THE ORBIT OF URANUS. 



57 



ing formulae. Express the usual developments of the longitude and logarithm of 

 radius vector in the form 



Put also 



v= Z -f- 2 Fi sin ij ; 

 p = » -J- 2 i^i cos ij. 



F',= 



dVt 

 de 



e 



R\ = 



dR, 



de 



e 

 Express any set of corresponding terms of the preceding perturbations in the form 



tl =Zj siniV-fX^cos A^: 



ell — ehn ■ 

 ^e : 



We shall then have 



F, sin ^+7-; cos A^: 

 A^ cos N-\- A^ sin N. 



25^ = 2 ( F", F, + V, E.) sin (N + ig) + 2 ( V'\ F, - F', E,) sin {N- if,) 

 + 2 ( F", F, - V\ £J cos (iV+ vj) + 2 ( V\ F, + T"', E,) cos {N - ig) 



^2 hi 



2.V = 2 {R\ E, - 7?", F^ cos (.V+ ig) + 2 (rv', E. + E% F,) cos (A^- irj) 



+ 2 («', i;, + E\ F,) sin (JV+ vj) + 2 (i?', £, 



+ 2.^» 



ii",i^,)sin(.V-iy) 



The numerical values of T", F, /?', and R' are as follows ; 



The final results of the entire computations are given in the following table: 

 In the columns hv^, we have the complete perturbations of the longitude computed 



dR 6R 



by the direct method from the values of , -^— , Q, etc., already given. Next 



^v op 



we have, under the caption h\, the perturbations of the true longitude deduced 

 from the long period perturbations of the elements, as set forth in the last para- 

 graph, omitting the constants added to the perturbations. Under hi\ we have the 



8 April, 1873. 



