98 



THP] ORBIT OF URANUS. 



Substituting for u its value, 12''45', and for <^>7 and hk their above values in the 

 expression for hiS, we find tliat the terms of 6,'i in question will add the following 

 terms to (b.c.O), (b.s.l), etc. 



(5.C.0) = — .010 ^>7 + .046 ,7.- = + 0".0'2 T {1 + ^) 

 (b.s.l) = + .975 ,^>7 + .221 ,V.- = — .727^(1 + ^i) 

 (A.c.l) = 4- .221 hn — .975 hk = — .44 7^(1 + /<) 

 (^,..5.2) = + .046 <'.ri + .011 a- = — .04r (1 + ^i) 

 (/,.e.2) = — (b.c.O) = — .027 (1 + ^i) 



These values will be employed in the construction of the provisional ephemeris, 

 but not in the tables. 



Collecting all tliree classes of terms discussed in this section, we have the 

 following constant and secular terms in (b.c.O), [b.s.l), etc. 



(b.c.O) = + 0".26 + (— 0".09 + O.OS:t)T 

 (•?,.s.l) = (_ o".92 — 0".6lu)T— 0".05T^ 

 (b.c.l) = (+ 1 .86 — 1 .687)7'+ .12T^ 

 {b.s.2) = —0.06 —O.OoT 

 [b.c.2) = — .01 + (0 .09 — .08/) T 



Positions of Uranus residtinfj from (he jjreceding theory. 



The next step in order is the preparation of an ephemeris of the planet for 

 comparison with observations. As this provisional theory is, for future use, super- 

 seded by the tables appended to the present work, it seems unnecessary to enter 

 very fully into the details of the computation of the ephemeris. The perturba- 

 tions of the longitude, logarithm of radius vector, and latitude, were first com- 

 puted by the formulse already given. 



Iv = (v.c.O) -\- (v.c.l) cos (J -\- (v.c.2) cos 2/7 -|- etc., 



-j- (v.s.l) sin g -\- (v.s.2) sin 2;/ -\- etc., 



M^p = (p.c.O) -\- (p.c.l)cos// -f (p.c.2) cos 2// -j- etc., 



+ (p.s.l) sin g -\- (p.s.2) sin 2g -{- etc., 



53 = {b.c.O) + (b.c.l) cosy + (^-^-l) sin g. 



Each coefficient (''.c.Oj, (r.e.l), etc., is composed at most of the following quan- 

 tities: 



1. The five classes of secular, long period, or constant terms, the separate values 

 of which, with the sum of all, are given on pages 89 to 93. 



2. Periodic terms due to the action of Jupiter, Saturn, and Neptune, given on 

 pages 83 to 87. 



3. Terms depending on the product of the masses of Jupiter and Saturn, given 

 on page 88, omitting those depending on iVg and N.,, because they are given in 

 column 5 of the terms of the first class. 



The sum of the perturbations thus computed is given in the third column of the 

 following ephemeris. 



An approximate value of the perturbations produced by Neptune alone is inde- 

 pendently computed for every fourth date, and the result is given in the fourth 



