According to Equations (11.138) and (11.150),

where we have neglected contributions. It follows from Equations (11.192), (11.193), and (11.204) that

(11.248) | ||||||

and | (11.249) |

To next order in , Equations (11.138) and (11.150) give

(11.250) | ||||||

and | (11.251) |

where use has been made of Equations (11.217) and (11.218), as well as the previous expressions for and . Hence, Equations (11.192), (11.193), and (11.204) yield

According to Equations (11.139) and (11.151),

(11.254) | ||||||

and | (11.255) |

Equations (11.192), (11.193), and (11.205) yield

(11.256) | ||||||

and | (11.257) |

It follows from Equations (11.76), (11.122)-(11.124), (11.170), (11.171), (11.182), and (11.183), as well as the previous expressions for , , , and , that the net perturbation to the lunar orbit due to terms in the solution of the lunar equations of motion that depend linearly on is

(11.258) | ||||||

(11.259) | ||||||

and | (11.260) |

Here, is the mass of the Earth, and the mass of the moon. The previous expressions are accurate to .

The first term on the right-hand side of Equation (11.259) is known as the *parallactic inequality*. The parallactic inequality attains its maximum amplitude when the Moon in half illuminated (i.e., when
or
). Conversely, the amplitude of the parallactic inequality is zero when the Moon is either fully illuminated or not illuminated at all (i.e., when
or
).
According to Equation (11.259), the parallactic inequality generates a perturbation in the lunar ecliptic longitude that oscillates with a period
of a synodic month, and has an amplitude (calculated using
,
, and
) of
arc seconds (Yoder 1995). As before, the oscillation period is in good agreement with
observations, whereas the amplitude is somewhat inaccurate [it should be
arc seconds (Chapront-Touzé and Chapront 1988)] because of the
omission of higher-order (in
) contributions.