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2, =Sum of Ordinates on Displacement Table. 2n =Sum of Moments (differences X lever) for Centers of Flotation.
17,45 8.25 4.80 5,313 561.50 110.60 186,691
32.35 32.30 31.45
33,850 33,700 31,150 128,094 25,200 16,390 13,310
184,257 7,370,280 558,767
W.L.5.32.50 32.50 31.45 ! 30.25 28.55 24.65 21.00
68.92 179,429 7,177,160 453,558 15.824
W.L.4. 32.50 32.50 31.40 30.20 27.85 24.15
174,811 6,992,440 350,152 19.998
W.L.3. 32.50 32.50 31.30 30.00 26.55 23.35
166,212 6,648,480 249,913 26.603
32.30 32.25 31:00 29.25 24.60 22.15 33,698 33,540 29,791 25,020 14,887 10,870
153,376 6,135,040 153,096 40.07
* See Table of Center of Buoyancy and Displacement, pp. 24 to 27.
EXPLANATION OF TABLE, GIVING EFFECT OF FORM OF WATER LINE ON POSITION
OF LONGITUDINAL METACENTER.
Longitudinal and Lateral Stability Compared. The first four lines are exactly the same as those in the other table ; and the last eight lines differ only in having length and breadth interchanged, so as to give pitching instead of rolling:
On comparing them with the following table, it will be noticed that, in the algebraic factor, the length and breadth always interchange; and that the numerical factor remains unchanged for forms (1), (3), and (A), namely, the square or rectangle, the circle or ellipse, and the wedge. Of the nine forms selected, these are obviously the only ones in which breadth and length are absolutely interchangeable.
With respect to the comparison of the different forms, one with another, if we disregard the wave-bow No. (8), the variation of stability follows much the same sequence for longitudinal as for lateral stability, but with a somewhat less absolute value. This result might be expected à priori, because the extreme breadth ordinate cuts the outline at right angles in all but the wedge form (9) ; while the extreme length ordinate meets the outline more sharply. In forms (2) and (4) this difference is only of the second order ; but, as the figures show, it is quite sufficient to be of practical importance even in these.
Differ Chiefly in Wave-Bow.- The wave-bow form (8) falls altogether out of its sequence, and its stability is less than the wedge form (9) as regards pitching. This is due to the sudden falling off of the extreme ordinate length, which meets the curve tangentially, instead of normally, as the extreme breadth ordinate.
Fine Bow · Affects Pitch More than Rolling. - If we consider rolling on any given axis, it is easily seen from geometrical considerations, and also from the algebraic form of the integral, that the instantaneous stability depends, firstly, on the length of the transverse axis, and, secondly, on the slowness of the rate of diminution of that axis, as we pass along that axis of motion. Hence sharp bows have less stability for pitching than bluff bows, while their lateral stability for rolling is not so very different.
Caution in Use of Table. — In the table of lateral stability, the element of length only appears as a simple factor ; therefore, as regards lateral stability, we may compound the moments by simple addition for a vessel built up in different lengths for the different forms. Thus, the values in lines 1 to 8 of column (2) are simply the means of the corresponding values in columns (1) and (3). We cannot apply this process to the longitudinal stability because here the length element enters as a cubic factor. If we were so to compound the moments of length, what we should really do would be equivalent to screwing together two longitudinal halves of different vessels ; in the case before mentioned, screwing half a box to half a tub; not introducing a flat midship length between two semicircular ends.
Explanation of Table Giving Effect of Form of Water
Line on Position of Metacenter. Explanation of Table. — By the preceding table we can at once make an approximate estimate of the value of any proposed form of water line, by selecting that form in the table to which it comes nearest. From this table we gather that the more nearly the water line approaches to a right parallelogram, the more it will contribute to the stability of a ship. No. 9, on the contrary, the straight line wedge form, is the least stable of these water lines, and from the comparison of the successive groups of lines on the table we shall see exactly how this comes about.
Areas on Water Lines. The first and second lines in the table give the measures simply of the areas of those water lines. From lines 3 and 4 we see that, Fig. 1 being taken as the standard of comparison, Fig. 2 only contains 89 per cent of the rectangular area, and this diminution is effected merely by rounding off the rectangular corners, the length and breadth remaining the same in both. In Fig. 3, when the curvature of the ends extends quite to the middle of the water line, its area is reduced to 69 per cent. In Fig. 6, by forming the water line of parabolic arcs, a favorite form of some builders, the area is reduced to twothirds of the rectangle. Figs. 7 and 8 are the lines used for a wave stern and a wave bow ; from which it appears at once how much more powerful the stern contributed to the stability of a ship than the bow; the stern line being 62 per cent, and the bow line only 50 per cent.
Metacentric Moments. Lines 5 and 6 are the actual measure of the stability (by its moments) for small inclinations. For example : in the rectangle, the moment is one-twelfth part of the product of the length by the cube of the breadth, or .08 of that product ; and as we pass along line 6 we find it gradually diminish, until, in the wedge form, it is only .02, showing that a sharp wedge form has only one-fourth part of the power to carry top weight that the rectangular form has, although its power of buoyancy, or power to carry absolute load, is one-half. This is set out more fully in lines 7 and 8 ; so that by carefully comparing together line 4 and line 8, the relative values of all those figures for carrying absolute weight and for carrying top weight may be clearly seen.
Metacentric Intervals. Lines 9 and 10 measure the powers of ships, formed on these water lines only to carry top weight without upsetting.