According to Equation (11.155),

(11.305) |

It follows from Equations (11.209), (11.213), and (11.217) that

According to Equation (11.156),

(11.307) |

It follows from Equations (11.209), (11.214), and (11.217) that

According to Equation (11.152),

(11.309) |

It follows from Equation (11.217), as well as the previous expression for , that

(11.310) |

The preceding expression can be combined with Equation (11.208) to give

(11.311) |

The non-trivial solution of this equation is such that

(11.312) |

It follows from Equation (11.210) that

(11.313) |

Thus, setting , we get

The arbitrary parameter is chosen such that the parameter , appearing in Equation (11.124), is the same as in the undisturbed motion. Thus, making use of Equation (11.184),

Hence, Equations (11.306) and (11.308) reduce to

(11.316) | ||||

and | (11.317) |

It follows from Equations (11.122)-(11.124), (11.184), (11.187), and (11.188), 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

(11.318) | ||||||

(11.319) | ||||||

and | (11.320) |

The previous expressions are accurate to .

The first term on the right-hand side of expression (11.320) is Keplerian in origin (i.e., it is independent of the perturbing
influence of the Sun).
The second term, which is known as *evection in latitude*, is due to the
combined action of the Sun and the inclination of the lunar orbit to the ecliptic. Evection in latitude can be thought of as causing a slight increase in the inclination of the lunar orbit
at the times of the first and last quarter moons (i.e., when
and
), and a slight decrease at the times of the new moon and
the full moon (i.e., when
and
). (See Exercise 4.) Evection in latitude generates a perturbation in the lunar ecliptic latitude that oscillates sinusoidally with a period of 32.3 days, and has
an amplitude (calculated with
and
) of 602 arc seconds. As before, the oscillation period is in good agreement with observations, but the
amplitude is somewhat inaccurate [it should be 624 arc seconds (Chapront-Touzé and Chapront 1988)] due to the omission of higher order (in
and
) contributions.