39 



^^ cos {2 ^— 2m,} 

 hence (^ + Vf^hr^=—-\ -{sW") ^ ''''^ (^ v-2m*-T) 



neglecUng quantities multiplied by Ji m^ as being of the order m* 



2. £±—.^ - I m"-e sin (2 u—2 sin (v— ^) (I— 3e cos (v— ^r)) 



3^. fe cos {v—2m k + t) 

 — "f7|^ — e cos (3 f — 2m V — t) 



3. Lastly - ^-^i^t = - ^+1^ + 2" " Va ^ *^«^ ("-- ) 



cos 2 V —2m I* 



S+.m 

 2a~ 



•a- 



3m« J — f+L^ g COS (v— 2mc+5r) 

 ^ _ 3^! ^ ^^g ^_^ v—2mv^ 

 Substituting these quantities in equation (c) it becomes 

 V^^M 7-17 + :r- -TT ecosfv— ^)+ 5-1 -. jcos (2»— Swif) 



-^] 2jli ^ ^os(.-2»o+T)-^j 2_2^iecos(3i— 3m.) 



(."''i— 2;n5 t+3_2»J 



= o. (d) 



Each of the terms after the four first terms are contained in the 

 form m^K cos (cc — «■) : it is therefore required to integrate an equa- 

 tion of the form 



^-T + M — G-^ mP- K. (cos cv— t) + ^'c =0 



G and K being constant quantities. 



