Froude considered that the surface friction and the head re sistance of keels and deadwood was responsible for the resistance varying as the second power of the angular velocity and that the creation of a small wave at each roll causing dissipation of energy was responsible for the resistance varying as the first power of the angular velocity. These views have been generally confirmed as the expression for the decrement, viz., a0+ fits the curve of extinction of practically any ship fairly well if the coefficients a and b are judiciously selected. The experiments on "Revenge" referred to above were carried out in the light and in the deep conditions and before and after bilge keels were fitted to the ship.
The outstanding result of these experiments was the unexpected large increase in the value of the "b" coefficient of ter bilge keels were fitted and other experiments confirmed the result. Mr. R. E.
Froude, after experimenting with a deeply submerged plane oscil lating in water, found that the head resistance of the bilge keels only accounted for about one-fourth of the energy lost in a single swing due to the increased value of the "b" coefficient; and some cause other than head resistance must be operating to account for the discrepancy. The matter has since been closely investigated in other directions and it is generally agreed that bilge keels become effective not only by virtue of the head resistance de veloped but by indirectly influencing the streamline motions which would exist if there were no bilge keels. At the same time during the early portion of the swing the bilge keels set in motion a considerable mass of water the energy from which is for the most part dissipated ; motion of the ship ahead intensifies this loss and combined with the increase of other resistances at high speeds produces a more rapid extinction. For detailed consider ation of this portion of the subject reference can be made to papers of the I.N.A. by Professor Bryan in 190o and Mr. A. W. Johns in 1905 and 1909.
The problem of resisted rolling among waves was investigated by Froude, who after considering other methods developed and adopted the process of "graphic integration" which is an exact method of determining the motion of a ship, having given the elements of the ship's rolling in still water and the wave-series acting upon her. Reference should be made to Froude's paper in Trans. I.N.A. 1875.
Analytical consideration of the equation of motion shows that the ship has a free oscillation which in time dies out, leaving a forced oscillation in the period of the waves. Actual observation of rolling, however, shows that a ship seldom completely f or sakes her own natural period of roll; this is due to departure from exact uniformity in the waves encountered. For considera
tion of special conditions reference can be made to papers by M. Bertin in Trans. I.N.A. for 1894 and 1895. In a paper read at I.N.A. 1896 Mr. R. E. Froude investigated the probable maximum amplitude of roll when the ship is under the influence of a non synchronous and non-harmonic swell. Further papers read at the I.N.A. in 1896 and 1898 by Professor Kriloff dealt exhaus tively with the whole motion of a ship including pitching and rolling.
Another method of reducing rolling in a seaway to which con siderable attention has been given is by the use of water chambers. The action is mainly automatic and the motion of the water in the tank is not to be controlled to any extent except that the tank can be put out of action by throttling the motion or by emptying. The first ship fitted was "Inflexible" in which was provided an open rectangular tank extending across the ship. It was tried out at sea by Sir Philip Watts who also experimented on rolling models of "Edinburgh." In essence the method consists in fitting across the ship one or more tanks of such shape that when filled to a suitable height with water the motion of the water from side to side as the ship rolls is such as to retard the rolling. The water chamber reduces the angle of roll chiefly by modifying the right ing couple acting upon the ship throughout her rolling ; it increases the righting couple which opposes the motion as the ship heels over, thereby reducing the amount of heel and on the return roll it lessens the righting couple and causes the ship to move more slowly than she otherwise would; her angular momentum on reaching the upright is consequently less and she tends to roll less the other way. (See papers by Sir P. Watts, Trans. I.N.A. 1883 and 1885.) The tanks were of some use in reducing rolling but were excessively noisy, and extensive use has not been made of them.