1292 



tan^fi=^/ j—^-tan^Ui ...... (4) 



\-\-eicosfi 



The true orbit-longitude then is 



wi:=fi-r ^j- ......... (5) 



This intermediary orbit is, also for the three inner satellites, not 

 the comi)lete periodic orbit, but only contains its leading terins. To 

 get this intermediary orbit we must 1 restrict the perturbative 

 function to a certain part of it (viz: the "secular" and the "critical" 

 parts), and 2 we must take initial values, or constants of integra- 

 tion, which satisfy certain conditions. The complete solution is then 

 derived by adding lo the intermediary orbit: 



1. "perturbations" which arise from the parts of the perturbative 

 function that were at first neglected; 



2. "variations" which are due to the fact that the actual constants 

 of integration do not exactly satisfy the conditions for the interme- 

 diary orbit. 



Of these the variations are the most important. To get these we 

 must form the variational equations. These lead to a system of 

 equations entirely similar to those which are used in the treatment 

 of secular perturbations by the method of Lagrangf,. The resulting 

 determinant has 5 roots /?,... /^J^, corresponding to the four own 

 perijoves t^i, and the argument of the libration -nr^ respectively. 

 The inequalities in longitude and radius-vector are then given by 

 formulas which, if we restrict ourselves to the first order, assume 

 the form 



öwi=. 2 Wij 6j sin {Xi — tcr^) -|- 2 w'.y fy sin (pj, \ 



^ ^ • • (6) 



öri = 2 Rij Sj cos (A,- — Wj), i 



where 



(fi = Pit + ffi, = ax + tü-,- 



as above, and j assumes the values 1 to 5. These formulas include 

 not only the free equations of the centre la, but also the inequalities 

 of group II (arguments (f\ — (p^) and the libration III (argument (p^). 

 As to the perturbations: by the introduction of Ci instead of iii we 

 have realised that thei^e are no small divisors. In the ordinary theory 

 small divisors appear in the inequalities I/>, II and III. Of these \b 

 is already included in the intermediary orbit; II arid III appear as 



