ivotal point or fulcrum for the moments of all weights placed orward or aft of this position. It will be obvious, therefore, that s location is of great value in determining the trim of the vessel, nd the various alterations thereof due to rearrangements of weights on board. Its position is calculated by taking the areas of he sections and putting them through the multipliers; these unctions of areas are in turn multiplied by the number of interals, (each one is forward or aft of the mid-ordinate,) and the ifference between these forward and after moments divided by he sum of the area functions. The quotient resulting is the umber (or fraction) of intervals that the centre of buoyancy is orward or aft of the length according as the moment preponlerates forward or aft respectively.

This centre should be calculated for various draughts, as of ourse it changes with different draughts and alterations of trim, wing to the changing relationship between the fineness of fore nd after bodies at different immersions and trims.

Transverse Metacentre (M.C.)

The position of this element is, in conjunction with the centre of gravity, the most vital in the design of the ship. As its name implies, it is the centre or point beyond which the centre of gravity of the ship may not be raised without producing unstable equilibrium in the upright position, or, otherwise stated, if the ship be inclined transversely to a small angle of heel, the centre of buoyancy which originally was on the centre line will move outboard to a new position; but, as it acts vertically upward, it must somewhere intersect the centre line. This point of intersection is known as the metacentre. One of the factors in the determination of its location above the centre of buoyancy has already been calculated, viz: the volume of displacement V; the other, the moment of inertia of the water plane about the centre line of ship, we shall proceed to compute. The height M above the C.B. or B.M. is found by :

Moment of Inertia of Water Plane

Volume of Displacement

I

9 or,

B.M.

The moment of inertia of the water plane is a geometrical measure of the resistance of that plane to " upsetting," or when taken about the centre line, as in the case of calculating for transverse metacentre, to "careening. So that the greater the waterline breadth the higher will be its value; for we must imagine the water plane as being divided into a great number of small areas, and each of these multiplied by the square of its distance from the

centre line of ship, when the sum of these products will equal the moment of inertia of half the water plane, about the middle line of vessel as an axis. As both sides of the water plane are sym metrical, the total I will be this result multiplied by 2. Applying this principle to W.L. 4 in the example with which we are con cerned, we get the following tabular arrangement:

ORDINATES.

Moment of Inertia of Water Plane (I).

CUBES OF
HALF-
BREADTHS.

SIMPSON'S
MUL-
TIPLIERS.

PRODUCTS.

HALF-
BREADTHS
OF W.L. 4.

The calculation for Moment of Inertia and Transverse Metacentre above C. B. may be more easily remembered if we treat the cubes of water line half-breadths as the ordinates of a curve twothirds the area of which will equal I, and this, in turn, divided by V will give B.M.

However, when we know a, or the coefficient of water line, we may arrive very accurately at the moment of inertia of the water

lane, and consequently at the B.M. without the labor of the oregoing calculation by multiplying the Length by the Breadth3 by coefficient, which coefficient will be determined by a and selected from the table given on page 48. By referring to this able, we find for a (value .694) that the coefficient "i" (inertia coefficient) is equal to .0414, whence we get I= L × B3 × i = 100 < 123 x .0414=7154 moment of inertia, which is sufficiently close for all purposes, and: :

By transposing and taking the calculated I, we find

Longitudinal Metacentre (L.M.C.)

From the definition given for the transverse metacentre it will be seen that if the ship be inclined longitudinally, instead of, as in the former case, transversely, through a small angle that the point in which the vertical through the altered C.B. intersects the original one will also give a metacentre known as the longitudinal, or L.M.C. Its principal use and value are in the determination of the moment to alter trim and the pitching qualities of the vessel, or longitudinal stability. It will be obvious that the moment of inertia of the water plane must be taken through an axis at right angles to the previous case, viz., at right angles to the centre line through the centre of gravity of water plane, which will be where the original and new water planes cross one another in a longitudinal view.

L.M.C. above C.B. =

I1 of Water Plane about its C.G.
Volume of Displacement

Therefore, to calculate the M I1, we must figure the moment of inertia with, say, ordinate 5 (or any other one) as an axis when the moment about a parallel axis through the centre of gravity plus the product of the area of water plane multiplied by the square of the distance between the two axes will equal the moment about ordinate 5.

The moment of inertia about the midship ordinate we shall call I, and the distance of the centre of gravity from this station = x. The moment of inertia about the centre of gravity of plane = 11. We then have I= I1+ Ax2, or I1=I-Ax2. A clearer conception of this will be obtained from the tabulated arrangement.

Area of water plane = 62.41 × (3 × 10) × 2.

=832.14 square feet.

Distance of centre of flotation abaft ordinate 5

=

(67.5752.41) 10
62.41

=2.42 feet.

Moment of inertia of water plane about ordinate 5

= 324.13 × (3 × 10) × 102 × 2 = 432,172 = I.

Moment of inertia of water plane about axis through its centre of flotation.

=

= 432,172 — (832.14 × 2.422) = 427,304 = 11.

Longitudinal metacentre above C. B.

I 427,304

2583.7

165 feet Longitudinal B.M.

=

An excellent approximate formula for the longitudinal B.M. is given by J. A. Normand in the 1882 transactions of the I.N.A. Taking the symbols we have been using :

Applying this formula to the vessel with which we are dealing, we find :

L.B.M.=.0735

832.142 × 100
12 x 2583.7

= 164.12 feet.

which is a very close approximation to the calculated result of 165 feet.

We may also use the approximate formula which we applied in the case of the transverse B.M. altered to suit the new axis with a modified coefficient, as :

L.B.M. = L3× B× i1.

Moment to Change Trim (M1).

As the centre of gravity of the displacement (or centre of buoyoncy), either in the vertical or the longitudinal direction may be an entirely different locus from the ship's centre of gravity, it is obvious that unless the moment of the weights of the ship and engines, with all equipment weights, balances about the centre of buoyancy we shall have a preponderating moment deflecting the head or stern, as the moment is forward or aft of the C.B., respectively, until the vessel shall have reached a trim in which the pivotal point or C. B. is in the same vertical line as the completed ship's centre of gravity. To determine the moment necessary to produce a change of trim (M1) in a given ship, it is necessary to know the vertical position of the centre of gravity of the vessel and the height of the longitudinal metacentre (L.M.C.). The former may be calculated in detail or preferably proportioned from a similar type ship whose centre of gravity has been found by experiment; although great accuracy in the location of this centre in calculating the moment is not as important as in the case of G.M. for initial stability, as small variations in its position can only affect the final result infinitesimally. To investigate the moment affecting the trim, let us move a weight P already on board of the 100foot steamer whose calculations are being figured.

D= Weight of ship including weight P = 73.82 tons.
P 5 Tons.

BM 165 feet.

=

GM = 160 feet.

1 = 50 feet (distance moved).

L= 100 feet (length of vessel).