25 



Elements V were deduced from Elements IV by the following process, which 

 is, perhaps, a new application of the method of mechanical quadratures to the 

 direct computation of a disturbed orbit from observations. The conditional 

 equation has the form, 



In which, 



9 = ax-\-by-\-cz-\-du-\-ev-\- &.c. -|- n 



a = (i — 146 years 340 clays) mean time Greenwich. 

 b = 1 



d r 

 d — [ac 2xAj 



e = [a- c ^ X B* '\ 



n = / o, — J \— computed less observed true orbital longitude. 



Also, 



«o = th<2 assumed true sidereal daily angular motion , for a = o 

 "o = " " orbital longitude , " 



'"o = " " radius vector , " 



X, ij, and z = A «^, , a u^ , a r^ = the required corrections 



J^'^ _ value of w for date t, from normal place, using r = r^ 

 „/(") ^ „^ _(_ a n„ 



r = r„ -\- z -\- a u -\- a "v -\- &c. 



K" = (r^ -\- z)- (?i„ 4- a?) = Neptune's daily area. 



n = KM r 



2 = value of w for conservation of areas. 

 A , B " , &c. = the part of the co-efficients of u and v, introduced by quadratures. 



These co-efficients were computed by Laplace's formula for mechanical 

 quadratures, Mcc. Cel. V, p. 207. The corrections of the residual errors are 

 stated above. The XIII conditional equations were 



I 



= 



— 0.6 X X 



+ 0.5 



X y —0.213 X 2 



+ 0.015 X u 



— 0.35 



II 



= 



— 0.7 



+ 1- 



+ 1.444 



— 0.216 



— 0.19 



III 



= 



— 0.6 



+ 1- 



+ 1.728 



— 0.212 



— 0.13 



IV 



= 



— 0.5 



+ 1- 



+ 1.954 



— 0.199 



+ 0.24 



V 



=. 



— 0.4 



+ !■ 



+ 2.121 



— 0.172 



+ 0.51 



VI 



= 



— 0..3 



+ 1- 



+ 2.228 



— 0.135 



+ 0.57 



VII 



= 



— 0.2 



+ 1- 



+ 2.266 



— 0.091 



+ 0.29 



VIII 



= 



— 0.1 



+ 1- 



+ 2.210 



— 0.044 



+ 0.88 



IX 



= 



+ 0.0 



+ 1- 



+ 2.112 



+ 0.000 



+ 1.20 



X 



= 



+ 0.1 



+ 1- 



+ 1.994 



+ 0.040 



+ 0.49 



XI 



= 



+ 0.2 



+ 1- 



+ 1.731 



+ 0.070 



+ 0.68 



XII 



= 



+ 0.4 



+ 1- 



+ 1.149 



+ 0.094 



+ 0.54 



XIII 



, = 



+ 1.2 



+ 1- 



— 1.696 



— 4.050 



+ 0.20 



