TAHLK OF CONTENTS. x 



PAH 



CIIAITKR IV. Terin8 of the Second Order due to the Action of Saturn. 



1'reliiiiiiiary investigation of the orbit of Suturn . . .65 



Perturbations of Saturn and Uranus . . . . 63 



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



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



Perturbations depending on the product of the masses of Jupiter and 



Saturn . ...... 77 



CHAPTER V. Collection and Transformation of the preceding Perturba- 

 tions of Uranus. 



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



Secnlar variations ....... 80 



Auxiliary expressions on which the perturbations depend . . 81 



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



Portions of Uranus resulting from the preceding theory .- . 98 



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



CHAPTER VI. Reduction of the Observations of Uranus, and their Com- 

 parison with the preceding Theory. 

 Reduction of the ancient observations . . . . .106 



Their comparison with the provisional theory . . . .110 



Discussion of the modern observations . . . . .111 



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



Adopted positions of fundamental stare . . . . .113 



Discussion of corrections to reduce the different observations to a liomo- 

 geneous system . . . . . . .115 



Table of these corrections ..... 120 



Results of the observations from 1781 to 1830 .... 123 



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 Jahr- 

 buch 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 provi- 

 sional theory in terms of the corrections to the heliocentric co-ordinates 158 

 Expressions of the same quantities in terms of the corrections to the ele- 

 ments 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 



CHAPTER VIII. Completion and Arrangement of the Theory to fit it for 



Permanent Use. 

 Correction of the coefficients of the long inequality between Uranus and 



Neptune for the terms of the second order . . . .178 



Concluded element*, or elements IV of Uranus . . . .181 



Long- period and secular perturbations of the elements . . .182 



Table of these perturbations from A.D. 1000 until A D 2200 . . 184 



Mean elements of Uranus .... .184 



Expressions for the concluded theory of Uranus . . .185 



