( 723 ) 



■ r= 2° 12' 8" 7 + 0".4231 T 

 iX = 336 24 24 + 48 .916 ï 

 The inclination and node of Jnpiter's orbit referred to the mean 

 eijnator are therefore (the node being connted "in the equator"): 

 co= 3° 6'55".l + 0".0243T 

 ^'= 135 46 44 + 50 .155 T 

 The position of the orbital phines of the satellites — excbiding 

 periodic, but including' secular perturbatioJis — referred to the mean 

 equator, are given by the formulas : 



iisin (6' — Sli)= Pi = -J ^y 7 J ^^"^ 

 ii cos {&' — Sli) = qi = ^j ^y 7; <^os Fj + ( ' — IM ) w 

 Referred to the orbit of Jupiter they are') 



lisin Ni^= 2j Oij Yj sill 6j -j- {lico sin 6 

 I [COS N{ = ^j öy yj cos 6 J -f- imo cos 8 

 where we have ') 



Fi— 180° ^ d — di 



If the periodic perturbations are represented by <Spi , dqi , dsi , we 

 have for the latitude of the satellite referred to the mean equator 

 l3i = {qi -f- öqi)sin {v{- 8') -^ (pi + (fpi) cos {vi— &) 



and referred to the orbital plane of Jupiter 



Si = lisin {vi— Ni) -\- (fsi . 



Here vi is the true orbit-longitude of the satellite. In both formulas 

 quantities of the third order in the inclinations are neglected. The 

 neglected terms in |-?i are thus of the order of magnitude of 0^^.00002 

 and in si of the order of 0°.01. 



The values of the coeflicients öy and fx,- are : 



(7,j = — 0.019 + .012 () — .019 ;ii 



a,, — — 0.001 -f .001 Q - .001 X, 



o,, — 0.000 



(j^, = 4- 0.0203 — .020 (> + .020 ;., 



(j„ = — 0.0347 4 .028 Q + .002 ;., — .035 ;., + .005 ;., - .0005 x^ 



o., = - O.OOIO — .001 Q — .001 2, + .001 /, 



1) Rigorously these formulas are true with reference to the fixed orbit of 

 Jupiter, and a correction must be applied to derive the latitude referred to the 

 moving orbit. It is, however, suificienlly accurate to use the same formulas for 

 the latitude referred to the moving orbit, provided we take for w and 6 the incli- 

 nation and node of the mean equator referred to this same moving orbit (as was 

 done here). For the motion of the node 5 referred to the moving orbit I adopted 

 — 0".0979 instead of — 0".07iO (Souillart II page 166). This in the value which 

 results if Souillart's final value of h^ is used instead of the approximate value 

 used by Souillart himself. 



~) The meaning of F, is thus here slightly different from what it was in the 

 subordinate investigation I. 



49* 



