that of the effect of squalls or gusts of wind, the danger of capsizing is evidently much increased. What was said respecting the action of gusts on a rigged ship rolling in still water does not exactly apply here, because the wave normal, with respect to which the instantaneous righting moment has to be estimated, is constantly changing its position, while the wind maintains its horizontal direction. Various attempts have been made to deal with this problem, but none of them are entirely satisfactory. For our present purpose it will therefore suffice to say generally that the sudden action of the wind upon the sails, coming, as it may, at the extreme of the roll to windward, must tend to increase the return roll to leeward, beyond the range which would be attained under the action of the waves alone, or of the waves and the steadily applied wind. A range of stability sufficient to provide against such impulsive actions ought to be secured in all rigged ships; and experience leads to the conclusion that, to secure a proper margin of safety, the range of the curves of stability for such ships ought to be at least 70 or 80 degrees. The Admiralty Committee on Designs fixed 50 degrees as the minimum range to be desired in the curves of stability for mastless ships; so that the range named for rigged ships is about 50 per cent. greater. It will be understood that these limiting values of the range of stability are not direct deductions from theory, but rules based upon experience, and probably providing a reasonable margin of safety.

It has been shown that a ship accompanying the motion of the waves, and heaving vertically, is subjected to accelerating forces which affect her apparent weight, just as similar accelerating forces affect the pressure of the water in the wave. Actual observations have proved the apparent weight of a ship to have varied about 20 per cent. above and below the true weight;* and the importance of this variation must not be overlooked, because the righting moment at

* See remarks made on page 155.

any instant involves as a factor the apparent weight, which may be above or below the true weight. Take the case where it is less than the true weight-i.e. the upper half of the orbits, or the upper half of the waves. Then, since the force of the wind is not affected by the wave motion, it must during this time have a greater inclining effect upon the vessel than the same force of wind would have in still water; and in vessels of low freeboard, such as the Captain, where the curve of stability is exceptionally small in area, and flat-topped (see Fig. 47), this virtual loss of righting moment due to the vertical heaving motion may prove another cause of danger. Of course, in the lower half of the orbit the apparent weights and the instantaneous righting moments are greater instead of less than in still water. And it must also be remembered that for a ship on a wave the vertical accelerating forces become zero, the true weight being the apparent weight, at nearly the same time that the wave normal reaches its maximum inclination (as in Fig. 72). Both these are circumstances telling in favour of the ship; but at the same time the subject now briefly mentioned is certainly one deserving attention in discussions of the safety of rigged ships rolling amongst waves.

Only a passing notice has been bestowed hitherto upon the very important effects of fluid resistance in modifying the rolling of ships among waves. This branch of the subject is, however, of great interest, and has attracted the attention of several able investigators: although they are not agreed in all points, there are many general considerations which command universal support; to some of these, brief reference will now be made.

The deductions from the hypothetical case of unresisted rolling, which is mainly discussed in the modern theory, can be regarded only as of a qualitative and not of a quantitative character. For example, one of these deductions is that a ship rolling unresistedly among waves having a period double her own natural period will accumulate great rolling motion, and infallibly upset. As a matter of fact, we know

that, while the assumed ratio of periods leads to the production of heavy rolling, ships do not commonly, nor in any but exceptional cases, upset under the condition of synchronism ; in other words, the character of the motion is well described by the deduction from the hypothetical case, but its extent is not thus to be measured. Similarly, in other cases, the effect of resistance must be considered when exact measures of the range of oscillation are required, as they may be in discussing the safety of ships. The problem, therefore, resolves itself into one of correcting the deductions from the case of unresisted rolling, by the consideration of resistance coming into play.

In accordance with the principles explained in Chapter IV. it is possible by means of still-water rolling experiments to ascertain the moment of resistance of a ship corresponding to any assigned arc of oscillation. If the ship herself has not been rolled for that purpose, but a sister ship or similar vessel has been so rolled, her coefficients of resistance may be estimated with close approximation, and the retarding effect of resistance may be determined. This is true within the limits of oscillation reached by the still-water experiments, say, 10 or 15 degrees on each side of the vertical, and in high-sided ships of ordinary form the limits may probably be extended. In fact, it may be assumed that the coefficients of resistance for most ships are or may be ascertained by these rolling experiments, for inclinations as great as are likely to be reached by the same ships when rolling in a seaway, in all but exceptional circumstances.

If a vessel rolls through a certain arc amongst waves, it appears reasonable to suppose that the effect of resistance will be practically the same as that experienced by the ship when rolling through an equal arc in still water. The intrusion of the vessel into the wave, as already remarked, must somewhat modify the internal molecular forces, and she must sustain certain reactions, but for practical purposes these may be disregarded, not being proportionally large.

Resistance is always a retarding force; in still water it

tends to extinguish oscillation; amongst waves it tends to limit the maximum range attained by the oscillating ship. This may be well seen in the critical case of synchronism; where a ship rolling unresistedly would have a definite addition made to her oscillation by the passage of each wave. The wave impulse may be measured by the added oscillation; the dynamical stability corresponding to the increased range expressing the "energy" of the wave impulse. At first the oscillations are of such moderate extent that the angular velocity is small, and the wave impulse more than overcomes the effect of the resistance; the rolling becoming heavier. As it becomes heavier, so does the angular velocity increase and with it the resistance; at length, therefore, the resist ance will have increased so much as to balance the increase of dynamical stability corresponding to the wave impulsethen the growth of oscillation ceases. As successive waves pass the ship after this result is attained, they each deliver their impulse as before, but their action is absorbed in counteracting the tendency of the resistance to retard and degrade the oscillations.

When a ship is rolling "permanently" amongst waves, her oscillations having a fixed range and period, a similar balance will probably have been established between the wave impulse and the resistance; and here also the actual limit of range will fall below the theoretical limit given by the formula for unresisted permanent rolling on page 201. Resistance may, in this case, be viewed as equivalent to a reduction in the steepness of the waves; this diminished slope taking the place of what has been termed the "effective slope" for unresisted rolling. Hence it will be seen why the general deductions from the theory of unresisted rolling are so well borne out by experience with actual ships whose behaviour is largely influenced by resistance.

Mr. Froude has approached in this way the problem of determining the maximum range likely to be attained by ships of known natural period rolling amongst waves of known dimensions; making an allowance from the actual steepness

of the wave, in order to provide what he terms the "maintaining power" required to balance the resistance, and using the remaining wave as that which for unresisted rolling would produce oscillations similar to those actually performed by the ship. These, however, are matters lying outside the scope of the present work, and they cannot be pursued further. The broad practical deduction is that increased resistance acts beneficially on a ship by limiting her maximum oscillations, and the correctness of this deduction, although formerly it was disputed by some high authorities in the science of naval architecture, has now been placed beyond doubt.

The Admiralty Committee on Designs took evidence in 1871 as to the advantages or otherwise of bilge-keels; this evidence was not unanimously favourable to the use of such keels, but its general tenour was so. Some of the Indian troopships had been fitted with deep bilge-keels at that time, and the reports on their effect on the behaviour of the ships were most definite. The captain of the Serapis reported that the bilge-keels, having been tried under all conditions of wind and sea, had proved a perfect success, and added, "I can con"fidently say her rolling has been lessened 10 degrees each

way." As regarded the Crocodile, no similarly severe tests had at that time been made, but the opinion was confidently expressed that "the rolling had been much checked by the "bilge pieces," the ship having often rolled heavily before they were fitted, and being considered "remarkably steady" afterwards. Mr. Froude also came forward with the reports of his experiments on models, and strongly recommended the use of deep bilge-keels, a recommendation which was endorsed by the committee in their report.

The experiments of Mr. Froude were made at Spithead with the same model of the Devastation as had previously been used to determine the effects of different depths of bilge-keels upon still-water oscillations.* At the time considerable doubt was entertained in some quarters as to

* See the accounts of these experiments at page 125.