5 28 SCIENCE PROGRESS 



does not present any particular difficulties, in so far as the 

 periodic inequalities are concerned ; it is similar to, but simpler 

 than the lunar theory, for the ratio of the month and year is 

 only about 1/260. The secular perturbations of the centre are, 

 however, intimately connected for the four satellites, so that 

 it is impossible to separate them, and the apparently simple 

 and obvious course of having one theory for the three inner 

 satellites and another theory for the fourth cannot be adopted. 



In the old theory of the motion of the satellites, the undis- 

 turbed Keplerian eclipses are taken as intermediary orbits for 

 the four satellites, and from these orbits by the addition of 

 perturbations and variations, the complete solution is derived. 

 This method of procedure is the best in the case of the fourth 

 satellite, but not for the inner three. The new theory of de 

 Sitter is chosen so as to be the best for the case of the inner 

 satellites, and this more than compensates for the somewhat 

 increased difficulties in the case of the fourth satellite. 



The motion of the satellites can be described as a uniform 

 motion in a circle with superposed inequalities. The inequalities 

 in longitude and radius-vector can be classified according to 

 period into four separate groups : (1) Short-period inequalities 

 with periods not exceeding seventeen days, and including the 

 " equations of the centre " and the " great inequalities " ; 

 (2) Inequalities with periods between 400 and 500 days : these 

 inequalities are negligible in the radius-vector and are zero 

 for the fourth satellite ; (3) Librations of the inner satellites 

 with a period of about seven years ; (4) Long-period inequalities 

 with periods longer than twelve years. The " great inequali- 

 ties " and those of groups (2) and (3) arise through the com- 

 mensurability of the mean motions and in the usual theory 

 have small divisors. 



In the new theory, intermediary orbits are used in which 

 the expressions for the mean longitudes, mean anomalies, and 

 longitudes of the perijoves of the satellites are rigorously 

 satisfied. The special features which make this solution a 

 good first approximation in the case of the three inner satellites 

 are the moving perijove and the fact that the induced equations 

 of the centre or " great inequalities " are larger than the free 

 or ordinary equations. For the fourth satellite the free eccen- 

 tricity is larger than the induced. 



In the new theory the equations of the centre are treated 



