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Astronomy. — “On the periodic solutions of a special case of the 
problem of four bodies’. By Prof. W. pr Sirrer. (Communi- 
cated by Prof. E. F. vAN DE SANDE BAKHUYZEN). 
The special case considered in this paper is that of a central body 
and three planets, or satellites, whose masses are small compared 
with the mass of the central body, and whose orbits are all situated 
in one and the same plane, the mean motions (in longitude) being 
roughly proportional to the numbers +, 2 and 1. This special case 
is realised in nature by the three inner Galilean satellites of Jupiter, 
if the inclinations, the influence of the sun and of the fourth satellite, 
and the compression of the planet are neglected. This latter restriction 
is not essential, since the compression does not disturb the periodicity, 
provided only the motions take place exclusively in the plane of the 
planet’s equator. 
Neglecting at first the relation between the mean motions, we will 
consider the periodic solutions of the problem thus generalized for 
the case that the masses of the satellites are zero, i.e. for the unper- 
turbed problem. These may be divided into two kinds, analogous 
to PorrcarÉ’s well known classification of the periodic solution of 
the problem of three bodies. In the solutions of the first kind (sorte 
première of Poincaré) the (unperturbed) orbits of the satellites are 
circles, in those of the second kind they are Keplerian ellipses with 
arbitrary excentricities. 
The solutions of the first kind exist, if the differences of the mean 
motions are commensurable, thus: 
vy, — TY; = pr, Yv, — Ps = qv, 
p and g being integers, mutually prime. This condition can also be 
expressed by saying that the mean motions must satisfy a linear 
equation of the form 
av, + Br, + yr, — 9, 
where a, 2 and y are mutually prime whole numbers, satisfying 
the relation 
atp+y=—0. 
The mean motions can then be expressed thus: 
hy So, De t= ¢, Ps VY, = CV — %, 
1 
where ¢,, ¢,, C,; are again whole numbers. We have then: 
Led 
OC p = Cs — ¢, y= CT; 
De ts gt 
Then, if we put 
