If N = 1, then aQ = 0, and the last term in Equation [ 1 5a' ] vanishes. 

 If N > 1 , 



• U N-1 27r\ . /N 27r\ 



1=1 i = sm — 



and again the last term in [l5a'] vanishes. 



Therefore, the equation of motion for xq is 



N 

 2(Mq + NM)xq + pNSj P = 2 pgkATQ ^ cos (ka^ cos 9- - ojt) [l6a] 



1=1 



By similar arguments, the equation of motion for y-Q, Equation 

 [ 1 5b] , becomes 



2(Mq + NM)yQ - pNSj (S = [l6b] 



The equation for z^,, ■written out, is 



N 

 (Mq + NM)(zq + g) = pgkATQ y sin (kaQ cos e. - ojt) + N pgSg 



1 = 1 



N 



- pgAS(O) y^ sin (kaQ cos 6- - wt) - pgNzQS(O) 



1 = 1 



N N 



- pgaagSfO) y sin e. + pgpaQS(O) ^^ cos 6- 



1=1 1=1 



Again we note that Mq + NM = p Sq N, which enables us to eliminate the 

 gravity term on the left. Also, 



N 



N > 1 



1 



1=1 



y^ cos e^ = ; 



18 



