in Aa and AS to a minimum, whatever the order or the limits of the groitji* info 

 which they may be divided. 



The method pursued in this investigation is the following : — 



(rt) The residuals are first arranged in the order of declination. The mean values 

 for any group represent nearly the corrections for the mean declination of that group 

 and for twelve hours of right ascension. 



(I) The residuals for each group minus the mean value for that group, are then 

 arranged in the order of right ascension, and a graphic curve is drawn through the 

 points representing these values. Each curve will give a series of values for the even 

 hours of right ascension, which are arranged in vertical columns, the horizontal argu- 

 ment beina: the mean declination. It will be seen that the values in the vertical 

 columns are derived from the same curve, while those in the horizontal columns are 

 derived from different curves. In order to connect the different groups in declination, 

 a curve is drawn through the points represented by the residuals in the horizontal 

 columns. This will, of course, disturb to a slight extent the values already found in 

 the vertical columns, but they can be rectified in the way already described. A 

 second approximation will ordinarily give smooth curves for both the vertical and the 

 horizontal arguments. In this investigation, however, I have preferred to retain the 

 values derived directly from the original curves without any attempt to smooth them. 

 (c) On account of the limited number of the residuals which usually compose the 

 groups arranged in the order of declination, there is danger of introducing systematic 

 errors in drawing the graphic curves. Hence after the sum of the two residuals 

 already found has been subtracted from the original residuals, the new values are 



