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' tan a. ds
/
P 'area of water-planefor 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
giving Leclert's first expression; also, since p = %-/,
dl d
W=p-l-V5";=P'.
which is Leclert's second expression for p'.
The value of p' at the upright can be obtained from the
meta centric diagram by the following simple construction. Let
M and B be the meta centre and the centre of buoyancy for a
water-line WL on the meta centric diagram (fig. I8); draw the
tangent to the B curve meeting WL at Q, and through Qdraw 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
f, PQ=4hcot0=h%l==§ ;
V also V
M f e. MR=BM-(BP+PR)=p-K (tan o+¢an ¢).
If D be the draught,
"I G H' tan 0-l-tan ¢>= -5?-5 = -A-g%,
6
Q whence
Q ' MR=p+Vg%=p
1' the curve of flota-F4
MI 5 tion being concave
upwards if R is
below M. 4|
B1 F. Fa g ” B IF or moclerate illi)
8 .
o . wH== s I .L sf' .L as L - I .°~"'=— §§ .'§§ i'€f'sth2°"§ u2y'3 ' ' ""
ancy of the added
F B layer due to a small
" M5 additional submersion
will act through s
the centre of curva~,
ture of the curve a
y of flotation; this Fm' 18
/ ~> point may be regarded as that at which any
V65 M additional weight will, onf being placed on a
M* l 5 ship, cause no difference to the values of the
55 gg righting momentat' moderate angles of inclination.
The curve of flotation, therefore, and its
evolute bear similar relations to the increase or
ax' decrease of the stability of a shipdue to altera-Fm
x6 tion of draught, as the curves of buoyancy and
through the centre of flotation. M. Emile Leclert, in a paper read
at the Institution of Naval Architects, 1870, proved the e uivalence
of the above formula to the two following, which are (lcnown as
Leclert's Theorem:
n'=P+V%andP'=%.
where I and V are respectively the moment of inertia of the water plane
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 (T fans. 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, (6g. 17) be the centres of buoyancy of a ship in two
consecutive inclined positions, and F, F, the corresponding centres
of flotation. Draw normals BM, BIM,
f meeting at the pro-meta centre M, and
FC, FiC, meeting at the centre of curvature
C. Produce FB, F1B, to meet at O;
" join OM, MC.
Then BM, CF and B1M, CF, are respectively
parallel, and ultimately also
BB, , FF, ; hence the triangles MBB1,
CF F 1 are similar and
§ LI § & 0B
CF 'FF1 "(W"
so that O, M and C are collinear.
° If the displacement V be now increased
FIG, 17, by a'V, changing B to B', and M to M',
then since the added displacement:IV 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'><V=BF ><dV
Draw MED parallel to BF, then
dV BB' M§ M'E dp
v W MD '(T”p'-p
i1
V >
xxlv. 30
of pro-meta centres do to the actual amount of
the stability, 3
>'I' he 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 p'¢g€7=O. These occur in an r
ordinary ship-shape body at positions: (1) 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, Fa, F4, FS, F1, Fg.
The following relations between the curves of buoyancy
and of pro-meta centres 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
correspond in to inclinations 0, 0 + dB, where dl?
vanishes in the limit; B', B", the centres of buoyancy,
meta centre; produce GZ' to meet, B”M' in U.
Then, neglecting squares of small quantities,
d(GZ')=Z'U=M'Z'.d6',
or vertical distance of M' above G=d 392).
If—..
U
8
U
FIG. 19.
M' the pro-Also
M'B'=M'B”;
hence
Z”B”-Z'B'=MZ'-MZ”=Z”U=GZ'.dB,
or
¢i(B'Z')
GZ *ir* V
i.e. the vertical distance (B'Z') of G over B is equal to fGZ.d0.
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'.

II