178 THEORY OF JUPITER'S SATELLITES. [7 



the first and second satellites, and also in the case of the second and 

 third, the quantity n 2n' is small. Hence in the case of the first satellite 

 disturbed by the second, and also in that of the second disturbed by the 

 third, a. 2 , g^ will be considerable, while a,, g 1 will be considerable for the 

 cases of the second satellite disturbed by the first, and -the third dis- 

 turbed by the second. 



The difference n 2n' is positive, and the term 2f/j./a 3 in the expression 

 for N*, which depends upon the ellipticity of Jupiter, increases the divisor 

 i' (n n')" N" algebraically ; hence the ellipticity of Jupiter will diminish 

 a 2 , #2 where these are considerable, and increase a n g^ 



Let us now compute the mutual perturbations of the first and second 

 satellites ; the foregoing formulae apply without distinction to the pertur- 

 bations of Satellite I by Satellites II, III, IV, and Satellite II by Satellites 

 I, III, IV ; but as we are conducting the computations simultaneously, let 

 us slightly change the notation, so that where heretofore a, a,-, g i &c. 

 referred always to the body whose motion was under consideration, they 

 shall now refer always to Satellite I, the corresponding quantities for Satellite 

 II being denoted by accented letters. 



Then we have fa" =/''-', 



4-&i-^ 



1 / A 1 / 7 1\ 



A, (ao, a~). A, at>, , 



a v a \ a) ' 



dA t _ 1 / . 2 db t \ dAl 



da a~ \ da) ' da' a 



dA^ 1 / ^db 1 .\ dA' ] 



7 ! I " 7 * I I 7 I ~ 



da a- \ da / da a 



f 2 / t \ __ / 2 I O ., ? \ ^ ft '2 / 0^ I 9. ) 



\cfct 2 a da i a' \ da 2 da) \da' 2 a' da')' 

 Now let us employ the following data : 



The mean motions of the several satellites and of Jupiter in 365 '25 

 days, expressed in centesimal seconds, are*, 



[* These are derived from the synodic periods of the satellites, the mean tropical motion 

 of Jupiter, and the precession of the equinoxes adopted in Damoiseau's ' Tables,' Introduction, 

 p. iii.] 



