where

(1090) |

Let be the value of at the Lagrange point, for . When ,
it is easily demonstrated that

(1091) | |||

(1092) | |||

(1093) | |||

(1094) | |||

(1095) |

Note that .

Figures 50-54 show the intersection of the zero-velocity
surface with the - plane for various different values of , and
illustrate how the region from which is dynamically excluded--which we shall term the *excluded region*--evolves as the value of
is varied. Of course, any point not in the excluded region is in the so-called *allowed region*.
For , the allowed region consists of two
separate oval regions centered on and , respectively, plus an
outer region which lies beyond a
large circle centered on the origin. All three allowed regions are separated
from one another by an excluded region--see Figure 50. When ,
the two inner allowed regions merge at the point--see Figure 51.
When , the inner and outer allowed regions merge at the point, forming a horseshoe-like excluded region--see Figure 52.
When , the excluded region splits in two at the point--see Figure 53.
For , the two excluded regions are localized about the
and
points--see Figure 54. Finally, for , there is no excluded
region.

Figure 55 shows the zero-velocity surfaces and Lagrange points
calculated for the case . It can be seen that, at very small values of
, the and Lagrange points are almost *equidistant* from mass .
Furthermore, mass , and the , , and Lagrange points all lie approximately
on a *unit circle*,
centered on mass . It follows that, when is small, the Lagrange points , and all
share the orbit of mass about (in the inertial frame) with being directly opposite ,
(by convention) ahead of , and behind.