Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 19, Scripture-SUG

S H I P-B U I L D I N G,
'Bezout's
Mecba-
tique, art.
lability of of gravity in each fe&ion is to be found. Let AB be
Ships, i;ne 0f floatation of the (hip when in an upright
» ftate, and ab the water line when inclined. Then, be-
caufe the weight of the fhip remains the fame, the
quantity of water difplaced will alfo be the fame in both
cafes, and therefore AEB—a E b, each fuftaining the
fame part of the whole weight of the fhip. From each
of thefe take the part AE b, which is common to
both, and the remainders AO a, BO b will be equal $
and which, becaufe the inclination is fuppofedveryfmall,
may be confidered as reftilineal triangles, and the point
O the middle of AB.
Now, let H, I, K, be the centres of gravity of the
fpaces AO«, AE and BO £, refpe&ively. From
thefe points draw the lines H /$, I /, and K k, perpen¬
dicular to AB, and let IL be drawn perpendicular to
EO. Now to afcertain the diftance y q o{ the centre
of gravity y of the part a E from the line AB, the
momentum of « E £ with refpeft to this line muft be
put equal to the difference of the momentums of the
parts AE Z>, AO a, which are upon different fides of
AB *. Hence aE^Xyy? °r AEBxy?—AE^>
XI i—AO rt X H /i. But fince g is the common centre
of gravity of the two parts AE£, BOZ», we have there¬
fore AEBx,? 0=AE£xI 2+BOZ>xK E Hence
by expunging the term AEZ»Xl 2 from each of thefe
equations, and comparing them, we obtain AEBXy?
=AEB xg O—BO t- X K k—AO a x H /$.
Now, flnce the triangles AO a, BO b, are fuppofed
infinitely fmall, their momentums or produfls, by the
infinitely little lines H h, K k, will alfo be infinitely
fmall wTith refpeft to AEB X.? O •, which therefore be¬
ing reje&ed, the former equation becomes ABXyy
“AEBx^O? and hence yq—gO. "Whence the
centres of gravity y, being at equal diftances below
AB, the infinitely little line y g is therefore perpendicu¬
lar to EO. For the fame reafon ^y, fig. 52. may be
confidered as an arch of a circle whofe centre is M.
To determine the value of gy, the momentum of
a E£ with refpeft to EO muft be taken, for the fame
reafon as before, and put equal to the momentums of
the two parts AO a, AE b; and we {hall then have
a EZ<X£y, or AEB XJT y—AEB x IL-{-AO axO //.
But fince g is the common centre of gravity of the two
fpaces AE b, BO£, we ftiall haveAE^xIL—BO £
X0£:=0, or AE^x IL=rBO£ X OE Hence AEB
X^y=:BO bxO k-^-KO axO h=2^S>ObxOk; be-
caufe the two triangles AO a, BO are equal, and that
the diftances Ok, Oh, are alfo evidently equal.
Let a? be the thicknefs of the feftion reprefented by
ABC. Then the momentum of this feftion will be
2 BO bxxxOk, which equation will alfo ferve for
each particular feflion.
Now let f reprefent the fum of the momentums of
all the feftions. Hence f, X x Y. g V—f 2
BOZ>XtfXO£. Noav the firft member being the
fum of the momentums of each fe&ion, in proportion
to a plane paffmg through the keel, ought therefore to
be equal to the fum of all the feftions, or to the volume
of the immerfed part of the bottom multiplied by the
diftance g y. Hence V reprefenting the volume, we
{hall have VXg Y—fi 2 BO £X^ X O L
In order to determine the value of the fecond member
of this equation, it may be remarked, that -when the
fhip is inclined, the orieiual plane of floatation CBPQ
Vol. XIX. Part I.
297
(fig. 54.) becomes C Now the triangles NI n, Stability of
BO b, being the fame as thofe in figures 52. and 53. j, S^]PS- t
and as each of thefe triangles has one angle equal, they j-g
may, upon account of their infinite fmallnefs, be confi¬
dered as fimilar j and hence BO b : NI n : : OB,*
: IN!*; whence BOxNI//. Moreover, we
1N|*
have (fig. 53.) O i'rrfOB, for the points K and k
may be confidered as equidiftant from the point O :
JO£|
whence BO £ X O ^
IN :
X NI n.
rOBjJ
Hence V X ,§■ v—f 4 X ^ X NI «. From this e-
IN|*
quation the value of ^ y is obtained.
To find the altitude g M (fig. 53.) of the meta-Fig. 55.
center above the centre of gravity of the immerfed part
of the bottom, let the arc NS be deferibed from the
INxNS
centre I Avith the radius IN; thenNI«—-
Now
fince the tAVro ftraight lines y M, g M are perpendicular
to a n and AN refpe£tr-ely, the angles M and NI «
are therefore equal; and the infinitely little portion gy,
which is perpendicular to g M, may be confidered as an
arch deferibed from the centre M. Hence the two fee-
tors NIS, <§-My are fimilar; and therefore ^ M : ^y : :
INx^y. and confeqUentl)r
IN : NS. Hence NS
NI n —
IN|* Xgy
2^ M
former equation, and reduced,
|OB|3xvX^y
g M
g M
Noav this being fubftituted in the
Ave have V Xg V—f
But fince g M and g y are the
Hence g Mz=l^ ^
Let
fame, Avhatever fe&ion may be under confideration, the
equation may therefore be expreffed thus, V xg y=
'■b r 3 -»
y—OB, and the equation becomes g M — -y —
Whence to have the altitude of the metacenter above
the centre of gravity of the immerfed part of the bot¬
tom, the length of the fetftion at the water-line muft be
divided by lines perpendicular to the middle line of this
fe&ion into a great number of equal parts, fo that the
portion of the curve contained between any tAvo adja¬
cent perpendiculars may be confidered as a ftraight line.
I hen the fum of the cubes of the half perpendiculars or
ordinates is to be multiplied by the diftance between
the perpendiculars, and tAvo-thirds of the produft is to
be divided by the volume of the immerfed part of the
bottom of the ftiip.
It is hence evident, that while the feftor at the Ava-
ter line is the fame, and the Volume of the immerfed part
of the bottom remains alfo the fame, the altitude of the
metacenter Avill remain the fame, Avhatever may be the
figure of the bottom.
Chap. IV. 0/~ the Centre of Gravity of the immer-
fed Part of the Bottom of a Ship.
The centre of gravity* of a fhip, fuppofed homo-#See ^
geneous, and in an upright pofition in the Avater, is in a chunks,
P p vertical