emptyin of water-ballast and oil-fuel tanks, and particularly in the
case of ships fitted to carry large quantities of oil in bulk.
Let fig. 23 represent the section of a vessel fitted with a tank
PQRS partly full of water. Let WL, 'wl be the upright water-lines

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Fic.. 23.

of the vessel and tank, G the centre of gravit 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 01, 02, . . . the centre of buoyancy B moves along the curve of buoyancy to Bl, B, ,... the normals at which are tangential to the meta centric curve M, M1, .. . those at small an les passin through the meta centre M. If the water in the tank could be kept from moving as the inclination roceeded, G would be fixed in the ship, and the righting levers would be GZ, , GZ;, those at small angles being equal to GM sin 0. Actually, if the inclination be slowly performed, the water-level in the tank changes successively to wlll, w¢l2, . .maintaining a level surface at all times; its centre of gravity moves to bl, b2, . . . thereby causing a corresponding alteration in the combined centre of gravit G. Drawing bn, bn, .. rpendicular to the verticals through bl, b¢, . . and calling w, Wpfhe weights of the water and of the water and ship combined, then at the angle 0; the line of action of the weight of the water w has moved through a distance bn and the righting moment of the ship is diminished by an amount wXbr¢. 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, will, wtlg, &c. The values of the levers bfi, bn. .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| be a small angle of inclination, the line of action of the weight blm will be such as to pass through the meta centre rn corresponding to the water-line wl, and determined by the formula bm =-:7where 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 therefore be replaced by an equal weight fixe 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 “L. l' .l L.

GG 'W '""“v v 'V

lf the tank contain a fluid of specific gravity p 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 rol ing; 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 amount calculated. To avoid danger of capsizing in still water, large tanks ina ship are Elled or emptied;in succession as far as possible, so that not more than one or two are partly full at the same time. Water-tight longitudinal partitions arealso 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 § Rollin .) Hitherto the stability of a ship has been considered only with reference to inclinations, about either a longitudinal or transverse axis. These are the only cases which it is necessary to stab” deal with in practice for the purpose of ascertaining the m W probable qualities as re rds stability of a vessel by Mm" comparing the elements (fits 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 floatin body, however, it is necessary to take account of the true line ofg the action of the buoyancy and not merel y of its pro'ection on the plane of inclination. The development of this part of the sub'ect has argely been due to M. Dupin in his Mémoire de la stability dies corps flvttants and to M. Guyou in his Théorie du mzvire. If a ship is inclined in all ssible positions, keeping the displacement constant, the locus ofxihe 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 sitions is known when (a) the forms and dimensions of the surface ofpguoyancy, 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 onl to the arrangement and magnitude of the component weights oflthe 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 p ane to the surface of uoyancy is therefore arallel to-the water plane, i.e. it is horizontal, and the normal to the surface is vertical. f 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 necessa, as in the particular inclinations already considered, to find therline of action of the buoyancy for adjacent positions, Le. to trace the normals to the surface of buoyancy. Consecutive normals to this surface will not, in general, intersect; but, from the pro rties of curvature of surfaces, there are two particular directions oixinclination for which adjacent” normals to the surface will

intersect the original normal,

these directions being < gl

perpendicular to on e ., »

another and parallel to the

principal axes of the indicatrix

of the surface of buoy-, ,

ane . s

Ififig. 24 be a plan of the

water-plane, Ox* the axis of

inclination passing through

O the centre of flotation, Og' and Oz perpendicular axes in and at right angles to the plane of otation, then, from a consideration of the wedges of immersion and emersion for a small inclination 0, the travel o the centre of buoyancy B becomes:- I',

FIG. 24.

%ffy'2.dx'.dy' (or BB, in fig. 24) parallel to Oy -3-7ffx'y'.dx' .dy' (or-B, B2) parallel to Ox and

fy”.dx'.dy' (or B2B') parallel to Oz.

These may be written:-

gi-I, '; gf-P; and;%I, ' respectively » V where Iz' 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:- x= -BIB; cos ¢-BB; sin ¢=%-(P cos ¢>-l, ' sin ¢); y= BB, cos da-BIB; sin ¢=%(I, ,' cos 4:-l-P sin ¢); 0” 1

Z= B2B'=w'Iz .,

Also,

l, ,' = I, cos' ¢>+I, , sin' ¢;

P =(I, - I,) sin ¢ cos ¢;