increased by an arbitrary constant. If from any point Bi' 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

5 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

s., /' . surface of buoyancy in~ its general " 3 u " ishape, and touches it when GB is normal, i.e. at positions ' of equilibrium Bl, BQ, Bi, B4, ; it has the

property that a radius GN drawn

from G is always vertical

corresponding to N, and

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 Emv2/2g, and the total energy by (WXGN)+§ }g. Emi/2; the latter is constant when there are no resistances, and steadil decreases if resistances are in operation. Neglecting resistance, when the body is momentarily at f€St, W XGN becomes W.l, where l is a linear quantity; and through»"~- (Pawn

ff Svauu w

~ Duma

ls

V I

¢

1

F1o.26.» ' I

when the body is in the 'position

has a length proportional to the

out the motion GN is less than I by $2mv'. The effect of.resistance is gradually to decrease 1 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 origin all entirely within the podaire and then to be capable of expanding untiiithe 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 inrf?- Y' . -fj

=-~"~» a.4/-ffl-r' creased or diminished. Let, the xj Ai ii? A radius GN to the podaire along the Y '*Q'~ edge of the water be represented by l; from the 'energy equation the

radius for any other position 'of the

body moving without-external forces

is less than l, and the position lies within the lake so bounded. The diminution of I 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 pgzition. Finally, if moving near a position of relative stability, the y will move freely from side to side until the drying has proceeded so far that separate branches XX' or YY' are obtained; whenthis occurs, the body will be fenced, as it were, on one side or the other, anchwég oscillate until a position of absolute stability is finally attain - » »-With

regard to the surface of flotation it has been shown that in order that the displacement shall remain constant, consecutive waterlmes 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, of 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 and may be synclastic in certain parts and anti-elastic or saddle-shaped in others. I ' ' Q ' "

The relation between the surface of flotation and the stability of the ship is similar to that established in thejtwo dimensional cases, i.e. the projection on the plane of inclination of the curve corresponding to the inclination has a centre of curvature who'se height is a measure of the increase or decrease of stability caused b 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. ' ' ' = A Rolling of ships. ' 'A

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 FIG. 27.

direction; also horizontal translationsiand 'rotations vertical axis which are not generally of an oscillatory character and will not materially affect the rolling. It is about a

convenient when considering rolling to neglect the fglzijsted influence of the other accompanying oscillations, whose effect in most cases is. slight in magnitude although complex in character. ,

The ship is in'rhe»firsf place conceived to, bevrolling in still water without any resistances operating todiniinish the motion. The equation of motion for moderate anglesqf inclination, within which the arm of the righting couple is approximately proportional to the angle of heel (Le. GZ=mX0), is" G ~' ' " ' Ng-3?-€%m.9, - ..... (I)

where e is the radius of gyration of the ship about the axis of rotation, m the meta centric height, "6 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 'Ir m g, , . . (2)

showing that the time of oscillation varies directly as the radius of gyration, and inversely as the square root of the meta centric height. The value of T is generally about 10 seconds in a large Atlantic liner, 7 to 8 seconds in -a battleship, ~and 5 to 6 seconds in second-Class cruiser sand shi s' of similar type. In at large modern warship e is about one-third the breadth of the ship. V ~ ' f F or unresisted rolling of ships among waves the theory generally accepted is that;due to, Froude, (see Trans. Inst. Nav. Arch., 186I 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 theoryiis 'based on the proposition that, when a ship is among waves, the impressed forces on er tend to place her normal 'to a wave sub-surface, which is assumed to be the surface passing throu h the ship's centre of buoyancy, and which is' regarded as the egective wave surface pas 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 b instant, to, the effective surface of the wave that is passingcher. %hen she at any instant deviates from this position, ths, ¢ff0rfby which she endeavours to return to, the normal depends son' the angle of deviation, in the same manner asthe effort to assume ian upright position, when forciblynnclined in still water, depends on the angle of inclination(Hence her stability (i e. her effort tQ become vertical) in still walter measures 'her effort to become normal to the wave at any instant on a wave." F roude 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 f given - by' T=1r ii as obtained theoretically. On these assumptions m.g

the equation of motion is obtained by substituting, for the angle of inclination in still water, the instantaneous angle between the ship and the dpormal to the wav;-slope, and thus becomes 0- - ~ '-E-P-=-'l%;1-(0-fH1)=-T2(0-e61), » . .- . (3)

where 0=angle=of shipls masts to the vertical, and Biéangle of normal to wave-slope to the vertical at the instant considered. 01 - ». ' ~ » lv

has to be expressed in terms of time, and is given by 01=®, sm %, where 9, is the maximum wave-'slope, T| 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 theregpre be v;fr1tten-¢-0 rr ' ' 1r ' ' '

~ ' W= -T2 Q-@1¢SlI1'E¢ i if ~ ' (4) which is the general differential equationiof the unresisteizl motion of a ship in regular Waves of constant period; Tile solution of' this equation isf i 6,

0:Ci.SiH%'fd' C2CO§ %ldT SiHi"I1%f: . where Ci and C2 are constants depending on- the initial movement and attitude of the ship.-The

last term of this expression” ' ~; . A - 5- 61 '~ lr. t

T-f.S1nT, -

Iii