The equations of motion for a ship pitching and rolling in a calm sea were found to be coupled by means of a second order term in the rolling equation of motion. When the pitching motion is simple harmonic the equation of motion in roll is of the Mathieu type, the two independent parameters being pitching amplitude and frequency. The stability diagram of the Mathieu equation including the effect of linear damping is determined theoretically and verified by solving the Mathieu equation on an analogue computer. In experiments with one ship model form, it was found that the transition from stable to unstable motion occurred for the value of the parameters predicted by the theory, and that the rolling motions obtained experimentally closely resembled the computer solutions. It was concluded that (1) unstable rolling motions are possible for very small pitching amplitudes, (2) when the natural frequency of pitch is twice that of roll the amplitude required to produce unstable motion is a minimum, and (3) for any pitching frequency, the greater the amount of damping, the greater the amplitude of pitching must become to produce unstable rolling motions. (Author).

An investigation was made of the effect of certain nonlinear coupling terms on the heave-pitchroll motions of a ship moving in longitudinal waves (either head or following waves). Linear damping terms were retained in the analysis because when forced ship motions under wave excitation are studied, the damping terms are essential elements of the problem. The analysis for the heave-pitch-roll system showed the existence of a type of instability such that the responses in pitch or roll may grow without bound even when the effect of damping was taken into account because of nonlinear coupling between heave and pitch and between heave and roll. This is in contrast to the results obtained from the linear theory in which, because of the presence of damping, all the responses are finite. (Author).