tion is found by subtracting the square of the draft of water from the square of half the breadth, and then dividing three-eighths of the remainder by the draft of water. 5. That the stability of vessels is augmented in proportion to the cubes of the breadths, provided that the centres of gravity of the vessel and of the displaced fluid coincide. 6. That vessels having the same length and breadth, but different drafts of water, have equal stability when their centre of gravity, and the centre of gravity of the displaced water, are in the same point. 7. That the altitude of the metacentre in different vessels above the centre of gravity of the displaced water is proportional to the squares of the breadths. The apparatus for making these experiments was simple, as appears from the perspective view (Plate XLIV. Fig. 1) of the whole together. In this A A represents a cistern filled with water, and mounted to a convenient height upon framed legs; B, a model on which the experiment was tried, by attaching a fine line, a a, to the top of the mast, D, and conducting it over a pulley, E. A scale, F, is suspended to the end of the line for the reception of the weights. These cause the model to incline, as the figure shows; and the degree of inclination of the mast from the perpendicular is shown by the plumb-line, b, upon a divided arch, d. To prevent the body being drawn away towards the pulley, E, by the draft of the line, a, it is retained by two small lines (shown dotted at g, g), which are made fast to sliders, s, s, at the side of the cistern, and have hooks at the opposite ends, which take hold of pins projecting from the stem and stern of the model, B; and these are previously adjusted, so that the centre of gravity of the model will be found in a line between them. The manner of making this adjustment is shown in Fig. 2, which represents a frame of wood, H, supporting two small uprights, h, h. These have pieces of brass plate at the upper ends, with notches to receive the pins or pivots of the model, B. These pivots are fitted into grooves in two pieces of brass plate attached to the ends of the model. One of these slips of brass is shown separately at Fig. 3, where k is the pivot, and a screw tapped into the brass slide to which the pivot is fixed, and passing through the same groove by means of this screw, the pivot k can be fastened at any part of the groove, and raised or lowered. The ballast is then raised or lowered till the model will barely rest on the pivots without overturning, as shown in Fig. 2. It is necessary, in order to know exactly what weight is applied to the top of the mast, D, that the line, a, draw in a direction at right angles thereto. To ascertain this, a ruler, m, is fixed upon the top of the mast, and the pulley, E, is attached to a cross rail, H, which applies against the uprights, and is suspended by a line, n, which passes over a pulley, and is made fast to a cleat, o. By this means the pulley, E, can at pleasure be raised or lowered until the direction of the line, a, corresponds with the ruler, m. The manner of conducting VOL. VII. No III. N 1 the experiments with this apparatus is as follows. The cistern is filled with water up to a certain mark; and the model being put in, loaded with ballast, the water is added or decreased till the edge of the gunwhale is exactly on a level with the edge of the cistern, as ascertained by looking across it, or applying a straight ruler. The plumb-line, d, cutting the zero of the divided arch, shows the vessel to be upright. In this state the model is ready for making the experiments. The hooks of the two strings, g, g, attached to the pivots, and the two sliders, s, s, are raised or lowered to make the strings, g, g, horizontal in the water. Weights being now put into the scale, F, will show what weight is requisite to incline the model. The pulley, E, being raised or lowered by the line n, as is found necessary to make the line, a, draw parallel to the ruler, m, or perpendicular to the mast. The inclination of the mast is shown by the plumb-line, b, cutting the divisions of the arch, d; but to counteract the weight of the plummet, b, which tends to incline the mast, another counterbalancing plummet and line, r, is applied on the opposite side of the model. For this purpose holes are made in the arch, d, at every division, and a peg is put in at the division opposite to that which is cut by the plumb-line. The experiment is tried with different weights, to produce the several inclinations at every 5o, until 30° from the perpendicular; and to verify the experiment, the model is changed end for end, the strings, g, g, being hooked on the pivots at the opposite ends. In this way the series of trials are made on the opposite side, by which means, if there is any difference in the two sides, or in the ballast, it will be detected, and allowed for by taking the mean of the different trials. Example with Model I. (Plate XLV.)—Exper. 1. The total depth of the model being 7.5 inches, the height of the centre of gravity is subtracted in each experiment, and the length of the mast is afterwards added, which gives the length of lever at which the weight is applied to produce the various inclinations of 5°, 10°, 15°, 20°, 25°. The centre of gravity in Exper. I being situated two inches above the bottom of the model, 2 oz. 10 dr. inclined it 5o. The model was turned end for end; and to incline it, the same angle required 3 oz. 5 dr. The mean of both is 2 oz. 15 dr., which is considered as the true power. This number (2 oz. 15 dr.) being reduced to the decimals of an oz., gives 2.9687 oz., and is set down on the right hand in Table I. It is evident that the effort of the water to restore the vessel to its original vertical position is exactly equal to the inclining power. If, therefore, the momentum of the effort to incline the vessel be divided by the weight of the displaced fluid, the quotient will be the length of lever on which the water acts. Exper. 1.-Model 1 inclined 30°; inclining power, 20.3125 oz. length of lever, 24.98 inches; and weight of displaced fluid, 559'06 = 0.90758 parts of an inch, the length of 20.312 x 24.98 559.06 lever on which the fluid acts, which, for the sake of distinction, call x. Then, to find the greatest height, the centre of gravity of the model can be raised above the bottom without oversetting (called by the French writers on naval architecture the metacentre), as 30°: x or 90758 :: radius: 1-8152, or the height of the metacentre above the centre of gravity, which call y. Then 2 inches, the height of the model's centre of gravity, added to 1.8152, the sum 3.8152 is the altitude of the metacentre, according to this experiment, above the bottom of the model. By the same process the height of the metacentre was obtained, when the experiment was made at 10°, 15°, 20°, 25°. To ascertain more exactly the height of the metacentre, the centre of gravity of the figure was raised. In Exper. 2, the height of the centre of gravity was raised to 2.25 inches above the under side of the figure; and in Exper. 3, to 3 inches. By these two, the accuracy of the first experiment was corroborated. Three heights are thus obtained. But from the unavoidable inaccuracies to which experiments are liable, these numbers differ; therefore take the mean of the heights of the metacentre thus determined. But as these means, when compared, are found neither to increase nor decrease in a regular manner, take the differences between each set of experiments; and then take the mean of all the differences, and examine which of the differences most nearly coincides with the mean difference. The two experiments whose difference is the nearest to the mean difference are to be considered as the best experiments. Then, by adding and subtracting the mean common difference, the whole will be brought into a regular series, and the irregularities corrected. The following example will more clearly explain the process. In Table 3, are set down the various heights of the metacentre at every angle of inclination, as determined by Exper. 1, 2, and 3; and the fourth horizontal line contains the mean results. In the eighth right hand vertical column are placed the differences of the mean heights. The mean of the five differences is 0538. The nearest difference to this number is 0584, which is the difference between the mean height of the metacentre at 15° and 20°. By taking the mean of the experiments at 15° and 20°, 3·5995 — 3.6579 = 3.6287; and adding and subtracting half of 0538 (which is 0269) to 3.6287, the numbers 3.6556 and 3.6018 are obtained, which are the corrected heights of the metacentre. Then by adding '0538 to 3.6556, and subtracting the same from 3.6018, the corrected altitudes of the metacentre are obtained, which are set down in the last column on the right. By subtracting the altitude of the model's centre of gravity from these numbers, a more accurate value of y is obtained. (See Table 4.) Then with the value of y, and the angle of inclination of the model, the length of x is obtained. This number multiplied by 55906, the weight of the displaced water, gives the momentum of the stability. This number divided by the length of lever which inclined the model, gives those numbers called the regular series; and according to the agreement or non-agreement |