1299 



longitudes of the fourth satellite and of the others, as there is in 



thé case of the three inner satellites. It is however easy to choose 



as origin an opposition of II and III at which the condition is very 



nearly satistied. ^) 



Then for all satellites tu and r, are constant in the intermediary 



orbit. We take 



vi = 0, and consequently oi=zai ; tn ^= -tji . . . (17) 



d[Ri] d[Ri] ^ ^ ^ 



Then also ai — — and — — are constants. These must be deter- 



Oai Oiii 



mined so that 



-—— = — icrii, . (18) 



Orii 



ai —^ — — — |xïjï% (19) 



Oai 



Then we have 



li ~ ar , JTi == — >«r + TT-i^ , Xi = (ci — >{) T -f Tii^ . (20) 

 The radius-vector n and the true oi-bit-longitude ivi are now 

 determined by (4t and (5), and we thus see that the intermediary 

 orbit is a Keplerian ellipse with the constant semi-axis a/, the 

 constant excentricity ei=^sin(pi, determined by 2 sin ^j, (ri=i]i , and 

 the mean anomaly li = CiT, and these ellipses rotate in their plane 

 with the angular velocity — a, common to all satellites. 



The conditions (J 8) and (19) serve to determine the two para- 

 meters ftj- and tii. For the inner satellites this intermediary orbit is, 

 as has already been pointed out, a very good approximation, better 

 than the fixed Keplerian ellipse. For IV the excentricity as deter- 

 mined from (18) is extremely small, and the intermediary orbit con- 

 sequently differs very little from a circle described with the uniform 

 velocity a — x. 



{To be continued next page.) 



1) Thus e.g. on 1899 June 28, llh 47"! 35s G. M. T. the longitudes counted 

 from the first point of Aries are: 



Ai = 193°,64 , A2 = 13°.64 , A3 =193.64 , A^ = 192°.75. 



