( 684 ) 
It is owing to this particular circumstance, that the motion of the 
satellites can also be considered as a periodic solution of the second 
kind, as will now be shown. 
In the periodic solutions of the second kind of the unperturbed 
problem the excentricities are arbitrary, and the mean motions (not 
only their differences) are mutually commensurable. In other words 
we have here x = 0. 
If the masses are not zero, these solutions may also remain periodic. 
In the perturbed motion we must then distinguish the mean motions 
in longitude and those in anomaly. Let 
nit + lio be the mean anomaly 
ASP t= As 
if then a; be the longitude of the pericentre,-we have 
nml + aij 
Stare , longitude, 
Inquiring into the conditions that these solutions shall remain 
periodic for small finite values of the masses, we find again that 
there must be a symmetrical conjunction at the beginning of the 
period, i.e. for t= 0. The angles 
im I I 
must all be 0° or 180°. One of the angles /;, (e.g. /;,) can always 
be made identically zero (or 180°) by a convenient choice of the 
zero epoch. There thus remain 4 angles, each of which can have 
one of two values. We have thus 16 combinations which may a 
priori be expected to give rise to periodic solutions. 
20 80 
CL uke ; A 2x 
Now if ie were zero, then at the end of the period 7 = — 
Yv 
the configuration for tO would be exactly restored, as it ts in the 
unperturbed problem. It is, however, sufficient to insure the periodicity 
de 
of the solution, that the value of en integrated over a complete 
period shall be the same for the three satellites. In addition to the 
conditions of symmetry we have therefore the conditions 
ge 1 
7 
as dt = le dt = Ts dt = a 1 
Eo ee 
0 0 
0 
After the completion of the period the whole system is then 
rotated through the angle — «7, as in the solutions of the first kind. 
