TWO DEGREES OP THE NORTH POLE. 29 
In determining the right ascensions and polar distances of the un- 
known stars the process employed was the reverse of that used for 
the standard stars and similar to that of Publication i, Sec. VIII, in 
each detail with one exception in the case of annual aberration. Here 
a second step was necessary, since the formulas given on p. 32, Publi- 
cation I, have in the second member the quantities «! and r, which 
are supposed to be free from the effects of aberration, whereas these 
quantities as derived from the rectangular coordinates have been 
freed from the effects of refraction only. The following differential 
forms derived by Jacoby for a similar case give the necessary correc- 
tions. (Columbia Contributions, no. 21, p. 25.) 
a — «! = 
2(7!- — 7Z-, )(« — «,) 
7z~7Ti= — n{a — a,)2sin^i". 
In combining the results from the eight plates into a single prelimi- 
nary catalogue, each plate was given weight unity. The means of 
the separate results from the four 90° plates were treated as if they 
also were obtained from a single plate and were given weight unity 
in the final combination. The right ascensions and polar distances 
derived from each of the four 90° plates had previously been reduced 
to the mean standard of the four plates by the method described in 
Publication i, Sec. X, pp. 68-72. Hereafter when reference is made 
to the 90° plate the mean result from the four plates is to be under- 
stood. 
Table VI contains the preliminary catalogue. The first column 
contains the catalogue number, the second the number of plates upon 
which each star appears, the third the mean right ascension from all 
the plates, and the remaining nine columns the residuals in the sense 
plate minus mean. The corresponding values in polar distance are 
given on the opposite pages. 
