IV THE THEORY OP THE LONG INEQUALITY 



P, and thence upon the orbit of m. If, therefore, we take PN = Pv , and take N as the 

 origin from which the angles are measured, and NPR for the plane of reference, the angles 

 9, e, and •ar will be the same as before. But 9', e, "&', when referred to the ecliptic, were 

 measured from t to Q, and thence upon the orbit of ml ; and when they are referred to the 

 plane NPR, they are measured from N to R, and thence upon the plane of the orbit of m ; 

 consequently, the values of these angles are less now than they were before by the quantity 

 PQ + QR- PR. Also n' = Si ' + PR - PQ. 



Now upon again changing the plane of reference from the orbit of m to that of m, no 

 further alteration needs to be made in the angles. For we may take RN= RN and take N 

 as the origin of angles, and recollecting that the node opposite to R is the ascending node of m 

 upon the orbit of m the angles 9, e, •ar will be altered by 360°, and the angles 9', e, nf are not 

 altered at all. Also <y = y, and II differs from II' by 180°, and therefore 2 II differs from 211' by 

 360° ; consequently the same value may be used for II as for II'. It appears, therefore, that 

 when e' and II' have received the above correction, no further alteration is required, whichever 

 orbit is taken for plane of reference. 



5. The values of e, "sr and Si at Greenwich mean noon, Jan. 1, 1801, given in Baily's 

 Astron. Tables and Formulas, are 



e = 177°4.8'23"-0 

 ■sr - 167 31 16 -1 

 Si = 72 59 35 -3 



The apparent annual increase of w = 52" , 50, 



that of SI = 14"-l6. 



The mean sidereal period = 30686*8208296 mean solar days. Hence the value of n is 



42"-233 11392. 



The number of days between mean noon, Jan. 1, 1801, and mean noon, Jan. 1, 1847, is 

 46 x 365 + It, Therefore the mean motion of Uranus during that interval is 197° 5' 58""55, 

 and using the value of precession given by Bessel 50"'2294, the precession in 46 years 

 = 38' 30"'55. Therefore adding these quantities to the above value of e, we have for the 

 epoch of Greenwich mean noon, Jan. 1, 1847, 



e = 15° 32' 52"- 10. 



Again, the increase of "sr in 46 years = 40' 15" •O, 



and that of fl, in the same period = 10' 5l"'36. 



Therefore adding these quantities to the values of ar and Si a t the former epoch, respectively, 

 we have at the new epoch 



■sr = l6S°ll'3l"-l, 



Si = 73 10 26 -66. 



The values of the inclination and eccentricity at the former epoch are (Baily's Astr. 

 Tables and Form.) 



