Again, to obtain the displacement below the hori Eontal section 6°, we have the solidity between the sections 5° and 6° 7046.80 6907.40 6977.10 2 Solidity between the sections 1° and 5° before determined == 29269.14 Solidity between the sections.1° and 6° = 36246.24 100977.93 — 36246.24 Hence — 1849.48 tons must 35 De the displacement below the horizontal section 6°. Therefore, by diminishing the draught of water 5 7eet, the displacement is decreased 1035.6 tons.
To determine the displacement below the horizon al 7°, we have Extreme Areas. Even Areas. Odd Areas, 74,38.64 7517.42 7204.54 7335.24 6617.16 6907.40 7046.8014134.58=2 21550.58 14382.04 4 286202.32=4S 28764.08=28 And since = we shall farther have 3 + 4 S + 2 s)— 3 = (14134.58 ± 86202.32 + 28764.08) X = 43033.66. And by subtracting this result from the total displacement, and dividing the difference by 35, we shall have 1655.55 tons for the displacement below the horizontal section Thus, by decreasing the draught of 6 feet, the dis placement will be diminished 1229.53 tons.
By obtaining in this manner the displacements be low the successive horizontal sections (applying the formula for equidistant ordinates, when the number of areas is odd, and when they are even, adopting the method above employed for ascertaining the displace ments below the sections and 6°) we shall obtain the numbers recorded in the first numerical column of the following table; and if we farther subtract each of these results from the total displacement, (2885.08 tons) we shall find the numbers entered in the second numerical column, and which denote respectively the successive alterations of displacement, produced by an uniform diminution of the draught of water.
To construct the desired scale of solidity, let the vertical line 1° B. Fig. 3, Plate CCCCXCIV. be drawn equal to 22.35 feet, the mean draught of water, measured from the under side of the false keel; and let this line be divided at equal intervals of a foot, at the points ;2°, 3', 6°, &c. corresponding to the intervals at which the Foregoing displacements were computed. At the point 1°, erect the perpendicular line 1°A equal to 2885.08 tons, the displacement be low the horizontal section 1°; and at the point the perpendicular equal to 2671.43 tons, the displace ment below the horizontal section 2°; at the point the perpendicular equal to 2460.25 tons; at the point 4°, the perpendicular 4°E equal to 2252.54 tons; and through the remaining points 5°, 6°, 7°, 9°, &c. perpendiculars corresponding respectively to the displacements at those points. Tnen, if through the extremities A, C, D, E, &c. a fair and uniform curve be drawn, the scale of solidity will be completed, and the uses of which we shall now proceed to exemplify.
Suppose in the first place, the vessel were floating at her load water line 1°A, and that it were necessa ry for some purpose to diminish her draught of wa ter four feet. Then since the line 1°A represents the displacement at the load water line, and 5'F the dis placement corresponding to the point 5, or when the draught of water is diminished four feet; through F draw Fa parallel to 1°B, and Act will represent on the scale of equal parts from which 1°A was laid off, the quantity of lading or stores to be removed, in or der to reduce the draught of water to the desired quantity. The value of Aa in the example selected
will be 836.26 tons.
Suppose, in the next place the converse case were to occur, to ascertain what alteration would take place in the draught of water, by diminishing the lading by any given quantity, say 419 tons, when the load wa ter line was denoted by 2°C. If on this water line. Cb be laid off equivalent to the 419 tons, and through b, the line bE be drawn to intersect the curve soli dity in E, and E4° be drawn parallel to Al°, inter secting 1°13 in 4', then will 2° 4°, measured on the scale of equal parts, he the diminution of the draught of water desired, and which in the present instance is two feet.
Again, suppose a vessel had received some un known portion of her lading, and it were required to ascertain by means of the scale of solidity, the exact quantity thereof'. I.et the mean draught of water he in the first place ascertained, and laid off front II to II; and through let III be drawn parallel to 1°A, meeting the curve of solidity in I. From I let IK be drawn perpendicular to 1°A. Then will HI, measur ed on the scale of i°A, represent the actual displace ment produced by the vessel and her present lading; and AK the quantity necessary to hring her down to the load water line 1°A. Also 1°H will denote the depth to which the vessel will sink, in order to re ceive the portion of the cargo denoted by AK.
Suppose, lastly, that a vessel had received a known portion of her cargo, and it were required to ascertain the draught of water produced thereby. On the load water line I°A, lay on. 1.1, equal to the given displace ment, and through L draw I.M perpendicular to 1°A, intersecting the curve of solidity in M. Through M draw AIN parallel to 1°A, intersecting i°13 in N. Then will NB denote the mean draught of water re quired.
The uses therefore of scales of solidity are very great, and in the mercantile navy in particular, arc of the highest importance. Every ship ought to have a scale of solidity constructed for it; and the more numerous the horizontal sections for which it is calculated, the better. It is probable, also, that if the properties of the curves of solidity of seN eral ves sels of each class of men-orwar were investigat ed, some analytical relations might be discovered, of great importance to naval architecture.
It may be proper to remark, that in consequence of the great numerical disproportion between the displacements and their correspohding water, it is convenient in practice, to adopt different scales for those elements. For example, if l°A, which in the present case is represented numerically by 2835, be measured by the same scale as that employed for the draught of water, the figure will he of an incon venient size, unless the scale for the latter be assum ed very minute, which will be attended also with dis advantage. The best way, therefore, is to adopt one convenient scale for the draught of water, and another for the displacement, but taking care in the applica tion of the figure, to measure each by its own proper scale.