176 THEORY OF JUPITER'S SATELLITES. [7 



Substitute in the equations of motion and write 



where M is the mean motion of Jupiter about the Sun, and equate co- 

 efficients of the various terms. 



We obtain the following equations : 

 constant term in the first equation : 



' '" r * -^ r -*- r ~. * 7 



a" " 2 2 a, da 



this gives the relation between a and n ; 

 coefficient of cos 2(1 L) in the first equation : 



P- I , -f\ 3 TLt. 



4n-c 4n-k + r - 3 (3 + 5/ ) = - M - ; 

 coefficient of sin 2 (l L) in the second equation: 



hence 



3 + 5/7 ,r ~\8 3 + 1 



these stand for the inequality which is called the Variation in Lunar 

 Theory. We now see that in omitting -j when forming the differential 



equations we have omitted quantities of the order of M 3 in the values 

 of c and k ; 



coefficient of cos i (I I') in the first equation : 



m ' d^. 



i" (n n')" a,- 2n (n n') iq, + , (3 + 5f) a,- = - 



a sv a da 



coefficient of sin i (I I') in the second equation : 



m . 



. 

 i" (n n')- g { + 2n (n n') ia { = -- - iA i . 



t/ 



Multiply the second of these equations by 2n/i(n n'), and subtract 

 from the first : 



[ / /\. V- 



i(n _ 7l0 -_4n- + g 



