( 771 ) 



find 1902. All otiier corrections Lp and L<i derived from the obser- 

 vations are inclnded in tiie following discussion, with weights in- 

 versely proportional to the squares of their probable errors and 

 corresponding to a p. e. of weight unity of ± 0.°0050. 



Before this discussion can be related the theoretical expressions 

 for i> and q must be developed. 



At the time when the analytical theory of the satellites was created 

 bv L.\(;range and Laplace, the eclipses were pi-actically tlie only 

 phenomena of the satellites which were observed. For these the 

 natural fnndamental plane is the plane containing the axis of the 

 shadow-cone, i. e. the plane of Jupiter's orbit. This was accordingly 

 used by them. Souillart, in his theory published in 1880, followed 

 their example. 



The first thing \Ahich must be done before the theory can be 

 compared with modern observations is thus to reduce the expressions 

 for the latitudes referred to Jupiter's orbit to latitudes referred to the 

 equator. This has already been done by Marth, who in 1891 

 published tables for the computation of the co-ordinates of the satel- 

 lites, based on Souillart's theory (Monthly Notices, June 1891, pages 

 505—539). 



Let / and N be the inclination and node ') of the orbital plane 

 of one of the satellites with reference to the orbit of Jupiter. 

 Souillart's theory then gives 



/,■ sin A',- = .5" bij sin 6j -\- \ii to sin 6^ 1 



^ ^7' . . • • • • (^' 



/,■ cos Ni = 2 bij sin 6j -f (i; w sin 6„ \ 

 j=\ ) 



In these formulae <o and (9„ are the inclination and node of Jupiter's 

 equator on its orbit. All longitudes are counted from the first point 

 of Aries. The quantities by are constants, and the angles &i vary 

 proportionally with the time. Of the constants hij four only are 

 mutually independent. If we put : 



bii = Yi 1^0 = <^ij "!] ! 



then the y, are constants. The multipliers Oy and m and the coeffi- 

 cients of the time in the expressions for Ö, are given by the theory 

 as functions of the masses, the compression of Jupiter and the mean 

 motions. The constants Oij are small numbers (the largest is a^^ = 0.1944) 

 with the exception, of course, of those in the diagonal, ff,,- = 1. The 

 value of m differs little from unity. The angles 7, and 6i are what 

 Laplace calls the "inclinaisons et noeuds propres" of the satellites. 



') With node I mean ascending node, unless ütherwise slated. 



