Prof. Challis m the Theory of the Moon's Motion, 523 



It may be remarked, that tlie method so far applies as well to 

 the planetary as the lunar theory. 



First Approximation. 



* 



Terms on the right-hand sides of the above equations involving 

 higher powers of r than the second will be omitted. Thus the 

 second equation may be put under the form 



^'Alt^'^) 3n'2sin2<f> ^, 

 — — _ i_ fjif^ 



«"■) Kf-') 



But by the hypothesis of the problem, the moon^s longitude 6 

 always differs by a small angle from a mean longitude nt-\-e. 



Hence -— +/i'=7i nearly, and dt= — ^. Substituting these 

 cLi n —^ n 



values in the above equation, integrating, and omitting the square 



of the disturbing force, we have 



Hence 

 and 



^ 37i'Vcos2(i) 



rd6 h , 37i'^r cos 2<f> 

 at r 4i{n—w) 



r^dffi^ h^ ^ ,, ,22/, , 3cos2<^\ , 

 -^ ^^-^nlh + n'^r^^l + —j-^) nearly. 



But by the equation (A), to the same approximation, 

 dr^ r^d6^ r, 2m' 2yu, Sti'V 



Hence by substitution, and altering the designation of the arbi- 

 trary constant, 



*V^.&-^,o.o ,0 



It may be observed, that in obtaining this equation it has not 

 been necessary to employ an approximate value of the radius- 

 vector. It follows, however, from the reasoning, that r differs 

 little from a mean value, because approximately 



r^(^^ni^^r^n = h. 



If the equation (C) be supposed to apply to a circular orbit, we 



2M2 



