r 2 2 



i = m(z/)(x + y )dz.' 



= 1 



.^ 2 J . 2 



m(zp aQ dz-' = ag M 



The integral terms in Equations [l5d] and [I5e] can also be written 

 explicitly by expanding x and y in the integrands. Thus 



m(z.'')y dz.' = J m (z/) [ a^ sin 0. + y^ + y ag cos 9. - a z.'] dz." 



= (aQsin9. +y„+ya„cos9.)M-Q' m(z.')z.'dz.'' 



I m(z.'')xdz.'= m(z.')[aQCOs9. +x^ + (3z.'-Ya^sin9.]dz.' 



i— I L_j 



= (aQ cos 9. + Xq - y a^ sin 9. ) M + p I m (z.') z.' dz.' 



We note that the final integral terms here ■would have vanished if z' had 

 been measured from the spar center of gravity. 



Now let us write the equations in full. Equation [I5a] becomes 



N 



(Mq + NM)xq = 2pgkATQ y cos(kaQCOs9.-ajt)-pNxQSQ 



i=l 



N 

 -pNpSj^ + paQ-ySQ ^ sin 0^ 



i=l 



Clearly, Mq + NM = pSgN, since at rest the buoyancy of the total raft 

 equals its weight. Also, we now impose the condition that the spars have 

 a regular angular spacing. Thus 



3 = + ^(i- 1) ; i = 1,2, . . .,N 

 1 1 N 



17 



