( 683 ) 
and if we count the time from the instant-of a conjunction of II 
and III, and the longitudes from the common longitude of these 
satellites for that instant, we have 
A= Ct —v 
A, == CT —V 
A, =cr—vtK 
aA, BA, + yA, =X. 
After the lapse of the period 
2% 
T= — 
v 
the relative positions of the four bodies are the same as for the 
instant ¢—= 0, the whole system being rotated in a retrograde direction 
through the angle x7’. 
By a reasoning entirely similar to that used by Porncars') for the 
solutions of the first kind of the problem of three bodies, we find 
easily that the condition, that these solutions shall remain periodic 
if the masses have small finite values, is 
KS 00° ar "4805. 
In other words, there must be a symmetrical conjunction or opposition 
of the three satellites at the beginning of the period. *) 
The reasoning by which the existence of these solutions for small 
values of the masses is proved, fails in only one case, viz. when 
= = 0 or a whole number. 
This exceptional case is analogous to the well known exceptional 
case for the periodic solutions of the first kind of the problem of 
three bodies. 
For the special case of Jupiter’s satellites we have 
tka 3,7 — 2, K = 180° 
A, = Art — v + 180° 
A, = 2t — v 
Ast 0 
Td, — 4, == A, aA, 
In the system of Jupiter we find that v is small compared with r. 
We have roughly (in degrees per day): 
po dl .0571 
Pe 0.1900. 
1) Les méthodes nouvelles de la mécanique céleste, tome I, § 40. 
3) See also Les methodes nouvelles, t. 1. 8 50. 
