1290 

 where x is a small quantity whose value is approximately 



1 



69 



We will consider x as a small quantity of the first order. The 

 masses of the satellites then are of the second order (the largest is 

 about 0-8 X 1<^~')- 



Generally we put, for all satellites including the fourth 



}.i=z(ci - x)T-f ;.,o ....... (1) 



We thus have, the value of 6\ being only given approximately, 

 ,-^ = 4 , c, = 2 , <•, r=z 1 , e^-= 0-437 . 



If we start with uniform motion in a circle as a first approxi- 

 mation, the inequalities can according to their periods be divided 

 into four sharply separated groups. ') 



I. The inequalities of the first group have periods not exceeding 

 17 days. They can be subdivided into three sub-groups: 



la. The equations of the centre, which are 

 (fu'i =2^ Tij 8j sill (P.,- — TCj) , 



(f I'i ==: — III ^ Tij Bj COS ().{ — tUj ), 



where the sums are to be taken for the values of j from 1 to 4, 

 and where f/ and tu",- are the "own" excentricities and perijoves. 



lb. The "great" inequalities. These are approximately : 



rr?<',-= 2eisin CiX 



(Sri ^== — ^i e/ «'OS Ci X 

 \c. Other inequalities of short periods. 



II. The inequalities of the second group have periods between 

 400 and 500 days. Their expressions are 



(flO{= 2 TCij shl <(j, 



j 

 with •) 



(pi =XT + 'UTi. 



III. The libration has a period of about 7 years. 



IV. The inequalities of the fourth group have periods of more 

 than 12 years. 



The inequalities 16, II and III arise out of the mutual commen- 

 surability of the mean motions. In a previous communication ") 



1) See also "Elements and Masses", p. 655, where however the libration (III) is 

 left outside the groups and the group IV is numbered III. 



^) These ^f differ 180^ from the angles so called in ''Elements and Masses"; so 

 the coefficients /.,y here have the other sign. 



^) "On the periodic solutions of a special case of the problem of four bodies", 

 these Proceedings, Feb. 19U9, Vol XI, p. 682. 



