Prof. Challis on the Theory of the Moon's Motion. 527 



The first approximations to the radius-vector and the true longi- 

 tude in terms of the mean longitude as given by this solution, are 



r—a—aecosc.nt + e + y 



6=:nt-^€ + 2eamc.nt + € + y. 



Second Approximation. 



The same terms of the equations (A) and (B) are to be used 

 for the second approximation as for the first, the next terms 

 being of a higher order by two degress. Thus the equation (B) 

 gives 



dSr''(^^+n')y=-'^ n'^r^ sin 2<t> dt. 



Putting, for shortness' sake, jo for the angle c(n^ + e + 7), and q 

 for [nt + e) — [nH + e'), and using the values of r and 6 given by 

 the first approximation, it will be found that 



r^mi2(f> = «2(sin2g' + esm{2q +p) — Sesin {2q — jo)). 



Substituting in the equation above, integrating, and omitting 

 terms of the fourth order. 



Hence by squaring, and substituting the approximate value of r 

 in the small term, 



r^^=!^-2n'h + n'^r^ 

 dt^ r^ 



+ 



(5e \ 



cos2g'+ -5-cos(2g'+jo)— 5ecos(25'— jo)j 



2 



Again, by a like process, 



r^ cos 2(^ = a^('cos 2g' — 3e cos(2g'— jo) + e cos(2g' +jo)). 

 Hence the equation (A) gives 



s^+^i^=^^+-«^+y-^-2- 



+ — 2 — (cos2g'— 3ecos(2g'— ;?) + ecos (2g'+p)). 



Consequently, by subtraction, 



£ + ^ -y -"P + C = «'V<3cos(3(?-i>) -cos(3g +;>)). (C) 



It will be seen that this approximation has introduced no new 

 term independent of the sun's longitude. Putting, for brevity, 



