Shipbuilding

SHIPBUILDING. When ships were built of wood and propelled by sails their possible size and proportions were limited by the nature of the structural material, while the type of structure had been evolved by long experience and was incapable of any radical modification. Speed depended so much on circumstances independent of the design of the vessel, such as the state of the wind and sea, that it was impossible to include a definite speed over a voyage or measured distance as one of the essential requirements of a design; and the speed actually obtainable was low even under the most favourable conditions when judged by modern standards. Stability depended principally on the amount of ballast carried, and this was determined experimentally after the completion of the vessel. Under these conditions there was no room for any striking originality of design. One vessel followed so closely on the lines of another, that the qualities of the new ship could be determined for all practical purposes by the performance of an almost identical vessel in the past. The theoretical science of shipbuilding, the object of which is to establish quantitative relations between the behaviour and performance. of the ship and the variations in design causing them, was generally neglected.

With the introduction of iron, and later of steel, as a structural material for the hulls of ships, and of heat engines for their propulsion, the possible variation of size, proportions and propelling power of ships was enormously increased. In order to make the fullest use of these new possibilities, and to adapt each ship, as closely as may be, to the special purpose for which it is intended, theoretic knowledge has become of paramount importance to the designer. He has been forced to investigate closely those branches of the abstract physical sciences that bear specially on ships and their behaviour, and these mathematical and experimental investigations constitute the study of Theoretical Shipbuilding. It embraces the consideration of problems and questions upon which the qualities of a ship depend and which determine the various features of the design, having regard to the particular services that the ship will be required to perform; i.e. the requirements that must be fulfilled in order that she may make her various passages economically and with safety in all conditions of wind and sea, the best form for the hull with regard to the resistance offered by the water and the engine power requisite in order to attain the speed desired, the nature of waves and their action upon the ship, and the structural arrangements necessary in order that she may be sufficiently strong to withstand the various stresses to which she will be subjected. The determination of the most suitable dimensions to fulfil certain conditions involves the consideration of a different set of circumstances for almost every service; and here the experience gained in vessels of similar type, together with the known effect of modifications made to fulfil new conditions of each particular design, can be used as a guide. The requirements of economical working, safety, &c., determine the length, breadth, depth and form. The length has a most important bearing on the economy of power with which the speed is obtained; and on the breadth, depth and height of side, or freeboard, depend to an important degree the stability and seaworthiness of the vessel.

While, however, the importance to the ship designer of mathematical theories based on first principles and experiment can hardly be overrated, it should be observed that the circumstances and conditions postulated are invariably much less complex than those which surround actual ships. The applicability of the theories depends on the closeness with which the assumed circumstances are realized in practice. The ultimate guide in the design of new ships must, therefore, still remain practical experience. To this experience theory is a powerful assistance, but can by no means replace it.

Theoretical Shipbuilding Stability. When a ship floats at rest in still water, the forces acting upon her must be in equilibrium. These consist of the weight of the ship acting vertically downwards through its centre of gravity and the resultant pressure of the water on the immersed hull. If the ship be supposed removed and the cavity thus is in equilibrium under the same system of fluid pressures, the resultant of these pressures must be equal and opposite to the weight of the water in the cavity and will therefore act vertically upwards through the centre of gravity of this portion of water. Defining the weight of water displaced by the ship as the displacement, and its centre of gravity as the centre of buoyancy, it is seen that the fundamental conditions for the equilibrium of a ship in still water are (a) that the weight of the ship must be equal to the displacement, and (b) that its centres of gravity and buoyancy must be in the same vertical line.

A floating ship is always subject to various external forces disturbing it from its position of equilibrium, and it is necessary to investigate the stability of such a position, i.e. to determine whether the ship, after receiving a small disturbance, will tend to return to its former position, in which case its equilibrium is termed stable, or whether, on the other hand, it will tend to move still farther from the original position, when the equilibrium is termed unstable. The intermediate case, when the ship tends to remain in its new position, is a third state of equilibrium, which is termed neutral. Of the modes of disturbance possible, it is evident that a bodily movement of the ship in a horizontal direction or a rotation about a vertical axis will not affect the conditions of equilibrium; the equilibrium is also stable for vertical displacements of a ship. The remaining movements, viz. rotations about a horizontal axis, can be resolved into rotations in which the displacement is unaltered, and vertical displacements, the effect of the latter being considered separately. Of the various horizontal axes about which a ship can rotate two are of particular importance, viz. (i) an axis parallel to the longitudinal plane of symmetry, (2) an axis at right angles to this plane, both axes being so chosen that the displacement remains constant; the stability of a ship with reference to rotations about these axes is known as the transverse stability and the longitudinal stability respectively. In the following account the consideration of stability is confined at first to these two cases; the general case of rotation about any horizontal axis whatever being dealt with later.

Let fig.' I represent a transverse section of a ship, WL y, being its water line when upright, and W'L' its water line when inclined to a small angle 0 as shown.

Assuming that the displacement is unaltered, if G be the position of the ship's centre of gravity and B, B' the positions of its centre of yv buoyancy in the upright A and inclined positions respectively, the forces acting on the ship consist of its weight W vertically downwards through G and the resultant water pressure equal to W acting vertically upwards through B'. These constitute a couple of momentWXGZ where Z is the foot of the perpendicular from G on to the vertical througn B'; the direction of the couple as drawn in the figure is such as would cause the ship to return to its original position, i.e. the equilibrium is stable for the inclination shown.

If M be the intersection of the vertical through B' with the original vertical, the moment of the restoring couple is equal to W XGM sin 0, and GM sin 0 is termed the righting lever. If, by moving weights on board, G be moved to a different position on the original vertical through B, the original position of the ship will remain one of equilibrium, but the moment of stability at the angle of inclination 0 will vary with GM. If G be brought to the position G' above M the moment W XG'Z' will tend to turn the ship away from the original position. It follows that the condition that the original position of equilibrium shall be stable for the given inclination is that the centre of gravity shall be below the intersection of the verticals through the upright and inclined centre of buoyancy; and the moment of stability is proportional to the distance between these two points.

When the inclination 0 is made smaller the point M approaches a definite position, which, in the limit when 0 is indefinitely small, is termed the metacentre. In ships of ordinary form it is found that for io to 15 degrees of inclination, the intersection of the verticals through the centres of buoyancy B and B' remains sensibly at the met a centre M; and therefore within these limits the moment of stability is approximately equal to W XGM sin 0. Since the angle on either side of the vertical within which a ship rolls in calm or moderate weather does not usually exceed the limit above stated, the stability and to a great extent the behaviour of a vessel in these circumstances are governed by the distance GM which is known as the metacentric height. The position of G can be calculated when the weights and positions of the component parts of the ship are known. This calculation is made for a new ship when the design is sufficiently advanced to enable these component weights and their positions to be determined with reasonable accuracy; in the initial stages of the design an approximation to the vertical position of G is made by comparison with previous vessels.

The position of the centre of gravity of a ship is entirely independent of the form or draught of water, except so far as they affect the amount and distribution of the component weights of the ship. The position of the metacentre, on the other hand, depends only on the geometrical properties of the immersed part of the ship; and it is determined as follows: Let WL, W'L' (fig. 2) be the traces of the upright and inclined water planes of a ship on the transverse plane; B, B' the corresponding position of the centre of buoyancy; 0 the angle of inclination supposed indefinitely small in the limit, and S the intersection of WL and W'L'; join BB'.

By supposition the displacement is unchanged, and the volumes WAL, W'AL' are equal; on subtracting W'AL it is seen that the two wedges WSW', LSL' are also equal. If dx represent an element of length at right angles to the plane of the figure, y,, y2, the halfbreadths one on each side at any point in the original water 1 ine, so that WS =y,, SL =y 2, the areas WSW', LSL' differ from Zy, 2. B, 2y2 2.0 by indefinitely small amounts, neglecting which the volumes of WSW', LSL' are equal to f 2 y, 2 0dx and f 2 y220dx. Since these are equal we have f ' y i 2 dx = z f y2 2 dx or fy i dx X 2 1 = f y2dx X 2; i.e. the moments of the two portions of the water plane about their line of intersection passing through S are equal. This line is also the axis of rotation, which therefore passes through the centre of gravity of the water plane. For vessels of the usual shape, having a middle line plane of symmetry and floating initially upright, for small inclinations consecutive water planes intersect on the middle line.

Again if g,, g 2 are the centres of gravity of the wedges WSW', LSL', and v the volume of either wedge, the moment of transference of the wedges vXg,g2 is equal to the moment of transference of the whole immersed volume Vxbb' where V is the volume of displacement.

But v X g,S = moment of wedge WSW' about S = a fy, 3 .0 .dx, and v X Sg t =moment of wedge LSL' about S = fy2 3 .0 .dx. Adding, 3f (y, 3 +y2 3)0.dx=vXg,g2=Vxbb'. But BB' =BM .0 to the same order of accuracy, and if (y 1 3 +y2 3). dx is the moment of inertia of the water plane about the axis of rotation; denoting the latter by I, it follows that BM =I V; i.e. the height of the metacentre above the centre of buoyancy is equal to the moment of inertia of the water plane about the axis of rotation divided by the volume of displacement. These quantities, and also the position of the centre of buoyancy can be obtained by the approximate methods of quadrature usual in ship calculations, and from them the position of the metacentre can be found.

If the ship is wholly immersed, or if the inertia of the water plane is negligible as in a submarine when diving, BM =0, and the condition for stability is that G should be below B; the righting lever at any angle of inclination is then equal to BG sin 0. During the process of design the position of the centre of gravity formed filled with water then, since this volume of water > 4,w FIG. I.

FIG. 2.

is determined by the disposition of hull material and fittings, machinery, coal and all other movable weights, the position of which is necessarily fixed by other considerations than those of stability; but the height of the metacentre above the centre of buoyancy varies approximately as the cube of the breadth, and any desired value of GM is readily obtained by a suitable modification in the beam.

Class of Ship.		Approximate GM in Ft.
First class battleship and cruiser .	.	31 to 5
Second and third class cruiser and scout		2 to 3
Torpedo boat destroyer. .	.	II to 21--
First class torpedo boat. .		to 11
Steam picket boat or launch.	.	8 to 12
River gunboat (shallow draught) .		8 to 20
Large mail and passenger steamer		5 to 2
Cargo steamer. .. .		I to 2
Sailing ship .		2 to 6
Tug. .		I to 2 2

The metacentric height in various typical classes of ships at " normal load " is as follows: - The metacentric height adopted in steamships is governed principally by the following considerations: (a) It should be sufficiently large to provide such a position of G as will give ample stability at considerable angles of inclination and sufficient range.

(b) Where ample stabilit y at large angles is obtained by other means, the stability at small angles, which is entirely due to the metacentric height, should be sufficient to prevent forces due to FIG. 3. - Metacentric Diagram of a Battleship.

wind on upper works, movement of weights athwartships, turning, &c., causing large and uncomfortable angles of heel.

(c) It should be sufficient to allow one or more compartments to become opened to the sea, through accidental damage, without risk of capsizing.

(d) It should, if possible, be sufficiently large in the normal condition of the ship to permit the greatest possible freedom in the stowage of a miscellaneous cargo without producing instability.

(e) On the other hand an excessive value causes rapid and uncomfortable rolling among waves.

A ship having small initial stability is said to be " crank," while one possessed of a large or excessive amount is termed " stiff." The former type is generally found to be steadier and easier in rolling among waves; and for this reason when other circumstances permit, the metacentric height is usually chosen as small as possible consistent with safety and comfort.

The metacentric height is affected by an alteration in displacement or in position of the centre of gravity caused by loading or unloading cargo, fuel and stores. In consequence the stability has to be investigated for a variety of conditions, particularly that in which the metacentric height is a minimum. The change in the position of the centre of gravity can be readily determined from an account of the weights removed, added or shifted; and the height of the metacentre is obtained by calculating its position at a number of water lines, and drawing a curve of heights of metacentre above keel on a base of the draught of water. The results are conveniently embodied in the form of a metacentric diagram; the curves of height of metacentres and vertical positions 28 '

3 ' P°.

FIG. 4. - Metacentric Diagram of a Merchant Vessel.

of centres of buoyancy being set up from a line intersecting the water lines at 45°.

Figs. 3, 4 and 5 are the metacentric diagrams for a battleship, a vessel sharply curved at the bilge typical of a large number of merchant steamers, and a sailing ship of " Symondite " (or peg top) section; it will be observed that in the first and second the M curve is slightly concave upwards, and in the third sharply convex.

The buoyancy curve in all cases is nearl y a straight line whose inclination at a particular water plane to the horizontal is equal to tan 'Ah/V; where A is the water plane area, and h the depth of the centre of buoyancy below the surface. The position of the metacentre at an intermediate water line is obtained from the diagram by drawing a horizontal line at the draught required, and squaring FIG. 5. - Metacentric Diagram of a Sailing Ship of " Symondite " section.

up from its intersection with the 45° line to meet the curve of metacentres.

MEAN DRAFT	TONS PER INCH	DISPLT: IN TONS
5'v;	10 49	1077
O%2		906
13- 7	8.55	631
9 . 8'/Z	5.74	295

With these curves are associated (though usually drawn separately) two others known as the curves of Displacement and of Tons per inch and expressed by AA and BB respectively in the above figures. These have the mean draught of water as abscissa (vertical), and 30' 6 21452 'a' Leve Deep C 14715 ' '8429 ' '2wi. 3w L 5532 ° 0 0 Tons Per Inch 49.8 50.4 Tons 15600 13190 Gm (Light), 2.99, Gm (Deep), 1 67.

21' 2" 346 10800 16.3" 48.5 ®11111111111 84 50 7780 12'-- 47.5 614-0 the displacement in tons and the number of tons required to increase the mean draught by I in., respectively, as ordinates (horizontal). The ordinate of the curve of displacement at any water line is clearly proportional to the area of the curve of tons per inch up to that water line.

The properties of the metacentric stability at small angles are used when determining the vertical position of the centre of gravity of a ship by an " inclining experiment "; this gives a expert- check on the calculations for this position made in the initial stages of the design, and enables the stability of m the completed ship in any condition to be ascertained with great accuracy.

The experiment is made in the following manner: Let fig. 6 represent the transverse section of a ship; let w, w be two weights on deck at the positions P, Q, chosen as far apart transversely as convenient; and let G be the combined centre of gravity of ship and weights.

When the weight at P is moved across the deck to Q', the centre of gravity of the whole moves from G to some point G' so that GG' is parallel to PQ' (assumed horizontal) and equal to hw/W where h is the distance moved through by W P, and W is the total dis placement. The ship in consequence heels to a small angle 8, the new vertical through G pass ing through the meta centre M; also GM = GG' cot 0= hw/W cot 0, the metacentric height being thereby determined and the position of G then FIG. 6. found from the meta centric diagram. In prac tice 0 is observed by means of plumb bobs or a short period pendulum recording angles on a cylinder; 1 the weight w at P, which is chosen so as to give a heel of from 3° to 5°, is divided into several portions moved separately to Q'. The weight at Q' is replaced at P, the angle heeled through again observed; and the weight at Q similarly moved to P' where P'Q =h= PQ', and the angle observed; GM is then taken as the mean of the various evaluations.

In the case of small transverse inclinations it has been assumed that the vertical through the upright and the inclined positions of the centre of buoyancy intersect, or, which is the same thing, that the centre of buoyancy remains in the same trans verse plane when the vessel is inclined. This assumption is not generally correct for large transverse inclinations, but is nevertheless usually made in practice, being sufficiently accurate for the purpose of estimating the righting moments and ranges of stability of different ships, calculated under the same conventional system; this is all that is necessary for practical purposes.

With this assumption, there will always be a point of intersection (M' in fig. 7) of the verticals through the upright and inclined centres of buoyancy; and the righting lever is, as before, GZ =GM' sin 0. In this case, however, there is no simple formula for BM' as there is for BM in the limiting case where 0 is infinitesimal; and other methods of calculation are necessary.

The development of this part of the subject was due originally to Atwood, who in the Philosophical Transactions of 1796 and 1798, advanced reasons for differing from the metacentric method which was published by Bouguer in his Traite du navire in 1746. Atwood's treatment of stability (which was the foundation of the modes of calculation adopted in England until about twenty years ago) was as follows: Let WL, W'L' (fig. 7) be respectively the water lines of a ship when 1 Such an instrument is described by Froude for recording the " relative " inclination of a ship amongst waves, Transactions of Institution of Naval Architects, 1873, p. 179. The pendulum should have sufficient weight and the arm carrying the pen may be about 4 ft. long. If the cylinder be fitted with a clock recording the time the natural period of the ship will also be obtained.

upright and inclined at an angle 0, S their point of intersection; B and B' the centres of buoyancy, g 1 and g 2 the centres of gravity of the equal wedges WSW', L'SL, and h 1, h 2 the feet of the perpendiculars from g1, g2 on the inclined water line. Draw GZ, BR parallel to W'L', meeting the vertical through B' in Z and R.

The righting lever is GZ as before; if V be the volume of displacement, and v that of either wedge, then Vxbr=vXhlh2 also GZ=BR - BG sin 0; whence the righting moment or Wxgz=W u' h2 --BG sin B .

This is termed Atwood's formula. Since BG, V and W are usually known, its application to the computation of stability at various angles and draughts involves only the determination of v X h1h2. A convenient method of obtaining this moment was introduced by F. K. Barnes and published in Trans. Inst. N.A. (1861). The steps in this method were as follows: (a) assume a series of trial water lines at equal angular intervals radiating from S' the intersection of the upright water line with the middle line plane; (b) calculate the volumes of the various immersed and emerged trial wedges by radial integration, using the formula 2 f°d4./r2dx, where r, ¢ are the polar co-ordinates of the ship's side, measured from S' as origin, and dx an element of length; (c) estimate the moment of transference of the same wedges parallel to the particular trial water line by the formula v X h l h 2 = l f °cos (0-4)d4fr3dx, 0 adding together the moments for both sides of the ship; and (d) add or subtract a parallel layer at the desired inclination to bring the result to the correct displacement. The true water line at any angle is obtained by dividing the difference of volume of the two wedges by the area of the water plane (equal to frdx, for both sides) and setting off the quotient as a distance above or below the assumed water line according as the emerged wedge is greater or less than the immersed wedge. The effect of this " layer correction " on the moment of transference is then allowed.

The righting moment and the value of GZ are thus determined for the displacement under consideration at any required angle of heel.

A different method of obtaining the righting moments of ships at large angles of inclination has prevailed in France, the standard investigation on the subject being that of M. Reech first published in his memoir on the " Construction of Metacentric Evolutes for a Vessel under different Condi tions of Lading" (1864). The principle of his method is dependent on the following geometrical properties: Let B', B" (fig. 8) be the centres of buoyancy corresponding to two water lines W'L', W"L" inclined at angles 0, 0+d9, to the original upright water line WL, dO being small; and let g1, g2 be the centres of gravity of the equal wedges W'TW", L'TL". The moment of either wedge about the line gig2 is zero, and the moments of W'L'A and of W"L"A about gig2 are therefore equal; since these volumes are also equal, the perpendicular distances of B' and B" from g i g 2 are equal, or B'B" is parallel to gig2.

The projection on the plane of inclination of the locus of the centre of buoyancy for varying inclinations with constant displacement is termed the curve of buoyancy, a portion BB'B" of which is shown in the figure. On diminishing the angle dO indefinitely so that B" approaches B' to coincidence, the line B'B" becomes, in the limit, the tangent to the curve BB'B", and g i g 2 coincides with the water line W'L'; hence the tangent to the curve of buoyancy is parallel to the water line.

Again, if the normals to the curve at B', B" (which are the verticals corresponding to these positions of the centre of buoyancy) intersect at M', and those at B", B"' (adjacent to B") at M", and so on, a curve may be passed through M', M',. ., commencing at M, the metacentre. This curve, which is the evolute of the curve of buoyancy, is known as the metacentric curve, and its properties were first ?- -- --?-F? ,--, L' FIG. 7.

FIG. 8.

investigated by Bouguer in his Traite du Navire. The points M'M", ... on the curve are now termed pro-metacentres.

If p represent the length of the normal B'M' or the radius of curvature of the curve of buoyancy at an angle B, then p.d6 =ds the length of an element of arc of the B curve. In the limit when dB is indefinitely small, B c -= p. Using Cartesian co-ordinates with B as origin and By, Bz, as horizontal and vertical axes, we have de cos = B =p cos 0, dz ds d6 s i n 6 = p sin 6; (2) whence ° Pc y= os B.d6; z= o p. sin 6.d6, and the righting lever GZ=y cos 6+(z - BG) sin 0. The radius p is (as for the upright position) equal to the moment of inertia of the corresponding water-plane about a longitudinal axis through its centre of gravity divided by the volume of displacement; the integration may be directly performed in the case of bodies of simple geometrical form, while a convenient method of approximation such as Simpson's Rules is employed with vessels of the usual ship-shaped type. As an example in the case of a box, or a ship with upright sides in the neighbourhood of the water-line, if BG =a and BM =po, then p = po sec 3 6; whence ? e Y = p cos 6 .d0= po tan 0, .o e z= p sin Ode = 2 p o tan g 6, .o and GZ = (po - a) sin 6 +1po tan g 6. sin e; which relations will also hold for a prismatic vessel of parabolic section. It is interesting to note that in these cases if the stability for infinitely small inclinations is neutral, if po=a, the vessel is stable for small finite inclinations, the righting lever varying approximately as the cube of the angle of heel.

The application of the preceding formulae to actual ships is troublesome and laborious on account of the necessity for finding by trial the positions of the inclined water-lines which cut off a constant volume of displacement. To avoid this difficulty the process was modified by Reech and Risbec in the following manner: - Multiply equations (I) and (2) by V.d6, V being the volume of displacement; we then have d(Vy)= I cos 6. d6, (3) d(Vz) =I sin 6.de, . (4) where I is the moment of inertia of the inclined water-line about a longitudinal axis passing through its centre of gravity. These formulae have been obtained on the supposition that the volume V is constant while 0 is varying; but by regarding the above equations as representing the moments of transference horizontally and vertically due to the wedges, it is evident that V may be allowed to vary in any manner provided that the moment of inertia I is taken about the longitudinal axis passing through the intersection of consecutive water-lines. In particular the water-lines may all be drawn through the point of intersection of the upright water-line with the middle line, and the moments of inertia are then equal to s fr 3 dx for both sides of the ship, r being the half-breadth along the inclined water-line; the increase in volume is the difference between the quantity f del 1r 2 dx for the two sides of the ship.

If Va., Vo be the volumes of displacement at angles a and o respectively, 1 r'x V. - Vo ja d6 [ f differ d ence] ' and substituting in (3) and (4) and integrating, V ay = J d6 [ J 3sumx] cos o, Vaz =.lade [ f 3 umx] sin 6.

On eliminating Va in (5), (6) and (7), y and z can be found.

This is repeated at different draughts, and thus Va, y and z are determined at a number of draughts at the same angle, enabling curves of y and z to be drawn at various constant angles with V for an abscissa; from these, curves may be obtained for y and z with the angle a as abscissa for various constant displacements; GZ being equal to y cos a+(z - a) sin a. From the foregoing it is evident that the elements of transverse stability, including the co-ordinates of the centre of buoyancy, position [[[Theoretical]] of pro-metacentre, values of righting lever and righting moment, depend on two variable quantities - the displacement and the angle of heel. The righting lever GZ is in England selected as the most useful criterion of the stability, and, after being evaluated for the various conditions, is plotted in a form of curves - (a) for various constant displacements on an abscissa of angle of inclination, (b) for a number of constant FIG. 9. - Cross Curves of Stability of a Battleship.

angles on an abscissa of displacement. These are known as curves of stability and cross curves of stability respectively; either of these can be readily constructed when the other has been obtained; which process is utilized in the method now almost universally adopted for obtaining GZ at large angles of inclination, a full description being given in papers by Merrifield and Amsler in Trans. I.N.A. (1880 and 1884). The procedure is as follows: I. The substitution of calculations at constant angle for those at constant volume. A number of water-lines at inclinations having a constant angular interval (generally 25°) are drawn passing through the intersection S' of the load water-line with the middle line on the body plan. Other water-lines are set off parallel to these at fixed distances above or below the original water-line passing through S'.

2. The volumes of displacement and the moments about an axis through S' perpendicular to the water-line are determined for each draught and inclination by means of the Amster-Laffon integrator, a oQ ? L OF Inclination Deep Condition Shewn Normal Light' Fig. io. - Curves of Stability of a Battleship.

the pointer of this instrument being taken in turn round the immersed part of each section.

3. On dividing the moments by the corresponding volumes, the perpendicular distance of the centre of buoyancy from the vertical through S' is obtained, i.e. the value of GZ, assuming G and S' to coincide.

4. For each angle in turn " cross curves " of GZ are drawn on a base of displacement.

w w z N	2

'1$0 ' s oo , ,eq00 Tons, Displacement 5° (5) 4-0,000 30,000 20,000 10,000 5. From the cross curves, curves of stability on a base of angle of inclination can be constructed for any required displacement, allowance being made for the position of G by adding to, or subtracting from, each ordinate, the quantity GS' sin a according as G is below or above S'.

A typical set of cross curves of stability for a battleship of about 18,000 tons displacement is shown in fig. 9. It will be observed that the righting levers decrease with an increase of displacement; and this is a general characteristic of the cross curves for ships of ordinary 40  ?5 w z N [[Angle Of Deep Condition Shewn Normal Light

Fig. I I]]. - Curves of Stability of a Merchant Vessel.

form. The additional weights that constitute the difference between light and deep load (i.e. cargo, coal, stores and water) are generally placed low down, and thus the position of the centre of gravity is usually lower when loaded than when light, causing an increase of stability which frequently more than compensates for the loss of stability indicated by the cross curves.

The stability curves for the same vessel are reproduced in fig. 10. It is customary in warships to draw separate curves for three conditions: (a) normal load, i.e. fully equipped with bunkers about half full, and reserve feed tanks empty; (b) deep load with all bunkers and tanks full; (c) light with all coal, water (except in boilers), ammunition, provisions and consumable stores removed.

The curves for a cargo or passenger ship are generally drawn for the condition when light, when fully laden with passengers or with a -45 FIG. 12. - Curves of Stability of a Box-shaped Vessel showing the influence of beam and freeboard.

homogeneous cargo, and sometimes for an intermediate condition; typical curves are given in fig. i 1.

Stability curves are obtained on the assumptions I. That all openings in the upper deck, forecastle and poop (if any) are covered in and made watertight; and the buoyancy of any erections above there decks is generally neglected.

2. That the side of the ship is intact up to the upper deck, all side scuttles, ports or other openings being closed.

3. That all weights in the ship are absolutely fixed.

4. That no changes of trim occur during the inclination.

In some cases curves are drawn (a) with forecastle and poop intact, (b) with these thrown open to the sea, the latter condition being more commonly considered.

The slope of the stability curve for small angles, the maximum righting lever with the angle at which it occurs, and the range or the inclination at which the stability vanishes are of particular interest, inasmuch as the curve depends principally on these features; and the effect on them, particulars of variation of freeboard, breadth and position of centre of gravity, is considered below.

The stability curve AA (fig. 12) is drawn for a box-shaped vessel of draught to ft., freeboard to ft. and beam 30 ft.; with C.G. in the water-plane. The curves EE, FF, GG are drawn for the same vessel, but with freeboard altered to 122 i 71 and 5 ft. f respectively; it will be observed that freeboard has no influence on the stability at small angles, but has a marked effect on the range and maximum righting lever. An increase of freeboard is generally accompanied by a rise in the position of the centre of gravity; this is not included in the curves, but would actually reduce 2.0 - ON 1R FIG. 13. - Curves of Stability of " Monarch " and " Captain." the stability to some extent. The effect of freeboard on the range and on the safety of ships is also illustrated by a comparison between the curves of stability (fig. 13) of the armoured turret ships " Monarch " and " Captain," the latter of which was lost at sea in 1870. These vessels were similar in construction and dimensions except that the freeboard of the " Monarch " was 14' o" and that of the " Captain " 6' 6"; the smaller freeboard of the " Captain " was associated with a slightly lower position of the centre of gravity and a greater metacentric height. The stability curve of the " Captain " in consequence rises rather more steeply than that of the " Monarch " up to about 14° when the deck edge is immersed; the righting lever then rapidly declines, and vanishes at 541°, in contrast to the " Monarch's," where the maximum righting lever is doubled and range augmented 1.3 times by the additional freeboard. For the influence of the range in enabling a ship to withstand a suddenly applied force see " Dynamical Stability." Again, for the box-shaped vessel previously considered, if the breadth is modified successively from 30 ft. to 35, 25 and 20 ft., other features remaining unaltered, the curves of stability then E ff ect of obtained are represented by BB, CC and DD in fig. 12. It is seen that alteration in beam affects principally the stability levers at moderate angles of inclination, while at 90° inclination the curves all intersect. Since at small angles GZ =GM.B (in circular measure) approximately, the initial slope of the curve is proportional to GM, and the tangent to this curve at the origin can be drawn by setting by the value of GM as an ordinate to an angle of one radian (57 3°) as abscissa, and joining the point to the origin. (See figs. to and 11.) The height of the metacentre above the centre of buoyancy will, caeteris paribus, vary with the cube of the breadth, and an increase of beam will result in a large increase of stability at moderate angles.

						75
	30		25	20	30	30	30

Finally the effect of an alteration in the vertical position of the centre of gravity is illustrated by the three stability curves of a steam yacht in fig. 14, where the centre of gravity is Effect o f successively raised I ft. In the condition corresponding of to the fourth and lowest curve, the GM is negative (- 2 ft.) and so also are the righting levers up to 15° when the curve C.G. crosses the axis; from 15° to about 52° the GZ is positive, but above ?

IS° 30° 45° Angle of t Oyy `N FIG. 14. - Curves of Stability of a Steam Yacht showing effect of variation in height of centre of gravity.

s75 gOo 35 3.0 ?0 _5 that value it again becomes negative. In this case the stability is unstable at the upright position, and the ship will roll to an angle of 15° on either side where the equilibrium is stable. This peculiarity is not uncommon in merchant steamers at light draught. Ample stability at large angles and good range is provided in such cases by high freeboard; but, apart from any considerations of safety, water ballast is used to lower the centre of gravity to a sufficient extent to avoid excessive tenderness.

The properties of the loci of centres of buoyancy and of prometacentres were fully investigated by Dupin in 1822, including also Geo- the surfaces into which these curves develop when admit metrical ting inclinations about transverse and " skew " axes. It m properties. has been shown that the tangent to the curve of buoyancy at any point is parallel to the corresponding water-line; and assuming that the ship is only free to turn in a plane perpendicular to the axis of inclination, the positions of equilibrium are found by drawing from the centre of gravity all possible normals to the buoyancy curve, or equally, all possible tangents to its evolute, the metacentric curve, since the condition to be satisfied is, that the centres of gravity and buoyancy shall lie in the same vertical. Again, clearness in fig. 16.1 It will be seen that the metacentric curve contains eight cusps, M 1, M ... M 8. Assuming the ship to heel to starboard, M 1 corresponds to the upright position, M2 to the immersion of the starboard topsides and emersion of the port bilge; 1VI 3 corresponds to 90° of heel, M4 to the complete immersion of the deck and the emersion of the starboard bilge. M5 corresponds to the bottom-up position and similarly for Ms, M 7 and M8. There are also 6 nodes, of which P and Q are on the middle line. By means of those curves, the effect of a rise or fall in the position of the ship's centre of gravity can readily be traced. The positions of equilibrium correspond to the normals that can be drawn from G to the buoyancy curve, or equally to the tangents drawn to its evolute the metacentric curve. For stable equilibrium G lies below M, i.e. generally between B and M; and for unstable equilibrium, similarly, B is between G and M.

the ship under consideration, G 1 was the actual centre of gravity, and G i M I corresponds to the upright position of stable equilibrium! As the vessel heels over, equilibrium (this time un stable) is again reached at about 90°, and a third position (stable) is obtained when the vessel is bottom up, G i M 5 being then the meta centric height. A fourth (unstable) position is obtained at about 270°, after which the original position G i M 1 is reached, the f vessel having turned completely round. For this position of G1 therefore, there are four positions of equilibrium, two of which are stable and two unstable; and this is also true for all positions of G between M 1 and M5.

If G lies at G4 between M5 and the point P, there are six positions of equilibrium, alternately stable and unstable. If G is below P as at G5, there are two positions of equilibrium of which the upright only is stable. A self-righting life-boat exactly corresponds to this condition, the vessel being capable of resting only in the original upright position. If G is above Q, on the other hand, as at G3, there are again only two positions of equilibrium, the vessel being unstable when upright. If G is at G2 there are again six positions of equilibrium; the upright position is unstable, but a stable position is reached at a certain angle on either side. This phase is often realised in merchant ships when light, as already stated (vide fig. 14). When G is exactly upon one of the branches of the metacentric curve, the equilibrium is neutral; if it is at M1 the ship is stable for finite inclinations, and if at Q unstable; similarly for M5 (except that the neutral state is then reached at 180°) and for P.

In all the above cases it will be observed that the positions of stable and unstable equilibrium are equal in number and occur alternately. There are two exceptions: I. When the moment of inertia of the water plane changes abruptly so that the B curve receives a sudden change of curvature. This is possible with bodies of peculiar geometrical forms, and two positions of M then correspond to one position of the body; if G lies between them, the equilibrium is stable for inclinations in one direction and unstable for those in the opposite direction, and is then termed " mixed." 2. When the equilibrium is neutral, this condition may be regarded as the coincidence of two or more positions of equilibrium alternately stable and unstable. The ship may then be either stable, unstable or neutral for finite inclinations; in exceptional cases she may be stable in one direction and unstable in the other, resembling to some extent the condition of " mixed equilibrium." Another curve whose properties were originally investigated by Dupin is the curve of flotation F 1 F 2 F 3 ... (fig. 15), which is the envelope of all the possible water-lines for the ship when inclined transversely at constant displacement. Since, as previously shown, consecutive water-planes intersect on a line passing through their The curves of buoyancy and flotation and the metacentric curve for various forms, including that of H.M.S. " Serapis," were obtained by practical investigation by the writer in 1871. The results showed that Dupin's investigations, which were apparently purely theoretical, had not fully disclosed certain features of the curves, such as the cusps, &c.

F S t -_ FIG. 15. - Metacentric, Buoyancy and Flotation Curves of " Serapis." when the curve of statical stability crosses the axis, making an acute positive angle as at P in fig. 14, the values of GZ on either side of P are such as to tend to move the ship towards the position at P, and the equilibrium at P is stable. Similarly, when the curve crosses the axis " negatively," as at the origin and Q, the equilibrium is unstable. Since the angle of intersection cannot be either positive or negative twice in succession, on considering rotation in one direction only, it follows that positions of stable and unstable equilibrium occur alternately and the total number of positions of equilibrium is even.

The radius of curvature of the curve of buoyancy is equal to I/V, and is always positive. The curve, therefore, has no re-entrant parts or cusps, is continuous and has no sudden changes in direction; parallel tangents (or normals) can be drawn through two points only (corresponding to inclinations separated by 180°), which property is shared by its evolute, the metacentric curve. On the other hand, the moment of inertia I varies continuously with the inclination, attaining maximum and minimum values alternately; and the metacentric curve, therefore, contains a series of cusps corresponding to the values of I when dI = o, which will generally occur at positions of symmetry (e.g. at o° and 180°), near the angles at which the deck edge is immersed or emerged, and at about 90° and 270°.

The curves of buoyancy and flotation and the metacentric curve for H.M. troopship " Serapis " are shown with reference to the section of the ship in fig. 15, and on an enlarged scale for greater centre of gravity, or, as it is termed, the centre of flotation, the curve of flotation will be the locus of the projections of the centres of flotation on the plane of the figure, which curve touches each waterline.

From consideration of the slope of a ship's side around the periphery of a water-line, Dupin obtained the following expression for p', the radius of curvature of the curve of flotation, fy 2 tan a. ds p' - area of water plane for both sides, where ds is an element of the perimeter, a the inclination of the ship's side to the vertical, and y its distance from the longitudinal axis through the centre of flotation. M. Emile Leclert, in a paper read at the Institution of Naval Architects, 1870, proved the equivalence of the above formula to the two following, which are known as Leclert's Theorem: p' =p +V dV andp' =dV' where I and V are respectively the moment of inertia of the waterplane and the volume of displacement, and p is the radius of the curve of buoyancy or B'M'. Independent analytical proofs of the formulae were given in the paper referred to; and (Trans. I.N.A., 1894) a number of elegant geometrical theorems in connexion with stability, given by Sir A. G. Greenhill, include a demonstration of Leclert's Theorem as follows (in abbreviated form): Let B, B 1 (fig. 17) be the centres of buoyancy of a ship in two consecutive inclined positions, and F, F 1 the corresponding centres of flotation. Draw normals BM, B1M, meeting at the pro-metacentre M, and FC, F 1 C, meeting at the centre of curvature C. Produce FB, F 1 B 1 to meet at 0; join OM, MC.

Then BM, CF and B 1 M, CF I are respectively parallel, and ultimately also BB ', FF 1; hence the triangles Mbbi, Cff I are similar and BM BB I OB CF = FF 1 = OF' so that 0, M and C are collinear.

If the displacement V be now increased by dV, changing B to B', and M to M', then since the added displacement dV may be supposed concentrated at F, B' will lie on OBF, and it may be shown similarly as before that M' lies on OC. Further, considering the transference of moments, BB' XV = BF XdV.

Draw MED parallel to BF, then dVBB'MEM'E dp _ _ V - BF - MD_ - CD _ - p' - p' dP

av - dv giving Leclert's first expression; also, since p = V, dV = p dv = p', which is Leclert's second expression for p'. The value of p' at the upright. can be obtained from the metacentric diagram by the following simple construction. Let M and B be the metacentre and the centre of buoyancy for a water-line WL on the metacentric diagram (fig. 18); draw the tangent to the B curve meeting WL at Q, and through Q draw QR to meet MB and parallel to the tangent to the M curve at M. Let BP=h, and area of water-line be A. Then PQ=h cot o=hAI=A; also MR =BM - (BP--PR)=p - A (tan 6d-tan 4)). If D be the draught, tan 0+tancp = - = - Adu, MR=p+VdV dp -p, the curve of flotation being concave upwards if R is below M.

For moderate in s clinations from the upright, the buoyancy of the added layer due to a small additional submersion will act through the centre of curvature of the curve of flotation; this point may be regarded as that at which any additional weight will, on being placed on a ship, cause no difference to the values of the righting moment at moderate angles of inclination. The curve of flotation, therefore, and its evolute bear similar relations to the increase or decrease of the stability of a ship due to alteration of draught, as the curves of buoyancy and of pro-metacentres do to the actual amount 'of the stability.

The curve of flotation resembles the curve of buoyancy in that not more than two tangents can be drawn to it in any given direction, but it differs in that its radius of curvature can become infinite or change sign. It contains a number of cusps determined by =O. These occur in an ordinary ship-shape body at positions: (I) at or near the angles at which the deck is immersed or emerged (four in number); and (2) at or near the angles 90° and 270°. There are, therefore, six cusps in the curve of flotation of an ordinary ship; they are shown in figs. 15 and 16 by the points F2, F3, F4, Fs, F7, F8.

The following relations between the curves of buoyancy and of pro-metacentres and the curve of statical stability are of interest, and enable the former curves to be constructed when the latter have been obtained. If GZ', GZ" (fig. 19) are the righting levers FIG. 19. corresponding to inclinations 0, 0 + do, where dO vanishes in the limit; B', B", the centres of buoyancy, M' the prometacentre; produce GZ' to meet B"M' in U.

Then, neglecting squares of small quantities, d(GZ')=Z'U=M'Z'.do, or vertical distance of M' above G = d(GZ') do Also M'B' = M'B "; hence Z"B" - Z'B'=MZ' - MZ "=Z"U= GZ'.de, or = d(B'Z') GZ do ' i.e. the vertical distance (B'Z') of G over B is equal to fGZ.do. It follows that by differentiating the levers of statical stability and finding the slope at each ordinate the vertical distance of M' over G is obtained, and M' may be plotted by setting up this value from Z' above GZ' drawn at the correct inclination; also that by integrating the curve of statical stability and finding its area up to any angle, the vertical separation of G and B' is obtained, and B' may be plotted by setting down this value increased by BG below Z'.

B1 FIG. 16.

FIG. 17.

whence FIG. 18.

The work done in inclining a ship slowly so as to maintain a constant displacement (and avoid communicating any unnecessary movement or disturbance to the water) is given by the expression f°M.dO where M is the moment resisting the inclination. This may be written W X f°GZ.dO; and it has been shown above that this is equal to the weight multiplied by the vertical separation of the centres of gravity and buoyancy. This is otherwise evident since the work is the sum of that done against the forces acting on the ship, viz. the weight and the buoyancy; these are respectively equal to WXrise of G, and W Xfall of B, giving the value W. (Z'B' - BG) as before.

The dynamical stability of a ship at any angle is defined as the work done in inclining the ship from the upright position; and its value is conveniently obtained by integrating the curve of statical stability as stated above. The dynamical stability can thus be calculated at various angles and a curve obtained, whose ordinates represent work done in foot-tons. The curve of dynamical stability is drawn for a battleship (normal condition) in fig. Io, and is there shown in relation to the curve of statical stability; it will be seen that the dynamical stability increases continuously until the righting moment vanishes, when it becomes a maximum.

A formula for the dynamical stability of a ship at any angle was given by Canon Moseley in a paper read before the Royal Society in 1850. Experiments on models made under his direction at Portsmouth Dockyard showed that the actual work in quickly inclining to a moderate angle agreed closely with that calculated in the case of a model of circular section; but considerable divergence was obtained with a model of triangular section owing to the motion of the water set up, and also, probably, to the variation in displacement during the roll.

The existence of large righting couples at moderate angles of heel is of greater importance in a sailing ship than in a steamship, since in the former it determines the amount of sail that can be safely carried under known weather conditions and thereby influences the speed. A sailing ship in motion is subjected to the wind-pressures on the sails and the upper works of the ship, and to the water-pressures on the hull. When the ship is in steady motion, these forces are equal and opposite; and, so far as the stability is concerned, it is sufficient to determine the transverse resultant of the wind-pressure on the sails, and its moment, the water-pressure on the hull affecting only the speed and leeway of the ship.

The pressure on the sails depends on their form and area, their position, and the apparent velocity of the wind, i.e. the velocity relative to the ship. The pressure of the wind on the hull is obtainable similarly to that on the sails, but is usually neglected as the heeling moment is small. Experiments have been made to determine the wind-pressure on plates by Dines, Langley, Eiffel, Stanton and others; and the results of the experiments are briefly as follows The normal pressure R in pounds on a plate of area A square feet exposed to face normally a wind of velocity V feet per second is given by the formula R = KAV 2, where K is a coefficient depending on the form and area of the plate. For a square or circular plate of about i sq. ft. in area K is about. 0014, corresponding to a pressure of 1 lb per sq. ft. at about 16 knots. The coefficient increases slightly for larger dimensions of the plate. It has also been found that a departure from the square or circular form involving an increase in perimeter for the same area causes an increase in the mean pressure. An alteration from the plane to the concave, analogous to the " bellying " of sails, is accompanied by a slight increase in the pressure per square foot of projected area; but for any large amount of concavity the increase is more than counterbalanced by the decrease in the projected area.

No simple law exists connecting the normal pressure on a plate exposed obliquely to the wind with the angle of incidence; it is found that the results for air exhibit a close agreement with those for water after allowing for the difference of density between the two fluids. At small angles of incidence up to about 20°, or even 40° (varying with the shape of the plate), the pressure varies directly as the angle; beyond this limit it is slightly diminished, afterwards increasing or decreasing to a value which is almost constant for the remaining angles up to and including 90°. The centre of pressure for oblique impact lies between the leading edge and the centre of gravity of the area. In a plate I ft. square, it lies 0.3 ft. from the leading edge at Io° inclination and 0.4 ft. at 30° inclination, gradually approaching the centre of the plate as the angle of inclination is increased. A slight curving or concavity of the plate does not appear to have much influence on the normal component of the windpressure.

The wind-pressure on the sails of a ship cannot be calculated with any degree of precision because existing information is insufficient to take account of (a) the variety in area and shape of the sails used; (b) the different positions in which the sails may be placed relative to the wind and to each other; and (c) the interference of adjacent sails with each other. On the other hand, conclusions based on these experiments are of value both in assisting in an intelligent appreciation of the effects of changes in the sail areas, sail positions, and in the form of rig, and in forming a comparison between the various qualities of speed, stability and general behaviour of vessels with which experience has been obtained.

The stability of a sailing vessel is usually estimated by assuming all plain sail to be placed in a fore and aft direction and to be subject to a normal pressure of i lb per sq, ft., corresponding to a wind of about 16 knots. The resultant pressure of the wind is supposed to act through the centre of gravity of the total sail area (termed the centre of effort). The resultant pressure of the water on the hull, which is equal and opposite to the wind-pressure, is assumed to pass through the centre of gravity of the area of the immersed middle line plane (termed the centre of lateral resistance). If h be the vertical distance between these points in feet, A the sail area in square feet, and a the angle of heel, the moment causing the heel is (on these assumptions) 2240 and o and the righting moment is approximately Wxgm sin a.

Hence A h sin a = 2240.W X GM' The reciprocal of this quantity or 2240 Wxgm Ala is a measure of the capability of the ship to stand up under her canvas and is termed the power to carry sail. Its value varies with different sizes and classes of ships and boats. It is relatively small in small boats and small yachts owing to the practicability of reducing the angle of heel by movable ballast; and a low value is also permissible in large yachts on account of their great range of stability. In boats and yachts it varies from 3 to 4 and in full-rigged sailing ships from 15 to 20.

The stability of sailing vessels at large angles of inclination varies considerably with the class of vessel. In racing yachts and other completely decked sailing boats whose ratios of beam to depth and draught are comparatively small, initial stability is obtained by lowering the centre of gravity with ballast fitted on the keel, and the range then extends to considerably over 90°; on the other hand, a number of half-decked or open sailing boats immerse their gunwales when inclined to a moderate angle. With reference to this, Mr Dixon Kemp in his Yacht Architecture remarks that the deck edge should not be immersed at an angle of heel less than 20°; some small centre-board boats whose gunwales are awash at 12° or 15° cause anxiety. With full-rigged sailing ships this angle is commonly 20° to 25°.

The effect of a sudden gust of wind on a sailing ship is obtained by equating the work done on the ship by the gust to her dynamical stability; and the angle at which this equality holds will be the extreme angle of heel, assuming the ship to be originally upright and at rest. Since the dynamical stability is represented by the area of the statical stability curve it is convenient to represent this angle in relation to this latter curve. The effects of the resistance and inertia of the water and any change of displacement are neglected; the wind-pressure is assumed constant during the roll, in accordance with the results of experiments on oblique plates (the maximum angle of roll being supposed less than 5 00); the modification of the pressure due to the motion of the sail is also neglected.

Let OPQ (fig. 20) be the curve of statical stability, the ordinates representing righting moments, and let the heeling couple due to the gust be represented by OS. If N be the extreme angle of heel, draw Spur parallel to the base, cutting the curve at P, R; and PM, NQ perpendicular. The work done by the wind is the area Osun and is equal to the dynamical stability of the ship or the area Opqn. Hence the areas OPS, PQU are equal, and the extreme angle of heel is determined by this equality. If P and Q lie on the initial and approximately straight portion of the curve, the extreme angle of heel ON is about twice that of the steady angle OM corresponding to the strength of the gust. The area QUR represents the reserve dynamical stability when the wind is blowing with strength corresponding to OS; the intercepts of the ordinates below Spur doing work against the force of the wind, leaving the segments above SPR available for absorbing the kinetic energy possessed by the vessel at the position of steady heel PM. As the strength of the gust is increased the points P and Q travel farther along the curve until P', Q' are reached, such that the areas P'Q'Q, OTP' are equal; the vessel will then come momentarily to rest at Q' and will be in unstable equilibrium, any increase in the wind-pressure causing her to capsize. It follows that a ship sailing in a wind of sufficient strength to cause a moderate angle of heel equal to OM' will be on the point of capsizing if the wind should happen to drop and afterwards return suddenly with its - FIG. 20.

former force. A more dangerous, though improbable, case in which a gust of wind strikes the ship just as she has completed a roll to windward can similarly be investigated; it is found that the safe angle of steady heel under this condition is considerably less than that represented by OM'. It thus appears that it is of the greatest importance that sailing vessels should possess large dynamical stability in order to provide against the risk of capsizing due to fluctuations in the wind :pressure. Although the neglect of the wind and water resistances in the above investigation materially modifies the quantitative results, the general conclusions point to the necessity for sufficient range and freeboard however large the righting levers may be at small inclinations.

The centres of effort and of lateral resistance have not the same longitudinal position, consequently a horizontal couple is produced which turns the vessel either into the wind or away from it. In the former condition the vessel is said to be " ardent," and in the latter to be " slack." In order that a vessel may be quick in going about and yet not require too large a helm angle on a straight course, she should be slightly " ardent," i.e. the true centre of effort should be slightly abaft the true centre of lateral resistance. The assumed and true positions of these centres differ to some extent, and on making allowance for this it is found that in the majority of vessels possessing slight ardency the assumed C.E. lies slightly before instead of abaft the assumed C.L.R. In small sailing boats the points are usually very near together; but in a large number of sailing ships, including H.M. sloops, their distance apart is about. 05 L, and in yachts about 02 L, where L is the length.

It may be noted in this connexion that the area of sail spread and the size of the ship are often connected by the coefficient W known as the Driving Power. The value for small sailing boats and for y achts is about 200, and for full-rigged sailing ships from 80 to too (including plain sail only).

The method of estimating the righting moment of a ship when ongl- inclined from a position of equilibrium through a small angle in the longitudinal plane is exactly analogous to that used in the case of small transverse inclination, and stab/l/ty similar propositions are true in both cases, viz.: I. Consecutive water-lines intersect about an axis passing through the centre of flotation.

2. The height of the longitudinal metacentre M above the centre of buoyancy is equal to the moment of inertia about this axis divided by the volume of displacement of the ship.

3. The righting moment at any small angle of inclination 0 (circular measure) is equal to W. G M .0.

In fig. 21 let WL be the water-line corresponding to the positions G and B, and conceive a longitudinal movement of a portion of M 0 L FIG. 21. the weights in the ship causing G to move horizontally to G'. If G' be abaft G the ship will alter trim by the stern until B moves to B' vertically beneath G' and the water - line changes to W'L', intersecting WL at the centre of flotation F.

If L be the length of the ship between the draught marks, the change of trim (WW' +LL') is equal to L .0, and the moment changing trim is W. GG' or W. GM .0; the change of trim in inches (other linear dimensions being in feet) is therefore Wxgg' =Wxgm 12l The change of trim due to any horizontal movement of weights is therefore equal to the moment of the shift of weight divided by the quantity Wxgm 12 Xl which is the moment required to change trim one inch. Since the longitudinal moment of inertia of the water-plane includes the cube of the length as a factor, the longitudinal BM is usually large compared with BG, and the moment to change trim I in. in foot-tons is nearly equal to Wxbm Wxi _ I 12xl - 12xlxv 420l' which is approximately constant for moderate variations of draught.

feet abaft the centre of flotation F, the bodily sinkage in inches is If a weight of moderate amount w tons be placed at a distance of a the moment changing trim by the stern is wa foot-tons, and the change of trim is therefore where T is the " tons per inch " and M the moment to change trim t in. If b be the distance of F abaft the middle of length, the draughts forward and aft are increased respectively by /[ a L+2b 1' M? 2L) w and wit T+1Vl L-21) 2 inches.

A ship provided with water-tight compartments is liable to have water admitted into any of them on account of damage received, or may require to carry water or other fluid in bulk as ballast or cargo. The effect of this addition on the draught and the stability is therefore of interest. There are three cases: I. When the water completely fills a compartment; 2. When the water partially fills a compartment up to the level of the water-line, remaining in free communication with the sea; and 3. When a compartment is partially filled with water without any communication with the sea.

In the first case the water is regarded as a weight added to the ship; the mean sinkage is obtained from the displacement curve, the change of trim from the " moment to change trim," and the angle of heel from the metacentric diagram, or (for large angles) the cross curves. In general, if the compartment filled is low in the ship, the stability is increased; if high, it is diminished.

In the second case, assume in the first place the compartment to be amidships, so that no heel or change of trim occurs, and to be moderate in size, so that the sinkage is moderate in amount.

Let ABCD (fig. 22) be such a compartment bounded by watertight bulkheads sufficiently high to prevent water reaching adjoining 6 FIG. 22.

compartments. Let the water-lines be Wefl, W'Ghl', before and after bilging; let A, a be the area of the whole water-plane Wefl and of the portion EF within the compartment respectively, in square feet; and let v be the volume contained in Ebcf diminished by the volume of any solid cargo in the compartment. The buoyancy is reduced by an amount v by bilging, and the amount added through sinking must be equal to the amount so lost. If x be the sinkage in feet, then v=x(A - a), so that the mean sinkage is equal to the buoyancy lost divided by the area of the intact water-plane. In the event of the compartment being so situated as to cause heel and change of trim, the mean sinkage is first determined as above, and the effect of heel and change of trim superposed.

To obtain the heel produced, the position of the centre of flotation for the intact portion of the water-plane is found, and thence the vertical and horizontal positions of the new centre of buoyancy are deduced by taking account of the buoyancy lost through bilging, and then regained by the layer between the two water-planes. The moment of inertia of the intact water-plane is found about an axis through the new centre of flotation and thence the height of the new metacentre M' determined. The heel 6 (assumed small) is found by equating the horizontal shift of B to sin 6 X the vertical distance of M' above G, both being equal to the moment causing heel divided by the displacement. In a similar manner the change of trim is obtained. If the compartment bilged is large so that considerable changes in its area and that of the ship at the water-line result, the sinkage and alteration in stability are found by a tentative process, closer approximations to the final water-line being successively made.

An investigation of the stability when bilged at or near the waterline is of special importance in warships owing to their liability to damage by gunfire in action, with the consequent opening up of a large number of compartments to the sea. Calculations are made of the sinkage and stability when the unarmoured or lightly armoured parts of the ship are completely riddled; the stability should be sufficient to provide for this contingency.

	11,
1,111,?11?Yz	bi b s

The third case, where the ship is intact but has compartments partially filled with water or other liquid, is of frequent occurrence. Practical illustrations occur in connexion with the filling and W' o s_ J L emptying of water-ballast and oil-fuel tanks, and particularly in the amount calculated. To avoid danger of capsizing in still water, case of ships fitted to carry large quantities of oil in bulk. large tanks in a ship are filled or emptied in succession as far as Let fig. 23 represent the section of a vessel fitted with a tank possible, so that not more than one or two are partly full at the same Pqrs partly full of water. Let WL, wl be the upright water-lines time. Water-tight longitudinal partitions are also fitted in wide tanks in order to reduce the moment of inertia of the free surface. On the other hand tanks, partly filled with water, have been fitted and found effective in certain ships in order to reduce the rolling oscillations among waves. (See § Rolling.) Hitherto the stability of a ship has been considered only with reference to inclinations about either a longitudinal or transverse FIG. 23.

of the vessel and tank, G the centre of gravity of the vessel and water combined, B the centre of buoyancy of the vessel, and b the centre of gravity of the water.

As the ship is inclined successively through angles 0 1, 0 2, .. the centre of buoyancy B moves along the curve of buoyancy to B1, B2, ... the normals at which are tangential to the metacentric curve M 1 M 1, ... those at small angles passing through the metacentre M. If the water in the tank could be kept from moving as the inclination proceeded, G would be fixed in the ship, and the righting levers would be GZI, GZ 2, ... those at small angles being equal to GM sin B. Actually, if the inclination be slowly performed, the water-level in the tank changes successively to w 1 1 1 , w212,... maintaining a level surface at all times; its centre of gravity moves to b1, b2,... thereby causing a corresponding alteration in the combined centre of gravity G. Drawing br1, br2, ... perpendicular to the verticals through b 2 ,.. . and calling w, W the weights of the water and of the water and ship combined, then at the angle 0 2 the line of action of the weight of the water w has moved through a distance br 2 and the righting moment of the ship is diminished by an amount w X br2. It is evident that the movement of the centre of gravity of the water in the tank is the same as would be the movement of the C.B. of a ship having the same form as the tank and water-lines corresponding to wl, w,1 1 , w212, &c. The values of the levers br1, br2... can therefore be obtained by a process similar to that used for obtaining the righting levers of the ship; cross curves and thence ordinary stability curves being drawn for various heights of water and inclinations. If 0 1 be a small angle of inclination, the line of action of the weight b,m will be such as to pass through the metacentre m corresponding to the water-line wl, and determined by the formula bm = v where i is the moment of inertia of the water-plane wl about a longitudinal axis through its centre of gravity and v the volume of water contained. The moving weight w at b may there fore be replaced by an equal weight fixed at m, which is the virtual centre of gravity of the water; and the centre of gravity G of ship and water is likewise raised to a virtual position G' where GG'=W bm= v V If the tank contain a fluid of specific gravity the virtual rise of the centre of gravity is The loss of stability at small angles due to the mobility of the water is thus independent of the quantity in the tank, but is proportional to the moment of inertia of its free surface. It is possible for a small quantity of water with an extensive free surface to render a ship unstable in the upright condition; the angle to which this large loss of stability extends depends, however, on the quantity of water in the tank, for the extent of the sideways movement of the centre of gravity G of ship and water is minute if the tank be either nearly empty or nearly full, and the loss of stability at all angles above a small amount will then be inappreciable; the loss at moderate angles is usually a maximum when the tanks are about half full.

The assumption made above, viz. that the ship is inclined so gradually as to maintain a level water surface in the tank, is by no means in accordance with the actual circumstances during rolling; waves are then set up in the water, causing it to wash from side to side, so that the loss of stability may be either more or less than the axis. These are the only cases which it is necessary to Stability deal with in practice for the purpose of ascertaining the probable qualities as regards stability of a vessel by direction. comparing the elements of its stability in the design stage with those of existing ships whose qualities have been tested by experience. For the exact theoretical consideration of the stability of a ship or any floating body, however, it is necessary to take account of the true line of the action of the buoyancy and not merely of its projection on the plane of inclination. The development of this part of the subject has largely been due to M. Dupin in his Memoire de la stabilite des corps flottants and to M. Guyou in his Theorie du navire. If a ship is inclined in all possible positions, keeping the displacement constant, the locus of the centre of buoyancy is a closed surface which is known as the surface of buoyancy; the curve of buoyancy for two-dimensional inclinations being the projection on the plane of rotation of the corresponding points on the surface of buoyancy. Similarly the envelope of all the water-planes is defined as the surface of flotation. The stability of a ship in all positions is known when (a) the forms ancl'dimensions of the surface of buoyancy, and (b) the position of the centre of gravity relative to it, have been obtained; the former depends entirely on the geometrical form of the ship and on the constant volume of displacement assumed, and the latter has reference only to the arrangement and magnitude of the component weights of the structure and lading. For an infinitesimal inclination the line joining the centres of buoyancy when upright and inclined is parallel to the water-plane, and the tangent plane to the surface of buoyancy is therefore parallel to the waterplane, i.e. it is horizontal, and the normal to the surface is vertical. If the initial position is one of equilibrium, the centre of gravity must lie on the normal. To determine the effect of a small disturbance from the position of equilibrium, it is necessary, as in the particular inclinations already considered, to find the line of action of the buoyancy for adjacent positions, i.e. to trace the normals to the surface of buoyancy. Consecutive normals to this surface will not, in general, intersect; but, from the properties of curvature of surfaces, there are two particular directions of inclination for which adjacent normals to the surface will intersect the original norY' mal, these directions being perpendicular to one another and parallel to the principal axes of the indicatrix of the surface of buoyancy.

If fig. 24 be a plan of the water-plane, Ox' the axis of FIG. 24.

THEORETICAL]

inclination passing through O the centre of flotation, Oy' and Oz perpendicular axes in and at right angles to the plane of flotation, then, from a consideration of the wedges of immersion and emersion for a small inclination 0, the travel of the centre of buoyancy B becomes: - 6 ' .dx'.d ' ' V y y (or BB,. fig. 2}) parallel to Oy x'y'.dx'.dy' (or - B,B2) parallel to Ox' 2 0'2 f f y'. dx'. dy' (or B 2 B') parallel to Oz. These may be written - Ix'; -e P; UI x ' respectively where I x ' is the moment of inertia of the water-plane about Ox', and P the product of inertia about Ox', Oy'. If the principal axes of inertia of the water-plane Ox, Oy make an angle with Ox', Oy', and if, from B as origin, axes Bx, By, Bz are drawn parallel to Ox, Oy, Oz, then the co-ordinates of B' are as follows: - B x= - B 1 B 2 cos 0 - BB, sin (A= V (P cos 4' y= BB, cos It.--BIB2 sin 4'=U(I x ' cos 4'-1-P sin 0); 2 z= B 2 B'= Ix'. Also Ix' = I x cos {-I y sin 4'; P = (I x - h) sin 4 cos and where I x, I,, are the principal moments of inertia of the water-plane. Hence x= - v h .sin 4); y= I x cos 02 z = 2 V (Ixcos 2 4)+I„ sin e 4)).

Eliminating 0 and 4, the locus of the centre of buoyancy for small inclinations of the ship becomes the elliptic paraboloid - x2 y2 22= I"/`7+Iz /V The equation to the indicatrix referred to axes parallel to B; By is therefore x2 y2 I"/V+Ix/V =constant; unaltered. The resultant couple can be readily found, but in this case it bears no simple relation to the indicatrix, as before; it may be shown, however, that the plane of the couple is conjugate to the axis of inclination with respect to the confocal ellipse x2 y- + 1 - constant.

V

a V - a In the case when GM =0, the ship being in neutral equilibrium for that direction of inclination, the resultant couple is parallel to the axis Ox', i.e. perpendicular to the plane of the indicatrix.

0

o°

I°

5°

to°

20°

30°

40°

50°

60°

70°

80°

90°

GM

4'

4 I

7'

16

50.4'

103'

168'

237

300

354

388

400'

a

0°

60°

78.5

76.8°

68 5°

59°

49'3°

39'5°

2 9'7°

1 9' 8 °

9'9°

0°

?

90°

?9'o°

6'5°

3' 2 °

1'5°

1'0°

0'7°

0'5°

0'3°

0'2°

0'1°

o°

Numerical values of the metacentric height GM, the angle of obliquity a or QOM (equal to tan l I x ' PaV) and the angle are given in the following table for a ship whose transverse GM is 4 ft., longitudinal GM 400 ft., and BG io ft.: - and the indicatrix is therefore similar and similarly situated to the momental ellipse of the water-plane, and the surface of buoyancy is everywhere synclastic and concave to all points within it. The quantities I " /V and I x /V are evidently equal to BM x and BM " (referring to inclinations about Oy and Ox respectively); and the indicatrix and momental ellipse become x 2 y2 B M5+BMx =constant.

The angle ,/i that BB 2 (the projection of BB' on the plane of the indicatrix) makes with xO is given by tan 4)= - I x. cot cp; hence the direction is conjugate to that of the axis of rotation with respect to the indicatrix.

° t I ' This is illustrated in fig. 25, where the ellipse shown is the indicatrix; OPx' the axis of inclination, OQ the conjugate radius, and ORMy' the perpendicular on the tangent. Draw QN parallel to OM to meet OP. The triangle OMQ is similar to BB 1 B 2; and they can be made equal by giving a suitable value to the constant in the indicatrix equation. In that case QN is the projection on the plane of the figure of the normal to the surface at 13 1, and the shortest distance between the normals at B and B 1 is equal to ON = MQ = B 1 B 2 = V, since ON or the axis of inclination is perpendicular to them both. Also, the length B'M of the normal at B' intercepted between B" and the foot of the common perpendicular is equal to Qe since 0 is the angle between the normals at B and B'; it follows that B'M' = B 1 = V, an expression analogous to that obtained before for the case of small inclinations in the direction of the principal axes of the waterplane. It is worthy of note that the radius of curvature p of the normal section of the surface of buoyancy through Oy' is, in general, 2 less than BM; the latter being equal to, and p being equal 22 to 2 2; p is also obtainable by Euler's equation - I _ cos 2 4) s11124) n - BMx+BM5' becoming equal to BM for inclinations about the principal axes. Similarly the radius of curvature of the normal section through Q is, in general, greater than BM.

If the centre of gravity G of the ship is coincident with B, the arm of the righting couple is OM or Ix B; and there is also a couple of lever ON or U 0 i n a perpendicular vertical plane. The resultant couple lies in a plane containing OQ, having a lever equal to OQ or VIIx'2+P2 or U. II x 2 cos 2 0+4 2 sin 20. In the general case when G is situated at a distance a above B, the righting lever becomes (ky-_a) 0, and the perpendicular couple is The greatest angle of obliquity (a) occurs in this case when is about 54° and the plane of the couple is nearly coincident with the middle line plane for all angles of 4) greater than about 30°. It follows that if a weight is moved obliquely across the ship the axis of rotation is approximately longitudinal, except when the line of movement is nearly fore and aft; and in the latter case a small deviation from a fore and aft direction produces a large change in the position of the axis of rotation.

The direction of the axis of rotation is above expressed with reference to the position of the inclining couple in relation to the indicatrix of the surface of buoyancy; as, however, the couple is assumed small, the direction of the axis and the amount of inclination may equally be obtained by resolving the couple in planes perpendicular to the principal axes and superposing the separate inclinations produced by its components.

It has been shown above that the positions of equilibrium are found by drawing all possible normals to the surface of the buoyancy, and the condition for stability for an inclination in any direction is that the centre of gravity shall lie below the corresponding metacentre. The height of the metacentre varies with the moment of inertia of the water-plane about the axis of inclination, and the maximum and minimum heights are associated with the maximum and minimum moments of inertia, which again correspond to inclinations about the least and greatest axes of inertia respectively. If the centre of gravity lies below the lowest position of the metacentre (the transverse metacentre in the case of a ship when upright) the equilibrium is stable for all inclinations, and the condition is referred to as one of absolute stability; if it lies above the highest metacentre, the condition is one of absolute instability; if it lies between the highest and lowest metacentres, the condition is one of relative stability; the ship being stable for inclinations about a certain set of axes, and unstable otherwise.

The foregoing remarks apply to a vessel whose axis of inclination is fixed so that the component couple perpendicular to the plane of inclination is resisted. If, on the other hand, the vessel is free to move in all directions the resultant couple does not in general tend to restore the original position of equilibrium, although the component in the plane of inclination complies with the conditions above stated for absolute stability. If m 1 and m 2 be the greatest and least values of GM, the ratio of the component couples perpendicular to and in the plane of inclination, or tan a (fig. 25), is greatest when tan 4) -= / -; and then tan a - ?n1, - m2 If m2/m1 be small, this m 2 m1m2 ratio is large, being equal to 4'95 in the numerical example above. In such cases the extent of the movement that can result from a small initial disturbance cannot be readily determined by a statical method, but the investigation of the work done in moving the vessel from one position to another appears to meet this difficulty.

This process is employed by M. Guyon in his Theorie du navire, the stability of a ship in any condition being treated throughout from the dynamical standpoint. He proved that: I. For changes of displacement, without change in inclination, the potential energy of a system consisting of a floating body and the water surrounding is a minimum when the weight of the body is equal to its displacement.

2. For changes of direction, without change of displacement, the potential energy of the system is equal to the weight of the body, multiplied by the vertical resolute of BG; when this distance is a minimum or a maximum the stability is respectively stable or unstable. A statical proof of this has been given in the twodimensional case.

The potential energy is thus equal to the dynamical stability FIG. 25.

increased by an arbitrary constant. If from any point B 1 ' of the surface of buoyancy (fig. 26) a tangent plane be drawn, the perpendicular upon it, GN, is proportional to the potential energy, and the stability of the body is thus the same as that of the surface of buoyancy regarded as a solid capable of rolling on a horizontal plane. The locus of the foot of the perpendicular N is called the " podaire " (shown dotted in the figure); this surface resembles the surface of buoyancy in its general shape, and touches it when GB is normal, i.e. at positions of equilibrium B 1, B2, 133, B 4 ,; it has the property that a radius GN drawn from G is always vertical when the body is in the position corresponding to N, and has a length proportional to the potential energy.

If the ship or body be supposed to move under no external forces, and the effect of any change in the displacement be neglected, the kinetic energy of the system can be expressed by /mv 2 /2g, and the total energy by (Wxgn)-}-Zg. Zmv; the latter is constant when there are no resistances, and steadily decreases if resistances are in operation. Neglecting resistance, when the body is momentarily at rest, W XGN becomes W.l, where 1 is a linear quantity; and through out the motion GN is less than 1 by 2 g Zinv . The effect of re sistance is gradually to decrease l or the maximum value of GN; and it may be exhibited graphically by the following conception. Imagine a sphere of water, with centre at G, to be originally entirely within the podaire and then to be capable of expanding until the whole surface is submerged. It will first touch the podaire at the minimum normal, and will then form a small lake round it; similar lakes will form later at all other positions of absolute stability. Positions of absolute instability will be touched externally by the sphere, and if the water recede a little, will form small islands. At positions of relative stability the water will in general divide the surface into two parts meeting at an angle (fig. 27), and become one or the other of the branches XX', YY' according as the size of the sphere is slightly increased or diminished. Let the radius GN to the podaire along the Y edge of the water be represented by 1; from the energy equation the FIG. 7 radius for any other position of the body moving without external forces is less than 1, and the position lies within the lake so bounded. The diminution of 1 due to resistances has the effect of gradually drying the lake. If the body is originally placed near a position of absolute stability, the small lake on drying will leave the body in or very near that position. On the other hand, if the body is placed at rest near a position of absolute instability, the water in drying will necessarily cause the body to move farther and farther from that position. Finally, if moving near a position of relative stability, the body will move freely from side to side until the drying has proceeded so far that separate branches XX' or YY' are obtained; when this occurs, the body will be fenced, as it were, on one side or the other, and will oscillate until a position of absolute stability is finally attained.

With regard to the surface of flotation it has been shown that in order that the displacement shall remain constant, consecutive waterlines must intersect on a line passing through the centre of gravity of the waterline or the centre of flotation. If the inclination take place from a given position in all possible directions, the lines of intersection with the original water-plane will all meet at the centre of flotation, which must, therefore, lie in the envelope of the water-planes, or the surface of flotation. The surface is therefore the locus of the centre of flotation for all possible inclinations. Since the curvature of the curve of flotation, which is the projection of the centre of flotation for inclinations about an axis perpendicular to the plane of projection, may change sign, the surface can also undergo similar changes in curvature acrd may be synclastic in certain parts and anti-clastic or saddle-shaped in others.

The relation between the surface of flotation and the stability of the ship is similar to that established in the two dimensional cases, i.e. the projection on the plane of inclination of the curve corresponding to the inclination has a centre of curvature whose height is a measure of the increase or decrease of stability caused by an alteration in displacement; the investigation, however, of the general case and the extension of Leclert's theorem to oblique inclinations contain no features of special interest or importance.

Rolling of Ships. The action of the waves upon a ship at sea is such as to produce rolling or angular oscillations about a horizontal longitudinal axis, pitching or angular oscillations about a horizontal transverse axis, and heaving or translational oscillations in a vertical influence of the other accompanying oscillations, whose effect in most cases is slight in magnitude although complex in character.

The ship is in the first place conceived to be rolling in still water without any resistances operating to diminish the motion. The equation of motion for moderate angles of inclination. within which the arm of the righting couple is approximately proportional to the angle of heel (i.e. GZ =m X6), is d20 g (1) dt - - el m . B, .

where e is the radius of gyration of the ship about the axis of rotation, m the metacentric height, 0 the angle of inclination and g the acceleration produced by gravity. From this the time deduced for a single oscillation, from port to starboard, or vice versa, is T -= 7r7V.. (2) m .g showing that the time of oscillation varies directly as the radius of gyration, and inversely as the square root of the metacentric height. The value of T is generally about to seconds in a large Atlantic liner, 7 to 8 seconds in a battleship, and 5 to 6 seconds in secondclass cruisers and ships of similar type. In a large modern warship e is about one-third the breadth of the ship.

For unresisted rolling of ships among waves the theory generally accepted is that due to Froude (see Trans. Inst. Nay. Arch., 1861 and 1862). Before his work, many eminent mathematicians had attempted to arrive at a solution of this most difficult problem, but for the most part their attempts met with scanty success; wave-motion and wave-structure were imperfectly understood, and the forces impressed on a ship by waves could not be even approximated to. Froude's theory is based on the proposition that, when a ship is among waves, the impressed forces on her tend to place her normal to a wave sub-surface, which is assumed to be the surface passing through the ship's centre of buoyancy, and which is regarded as the effective wave surface, as far as the rolling is concerned. As in water at rest the ship is in equilibrium when her masts are normal to the surface of the water, so in waves she is in equilibrium when her masts are normal, instant by instant, to the effective surface of the wave that is passing her. When she at any instant deviates from this position, the effort by which she endeavours to return to the normal depends on the angle of deviation, in the same manner as the effort to assume an upright position, when forcibly inclined in still water, depends on the angle of inclination. Hence her stability (i.e. her effort to become vertical) in still water measures her effort to become normal to the wave at any instant on a wave. Froude made the assumptions that the profile of the wave was a curve of sines, and that the ship was rolling broadside on in a regular series of similar waves of given dimensions and of given period of recurrence. He was aware that the profile of the wave would be better represented by a trochoid, but in his first paper he gave several reasons why he preferred the curve of sines. He also assumed that the ship's rolling in still water was isochronous, and that the period of the rolling was given by T= ' i V z g , as obtained theoretically. On these assumptions n the equation of motion is obtained by substituting, for the angle of inclination in still water, the instantaneous angle between the ship and the normal to the wave-slope, and thus becomes = = - 72 (0 - 61),. .. (3) where 0=angle of ship's masts to the vertical, and 0 1 =angle of normal to wave-slope to the vertical at the instant considered. 01 has to be expressed in terms of time, and is given by 0 1 =0 1 sin TI' where e i is the maximum wave-slope, T 1 is the half period of the wave, i.e. half the time the wave takes to travel a distance equal to its length, and t is the time dating from the mid-trough of the wave. Equation (3) can therefore be written - d2B '?" (dt? = -'- 0 - 0 1. sin+i)t which is the general differential equation of the unresisted motion of a ship in regular waves of constant period. The solution of this equation is 0 1 7 T2. sinTit, I --112 direction; also horizontal translations and rotations about a vertical axis which are not generally of an oscillatory character and will not materially affect the rolling. It is convenient when considering rolling to neglect the. (4) B = C I. sin I d-C 2 cosTt {- O` I 2 sin I - ' I -t, (5) I - T12 where C 1 and C2 are constants depending on the initial movement and attitude of the ship.

The last term of this expression, represents the forced oscillations imposed on the ship by the passage of the series of waves during the time t; and the first and second terms, C,. sin Z ' r - t-I-C2. cos Lt, are the same as the free oscillations of the ship in still water. Equation (5) indicates, therefore, that the ship performs oscillations as in still water, but has superposed on these a series of oscillations, governed by the wave-slope and the relation existing between the period of the ship and that of the wave. The equation shows that there will be innumerable phases, and of these three are worthy of notice.

(a) In the case in which the ship's period T is equal to the semiperiod T 1 of the wave, equation (5) becomes indeterminate. The correct solution to equation (4) is then 0 = C1 sin I.t+C2cos4t -2, I OE cos rt.. (6) It is seen that at each successive wave crest and hollow the range of the oscillation is increased, so that the ship under these conditions would inevitably capsize but for the effect of the resistances and the departure from synchronism at large angles of roll.

(b) When T1 =0, in which case the ship is assumed to be quick in her movements, or the period of the wave is infinitely long as compared with that of the ship, the equation (5) becomes - 9 = 0, sin I 1t, that is to say, the ship will behave very much as a thin flat board does on the surface of a wave, her masts being always perpendicular to the surface.

(c) If we choose the initial conditions in equation (5) so that the coefficients C 1 and C2 are zero, then the equation will become - B = O . I T2 sin Tit. I T12 Since 9 1, the slope of the wave, is equal to 0 1 sin T 1 t, the ratio of the ship's angle to the vertical to the angle that the normal to the wave-slope makes with the vertical, or 0/011 I T2 - constant.

I

T12 That is to say, the ship forsakes her own period and takes up "forced" oscillations in the period of the wave. Under these conditions the ship's masts will lean towards the wave-crest if T is greater than T1, and from the wave-crest if T is less than T1.

Froude in his first paper further showed how the successive angles of a ship's rolling may be exhibited graphically, and he touched on the influence of resistance in reducing rolling. The following is the summary he gave in 1862 of the conclusions he had reached: " (i.) All ships having the same ' periodic time,' or period of natural roll, when artificially put in motion in still water, will go through the same series of movements when subjected to the same series of waves, whether this stability in still water (one of the conditions which govern the periodic time) be due to breadth of beam, or to deeply stowed ballast, or to any such peculiarity of form as is in practical use.

" This statement would be almost rigorously true if the oscillations were performed in a non-resisting medium, or if the surfacefriction and keel-resistance, by which the medium operates to destroy motion, were of the same equivalent value for all the ships thus compared. It requires, however, to be modified in reference to the circumstance that of two ships having the same periodic time in still water, the comparative forms may be such that the one shall experience such resistance in a higher proportionate degree than the other, and the necessary modification may be expressed in terms of their relative behaviour when set in motion in still water. The vessel which is the more rapidly brought to rest by resistance in still water will in the greater degree resist the accumulations of angle imposed on her by consecutive wave-impulses, and will the more fall short of the maximum angle which both would alike attain if oscillating in a non-resisting medium.

" (ii.) The condition which develops the largest angles of rolling is equality in the periodic times of the ship and of the waves; and this is true alike for all ships, whether their scale of resistance, as above referred to, be large or small.

" (iii.) That ship will fare the best which, caeteris paribus, has the slowest periodic time.

" (a) The waves which have a periodic time as slow as hers will have a greater length from crest to crest than those of quicker period; and, on the whole, long waves are relatively less steep than short ones. Now it is the steepness of the waves in a wave-series, not their height simply, which governs the rate at which angles of rolling will accumulate in a given ship when exposed to it.

" (b) Of two ships one of which has periodic time rather slower than the waves in a given ratio, the quicker ship will accumulate the larger angles.

" (c) It will require a heavier or a more continued gale to rear waves which have the lengthened period.

" (d) When the gale has continued so long that the largest waves have outgrown the period of the ship, she will not thereby have been released from the operation of waves having her own period, since the larger waves carry on their surface smaller waves of every intermediate period (this, at least, I believe to be the case).

" (e) When the gale has ceased and the sea is going down, the slower the period of the ship the sooner she will be released from waves of as slow a period.

" (iv.) There are two, and only two, methods of giving a slow period to a ship: " (a) By increasing her ' moment of inertia,' as by removing her weights as far as possible from her centre of gravity; an arrangement which for the most part can only be accomplished to a limited extent.

" (b) By diminishing her stability under canvas. This can always be accomplished in the construction of a ship, and generally in her stowage, to any degree consistent with her performance of her regular duties, by simply raising her weights. Were we to raise these so high as to render her incapable of standing up against the action of the wind on her sails, the steepest waves would pass under her without putting her in motion.

" Thus the enormous weights carried by the armour-plated ships, extended laterally to the greatest possible distance from the centre of gravity, and raised high above it, serve in both respects to moderate, not to enhance, this tendency to roll; and when it is said that with the weights thus placed, and once put in motion, a ship ' must roll deep (deep, though easy),' it should be remembered that those very relations of force and momentum, which show how difficult it must be to check her motion when once it has been impressed on her, show also that it must be equally difficult to impart that motion to her in the first instance. The difficulty of starting her has a priority in point of time over the difficulty of stopping her, and prevents it from being felt by limiting the motion which would have called it into play.

" (v.) The conditions which govern pitching may be noticed here, though they have not been discussed in the paper.

" Were it possible, by concentrating her weights or by extending her plane of flotation, to give to the ship a period indefinitely quick for both longitudinal and transverse oscillations, as compared with that of such waves as are large enough to put her in motion, she would acquire no cumulative oscillation, but would float always conformably to the mean surface of the wave which passes under her.

" But this condition, which is so unapproachable in practice in reference to transverse oscillations that the attempt to approach it will but develop the evils pointed out in (iii.), is of necessity so closely approached in practice in reference to longitudinal oscillations, that those evils can only be escaped by approaching it as closely as is possible. The plunging of a ship whose weights are extended far fore and aft is but an incipient development of those phases of oscillation which have their proper development in transverse motion only. The best that can be desired in reference to longitudinal motion is that the ship's period, for longitudinal oscillation, shall be as quick as possible, and her position always as conformable as possible to the mean surface of the passing waves.

" I have insisted here, more prominently than in the body of the paper, on the circumstance that a total loss of stability, using that word in the ordinary sense of power of carrying sail, implies the possession of absolute stability, as regards rolling motion due to wave-impulse, because it has been pointed out to me that the attention of readers should be more strongly directed to it, not indeed as representing a practically available possibility, but as serving best to force the mind, by contact with an extreme conclusion immediately deducible from the theory, to appreciate its fundamental principles. And the proposition thus certainly furnishes a crucial test of whether the principles have been appreciated or not, and it supplies also a ready means of testing the theory by a crucial experiment. I must, in addition, express my own confident belief that any one who will try the experiment fairly will find the proposition so fully verified that he will feel obliged to admit that the theory which leads to so paradoxical yet true a conclusion deserves at least a careful study. But the more practically useful aspect of the theory is that which presents to view the varying phases of cumulative oscillation which a ship tends to undergo when exposed to various types of wave-series; the phases depending on the relation which her natural period of rolling, when set in motion in still water, bears to the period of wave-recurrence, and on the maximum steepness of each individual wave of the series - phases, in fact, which she would actually undergo but for the effect of surface-friction and keel-resistance; the nature and value of which conditions, as well as the nature and necessity of experiments for their determination, have been pretty fully dealt with in the body of the paper.

" I will here only add a synoptical statement of the principal features of those phases, given in a rather more complete form than in that part of the paper which referred to them, though they are pretty fully exhibited by the diagrams.

" By a ' complete phase ' is meant that series of oscillations which the ship undergoes counting from the time when, for a moment, she is stationary and upright in a similar position, and is about to recommence an identical repetition of the movements she has just completed.

" For the benefit of those who may glance at the appendix before they read the paper, I will mention that T is the number of seconds occupied by the ship in performing a single oscillation in still water, starboard to port, or vice versa. T 1 is the number of seconds occupied by the wave in passing from hollow to crest, or crest to hollow. 61 is the number of degrees in slope of the steepest part of the wave; and p/q is the ratio T/T 1, with the numerator and denominator converted into the lowest whole numbers that will express the ratio, where, however, it must be noticed that for T/T 1 =1, p/q must be taken as the limit of such a form as o o 9 o o o o Then " (i.) The ship will complete the phase in the time =2qT.

" (ii.) In completing the phase the ship will pass through the vertical position 2 p times, or 2 q times, according as p or q is the smaller number.

" (iii.) The ship will pass through the vertical position at the middle of the phase.

°	ti ti aF -4:,	?'	- g b?; ~ F	9 ?i? a	?a+ ?	b? 7?d ? ? a ? a
	5"	I		Infinite.	Infinite.	Infinite.
5"	6'25»	0 8		50»	8	45 deg.
5 »	4"	I 2	?		8	36
5		0 5		20"	2	18
5„	2 5"	2			2	9„
5"		0.55		90"		20 „
5"	2 77"	1 8		50"	to	II „

" (iv.) On either side of the middle of the phase there must occur, as equal maximum oscillation, the maximum in the phase, say 0, which will approximately (but never in excess) =- q " (v.) From these propositions it appears that if we compare two cases, in one of which the value of T/T I is the reciprocal of its value in the other, the phase will in each case consist of the same number of oscillations similarly placed; but in that one in which the period of the wave is slower than the period of the ship, the angles of oscillation will be the larger in the ratio p/q or q/p, whichever is the greater. The following table expresses the results of the above propositions, as exhibited in the diagrams, based on the assumption that the period of the ship is in every case T =5", and that the maximum slope of the wave e1=9 degrees: The assumption made in equation (I) that Gz=m .6 is true if the sections of the ship in the vicinity of the water-line are concentric circular arcs; and is approximately true generally for small angles of inclination as long as m is not small. If m be small, the relation does not generally hold.

In a wall-sided ship, GZ=sin 0(m +-1-a tan g 6), where the BM is denoted by a; whence the equation for rolling through small angles becomes d 2 6 ag 3 dt 2+ 2 6+2E2 6 where 6 5 and higher powers of 6 are neglected.

a	16 ft.	16 ft.	16 ft.
m	3 ft.	4 in.	a in.
Tm T			2.98

Sections of other forms lead to a similar equation, but with different coefficients of 6 3; the above equation is therefore typical of all others. This condition has been worked out fully by Professor Scribanti,' who obtained a solution in the following form: T - O?IJ [l - ()2()2 ] where is the maximum angle of roll. J is defined as the moment of inertia of the water-plane expressed in foot-ton units, i.e. is equal to W.a, where W is the displacement in tons. I is the mass moment of inertia of the ship about its axis of oscillation, and 2. Some numerical results for - T„, where T„? is the period 0 2 +4T T found by the usual " metacentric " formula and 0 is 12°, are: 1 Trans. Inst. Naval Arch., 1904.

When the metacentric height is zero, the formula becomes T= 167 r JI =5.25,.j ?. V J V g It has been assumed in the foregoing that the rolling in still water and among waves is unresisted; it remains to take into account the resistances which always operate during rolling. In still water these cause a degradation of the amplitude until the ship finally comes to a position of rest; and when a vessel is rolling among waves they cause a similar degradation of amplitude. The earliest investigations of resisted rolling in still water were made by Froude in England, and by Bertin, Duhil de Benaze, Risbec and Antoine in France. The method adopted was actually to roll the ship in still water and observe how the amplitude decreased roll by roll. Men were caused to run from side to side of the ship, their runs being so timed as to add to the angle of roll on each successive swing until the maximum angle obtainable was reached, when all movement on board was stopped, and the ship allowed to roll freely of herself until she came to rest. During this free movement a complete record of her angular motion was registered by means of a short-period pendulum and an electric timer, and from this a curve of " declining angles " was constructed, in which abscissae represented number of rolls and ordinates extreme angles of roll to one side of the vertical. From this curve another curve was constructed, which was termed a " curve of extinction," in which the abscissae represented angles of roll and the ordinates the angle lost per swing. Figs. 28 and 29 give examples of these curves obtained from experiments with H.M.S. " Revenge." 2 Having obtained such curves, Froude proceeded to investigate the relation between the degrad