According to Equation (11.153),

(11.321) |

It follows from Equations (11.209), (11.211), (11.290), and (11.315) that

According to Equation (11.154),

(11.323) |

It follows from Equations (11.209), (11.212), (11.290), and (11.315) that

According to Equation (11.127),

(11.325) |

where use has been made of Equation (11.315). Equation (11.189) yields

(11.326) |

Finally, according to Equations (11.130) and (11.142),

(11.327) | ||||||

and | (11.328) |

where use has been made of Equation (11.315). Hence, Equations (11.192), (11.193), and (11.196) yield

It follows from Equations (11.122)-(11.124), (11.159), (11.162), (11.174), (11.185), and (11.186), as well as the previous expressions for , , , , and , that the net perturbation to the lunar orbit due to the remaining terms in the solution of the lunar equations of motion is

These expressions are accurate to and .

All of the terms on the right-hand sides of Equations (11.332) and (11.333) are Keplerian in origin (i.e., they
are independent of the perturbing influence of the Sun).
The term on the right-hand side of Equation (11.332) is due to the slight inclination of the lunar orbit to the ecliptic plane, and is
known as
the *reduction to the ecliptic*.