142 THEORY OF JUPITER'S SATELLITES. [7 



Also 



_l/V + i/.o. + 1/V + |A'-1/V= -ii/V, 



11 ft 4 



'1152- 7 ' 



Finally substituting for a,, Cj in the equation which gives the relation 



between q- and , we have 

 a 



or 



Hence the particular case of our problem is solved to the second 

 order in f. 



Now suppose new terms be added to ; and to z, which involve two 

 new arbitrary constants. If the principal periodic term in be taken as 



-ecos(pt + ft) these constants may be supposed to be e, ft. We will 

 ct/ 



suppose e small, so that its square may be neglected, and we will omit ft 

 in writing, remembering that it always accompanies pt. 



Let Sr and Sz be the increments of the former values of r and z, 

 due to these terms involving e. 



Then since the new values must satisfy the same differential equations, 

 we must have 



/ * \ P- * , " 5. 5l/z2 * , 2l/2 

 fit (*") =-^8r+-8r- - .- 8r + -^ Sz, 



- 



r 3 r 



3/iz 5, 

 - Sr + 



