From (3.8) we obtain the second equation of the center of gravity for the first sub-system...
It follows, from the equations for the general center of gravity, that the
values of the sums
are independent of the (moment) t0 at which we are
observing the system.
This (consists of) a body of mass m2 and a general center
of gravity for n-2 bodies. The line of action of forces of the
second sub-system is othogonal to the line of the first, and the resultant
of forces of the interaction of a body of mass m1 with
bodies m2 and
is applied exactly (to) the general center of gravity of the latter.
are calculated semi-axes of Kepler's orbit aibi, on
which μi bodies are moving. Orbits of bodies with masses
m1, m2 are built as free
(multiplications) of n-1 Keplerian orbits.
The problem of three bodies in a straight line was examined first by Euler
 and then by Lagrange , and is characterized by coefficients
that give the (transformation) of a system of three bodies into two free
sub-systems, (which) are represented as non-real (one line of forces
(acting)). Lagrange's system of three bodies with
 during (diagonalization) of the quadratic forms are divided (into)
free bodies which don't act on one another, the same, as with similar
systems of four and six bodies.
4. Comparison of obtained results with (classical)
Let us do once again the separation of n-1 minima of quadratic form (3.8) on the axis of Descartes coordinates, according to Jacobi. Each of the component-minima, in expression (2.3), is related to the periods of the θ-functions.
Their free multiplication defines a structure of general and of particular
solutions of the problem. In this way a number of constants, shown by the
Bruns-Poincaré theorem, is exactly the same with any
as (the) number of constants, which define the obtained solutions. So, with any
all 6n coordinates of the centers of gravity, 3n of Kepler's
and n in particular integrals of energy, become linear transformations
(of) one from ten general, and the n-body problem is (for) any
a problem of 6th order (look at ). These particular integrals are
obtained as a result of bringing the quadratic (forms) (activity) toward the
main axes and represent, by themselves, solutions of canonical equations,
which totally correspond to the Hamilton-Jacobi theorem .
are solvable explicitly in quadrature.
Let us look at isoparametric equations (see )
Let us notice that an increase in the dimensions of space, for example, a change in phase , will cause the creation of integrals of motion that do not appear in any of the three classes that are given by the Bruns-Poincaré theorem, as well as changing of the physical dimensions of the integrals of motion,and this will cause the (condition) in which it will be impossible to integrate the problem.