THE ORBIT OF URANUS. . • 187 



Among the elliptic terms may be included the effect of the following minute 

 constants introduced by the perturbations. 



(t,.s.2) = — ()". 144 



(u.c.2) = + 0.l;30 

 0.4343 (p.c.O) = + 19T2 in units of the 7th place 

 0.4343 (p..9.1) = + 63 of decimals. 



0.4343 (p.c.l) = + 73 

 0.4343 (p.s.2) = + 5 

 0.4343 (p.c.2) = + 4 



This term (p.c.O) is that added as a correction to the logarithm of the mean 

 distance. 



To the coefficients (^v.s.l), (r.t.l), etc., are still to be added the following periodic 

 terms : — 



1. The periodic terms due to the action of Jupiter, given in Chapter V, omitting 

 the terms multiplied by T, which are included in the perturbations of the 

 elements. 



2. The periodic terms produced ])y Saturn, including those terms midtiplicd 

 both by T and by sin A., or cos A.,, but omitting those multiplied by T oidy for 

 the same reason as in the case of Ju[)iter. 



3. The periodic terms produced by Neptune, multiplied by the factor 0.86294 

 on account of the correction to the mass of that planet, and omitting the terms 

 multiplied by SI, Se, and eSg. 



4. The periodic terms multiplied by the product of the masses of Jupiter and 

 Saturn, given on page 88, omitting the terms multiplied by the sine and cosine of 

 JVj and N-, because they are replaced by the terms of <)l, bh, and bk, given on page 

 183, and tabulated in the columns headed (2) on page 184. The result will be 

 the same whether we employ the terms of {v.c.O), (r.s.l), etc., given at the bottom 

 of page 88 and the top of page 89, omitting tlie numbers in the columns 2 ou 

 page 184 from the expressions on page 186, or whether we include the latter and 

 omit the former. 



The true anomaly of Uranus will then be : 



i/o + ^^ + (equation of centre from elements IV, using for mean anomaly ffc, + ^0 



-\- 2 (v.s.i) sin itj -\- S (i-\c.i) cos ij. 



The logarithm of the radius vector will be: 



log r in elliptic orbit from elements IV. 

 -\~ 2 {p.s.i) sin ig -\- 2 (p.c.i) cos i(/ 



care being taken to multiply the coefficients by the modulus where that has not 

 already been done. All the terms in Chapter V are so multiplied. 



To pass from the true anomaly to the true longitude we must investigate the 

 secular motion of the planes of the orbit and of tlie ecliptic. The efl'ect of this 

 motion on <p, 0, and r will be found by successive approximations from the formulic 



