INTRODUCTION 
XXXIII 
Probably the most fruitful source of mistakes will be found to have arisen in the 
process of assembling the data arising from the computations in order to form 
the Catalogue as printed. In preparation for the printer’s copy of the Catalogue the 
needed material from various sets of computations had been assembled on cards 
arranged in the order of right-ascension. The solution of the normal equations for 
each star had given corrections applicable to the assumed positions and annual 
variations for 1875. Another process gave those to be applied to the assumed position 
and annual variation for 1900. The corrected positions and annual variations for 
both dates were copied on the catalogue-cards, together with other data relating 
to precession, proper-motion, etc. Then the corrected position for 1900 was rigor¬ 
ously reduced to that of 1875, with the elements provided for that purpose in this 
Catalogue. This served for the detection of several small errors in the original 
ephemerides, or in forming the corrected places. This process not only resulted 
in a satisfactory verification of the relative positions for 1875 an d I 9co, but at the 
same time it afforded a good check on the annual variations and a check against 
gross errors in the secular variation; but it was, of course, ineffective as to errors 
common to the data for both 1875 and 1900. Various tests and checks have been 
employed to guard against any very important error of this kind, like errors of 
i' or I s . 
METHOD OF CORRECTING THE CATALOGUE-POSITIONS AND MOTIONS 
BY MEANS OF ADDITIONAL OBSERVATIONS. 
The probable errors already described may be made to serve a useful function 
additional to that of affording a criterion of the precision attained. From these 
probable errors may be recovered normal equations which, though not the same, 
are sufficiently equivalent to those from which the catalogue-positions and motions 
resulted. As an example, take No. 1, Br 3208, or 33 Piscium, in right-ascension. 
We have: e 0 = ± '.'05, the probable error of right-ascension at the mean epoch, 
1870.4; = ± '.'27, the probable error of centennial /x. If now we put for the prob¬ 
able error of the unit of weight, ± '.'30, and if we make a quarter of a century the 
unit of time from the mean date, we shall have as normal equations for correction of 
the Catalogue for 1870.4: 
30.0 Aa 0 4 - .0 A/x 0 = *000 1 ^ 
o Aa 0 +19.8 A/x 0 = .000 J 
Aa 0 is the required correction of the right-ascension for 1870.4, the catalogue mean 
epoch, and its coefficient is the weight corresponding to e 0 . A/x 0 is the required 
correction of the catalogue-value of 25 [x, and its coefficient is the weight corre¬ 
sponding to Now suppose that a determination of the right-ascension of this 
4 
star is to be made in 1909.9, and that the weight of that determination is to be 
6.0. Suppose this determination corrects the catalogue right-ascension by + *030. 
Then we shall have as a conditional equation from this determination: 
Aa 0 + T 9 ? 9 : 9 — I^ 7 ° i 4 Ap, 0 == -K030, or Aa 0 -f 1.58 A/a 0 = +^030 
2 5 
