( 656 ) 



övi := ^j Hjj sin (fj 



The coefficients Xjj are proportional to f?/, and also depend on ihe 

 masses. 



///. lnc(|ualitips with very long periods (exceeding 12 years). 

 These also are negligible in the i-adii-vectores. 



The latitudes of the satellites over the plane of .Tnpiter's orbit are 

 given by the forninla : 



,<!/ = /,• .mi (vi—Ni). 

 The inclinations and nodes ^) are in Souii.lakt's theory determined 

 by the formulas.- 



/,• sin A^i = ^ j dij 7/ sin 6 j -\- [ino sin 6 + periodic terms 

 /, cos Ni := 2 j (lij Yj cos 6^ -f- niiM cos 8 4- periodic terms 

 Mr. CooKSON and 1 have in all our work on the satellites referred 

 the latitudes to a fundamental plane, which is defined by its incli- 

 nation and node referred to the ecliptic and mean equinox of date. 

 For these Marth's values have been adopted, which are for 1900.0 : 

 -7 = 2^ 9' 3".94 :Y = 336° 21' 28",4 

 The longitude of the node of this plane on Leverrier's orbit of 

 Jupiter, counted in. the orbit, and the inclination on that orbit are : 

 ^„ = 315° 25' 48"4. to„ = 3° 4' 4".75. 

 The longitude of the node of the orbital plane on the fundamental 

 plane, counted m the furulainenUil /jlane, is therefore 



8\ = 135" 24' 34". 3. 

 The longitudes in the tundamental plane have been counted from 

 the point O, of which the longitude is '■^) 



r> = 135°27'2".5. 

 If the inclination and (lesceiidiiui node of the fundamental plane on 

 the orbit of 1850.0 are represented by cj» and i|'„, (thus if'o = '^o + 180°) 

 and if the longitudes of the nodes Sl^i fn'c reckoned from this descending 

 node, we have : 



Pi = — H ''<ifi Jl/ = — ^f -^"i (^'i— V'o) 

 qi = ii COS SI; = li cos {Ni — 1|%) + a>o 

 from which 



Pi =: 2£ I (Ï,; Yi sin (if'o — '^z) + f*!*'^ ■'*"* (V'o~~^) ~i" periodic terms 

 q, = ^j Oij Yj cos {i]\ - 6j) + imo cos (if'g — <9) 4- tf^o -f periodic terms. 



') By node 1 always mean ascending node, unless otherwise staled. 

 -) Mauth evidently intended to adopt 0=iV I^robably he has applied the cor 

 rection, needed to derive ' from 6 + 180°, with the wrong sign. 



