vi TABLE OF CON TENTS. 



CHAPTER IV. 



TERMS OF THE SECOND ORDER DUE TO THE ACTION OF SATURN. 



PAOB 



Preliminary investigation of the orbit of Saturn . . 65 



Perturbations of Saturn and Uranus . . .68 



Formation of the expressions for the terms of the second order . . .69 



Perturbations depending on the square of the mass of Saturn . . .76 



Perturbations depending on the product of the masses of Jupiter and Saturn . 77 



CHAPTER- V. 



COLLECTION AND TRANSFORMATION OF THE PRECEDING PERTURBATIONS OF URANUS. 



Terms independent of the position of the disturbing planet . . 79 

 Secular variations 



Auxiliary expressions on which the perturbations depend . . . .81 



Reduced expressions for the latitude of Uranus . . ... 93 



Positions of Uranus resulting from the preceding theory ..... 98 



Elements III of Uranus . . . . . . . .99 



CHAPTER VI. 



REDUCTION OF THE OBSERVATIONS OF URANUS, AND THEIR COMPARISON WITH THE 



PRECEDING THEORY. 



Reduction of the ancient observations ... . 106 



Their comparison with the provisional theory . . .110 



Discussion of the modern observations 



Reduction of the results to a uniform system . . . . . .111 



Adopted positions of fundamental, stars .... .113 



Discussion of corrections to reduce the different observations to a homogeneous system . 115 



Table of these corrections 



Results of the observations from 1781 to 1830 . . . . . .122 



Observations from 1830 to 1872 ... . .126 



Table to convert errors of right ascension and declination of Uranus into errors of longitude 



and latitude .... ... 127 



Tabular summary of results of observations, 1830 to 1872 . . 131 



Corrections to be applied to the positions of Uranus in, the Berlin Jahrbnch and the Nautical 



Almanac to reduce them to positions from the provisional' theory . . . 151 



CHAPTER. VII. 



FORMATION AND SOLUTION OF THE EQUATIONS OF CONDITION RESULTING FROM THE 



PRECEDING COMPARISONS. 



Expressions of the observed corrections to the longitudes of the provisional theory in terms 



of the corrections to the heliocentric co-ordinates ..... 158 



Expressions of the same quantities in terms of the corrections to the elements of Uranus and 



the mass of Neptune . . . . . . . . .161 



Table to express errors of heliocentric co-ordinates as errors of elements . . . . . 162 



Discussions and solutions of the equations thus formed ..... 165 



Concluded corrections to the elements of longitude . . . . . .173 



Corrections to the inclination and node of Uranus 173 



