1298 



a/ rti 



(14) 



Then the terms of Ri which are independent of the latitudes 

 become 



Qi 



Ji 



> — 4- ^L 



^ 10 



" 0^' 





+ 



+ 



(C— 3<) (\ — lli) 



\-\-mi 

 where we have put 



a/ dii Qi 





Jj Kj 



• (15) 



J6' Kh* 



Ji ^ — - , • Ki ^=1 — - 

 a^* a,' 



3. The intermediary orbit. 



The pertiirbative function Ui consists of a series of terms of the 

 form 



K cos Di 



^ = P\i — P^' + 9^> + q'^'j 

 To get the intermediary orbit we take 



R, = [Ri\. 

 Hence the argument /) must satisfy the condition 



{p -{- g')cj -\(q — p)<^c = ^- I 



The function [R; \ inchides the "secular" part of R;, for which 

 q = p, q' = — />, and the "critical" part, which becomes non- 

 periodic as a consequence of the commensurability of a and Cj. 



^Ri , ^Ri . . • . ,. I 



Since — and contain only sines, we shall have 



^ = . *i' = o 



dr dx 



if 



or 



^;o--^<o = ^X180^ 



(16) 



.T/„=rA;X 180°, 



k being any integer number. 



If we count the time from the epoch of an opposition of II and 

 III, and the longitudes from the longitude of III at that epoch, then 

 we have 



^,, = , ^,„-180° , :rr,,=:0. 



As to the fourth satellite, the condition A,, := or 180° is 

 generally not satistied, since tliere is no relation between the 



