29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 233 



eccentricity and inclination, I find 



8s = - l"-424 cos (nt -C) + 0"'048 cos ( - nt + 2Nt -C)- 0"'007 cos (3nt - 2Nt - C). 



The reason why, in the result found by the Astronomer Royal, the 

 terms which are multiplied by t do not completely destroy each other, as 

 they ought to do, appears to be the following. 



It is at once seen, from the form of the periodic terms to which the 

 Astronomer Royal confines his attention, that his investigation is only com- 

 plete with respect to the terms which are independent of the eccentricity 

 and inclination of the Moon's orbit. In order to take the eccentricity and 

 inclination into account, other periodic terms must be included, the argu- 

 ments of which involve the Moon's mean anomaly and its mean distance 

 from the node. From the combination of these terms with each other 

 will arise terms with the same arguments as those which are independent 

 of the eccentricity and inclination, while each of their coefficients contains 

 the square of one of these elements as a factor. Hence it is clear that 

 terms of this order are omitted in the investigation. 



On the other hand, a slight examination shews that the coefficients in 

 the Astronomer Royal's expressions for 



- cos I and v. 

 a 



as well as in the quantities taken from his Factorial Table, include very 

 sensible portions depending on the squares of the eccentricity and inclination. 



In fact, it is plain that this must necessarily be the case since the 

 quantities in question are functions of the Moon's actual coordinates, in 

 which the numerical values of those elements are essentially involved. 



Now, if terms depending on the squares of the eccentricity and incli- 

 nation were either wholly neglected, or completely taken into account, the 

 terms which are multiplied by t would be found identically to destroy each 

 other ; but if, as in the present case, such terms are taken into account 

 in one part of the investigation, and omitted in another part, it will follow 

 that some of the terms multiplied by t will remain outstanding. 



A curious circumstance relating to this inequality of latitude remains 

 to be noticed. 



In the Mecanique Celeste, tome in. p. 185, Laplace proves that the 

 plane of the Earth's orbit in its secular motion carries the plane of the 

 A. 30 



