PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 123 
transformation, although progress has not been as marked here as in the 
other investigations connected with the problem of three bodies. Besides 
the general problem of three bodies and of 7 bodies, several problems of a 
more special character are often considered, such as the problem of three 
bodies in a plane, and the restricted problem of three bodies. The last- 
named, which has occupied a prominent place in recent researches, may 
be described as follows : Two bodies, S and J, revolve round their centre 
of gravity in circular orbits, under the influence of their mutual attrac- 
tion. <A third body P without mass (i.e. such that it is attracted by S 
and J, but does not disturb their motion) moves in the same plane as S 
and J. ‘The restricted problem of three bodies is to determine the motion 
of P. This problem was first discussed by Jacobi! in 1836, who showed 
that it depends on a system of differential equations of the fourth order, 
one integral of which can be written down. This is now generally called 
the Jacobian integral of the restricted problem of three bodies. 
The most satisfactory reduction of the differential equations of the 
problem of three bodies, previous to 1868, was that of Bour? Bour first 
applies a theorem.due to Jacobi* and Bertrand,‘ in which, by making 
use of the integrals of motion of the centre of gravity, the problem of 
three bodies is made to depend on the motion of two fictitious masses 
m, and m2, whose potential energy depends only on the lengths of the 
lines joining them to each other and to the origin. our takes as his co- 
ordinates q, and g», the distances of m, and mz, respectively from the 
origin ; g; and q,, the angles made by q, and q, respectively with the 
intersection of the plane through the bodies and the origin with the 
invariable plane ; p, and p,, which denote ms and m, it respectively ; 
a 17 
and p; and p,, which are the components of angular momentum of m, 
and mz, respectively, in the plane through the bodies and the origin. 
With these coordinates the equations become 
dp;_ cH dq; _ 
ats) ight tae 
cH . 
Sar) (i=1, 2, 3, 4) 
where H is a certain function of the quantities p and g, and H=constant 
is an integral of the system. 
For the rectification of an error in Bour’s paper, see Mathieu’s paper 
of 1874, referred to later in this section. 
For the problem of three bodies in a plane, Bour’s system becomes 
IN Ea RS ME AE 
dt. dq) at Cp; 
where 7, P2; 91, 72 are defined as before, but 3 is now the angle between 
q and qo, and p; is the difference of the angular momenta of m, and m, 
round the origin. 
The problem was reduced in various ways to systems equivalent to, or 
' Comptes Rendus, iii. pp. 59-61. 3 
* ‘Mémoire sur le probléme des trois corps,’ Journal de UV Lcole Polytechnique, 
xxi. pp. 35-8, 1856, 
8 «Sur lélimination des noeuds dans le probléme des trois corps,’ Crelle, xxvi. 
pp. 115-131, 1843. 
* Mémoire sur l'intégration des équations différentielles de la mécanique,’ Ziou- 
ville, xvii. pp. 393-436, 1852. 
