( 655 ) 



The quantities if, are llie lihralioii, which is delermiiicd h_v the 

 formulas : 



0-=l^ — 3/, -f 2/, - 180° = k sin ~ " = k sin (^ {l-t^) 



The (|iiaiili(ies ^^^, (and therefore |i' and 7') depend on the masses, 

 and have heen given in (rnni. I^nhl. 17, Art. IS, ^^\^ to terms of 

 the third order. 



The ine(inalities ean he divided into three «-ronps, aeeordijig to 

 their periods, of whieh the tirst gi'on[) mav he divided into three 

 subdivisions. These are : 



lit. Eq/iaf/Ot/s of the centre. The osenlating excentricities and 

 perijoxes — exelnding their |)eriodie perturbations (whieh arc taken 

 into aeeonid se|>arately as inequalities of the longitudes and radii- 

 vectores) — are determined by the formulas: 



hi = 2Ei sin i2,- -= -^ j "^ij ^j sin toy 



ki = '2Ei cos Hi =z 2^j tij e.j cos lOj. 



Here *^i and to, are the oirn excentricities and perijoves of the four 



satellites. The angles (o/ vary proportionally to the time, and the 



coefficients xtj depend on the masses, tü being unity. We have then 



övi =z: — cos li hi -\- sin li ki 



ÖQi =:^ — I sm 1° {si?i li hi -\- cos li /•/). 



The s(|uares of E are negligible, except for the fourth satellite. 

 The corresponding term is mentioned under Ic. 



Ih. 7 he (jreat inequalities. These arise (as perturbations in A/ and Z,) 

 through the commensurability of the mean motions of the three 

 inner satellites. They are : 



ffrj = .Pj sin 2(/j — /J rfo^ z=z — | sin 1° x^ cos 2{l^ — l^) 



(fv, = — .r, sin {l^ — /J d(>, = ^ sin 1° .p, cos {l■^—l,) 



ffj-, = - .Cj sin (/, — /,) d(.>, = è -^"i 1° '''5 '■'^^' (^3 — ^.0 



Ic. Minor ineqwditie.'< of shm-t perioiU. The periods of all the 

 inequalities of group / are short (not exceeding 17 days). 



II. Tnecpialities arising through the commensurability of the mean 

 motions, and having |)eriods l)etvveen 400 and 500 days. These only 

 exist for the satellites /, // and ///. In the radii-vectores they are 

 negligible. Their expressions are: 



