gg THE ORBIT OF UK A N U S. 



Substituting for o> its value, 1245', arid for fy and bk their above values in the 

 expression for &j3, we find that the terras of &j in question will add the following 

 terms to (fi.c.O), (fe.s.l), etc. 



(j.c.O) = .010 fy + .046 M = + 0".027 7 (1 -f p) 

 (&..!) = + .975 bv, -f .221 M = .727 (1 + p) 

 (A.c.l) = + .221 ^ .975 tk = .447(1 + p) 

 (1.8.2) = + .046 fy + .011 M = .047 7 (1 + p) 

 (6. c .2) = - (6.C.O) - .027 (1 + p) 



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

 but not in the tables. 



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

 following constant and secular terms in (i.c.O), (&..!), etc. 



(j.c.0) = -f 0".26 -f ( 0".09 -f 0.08:/)7 

 (b.s.l) = ( 0".92 0".61//)7 O'.OST 2 

 .(6.c.l) = (-[- 1 .86 -- 1 .68p)T+ .127 72 

 (b.s.2) = -0.06 0.05 T 

 (6.c.2) = - .01 + (0 .09 m-i] T 



Positions of Uranus resulting from tJie preceding flieory. 



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 formulae already given. 



lv = (v.c.O) -f- (v.c.l) cos g -f- (w.c.2) cos 2g -f- etc., 

 -4- (v.s.l) sin g -4- (v.s.2) sin 2>j -f- etc., 



= (p.c.O) 4- (p- c -l) cos ff + (p- c -2) cos 2g -4- etc., 

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



= (ft.c.0) + (6-c.l) cos^r -f- (6.5.1) sin g. 



Each coeiRcient (v.c.O), (v.c.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 N 6 and JV 7 , 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 



