Equations (11.41)–(11.43) and (11.47)–(11.49) yield

It is also easily demonstrated thatThe Cartesian components of the lunar equation of motion, (11.33), are

(11.54) | ||

(11.55) | ||

(11.56) |

It is convenient, at this stage, to adopt the following normalization scheme:

with , , and . In normalized form, Equation (11.50)–(11.53) become whereas Equations (11.57)–(11.59) yield Here, where(11.75) |

Equations (11.62)–(11.65) and (11.69)–(11.71) yield

Likewise, (11.62)–(11.65) and (11.72)–(11.74) give(11.80) | ||

(11.81) | ||

(11.82) |

Finally, let us write

Here, is a constant, and , , , . Expanding Equations (11.66)–(11.68) and (11.77)–(11.82), and neglecting terms that are third order, or greater, in the small parameters , , , , , , and , we obtain where(11.89) | ||

(11.90) | ||

(11.91) | ||

(11.92) | ||

(11.93) |

After Equations (11.86)–(11.93) have been solved for , , , and , the geocentric Cartesian coordinates, (, , ), of the Moon in the non-rotating reference frame are obtained from Equations (11.44)–(11.46), (11.60)–(11.61), and (11.83)–(11.85). However, it is more convenient to write , , and , where is the radial distance between the Earth and Moon, and and are termed the Moon's geocentric (i.e., centered on the Earth) ecliptic longitude and ecliptic latitude, respectively. Moreover, it is easily seen that, neglecting terms that are third order, or greater, in the small parameters , , , and ,