points which determine it, viz. the transverse metacentre and the centre of gravity.

Position of the Transverse Metacentre of a Ship when floating at any given Waterline.-The point M depends solely upon the geometrical shape of the underwater body, and its position can be determined for the ship when floating at any particular waterline.1 It is determined with reference to the C.B., and because of this the position of the C.B. has a distinct influence on the stability. The distance BM is given by (Moment of Inertia of waterplane about the centre line) (Volume of

The moment of a figure about any axis is obtained by dividing the area into a large number of small areas, and multiplying each by its distance from the axis. The addition of all such products is called the moment.

The moment of inertia of a figure about any axis goes one step further. The area is divided into a large number of small areas, and each of these is multiplied by the square of its distance from the axis. The addition of all such products is a moment of the second degree, and is called the moment of inertia.2

I

The expression may be approximately found as follows:—

See the author's "Text-book of Theoretical Naval Architecture."

2 This definition and the value of the moment of inertia given for a rectangle may be illustrated by the following approximation to the I of a rectangle 100 ft. × 20 ft. about the centre line. Divide the breadth into ten strips, each having an area of 100 × 2 = 200 square ft.; the centre of the strips from the centre line are 9, 7, 5, 3, and 1 ft. respectively. The I of the half of rectangle about the centre

line is therefore nearly

[200 × (9)3] + [200 × (7)3] + [200 × (5)3] + [200 × (3)2] + [200 × (1)2]

=200 × 165 = 33,000

and for both sides I = 66,000. The exact value is (100)(20)3 = 66,666. If the strips had been taken 1 ft. wide a closer approximation still would have been obtained.

(2) V can be written Vk.L.B.D

where D = mean draught (ex keel projection if any)

k is a coefficient of fineness

where a is a new coefficient obtained from n and k. It is found that a varies between narrow limits, being about in battle-ship forms and in cruiser forms. The important point that this approximate formula emphasizes is, the great influence of beam on the distance BM, and therefore on the position of M. This is the reason why battle-ships have to be so broad. It is necessary to have M above G for stable equilibrium, and in battleships the point G is high owing to the great weights of guns and

BATTLESHIP

CRUISER

FIG. 163.-Shapes of load water plane.

armour carried high up, so that M must be high also, and this is obtained by giving such ships a large beam.

Thus Drake, a cruiser of 14,100 tons, carrying two 9.2-in. guns in shallow barbettes and sixteen 6-in. guns in casemates, is 71 ft. beam.

But Duncan, a battle-ship of 14,000 tons, carrying four 12-in. guns in massive barbettes, with twelve 6-in. guns in casemates, has to be 75 ft. beam to obtain sufficient stability. (See Fig. 163 for comparison of shapes of waterplane of a battle-ship and a cruiser.)

A similar cause was in operation resulting in the King Edward VII. being made 78 ft. beam. In this ship the adoption of an armoured battery instead of casemates, and the provision of four 9-2-in. guns on the upper deck instead of 6-in. guns, caused the C.G. to be higher than in previous ships, and M had to be made higher by increasing the beam.

EXAMPLES OF STABLE AND UNSTABLE EQUILIBRIUM.

An example of the above principles is seen in the case of a log floating with one half its bulk immersed-say 20 ft. long and 18 in. square. It is a matter of experience that such a log will never float with one face horizontal, as Fig. 164, but always with a corner of the section downwards, as Fig. 165. The following application of the foregoing principles will explain the

reason.

If the log is placed as in Fig. 164, we have the C.B. 4 in. from W.L., and the C.G. at half depth. To find the position of the metacentre we use the formula

I

BM = √, and I = 11⁄2 × 240 × 183 and V = 240 × 18 × 9

12

all dimensions being in inches, so that

BM = (1 × 240 x 183)÷(240 x 18 x 9) = 3 in.

so that M is 11 in. below G and the log is in unstable equilibrium, and cannot float with one face horizontal.

If, however, the log be placed as in Fig. 165, with one corner downwards, we have the C.B. 4:24 in. from the C.G. and

so that M is 4.24 in. above G, and the log is in stable equilibrium.

We notice that there are two influences at work in obtaining the stability in the second case, viz. (1) C.B. rises, owing to the new shape of the displacement, carrying M with it, and (2) BM is increased owing to the greater breadth at the waterplane.

Another interesting example of stable and unstable equilibrium in a floating body is seen in the case of a duck or a swan. Under ordinary circumstances

the shape of the waterplane is full, and the bird floats in stable equilibrium. If a swan, however, reaches down to the bottom of the water with her tail in the air, the waterplane is considerably reduced in area, being approximately circular, and this pulls down the M sufficiently to cause unstable equilibrium. In order to remain in this position it will be noticed that a swan has to work against the water with her feet to counteract the instability.

Metacentric Diagram.-The transverse metacentre, as we have seen, is a fixed point for a vessel floating at a definite waterline.

It will, however, be a different point for any other waterline, because B will shift, and the distance BM will be different. It is desirable to have some ready means of determining the position of M for any waterline at which a ship happens to be floating. This is done by constructing a metacentric diagram. Four or more parallel waterlines are taken, and for the ship as floating at each

of these lines the position of the C.B. is calculated. The waterlines are set out at the proper distance apart, as Fig. 166, and a line drawn across at 45°. Through the intersections verticals are drawn and on these verticals are set down the positions of the respective centres of buoyancy, B1, B2, B, B. Thus for the ship shown in Fig. 166 the following were results obtained :

เ