130 THE ORBIT OF URANUS. 



The values of 1 and &b are printed in the last two columns of the table. The 

 formula for 56 might have contained the additional term 



= sn 



&o being the correction to the obliquity of the ecliptic adopted in the ephemeris 

 to reduce it to that employed in the provisional theory. This correction is, how- 

 ever, deferred until we come to form the equations of condition. 



From the values of Z and 56 thus obtained we are to find the mean values 

 during each group of observations. If these quantities varied uniformly, the 

 proper value would be that corresponding to the mean date of each group. But 

 the second differences are so large that this value would generally be in error by 

 one- or two-tenths of a second. Owing to the minuteness of this difference, it has 

 been considered that when the mean date was near the middle of a twenty-day 

 interval, the correction 5Z interpolated to that date without regard to second 

 differences would furnish a sufficient approximation to the required mean value of 

 1 during an interval of about 30 days. In other case the value of II was inter- 

 polated to 5-day intervals through the period of each group of observations, and the 

 mean value taken. 



During the years 1850-1863 the sun's longitude employed in the ephemeris 

 required a gradually increasing correction, amounting at the latter date to about 

 3". A small correction of which the maximum value is about 0",15 was applied 

 to cl to reduce it to the value it would have had if Hansen's tables had been 

 employed. 



The corrected mean values of il and e& thus obtained are given in the last two 

 , columns of the following table, being inclosed in brackets and printed immediately 

 above the values of Al and Ab derived from observation. 



I deem it proper to mention that the mechanical labor of constructing these tables 

 of comparisons, in the manner just described, was in great part performed by Dr. 

 C. L. F. Kampf, who was employed by the Smithsonian Institution to assist me in 

 the work. Before using it I subjected the whole of the work to a careful revision, 

 altering especially the relative weights of the corrected means in many cases. As 

 the assigned weights now stand, each set of results which are combined into a 

 single mean has its own unit of weight, which does not necessarily coincide with 

 that of any other set. The use of a uniform scale of weights through this series 

 of observations, and the assignment to every final mean of a weight equal to the 

 sum of the weights of the quantities whose mean was taken, would have led to 

 weights in many cases quite fictitious, owing to the obvious presence of systematic 

 errors in the results. For this reason I have made no further use of the weights 

 found in this table, and their lack of homogeneousness therefore does no harm. 



