THE OKBIT OF NEPTUNE. 3 



which would result from arbitrary changes in the elliptic orbit, there is left a 

 small residual, mostly developed in powers of the time, and only amounting to a 

 few seconds in a century, which alone is retained. 



With the new perturbations, and revised normal places of Neptune, Mr. Walker 

 obtained the following final set of elements, which he denominated Elliptic Ele- 

 ments II.: 



7i = 47 12' 6".50 

 1= 130 4 20 .81 

 = 328 32 44 .20 

 1 46 58 .97. 

 e =.00871946. 

 It 21".55448. 

 Epoch, Jan.l, 1847. 



From these elements and perturbations we have a continuous ephemeris of 

 Neptune since the time of its optical discovery. From 1846 till 1851 inclusive, 

 this ephemeris is found in the Appendix to Vol. II of the Smithsonian Contri- 

 butions to Knowledge ; for 1852, in Vol. Ill of the same series, and also in the 

 Astronomical Journal; and for subsequent years, in the American Ephemeris and 

 Nautical Almanac. 



All the modern observations on which these elements were founded were made 

 in the ye t ars 1846-47, while the planet was moving over an arc of only two and 

 a half degrees. Considering that the complete determination of the elements 

 requires, effectively, four observed longitudes, all in different parts of the orbit, 

 and that three of these positions are included in a space of less than three degrees, 

 it must be admitted that an accurate determination of the elements was, under 

 the circumstances, impossible, owing to the imperfections of the observations. As 

 already remarked, the errors of observation would be multiplied several hundred 

 times in the elements. Hence, with the best possible observations, the elements 

 would be uncertain by one or more minutes. But the observations themselves 

 were mainly differential ones; and it is very doubtful whether the positions of the 

 stars of comparison were as well determined as the position of the planet itself 

 could be determined by a series of good meridian observations. 



3. The theory of Neptune was next taken up by Professor Kowalski, of the 

 University of Kasan. His work was published under the title of " Eeclierc1t.es 

 sur les mouvements de Neptune, suivSes des tables de cette planete, Kasan, 1855." 

 The long-period perturbations of the elements are here developed, in their general 

 form, as perturbations of the co-ordinates. There are, therefore, a much larger 

 number of terms having large coefficients in this theory than in that of Professor 

 Peirce. 



Owing to this change in the form of the perturbations, the two theories cannot 

 be directly compared. But the ephemerides resulting from each theory can be 

 compared directly with observation, and corrections of the elements thence ob- 

 tained. It is thus found that the elements in question require, approximately, 

 the following corrections in order that the ephemerides may agree with obser- 

 vations to 1863 : 



