170 THEORBITOFURANUS. 



A simple glance at the residuals hi shows that they are much greater than the 

 purely accidental residuals resulting from the theory of least squares. We may 

 divide the possible causes of these systematic errors into three classes. 



1. Sijdrmatic Errors of Observaiion. — These may result from deviation of the 

 line of collimation of the instrument from a true great circle, or from any pecu- 

 liarity of the observer which leads to his registering the transit of Uranus earhcr 

 or later than that of a fixed star. If we compare the corrections derived from the 

 work of different observatories as given in the last chapter, we shall find frequent 

 cases not only of systematic difierences between the results of different observa- 

 tories, but between those of the same observatory in two successive years. An 

 instance which particularly attracted my attention on first preparing the com- 

 parisons of theory and observation is that of the Greenwich observations for 1831, 

 which, as compared with observations at the same observatory during the years 

 preceding and following, seem to be affected Avith some constant error in R. A. of 

 about 2". I find that this discrepancy can be attributed only to the original 

 observations. 



2. Errors in the Theory comjytred. — These may arise from errors in the preceding 

 theoretical computations, from tlie omission of the terms of the second order pro- 

 duced by Neptune, from the adoption of an erroneous mass of Saturn, or from the 

 attraction of an unknown planet. "With regard to the probability of these different 

 sources of error it may be remarked that errors of computation seem possible only 

 in the terms of the second order, that the mass of Saturn is taken from the 

 exhaustive discussion of the Saturnian system by Bessel, in which an error sufficient 

 to influence the theory of Uranus seems highly improbable, and that a trans- 

 Neptunian planet large enough to produce a sensible deviation of the orbit of 

 Uranus from an ellipse in the course of a century would be too large to have escaped 

 detection. The choice of the elliptic elements of Uranus and Neptune is such 

 that the terms of the second order, due to the action of Neptune, can scarcely 

 become sensible witliin a century of the epoch. 



3. Errors in the various Reductions hij which Theory and Ohserraiion are com- 

 pared. — In the method adopted for comparing theory and observation a number of 

 small uncertainties incident to the imperfections of the older data of reduction 

 necessarily creep in. In the early observations the imperfections arise principally 

 from the uncertainty of the instrumental corrections, and the errors in the adopted 

 positions of the fundamental stars, and indeed in nearly all the data of reduction. 

 In the late years they arise principally from the great magnitude of the correction 

 to Bouvard's tables, and the consequent rapid change of the corrections to the 

 geocentric ephemerides, which make the determination of the corrections Al and 

 U from theory and observation soraeAvhat uncertain. Errors from this source will 

 necessarily be in part of a systematic character, and, in view of their possibihty, 

 I regret not having been able to completely re-reduce all the observations before 

 1840, and to compare all since directly with ephemerides computed from the 

 provisional theory. In order, however, to test the question whether they are 

 sensible, I have prepared an ephemeris from the provisional theory for the three 

 recent oppositions of 1861-2, 1862-3, and 1872, and compared it directly with 



