The writers, with the aid of mathematics, succeeded in obtaining some insight into the question whether, from the point of view of ship vibrations, the stiffness of the engine Beds should be rigid or elastic. It was found that, if the frequency of the natural vibration of the engine bed is higher than that of the ship's body, the amplitude under resonance that corresponds to the ship's fundamental vibration does not diminish, and that at higher engine speeds, another resonance that corresponds to the vibration of the engine system is likely to occur. If, on the other hand, the frequency of the natural vibration of the engine system were lower than that of the ship's body, the amplitude under resonance that corresponds to the ship's fundamental vibration will be greatly reduced and, moreover, that the resonance condition that corresponds to the engine system, will scarcely be encountered. Naturally, vibration damping should exist in the ship's body, and, possibly, also in the engine system. In the case of high speed engines, it does not follow that it is always safe to stiffen the engine bed as rigidly as possible.It was also found that, in the case of ship vibration, coupled with vibration of the engine system, even should the position of the engine room be at the loop of the ship's vibrations, the amplitude at any point under resonance vibration of the ship's body, is not very marked unless the mass of the engine system is negligible, and also that the damping coefficients for ship vibration combined with the vibration of the engine system, are intermediate between the coefficient that is proper to the ship's body alone and the one that is proper to the engine system alone.