-A CATALOGUE OF STANDARD STARS. 541 
The secular variation in declination is 
— 0/.1950 tg Ai sin a” + 0.4481 cos (a! + 91° 6^ — 0".01944 Ai sin a. 
This, however, it was not necessary to compute, as existing catalogues furnished 
by interpolation sufficiently accurate values for our immediate purpose, which was to 
test the computed right-ascensions and declinations. 
Table II. contains the comparison of the theoretical and observed positions; the 
observed positions having been corrected so as to correspond to the same proper 
motions as had been previously computed. 
In certain cases it was necessary to form the “observed ” positions by combining the 
results of several years’ observations. For Henderson's and Le Verriers AR. this 
was done by means of the annual variations derived from our computed places: for 
the Greenwich observations, by the annual variations given in the separate volumes. 
In either case care was taken to have the * mean epoch" adopted correspond very 
precisely to the mean of the times of observation. 
The computed right-ascension and declination of a star for any time depend strictly . 
each upon four elements (besides precession); upon the AR. and proper motion in 
AR. for some fixed epoch, and also the Decl. and P. M. in Decl. for the same epoch. 
Let a, ð, be the star's right-ascension and declination at the time ¢ + 1855, referred 
to the co-ordinate planes of 1855; e and A being the right-ascension and declination 
for the same time, referred to its own co-ordinate-planes. 
Then, if we vary the assumed AR. for 1855 by » and the assumed proper motion 
in AR. forn years by vg: the assumed Dec. for 1855 by 1, and the assumed proper 
motion in Dec. for n years by /; we shall have 
4a s E, s=: 4. 
[In our calculations we shall make n= 30.] 
Again (Fundamenta Astronomie, p. 304 (15) ). 
48 sin 8 sin (a! F i! — 2’) 
Ge t / D eo Y d E 
4 u' = A a (cos 6 + sin 8 tg ' cos (o! + à 2b E cos à' : 
4 ô = — 4a sin 6 sin (a! + — 2) + A. cos Ai (cos 6 + sin 8 tg 9' cos (a! + i’ — 2^). 
The notation here used is that of the Fundamenta. 
Let | 
sin d sin (a + à — 2) = m sin E, 
cos d cos Ai + sin 6 sin Ai cos Loi + 4! — z^) = m cos E, 
