132 Royal Astronomical Society. 



In 1712 +9§7 In 1804 +24-2 



1750 —47-6 1840 -66-6 



Tliese are then converted into corresponding errors of mean longi- 

 tude, wliich the author finds more convenient. 



Then, formulae are investigated for the effects of small corrections 

 of the elements of the orbit of Uranus, and for the perturbations of 

 mean longitude produced by a disturbing planet, expressed in the 

 notation of Pontecoulant. These are expanded as far as the second 

 order of eccentricities (involving only the first power of the eccen- 

 tricity of the unknown planet), and the whole is reduced to numbers, 

 with no symbols remaining, except for functions of the corrections 

 of the elements of Uranus, and functions of the epoch, longitude of 

 perihelion, eccentricity and mass, of the disturbing planet. All the 

 numerical quantities are computed on the supposition that the mean 

 distance is double that of Uranus. Any one of these expressions, 

 adapted to a certain time, being made equal to the error in the ta- 

 bular place of Uranus for the same time, furnishes an equation of 

 condition. 



These equations of condition are treated by the method of least 

 squares ; and the successive steps of elimination are given. The 

 author considers that the modern observations are scarcely sufficient 

 to give the eccentricity and longitude of perihelion of the disturbing 

 planet ; but when the ancient observations (always omitting that of 

 1690 as uncertain) are combined, there are ample means for deter- 

 mining these elements. The equations, after the elimination had 

 proceeded to a certain degree, were solved by successive substitu- 

 tion. The results thus obtained were — 



Hypothesis I. 

 Assumed mean distance = 2 X that of Uranus. 



Mean longitude, October 6, 1846 325° 7' 



Longitude of perihelion 315 57 



Eccentricity of the orbit 0-16103 



Mass (that of the sun being 1) 0'0001656 



which were communicated to the Astronomer Royal in October 1845. 



The author then states that he made a second investigation, on 



the supj)osition that the mean distance of the disturbing planet 



= mean distance of Uranus x The process, with very little 



0'515 



difference, is the same as that for the former assumption of mean di- 

 stance. The formulae, the equations, &c., are given in the same man- 

 ner as before. The elements obtained thus are as follows : — 

 Hypothesis II, 

 Assumed mean distance = 1'942 X that of Uranus. 



Mean longitude, October 6, 1846 323° 2' 



Longitude of perihelion 299 11 



Eccentricity of the orbit 0-12 0615 



Mass (that of the sun being 1) 000 015003 



The corrections to the elements of the orbit of Uranus are inves- 

 tigated on both hypotheses. Then on substituting the effects of the 



