Let us define a Cartesian coordinate system
in an inertial reference frame whose
origin coincides with the center of mass, , of the two orbiting masses. Furthermore, let the orbital plane of these masses
coincide with the - plane, and let them both lie on the -axis at time --see Figure 47.
Suppose that is the constant distance between the two orbiting masses, the constant distance
between mass and the origin, and the constant distance between mass and the origin. Moreover,
let be the constant orbital angular velocity. It follows, from Section 6.3,
that

where .

It is convenient to choose our unit of length such that , and our unit of
mass such that . It follows, from Equation (1030), that . However, we shall continue to
retain in our equations, for the sake of clarity. Let , and
. It is easily demonstrated
that , and
. Hence, the two orbiting masses, and , have position
vectors
and
, respectively, where (see Figure 47)

Let the third mass have position vector . The Cartesian components of the equation of motion of this mass are thus

where