The centre of buoyancy is found by calculation, and as this centre is the centre of gravity of the displaced liquid, we may find the mean centre of gravity of the whole liquid displaced by the vessel by dividing the length, or load water-line, into sections, and finding the mean of their various centres of gravity.

To do this in a simple manner, it has been suggested to take various immersed sections at equal distance apart, such as EF K, Fig. 259, and, after drawing these on paper, to cut them out and paste them together; thereafter, by suspending these combined sections from two points, the position of intersection of two vertical plumb lines, also suspended from the same points, will give the position of the mean centre of buoyancy on the combined section.

The distance G M is what is known as the Metacentric Height, and is the distance between the metacentre and the centre of 2

3 D

gravity, and may be found by integration thus: dx, where y = half-breadth, dx the interval, and D the displacement in cubic feet, and which may be stated in the form of an approximate rule, thus:

Divide the length of load water-line into equal intervals, at which measure the half-breadths at the load water-line; cube each of these half-breadths, and regard the cubes as the ordinates of a plane figure having the length of the load water-line as its base. Find the area of that figure by Simpson's rules. Divide twothirds of that area by the volume of displacement. The quotient is equal to the height of metacentre above centre of buoyancy, from which deduct height of centre of gravity above centre of buoyancy, and the difference is the metacentric height.

Approximate methods of obtaining these results for various ships from the exact determination of them for one ship have been proposed, the vessels considered being of the same type.

ARTICLE 269, p. 288.

The ultimate strength of leather belts is about 3,200 lbs. per square inch; the working strength being about one-eighth of this. Ultimate strength of steel wire, 90 to 140 tons per sq. in. Phosphor bronze wire,

Copper wire,

1700

45

18

APPENDIX.

ARTICLE 269, p. 288.

647

The ultimate strength of sheet copper appears to vary from 14 to 17 tons per square inch, the elastic limit being reached at about one-half of this. When annealed the ultimate strength is reduced to about 15 tons, with an elastic limit of about 5 tons. At temperatures of about 400° F. there is a loss of strength of about 20 per cent. The strength of brazed joints appears to be about 75 per cent. of that of the metal. Solid drawn copper reaches about 18 tons, the elastic limit being reached about 16 tons.*

Exhaustive tests were made by Messrs. Platt and Hayward to ascertain the relation which exists between compression, tension, and shearing stresses. Minutes of Proc. Inst. C.E., vol. xc., 1887, p. 408.

Aluminium bronze, an alloy of copper and aluminium, has an ultimate tensile strength of from 30 to 40 tons per square inch.

ARTICLE 646, p. 590.

The loss of head due to the resistance to the flow of water in a pipe arises mainly from friction, and is found to vary directly as the length of the pipe, inversely as the diameter, and nearly directly as the square of the velocity, all multiplied by a coefficient, whose value may be taken approximately as 02.

For long lengths of pipes the velocity may be expressed by

and for short lengths, such as where a pipe passes through a wall or embankment, and the discharge taking place within a short distance of the supply, the velocity may be expressed by

In the above the resistance at entrance of pipe is supposed to be insensible, which may be considered correct when the form of the mouthpiece is made that of the "contracted vein."

* Trans. Inst. Engineers and Shipbuilders in Scotland, 1888.

ARTICLE 653, p. 598.

The resistance to vessels passing through the water has been carefully investigated from time to time, and it seems now established that in a vessel of fair lines the principal source of resistance lies in the skin friction, very much the same as in the case of the passage of water through a pipe or channel where there is a certain loss of head due to the friction arising from the resistance offered to the flow of the water by the bounding surface. In the ship a further loss of energy arises from eddies and waves, but these in a vessel of fair lines are small in comparison with the loss from friction of the sides and bottom. These deductions are derived from experiments with models and vessels moving at speeds suitable for these forms, as it has been found that the amount of resistance varies somewhat at different speeds.

The late Dr. Froude devoted much attention to this subject, and by means of models moving in his large tank at Torquay he deduced much valuable information regarding the comparative law of resistance in the model and ship.

Taking a ship and her model, the law enables us to find the comparative speeds at which these should be driven, so that the resistance may be compared and the power estimated.

This law of comparison for vessels of the same forms or lines may be stated generally as follows:-The speeds are to each other as the square roots of the linear dimensions, and the corresponding resistances are as the cubes of these dimensions.

If V and v be the speeds of the ship and model,

L and the respective lengths,

and R and r the corresponding resistances,

so that, if the model advance with a velocity of 1, the corresponding speed of the ship will be 5. Again, the corresponding resistances at these speeds will be

APPENDIX.

=

649

Since velocity x resistance energy expended, or power developed, such as a horse-power (see Article 661, p. 610), we may find the comparative powers required to drive the ship and model, thus,

But, since V x R may be called P, or the power required by the ship, and since v × r may be also called p, or the power require to move the model at its corresponding speed, then we may write for (3)

Por p = power, being here used as energy expended, then

P

Ρ

may be considered as the ratio of the effective horse-power required in each case to overcome the fluid resistance of the ship or model.

It must be noted that in a steam vessel there is, over and above this, the friction of the machinery to be considered in calculating the gross power required. In reference to these other resistances Dr. Froude finds that the thrust, or effective horse-power, is only 37 per cent. of the indicated horse-power.

ARTICLES 607, 653, pp. 549 and 599.

The muzzle velocities of the projectiles from heavy guns average about 2,000 feet per second; the weight of the charge being about half the weight of the projectile, and the energy per ton weight of gun being about 500 foot-tons. The penetrating power of the shot is approximately:

where E is the energy of the shot in foot-tons; f the ultimate shearing strength of the material of plate in tons per square inch; and c the circumference of projectile in inches; t being the thickness of plate penetrated in inches. When cordite is used the weight of the charge can be greatly reduced, and higher muzzle velocities obtained.

ARTICLE 287, p. 306.

From experiments made by Lloyd's, the following are the relations of strength and thickness in corrugated steel flues:

where 8 is the working strength in lbs. per square inch; T the thickness of the plate in sixteenths of an inch; and D the greatest diameter of the furnace in inches.

ARTICLE 651, p. 597.

Centrifugal pumps are much used for the discharge of water. They also form convenient circulating pumps for the condensers of marine engines.

The arrangement of such pumps consists mainly of a series of curved blades, radial at their outer extremities, revolving in a case. When a rapid motion is imparted to the vanes the water in the casing is acted upon, and by the centrifugal action set up, tends to form a vacuum at the centre, causing the water in the suction pipe to rise and enter the casing, at the same time the water at the outer part of the casing is being discharged by the discharge pipe.

As in the case of the turbine it is important that the arrangements of the vanes, casing, and pipes should be such as to lead to as little agitation of the water as possible, whereby loss of energy by frictional resistance and otherwise may be avoided.

From 350 to 400 feet per minute is about the speed allowed for the water in the pipes; the speed of periphery of wheel being from five to six times that.

The diameter of wheel is generally from two to three times the diameter of the pipes.

The efficiency is about 64 per cent.

SECTION 4.-Resistance to Shearing, p. 298.

The pressure required for shearing or for punching may be calculated by means of the following formula :

in which t is the thickness of the plate to be shorn or punched; e, the length of the cut; that is, the breadth of the plate, for shearing, or the circumference of the hole, for punching; so that ct is the