14 LECTURES ON THE LUNAR THEORY. [LECT. III. 



These quantities are to be substituted where they occur in the ex- 

 pressions for the forces found in Lecture II. 



Let us now make a few general remarks upon the result of the 

 substitution. 



It will be observed that the disturbing forces all involve the coeffi- 

 cient m'a'~ 3 . It is very important to notice that the Sun's parallax is 

 not required for the evaluation of this quantity. By Kepler's Third Law 

 it is derivable from observations of the Sun's mean motion alone. Other 

 terms however, namely those with the coefficient m'/a'^u, involve the 

 Sun's parallax directly ; and that constant may be obtained by comparing 

 the observed with the theoretical values of the coefficients of those in- 

 equalities, with an accuracy probably greater than that of any other 

 method. 



The mean disturbing force is radial, and is equal to 



2 a* 



or the mean effect of the Sun's disturbance is to diminish the Moon's 

 gravity towards the Earth ; and to diminish it more, the greater is the 

 eccentricity of the Sun's orbit. Now e' has been diminishing for ages ; 

 hence the Moon's gravity towards the Earth has been increasing, and its 

 average time for accomplishing a revolution about the Earth has been 

 diminishing. 



This is one cause of the Secular Acceleration of the Moon's mean 

 motion which Halley derived from the records of ancient eclipses. 



It may also be noticed that the coefficient of the chief periodic part 

 of the disturbing force, which involves 1 - e'-, increases as e' diminishes. 



2i 



Finally let it be observed that the term with argument 



2 (6 -n't) + 2 (n't -m'), 

 which does not involve the Sun's Mean Longitude, is absent from the 



development of (-7) 2(8 8'). 

 \r / sin v 



