THE ORBIT OF URANUS. 129 



We thus have, for the interval occupied bj' each group of observations, a mean 

 correction to the geocentric longitude and latitude of the planet given by obser- 

 vations, which are found in the ninth and tenth columns of the table, on the same 

 horizontal line with tlie mean corrections in right ascension and declination from 

 which they are derived. The next step is to obtain the corresponding corrections 

 given by the provisional ephemeris. 



This correction has been first obtained for every twentieth day of each of the 

 forty-two oppositions included in tlie table. The heliocentric longitude, latitude, 

 and radius vector were interpolated to the most convenient twenty-day intervals, 

 and compared with the corresponding co-ordinates in the lieliocentric ephemeris. 

 Tliis ephemeris was of course the one corresponding to that with which the 

 observations were compared, namely, the Berliner Jahrbuch for the years 1830-33, 

 and the Nautical Almanac for subsequent years. These comparisons are fully 

 given at the end of this chapter, and the resulting corrections to the printed 

 ephemeris are given in the proper columns of the table. 



These corrections to the heliocentric co-ordinates were then changed to corrections 

 of geocentric longitude and latitude by tlie following formuLx. Put 



r\ the projection of the planet's radius vector on the ecliptic ; 



o\ the projection of the planet's distance from the earth on the same plane ; 



p, this distance itself; 



?.,/3, the planet's heliocentric longitude and latitude ; 



X, the sun's geocentric longitude ; 



R, its radius vector; 



M, the modulus of the common logarithms; 



hi, hb, the corrections to the geocentric longitude and latitude ; 



8p, the correction to the common logarithm of the radius vector. 



Then 



gz = ':i 1 1 -f ^ cos (i — X) I hX 



=- sm [L — /) vf-^ — r» 



f ^ ^ MainV 



Rf 



-4- -7T sin (Z — X) tan /55/3 



• P" 



lb= -C I 1 + ^ tan^ cosrX — X) 1 5/? 

 pip" i 



_ LJ^ tan 8 sin (i — ?.) hX 

 PP 



+-Z{' + i^''^^'-'^\'^^M^ 



PP 



The last term in hi and tlie last two terms of hh have been omitted in the com- 

 putation, as they scarcely ever exceed a few hundredths of a second. 



17 May. 1873. 



