The natural period of roll of ships in any condition can in general be obtained by the well known formula T =2π K √< gm >, where K is the transverse radius of gyration of ship, m the metacentric height and g the acceleration due to gravity. The radius of gyration has hitherto been expressed by the breadth moulded and recently by the breadth and depth moulded, but these simple methods can not afford satisfactory results. After investigations of rolling materials of many vessels, new formulae for K have been found to give sufficiently accurate values. For merchant vessels the formula is written as ( K/B )²= f [ C_bc_u +1.10 C_u (1- C_b )( H/d -2.20)+ H/B² ]. where B is the breadth moulded, C _b the block coefficient, C _u the upper deck area coefficient, H the effective depth, i e , the depth moulded plus mean height of projected areas on Profile of erections and deckhouses, d the mean draught and f the coefficient depending on the type of ship. The value of f is 0.125 for passenger, passenger and cargo, and cargo vessels. The formula for K of warships is of the same form as for merchant vessels. The errors of K -values obtained by the new formulae have been shown within the limits of about 3 per cent.