The Rev. A. Thacker on a case of disturbed Elliptic Motion. 53 

 and, eliminating r and restoring the value of k, we have 



J..CB) 



(*+£) 



+ - • : Jg£{(93« 9 +198«i8-t ■93/3 9 )sintt-15(a s -£ 8 )siii2« [ 



+ (« -/3) 2 sin 3m}. J 



Again, 



f \/(*-r){r-fi {^g + /^ + «)(r + /3)} 



Integrating this in the same way, and to the same degree of 

 approximation, we shall have 



Lct cos (u-B)r = v > and cos «-/s =w ; then ifc 



is easily proved that 



Kr\Z£*4 (C) 



and, consequently, 



0=*>-g--^{3(a+/3) 2 u-(« 2 -/3 2 )sin M }. . (D) 



The equations (A), (B), (C), (D) determine the motion ; and 

 as there is no limitation with regard to the relative magnitudes 

 of a. and B, it follows that those equations arc true, however 

 much the orbit differs from a circle. 



The apsidal angle is the value of 6 corresponding to r=a, and 



1""5 •v / «/3(a + /3) 2 \, or 



^ » 2 « + £' 



if « be the mean angular motion of the body. 



In the lunar theory, the expression for the radial disturbing 



force contains a term —^n' 2 r, where n' is the mean motion of 



the earth, and the effect of this on the moon's motion may be 



found by substituting — - n' 2 for jjj in the preceding equations. 



For example, if we denote by m the ratio of the earth's mean 

 motion to that of the moon, the apsidal angle becomes 



