( 691 ) 
of a,a’,3 and 2, we find the following summary. Only those com- 
binations can give rise to periodic solutions, in which 2, , 2, and &, 
are of the same sign. The letter OU stands for undetermined. 
Out of these 16 combinations there are only two, (6) and (16), 
for which the perturbed orbit can remain periodic for all values of 
the masses. There are four: (2), (7), (12) and (13), for which the 
periodicity is only possible if a certain condition involving the masses 
is satisfied. 
For all solutions we find from the equation y, = 0 
el pe en 6.225. 
It needs hardly be pointed out, that this is only a rough approxi- 
mation, the higher orders of ¢; having been neglected. In the system 
of Jupiter’s satellites we find actually (see these Proceedings, March 
1909 ele, — 0,11. 
Further if we put 
EN 
Ze 
then we find, for the solutions (6) and (16), from w,, == 0 
ay B 
&, B €, 2nA 
If the longitudes are counted from the apocentre of III, and the 
time from a passage of [IL through this apocentre, we have, for 
t= 0, 7, = 180°, /, = 180’, therefore 2, = 0°. For the corresponding 
values for II and I we find, for t=O, for solution (6): 
RIN ne, == 1807 
l= 0 bl 
d=0 A == 160 
K =a, — 34, + 2A, = 180° 
and x is positive: the mean motion in anomaly exceeds the mean 
motion in longitude. This is the case of nature. 
For the solution (16) we find: 
mr 0° t, = 150 
b = 180 i = 180 
4, == 180 A= 0 
== 180° x negative. 
1) The expressions there given are based on Sovuittarr’s theory. The quantities 
s;‚ Which here appear as excentricities, are thus there considered as perturbations, 
and are called 2), 2, 2g. 
