Since the motions are assumed to be small enough that squares and prod- 

 ucts of these variables are negligible, the location of the raft is complete- 

 ly specified, and the three angular displacennent variables can be treated 

 as the components of a vector. 



Let r ' be the position vector of a point fixed in the raft, where 

 r' = (x',y',z''). In terms of the space -fixed coordinates, r = (x, y, z), 

 ■we have, to first order in small quantities: 



rz^r'+rpj+axr' [l] 



where 



a = (a, (3,\ ) 



In terms of components, this equation is equivalent to 



X = x'+ Xq+ pz'-'Yy' 



y = y'+yQ+yx'-az' [1"] 



z = z'+ZQ+Q-y'-px' 



To first order also, it follows that 



x' = x-XQ-(3z+Yy 



y' = y-yQ-yx+Q'z [1"] 



z' = z— ZQ-ay+Px 



The unprimed coordinates are related by 



X = aQ cos 9. + x^ 



y = ag sin 9^ + y. [2] 



