Beck and Tuck 
which dictate the ultimate motions are complicated, but in most cases 
there are pairs of forces which contribute most to this balance, other 
forces being formally of a smaller order of magnitude with respect 
to a small parameter such as beam/length ratio. 
For instance, in heave and pitch the dominant force balance 
is between hydrostatic restoring force and the pressure of the inci- 
dent wave (so-called Froude-Krylov exciting force). Inertia, both 
natural and hydrodynamic (added mass), damping, and diffraction of 
the incident wave are all effects of lesser significance in the range 
of wave periods considered. Indeed, remarkably in shallow water the 
natural inertia or mass of the ship has the least influence of all these 
forces. Similar simplifications can be made to the other modes of 
motion, leading to ''first-order) theories involving only the dominant 
forces. 
However, the computations presented in Section 3 for coupled 
surge, heave and pitch do include all forces, not only those of first 
order. The first order computations are verified as numerically rea- 
sonable, and information is obtained about the most significant second 
order effects. For example, the diffraction exciting force (unfortunat- 
ely neglected by Beck and Tuck, 1971, in making similar comparisons) 
appears to be the most significant second-order contribution to heave, 
whereas pitch is affected more by added hydrodynamic inertia than 
by diffraction effects. 
In the case of surge the first order balance is between natural 
inertia and Froude-Krylov exciting force, and this first order result 
appears to be remarkably accurate. In particular, there appears 
little need to worry about coupling with the other modes, Some moor- 
ing force considerations are discussed in Section 4, the conclusion 
being that for large ships only surge is likely to be affected, and then 
perhaps only marginally. The general theory of surging of moored 
ships has been thoroughly treated in the Civil Engineering literature 
(see Wilson, 1967, for a bibliography), and perhaps the only new 
contribution we can make here concerns the correct evaluation of the 
surge exciting force as a function of hull geometry. This question is 
given some attention in Section 3 and Appendix III. 
In Section 5 we continue the theoretical treatment of the very 
difficult problem of horizontal plane motions, clearing up most but 
not all of the loose ends left by Tuck (1970) for sway, roll and yaw. 
The appropriate integral equations which determine the hydrodynamic 
coefficients in these modes have been set up, but the roll equations 
have not yet been solved. 
