investigated by Bouguer in his Traité du Navire. The points
M'M', on the curve are now termed prwmetacentres.
If p represent the length of the normal B'M' or the radius of
curvature of the curve of buoyancy at an angle 0, then p.d6 =ds the
length of an element of arc of the B curve. In the limit when dt? is
indefinitely small, £=p. Using Cartesian coordinates with B as
origin and By, Bz, as horizontal and vertical axes,
we have
%=%cos0=pcos0, . . (1)%=%sin
0=p sin 6; . . (2)
whence 6
y= p.cos 6419; z= 0 p.sin 0.d0, and the righting lever GZ =y cos 0+(zBG) sin 0. The radius p is (as for the upright position) equal to the moment of inertia of the corresponding waterplane 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 shipshaped type. As an example in the case of a box, or a ship with upright sides in the neighbourhood of the waterline, if BG =a and BM =p0, then p=p0 sec” 6; whence
a
y= p cos 0.d6=p0' tan 0,
o
z= p sin 6.d0 = épo tan” 0,
and
GZ=(Poa) sin 0+%p0 tanz 6.sin 0; 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, i.e. if p0=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 waterlines which cut off a constant volume of displacement. To avoid this difficulty the process was modified by Reech and Risbec in the following manner zMultiply equations (I) and (2) by V.d0, V being the volume of displacement; we then haved(Vy)
=I cos 0.d0, .... (3)
d(Vz) =I sin 0.d0, .... (4)
where I is the moment of inertia of the inclined waterline about a longitudinal axis passing through its centre of gravity. These of prometa centre, values of righting lever and righting moment. depend on two variable quantitiesthe displacement an the angle of heel. The righting lever' GZ is in England selected as the most useful criterion of the stability, and, after c;"'L"l;l° being evaluated for the various conditions, is plotted 'sa 'Yi in a form of curves~(a) for various constant displacements on an abscissa of angle of inclination, (b) for a number of constant Q
15°
309
nn I
I
I
3 p:
5'Oo,
5 1 *§  »
ua  ' .:
U. 1 ».
E L 1 ' i '*"
I
5, L » ' I
M .', °° J L, 'SPN, . 16900 2/600 rous osP.Ac i * ee
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1 ' J*
l~
B. A
FIG. 9.Cross Curves of Stability of a Battleship. angles on an abscissa of displacement. These are known as curves of stability and crass 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 I 884). The procedure is as follows: I. The substitution of calculations at constant angle for those at constant volume. A number of waterlines at inclinations having a constant angular interval (generally I5°) are drawn passing through the intersection S' of the load waterline with the middle /line on the body plan. Other waterlines are set off parallel to these at fixed distances above or below the original waterline passing through S'. 2. The volumes of displacement and the moments about an axis throug h S' perpendicular to the waterline are determined for each draug t and inclination by means of the AmslerLaffon integrator, 1

I
I I
f A
4., ,
ff/' <“»
99/ ¢', ' 4
I ' I 7
Q . 1
§ ?'§ 9>'i*¢
Q/ Q" gr),
441.900
 @'o°"'./
/ f 4,
formulae have been obtained on the supposition that the volume V V <o§ ;r$E$>f' is constant while 0 is varying; but by regarding the above equa /'§§§ f;', Q '~" moon tions as representing the moments of transference horizontally and '“' ' +° 'W ve ica y ue o e we ges, 1 is evi en a may e a owe, . ., n'1ld tth d 't' 'd tthtV b ll d k to vary in any manner provided that the moment of inertia I is Z', ff/ ~ '~, taken about the longitudinal axis passing through the intersection 1, ' »'» ' . » of consecutive water 1' I rt' lar th w ter lines may all be, '° ' " ' " ' " "§ ' 29900 * mes. n pa 1cu e a  u hy, ,<' I fx i 1 drawn through the point of intersection of the upright waterline '§ '. ' nj/ y xgith the middle line, and the moments of inertia are then equal t o;' 5 I/ V fr3dx for both sides of the ship, r being the halfbreadth along the N,  o 00Q inclined waterline; the increase in volume is the difference between £9 'D 1 i ", "' the quantity fd0f§ r'dx for the two sides of the ship. 5 ff' ig* f If Va, V0 be the volumes of displacement at angles a and 0 re, ';¢ 'Q ' spectively,  I 1 ', ', i,  ' = , 1 l,2dx o' i5° . ao' ' 46* '  ~ 15° V..V0=£ d0[fdi§ fe, enCe:l,  (5) Anas.: or lnctimriou '» ' (T and substituting in () and () and inte ratin g 3 4 I advg gi DEEP Couomou snswu —~— ~~»~ B ' »,
v..y= L .zelfgum ]¢0.@, (6) £1:> § f; gL 3, 1. a I
Vaz = £210 U °; 'f "] sin 0. . . (7) On eliminating .Vain (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 al(za) sin a.
From the foregoing it is evident that the elements of transverse stability, including the coordinates of the centre of buoyancy, position 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 throug h S' is obtained, i.e. the value of GZ, assuming G and S' to comci e.
4. For each angle in turn “ cross curves ” of GZ are drawn on a
base of displacement.