396 COMPARISON OF CATALOGUES OF STARS. 



Avill be seen from the values of Aa under Auwers minus P from +70° to +85° that 

 the right ascensions of the Polar stars between h and G 1 ' are about .10 s too small, while 

 those from 6 h to 18 b are nearly .10 s too great. Hence, in the determination of the 

 instrumental constant n, the values derived from observations made between January 

 and April will be too large, while the values derived from observations made between 

 April and September will be too small. For stars beyond 10° south declination there- 

 fore, systematic errors in right ascension will be introduced through n, which will not 

 be corrected by a close adherence to the fundamental system. 



In the conversion from one fundamental system to another, two methods have 

 been employed : — 



(a) The Analytical Mctliod. 



If we have a series of residuals Aa, arranged in the order of right ascension, 

 which are strictly circular functions we may always have : — 



A« = a constant + m sin a + n cos a + m' sin2« + n' cos 2 a, etc. (1) 



If Aa is a function of the declination we may have approximately the additional 



equation : — 



\a = a constant + a sin 8 -+- b cos8. (2) 



In practice it will ordinarily be sufficient to assume : — 



Aa = a constant + m sin a + n cos a + a sin 8 + b cos 8. (3) 



In Volume X. of the Annals of the Observatory the values of Aa were first com- 

 puted from equation (1). The computed values having been subtracted from the 

 observed values, a new series of residuals was obtained which were assumed to be 

 functions of the declination. New values of Aa were then computed from equations 

 of the form (2). The total correction was then assumed to be the sum of these two 

 computed partial residuals. 



For the residuals in declination we may with less exactness assume : — 



A8 = a constant + c sin 8 + d cos 8. 



(b) The Graphical Method. 



If we assume any given unit in right ascension as a horizontal argument, and an 

 aliquot part of Aa as a vertical argument, it is obvious that points representing Aa 

 may be laid off which will bear a definite relation to a fixed horizontal line. If a 

 smooth curve is drawn through these points, values of Aa nearly representing obser- 

 vation may be derived for any right ascension by reading off the vertical co-ordinate 

 passing through the curve at this point. By a similar method of procedure we may 

 obtain the values of AS which represent the observations. 



