519 



giving 



B=-ll; Cr+TV (42) 



Thus, if we neglect eccentricities and inclinations, and con- 

 fine ourselves to the first power of the disturbing force, or to 

 the second power of »?, the perturbation of the moon's vector, 

 produced by the sun's attraction, is composed of the three fol- 

 lowing terms : 



S/3= -^>-l^\-/37 + ^^-/3-'7-'/37/3. (43) 



The first of these three terms expresses that the sun's abla- 

 titious force, by partially counteracting the earth's attractive 

 force on the moon, allows our satellite to revolve in a some- 

 what smaller orbit than would otherwise be consistent with 

 the observed periodic time : the ratio of the diminished to the 



7)1 



undiminished radius of the orbit being that of 1 — to 1. 



The second term expresses a displacement of the moon, through 

 perturbation, from its diminished circular orbit, of which dis- 

 placement the constant magnitude or length bears to the radius 



of the undiminished orbit the ratio of — — to unity ; while 



lb *' 



the direction of this displacement is always J/'om that fictitious 

 moon, to which it has been seen that one of the two principal 

 components of the sun's disturbing force is directed : an oppo- 

 sition of sign which may at first surprise, but which is exactly 

 analogous to the contraction of the orbit produced by the ab- 

 latitious force (when the periodic time is given), and is to be 

 explained upon similar principles. Finally, the third term of 

 the formula (43) for Sj3, expresses that with the two foregoing 

 displacements a third is to be combined, which is, like them, of 

 constant amount, being equal to -/^ths of the second displace- 

 ment, or bearing to the radius of the moon's orbit the ratio of 



-rrr to Unity ; but being always directed to what, by an exten- 

 sion of a recently employed phraseology, might be called the. 



