Let , , and , , be the Cartesian coordinates
of the Moon in and , respectively.
It is easily demonstrated that (see Section A.16)

Moreover, if , , are the Cartesian components of the Sun in then (see Section A.5)

(1131) | |||

(1132) | |||

(1133) |

giving

where use has been made of Equations (1125)-(1127).

Now, in the rotating frame , the lunar equation of motion (1123) transforms to (see Chapter 7)

(1138) |

where , and use has been made of Equations (1134)-(1136).

It is convenient, at this stage, to normalize all lengths to , and all times to . Accordingly, let

and , and . In normalized form, Equations (1139)-(1141) become

respectively, where is a measure of the perturbing influence of the Sun on the lunar orbit. Here, and .

Finally, let us write

where , and , , . Thus, if the lunar orbit were a circle, centered on the Earth, and lying in the ecliptic plane, then, in the rotating frame , the Moon would appear stationary at the point , . Expanding Equations (1145)-(1147) to

Now, once the above three equations have been solved for , , and , the Cartesian coordinates, , , , of the Moon in the non-rotating geocentric
frame are obtained from Equations (1128)-(1130), (1142)-(1144), and (1148)-(1150). 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 *ecliptic
longitude* and *ecliptic latitude*, respectively. Moreover, it is easily seen that, to second-order in , ,
, and neglecting terms of order ,