and since each oscillation is performed in the same period, each of these spaces also represents a certain number of seconds. Any ordinate, drawn at right angles to OX, through the points marking these equal spaces, shows the extreme angle of heel reached at that particular oscillation; and the difference between any two ordinates so drawn shows the loss of range, or extinction of the rolling, in the corresponding number of oscillations. For example, after making

twelve oscillations from the extreme angle (133 degrees) where the record of observations begins, the Sultan only reached an extreme angle of 8 degrees, the loss of range in that number of rolls being 53 degrees. Here the rate of extinction was slow, the vessel having a large moment of inertia, no keel, and only shallow bilge-keels, to assist the extremities in developing resistance to the motion. If there were deeper bilge-keels, the rate of extinction would be much more rapid.

Experiments have been made by Mr. Froude to show how

rapidly the rate of extinction may be increased by deepening bilge-keels. A model of the Devastation was used for this purpose, and fitted with bilge-keels which, on the fullsized ships, would represent the various depths given in the following table. The model was one-thirty-sixth of the full size of the ship, and was weighted so as to float at the proper water-line, to have its centre of gravity in the same relative position as that of the ship, and to oscillate in a period proportional to the period of the ship. In smooth water it was heeled to an angle of 8 degrees, and was then set free and allowed to oscillate until it came practically to rest, the number of oscillations and their period being observed. The following results were obtained :

The great advantages resulting from the use of bilgekeels are obvious from this table. It will be noted that the period of oscillation is changed but very little as the resistance becomes increased; this being a result which theory had predicted, and one which justifies the use of the hypothesis of unresisted rolling in approximating to the period of a ship. Increased resistance, however caused, is equivalent to an increase in the moment of inertia, and therefore tends to lengthen the period somewhat from that for unresisted rolling, but the difference is not so great as to require attention in the broad practical deductions with. which we are chiefly concerned. The rate of extinction of the oscillations depends upon the proportionate effect of the resistance and the moment of inertia of the ship; and

other things being equal, the ship which has the greater moment of inertia will be the more difficult to set in motion, but afterwards her motion is likely to be longer sustained.

The investigations by which the value of the resistance. is deduced from curves of extinction are of such a character that they cannot here be reproduced, but they proceed on the principle that the loss of range per oscillation represents an amount of "work" done by the resistance, and this amount can be ascertained by calculating the dynamical stability corresponding to the loss of range. Mr. Froude has been the chief investigator in this field, and his published analyses of numerous experiments are full of interest and instruction. Not content with obtaining the aggregate value of the resistances for ships, he has separated them into their component parts, assigning values to frictional and keel resistances, as well as to surface disturbance. In doing so, Mr. Froude has been led to the conclusion that surface disturbance is by far the most important part of resistance, as the following figures given by him for a few ships will show.

Frictional and bilge-keel resistances in this table have been obtained by calculation from the drawings of the ship, Mr. Froude making use of data as to coefficients for friction and for head resistance which he had previously obtained by independent experiments, and which may therefore be regarded as leading to thoroughly trustworthy results. The total resistance in each case was deduced from the curves of extinction obtained from still-water rolling ex

The formulæ for dynamical stability will be given and explained farther on in this chapter.

periments; and this also must be regarded as accurate. But it will be noticed that in no case does the sum of the frictional and keel resistances much exceed one-fourth of the total resistance, while it is much less than one-fourth in other cases. The consequence is that surface disturbance. must be credited with the contribution of three-fourths or thereabouts of the total resistance, a result which could scarcely have been predicted. Waves are constantly being created as the vessel rolls, and as constantly moving away, and the mechanical work done in this way reacts in a reduction of the amplitude of successive oscillations. Very low waves, so low as to be almost imperceptible, owing to their great length in proportion to their height, would suffice to account even for this large proportionate effect. For example, Mr. Froude estimates that a wave 320 feet long and only 14 inch in height would fully account for all the work credited to surface disturbance in the fourth case of the preceding table.

Another important deduction from the figures in the table is the large proportionate effect of "keel" resistance, as compared with frictional resistance, thus confirming what was said above as to the advantages of deep bilge-keels. Ships of the Royal Navy recently constructed have been furnished with much deeper bilge-keels than were formerly in use, but a limit to the depths that can be fitted is often reached, because of the necessity for compliance with certain conditions and extreme dimensions in order that the vessels may be able to enter existing docks. The evidence in favour of the use of bilge-keels is now considered unquestionable, and hereafter examples will be given of their usefulness as regards limitation of the rolling of ships in a seaway; but only a few years have elapsed since eminent naval architects, like M. Dupuy de Lome, of the French navy, regarded bilge-keels with suspicion. The change of opinion that has taken place is mainly the consequence of direct experiment and careful observation; and it furnishes another instance of the use of theoretical investigations in giving a

direction to practice; for these investigations had led to the conclusions now generally accepted before experimental knowledge had reached its present stage.

Exception has been taken by some eminent French writers to the views propounded by Mr. Froude as to the relative influence of the several component parts of the fluid resistance; but their objections are in no way directed against the experimental data obtained by Mr. Froude, these data being matters of actual observation, and not of theoretical calculation. We are bound to say, however, that, after carefully considering these objections and the accounts of French rolling experiments which have been published, we are strongly of opinion that Mr. Froude has the best of the argument; and his view of the importance of surface disturbance derives considerable support from experiments made on very special forms of ships. For example, in

experimenting upon the model of the Devastation, Mr. Froude found that, when the deck-edge amidships was considerably immersed before the model was set free to roll, the deck appeared to act like a very powerful bilge piece, rapidly extinguishing oscillations. MM. Risbec and de Benazé, of the French navy, also found by experiment that, when bilge-keels were moved high up the sides of a vessel, so that, as she rolled, the bilge-keels emerged from the water and entered it again abruptly, their effect became much greater than when they were more deeply immersed; as one would anticipate from the increased surface disturbance that must exist when the bilge-keels are so high on the sides. Experience with the low-freeboard American monitors furnishes further support to this view; immersion of the deck and the existence of projecting armour developing greatly increased resistance-a circumstance which undoubtedly tells much in favour of these vessels, and assists in preventing the accumulation of great rolling motions.

Fluid resistance to the motion of a floating body, or of a body immersed in it, depends upon the rate of motion. When a flat surface is pushed forwards, the direct or head