56 THEORBITOFURANtrS." 



The corresponding perturbations of the elements may then be put into the form 



h log a = Mr A cos N-\- 5»o, 

 hi =^ 3Ii'L sin N -i- Uo, 



ehn = Mve Wsin N-\- ehn^, 

 h = 3IrE cos N-\- he,, 



hy = Mvl cos N-\-hy„ 

 tan yhr^ Mv T sin N -\- tan yhto. 



Here, 5»o, hl^, etc., are arbitrary constants so taken that h log a, hi, etc., shall 

 yanish at the fundamental epoch. 



All the terms depending on the same values of i' and i are to be combined into 

 a single one. And it will save labor to make this combination at as early a 

 stage as possible in the computation; that is, to multiply the various values of h, 



. L, — L, — L, and E by the sines and cosines of /'(./-J-yu, and afterward proceed 



6'» 6e da 



with the sums of the products according to the proper modification of the formulae. 

 Tlius are obtained the following long period perturbations of the elements of 

 Uranus : 



^Z= — 3474.32 sin (2Z'— j/) + 180.10 cos (2?'— g) 



+ 146.72 sin (4?— 2r/)— 54.10 cos (4?'— 2^) 



. _ 8.97 sin (6r— 3(7) + 5.03 cos (6/'— 3r/) 



+ ■ 0.64 sin (8Z'— Ag) — 0.53 cos (8/'— Ag) 



-j- constant = 3320". 18. 



// // 



ehn=— 484.96 sin (2r— .^) -f 0.73 cos (2?— g) 



+ 38.06 sin (4^—2^7)— 7.06 cos (4?'— 2r/) 



— 3.61 sin (6r— 3r/) + 1.38 cos (6?'— 3^) 

 + 0.33 sin (8Z'— 4^/) — 0. 15 cos (8^— 4r/) 

 + constant = 465". 23. 



// // 



= _ 484.21 cos (2/'— g)— 0.29 sin (2Z'— g) 



+ 38.21 cos (4/'— 2//)+ 7.16 sin (4?'— 2(7) 



— 3.61 cos (6^—3(7) — 1.40 sin {Ql'—^g) 

 + 0.33 cos {SI'— 4g) + 0. 15 sin (8?'— ig) 



— constant = 158". 59. 



S» = -f 2277 cos (2r— g) + 120 sin {21'— g) 



— 198 cos (4?'— 2g) — 78 sin (4/'— 2g) 

 + 18 cos (6r— 3^)+ 7sin(6Z'— 3j) 

 -|- constant = 630. 



The variations of the elements which fix the position of the plane of the orbit 

 are here omitted, because their nature is such that it is indifferent in which form 

 they are developed. 



These expressions are reduced to perturbations of the co-ordinates by the follow- 



