5^ Prof. Challis on the Theory of t/ie Moon's Motion, 



h^ 2/L6 w'^r* 

 Q for 2 H 1 — ^ C, the approximate value of dt may 



be thus expressed : 



(tv n u e 



'^^^ VS "" ^Q* ^^ ^^^ ^^^""-^^ ~^°^ ^^^ '''^^'^^^' 



The first term of the right-hand side of the equation is to be 

 integrated just as in the first approximation. In the other term 



we may substitute for Q* the approximate value of -p, viz. 



n^a^e^ sin*/?. Then observing that 



r&cos{2q-p)-co,{2q+p)^^ 2co8 2g 



J sin*j9 wsinja ^ 



we obtain by integration, 



> ^ . . ,«— r 1 , n'^ cos 20 . 



c(n/ + e+y)=cos-^ .v/flV— (a— r)2+ -2 ^ — -^ 



" ae a ^ ' ^ n^e sinjo 



^^2 



But \/«V--(a— r)*=flesin;3 nearly, and — =m*. Hence it 



will be readily seen that 



1 — e C0SJ9 + X- — -^ cos 2/>— m^ cos 2q ) . 



Again, by what is proved above, we have to quantities of the 

 second order, 



de h 3»'2 



Hence 



W = 7^ + ^'"''^9- 



'hdt . 3m^ 



Putting for r the value just obtained, and for h its value 

 wa'( 1— »- j, and integrating, the result is 



5g2 11m* . 

 ^ = 71^4- e + 2e sin »-|- —r- sin %p-\ ^— sin2^. 



We have thus arrived at the values of r and d given by the 

 second approximation. But it does not follow that these are 

 the complete values to the second order of small quantities; 

 because, on proceeding to the next approximation, terms of the 

 third order may rise to the second order by integration. This 

 will be the case with terms of which the circular function con- 

 tains the arc 2{q—p). I propose, therefore, to enter upon the 

 third approximation so far as may be necessary to discover terms 

 which rise to the second order by integration. For this purpose 



