(t' being a small fraction). And if G be the centre of gravity of the surface, and y its distance from the line WL, and if L be the length of pendulum corresponding to that time, then

where fda is the area of the semi-water line; and

g

π

= 3.26. But this gives us only the stability of

=

a vessel the size of the model: unless the quantities x and fdx, are measured by the scale for the whole vessel. And, in this case, t' must also be calculated for the full-sized water-line. The time, t, of an oscillation of a load-water-line the full size of the vessel, is thus deduced from t. Let the dimensions of the vessel be m times that of the model, then

The stability of a vessel depends upon the difference between this quantity, and the product of the volume of displacement into the distance between the centre of gravity of the ship and that of the displacement. But for general comparison, the calculation of this quantity will be sufficient; if it be borne in mind that the same value for this quantity represents a greater amount of stability in a vessel which is lean below, than in one which is narrower and full below.

Again we observe that if the stability and area of canvass for a vessel of a certain size be calculated; then the stability of a vessel, built on the same lines, but to a scale which makes her n times as great, will be increased in a much greater ratio. For let the stability of the former be fldx-Va

t2g

(where fldx=g. afdx)

and the area of the sails A

= square feet.

and T2=nt2; X=nx ; fdλ=n2fdx; Y=n3V; a=na

Therefore the expression for the stability is increased n1 times while the area of canvass which would be given to her, if the sail-draught were measured by the same scale, would be increased only n2 times.

But the height of the centre of effort of the sails is increased n times, and therefore the moment of the sails is only increased n3 times while the stability has increased n1 times. And hence the sail-draught should be measured by such a scale that the area of the sails be increased n3 times. Or

1

the scale instead of being a th scale should be a

n

For the same reason small boats, having so much less stability in comparison to large ones, and yet being furnished with a disproportionate area of canvass, meet with so many accidents.

And hence also the advantage of a large vessel; for the resistance increases as the square, the propelling

force as the fourth power, and the momentum as the third power.

Again, if the centre of gravity of the area of the load-water-line be not in the same vertical section as the centre of gravity of the displacement, but before it; suppose, then fl2dx limited to the portion of the load-water-line behind the centre of gravity of the displacement is not so great as the same expression limited to the rest of the area. Or the vessel is not so much supported against heeling in the after as in the fore part. Hence her stern will both be depressed, and fall off from the wind.

CHAPTER V.

ON UNEASY MOTION AND ROLLING.

WHEN a ship is inclined, she revolves round her centre of gravity. Then a part of the ship is immersed, which was formerly out of water, and a part is emersed, which was formerly below the surface. The solid immersed may consequently be of a different form from that emersed. Now there are two things to be remarked as possibly taking place co-instantaneously with this motion:

1. If the centre of gravity of the part immersed, and that of the part emersed, be not in the same vertical transverse section of the ship, then the centre of gravity of the new displacement is either before or abaft of the centre of gravity of the ship; and either the bows or stern are raised. The axis of rotation of the ship is, in fact, inclined to the