74 MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 



These quantities gave immediately the iiual perturbations by means of the following equations : 



nz = (>j) + nt + ndz. 

 n6s=[lt (<>,0,c) +k-^] Cl ~\ nt 



r- 5 e 2 -l 



+ 1-22 (1,0,*)+ JET (0,0,*) — g-H(0,0,s)-k. 2 cos 5 



+ j~22 (l,0,c) + e R (0,0,c) -H (0,0,c) +| 22 (0,0,c) + A-,- ^ fci~j sin £ 



+27 (0,0,c) «£ cos £ 



+ \H (0,0,s)— g -2" (0,0,*)1 nt sin £ 



+ T- 22 (2,0,*)-| (5-2e 2 ) fl" (0,0,*)+ 1 fc 2 ~j cos 2 * 



+ ^22 (2,0,c)+-g- 21 (0,0,0)-^ J sin 2 £ 



-| 22 (0,0,c) w* cos 2 e 



— |-ff(0,0,s) »f sin 2 s 



+ r_i?(3,0,s)+-g 2I(0,0,s)~] cos 3f +j~E (3,0,c)-|-2J(0,0,c)1sin 3 s 



—22 (4,0,*) cos 4 f + 22 (4,0,e) sin 4 £ 



- + 



— 22R {i,i',s) cos [(i — i'/j)e — i'(c' — /.w)] + 222R (i,i',c) sin [(* — V n) i—i'(c'—/.<c)] 



i'>0 i'>0 



2 v=2 C-e R (0,0,*) n< 



+ [#(l,0,c) + 22(0,0,c)-fr 1 ] cos s +[Q (l,0,s)+27(0,0,s)-e 2 21 (0,0,*)— 1%] sin £ 



— i? (0,0,s) nt cos e + J3" (0,0,c) «i sin e 



+ i [Q{2,0,c)-e H (0,0,c)] cos 2 e +% [Q (2,0,*)— e fl (0,0,*) sin 2 e 



+ J (? (3,0,(0 cos 3 f +i- § (3,0,s) sin 3 £ 



+ + 



+22S(v',«) cos \(i— i'/j) £-i'(c'—/uc)] + 22S (i,i',s) sin [{i—i'/n) £— **'(C— /«>)] 

 i'>0 i'>0 



JfL=C"(l,0,c)-£ F (0,0,0) -e?!—e F(0,0,*).w« 

 cost 2 



+ [.r(l,0,c) + /,] cos £ +[r(l,0,*)-e 4 F(0,0,s)+J] sin * 



+ T r (0,0,*) w* cos £ — V (0,0,c) m< sin s 



+ [Y(2,0,c)+ % V (0,0,o)] cos 2 £ + [F(2,0,*) + | V (0,0,*)] sin 2 £ 



+ 1' (3,0,c) cos 3 £ + r (3,0,*) sin 3 £ 



+ + 



+ 22Y (i,i',c) cos \(i-i'n) e-i'(c'-/.ic)} +22Y(i,i',s) sin [(i— i'/i) s—i'(o'—pic)] 

 i'>0 i'>0 



•where (</), O, I; h h fcj, I, and /, are constants of integration, connected by the relation 



G = —i[k + eh] 



leaving six independent constants. 



