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Below the calculations of moments just described, are written the calculations of the displacement up to the several water-sections between the load-water-section and the keel. The calculator here employs various rules according to his judgment, so as to save labor as much as possible. In the present case the displacement up to the 2d, 4th, and 6th water-sections is computed by Simpson's First Rule.
The area of each water-section in square feet being divided by 35, gives the tons displacement per foot of immersion, which is divided by 12 for the tons displacement per inch of immersion.
The areas of the midship-section (No. 9) up to the several water-lines, are computed from its ordinates, just as the displacements are computed from the water-sections; and those areas are written at the foot of the Table.
The two columns at the right-hand side of the Table, headed "Metacentre," contain the calculations of stability, estimated
y=The ordinates of the half-breadth, load-water-section. dx=The increment of the length of the load-water-section. D=Displacement of the immersed portion of the body in cubic
The first of those columns, headed "Cubes," contains the cubes of the ordinates, or half-breadths, of the load-water-section.
Each of those cubes is multiplied by the proper Simpson's multiplier (found in the second column from the left of the Table), and the products are written in the column headed "Multiples of Cubes." Those products having been added up, their sum (326006-757804) is multiplied by one-third of the horizontal H.I. interval (H. 14,5) giving the area of the curve whose ordinates
are the cubes of the half-breadths (1575699-329386). Two-thirds of that area is the coefficient of surface stability (1050466·219590); which, being divided by the displacement in cubic feet, (113145-05922) gives the height of the metacentre above the centre of buoyancy (9.28) feet.
From that height, at the lower left-hand corner of the Table, is subtracted the depth of the centre of buoyancy below L. W. L. (7.39 feet); leaving the height of the metacentre above L. W. L. (1.89 feet).
Practical Method of ascertaining the Height of the Centre of Gravity of a Vessel Equipped and Ready for Sea.
THE position of the centre of gravity of a vessel may be found by direct calculation. Still the results cannot be regarded with much confidence. When the difficulties which beset this method of ascertaining the centre of gravity of a ship were understood, it was seen that, as in any small body which is symmetrical with respect to a plane passing through it, its centre of gravity may be found by suspending it from two points in that plane by means of a string, and obtaining the point in which two vertical lines drawn through the points of suspension intersect, so, by altering the line of support of a ship, i.e., the vertical lines through the centres of gravity and buoyancy, and obtaining the point in which two such verticals intersect, the position of the centre of gravity of the ship may be found. Any person possessed of a small amount of mathematical knowledge, and having the drawings of the ship, could ascertain for himself the position of the centre of gravity of the ship in a very short time.
The rationale of the method may be briefly given as follows:
Let ACD, Fig. 14, represent the transverse section of a ship through G, the common centre of gravity of the hull and every article on board; WL the load-water-line when the ship is floating in the upright position; CBGM the middle line, which is therefore perpendicular to WL, and also contains G, the centre of gravity of the ship, and B, the centre of buoyancy (or centre of gravity of the displacement); let, also, P represent a weight or weights on any or all of the decks, such as the guns, shot, ballast, &c., capable of being readily transported to the opposite side of the deck or decks.
If the weight or weights P be moved across the deck to P' the ship will incline through an angle WSW', the amplitude of which will depend, cæteris paribus, upon the weight or weights moved, and the distance through which they have been moved.
When the ship has taken up the new position of equilibrium, the centre of buoyancy will have moved from B to B', and the
centre of gravity of the ship from G to G'; so that the line joining B' and G' will be vertical, and therefore perpendicular to W'L,' the new water-line, and will make the same angle BMB' with the middle line BGM as the water-lines do with each other; and B'G' produced will meet the middle line in a point M. This point, in ships of the usual form, may, without any appreciable error, be assumed to coincide with the metacentre when the inclination does not exceed 4° or 5°.
From a general and well-known property of the centre of gravity of a system of bodies, such as a ship, we know that since the weight or weights P have been moved in a horizontal direction to P', the centre of gravity has also moved in the same direction; therefore GG', the line joining the original and the new centres of gravity, will be horizontal; and from another property of the centre of gravity we have, that the weight of the ship × GG′ = P x distance through which it has been moved; or, if W represent the total weight of the ship, and c the distance through which the centre of gravity of the weight or weights P has been moved,.
Now by trigonometry GG'= GMx tangent of the angle between the middle line and the new vertical line B'G'M, i.e., the angle of the ship's inclination from the upright; or representing the angle of inclination by 0,
GG' GM tan 0
Equating the two values of GG' thus obtained,
The right-hand member of this equation (1) will contain all known quantities after the ship has been inclined; and since the metacentre corresponding to any draught of water is easily obtained by calculation from the drawings of the ship, and its position fixed, the distance GM set off below it will give the position of the centre of gravity of the ship.
Should the inclination obtained by the movements of the