7] THEORY OF JUPITER'S SATELLITES. 143 



or making r$r and 82 our new variables, the equations are 



d* i * \ / !" , v 5vz2 \ s , 2vz ? 

 -j-. (rSr) = (<-+ - rSr H - 82, 



jfttV / ^ 7 J> r 5 r ? , ,,5 



c/' 2 /o x / /A 3v 15v2 2 \ ? /3/AZ 15^ 35vz a \ ,-, 

 T-5 (Sz)= -5 -- - + 82+ - r + --- - r&r, 



rff 7 \ r 3 r 5 r 7 / \ r 5 r 7 i f ] 



in which the values of r and z already determined are to be substituted 

 in the coefficients of r8r and 82. 



To find the coefficients in our equations : 



tO T 



O / \ \ 



I + 3a, cos 2ot + 3a e> cos 4qt + - a '" ( 1 4- cos 4o) 

 2 z 7 J 



- = . [1 + 5a, cos 

 r or 





j cos 2g] - (1 - cos 2^^) + c, (cos 2^< - cos 4</) 

 - (l cos 2<) + Cj (cos 2^< cos 4^) + - ! cos 2^^ - j 



Ct I ^ -" 



- ttj COS 



i ... 7 



~ + 



